LMFDB - Embedded newform 147.6.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$147 = 3 \cdot 7^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	147.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:	$23.5764215125$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 97x^{2} + 7x + 294$ x^4 - x^3 - 97x^2 + 7x + 294
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$7$
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$10.1812$ of defining polynomial
Character	$\chi$	$=$	147.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.18123 q^{2} +9.00000 q^{3} +52.2950 q^{4} +22.0716 q^{5} -82.6311 q^{6} -186.333 q^{8} +81.0000 q^{9} -202.644 q^{10} +416.710 q^{11} +470.655 q^{12} +797.918 q^{13} +198.644 q^{15} +37.3245 q^{16} -1375.55 q^{17} -743.680 q^{18} +2313.03 q^{19} +1154.23 q^{20} -3825.91 q^{22} -955.402 q^{23} -1676.99 q^{24} -2637.84 q^{25} -7325.87 q^{26} +729.000 q^{27} -7035.29 q^{29} -1823.80 q^{30} +1261.19 q^{31} +5619.96 q^{32} +3750.39 q^{33} +12629.2 q^{34} +4235.89 q^{36} +9776.44 q^{37} -21236.4 q^{38} +7181.26 q^{39} -4112.66 q^{40} -5400.95 q^{41} +19686.6 q^{43} +21791.8 q^{44} +1787.80 q^{45} +8771.76 q^{46} +2056.56 q^{47} +335.921 q^{48} +24218.7 q^{50} -12380.0 q^{51} +41727.1 q^{52} +18022.7 q^{53} -6693.12 q^{54} +9197.45 q^{55} +20817.2 q^{57} +64592.6 q^{58} +7435.68 q^{59} +10388.1 q^{60} +3495.38 q^{61} -11579.3 q^{62} -52792.5 q^{64} +17611.3 q^{65} -34433.2 q^{66} +15856.4 q^{67} -71934.4 q^{68} -8598.62 q^{69} +58133.5 q^{71} -15093.0 q^{72} +39110.7 q^{73} -89759.8 q^{74} -23740.6 q^{75} +120960. q^{76} -65932.8 q^{78} +9760.69 q^{79} +823.812 q^{80} +6561.00 q^{81} +49587.4 q^{82} -70395.7 q^{83} -30360.6 q^{85} -180747. q^{86} -63317.6 q^{87} -77646.6 q^{88} +144306. q^{89} -16414.2 q^{90} -49962.7 q^{92} +11350.7 q^{93} -18881.8 q^{94} +51052.2 q^{95} +50579.7 q^{96} -79328.7 q^{97} +33753.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9} + 283 q^{10} + 402 q^{11} + 621 q^{12} + 462 q^{13} + 3273 q^{16} + 276 q^{17} + 243 q^{18} + 510 q^{19} + 4719 q^{20} + 1375 q^{22}+ \cdots + 32562 q^{99}+O(q^{100})$ 4 * q + 3 * q^2 + 36 * q^3 + 69 * q^4 + 27 * q^6 + 123 * q^8 + 324 * q^9 + 283 * q^10 + 402 * q^11 + 621 * q^12 + 462 * q^13 + 3273 * q^16 + 276 * q^17 + 243 * q^18 + 510 * q^19 + 4719 * q^20 + 1375 * q^22 + 6900 * q^23 + 1107 * q^24 + 2814 * q^25 - 15138 * q^26 + 2916 * q^27 + 540 * q^29 + 2547 * q^30 - 6410 * q^31 + 15519 * q^32 + 3618 * q^33 + 21144 * q^34 + 5589 * q^36 + 15250 * q^37 - 41250 * q^38 + 4158 * q^39 - 8547 * q^40 + 4308 * q^41 + 29198 * q^43 + 70743 * q^44 + 61800 * q^46 - 15060 * q^47 + 29457 * q^48 - 7302 * q^50 + 2484 * q^51 - 47476 * q^52 + 13692 * q^53 + 2187 * q^54 + 73124 * q^55 + 4590 * q^57 + 52309 * q^58 + 34830 * q^59 + 42471 * q^60 - 5364 * q^61 + 16029 * q^62 - 73487 * q^64 + 66864 * q^65 + 12375 * q^66 - 5994 * q^67 - 58272 * q^68 + 62100 * q^69 + 89268 * q^71 + 9963 * q^72 + 59638 * q^73 - 185442 * q^74 + 25326 * q^75 + 21308 * q^76 - 136242 * q^78 - 44062 * q^79 - 33381 * q^80 + 26244 * q^81 + 57596 * q^82 - 208446 * q^83 + 36324 * q^85 - 136968 * q^86 + 4860 * q^87 + 87597 * q^88 - 77520 * q^89 + 22923 * q^90 + 158256 * q^92 - 57690 * q^93 - 73722 * q^94 - 221376 * q^95 + 139671 * q^96 - 188630 * q^97 + 32562 * q^99

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

)

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

$2$	−9.18123	−1.62303	−0.811514	−	0.584333i	$-0.801356\pi$
$2$	−9.18123	−1.62303	−0.811514	+	0.584333i	$0.801356\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	52.2950	1.63422
$5$	22.0716	0.394829	0.197414	−	0.980320i	$-0.436746\pi$
$5$	22.0716	0.394829	0.197414	+	0.980320i	$0.436746\pi$
$6$	−82.6311	−0.937055
$7$	0	0
$7$	0	0
$8$	−186.333	−1.02935
$9$	81.0000	0.333333
$10$	−202.644	−0.640818
$11$	416.710	1.03837	0.519184	−	0.854662i	$-0.326236\pi$
$11$	416.710	1.03837	0.519184	+	0.854662i	$0.326236\pi$
$12$	470.655	0.943516
$13$	797.918	1.30948	0.654742	−	0.755853i	$-0.272777\pi$
$13$	797.918	1.30948	0.654742	+	0.755853i	$0.272777\pi$
$14$	0	0
$15$	198.644	0.227955
$16$	37.3245	0.0364497
$17$	−1375.55	−1.15439	−0.577197	−	0.816605i	$-0.695854\pi$
$17$	−1375.55	−1.15439	−0.577197	+	0.816605i	$0.695854\pi$
$18$	−743.680	−0.541009
$19$	2313.03	1.46993	0.734965	−	0.678105i	$-0.237199\pi$
$19$	2313.03	1.46993	0.734965	+	0.678105i	$0.237199\pi$
$20$	1154.23	0.645236
$21$	0	0
$22$	−3825.91	−1.68530
$23$	−955.402	−0.376588	−0.188294	−	0.982113i	$-0.560296\pi$
$23$	−955.402	−0.376588	−0.188294	+	0.982113i	$0.560296\pi$
$24$	−1676.99	−0.594297
$25$	−2637.84	−0.844110
$26$	−7325.87	−2.12533
$27$	729.000	0.192450
$28$	0	0
$29$	−7035.29	−1.55341	−0.776707	−	0.629862i	$-0.783111\pi$
$29$	−7035.29	−1.55341	−0.776707	+	0.629862i	$0.783111\pi$
$30$	−1823.80	−0.369976
$31$	1261.19	0.235709	0.117855	−	0.993031i	$-0.462398\pi$
$31$	1261.19	0.235709	0.117855	+	0.993031i	$0.462398\pi$
$32$	5619.96	0.970194
$33$	3750.39	0.599503
$34$	12629.2	1.87361
$35$	0	0
$36$	4235.89	0.544739
$37$	9776.44	1.17402	0.587012	−	0.809579i	$-0.300304\pi$
$37$	9776.44	1.17402	0.587012	+	0.809579i	$0.300304\pi$
$38$	−21236.4	−2.38574
$39$	7181.26	0.756030
$40$	−4112.66	−0.406418
$41$	−5400.95	−0.501777	−0.250888	−	0.968016i	$-0.580723\pi$
$41$	−5400.95	−0.501777	−0.250888	+	0.968016i	$0.580723\pi$
$42$	0	0
$43$	19686.6	1.62367	0.811837	−	0.583885i	$-0.198468\pi$
$43$	19686.6	1.62367	0.811837	+	0.583885i	$0.198468\pi$
$44$	21791.8	1.69692
$45$	1787.80	0.131610
$46$	8771.76	0.611213
$47$	2056.56	0.135799	0.0678997	−	0.997692i	$-0.478370\pi$
$47$	2056.56	0.135799	0.0678997	+	0.997692i	$0.478370\pi$
$48$	335.921	0.0210443
$49$	0	0
$50$	24218.7	1.37001
$51$	−12380.0	−0.666490
$52$	41727.1	2.13998
$53$	18022.7	0.881315	0.440658	−	0.897675i	$-0.354745\pi$
$53$	18022.7	0.881315	0.440658	+	0.897675i	$0.354745\pi$
$54$	−6693.12	−0.312352
$55$	9197.45	0.409978
$56$	0	0
$57$	20817.2	0.848664
$58$	64592.6	2.52123
$59$	7435.68	0.278093	0.139047	−	0.990286i	$-0.455596\pi$
$59$	7435.68	0.278093	0.139047	+	0.990286i	$0.455596\pi$
$60$	10388.1	0.372527
$61$	3495.38	0.120274	0.0601368	−	0.998190i	$-0.480846\pi$
$61$	3495.38	0.120274	0.0601368	+	0.998190i	$0.480846\pi$
$62$	−11579.3	−0.382563
$63$	0	0
$64$	−52792.5	−1.61110
$65$	17611.3	0.517022
$66$	−34433.2	−0.973009
$67$	15856.4	0.431537	0.215769	−	0.976445i	$-0.430774\pi$
$67$	15856.4	0.431537	0.215769	+	0.976445i	$0.430774\pi$
$68$	−71934.4	−1.88653
$69$	−8598.62	−0.217423
$70$	0	0
$71$	58133.5	1.36861	0.684306	−	0.729195i	$-0.260105\pi$
$71$	58133.5	1.36861	0.684306	+	0.729195i	$0.260105\pi$
$72$	−15093.0	−0.343118
$73$	39110.7	0.858990	0.429495	−	0.903069i	$-0.358692\pi$
$73$	39110.7	0.858990	0.429495	+	0.903069i	$0.358692\pi$
$74$	−89759.8	−1.90547
$75$	−23740.6	−0.487347
$76$	120960.	2.40218
$77$	0	0
$78$	−65932.8	−1.22706
$79$	9760.69	0.175960	0.0879798	−	0.996122i	$-0.471959\pi$
$79$	9760.69	0.175960	0.0879798	+	0.996122i	$0.471959\pi$
$80$	823.812	0.0143914
$81$	6561.00	0.111111
$82$	49587.4	0.814397
$83$	−70395.7	−1.12163	−0.560816	−	0.827940i	$-0.689513\pi$
$83$	−70395.7	−1.12163	−0.560816	+	0.827940i	$0.689513\pi$
$84$	0	0
$85$	−30360.6	−0.455788
$86$	−180747.	−2.63527
$87$	−63317.6	−0.896864
$88$	−77646.6	−1.06885
$89$	144306.	1.93112	0.965562	−	0.260173i	$-0.0837796\pi$
$89$	144306.	1.93112	0.965562	+	0.260173i	$0.0837796\pi$
$90$	−16414.2	−0.213606
$91$	0	0
$92$	−49962.7	−0.615427
$93$	11350.7	0.136087
$94$	−18881.8	−0.220406
$95$	51052.2	0.580371
$96$	50579.7	0.560142
$97$	−79328.7	−0.856053	−0.428027	−	0.903766i	$-0.640791\pi$
$97$	−79328.7	−0.856053	−0.428027	+	0.903766i	$0.640791\pi$
$98$	0	0
$99$	33753.5	0.346123
$100$	−137946.	−1.37946
$101$	−84833.7	−0.827495	−0.413747	−	0.910392i	$-0.635780\pi$
$101$	−84833.7	−0.827495	−0.413747	+	0.910392i	$0.635780\pi$
$102$	113663.	1.08173
$103$	20332.3	0.188839	0.0944197	−	0.995532i	$-0.469900\pi$
$103$	20332.3	0.188839	0.0944197	+	0.995532i	$0.469900\pi$
$104$	−148678.	−1.34792
$105$	0	0
$106$	−165471.	−1.43040
$107$	6962.19	0.0587877	0.0293938	−	0.999568i	$-0.490642\pi$
$107$	6962.19	0.0587877	0.0293938	+	0.999568i	$0.490642\pi$
$108$	38123.0	0.314505
$109$	112651.	0.908177	0.454088	−	0.890957i	$-0.349965\pi$
$109$	112651.	0.908177	0.454088	+	0.890957i	$0.349965\pi$
$110$	−84443.9	−0.665406
$111$	87988.0	0.677823
$112$	0	0
$113$	112005.	0.825167	0.412583	−	0.910920i	$-0.364627\pi$
$113$	112005.	0.825167	0.412583	+	0.910920i	$0.364627\pi$
$114$	−191128.	−1.37741
$115$	−21087.3	−0.148688
$116$	−367910.	−2.53862
$117$	64631.4	0.436494
$118$	−68268.7	−0.451353
$119$	0	0
$120$	−37014.0	−0.234646
$121$	12595.8	0.0782102
$122$	−32091.9	−0.195207
$123$	−48608.5	−0.289701
$124$	65954.0	0.385200
$125$	−127195.	−0.728108
$126$	0	0
$127$	82224.5	0.452368	0.226184	−	0.974085i	$-0.427375\pi$
$127$	82224.5	0.452368	0.226184	+	0.974085i	$0.427375\pi$
$128$	304862.	1.64467
$129$	177179.	0.937428
$130$	−161694.	−0.839140
$131$	−175812.	−0.895097	−0.447548	−	0.894260i	$-0.647703\pi$
$131$	−175812.	−0.895097	−0.447548	+	0.894260i	$0.647703\pi$
$132$	196126.	0.979718
$133$	0	0
$134$	−145581.	−0.700397
$135$	16090.2	0.0759849
$136$	256310.	1.18828
$137$	31330.3	0.142614	0.0713072	−	0.997454i	$-0.477283\pi$
$137$	31330.3	0.142614	0.0713072	+	0.997454i	$0.477283\pi$
$138$	78945.9	0.352884
$139$	152234.	0.668305	0.334152	−	0.942519i	$-0.391550\pi$
$139$	152234.	0.668305	0.334152	+	0.942519i	$0.391550\pi$
$140$	0	0
$141$	18509.1	0.0784038
$142$	−533737.	−2.22129
$143$	332500.	1.35973
$144$	3023.29	0.0121499
$145$	−155280.	−0.613332
$146$	−359084.	−1.39416
$147$	0	0
$148$	511259.	1.91861
$149$	362860.	1.33898	0.669489	−	0.742822i	$-0.266513\pi$
$149$	362860.	1.33898	0.669489	+	0.742822i	$0.266513\pi$
$150$	217968.	0.790978
$151$	205126.	0.732112	0.366056	−	0.930593i	$-0.380708\pi$
$151$	205126.	0.732112	0.366056	+	0.930593i	$0.380708\pi$
$152$	−430992.	−1.51308
$153$	−111420.	−0.384798
$154$	0	0
$155$	27836.5	0.0930648
$156$	375544.	1.23552
$157$	−77272.0	−0.250192	−0.125096	−	0.992145i	$-0.539924\pi$
$157$	−77272.0	−0.250192	−0.125096	+	0.992145i	$0.539924\pi$
$158$	−89615.1	−0.285587
$159$	162205.	0.508828
$160$	124042.	0.383060
$161$	0	0
$162$	−60238.0	−0.180336
$163$	184931.	0.545182	0.272591	−	0.962130i	$-0.412119\pi$
$163$	184931.	0.545182	0.272591	+	0.962130i	$0.412119\pi$
$164$	−282442.	−0.820012
$165$	82777.0	0.236701
$166$	646319.	1.82044
$167$	129262.	0.358657	0.179329	−	0.983789i	$-0.442607\pi$
$167$	129262.	0.358657	0.179329	+	0.983789i	$0.442607\pi$
$168$	0	0
$169$	265380.	0.714746
$170$	278748.	0.739757
$171$	187355.	0.489977
$172$	1.02951e6	2.65344
$173$	−507867.	−1.29013	−0.645067	−	0.764126i	$-0.723171\pi$
$173$	−507867.	−1.29013	−0.645067	+	0.764126i	$0.723171\pi$
$174$	581334.	1.45563
$175$	0	0
$176$	15553.5	0.0378483
$177$	66921.1	0.160557
$178$	−1.32491e6	−3.13427
$179$	−132589.	−0.309296	−0.154648	−	0.987970i	$-0.549424\pi$
$179$	−132589.	−0.309296	−0.154648	+	0.987970i	$0.549424\pi$
$180$	93492.9	0.215079
$181$	−740060.	−1.67908	−0.839538	−	0.543301i	$-0.817174\pi$
$181$	−740060.	−1.67908	−0.839538	+	0.543301i	$0.817174\pi$
$182$	0	0
$183$	31458.5	0.0694400
$184$	178023.	0.387642
$185$	215782.	0.463538
$186$	−104214.	−0.220873
$187$	−573205.	−1.19869
$188$	107548.	0.221926
$189$	0	0
$190$	−468722.	−0.941957
$191$	−582732.	−1.15581	−0.577904	−	0.816105i	$-0.696129\pi$
$191$	−582732.	−1.15581	−0.577904	+	0.816105i	$0.696129\pi$
$192$	−475133.	−0.930169
$193$	−400904.	−0.774725	−0.387362	−	0.921928i	$-0.626614\pi$
$193$	−400904.	−0.774725	−0.387362	+	0.921928i	$0.626614\pi$
$194$	728335.	1.38940
$195$	158502.	0.298503
$196$	0	0
$197$	671589.	1.23293	0.616464	−	0.787383i	$-0.288564\pi$
$197$	671589.	1.23293	0.616464	+	0.787383i	$0.288564\pi$
$198$	−309898.	−0.561767
$199$	−455022.	−0.814515	−0.407258	−	0.913313i	$-0.633515\pi$
$199$	−455022.	−0.814515	−0.407258	+	0.913313i	$0.633515\pi$
$200$	491517.	0.868887
$201$	142708.	0.249148
$202$	778878.	1.34305
$203$	0	0
$204$	−647409.	−1.08919
$205$	−119208.	−0.198116
$206$	−186675.	−0.306492
$207$	−77387.5	−0.125529
$208$	29781.9	0.0477303
$209$	963860.	1.52633
$210$	0	0
$211$	−1.19545e6	−1.84852	−0.924260	−	0.381764i	$-0.875317\pi$
$211$	−1.19545e6	−1.84852	−0.924260	+	0.381764i	$0.875317\pi$
$212$	942499.	1.44026
$213$	523201.	0.790168
$214$	−63921.4	−0.0954140
$215$	434514.	0.641073
$216$	−135837.	−0.198099
$217$	0	0
$218$	−1.03428e6	−1.47400
$219$	351996.	0.495938
$220$	480980.	0.669993
$221$	−1.09758e6	−1.51166
$222$	−807838.	−1.10012
$223$	296529.	0.399305	0.199653	−	0.979867i	$-0.436019\pi$
$223$	296529.	0.399305	0.199653	+	0.979867i	$0.436019\pi$
$224$	0	0
$225$	−213665.	−0.281370
$226$	−1.02834e6	−1.33927
$227$	218146.	0.280985	0.140492	−	0.990082i	$-0.455131\pi$
$227$	218146.	0.280985	0.140492	+	0.990082i	$0.455131\pi$
$228$	1.08864e6	1.38690
$229$	−1.22920e6	−1.54894	−0.774471	−	0.632609i	$-0.781984\pi$
$229$	−1.22920e6	−1.54894	−0.774471	+	0.632609i	$0.781984\pi$
$230$	193607.	0.241324
$231$	0	0
$232$	1.31090e6	1.59901
$233$	−62944.2	−0.0759567	−0.0379784	−	0.999279i	$-0.512092\pi$
$233$	−62944.2	−0.0759567	−0.0379784	+	0.999279i	$0.512092\pi$
$234$	−593395.	−0.708442
$235$	45391.7	0.0536175
$236$	388849.	0.454465
$237$	87846.2	0.101590
$238$	0	0
$239$	219330.	0.248372	0.124186	−	0.992259i	$-0.460368\pi$
$239$	219330.	0.248372	0.124186	+	0.992259i	$0.460368\pi$
$240$	7414.31	0.00830888
$241$	433864.	0.481183	0.240592	−	0.970626i	$-0.422659\pi$
$241$	433864.	0.481183	0.240592	+	0.970626i	$0.422659\pi$
$242$	−115645.	−0.126937
$243$	59049.0	0.0641500
$244$	182791.	0.196553
$245$	0	0
$246$	446286.	0.470193
$247$	1.84561e6	1.92485
$248$	−235001.	−0.242628
$249$	−633561.	−0.647575
$250$	1.16781e6	1.18174
$251$	−1.71109e6	−1.71431	−0.857155	−	0.515059i	$-0.827770\pi$
$251$	−1.71109e6	−1.71431	−0.857155	+	0.515059i	$0.827770\pi$
$252$	0	0
$253$	−398125.	−0.391037
$254$	−754922.	−0.734206
$255$	−273246.	−0.263150
$256$	−1.10964e6	−1.05824
$257$	984057.	0.929368	0.464684	−	0.885477i	$-0.346168\pi$
$257$	984057.	0.929368	0.464684	+	0.885477i	$0.346168\pi$
$258$	−1.62672e6	−1.52147
$259$	0	0
$260$	920984.	0.844926
$261$	−569859.	−0.517804
$262$	1.61417e6	1.45277
$263$	547095.	0.487724	0.243862	−	0.969810i	$-0.421586\pi$
$263$	547095.	0.487724	0.243862	+	0.969810i	$0.421586\pi$
$264$	−698820.	−0.617100
$265$	397791.	0.347969
$266$	0	0
$267$	1.29876e6	1.11493
$268$	829211.	0.705226
$269$	−641112.	−0.540198	−0.270099	−	0.962833i	$-0.587056\pi$
$269$	−641112.	−0.540198	−0.270099	+	0.962833i	$0.587056\pi$
$270$	−147728.	−0.123325
$271$	−548012.	−0.453280	−0.226640	−	0.973979i	$-0.572774\pi$
$271$	−548012.	−0.453280	−0.226640	+	0.973979i	$0.572774\pi$
$272$	−51341.8	−0.0420774
$273$	0	0
$274$	−287651.	−0.231467
$275$	−1.09921e6	−0.876498
$276$	−449664.	−0.355317
$277$	1.67414e6	1.31097	0.655485	−	0.755208i	$-0.272464\pi$
$277$	1.67414e6	1.31097	0.655485	+	0.755208i	$0.272464\pi$
$278$	−1.39769e6	−1.08468
$279$	102157.	0.0785698
$280$	0	0
$281$	1.81078e6	1.36804	0.684021	−	0.729462i	$-0.260230\pi$
$281$	1.81078e6	1.36804	0.684021	+	0.729462i	$0.260230\pi$
$282$	−169936.	−0.127252
$283$	2.51315e6	1.86531	0.932657	−	0.360764i	$-0.117484\pi$
$283$	2.51315e6	1.86531	0.932657	+	0.360764i	$0.117484\pi$
$284$	3.04009e6	2.23661
$285$	459470.	0.335077
$286$	−3.05276e6	−2.20687
$287$	0	0
$288$	455217.	0.323398
$289$	472283.	0.332627
$290$	1.42566e6	0.995455
$291$	−713958.	−0.494243
$292$	2.04529e6	1.40378
$293$	107228.	0.0729691	0.0364845	−	0.999334i	$-0.488384\pi$
$293$	107228.	0.0729691	0.0364845	+	0.999334i	$0.488384\pi$
$294$	0	0
$295$	164117.	0.109799
$296$	−1.82167e6	−1.20848
$297$	303781.	0.199834
$298$	−3.33150e6	−2.17320
$299$	−762332.	−0.493136
$300$	−1.24151e6	−0.796431
$301$	0	0
$302$	−1.88330e6	−1.18824
$303$	−763504.	−0.477754
$304$	86332.6	0.0535785
$305$	77148.7	0.0474875
$306$	1.02297e6	0.624538
$307$	1.49622e6	0.906042	0.453021	−	0.891500i	$-0.350346\pi$
$307$	1.49622e6	0.906042	0.453021	+	0.891500i	$0.350346\pi$
$308$	0	0
$309$	182990.	0.109027
$310$	−255573.	−0.151047
$311$	861202.	0.504899	0.252449	−	0.967610i	$-0.418764\pi$
$311$	861202.	0.504899	0.252449	+	0.967610i	$0.418764\pi$
$312$	−1.33810e6	−0.778222
$313$	−503937.	−0.290747	−0.145374	−	0.989377i	$-0.546438\pi$
$313$	−503937.	−0.290747	−0.145374	+	0.989377i	$0.546438\pi$
$314$	709452.	0.406068
$315$	0	0
$316$	510435.	0.287556
$317$	−480009.	−0.268288	−0.134144	−	0.990962i	$-0.542828\pi$
$317$	−480009.	−0.268288	−0.134144	+	0.990962i	$0.542828\pi$
$318$	−1.48924e6	−0.825841
$319$	−2.93167e6	−1.61302
$320$	−1.16522e6	−0.636109
$321$	62659.7	0.0339411
$322$	0	0
$323$	−3.18168e6	−1.69688
$324$	343107.	0.181580
$325$	−2.10478e6	−1.10535
$326$	−1.69790e6	−0.884845
$327$	1.01386e6	0.524336
$328$	1.00637e6	0.516505
$329$	0	0
$330$	−759995.	−0.384172
$331$	−2.19922e6	−1.10331	−0.551656	−	0.834072i	$-0.686004\pi$
$331$	−2.19922e6	−1.10331	−0.551656	+	0.834072i	$0.686004\pi$
$332$	−3.68134e6	−1.83299
$333$	791892.	0.391341
$334$	−1.18678e6	−0.582110
$335$	349977.	0.170383
$336$	0	0
$337$	−1.35725e6	−0.651008	−0.325504	−	0.945541i	$-0.605534\pi$
$337$	−1.35725e6	−0.651008	−0.325504	+	0.945541i	$0.605534\pi$
$338$	−2.43652e6	−1.16005
$339$	1.00805e6	0.476410
$340$	−1.58771e6	−0.744857
$341$	525550.	0.244753
$342$	−1.72015e6	−0.795245
$343$	0	0
$344$	−3.66825e6	−1.67133
$345$	−189785.	−0.0858449
$346$	4.66285e6	2.09392
$347$	3.79941e6	1.69392	0.846959	−	0.531659i	$-0.178431\pi$
$347$	3.79941e6	1.69392	0.846959	+	0.531659i	$0.178431\pi$
$348$	−3.31119e6	−1.46567
$349$	−1.31753e6	−0.579024	−0.289512	−	0.957174i	$-0.593493\pi$
$349$	−1.31753e6	−0.579024	−0.289512	+	0.957174i	$0.593493\pi$
$350$	0	0
$351$	581682.	0.252010
$352$	2.34189e6	1.00742
$353$	−3.42505e6	−1.46295	−0.731477	−	0.681866i	$-0.761168\pi$
$353$	−3.42505e6	−1.46295	−0.731477	+	0.681866i	$0.761168\pi$
$354$	−614418.	−0.260589
$355$	1.28310e6	0.540367
$356$	7.54649e6	3.15588
$357$	0	0
$358$	1.21733e6	0.501997
$359$	1.47084e6	0.602322	0.301161	−	0.953573i	$-0.402626\pi$
$359$	1.47084e6	0.602322	0.301161	+	0.953573i	$0.402626\pi$
$360$	−333126.	−0.135473
$361$	2.87399e6	1.16069
$362$	6.79466e6	2.72519
$363$	113362.	0.0451547
$364$	0	0
$365$	863235.	0.339154
$366$	−288827.	−0.112703
$367$	−4.98079e6	−1.93034	−0.965168	−	0.261630i	$-0.915740\pi$
$367$	−4.98079e6	−1.93034	−0.965168	+	0.261630i	$0.915740\pi$
$368$	−35659.9	−0.0137265
$369$	−437477.	−0.167259
$370$	−1.98114e6	−0.752335
$371$	0	0
$372$	593586.	0.222396
$373$	3.96047e6	1.47392	0.736962	−	0.675934i	$-0.236260\pi$
$373$	3.96047e6	1.47392	0.736962	+	0.675934i	$0.236260\pi$
$374$	5.26273e6	1.94550
$375$	−1.14476e6	−0.420373
$376$	−383205.	−0.139785
$377$	−5.61359e6	−2.03417
$378$	0	0
$379$	−1.75155e6	−0.626359	−0.313179	−	0.949694i	$-0.601394\pi$
$379$	−1.75155e6	−0.626359	−0.313179	+	0.949694i	$0.601394\pi$
$380$	2.66977e6	0.948452
$381$	740020.	0.261175
$382$	5.35020e6	1.87591
$383$	3.13834e6	1.09321	0.546604	−	0.837391i	$-0.315920\pi$
$383$	3.13834e6	1.09321	0.546604	+	0.837391i	$0.315920\pi$
$384$	2.74375e6	0.949549
$385$	0	0
$386$	3.68079e6	1.25740
$387$	1.59461e6	0.541224
$388$	−4.14849e6	−1.39898
$389$	−1.05252e6	−0.352661	−0.176331	−	0.984331i	$-0.556423\pi$
$389$	−1.05252e6	−0.352661	−0.176331	+	0.984331i	$0.556423\pi$
$390$	−1.45524e6	−0.484478
$391$	1.31420e6	0.434731
$392$	0	0
$393$	−1.58231e6	−0.516784
$394$	−6.16601e6	−2.00108
$395$	215434.	0.0694739
$396$	1.76514e6	0.565640
$397$	454724.	0.144801	0.0724005	−	0.997376i	$-0.476934\pi$
$397$	454724.	0.144801	0.0724005	+	0.997376i	$0.476934\pi$
$398$	4.17766e6	1.32198
$399$	0	0
$400$	−98456.3	−0.0307676
$401$	−2.88431e6	−0.895739	−0.447870	−	0.894099i	$-0.647817\pi$
$401$	−2.88431e6	−0.895739	−0.447870	+	0.894099i	$0.647817\pi$
$402$	−1.31023e6	−0.404374
$403$	1.00633e6	0.308657
$404$	−4.43638e6	−1.35231
$405$	144812.	0.0438699
$406$	0	0
$407$	4.07394e6	1.21907
$408$	2.30679e6	0.686053
$409$	−225914.	−0.0667782	−0.0333891	−	0.999442i	$-0.510630\pi$
$409$	−225914.	−0.0667782	−0.0333891	+	0.999442i	$0.510630\pi$
$410$	1.09447e6	0.321548
$411$	281973.	0.0823385
$412$	1.06328e6	0.308605
$413$	0	0
$414$	710513.	0.203738
$415$	−1.55375e6	−0.442853
$416$	4.48427e6	1.27045
$417$	1.37011e6	0.385846
$418$	−8.84942e6	−2.47727
$419$	4.31027e6	1.19941	0.599707	−	0.800220i	$-0.295284\pi$
$419$	4.31027e6	1.19941	0.599707	+	0.800220i	$0.295284\pi$
$420$	0	0
$421$	1.25088e6	0.343962	0.171981	−	0.985100i	$-0.444983\pi$
$421$	1.25088e6	0.343962	0.171981	+	0.985100i	$0.444983\pi$
$422$	1.09757e7	3.00020
$423$	166582.	0.0452665
$424$	−3.35823e6	−0.907184
$425$	3.62849e6	0.974436
$426$	−4.80363e6	−1.28246
$427$	0	0
$428$	364087.	0.0960719
$429$	2.99250e6	0.785039
$430$	−3.98937e6	−1.04048
$431$	−4.40793e6	−1.14299	−0.571494	−	0.820606i	$-0.693636\pi$
$431$	−4.40793e6	−1.14299	−0.571494	+	0.820606i	$0.693636\pi$
$432$	27209.6	0.00701475
$433$	−1.60951e6	−0.412549	−0.206274	−	0.978494i	$-0.566134\pi$
$433$	−1.60951e6	−0.412549	−0.206274	+	0.978494i	$0.566134\pi$
$434$	0	0
$435$	−1.39752e6	−0.354108
$436$	5.89110e6	1.48416
$437$	−2.20987e6	−0.553558
$438$	−3.23176e6	−0.804921
$439$	−4.42452e6	−1.09573	−0.547867	−	0.836566i	$-0.684560\pi$
$439$	−4.42452e6	−1.09573	−0.547867	+	0.836566i	$0.684560\pi$
$440$	−1.71379e6	−0.422012
$441$	0	0
$442$	1.00771e7	2.45347
$443$	−3.61438e6	−0.875034	−0.437517	−	0.899210i	$-0.644142\pi$
$443$	−3.61438e6	−0.875034	−0.437517	+	0.899210i	$0.644142\pi$
$444$	4.60133e6	1.10771
$445$	3.18507e6	0.762464
$446$	−2.72250e6	−0.648084
$447$	3.26574e6	0.773060
$448$	0	0
$449$	−467024.	−0.109326	−0.0546630	−	0.998505i	$-0.517408\pi$
$449$	−467024.	−0.109326	−0.0546630	+	0.998505i	$0.517408\pi$
$450$	1.96171e6	0.456671
$451$	−2.25063e6	−0.521029
$452$	5.85730e6	1.34850
$453$	1.84613e6	0.422685
$454$	−2.00285e6	−0.456046
$455$	0	0
$456$	−3.87893e6	−0.873575
$457$	−601252.	−0.134668	−0.0673342	−	0.997730i	$-0.521449\pi$
$457$	−601252.	−0.134668	−0.0673342	+	0.997730i	$0.521449\pi$
$458$	1.12856e7	2.51398
$459$	−1.00278e6	−0.222163
$460$	−1.10276e6	−0.242988
$461$	2.87193e6	0.629392	0.314696	−	0.949192i	$-0.398097\pi$
$461$	2.87193e6	0.629392	0.314696	+	0.949192i	$0.398097\pi$
$462$	0	0
$463$	−2.91502e6	−0.631959	−0.315979	−	0.948766i	$-0.602333\pi$
$463$	−2.91502e6	−0.631959	−0.315979	+	0.948766i	$0.602333\pi$
$464$	−262589.	−0.0566215
$465$	250529.	0.0537310
$466$	577906.	0.123280
$467$	−7.19376e6	−1.52638	−0.763192	−	0.646172i	$-0.776369\pi$
$467$	−7.19376e6	−1.52638	−0.763192	+	0.646172i	$0.776369\pi$
$468$	3.37989e6	0.713327
$469$	0	0
$470$	−416751.	−0.0870227
$471$	−695448.	−0.144448
$472$	−1.38551e6	−0.286256
$473$	8.20358e6	1.68597
$474$	−806536.	−0.164884
$475$	−6.10140e6	−1.24078
$476$	0	0
$477$	1.45984e6	0.293772
$478$	−2.01372e6	−0.403115
$479$	−2.79649e6	−0.556896	−0.278448	−	0.960451i	$-0.589820\pi$
$479$	−2.79649e6	−0.556896	−0.278448	+	0.960451i	$0.589820\pi$
$480$	1.11637e6	0.221160
$481$	7.80080e6	1.53736
$482$	−3.98340e6	−0.780974
$483$	0	0
$484$	658698.	0.127812
$485$	−1.75091e6	−0.337995
$486$	−542142.	−0.104117
$487$	−2.93247e6	−0.560289	−0.280144	−	0.959958i	$-0.590382\pi$
$487$	−2.93247e6	−0.560289	−0.280144	+	0.959958i	$0.590382\pi$
$488$	−651305.	−0.123804
$489$	1.66438e6	0.314761
$490$	0	0
$491$	−3.06121e6	−0.573046	−0.286523	−	0.958073i	$-0.592499\pi$
$491$	−3.06121e6	−0.573046	−0.286523	+	0.958073i	$0.592499\pi$
$492$	−2.54198e6	−0.473434
$493$	9.67740e6	1.79325
$494$	−1.69449e7	−3.12408
$495$	744993.	0.136659
$496$	47073.4	0.00859154
$497$	0	0
$498$	5.81687e6	1.05103
$499$	−6.55154e6	−1.17786	−0.588928	−	0.808186i	$-0.700450\pi$
$499$	−6.55154e6	−1.17786	−0.588928	+	0.808186i	$0.700450\pi$
$500$	−6.65167e6	−1.18989
$501$	1.16336e6	0.207071
$502$	1.57099e7	2.78237
$503$	−1.58524e6	−0.279367	−0.139684	−	0.990196i	$-0.544609\pi$
$503$	−1.58524e6	−0.279367	−0.139684	+	0.990196i	$0.544609\pi$
$504$	0	0
$505$	−1.87242e6	−0.326719
$506$	3.65528e6	0.634664
$507$	2.38842e6	0.412659
$508$	4.29993e6	0.739268
$509$	6.77981e6	1.15991	0.579953	−	0.814650i	$-0.303071\pi$
$509$	6.77981e6	1.15991	0.579953	+	0.814650i	$0.303071\pi$
$510$	2.50873e6	0.427099
$511$	0	0
$512$	432315.	0.0728829
$513$	1.68620e6	0.282888
$514$	−9.03486e6	−1.50839
$515$	448766.	0.0745593
$516$	9.26557e6	1.53196
$517$	856990.	0.141010
$518$	0	0
$519$	−4.57081e6	−0.744859
$520$	−3.28157e6	−0.532198
$521$	−1.01384e7	−1.63634	−0.818170	−	0.574977i	$-0.805011\pi$
$521$	−1.01384e7	−1.63634	−0.818170	+	0.574977i	$0.805011\pi$
$522$	5.23200e6	0.840411
$523$	99997.3	0.0159858	0.00799289	−	0.999968i	$-0.497456\pi$
$523$	99997.3	0.0159858	0.00799289	+	0.999968i	$0.497456\pi$
$524$	−9.19408e6	−1.46278
$525$	0	0
$526$	−5.02301e6	−0.791589
$527$	−1.73483e6	−0.272102
$528$	139981.	0.0218517
$529$	−5.52355e6	−0.858181
$530$	−3.65221e6	−0.564763
$531$	602290.	0.0926978
$532$	0	0
$533$	−4.30952e6	−0.657068
$534$	−1.19242e7	−1.80957
$535$	153667.	0.0232111
$536$	−2.95457e6	−0.444204
$537$	−1.19330e6	−0.178572
$538$	5.88620e6	0.876756
$539$	0	0
$540$	841436.	0.124176
$541$	−119743.	−0.0175896	−0.00879481	−	0.999961i	$-0.502800\pi$
$541$	−119743.	−0.0175896	−0.00879481	+	0.999961i	$0.502800\pi$
$542$	5.03143e6	0.735687
$543$	−6.66054e6	−0.969415
$544$	−7.73054e6	−1.11999
$545$	2.48640e6	0.358574
$546$	0	0
$547$	236568.	0.0338056	0.0169028	−	0.999857i	$-0.494619\pi$
$547$	236568.	0.0338056	0.0169028	+	0.999857i	$0.494619\pi$
$548$	1.63842e6	0.233063
$549$	283126.	0.0400912
$550$	1.00921e7	1.42258
$551$	−1.62728e7	−2.28341
$552$	1.60220e6	0.223805
$553$	0	0
$554$	−1.53707e7	−2.12774
$555$	1.94204e6	0.267624
$556$	7.96107e6	1.09216
$557$	−4.83666e6	−0.660553	−0.330277	−	0.943884i	$-0.607142\pi$
$557$	−4.83666e6	−0.660553	−0.330277	+	0.943884i	$0.607142\pi$
$558$	−937922.	−0.127521
$559$	1.57083e7	2.12617
$560$	0	0
$561$	−5.15885e6	−0.692063
$562$	−1.66252e7	−2.22037
$563$	−5.48295e6	−0.729026	−0.364513	−	0.931198i	$-0.618765\pi$
$563$	−5.48295e6	−0.729026	−0.364513	+	0.931198i	$0.618765\pi$
$564$	967932.	0.128129
$565$	2.47213e6	0.325800
$566$	−2.30738e7	−3.02746
$567$	0	0
$568$	−1.08322e7	−1.40878
$569$	−6.97660e6	−0.903364	−0.451682	−	0.892179i	$-0.649176\pi$
$569$	−6.97660e6	−0.903364	−0.451682	+	0.892179i	$0.649176\pi$
$570$	−4.21850e6	−0.543839
$571$	−1.17715e7	−1.51092	−0.755458	−	0.655197i	$-0.772585\pi$
$571$	−1.17715e7	−1.51092	−0.755458	+	0.655197i	$0.772585\pi$
$572$	1.73881e7	2.22209
$573$	−5.24459e6	−0.667306
$574$	0	0
$575$	2.52020e6	0.317882
$576$	−4.27620e6	−0.537034
$577$	6.77292e6	0.846909	0.423454	−	0.905917i	$-0.360817\pi$
$577$	6.77292e6	0.846909	0.423454	+	0.905917i	$0.360817\pi$
$578$	−4.33614e6	−0.539863
$579$	−3.60814e6	−0.447288
$580$	−8.12037e6	−1.00232
$581$	0	0
$582$	6.55501e6	0.802169
$583$	7.51025e6	0.915130
$584$	−7.28760e6	−0.884203
$585$	1.42652e6	0.172341
$586$	−984484.	−0.118431
$587$	−1.05020e7	−1.25799	−0.628996	−	0.777408i	$-0.716534\pi$
$587$	−1.05020e7	−1.25799	−0.628996	+	0.777408i	$0.716534\pi$
$588$	0	0
$589$	2.91717e6	0.346476
$590$	−1.50680e6	−0.178207
$591$	6.04430e6	0.711832
$592$	364901.	0.0427928
$593$	−7.59074e6	−0.886436	−0.443218	−	0.896414i	$-0.646163\pi$
$593$	−7.59074e6	−0.886436	−0.443218	+	0.896414i	$0.646163\pi$
$594$	−2.78909e6	−0.324336
$595$	0	0
$596$	1.89758e7	2.18818
$597$	−4.09519e6	−0.470261
$598$	6.99915e6	0.800373
$599$	−1.30198e7	−1.48265	−0.741325	−	0.671146i	$-0.765802\pi$
$599$	−1.30198e7	−1.48265	−0.741325	+	0.671146i	$0.765802\pi$
$600$	4.42365e6	0.501652
$601$	1.41821e7	1.60160	0.800801	−	0.598931i	$-0.204408\pi$
$601$	1.41821e7	1.60160	0.800801	+	0.598931i	$0.204408\pi$
$602$	0	0
$603$	1.28437e6	0.143846
$604$	1.07270e7	1.19643
$605$	278010.	0.0308796
$606$	7.00990e6	0.775408
$607$	1.20772e7	1.33044	0.665221	−	0.746646i	$-0.268337\pi$
$607$	1.20772e7	1.33044	0.665221	+	0.746646i	$0.268337\pi$
$608$	1.29991e7	1.42612
$609$	0	0
$610$	−708320.	−0.0770735
$611$	1.64097e6	0.177827
$612$	−5.82668e6	−0.628844
$613$	−2.72530e6	−0.292929	−0.146465	−	0.989216i	$-0.546789\pi$
$613$	−2.72530e6	−0.292929	−0.146465	+	0.989216i	$0.546789\pi$
$614$	−1.37371e7	−1.47053
$615$	−1.07287e6	−0.114382
$616$	0	0
$617$	8.47094e6	0.895816	0.447908	−	0.894080i	$-0.352169\pi$
$617$	8.47094e6	0.895816	0.447908	+	0.894080i	$0.352169\pi$
$618$	−1.68008e6	−0.176953
$619$	1.73835e7	1.82352	0.911759	−	0.410726i	$-0.134725\pi$
$619$	1.73835e7	1.82352	0.911759	+	0.410726i	$0.134725\pi$
$620$	1.45571e6	0.152088
$621$	−696488.	−0.0724744
$622$	−7.90690e6	−0.819464
$623$	0	0
$624$	268037.	0.0275571
$625$	5.43586e6	0.556632
$626$	4.62676e6	0.471891
$627$	8.67474e6	0.881227
$628$	−4.04093e6	−0.408868
$629$	−1.34480e7	−1.35529
$630$	0	0
$631$	−6.45149e6	−0.645040	−0.322520	−	0.946563i	$-0.604530\pi$
$631$	−6.45149e6	−0.645040	−0.322520	+	0.946563i	$0.604530\pi$
$632$	−1.81874e6	−0.181124
$633$	−1.07590e7	−1.06724
$634$	4.40707e6	0.435438
$635$	1.81483e6	0.178608
$636$	8.48249e6	0.831535
$637$	0	0
$638$	2.69164e7	2.61797
$639$	4.70881e6	0.456204
$640$	6.72879e6	0.649362
$641$	1.77716e7	1.70837	0.854185	−	0.519969i	$-0.174057\pi$
$641$	1.77716e7	1.70837	0.854185	+	0.519969i	$0.174057\pi$
$642$	−575293.	−0.0550873
$643$	−9.34806e6	−0.891649	−0.445825	−	0.895120i	$-0.647090\pi$
$643$	−9.34806e6	−0.891649	−0.445825	+	0.895120i	$0.647090\pi$
$644$	0	0
$645$	3.91063e6	0.370124
$646$	2.92118e7	2.75408
$647$	5.34386e6	0.501874	0.250937	−	0.968003i	$-0.419261\pi$
$647$	5.34386e6	0.501874	0.250937	+	0.968003i	$0.419261\pi$
$648$	−1.22253e6	−0.114373
$649$	3.09852e6	0.288764
$650$	1.93245e7	1.79401
$651$	0	0
$652$	9.67098e6	0.890946
$653$	−1.19701e6	−0.109854	−0.0549270	−	0.998490i	$-0.517493\pi$
$653$	−1.19701e6	−0.109854	−0.0549270	+	0.998490i	$0.517493\pi$
$654$	−9.30851e6	−0.851012
$655$	−3.88045e6	−0.353410
$656$	−201588.	−0.0182896
$657$	3.16796e6	0.286330
$658$	0	0
$659$	−1.17541e7	−1.05433	−0.527163	−	0.849764i	$-0.676744\pi$
$659$	−1.17541e7	−1.05433	−0.527163	+	0.849764i	$0.676744\pi$
$660$	4.32882e6	0.386821
$661$	−1.80115e7	−1.60341	−0.801706	−	0.597719i	$-0.796074\pi$
$661$	−1.80115e7	−1.60341	−0.801706	+	0.597719i	$0.796074\pi$
$662$	2.01915e7	1.79071
$663$	−9.87819e6	−0.872758
$664$	1.31170e7	1.15456
$665$	0	0
$666$	−7.27054e6	−0.635157
$667$	6.72153e6	0.584997
$668$	6.75975e6	0.586124
$669$	2.66876e6	0.230539
$670$	−3.21322e6	−0.276537
$671$	1.45656e6	0.124888
$672$	0	0
$673$	1.40977e7	1.19981	0.599904	−	0.800072i	$-0.295206\pi$
$673$	1.40977e7	1.19981	0.599904	+	0.800072i	$0.295206\pi$
$674$	1.24613e7	1.05660
$675$	−1.92299e6	−0.162449
$676$	1.38780e7	1.16805
$677$	7.65587e6	0.641982	0.320991	−	0.947082i	$-0.395984\pi$
$677$	7.65587e6	0.641982	0.320991	+	0.947082i	$0.395984\pi$
$678$	−9.25510e6	−0.773227
$679$	0	0
$680$	5.65718e6	0.469167
$681$	1.96331e6	0.162227
$682$	−4.82520e6	−0.397241
$683$	−1.08676e7	−0.891415	−0.445708	−	0.895179i	$-0.647048\pi$
$683$	−1.08676e7	−0.891415	−0.445708	+	0.895179i	$0.647048\pi$
$684$	9.79773e6	0.800728
$685$	691510.	0.0563083
$686$	0	0
$687$	−1.10628e7	−0.894283
$688$	734791.	0.0591825
$689$	1.43807e7	1.15407
$690$	1.74246e6	0.139329
$691$	−1.46118e7	−1.16415	−0.582073	−	0.813136i	$-0.697758\pi$
$691$	−1.46118e7	−1.16415	−0.582073	+	0.813136i	$0.697758\pi$
$692$	−2.65589e7	−2.10836
$693$	0	0
$694$	−3.48832e7	−2.74927
$695$	3.36005e6	0.263866
$696$	1.17981e7	0.923189
$697$	7.42928e6	0.579248
$698$	1.20965e7	0.939772
$699$	−566498.	−0.0438536
$700$	0	0
$701$	1.90104e7	1.46115	0.730577	−	0.682830i	$-0.239251\pi$
$701$	1.90104e7	1.46115	0.730577	+	0.682830i	$0.239251\pi$
$702$	−5.34056e6	−0.409019
$703$	2.26132e7	1.72573
$704$	−2.19992e7	−1.67292
$705$	408525.	0.0309561
$706$	3.14462e7	2.37441
$707$	0	0
$708$	3.49964e6	0.262386
$709$	−1.15929e7	−0.866116	−0.433058	−	0.901366i	$-0.642565\pi$
$709$	−1.15929e7	−0.866116	−0.433058	+	0.901366i	$0.642565\pi$
$710$	−1.17804e7	−0.877031
$711$	790616.	0.0586532
$712$	−2.68890e7	−1.98781
$713$	−1.20494e6	−0.0887653
$714$	0	0
$715$	7.33881e6	0.536859
$716$	−6.93374e6	−0.505458
$717$	1.97397e6	0.143398
$718$	−1.35041e7	−0.977585
$719$	−1.02861e7	−0.742040	−0.371020	−	0.928625i	$-0.620992\pi$
$719$	−1.02861e7	−0.742040	−0.371020	+	0.928625i	$0.620992\pi$
$720$	66728.8	0.00479713
$721$	0	0
$722$	−2.63868e7	−1.88384
$723$	3.90477e6	0.277811
$724$	−3.87014e7	−2.74398
$725$	1.85580e7	1.31125
$726$	−1.04081e6	−0.0732873
$727$	1.00970e7	0.708526	0.354263	−	0.935146i	$-0.384732\pi$
$727$	1.00970e7	0.708526	0.354263	+	0.935146i	$0.384732\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	−7.92556e6	−0.550456
$731$	−2.70799e7	−1.87436
$732$	1.64512e6	0.113480
$733$	1.92402e6	0.132266	0.0661332	−	0.997811i	$-0.478934\pi$
$733$	1.92402e6	0.132266	0.0661332	+	0.997811i	$0.478934\pi$
$734$	4.57298e7	3.13299
$735$	0	0
$736$	−5.36932e6	−0.365363
$737$	6.60752e6	0.448095
$738$	4.01658e6	0.271466
$739$	4.20273e6	0.283087	0.141543	−	0.989932i	$-0.454793\pi$
$739$	4.20273e6	0.283087	0.141543	+	0.989932i	$0.454793\pi$
$740$	1.12843e7	0.757522
$741$	1.66104e7	1.11131
$742$	0	0
$743$	−1.99659e7	−1.32684	−0.663418	−	0.748249i	$-0.730895\pi$
$743$	−1.99659e7	−1.32684	−0.663418	+	0.748249i	$0.730895\pi$
$744$	−2.11501e6	−0.140081
$745$	8.00891e6	0.528667
$746$	−3.63620e7	−2.39222
$747$	−5.70205e6	−0.373878
$748$	−2.99757e7	−1.95892
$749$	0	0
$750$	1.05103e7	0.682277
$751$	−3.63975e6	−0.235490	−0.117745	−	0.993044i	$-0.537567\pi$
$751$	−3.63975e6	−0.235490	−0.117745	+	0.993044i	$0.537567\pi$
$752$	76760.3	0.00494985
$753$	−1.53998e7	−0.989757
$754$	5.15396e7	3.30151
$755$	4.52745e6	0.289059
$756$	0	0
$757$	1.73429e7	1.09997	0.549986	−	0.835174i	$-0.314633\pi$
$757$	1.73429e7	1.09997	0.549986	+	0.835174i	$0.314633\pi$
$758$	1.60813e7	1.01660
$759$	−3.58313e6	−0.225765
$760$	−9.51270e6	−0.597406
$761$	−1.03568e7	−0.648284	−0.324142	−	0.946009i	$-0.605076\pi$
$761$	−1.03568e7	−0.648284	−0.324142	+	0.946009i	$0.605076\pi$
$762$	−6.79430e6	−0.423894
$763$	0	0
$764$	−3.04740e7	−1.88884
$765$	−2.45921e6	−0.151929
$766$	−2.88138e7	−1.77431
$767$	5.93306e6	0.364159
$768$	−9.98679e6	−0.610974
$769$	−1.81548e7	−1.10707	−0.553534	−	0.832826i	$-0.686721\pi$
$769$	−1.81548e7	−1.10707	−0.553534	+	0.832826i	$0.686721\pi$
$770$	0	0
$771$	8.85652e6	0.536571
$772$	−2.09653e7	−1.26607
$773$	−1.63566e7	−0.984565	−0.492282	−	0.870435i	$-0.663837\pi$
$773$	−1.63566e7	−0.984565	−0.492282	+	0.870435i	$0.663837\pi$
$774$	−1.46405e7	−0.878422
$775$	−3.32683e6	−0.198965
$776$	1.47815e7	0.881181
$777$	0	0
$778$	9.66346e6	0.572379
$779$	−1.24925e7	−0.737576
$780$	8.28886e6	0.487818
$781$	2.42248e7	1.42112
$782$	−1.20660e7	−0.705581
$783$	−5.12873e6	−0.298955
$784$	0	0
$785$	−1.70552e6	−0.0987829
$786$	1.45275e7	0.838755
$787$	−4.44728e6	−0.255952	−0.127976	−	0.991777i	$-0.540848\pi$
$787$	−4.44728e6	−0.255952	−0.127976	+	0.991777i	$0.540848\pi$
$788$	3.51207e7	2.01487
$789$	4.92386e6	0.281587
$790$	−1.97795e6	−0.112758
$791$	0	0
$792$	−6.28938e6	−0.356283
$793$	2.78903e6	0.157496
$794$	−4.17493e6	−0.235016
$795$	3.58012e6	0.200900
$796$	−2.37953e7	−1.33110
$797$	−9.15303e6	−0.510410	−0.255205	−	0.966887i	$-0.582143\pi$
$797$	−9.15303e6	−0.510410	−0.255205	+	0.966887i	$0.582143\pi$
$798$	0	0
$799$	−2.82891e6	−0.156766
$800$	−1.48246e7	−0.818950
$801$	1.16888e7	0.643708
$802$	2.64815e7	1.45381
$803$	1.62978e7	0.891948
$804$	7.46290e6	0.407162
$805$	0	0
$806$	−9.23932e6	−0.500959
$807$	−5.77001e6	−0.311884
$808$	1.58073e7	0.851784
$809$	2.51923e7	1.35331	0.676654	−	0.736301i	$-0.263429\pi$
$809$	2.51923e7	1.35331	0.676654	+	0.736301i	$0.263429\pi$
$810$	−1.32955e6	−0.0712020
$811$	−8.90585e6	−0.475470	−0.237735	−	0.971330i	$-0.576405\pi$
$811$	−8.90585e6	−0.475470	−0.237735	+	0.971330i	$0.576405\pi$
$812$	0	0
$813$	−4.93211e6	−0.261702
$814$	−3.74038e7	−1.97858
$815$	4.08173e6	0.215254
$816$	−462076.	−0.0242934
$817$	4.55355e7	2.38669
$818$	2.07417e6	0.108383
$819$	0	0
$820$	−6.23396e6	−0.323765
$821$	3.35490e7	1.73709	0.868543	−	0.495613i	$-0.165057\pi$
$821$	3.35490e7	1.73709	0.868543	+	0.495613i	$0.165057\pi$
$822$	−2.58886e6	−0.133638
$823$	−1.46001e7	−0.751376	−0.375688	−	0.926746i	$-0.622593\pi$
$823$	−1.46001e7	−0.751376	−0.375688	+	0.926746i	$0.622593\pi$
$824$	−3.78857e6	−0.194382
$825$	−9.89293e6	−0.506046
$826$	0	0
$827$	1.97530e7	1.00431	0.502156	−	0.864777i	$-0.332540\pi$
$827$	1.97530e7	1.00431	0.502156	+	0.864777i	$0.332540\pi$
$828$	−4.04698e6	−0.205142
$829$	−4.06255e6	−0.205311	−0.102655	−	0.994717i	$-0.532734\pi$
$829$	−4.06255e6	−0.205311	−0.102655	+	0.994717i	$0.532734\pi$
$830$	1.42653e7	0.718762
$831$	1.50673e7	0.756889
$832$	−4.21241e7	−2.10971
$833$	0	0
$834$	−1.25793e7	−0.626239
$835$	2.85302e6	0.141608
$836$	5.04050e7	2.49435
$837$	919409.	0.0453623
$838$	−3.95735e7	−1.94668
$839$	−1.60509e7	−0.787217	−0.393609	−	0.919278i	$-0.628773\pi$
$839$	−1.60509e7	−0.787217	−0.393609	+	0.919278i	$0.628773\pi$
$840$	0	0
$841$	2.89842e7	1.41309
$842$	−1.14846e7	−0.558260
$843$	1.62970e7	0.789839
$844$	−6.25159e7	−3.02088
$845$	5.85737e6	0.282202
$846$	−1.52943e6	−0.0734687
$847$	0	0
$848$	672690.	0.0321237
$849$	2.26183e7	1.07694
$850$	−3.33140e7	−1.58154
$851$	−9.34043e6	−0.442123
$852$	2.73608e7	1.29131
$853$	−1.23887e7	−0.582979	−0.291489	−	0.956574i	$-0.594151\pi$
$853$	−1.23887e7	−0.582979	−0.291489	+	0.956574i	$0.594151\pi$
$854$	0	0
$855$	4.13523e6	0.193457
$856$	−1.29728e6	−0.0605133
$857$	2.63519e7	1.22563	0.612816	−	0.790226i	$-0.290037\pi$
$857$	2.63519e7	1.22563	0.612816	+	0.790226i	$0.290037\pi$
$858$	−2.74748e7	−1.27414
$859$	−1.21249e7	−0.560654	−0.280327	−	0.959905i	$-0.590443\pi$
$859$	−1.21249e7	−0.560654	−0.280327	+	0.959905i	$0.590443\pi$
$860$	2.27229e7	1.04765
$861$	0	0
$862$	4.04702e7	1.85510
$863$	2.55952e6	0.116985	0.0584927	−	0.998288i	$-0.481371\pi$
$863$	2.55952e6	0.116985	0.0584927	+	0.998288i	$0.481371\pi$
$864$	4.09695e6	0.186714
$865$	−1.12094e7	−0.509382
$866$	1.47773e7	0.669578
$867$	4.25055e6	0.192042
$868$	0	0
$869$	4.06737e6	0.182711
$870$	1.28310e7	0.574726
$871$	1.26521e7	0.565091
$872$	−2.09906e7	−0.934834
$873$	−6.42562e6	−0.285351
$874$	2.02893e7	0.898439
$875$	0	0
$876$	1.84076e7	0.810471
$877$	2.65820e7	1.16705	0.583523	−	0.812096i	$-0.301674\pi$
$877$	2.65820e7	1.16705	0.583523	+	0.812096i	$0.301674\pi$
$878$	4.06225e7	1.77841
$879$	965051.	0.0421287
$880$	343290.	0.0149436
$881$	−8.32262e6	−0.361260	−0.180630	−	0.983551i	$-0.557814\pi$
$881$	−8.32262e6	−0.361260	−0.180630	+	0.983551i	$0.557814\pi$
$882$	0	0
$883$	1.81133e7	0.781798	0.390899	−	0.920434i	$-0.372164\pi$
$883$	1.81133e7	0.781798	0.390899	+	0.920434i	$0.372164\pi$
$884$	−5.73977e7	−2.47038
$885$	1.47706e6	0.0633927
$886$	3.31845e7	1.42020
$887$	−2.04255e7	−0.871694	−0.435847	−	0.900021i	$-0.643551\pi$
$887$	−2.04255e7	−0.871694	−0.435847	+	0.900021i	$0.643551\pi$
$888$	−1.63950e7	−0.697718
$889$	0	0
$890$	−2.92429e7	−1.23750
$891$	2.73403e6	0.115374
$892$	1.55070e7	0.652552
$893$	4.75689e6	0.199616
$894$	−2.99835e7	−1.25470
$895$	−2.92645e6	−0.122119
$896$	0	0
$897$	−6.86099e6	−0.284712
$898$	4.28785e6	0.177439
$899$	−8.87285e6	−0.366154
$900$	−1.11736e7	−0.459820
$901$	−2.47912e7	−1.01739
$902$	2.06635e7	0.845645
$903$	0	0
$904$	−2.08702e7	−0.849388
$905$	−1.63343e7	−0.662948
$906$	−1.69497e7	−0.686029
$907$	−2.01197e7	−0.812089	−0.406044	−	0.913853i	$-0.633092\pi$
$907$	−2.01197e7	−0.812089	−0.406044	+	0.913853i	$0.633092\pi$
$908$	1.14079e7	0.459190
$909$	−6.87153e6	−0.275832
$910$	0	0
$911$	3.17075e7	1.26580	0.632902	−	0.774232i	$-0.281863\pi$
$911$	3.17075e7	1.26580	0.632902	+	0.774232i	$0.281863\pi$
$912$	776993.	0.0309336
$913$	−2.93345e7	−1.16467
$914$	5.52023e6	0.218571
$915$	694339.	0.0274169
$916$	−6.42812e7	−2.53131
$917$	0	0
$918$	9.20672e6	0.360577
$919$	−8.70727e6	−0.340090	−0.170045	−	0.985436i	$-0.554391\pi$
$919$	−8.70727e6	−0.340090	−0.170045	+	0.985436i	$0.554391\pi$
$920$	3.92924e6	0.153052
$921$	1.34659e7	0.523104
$922$	−2.63678e7	−1.02152
$923$	4.63857e7	1.79217
$924$	0	0
$925$	−2.57887e7	−0.991005
$926$	2.67634e7	1.02569
$927$	1.64691e6	0.0629465
$928$	−3.95381e7	−1.50711
$929$	3.24042e7	1.23186	0.615931	−	0.787800i	$-0.288780\pi$
$929$	3.24042e7	1.23186	0.615931	+	0.787800i	$0.288780\pi$
$930$	−2.30016e6	−0.0872069
$931$	0	0
$932$	−3.29167e6	−0.124130
$933$	7.75082e6	0.291503
$934$	6.60475e7	2.47736
$935$	−1.26516e7	−0.473277
$936$	−1.20429e7	−0.449307
$937$	−2.19155e7	−0.815458	−0.407729	−	0.913103i	$-0.633679\pi$
$937$	−2.19155e7	−0.815458	−0.407729	+	0.913103i	$0.633679\pi$
$938$	0	0
$939$	−4.53544e6	−0.167863
$940$	2.37376e6	0.0876227
$941$	1.46019e7	0.537571	0.268785	−	0.963200i	$-0.413378\pi$
$941$	1.46019e7	0.537571	0.268785	+	0.963200i	$0.413378\pi$
$942$	6.38506e6	0.234443
$943$	5.16008e6	0.188963
$944$	277533.	0.0101364
$945$	0	0
$946$	−7.53189e7	−2.73638
$947$	−2.08378e7	−0.755053	−0.377527	−	0.925999i	$-0.623225\pi$
$947$	−2.08378e7	−0.755053	−0.377527	+	0.925999i	$0.623225\pi$
$948$	4.59391e6	0.166021
$949$	3.12071e7	1.12483
$950$	5.60184e7	2.01382
$951$	−4.32008e6	−0.154896
$952$	0	0
$953$	−942012.	−0.0335988	−0.0167994	−	0.999859i	$-0.505348\pi$
$953$	−942012.	−0.0335988	−0.0167994	+	0.999859i	$0.505348\pi$
$954$	−1.34031e7	−0.476800
$955$	−1.28618e7	−0.456346
$956$	1.14698e7	0.405894
$957$	−2.63851e7	−0.931275
$958$	2.56752e7	0.903857
$959$	0	0
$960$	−1.04869e7	−0.367258
$961$	−2.70385e7	−0.944441
$962$	−7.16209e7	−2.49518
$963$	563937.	0.0195959
$964$	2.26889e7	0.786359
$965$	−8.84860e6	−0.305884
$966$	0	0
$967$	−2.56570e7	−0.882346	−0.441173	−	0.897422i	$-0.645438\pi$
$967$	−2.56570e7	−0.882346	−0.441173	+	0.897422i	$0.645438\pi$
$968$	−2.34701e6	−0.0805059
$969$	−2.86352e7	−0.979694
$970$	1.60755e7	0.548575
$971$	−2.14274e7	−0.729324	−0.364662	−	0.931140i	$-0.618816\pi$
$971$	−2.14274e7	−0.729324	−0.364662	+	0.931140i	$0.618816\pi$
$972$	3.08797e6	0.104835
$973$	0	0
$974$	2.69237e7	0.909364
$975$	−1.89431e7	−0.638173
$976$	130464.	0.00438394
$977$	−2.04841e7	−0.686562	−0.343281	−	0.939233i	$-0.611538\pi$
$977$	−2.04841e7	−0.686562	−0.343281	+	0.939233i	$0.611538\pi$
$978$	−1.52811e7	−0.510866
$979$	6.01338e7	2.00522
$980$	0	0
$981$	9.12476e6	0.302726
$982$	2.81057e7	0.930069
$983$	2.77650e7	0.916459	0.458230	−	0.888834i	$-0.348484\pi$
$983$	2.77650e7	0.916459	0.458230	+	0.888834i	$0.348484\pi$
$984$	9.05736e6	0.298204
$985$	1.48230e7	0.486796
$986$	−8.88504e7	−2.91050
$987$	0	0
$988$	9.65159e7	3.14562
$989$	−1.88086e7	−0.611456
$990$	−6.83995e6	−0.221802
$991$	3.46832e7	1.12185	0.560926	−	0.827866i	$-0.310445\pi$
$991$	3.46832e7	1.12185	0.560926	+	0.827866i	$0.310445\pi$
$992$	7.08785e6	0.228684
$993$	−1.97930e7	−0.636998
$994$	0	0
$995$	−1.00431e7	−0.321594
$996$	−3.31320e7	−1.05828
$997$	−1.48572e7	−0.473369	−0.236685	−	0.971587i	$-0.576061\pi$
$997$	−1.48572e7	−0.473369	−0.236685	+	0.971587i	$0.576061\pi$
$998$	6.01512e7	1.91169
$999$	7.12703e6	0.225941

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.6.a.m.1.1		4
3.2	odd	2		441.6.a.w.1.4		4
7.2	even	3		21.6.e.c.4.4	✓	8
7.3	odd	6		147.6.e.o.79.4		8
7.4	even	3		21.6.e.c.16.4	yes	8
7.5	odd	6		147.6.e.o.67.4		8
7.6	odd	2		147.6.a.l.1.1		4
21.2	odd	6		63.6.e.e.46.1		8
21.11	odd	6		63.6.e.e.37.1		8
21.20	even	2		441.6.a.v.1.4		4
28.11	odd	6		336.6.q.j.289.3		8
28.23	odd	6		336.6.q.j.193.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.e.c.4.4	✓	8	7.2	even	3
21.6.e.c.16.4	yes	8	7.4	even	3
63.6.e.e.37.1		8	21.11	odd	6
63.6.e.e.46.1		8	21.2	odd	6
147.6.a.l.1.1		4	7.6	odd	2
147.6.a.m.1.1		4	1.1	even	1	trivial
147.6.e.o.67.4		8	7.5	odd	6
147.6.e.o.79.4		8	7.3	odd	6
336.6.q.j.193.3		8	28.23	odd	6
336.6.q.j.289.3		8	28.11	odd	6
441.6.a.v.1.4		4	21.20	even	2
441.6.a.w.1.4		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	147.6.a.m.1.1

Level	$147$
Weight	$6$
Character	147.1
Self dual	yes
Analytic conductor	$23.576$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$