LMFDB - Embedded newform 819.6.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,6,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, 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("819.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	819.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:	$131.354348427$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 148x^{4} - 156x^{3} + 4763x^{2} + 5973x - 23760 $ x^6 - x^5 - 148x^4 - 156x^3 + 4763x^2 + 5973x - 23760
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.45521$ of defining polynomial
Character	$\chi$	$=$	819.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.45521 q^{2} -12.1511 q^{4} -39.9280 q^{5} -49.0000 q^{7} -196.702 q^{8} +O(q^{10})$	\(q+4.45521 q^{2} -12.1511 q^{4} -39.9280 q^{5} -49.0000 q^{7} -196.702 q^{8} -177.888 q^{10} -775.077 q^{11} -169.000 q^{13} -218.305 q^{14} -487.516 q^{16} +667.181 q^{17} -998.815 q^{19} +485.169 q^{20} -3453.13 q^{22} -3701.16 q^{23} -1530.75 q^{25} -752.931 q^{26} +595.403 q^{28} -4020.48 q^{29} +1214.77 q^{31} +4122.49 q^{32} +2972.43 q^{34} +1956.47 q^{35} -10629.6 q^{37} -4449.93 q^{38} +7853.94 q^{40} -6442.35 q^{41} -9841.27 q^{43} +9418.03 q^{44} -16489.5 q^{46} -8507.75 q^{47} +2401.00 q^{49} -6819.83 q^{50} +2053.53 q^{52} -31042.8 q^{53} +30947.3 q^{55} +9638.42 q^{56} -17912.1 q^{58} +43092.2 q^{59} +31520.0 q^{61} +5412.07 q^{62} +33967.1 q^{64} +6747.84 q^{65} +7199.44 q^{67} -8106.97 q^{68} +8716.50 q^{70} +35458.2 q^{71} +75921.6 q^{73} -47357.1 q^{74} +12136.7 q^{76} +37978.8 q^{77} -5278.91 q^{79} +19465.6 q^{80} -28702.0 q^{82} -65825.0 q^{83} -26639.2 q^{85} -43845.0 q^{86} +152460. q^{88} -113052. q^{89} +8281.00 q^{91} +44973.2 q^{92} -37903.8 q^{94} +39880.7 q^{95} +33688.3 q^{97} +10697.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{2} + 109 q^{4} - 112 q^{5} - 294 q^{7} - 339 q^{8}+O(q^{10}) $ 6 * q + 5 * q^2 + 109 * q^4 - 112 * q^5 - 294 * q^7 - 339 * q^8	$ 6 q + 5 q^{2} + 109 q^{4} - 112 q^{5} - 294 q^{7} - 339 q^{8} - 417 q^{10} - 360 q^{11} - 1014 q^{13} - 245 q^{14} + 1353 q^{16} - 2708 q^{17} + 6892 q^{19} - 3261 q^{20} - 33 q^{22} - 9744 q^{23} + 2526 q^{25} - 845 q^{26} - 5341 q^{28} + 70 q^{29} - 130 q^{31} - 31303 q^{32} + 3917 q^{34} + 5488 q^{35} + 11512 q^{37} - 17799 q^{38} + 19339 q^{40} + 912 q^{41} - 34918 q^{43} - 6503 q^{44} + 19333 q^{46} - 8594 q^{47} + 14406 q^{49} + 83560 q^{50} - 18421 q^{52} + 10102 q^{53} - 14552 q^{55} + 16611 q^{56} - 8989 q^{58} + 43260 q^{59} + 137538 q^{61} + 19690 q^{62} + 16129 q^{64} + 18928 q^{65} + 8280 q^{67} + 42295 q^{68} + 20433 q^{70} - 62380 q^{71} + 76670 q^{73} - 10387 q^{74} + 287197 q^{76} + 17640 q^{77} + 21154 q^{79} - 182689 q^{80} + 223876 q^{82} - 223186 q^{83} + 277084 q^{85} + 64747 q^{86} + 126093 q^{88} - 235190 q^{89} + 49686 q^{91} + 121233 q^{92} + 8238 q^{94} - 113200 q^{95} + 108602 q^{97} + 12005 q^{98}+O(q^{100}) $ 6 * q + 5 * q^2 + 109 * q^4 - 112 * q^5 - 294 * q^7 - 339 * q^8 - 417 * q^10 - 360 * q^11 - 1014 * q^13 - 245 * q^14 + 1353 * q^16 - 2708 * q^17 + 6892 * q^19 - 3261 * q^20 - 33 * q^22 - 9744 * q^23 + 2526 * q^25 - 845 * q^26 - 5341 * q^28 + 70 * q^29 - 130 * q^31 - 31303 * q^32 + 3917 * q^34 + 5488 * q^35 + 11512 * q^37 - 17799 * q^38 + 19339 * q^40 + 912 * q^41 - 34918 * q^43 - 6503 * q^44 + 19333 * q^46 - 8594 * q^47 + 14406 * q^49 + 83560 * q^50 - 18421 * q^52 + 10102 * q^53 - 14552 * q^55 + 16611 * q^56 - 8989 * q^58 + 43260 * q^59 + 137538 * q^61 + 19690 * q^62 + 16129 * q^64 + 18928 * q^65 + 8280 * q^67 + 42295 * q^68 + 20433 * q^70 - 62380 * q^71 + 76670 * q^73 - 10387 * q^74 + 287197 * q^76 + 17640 * q^77 + 21154 * q^79 - 182689 * q^80 + 223876 * q^82 - 223186 * q^83 + 277084 * q^85 + 64747 * q^86 + 126093 * q^88 - 235190 * q^89 + 49686 * q^91 + 121233 * q^92 + 8238 * q^94 - 113200 * q^95 + 108602 * q^97 + 12005 * q^98

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 $

$2$	4.45521	0.787578	0.393789	−	0.919201i	$-0.371164\pi$
$2$	4.45521	0.787578	0.393789	+	0.919201i	$0.371164\pi$
$3$	0	0
$3$	0	0
$4$	−12.1511	−0.379721
$5$	−39.9280	−0.714254	−0.357127	−	0.934056i	$-0.616244\pi$
$5$	−39.9280	−0.714254	−0.357127	+	0.934056i	$0.616244\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−196.702	−1.08664
$9$	0	0
$10$	−177.888	−0.562531
$11$	−775.077	−1.93136	−0.965680	−	0.259735i	$-0.916365\pi$
$11$	−775.077	−1.93136	−0.965680	+	0.259735i	$0.916365\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−218.305	−0.297676
$15$	0	0
$16$	−487.516	−0.476090
$17$	667.181	0.559914	0.279957	−	0.960013i	$-0.409680\pi$
$17$	667.181	0.559914	0.279957	+	0.960013i	$0.409680\pi$
$18$	0	0
$19$	−998.815	−0.634748	−0.317374	−	0.948300i	$-0.602801\pi$
$19$	−998.815	−0.634748	−0.317374	+	0.948300i	$0.602801\pi$
$20$	485.169	0.271218
$21$	0	0
$22$	−3453.13	−1.52110
$23$	−3701.16	−1.45888	−0.729439	−	0.684046i	$-0.760219\pi$
$23$	−3701.16	−1.45888	−0.729439	+	0.684046i	$0.760219\pi$
$24$	0	0
$25$	−1530.75	−0.489841
$26$	−752.931	−0.218435
$27$	0	0
$28$	595.403	0.143521
$29$	−4020.48	−0.887733	−0.443867	−	0.896093i	$-0.646394\pi$
$29$	−4020.48	−0.887733	−0.443867	+	0.896093i	$0.646394\pi$
$30$	0	0
$31$	1214.77	0.227034	0.113517	−	0.993536i	$-0.463788\pi$
$31$	1214.77	0.227034	0.113517	+	0.993536i	$0.463788\pi$
$32$	4122.49	0.711680
$33$	0	0
$34$	2972.43	0.440976
$35$	1956.47	0.269963
$36$	0	0
$37$	−10629.6	−1.27648	−0.638238	−	0.769839i	$-0.720336\pi$
$37$	−10629.6	−1.27648	−0.638238	+	0.769839i	$0.720336\pi$
$38$	−4449.93	−0.499913
$39$	0	0
$40$	7853.94	0.776136
$41$	−6442.35	−0.598528	−0.299264	−	0.954170i	$-0.596741\pi$
$41$	−6442.35	−0.598528	−0.299264	+	0.954170i	$0.596741\pi$
$42$	0	0
$43$	−9841.27	−0.811671	−0.405836	−	0.913946i	$-0.633019\pi$
$43$	−9841.27	−0.811671	−0.405836	+	0.913946i	$0.633019\pi$
$44$	9418.03	0.733379
$45$	0	0
$46$	−16489.5	−1.14898
$47$	−8507.75	−0.561785	−0.280892	−	0.959739i	$-0.590630\pi$
$47$	−8507.75	−0.561785	−0.280892	+	0.959739i	$0.590630\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	−6819.83	−0.385788
$51$	0	0
$52$	2053.53	0.105316
$53$	−31042.8	−1.51800	−0.758999	−	0.651092i	$-0.774311\pi$
$53$	−31042.8	−1.51800	−0.758999	+	0.651092i	$0.774311\pi$
$54$	0	0
$55$	30947.3	1.37948
$56$	9638.42	0.410710
$57$	0	0
$58$	−17912.1	−0.699159
$59$	43092.2	1.61164	0.805821	−	0.592159i	$-0.201724\pi$
$59$	43092.2	1.61164	0.805821	+	0.592159i	$0.201724\pi$
$60$	0	0
$61$	31520.0	1.08458	0.542291	−	0.840191i	$-0.317557\pi$
$61$	31520.0	1.08458	0.542291	+	0.840191i	$0.317557\pi$
$62$	5412.07	0.178807
$63$	0	0
$64$	33967.1	1.03659
$65$	6747.84	0.198098
$66$	0	0
$67$	7199.44	0.195935	0.0979675	−	0.995190i	$-0.468766\pi$
$67$	7199.44	0.195935	0.0979675	+	0.995190i	$0.468766\pi$
$68$	−8106.97	−0.212611
$69$	0	0
$70$	8716.50	0.212617
$71$	35458.2	0.834778	0.417389	−	0.908728i	$-0.362945\pi$
$71$	35458.2	0.834778	0.417389	+	0.908728i	$0.362945\pi$
$72$	0	0
$73$	75921.6	1.66747	0.833735	−	0.552165i	$-0.186198\pi$
$73$	75921.6	1.66747	0.833735	+	0.552165i	$0.186198\pi$
$74$	−47357.1	−1.00532
$75$	0	0
$76$	12136.7	0.241027
$77$	37978.8	0.729985
$78$	0	0
$79$	−5278.91	−0.0951648	−0.0475824	−	0.998867i	$-0.515152\pi$
$79$	−5278.91	−0.0951648	−0.0475824	+	0.998867i	$0.515152\pi$
$80$	19465.6	0.340049
$81$	0	0
$82$	−28702.0	−0.471387
$83$	−65825.0	−1.04881	−0.524403	−	0.851470i	$-0.675712\pi$
$83$	−65825.0	−1.04881	−0.524403	+	0.851470i	$0.675712\pi$
$84$	0	0
$85$	−26639.2	−0.399921
$86$	−43845.0	−0.639254
$87$	0	0
$88$	152460.	2.09869
$89$	−113052.	−1.51287	−0.756435	−	0.654069i	$-0.773061\pi$
$89$	−113052.	−1.51287	−0.756435	+	0.654069i	$0.773061\pi$
$90$	0	0
$91$	8281.00	0.104828
$92$	44973.2	0.553967
$93$	0	0
$94$	−37903.8	−0.442449
$95$	39880.7	0.453371
$96$	0	0
$97$	33688.3	0.363538	0.181769	−	0.983341i	$-0.441818\pi$
$97$	33688.3	0.363538	0.181769	+	0.983341i	$0.441818\pi$
$98$	10697.0	0.112511
$99$	0	0
$100$	18600.3	0.186003
$101$	−91261.1	−0.890189	−0.445094	−	0.895484i	$-0.646830\pi$
$101$	−91261.1	−0.890189	−0.445094	+	0.895484i	$0.646830\pi$
$102$	0	0
$103$	84766.1	0.787280	0.393640	−	0.919265i	$-0.371216\pi$
$103$	84766.1	0.787280	0.393640	+	0.919265i	$0.371216\pi$
$104$	33242.7	0.301379
$105$	0	0
$106$	−138302.	−1.19554
$107$	−153559.	−1.29663	−0.648315	−	0.761373i	$-0.724526\pi$
$107$	−153559.	−1.29663	−0.648315	+	0.761373i	$0.724526\pi$
$108$	0	0
$109$	−43332.3	−0.349338	−0.174669	−	0.984627i	$-0.555885\pi$
$109$	−43332.3	−0.349338	−0.174669	+	0.984627i	$0.555885\pi$
$110$	137877.	1.08645
$111$	0	0
$112$	23888.3	0.179945
$113$	−159162.	−1.17258	−0.586289	−	0.810102i	$-0.699412\pi$
$113$	−159162.	−1.17258	−0.586289	+	0.810102i	$0.699412\pi$
$114$	0	0
$115$	147780.	1.04201
$116$	48853.2	0.337091
$117$	0	0
$118$	191985.	1.26929
$119$	−32691.9	−0.211628
$120$	0	0
$121$	439693.	2.73015
$122$	140428.	0.854192
$123$	0	0
$124$	−14760.8	−0.0862097
$125$	185895.	1.06413
$126$	0	0
$127$	−227290.	−1.25046	−0.625231	−	0.780440i	$-0.714995\pi$
$127$	−227290.	−1.25046	−0.625231	+	0.780440i	$0.714995\pi$
$128$	19410.9	0.104718
$129$	0	0
$130$	30063.0	0.156018
$131$	250299.	1.27433	0.637164	−	0.770728i	$-0.280107\pi$
$131$	250299.	1.27433	0.637164	+	0.770728i	$0.280107\pi$
$132$	0	0
$133$	48941.9	0.239912
$134$	32075.0	0.154314
$135$	0	0
$136$	−131236.	−0.608424
$137$	−425918.	−1.93876	−0.969382	−	0.245558i	$-0.921029\pi$
$137$	−425918.	−1.93876	−0.969382	+	0.245558i	$0.921029\pi$
$138$	0	0
$139$	347532.	1.52566	0.762830	−	0.646599i	$-0.223809\pi$
$139$	347532.	1.52566	0.762830	+	0.646599i	$0.223809\pi$
$140$	−23773.3	−0.102511
$141$	0	0
$142$	157974.	0.657453
$143$	130988.	0.535663
$144$	0	0
$145$	160530.	0.634067
$146$	338247.	1.31326
$147$	0	0
$148$	129161.	0.484705
$149$	411985.	1.52025	0.760126	−	0.649776i	$-0.225137\pi$
$149$	411985.	1.52025	0.760126	+	0.649776i	$0.225137\pi$
$150$	0	0
$151$	212094.	0.756983	0.378492	−	0.925605i	$-0.376443\pi$
$151$	212094.	0.756983	0.378492	+	0.925605i	$0.376443\pi$
$152$	196469.	0.689741
$153$	0	0
$154$	169203.	0.574920
$155$	−48503.5	−0.162160
$156$	0	0
$157$	−307320.	−0.995044	−0.497522	−	0.867451i	$-0.665757\pi$
$157$	−307320.	−0.995044	−0.497522	+	0.867451i	$0.665757\pi$
$158$	−23518.7	−0.0749497
$159$	0	0
$160$	−164603.	−0.508320
$161$	181357.	0.551404
$162$	0	0
$163$	−293664.	−0.865728	−0.432864	−	0.901459i	$-0.642497\pi$
$163$	−293664.	−0.865728	−0.432864	+	0.901459i	$0.642497\pi$
$164$	78281.5	0.227274
$165$	0	0
$166$	−293264.	−0.826016
$167$	−666494.	−1.84929	−0.924645	−	0.380830i	$-0.875638\pi$
$167$	−666494.	−1.84929	−0.924645	+	0.380830i	$0.875638\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	−118683.	−0.314969
$171$	0	0
$172$	119582.	0.308209
$173$	480941.	1.22173	0.610866	−	0.791734i	$-0.290821\pi$
$173$	480941.	1.22173	0.610866	+	0.791734i	$0.290821\pi$
$174$	0	0
$175$	75006.9	0.185142
$176$	377863.	0.919501
$177$	0	0
$178$	−503669.	−1.19150
$179$	344304.	0.803173	0.401587	−	0.915821i	$-0.368459\pi$
$179$	344304.	0.803173	0.401587	+	0.915821i	$0.368459\pi$
$180$	0	0
$181$	418553.	0.949630	0.474815	−	0.880086i	$-0.342515\pi$
$181$	418553.	0.949630	0.474815	+	0.880086i	$0.342515\pi$
$182$	36893.6	0.0825606
$183$	0	0
$184$	728028.	1.58527
$185$	424419.	0.911728
$186$	0	0
$187$	−517117.	−1.08140
$188$	103378.	0.213322
$189$	0	0
$190$	177677.	0.357065
$191$	584397.	1.15911	0.579555	−	0.814933i	$-0.303227\pi$
$191$	584397.	1.15911	0.579555	+	0.814933i	$0.303227\pi$
$192$	0	0
$193$	−652504.	−1.26093	−0.630463	−	0.776219i	$-0.717135\pi$
$193$	−652504.	−1.26093	−0.630463	+	0.776219i	$0.717135\pi$
$194$	150089.	0.286315
$195$	0	0
$196$	−29174.8	−0.0542459
$197$	278348.	0.511003	0.255501	−	0.966809i	$-0.417759\pi$
$197$	278348.	0.511003	0.255501	+	0.966809i	$0.417759\pi$
$198$	0	0
$199$	−624149.	−1.11726	−0.558632	−	0.829416i	$-0.688673\pi$
$199$	−624149.	−1.11726	−0.558632	+	0.829416i	$0.688673\pi$
$200$	301103.	0.532280
$201$	0	0
$202$	−406587.	−0.701093
$203$	197003.	0.335532
$204$	0	0
$205$	257230.	0.427501
$206$	377651.	0.620044
$207$	0	0
$208$	82390.3	0.132044
$209$	774159.	1.22593
$210$	0	0
$211$	−764939.	−1.18283	−0.591413	−	0.806369i	$-0.701430\pi$
$211$	−764939.	−1.18283	−0.591413	+	0.806369i	$0.701430\pi$
$212$	377203.	0.576416
$213$	0	0
$214$	−684138.	−1.02120
$215$	392943.	0.579740
$216$	0	0
$217$	−59523.9	−0.0858108
$218$	−193055.	−0.275131
$219$	0	0
$220$	−376043.	−0.523819
$221$	−112754.	−0.155292
$222$	0	0
$223$	696622.	0.938069	0.469035	−	0.883180i	$-0.344602\pi$
$223$	696622.	0.938069	0.469035	+	0.883180i	$0.344602\pi$
$224$	−202002.	−0.268990
$225$	0	0
$226$	−709098.	−0.923497
$227$	−260906.	−0.336062	−0.168031	−	0.985782i	$-0.553741\pi$
$227$	−260906.	−0.336062	−0.168031	+	0.985782i	$0.553741\pi$
$228$	0	0
$229$	−61198.8	−0.0771178	−0.0385589	−	0.999256i	$-0.512277\pi$
$229$	−61198.8	−0.0771178	−0.0385589	+	0.999256i	$0.512277\pi$
$230$	658392.	0.820663
$231$	0	0
$232$	790838.	0.964645
$233$	305420.	0.368559	0.184280	−	0.982874i	$-0.441005\pi$
$233$	305420.	0.368559	0.184280	+	0.982874i	$0.441005\pi$
$234$	0	0
$235$	339698.	0.401257
$236$	−523617.	−0.611975
$237$	0	0
$238$	−145649.	−0.166673
$239$	−1.16258e6	−1.31652	−0.658259	−	0.752791i	$-0.728707\pi$
$239$	−1.16258e6	−1.31652	−0.658259	+	0.752791i	$0.728707\pi$
$240$	0	0
$241$	458560.	0.508573	0.254286	−	0.967129i	$-0.418159\pi$
$241$	458560.	0.508573	0.254286	+	0.967129i	$0.418159\pi$
$242$	1.95893e6	2.15021
$243$	0	0
$244$	−383003.	−0.411839
$245$	−95867.2	−0.102036
$246$	0	0
$247$	168800.	0.176047
$248$	−238949.	−0.246704
$249$	0	0
$250$	828202.	0.838081
$251$	−1.76457e6	−1.76789	−0.883944	−	0.467593i	$-0.845121\pi$
$251$	−1.76457e6	−1.76789	−0.883944	+	0.467593i	$0.845121\pi$
$252$	0	0
$253$	2.86869e6	2.81762
$254$	−1.01262e6	−0.984836
$255$	0	0
$256$	−1.00047e6	−0.954120
$257$	−132315.	−0.124962	−0.0624809	−	0.998046i	$-0.519901\pi$
$257$	−132315.	−0.124962	−0.0624809	+	0.998046i	$0.519901\pi$
$258$	0	0
$259$	520850.	0.482463
$260$	−81993.5	−0.0752222
$261$	0	0
$262$	1.11514e6	1.00363
$263$	−2.02099e6	−1.80166	−0.900832	−	0.434168i	$-0.857042\pi$
$263$	−2.02099e6	−1.80166	−0.900832	+	0.434168i	$0.857042\pi$
$264$	0	0
$265$	1.23948e6	1.08424
$266$	218047.	0.188949
$267$	0	0
$268$	−87481.0	−0.0744007
$269$	824851.	0.695016	0.347508	−	0.937677i	$-0.387028\pi$
$269$	824851.	0.695016	0.347508	+	0.937677i	$0.387028\pi$
$270$	0	0
$271$	−856153.	−0.708155	−0.354077	−	0.935216i	$-0.615205\pi$
$271$	−856153.	−0.708155	−0.354077	+	0.935216i	$0.615205\pi$
$272$	−325262.	−0.266570
$273$	0	0
$274$	−1.89756e6	−1.52693
$275$	1.18645e6	0.946059
$276$	0	0
$277$	−1.06348e6	−0.832777	−0.416389	−	0.909187i	$-0.636704\pi$
$277$	−1.06348e6	−0.832777	−0.416389	+	0.909187i	$0.636704\pi$
$278$	1.54833e6	1.20158
$279$	0	0
$280$	−384843.	−0.293352
$281$	7462.61	0.00563800	0.00281900	−	0.999996i	$-0.499103\pi$
$281$	7462.61	0.00563800	0.00281900	+	0.999996i	$0.499103\pi$
$282$	0	0
$283$	399064.	0.296194	0.148097	−	0.988973i	$-0.452685\pi$
$283$	399064.	0.296194	0.148097	+	0.988973i	$0.452685\pi$
$284$	−430856.	−0.316983
$285$	0	0
$286$	583579.	0.421876
$287$	315675.	0.226222
$288$	0	0
$289$	−974726.	−0.686496
$290$	715194.	0.499377
$291$	0	0
$292$	−922529.	−0.633174
$293$	−764049.	−0.519938	−0.259969	−	0.965617i	$-0.583712\pi$
$293$	−764049.	−0.519938	−0.259969	+	0.965617i	$0.583712\pi$
$294$	0	0
$295$	−1.72059e6	−1.15112
$296$	2.09087e6	1.38707
$297$	0	0
$298$	1.83548e6	1.19732
$299$	625497.	0.404620
$300$	0	0
$301$	482222.	0.306783
$302$	944924.	0.596183
$303$	0	0
$304$	486939.	0.302197
$305$	−1.25853e6	−0.774667
$306$	0	0
$307$	1.53225e6	0.927860	0.463930	−	0.885872i	$-0.346439\pi$
$307$	1.53225e6	0.927860	0.463930	+	0.885872i	$0.346439\pi$
$308$	−461483.	−0.277191
$309$	0	0
$310$	−216093.	−0.127714
$311$	−618970.	−0.362885	−0.181442	−	0.983402i	$-0.558077\pi$
$311$	−618970.	−0.362885	−0.181442	+	0.983402i	$0.558077\pi$
$312$	0	0
$313$	−2.84236e6	−1.63990	−0.819951	−	0.572434i	$-0.805999\pi$
$313$	−2.84236e6	−1.63990	−0.819951	+	0.572434i	$0.805999\pi$
$314$	−1.36918e6	−0.783675
$315$	0	0
$316$	64144.4	0.0361361
$317$	3.48154e6	1.94591	0.972957	−	0.230986i	$-0.0741951\pi$
$317$	3.48154e6	1.94591	0.972957	+	0.230986i	$0.0741951\pi$
$318$	0	0
$319$	3.11618e6	1.71453
$320$	−1.35624e6	−0.740391
$321$	0	0
$322$	807984.	0.434273
$323$	−666391.	−0.355404
$324$	0	0
$325$	258697.	0.135857
$326$	−1.30833e6	−0.681828
$327$	0	0
$328$	1.26723e6	0.650383
$329$	416880.	0.212335
$330$	0	0
$331$	197744.	0.0992051	0.0496026	−	0.998769i	$-0.484205\pi$
$331$	197744.	0.0992051	0.0496026	+	0.998769i	$0.484205\pi$
$332$	799845.	0.398254
$333$	0	0
$334$	−2.96937e6	−1.45646
$335$	−287460.	−0.139947
$336$	0	0
$337$	1.10343e6	0.529259	0.264630	−	0.964350i	$-0.414750\pi$
$337$	1.10343e6	0.529259	0.264630	+	0.964350i	$0.414750\pi$
$338$	127245.	0.0605829
$339$	0	0
$340$	323696.	0.151859
$341$	−941543.	−0.438484
$342$	0	0
$343$	−117649.	−0.0539949
$344$	1.93580e6	0.881993
$345$	0	0
$346$	2.14269e6	0.962209
$347$	−2.33287e6	−1.04008	−0.520041	−	0.854141i	$-0.674083\pi$
$347$	−2.33287e6	−1.04008	−0.520041	+	0.854141i	$0.674083\pi$
$348$	0	0
$349$	1.92855e6	0.847552	0.423776	−	0.905767i	$-0.360704\pi$
$349$	1.92855e6	0.847552	0.423776	+	0.905767i	$0.360704\pi$
$350$	334172.	0.145814
$351$	0	0
$352$	−3.19525e6	−1.37451
$353$	2.64900e6	1.13147	0.565737	−	0.824586i	$-0.308592\pi$
$353$	2.64900e6	1.13147	0.565737	+	0.824586i	$0.308592\pi$
$354$	0	0
$355$	−1.41578e6	−0.596244
$356$	1.37370e6	0.574469
$357$	0	0
$358$	1.53395e6	0.632561
$359$	−1.67301e6	−0.685113	−0.342556	−	0.939497i	$-0.611293\pi$
$359$	−1.67301e6	−0.685113	−0.342556	+	0.939497i	$0.611293\pi$
$360$	0	0
$361$	−1.47847e6	−0.597095
$362$	1.86474e6	0.747907
$363$	0	0
$364$	−100623.	−0.0398056
$365$	−3.03140e6	−1.19100
$366$	0	0
$367$	−3.38359e6	−1.31133	−0.655665	−	0.755052i	$-0.727612\pi$
$367$	−3.38359e6	−1.31133	−0.655665	+	0.755052i	$0.727612\pi$
$368$	1.80438e6	0.694557
$369$	0	0
$370$	1.89088e6	0.718057
$371$	1.52110e6	0.573749
$372$	0	0
$373$	1.50359e6	0.559575	0.279787	−	0.960062i	$-0.409736\pi$
$373$	1.50359e6	0.559575	0.279787	+	0.960062i	$0.409736\pi$
$374$	−2.30386e6	−0.851683
$375$	0	0
$376$	1.67350e6	0.610457
$377$	679461.	0.246213
$378$	0	0
$379$	−2.75436e6	−0.984970	−0.492485	−	0.870321i	$-0.663911\pi$
$379$	−2.75436e6	−0.984970	−0.492485	+	0.870321i	$0.663911\pi$
$380$	−484594.	−0.172155
$381$	0	0
$382$	2.60361e6	0.912888
$383$	−2.01106e6	−0.700532	−0.350266	−	0.936650i	$-0.613909\pi$
$383$	−2.01106e6	−0.700532	−0.350266	+	0.936650i	$0.613909\pi$
$384$	0	0
$385$	−1.51642e6	−0.521395
$386$	−2.90704e6	−0.993078
$387$	0	0
$388$	−409350.	−0.138043
$389$	−167787.	−0.0562191	−0.0281096	−	0.999605i	$-0.508949\pi$
$389$	−167787.	−0.0562191	−0.0281096	+	0.999605i	$0.508949\pi$
$390$	0	0
$391$	−2.46935e6	−0.816846
$392$	−472283.	−0.155234
$393$	0	0
$394$	1.24010e6	0.402454
$395$	210776.	0.0679719
$396$	0	0
$397$	−2.40238e6	−0.765007	−0.382503	−	0.923954i	$-0.624938\pi$
$397$	−2.40238e6	−0.765007	−0.382503	+	0.923954i	$0.624938\pi$
$398$	−2.78072e6	−0.879931
$399$	0	0
$400$	746267.	0.233208
$401$	411329.	0.127741	0.0638703	−	0.997958i	$-0.479656\pi$
$401$	411329.	0.127741	0.0638703	+	0.997958i	$0.479656\pi$
$402$	0	0
$403$	−205297.	−0.0629679
$404$	1.10892e6	0.338024
$405$	0	0
$406$	877692.	0.264257
$407$	8.23876e6	2.46533
$408$	0	0
$409$	238011.	0.0703540	0.0351770	−	0.999381i	$-0.488800\pi$
$409$	238011.	0.0703540	0.0351770	+	0.999381i	$0.488800\pi$
$410$	1.14602e6	0.336690
$411$	0	0
$412$	−1.03000e6	−0.298947
$413$	−2.11152e6	−0.609143
$414$	0	0
$415$	2.62826e6	0.749114
$416$	−696701.	−0.197384
$417$	0	0
$418$	3.44904e6	0.965512
$419$	271522.	0.0755563	0.0377781	−	0.999286i	$-0.487972\pi$
$419$	271522.	0.0755563	0.0377781	+	0.999286i	$0.487972\pi$
$420$	0	0
$421$	2.37871e6	0.654089	0.327044	−	0.945009i	$-0.393947\pi$
$421$	2.37871e6	0.654089	0.327044	+	0.945009i	$0.393947\pi$
$422$	−3.40796e6	−0.931567
$423$	0	0
$424$	6.10619e6	1.64951
$425$	−1.02129e6	−0.274269
$426$	0	0
$427$	−1.54448e6	−0.409933
$428$	1.86591e6	0.492358
$429$	0	0
$430$	1.75064e6	0.456590
$431$	37191.4	0.00964383	0.00482192	−	0.999988i	$-0.498465\pi$
$431$	37191.4	0.00964383	0.00482192	+	0.999988i	$0.498465\pi$
$432$	0	0
$433$	−258394.	−0.0662313	−0.0331157	−	0.999452i	$-0.510543\pi$
$433$	−258394.	−0.0662313	−0.0331157	+	0.999452i	$0.510543\pi$
$434$	−265192.	−0.0675827
$435$	0	0
$436$	526535.	0.132651
$437$	3.69678e6	0.926019
$438$	0	0
$439$	1.30469e6	0.323106	0.161553	−	0.986864i	$-0.448350\pi$
$439$	1.30469e6	0.323106	0.161553	+	0.986864i	$0.448350\pi$
$440$	−6.08741e6	−1.49900
$441$	0	0
$442$	−502341.	−0.122305
$443$	4.46886e6	1.08190	0.540950	−	0.841055i	$-0.318065\pi$
$443$	4.46886e6	1.08190	0.540950	+	0.841055i	$0.318065\pi$
$444$	0	0
$445$	4.51393e6	1.08057
$446$	3.10360e6	0.738803
$447$	0	0
$448$	−1.66439e6	−0.391795
$449$	−5.64527e6	−1.32150	−0.660752	−	0.750604i	$-0.729763\pi$
$449$	−5.64527e6	−1.32150	−0.660752	+	0.750604i	$0.729763\pi$
$450$	0	0
$451$	4.99332e6	1.15597
$452$	1.93399e6	0.445253
$453$	0	0
$454$	−1.16239e6	−0.264675
$455$	−330644.	−0.0748742
$456$	0	0
$457$	−2.00192e6	−0.448390	−0.224195	−	0.974544i	$-0.571975\pi$
$457$	−2.00192e6	−0.448390	−0.224195	+	0.974544i	$0.571975\pi$
$458$	−272654.	−0.0607362
$459$	0	0
$460$	−1.79569e6	−0.395673
$461$	−1.56304e6	−0.342544	−0.171272	−	0.985224i	$-0.554788\pi$
$461$	−1.56304e6	−0.342544	−0.171272	+	0.985224i	$0.554788\pi$
$462$	0	0
$463$	3.56717e6	0.773341	0.386670	−	0.922218i	$-0.373625\pi$
$463$	3.56717e6	0.773341	0.386670	+	0.922218i	$0.373625\pi$
$464$	1.96005e6	0.422641
$465$	0	0
$466$	1.36071e6	0.290269
$467$	−7.38430e6	−1.56681	−0.783406	−	0.621510i	$-0.786520\pi$
$467$	−7.38430e6	−1.56681	−0.783406	+	0.621510i	$0.786520\pi$
$468$	0	0
$469$	−352773.	−0.0740564
$470$	1.51343e6	0.316021
$471$	0	0
$472$	−8.47634e6	−1.75127
$473$	7.62774e6	1.56763
$474$	0	0
$475$	1.52894e6	0.310925
$476$	397242.	0.0803596
$477$	0	0
$478$	−5.17953e6	−1.03686
$479$	4.15087e6	0.826609	0.413305	−	0.910593i	$-0.364374\pi$
$479$	4.15087e6	0.826609	0.413305	+	0.910593i	$0.364374\pi$
$480$	0	0
$481$	1.79640e6	0.354031
$482$	2.04298e6	0.400541
$483$	0	0
$484$	−5.34275e6	−1.03670
$485$	−1.34511e6	−0.259659
$486$	0	0
$487$	−6.19691e6	−1.18400	−0.592001	−	0.805937i	$-0.701662\pi$
$487$	−6.19691e6	−1.18400	−0.592001	+	0.805937i	$0.701662\pi$
$488$	−6.20007e6	−1.17855
$489$	0	0
$490$	−427109.	−0.0803615
$491$	547411.	0.102473	0.0512365	−	0.998687i	$-0.483684\pi$
$491$	547411.	0.102473	0.0512365	+	0.998687i	$0.483684\pi$
$492$	0	0
$493$	−2.68239e6	−0.497055
$494$	752039.	0.138651
$495$	0	0
$496$	−592222.	−0.108089
$497$	−1.73745e6	−0.315517
$498$	0	0
$499$	−7.72836e6	−1.38943	−0.694714	−	0.719286i	$-0.744469\pi$
$499$	−7.72836e6	−1.38943	−0.694714	+	0.719286i	$0.744469\pi$
$500$	−2.25883e6	−0.404071
$501$	0	0
$502$	−7.86154e6	−1.39235
$503$	−4.69163e6	−0.826806	−0.413403	−	0.910548i	$-0.635660\pi$
$503$	−4.69163e6	−0.826806	−0.413403	+	0.910548i	$0.635660\pi$
$504$	0	0
$505$	3.64387e6	0.635821
$506$	1.27806e7	2.21909
$507$	0	0
$508$	2.76182e6	0.474827
$509$	−840589.	−0.143810	−0.0719050	−	0.997411i	$-0.522908\pi$
$509$	−840589.	−0.143810	−0.0719050	+	0.997411i	$0.522908\pi$
$510$	0	0
$511$	−3.72016e6	−0.630244
$512$	−5.07844e6	−0.856161
$513$	0	0
$514$	−589493.	−0.0984172
$515$	−3.38454e6	−0.562318
$516$	0	0
$517$	6.59416e6	1.08501
$518$	2.32050e6	0.379977
$519$	0	0
$520$	−1.32732e6	−0.215261
$521$	−1.03766e7	−1.67479	−0.837393	−	0.546601i	$-0.815922\pi$
$521$	−1.03766e7	−1.67479	−0.837393	+	0.546601i	$0.815922\pi$
$522$	0	0
$523$	−2.14085e6	−0.342241	−0.171121	−	0.985250i	$-0.554739\pi$
$523$	−2.14085e6	−0.342241	−0.171121	+	0.985250i	$0.554739\pi$
$524$	−3.04141e6	−0.483890
$525$	0	0
$526$	−9.00392e6	−1.41895
$527$	810474.	0.127120
$528$	0	0
$529$	7.26227e6	1.12832
$530$	5.52213e6	0.853920
$531$	0	0
$532$	−594698.	−0.0910998
$533$	1.08876e6	0.166002
$534$	0	0
$535$	6.13131e6	0.926123
$536$	−1.41615e6	−0.212910
$537$	0	0
$538$	3.67488e6	0.547379
$539$	−1.86096e6	−0.275909
$540$	0	0
$541$	3.98470e6	0.585333	0.292666	−	0.956215i	$-0.405457\pi$
$541$	3.98470e6	0.585333	0.292666	+	0.956215i	$0.405457\pi$
$542$	−3.81434e6	−0.557727
$543$	0	0
$544$	2.75045e6	0.398480
$545$	1.73017e6	0.249516
$546$	0	0
$547$	−1.09295e7	−1.56182	−0.780912	−	0.624641i	$-0.785246\pi$
$547$	−1.09295e7	−1.56182	−0.780912	+	0.624641i	$0.785246\pi$
$548$	5.17537e6	0.736190
$549$	0	0
$550$	5.28589e6	0.745095
$551$	4.01571e6	0.563487
$552$	0	0
$553$	258666.	0.0359689
$554$	−4.73802e6	−0.655877
$555$	0	0
$556$	−4.22289e6	−0.579326
$557$	−9.02334e6	−1.23234	−0.616168	−	0.787615i	$-0.711316\pi$
$557$	−9.02334e6	−1.23234	−0.616168	+	0.787615i	$0.711316\pi$
$558$	0	0
$559$	1.66318e6	0.225117
$560$	−953813.	−0.128527
$561$	0	0
$562$	33247.5	0.00444036
$563$	3.60648e6	0.479527	0.239763	−	0.970831i	$-0.422930\pi$
$563$	3.60648e6	0.479527	0.239763	+	0.970831i	$0.422930\pi$
$564$	0	0
$565$	6.35501e6	0.837519
$566$	1.77791e6	0.233276
$567$	0	0
$568$	−6.97472e6	−0.907102
$569$	5.51282e6	0.713827	0.356913	−	0.934137i	$-0.383829\pi$
$569$	5.51282e6	0.713827	0.356913	+	0.934137i	$0.383829\pi$
$570$	0	0
$571$	−9.37449e6	−1.20325	−0.601627	−	0.798777i	$-0.705481\pi$
$571$	−9.37449e6	−1.20325	−0.601627	+	0.798777i	$0.705481\pi$
$572$	−1.59165e6	−0.203403
$573$	0	0
$574$	1.40640e6	0.178168
$575$	5.66557e6	0.714618
$576$	0	0
$577$	−1.05404e7	−1.31801	−0.659004	−	0.752140i	$-0.729022\pi$
$577$	−1.05404e7	−1.31801	−0.659004	+	0.752140i	$0.729022\pi$
$578$	−4.34261e6	−0.540669
$579$	0	0
$580$	−1.95061e6	−0.240769
$581$	3.22542e6	0.396412
$582$	0	0
$583$	2.40605e7	2.93180
$584$	−1.49340e7	−1.81194
$585$	0	0
$586$	−3.40400e6	−0.409492
$587$	1.34945e7	1.61645	0.808223	−	0.588877i	$-0.200430\pi$
$587$	1.34945e7	1.61645	0.808223	+	0.588877i	$0.200430\pi$
$588$	0	0
$589$	−1.21333e6	−0.144109
$590$	−7.66558e6	−0.906598
$591$	0	0
$592$	5.18210e6	0.607718
$593$	5.79392e6	0.676606	0.338303	−	0.941037i	$-0.390147\pi$
$593$	5.79392e6	0.676606	0.338303	+	0.941037i	$0.390147\pi$
$594$	0	0
$595$	1.30532e6	0.151156
$596$	−5.00606e6	−0.577272
$597$	0	0
$598$	2.78672e6	0.318670
$599$	−378674.	−0.0431219	−0.0215610	−	0.999768i	$-0.506864\pi$
$599$	−378674.	−0.0431219	−0.0215610	+	0.999768i	$0.506864\pi$
$600$	0	0
$601$	−1.01224e7	−1.14314	−0.571568	−	0.820555i	$-0.693665\pi$
$601$	−1.01224e7	−1.14314	−0.571568	+	0.820555i	$0.693665\pi$
$602$	2.14840e6	0.241615
$603$	0	0
$604$	−2.57717e6	−0.287443
$605$	−1.75561e7	−1.95002
$606$	0	0
$607$	2.22802e6	0.245441	0.122721	−	0.992441i	$-0.460838\pi$
$607$	2.22802e6	0.245441	0.122721	+	0.992441i	$0.460838\pi$
$608$	−4.11761e6	−0.451737
$609$	0	0
$610$	−5.60703e6	−0.610110
$611$	1.43781e6	0.155811
$612$	0	0
$613$	2.76839e6	0.297561	0.148781	−	0.988870i	$-0.452465\pi$
$613$	2.76839e6	0.297561	0.148781	+	0.988870i	$0.452465\pi$
$614$	6.82648e6	0.730762
$615$	0	0
$616$	−7.47052e6	−0.793230
$617$	2.44135e6	0.258177	0.129088	−	0.991633i	$-0.458795\pi$
$617$	2.44135e6	0.258177	0.129088	+	0.991633i	$0.458795\pi$
$618$	0	0
$619$	1.02459e6	0.107479	0.0537395	−	0.998555i	$-0.482886\pi$
$619$	1.02459e6	0.107479	0.0537395	+	0.998555i	$0.482886\pi$
$620$	589370.	0.0615756
$621$	0	0
$622$	−2.75764e6	−0.285800
$623$	5.53953e6	0.571811
$624$	0	0
$625$	−2.63882e6	−0.270215
$626$	−1.26633e7	−1.29155
$627$	0	0
$628$	3.73428e6	0.377840
$629$	−7.09187e6	−0.714717
$630$	0	0
$631$	4.26554e6	0.426482	0.213241	−	0.977000i	$-0.431598\pi$
$631$	4.26554e6	0.426482	0.213241	+	0.977000i	$0.431598\pi$
$632$	1.03837e6	0.103410
$633$	0	0
$634$	1.55110e7	1.53256
$635$	9.07523e6	0.893148
$636$	0	0
$637$	−405769.	−0.0396214
$638$	1.38832e7	1.35033
$639$	0	0
$640$	−775040.	−0.0747953
$641$	4.74556e6	0.456186	0.228093	−	0.973639i	$-0.426751\pi$
$641$	4.74556e6	0.456186	0.228093	+	0.973639i	$0.426751\pi$
$642$	0	0
$643$	1.44003e7	1.37355	0.686777	−	0.726869i	$-0.259025\pi$
$643$	1.44003e7	1.37355	0.686777	+	0.726869i	$0.259025\pi$
$644$	−2.20369e6	−0.209380
$645$	0	0
$646$	−2.96891e6	−0.279908
$647$	−1.56859e7	−1.47316	−0.736578	−	0.676353i	$-0.763560\pi$
$647$	−1.56859e7	−1.47316	−0.736578	+	0.676353i	$0.763560\pi$
$648$	0	0
$649$	−3.33998e7	−3.11266
$650$	1.15255e6	0.106998
$651$	0	0
$652$	3.56833e6	0.328735
$653$	1.23634e7	1.13463	0.567317	−	0.823499i	$-0.307981\pi$
$653$	1.23634e7	1.13463	0.567317	+	0.823499i	$0.307981\pi$
$654$	0	0
$655$	−9.99395e6	−0.910194
$656$	3.14075e6	0.284953
$657$	0	0
$658$	1.85729e6	0.167230
$659$	1.22074e7	1.09499	0.547494	−	0.836810i	$-0.315582\pi$
$659$	1.22074e7	1.09499	0.547494	+	0.836810i	$0.315582\pi$
$660$	0	0
$661$	4.01536e6	0.357455	0.178727	−	0.983899i	$-0.442802\pi$
$661$	4.01536e6	0.357455	0.178727	+	0.983899i	$0.442802\pi$
$662$	880993.	0.0781317
$663$	0	0
$664$	1.29479e7	1.13967
$665$	−1.95416e6	−0.171358
$666$	0	0
$667$	1.48804e7	1.29509
$668$	8.09863e6	0.702215
$669$	0	0
$670$	−1.28069e6	−0.110219
$671$	−2.44305e7	−2.09472
$672$	0	0
$673$	−1.59346e7	−1.35614	−0.678070	−	0.734997i	$-0.737183\pi$
$673$	−1.59346e7	−1.35614	−0.678070	+	0.734997i	$0.737183\pi$
$674$	4.91600e6	0.416833
$675$	0	0
$676$	−347047.	−0.0292093
$677$	−6.71044e6	−0.562703	−0.281352	−	0.959605i	$-0.590783\pi$
$677$	−6.71044e6	−0.562703	−0.281352	+	0.959605i	$0.590783\pi$
$678$	0	0
$679$	−1.65073e6	−0.137405
$680$	5.24000e6	0.434569
$681$	0	0
$682$	−4.19477e6	−0.345341
$683$	−5.42135e6	−0.444688	−0.222344	−	0.974968i	$-0.571371\pi$
$683$	−5.42135e6	−0.444688	−0.222344	+	0.974968i	$0.571371\pi$
$684$	0	0
$685$	1.70061e7	1.38477
$686$	−524151.	−0.0425252
$687$	0	0
$688$	4.79778e6	0.386429
$689$	5.24623e6	0.421017
$690$	0	0
$691$	1.75088e7	1.39496	0.697480	−	0.716604i	$-0.254305\pi$
$691$	1.75088e7	1.39496	0.697480	+	0.716604i	$0.254305\pi$
$692$	−5.84395e6	−0.463918
$693$	0	0
$694$	−1.03935e7	−0.819146
$695$	−1.38763e7	−1.08971
$696$	0	0
$697$	−4.29821e6	−0.335124
$698$	8.59208e6	0.667513
$699$	0	0
$700$	−911415.	−0.0703025
$701$	1.63957e6	0.126019	0.0630094	−	0.998013i	$-0.479930\pi$
$701$	1.63957e6	0.126019	0.0630094	+	0.998013i	$0.479930\pi$
$702$	0	0
$703$	1.06170e7	0.810240
$704$	−2.63271e7	−2.00203
$705$	0	0
$706$	1.18018e7	0.891123
$707$	4.47179e6	0.336460
$708$	0	0
$709$	−6.73323e6	−0.503046	−0.251523	−	0.967851i	$-0.580931\pi$
$709$	−6.73323e6	−0.503046	−0.251523	+	0.967851i	$0.580931\pi$
$710$	−6.30759e6	−0.469588
$711$	0	0
$712$	2.22375e7	1.64394
$713$	−4.49608e6	−0.331215
$714$	0	0
$715$	−5.23009e6	−0.382599
$716$	−4.18366e6	−0.304982
$717$	0	0
$718$	−7.45361e6	−0.539579
$719$	2.57923e7	1.86066	0.930330	−	0.366722i	$-0.119520\pi$
$719$	2.57923e7	1.86066	0.930330	+	0.366722i	$0.119520\pi$
$720$	0	0
$721$	−4.15354e6	−0.297564
$722$	−6.58688e6	−0.470259
$723$	0	0
$724$	−5.08588e6	−0.360595
$725$	6.15436e6	0.434848
$726$	0	0
$727$	−1.53461e7	−1.07687	−0.538434	−	0.842668i	$-0.680984\pi$
$727$	−1.53461e7	−1.07687	−0.538434	+	0.842668i	$0.680984\pi$
$728$	−1.62889e6	−0.113911
$729$	0	0
$730$	−1.35055e7	−0.938003
$731$	−6.56591e6	−0.454466
$732$	0	0
$733$	−1.39651e7	−0.960029	−0.480015	−	0.877260i	$-0.659369\pi$
$733$	−1.39651e7	−0.960029	−0.480015	+	0.877260i	$0.659369\pi$
$734$	−1.50746e7	−1.03277
$735$	0	0
$736$	−1.52580e7	−1.03825
$737$	−5.58012e6	−0.378421
$738$	0	0
$739$	−6.24341e6	−0.420543	−0.210272	−	0.977643i	$-0.567435\pi$
$739$	−6.24341e6	−0.420543	−0.210272	+	0.977643i	$0.567435\pi$
$740$	−5.15715e6	−0.346203
$741$	0	0
$742$	6.77681e6	0.451872
$743$	−2.68345e7	−1.78329	−0.891645	−	0.452735i	$-0.850448\pi$
$743$	−2.68345e7	−1.78329	−0.891645	+	0.452735i	$0.850448\pi$
$744$	0	0
$745$	−1.64497e7	−1.08585
$746$	6.69882e6	0.440709
$747$	0	0
$748$	6.28353e6	0.410629
$749$	7.52439e6	0.490080
$750$	0	0
$751$	−1.80463e7	−1.16759	−0.583794	−	0.811902i	$-0.698432\pi$
$751$	−1.80463e7	−1.16759	−0.583794	+	0.811902i	$0.698432\pi$
$752$	4.14767e6	0.267460
$753$	0	0
$754$	3.02714e6	0.193912
$755$	−8.46850e6	−0.540678
$756$	0	0
$757$	1.25861e7	0.798271	0.399136	−	0.916892i	$-0.369310\pi$
$757$	1.25861e7	0.798271	0.399136	+	0.916892i	$0.369310\pi$
$758$	−1.22713e7	−0.775741
$759$	0	0
$760$	−7.84464e6	−0.492650
$761$	2.99176e7	1.87269	0.936343	−	0.351086i	$-0.114188\pi$
$761$	2.99176e7	1.87269	0.936343	+	0.351086i	$0.114188\pi$
$762$	0	0
$763$	2.12328e6	0.132037
$764$	−7.10105e6	−0.440139
$765$	0	0
$766$	−8.95970e6	−0.551723
$767$	−7.28258e6	−0.446989
$768$	0	0
$769$	1.97249e7	1.20282	0.601408	−	0.798942i	$-0.294607\pi$
$769$	1.97249e7	1.20282	0.601408	+	0.798942i	$0.294607\pi$
$770$	−6.75596e6	−0.410639
$771$	0	0
$772$	7.92863e6	0.478801
$773$	−2.30550e7	−1.38777	−0.693884	−	0.720087i	$-0.744102\pi$
$773$	−2.30550e7	−1.38777	−0.693884	+	0.720087i	$0.744102\pi$
$774$	0	0
$775$	−1.85952e6	−0.111211
$776$	−6.62658e6	−0.395034
$777$	0	0
$778$	−747526.	−0.0442769
$779$	6.43472e6	0.379914
$780$	0	0
$781$	−2.74829e7	−1.61226
$782$	−1.10015e7	−0.643330
$783$	0	0
$784$	−1.17053e6	−0.0680129
$785$	1.22707e7	0.710715
$786$	0	0
$787$	1.58327e7	0.911209	0.455605	−	0.890182i	$-0.349423\pi$
$787$	1.58327e7	0.911209	0.455605	+	0.890182i	$0.349423\pi$
$788$	−3.38224e6	−0.194039
$789$	0	0
$790$	939053.	0.0535331
$791$	7.79892e6	0.443193
$792$	0	0
$793$	−5.32689e6	−0.300809
$794$	−1.07031e7	−0.602502
$795$	0	0
$796$	7.58409e6	0.424249
$797$	−5.43644e6	−0.303158	−0.151579	−	0.988445i	$-0.548436\pi$
$797$	−5.43644e6	−0.303158	−0.151579	+	0.988445i	$0.548436\pi$
$798$	0	0
$799$	−5.67621e6	−0.314551
$800$	−6.31051e6	−0.348610
$801$	0	0
$802$	1.83256e6	0.100606
$803$	−5.88451e7	−3.22048
$804$	0	0
$805$	−7.24123e6	−0.393843
$806$	−914640.	−0.0495921
$807$	0	0
$808$	1.79513e7	0.967313
$809$	−8.13186e6	−0.436836	−0.218418	−	0.975855i	$-0.570090\pi$
$809$	−8.13186e6	−0.436836	−0.218418	+	0.975855i	$0.570090\pi$
$810$	0	0
$811$	−2.80268e7	−1.49631	−0.748153	−	0.663526i	$-0.769059\pi$
$811$	−2.80268e7	−1.49631	−0.748153	+	0.663526i	$0.769059\pi$
$812$	−2.39380e6	−0.127409
$813$	0	0
$814$	3.67054e7	1.94164
$815$	1.17254e7	0.618350
$816$	0	0
$817$	9.82961e6	0.515206
$818$	1.06039e6	0.0554093
$819$	0	0
$820$	−3.12563e6	−0.162331
$821$	2.86993e7	1.48598	0.742990	−	0.669302i	$-0.233407\pi$
$821$	2.86993e7	1.48598	0.742990	+	0.669302i	$0.233407\pi$
$822$	0	0
$823$	−2.53155e7	−1.30283	−0.651414	−	0.758723i	$-0.725824\pi$
$823$	−2.53155e7	−1.30283	−0.651414	+	0.758723i	$0.725824\pi$
$824$	−1.66737e7	−0.855488
$825$	0	0
$826$	−9.40726e6	−0.479748
$827$	−3.35726e7	−1.70695	−0.853476	−	0.521133i	$-0.825510\pi$
$827$	−3.35726e7	−1.70695	−0.853476	+	0.521133i	$0.825510\pi$
$828$	0	0
$829$	−2.20043e7	−1.11204	−0.556020	−	0.831169i	$-0.687672\pi$
$829$	−2.20043e7	−1.11204	−0.556020	+	0.831169i	$0.687672\pi$
$830$	1.17095e7	0.589986
$831$	0	0
$832$	−5.74044e6	−0.287499
$833$	1.60190e6	0.0799877
$834$	0	0
$835$	2.66118e7	1.32086
$836$	−9.40687e6	−0.465510
$837$	0	0
$838$	1.20969e6	0.0595064
$839$	−1.61324e7	−0.791217	−0.395608	−	0.918419i	$-0.629466\pi$
$839$	−1.61324e7	−0.791217	−0.395608	+	0.918419i	$0.629466\pi$
$840$	0	0
$841$	−4.34691e6	−0.211929
$842$	1.05977e7	0.515146
$843$	0	0
$844$	9.29484e6	0.449144
$845$	−1.14038e6	−0.0549426
$846$	0	0
$847$	−2.15450e7	−1.03190
$848$	1.51339e7	0.722704
$849$	0	0
$850$	−4.55006e6	−0.216008
$851$	3.93419e7	1.86222
$852$	0	0
$853$	3.69299e7	1.73782	0.868912	−	0.494967i	$-0.164820\pi$
$853$	3.69299e7	1.73782	0.868912	+	0.494967i	$0.164820\pi$
$854$	−6.88100e6	−0.322854
$855$	0	0
$856$	3.02054e7	1.40897
$857$	1.94519e7	0.904710	0.452355	−	0.891838i	$-0.350584\pi$
$857$	1.94519e7	0.904710	0.452355	+	0.891838i	$0.350584\pi$
$858$	0	0
$859$	−6.89608e6	−0.318874	−0.159437	−	0.987208i	$-0.550968\pi$
$859$	−6.89608e6	−0.318874	−0.159437	+	0.987208i	$0.550968\pi$
$860$	−4.77468e6	−0.220140
$861$	0	0
$862$	165696.	0.00759527
$863$	1.83769e7	0.839933	0.419966	−	0.907540i	$-0.362042\pi$
$863$	1.83769e7	0.839933	0.419966	+	0.907540i	$0.362042\pi$
$864$	0	0
$865$	−1.92030e7	−0.872628
$866$	−1.15120e6	−0.0521623
$867$	0	0
$868$	723280.	0.0325842
$869$	4.09156e6	0.183797
$870$	0	0
$871$	−1.21671e6	−0.0543426
$872$	8.52357e6	0.379604
$873$	0	0
$874$	1.64699e7	0.729312
$875$	−9.10886e6	−0.402202
$876$	0	0
$877$	−2.14920e6	−0.0943579	−0.0471789	−	0.998886i	$-0.515023\pi$
$877$	−2.14920e6	−0.0943579	−0.0471789	+	0.998886i	$0.515023\pi$
$878$	5.81266e6	0.254471
$879$	0	0
$880$	−1.50873e7	−0.656758
$881$	1.55372e7	0.674422	0.337211	−	0.941429i	$-0.390516\pi$
$881$	1.55372e7	0.674422	0.337211	+	0.941429i	$0.390516\pi$
$882$	0	0
$883$	3.93951e7	1.70036	0.850179	−	0.526494i	$-0.176494\pi$
$883$	3.93951e7	1.70036	0.850179	+	0.526494i	$0.176494\pi$
$884$	1.37008e6	0.0589678
$885$	0	0
$886$	1.99097e7	0.852081
$887$	−1.41976e7	−0.605907	−0.302954	−	0.953005i	$-0.597973\pi$
$887$	−1.41976e7	−0.605907	−0.302954	+	0.953005i	$0.597973\pi$
$888$	0	0
$889$	1.11372e7	0.472630
$890$	2.01105e7	0.851036
$891$	0	0
$892$	−8.46471e6	−0.356205
$893$	8.49767e6	0.356592
$894$	0	0
$895$	−1.37474e7	−0.573670
$896$	−951136.	−0.0395797
$897$	0	0
$898$	−2.51509e7	−1.04079
$899$	−4.88397e6	−0.201546
$900$	0	0
$901$	−2.07112e7	−0.849948
$902$	2.22463e7	0.910418
$903$	0	0
$904$	3.13075e7	1.27417
$905$	−1.67120e7	−0.678277
$906$	0	0
$907$	9.02514e6	0.364280	0.182140	−	0.983273i	$-0.441698\pi$
$907$	9.02514e6	0.364280	0.182140	+	0.983273i	$0.441698\pi$
$908$	3.17029e6	0.127610
$909$	0	0
$910$	−1.47309e6	−0.0589692
$911$	4.27532e7	1.70676	0.853380	−	0.521289i	$-0.174549\pi$
$911$	4.27532e7	1.70676	0.853380	+	0.521289i	$0.174549\pi$
$912$	0	0
$913$	5.10194e7	2.02562
$914$	−8.91898e6	−0.353142
$915$	0	0
$916$	743632.	0.0292833
$917$	−1.22647e7	−0.481651
$918$	0	0
$919$	6.52256e6	0.254759	0.127379	−	0.991854i	$-0.459343\pi$
$919$	6.52256e6	0.254759	0.127379	+	0.991854i	$0.459343\pi$
$920$	−2.90687e7	−1.13229
$921$	0	0
$922$	−6.96365e6	−0.269780
$923$	−5.99244e6	−0.231526
$924$	0	0
$925$	1.62713e7	0.625270
$926$	1.58925e7	0.609066
$927$	0	0
$928$	−1.65744e7	−0.631782
$929$	808769.	0.0307458	0.0153729	−	0.999882i	$-0.495106\pi$
$929$	808769.	0.0307458	0.0153729	+	0.999882i	$0.495106\pi$
$930$	0	0
$931$	−2.39816e6	−0.0906783
$932$	−3.71118e6	−0.139950
$933$	0	0
$934$	−3.28986e7	−1.23399
$935$	2.06475e7	0.772391
$936$	0	0
$937$	−1.85110e7	−0.688780	−0.344390	−	0.938827i	$-0.611914\pi$
$937$	−1.85110e7	−0.688780	−0.344390	+	0.938827i	$0.611914\pi$
$938$	−1.57168e6	−0.0583252
$939$	0	0
$940$	−4.12770e6	−0.152366
$941$	2.00489e7	0.738101	0.369051	−	0.929409i	$-0.379683\pi$
$941$	2.00489e7	0.738101	0.369051	+	0.929409i	$0.379683\pi$
$942$	0	0
$943$	2.38442e7	0.873179
$944$	−2.10081e7	−0.767287
$945$	0	0
$946$	3.39832e7	1.23463
$947$	4.12068e7	1.49312	0.746558	−	0.665320i	$-0.231705\pi$
$947$	4.12068e7	1.49312	0.746558	+	0.665320i	$0.231705\pi$
$948$	0	0
$949$	−1.28307e7	−0.462473
$950$	6.81175e6	0.244878
$951$	0	0
$952$	6.43057e6	0.229963
$953$	332358.	0.0118543	0.00592713	−	0.999982i	$-0.498113\pi$
$953$	332358.	0.0118543	0.00592713	+	0.999982i	$0.498113\pi$
$954$	0	0
$955$	−2.33338e7	−0.827899
$956$	1.41266e7	0.499910
$957$	0	0
$958$	1.84930e7	0.651019
$959$	2.08700e7	0.732784
$960$	0	0
$961$	−2.71535e7	−0.948456
$962$	8.00335e6	0.278827
$963$	0	0
$964$	−5.57200e6	−0.193116
$965$	2.60532e7	0.900622
$966$	0	0
$967$	5.02387e6	0.172771	0.0863857	−	0.996262i	$-0.472468\pi$
$967$	5.02387e6	0.172771	0.0863857	+	0.996262i	$0.472468\pi$
$968$	−8.64888e7	−2.96668
$969$	0	0
$970$	−5.99274e6	−0.204501
$971$	2.49837e6	0.0850372	0.0425186	−	0.999096i	$-0.486462\pi$
$971$	2.49837e6	0.0850372	0.0425186	+	0.999096i	$0.486462\pi$
$972$	0	0
$973$	−1.70291e7	−0.576645
$974$	−2.76085e7	−0.932494
$975$	0	0
$976$	−1.53665e7	−0.516359
$977$	1.93338e7	0.648009	0.324004	−	0.946056i	$-0.394971\pi$
$977$	1.93338e7	0.648009	0.324004	+	0.946056i	$0.394971\pi$
$978$	0	0
$979$	8.76237e7	2.92190
$980$	1.16489e6	0.0387454
$981$	0	0
$982$	2.43883e6	0.0807055
$983$	−4.54905e6	−0.150154	−0.0750769	−	0.997178i	$-0.523920\pi$
$983$	−4.54905e6	−0.150154	−0.0750769	+	0.997178i	$0.523920\pi$
$984$	0	0
$985$	−1.11139e7	−0.364986
$986$	−1.19506e7	−0.391469
$987$	0	0
$988$	−2.05110e6	−0.0668489
$989$	3.64242e7	1.18413
$990$	0	0
$991$	3.81017e7	1.23243	0.616213	−	0.787580i	$-0.288666\pi$
$991$	3.81017e7	1.23243	0.616213	+	0.787580i	$0.288666\pi$
$992$	5.00789e6	0.161576
$993$	0	0
$994$	−7.74072e6	−0.248494
$995$	2.49210e7	0.798010
$996$	0	0
$997$	−3.23202e7	−1.02976	−0.514880	−	0.857262i	$-0.672164\pi$
$997$	−3.23202e7	−1.02976	−0.514880	+	0.857262i	$0.672164\pi$
$998$	−3.44315e7	−1.09428
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.6.a.c.1.4		6
3.2	odd	2		273.6.a.d.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.6.a.d.1.3	✓	6	3.2	odd	2
819.6.a.c.1.4		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	819.a (trivial)

\(f(q)\)	\(=\)	\(q+4.45521 q^{2} -12.1511 q^{4} -39.9280 q^{5} -49.0000 q^{7} -196.702 q^{8} +O(q^{10})\)	\(q+4.45521 q^{2} -12.1511 q^{4} -39.9280 q^{5} -49.0000 q^{7} -196.702 q^{8} -177.888 q^{10} -775.077 q^{11} -169.000 q^{13} -218.305 q^{14} -487.516 q^{16} +667.181 q^{17} -998.815 q^{19} +485.169 q^{20} -3453.13 q^{22} -3701.16 q^{23} -1530.75 q^{25} -752.931 q^{26} +595.403 q^{28} -4020.48 q^{29} +1214.77 q^{31} +4122.49 q^{32} +2972.43 q^{34} +1956.47 q^{35} -10629.6 q^{37} -4449.93 q^{38} +7853.94 q^{40} -6442.35 q^{41} -9841.27 q^{43} +9418.03 q^{44} -16489.5 q^{46} -8507.75 q^{47} +2401.00 q^{49} -6819.83 q^{50} +2053.53 q^{52} -31042.8 q^{53} +30947.3 q^{55} +9638.42 q^{56} -17912.1 q^{58} +43092.2 q^{59} +31520.0 q^{61} +5412.07 q^{62} +33967.1 q^{64} +6747.84 q^{65} +7199.44 q^{67} -8106.97 q^{68} +8716.50 q^{70} +35458.2 q^{71} +75921.6 q^{73} -47357.1 q^{74} +12136.7 q^{76} +37978.8 q^{77} -5278.91 q^{79} +19465.6 q^{80} -28702.0 q^{82} -65825.0 q^{83} -26639.2 q^{85} -43845.0 q^{86} +152460. q^{88} -113052. q^{89} +8281.00 q^{91} +44973.2 q^{92} -37903.8 q^{94} +39880.7 q^{95} +33688.3 q^{97} +10697.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} + 109 q^{4} - 112 q^{5} - 294 q^{7} - 339 q^{8}+O(q^{10}) \) 6 * q + 5 * q^2 + 109 * q^4 - 112 * q^5 - 294 * q^7 - 339 * q^8	\( 6 q + 5 q^{2} + 109 q^{4} - 112 q^{5} - 294 q^{7} - 339 q^{8} - 417 q^{10} - 360 q^{11} - 1014 q^{13} - 245 q^{14} + 1353 q^{16} - 2708 q^{17} + 6892 q^{19} - 3261 q^{20} - 33 q^{22} - 9744 q^{23} + 2526 q^{25} - 845 q^{26} - 5341 q^{28} + 70 q^{29} - 130 q^{31} - 31303 q^{32} + 3917 q^{34} + 5488 q^{35} + 11512 q^{37} - 17799 q^{38} + 19339 q^{40} + 912 q^{41} - 34918 q^{43} - 6503 q^{44} + 19333 q^{46} - 8594 q^{47} + 14406 q^{49} + 83560 q^{50} - 18421 q^{52} + 10102 q^{53} - 14552 q^{55} + 16611 q^{56} - 8989 q^{58} + 43260 q^{59} + 137538 q^{61} + 19690 q^{62} + 16129 q^{64} + 18928 q^{65} + 8280 q^{67} + 42295 q^{68} + 20433 q^{70} - 62380 q^{71} + 76670 q^{73} - 10387 q^{74} + 287197 q^{76} + 17640 q^{77} + 21154 q^{79} - 182689 q^{80} + 223876 q^{82} - 223186 q^{83} + 277084 q^{85} + 64747 q^{86} + 126093 q^{88} - 235190 q^{89} + 49686 q^{91} + 121233 q^{92} + 8238 q^{94} - 113200 q^{95} + 108602 q^{97} + 12005 q^{98}+O(q^{100}) \) 6 * q + 5 * q^2 + 109 * q^4 - 112 * q^5 - 294 * q^7 - 339 * q^8 - 417 * q^10 - 360 * q^11 - 1014 * q^13 - 245 * q^14 + 1353 * q^16 - 2708 * q^17 + 6892 * q^19 - 3261 * q^20 - 33 * q^22 - 9744 * q^23 + 2526 * q^25 - 845 * q^26 - 5341 * q^28 + 70 * q^29 - 130 * q^31 - 31303 * q^32 + 3917 * q^34 + 5488 * q^35 + 11512 * q^37 - 17799 * q^38 + 19339 * q^40 + 912 * q^41 - 34918 * q^43 - 6503 * q^44 + 19333 * q^46 - 8594 * q^47 + 14406 * q^49 + 83560 * q^50 - 18421 * q^52 + 10102 * q^53 - 14552 * q^55 + 16611 * q^56 - 8989 * q^58 + 43260 * q^59 + 137538 * q^61 + 19690 * q^62 + 16129 * q^64 + 18928 * q^65 + 8280 * q^67 + 42295 * q^68 + 20433 * q^70 - 62380 * q^71 + 76670 * q^73 - 10387 * q^74 + 287197 * q^76 + 17640 * q^77 + 21154 * q^79 - 182689 * q^80 + 223876 * q^82 - 223186 * q^83 + 277084 * q^85 + 64747 * q^86 + 126093 * q^88 - 235190 * q^89 + 49686 * q^91 + 121233 * q^92 + 8238 * q^94 - 113200 * q^95 + 108602 * q^97 + 12005 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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