LMFDB - Embedded newform 729.6.a.c.1.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,6,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

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

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

chi := DirichletCharacter("729.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	729.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:	$116.919804644$
Analytic rank:	$1$
Dimension:	$42$
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	729.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.21452 q^{2} -30.5249 q^{4} +59.2856 q^{5} +188.415 q^{7} -75.9377 q^{8} +O(q^{10})$	\(q+1.21452 q^{2} -30.5249 q^{4} +59.2856 q^{5} +188.415 q^{7} -75.9377 q^{8} +72.0035 q^{10} -476.837 q^{11} +1035.66 q^{13} +228.834 q^{14} +884.570 q^{16} -1526.74 q^{17} -1167.90 q^{19} -1809.69 q^{20} -579.128 q^{22} -3066.57 q^{23} +389.781 q^{25} +1257.83 q^{26} -5751.36 q^{28} -4311.80 q^{29} -551.599 q^{31} +3504.33 q^{32} -1854.25 q^{34} +11170.3 q^{35} +805.111 q^{37} -1418.44 q^{38} -4502.01 q^{40} -1097.69 q^{41} +3279.57 q^{43} +14555.4 q^{44} -3724.41 q^{46} -9281.20 q^{47} +18693.3 q^{49} +473.396 q^{50} -31613.5 q^{52} +30203.0 q^{53} -28269.6 q^{55} -14307.8 q^{56} -5236.77 q^{58} -19398.6 q^{59} +7150.63 q^{61} -669.927 q^{62} -24050.2 q^{64} +61399.7 q^{65} -26667.7 q^{67} +46603.5 q^{68} +13566.5 q^{70} -63678.3 q^{71} +6885.70 q^{73} +977.822 q^{74} +35650.1 q^{76} -89843.4 q^{77} -30007.2 q^{79} +52442.3 q^{80} -1333.17 q^{82} +32814.1 q^{83} -90513.4 q^{85} +3983.10 q^{86} +36209.9 q^{88} +6730.71 q^{89} +195134. q^{91} +93606.9 q^{92} -11272.2 q^{94} -69239.7 q^{95} +43659.2 q^{97} +22703.4 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 12 q^{2} + 624 q^{4} - 150 q^{5} - 573 q^{8}+O(q^{10}) $ 42 * q - 12 * q^2 + 624 * q^4 - 150 * q^5 - 573 * q^8	$ 42 q - 12 q^{2} + 624 q^{4} - 150 q^{5} - 573 q^{8} + 3 q^{10} - 1452 q^{11} - 2256 q^{14} + 8448 q^{16} - 3465 q^{17} + 3 q^{19} - 4128 q^{20} + 96 q^{22} - 5019 q^{23} + 18750 q^{25} - 3903 q^{26} - 6 q^{28} - 13008 q^{29} - 24273 q^{32} - 35868 q^{35} + 3 q^{37} - 51801 q^{38} + 96 q^{40} - 55833 q^{41} - 110757 q^{44} + 3 q^{46} - 90129 q^{47} + 57624 q^{49} - 145362 q^{50} + 3072 q^{52} - 103203 q^{53} - 6 q^{55} - 227154 q^{56} - 192 q^{58} - 176856 q^{59} - 31851 q^{61} - 246066 q^{62} + 86019 q^{64} - 167160 q^{65} - 801 q^{67} - 374589 q^{68} + 9375 q^{70} - 279531 q^{71} + 27012 q^{73} - 413970 q^{74} + 96 q^{76} - 185190 q^{77} - 462057 q^{80} - 6 q^{82} - 295536 q^{83} - 319803 q^{86} + 3072 q^{88} - 154827 q^{89} + 91002 q^{91} - 330558 q^{92} + 96 q^{94} - 353244 q^{95} - 463410 q^{98}+O(q^{100}) $ 42 * q - 12 * q^2 + 624 * q^4 - 150 * q^5 - 573 * q^8 + 3 * q^10 - 1452 * q^11 - 2256 * q^14 + 8448 * q^16 - 3465 * q^17 + 3 * q^19 - 4128 * q^20 + 96 * q^22 - 5019 * q^23 + 18750 * q^25 - 3903 * q^26 - 6 * q^28 - 13008 * q^29 - 24273 * q^32 - 35868 * q^35 + 3 * q^37 - 51801 * q^38 + 96 * q^40 - 55833 * q^41 - 110757 * q^44 + 3 * q^46 - 90129 * q^47 + 57624 * q^49 - 145362 * q^50 + 3072 * q^52 - 103203 * q^53 - 6 * q^55 - 227154 * q^56 - 192 * q^58 - 176856 * q^59 - 31851 * q^61 - 246066 * q^62 + 86019 * q^64 - 167160 * q^65 - 801 * q^67 - 374589 * q^68 + 9375 * q^70 - 279531 * q^71 + 27012 * q^73 - 413970 * q^74 + 96 * q^76 - 185190 * q^77 - 462057 * q^80 - 6 * q^82 - 295536 * q^83 - 319803 * q^86 + 3072 * q^88 - 154827 * q^89 + 91002 * q^91 - 330558 * q^92 + 96 * q^94 - 353244 * q^95 - 463410 * 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 $

$2$	1.21452	0.214699	0.107349	−	0.994221i	$-0.465764\pi$
$2$	1.21452	0.214699	0.107349	+	0.994221i	$0.465764\pi$
$3$	0	0
$3$	0	0
$4$	−30.5249	−0.953905
$5$	59.2856	1.06053	0.530266	−	0.847831i	$-0.322092\pi$
$5$	59.2856	1.06053	0.530266	+	0.847831i	$0.322092\pi$
$6$	0	0
$7$	188.415	1.45335	0.726676	−	0.686980i	$-0.241064\pi$
$7$	188.415	1.45335	0.726676	+	0.686980i	$0.241064\pi$
$8$	−75.9377	−0.419501
$9$	0	0
$10$	72.0035	0.227695
$11$	−476.837	−1.18820	−0.594098	−	0.804392i	$-0.702491\pi$
$11$	−476.837	−1.18820	−0.594098	+	0.804392i	$0.702491\pi$
$12$	0	0
$13$	1035.66	1.69965	0.849824	−	0.527066i	$-0.176708\pi$
$13$	1035.66	1.69965	0.849824	+	0.527066i	$0.176708\pi$
$14$	228.834	0.312033
$15$	0	0
$16$	884.570	0.863838
$17$	−1526.74	−1.28127	−0.640636	−	0.767844i	$-0.721329\pi$
$17$	−1526.74	−1.28127	−0.640636	+	0.767844i	$0.721329\pi$
$18$	0	0
$19$	−1167.90	−0.742202	−0.371101	−	0.928593i	$-0.621020\pi$
$19$	−1167.90	−0.742202	−0.371101	+	0.928593i	$0.621020\pi$
$20$	−1809.69	−1.01165
$21$	0	0
$22$	−579.128	−0.255104
$23$	−3066.57	−1.20874	−0.604371	−	0.796703i	$-0.706576\pi$
$23$	−3066.57	−1.20874	−0.604371	+	0.796703i	$0.706576\pi$
$24$	0	0
$25$	389.781	0.124730
$26$	1257.83	0.364912
$27$	0	0
$28$	−5751.36	−1.38636
$29$	−4311.80	−0.952059	−0.476030	−	0.879429i	$-0.657925\pi$
$29$	−4311.80	−0.952059	−0.476030	+	0.879429i	$0.657925\pi$
$30$	0	0
$31$	−551.599	−0.103091	−0.0515453	−	0.998671i	$-0.516415\pi$
$31$	−551.599	−0.103091	−0.0515453	+	0.998671i	$0.516415\pi$
$32$	3504.33	0.604966
$33$	0	0
$34$	−1854.25	−0.275087
$35$	11170.3	1.54133
$36$	0	0
$37$	805.111	0.0966833	0.0483416	−	0.998831i	$-0.484606\pi$
$37$	805.111	0.0966833	0.0483416	+	0.998831i	$0.484606\pi$
$38$	−1418.44	−0.159350
$39$	0	0
$40$	−4502.01	−0.444894
$41$	−1097.69	−0.101982	−0.0509908	−	0.998699i	$-0.516238\pi$
$41$	−1097.69	−0.101982	−0.0509908	+	0.998699i	$0.516238\pi$
$42$	0	0
$43$	3279.57	0.270487	0.135243	−	0.990812i	$-0.456818\pi$
$43$	3279.57	0.270487	0.135243	+	0.990812i	$0.456818\pi$
$44$	14555.4	1.13343
$45$	0	0
$46$	−3724.41	−0.259515
$47$	−9281.20	−0.612858	−0.306429	−	0.951894i	$-0.599134\pi$
$47$	−9281.20	−0.612858	−0.306429	+	0.951894i	$0.599134\pi$
$48$	0	0
$49$	18693.3	1.11223
$50$	473.396	0.0267793
$51$	0	0
$52$	−31613.5	−1.62130
$53$	30203.0	1.47693	0.738466	−	0.674291i	$-0.235551\pi$
$53$	30203.0	1.47693	0.738466	+	0.674291i	$0.235551\pi$
$54$	0	0
$55$	−28269.6	−1.26012
$56$	−14307.8	−0.609682
$57$	0	0
$58$	−5236.77	−0.204406
$59$	−19398.6	−0.725504	−0.362752	−	0.931886i	$-0.618163\pi$
$59$	−19398.6	−0.725504	−0.362752	+	0.931886i	$0.618163\pi$
$60$	0	0
$61$	7150.63	0.246048	0.123024	−	0.992404i	$-0.460741\pi$
$61$	7150.63	0.246048	0.123024	+	0.992404i	$0.460741\pi$
$62$	−669.927	−0.0221334
$63$	0	0
$64$	−24050.2	−0.733953
$65$	61399.7	1.80253
$66$	0	0
$67$	−26667.7	−0.725769	−0.362884	−	0.931834i	$-0.618208\pi$
$67$	−26667.7	−0.725769	−0.362884	+	0.931834i	$0.618208\pi$
$68$	46603.5	1.22221
$69$	0	0
$70$	13566.5	0.330921
$71$	−63678.3	−1.49915	−0.749576	−	0.661918i	$-0.769743\pi$
$71$	−63678.3	−1.49915	−0.749576	+	0.661918i	$0.769743\pi$
$72$	0	0
$73$	6885.70	0.151231	0.0756155	−	0.997137i	$-0.475908\pi$
$73$	6885.70	0.151231	0.0756155	+	0.997137i	$0.475908\pi$
$74$	977.822	0.0207578
$75$	0	0
$76$	35650.1	0.707989
$77$	−89843.4	−1.72687
$78$	0	0
$79$	−30007.2	−0.540951	−0.270475	−	0.962727i	$-0.587181\pi$
$79$	−30007.2	−0.540951	−0.270475	+	0.962727i	$0.587181\pi$
$80$	52442.3	0.916129
$81$	0	0
$82$	−1333.17	−0.0218953
$83$	32814.1	0.522835	0.261418	−	0.965226i	$-0.415810\pi$
$83$	32814.1	0.522835	0.261418	+	0.965226i	$0.415810\pi$
$84$	0	0
$85$	−90513.4	−1.35883
$86$	3983.10	0.0580732
$87$	0	0
$88$	36209.9	0.498449
$89$	6730.71	0.0900712	0.0450356	−	0.998985i	$-0.485660\pi$
$89$	6730.71	0.0900712	0.0450356	+	0.998985i	$0.485660\pi$
$90$	0	0
$91$	195134.	2.47019
$92$	93606.9	1.15302
$93$	0	0
$94$	−11272.2	−0.131580
$95$	−69239.7	−0.787129
$96$	0	0
$97$	43659.2	0.471136	0.235568	−	0.971858i	$-0.424305\pi$
$97$	43659.2	0.471136	0.235568	+	0.971858i	$0.424305\pi$
$98$	22703.4	0.238795
$99$	0	0
$100$	−11898.0	−0.118980
$101$	37762.3	0.368346	0.184173	−	0.982894i	$-0.441039\pi$
$101$	37762.3	0.368346	0.184173	+	0.982894i	$0.441039\pi$
$102$	0	0
$103$	19042.1	0.176857	0.0884283	−	0.996083i	$-0.471816\pi$
$103$	19042.1	0.176857	0.0884283	+	0.996083i	$0.471816\pi$
$104$	−78645.7	−0.713003
$105$	0	0
$106$	36682.1	0.317095
$107$	−48395.2	−0.408642	−0.204321	−	0.978904i	$-0.565499\pi$
$107$	−48395.2	−0.408642	−0.204321	+	0.978904i	$0.565499\pi$
$108$	0	0
$109$	−30446.4	−0.245453	−0.122727	−	0.992441i	$-0.539164\pi$
$109$	−30446.4	−0.245453	−0.122727	+	0.992441i	$0.539164\pi$
$110$	−34333.9	−0.270546
$111$	0	0
$112$	166667.	1.25546
$113$	−17555.5	−0.129335	−0.0646675	−	0.997907i	$-0.520599\pi$
$113$	−17555.5	−0.129335	−0.0646675	+	0.997907i	$0.520599\pi$
$114$	0	0
$115$	−181804.	−1.28191
$116$	131618.	0.908174
$117$	0	0
$118$	−23559.9	−0.155765
$119$	−287660.	−1.86214
$120$	0	0
$121$	66322.7	0.411812
$122$	8684.58	0.0528262
$123$	0	0
$124$	16837.5	0.0983385
$125$	−162159.	−0.928253
$126$	0	0
$127$	229286.	1.26144	0.630722	−	0.776009i	$-0.282759\pi$
$127$	229286.	1.26144	0.630722	+	0.776009i	$0.282759\pi$
$128$	−141348.	−0.762544
$129$	0	0
$130$	74571.1	0.387001
$131$	−52039.7	−0.264945	−0.132473	−	0.991187i	$-0.542292\pi$
$131$	−52039.7	−0.264945	−0.132473	+	0.991187i	$0.542292\pi$
$132$	0	0
$133$	−220050.	−1.07868
$134$	−32388.4	−0.155822
$135$	0	0
$136$	115937.	0.537495
$137$	−126086.	−0.573938	−0.286969	−	0.957940i	$-0.592648\pi$
$137$	−126086.	−0.573938	−0.286969	+	0.957940i	$0.592648\pi$
$138$	0	0
$139$	−99040.6	−0.434787	−0.217393	−	0.976084i	$-0.569755\pi$
$139$	−99040.6	−0.434787	−0.217393	+	0.976084i	$0.569755\pi$
$140$	−340973.	−1.47028
$141$	0	0
$142$	−77338.5	−0.321866
$143$	−493841.	−2.01952
$144$	0	0
$145$	−255628.	−1.00969
$146$	8362.81	0.0324691
$147$	0	0
$148$	−24576.0	−0.0922266
$149$	−436465.	−1.61059	−0.805293	−	0.592877i	$-0.797992\pi$
$149$	−436465.	−1.61059	−0.805293	+	0.592877i	$0.797992\pi$
$150$	0	0
$151$	−396131.	−1.41383	−0.706914	−	0.707299i	$-0.749913\pi$
$151$	−396131.	−1.41383	−0.706914	+	0.707299i	$0.749913\pi$
$152$	88687.7	0.311354
$153$	0	0
$154$	−109116.	−0.370756
$155$	−32701.8	−0.109331
$156$	0	0
$157$	−486897.	−1.57648	−0.788238	−	0.615370i	$-0.789007\pi$
$157$	−486897.	−1.57648	−0.788238	+	0.615370i	$0.789007\pi$
$158$	−36444.3	−0.116141
$159$	0	0
$160$	207757.	0.641586
$161$	−577789.	−1.75673
$162$	0	0
$163$	230832.	0.680498	0.340249	−	0.940335i	$-0.389489\pi$
$163$	230832.	0.680498	0.340249	+	0.940335i	$0.389489\pi$
$164$	33507.0	0.0972806
$165$	0	0
$166$	39853.3	0.112252
$167$	−251690.	−0.698354	−0.349177	−	0.937057i	$-0.613539\pi$
$167$	−251690.	−0.698354	−0.349177	+	0.937057i	$0.613539\pi$
$168$	0	0
$169$	701300.	1.88880
$170$	−109930.	−0.291739
$171$	0	0
$172$	−100109.	−0.258019
$173$	126132.	0.320413	0.160206	−	0.987084i	$-0.448784\pi$
$173$	126132.	0.320413	0.160206	+	0.987084i	$0.448784\pi$
$174$	0	0
$175$	73440.6	0.181276
$176$	−421796.	−1.02641
$177$	0	0
$178$	8174.57	0.0193382
$179$	397620.	0.927545	0.463773	−	0.885954i	$-0.346495\pi$
$179$	397620.	0.927545	0.463773	+	0.885954i	$0.346495\pi$
$180$	0	0
$181$	−845309.	−1.91787	−0.958935	−	0.283627i	$-0.908462\pi$
$181$	−845309.	−1.91787	−0.958935	+	0.283627i	$0.908462\pi$
$182$	236994.	0.530346
$183$	0	0
$184$	232869.	0.507068
$185$	47731.5	0.102536
$186$	0	0
$187$	728004.	1.52240
$188$	283308.	0.584608
$189$	0	0
$190$	−84092.9	−0.168996
$191$	548829.	1.08856	0.544282	−	0.838903i	$-0.316802\pi$
$191$	548829.	1.08856	0.544282	+	0.838903i	$0.316802\pi$
$192$	0	0
$193$	83240.3	0.160857	0.0804286	−	0.996760i	$-0.474371\pi$
$193$	83240.3	0.160857	0.0804286	+	0.996760i	$0.474371\pi$
$194$	53024.9	0.101152
$195$	0	0
$196$	−570612.	−1.06096
$197$	546117.	1.00258	0.501291	−	0.865279i	$-0.332859\pi$
$197$	546117.	1.00258	0.501291	+	0.865279i	$0.332859\pi$
$198$	0	0
$199$	−665183.	−1.19072	−0.595358	−	0.803460i	$-0.702990\pi$
$199$	−665183.	−1.19072	−0.595358	+	0.803460i	$0.702990\pi$
$200$	−29599.1	−0.0523243
$201$	0	0
$202$	45863.1	0.0790833
$203$	−812410.	−1.38368
$204$	0	0
$205$	−65077.4	−0.108155
$206$	23127.0	0.0379709
$207$	0	0
$208$	916115.	1.46822
$209$	556898.	0.881882
$210$	0	0
$211$	−683037.	−1.05618	−0.528090	−	0.849188i	$-0.677092\pi$
$211$	−683037.	−1.05618	−0.528090	+	0.849188i	$0.677092\pi$
$212$	−921945.	−1.40885
$213$	0	0
$214$	−58776.9	−0.0877349
$215$	194431.	0.286860
$216$	0	0
$217$	−103930.	−0.149827
$218$	−36977.7	−0.0526985
$219$	0	0
$220$	862927.	1.20204
$221$	−1.58118e6	−2.17771
$222$	0	0
$223$	−616634.	−0.830358	−0.415179	−	0.909740i	$-0.636281\pi$
$223$	−616634.	−0.830358	−0.415179	+	0.909740i	$0.636281\pi$
$224$	660270.	0.879228
$225$	0	0
$226$	−21321.4	−0.0277680
$227$	−1.10752e6	−1.42654	−0.713272	−	0.700888i	$-0.752787\pi$
$227$	−1.10752e6	−1.42654	−0.713272	+	0.700888i	$0.752787\pi$
$228$	0	0
$229$	334680.	0.421737	0.210868	−	0.977514i	$-0.432371\pi$
$229$	334680.	0.421737	0.210868	+	0.977514i	$0.432371\pi$
$230$	−220804.	−0.275224
$231$	0	0
$232$	327429.	0.399390
$233$	1.50632e6	1.81772	0.908858	−	0.417105i	$-0.136955\pi$
$233$	1.50632e6	1.81772	0.908858	+	0.417105i	$0.136955\pi$
$234$	0	0
$235$	−550241.	−0.649956
$236$	592141.	0.692062
$237$	0	0
$238$	−349369.	−0.399799
$239$	−331457.	−0.375346	−0.187673	−	0.982232i	$-0.560095\pi$
$239$	−331457.	−0.375346	−0.187673	+	0.982232i	$0.560095\pi$
$240$	0	0
$241$	−588241.	−0.652398	−0.326199	−	0.945301i	$-0.605768\pi$
$241$	−588241.	−0.652398	−0.326199	+	0.945301i	$0.605768\pi$
$242$	80550.2	0.0884155
$243$	0	0
$244$	−218273.	−0.234706
$245$	1.10824e6	1.17956
$246$	0	0
$247$	−1.20955e6	−1.26148
$248$	41887.1	0.0432465
$249$	0	0
$250$	−196945.	−0.199295
$251$	−517920.	−0.518894	−0.259447	−	0.965757i	$-0.583540\pi$
$251$	−517920.	−0.518894	−0.259447	+	0.965757i	$0.583540\pi$
$252$	0	0
$253$	1.46226e6	1.43622
$254$	278472.	0.270830
$255$	0	0
$256$	597936.	0.570236
$257$	−1.33443e6	−1.26027	−0.630133	−	0.776487i	$-0.717000\pi$
$257$	−1.33443e6	−1.26027	−0.630133	+	0.776487i	$0.717000\pi$
$258$	0	0
$259$	151695.	0.140515
$260$	−1.87422e6	−1.71944
$261$	0	0
$262$	−63203.2	−0.0568834
$263$	−1.50851e6	−1.34480	−0.672401	−	0.740187i	$-0.734737\pi$
$263$	−1.50851e6	−1.34480	−0.672401	+	0.740187i	$0.734737\pi$
$264$	0	0
$265$	1.79060e6	1.56633
$266$	−267255.	−0.231591
$267$	0	0
$268$	814030.	0.692314
$269$	588362.	0.495751	0.247876	−	0.968792i	$-0.420268\pi$
$269$	588362.	0.495751	0.247876	+	0.968792i	$0.420268\pi$
$270$	0	0
$271$	818157.	0.676727	0.338363	−	0.941016i	$-0.390127\pi$
$271$	818157.	0.676727	0.338363	+	0.941016i	$0.390127\pi$
$272$	−1.35051e6	−1.10681
$273$	0	0
$274$	−153134.	−0.123224
$275$	−185862.	−0.148204
$276$	0	0
$277$	2.26193e6	1.77125	0.885624	−	0.464403i	$-0.153731\pi$
$277$	2.26193e6	1.77125	0.885624	+	0.464403i	$0.153731\pi$
$278$	−120287.	−0.0933481
$279$	0	0
$280$	−848248.	−0.646588
$281$	852503.	0.644066	0.322033	−	0.946728i	$-0.395634\pi$
$281$	852503.	0.644066	0.322033	+	0.946728i	$0.395634\pi$
$282$	0	0
$283$	−2.33120e6	−1.73027	−0.865134	−	0.501542i	$-0.832766\pi$
$283$	−2.33120e6	−1.73027	−0.865134	+	0.501542i	$0.832766\pi$
$284$	1.94378e6	1.43005
$285$	0	0
$286$	−599780.	−0.433587
$287$	−206822.	−0.148215
$288$	0	0
$289$	911065.	0.641660
$290$	−310465.	−0.216779
$291$	0	0
$292$	−210186.	−0.144260
$293$	−2.52778e6	−1.72016	−0.860082	−	0.510155i	$-0.829588\pi$
$293$	−2.52778e6	−1.72016	−0.860082	+	0.510155i	$0.829588\pi$
$294$	0	0
$295$	−1.15006e6	−0.769421
$296$	−61138.3	−0.0405587
$297$	0	0
$298$	−530095.	−0.345790
$299$	−3.17593e6	−2.05444
$300$	0	0
$301$	617922.	0.393113
$302$	−481109.	−0.303547
$303$	0	0
$304$	−1.03309e6	−0.641142
$305$	423929.	0.260942
$306$	0	0
$307$	2.76583e6	1.67487	0.837433	−	0.546540i	$-0.184056\pi$
$307$	2.76583e6	1.67487	0.837433	+	0.546540i	$0.184056\pi$
$308$	2.74246e6	1.64727
$309$	0	0
$310$	−39717.0	−0.0234732
$311$	−1.78630e6	−1.04726	−0.523628	−	0.851947i	$-0.675422\pi$
$311$	−1.78630e6	−1.04726	−0.523628	+	0.851947i	$0.675422\pi$
$312$	0	0
$313$	1.46256e6	0.843826	0.421913	−	0.906636i	$-0.361359\pi$
$313$	1.46256e6	0.843826	0.421913	+	0.906636i	$0.361359\pi$
$314$	−591345.	−0.338467
$315$	0	0
$316$	915968.	0.516015
$317$	−2.25919e6	−1.26271	−0.631356	−	0.775493i	$-0.717501\pi$
$317$	−2.25919e6	−1.26271	−0.631356	+	0.775493i	$0.717501\pi$
$318$	0	0
$319$	2.05603e6	1.13123
$320$	−1.42583e6	−0.778381
$321$	0	0
$322$	−701736.	−0.377167
$323$	1.78308e6	0.950963
$324$	0	0
$325$	403681.	0.211997
$326$	280350.	0.146102
$327$	0	0
$328$	83356.3	0.0427813
$329$	−1.74872e6	−0.890698
$330$	0	0
$331$	2.08370e6	1.04536	0.522680	−	0.852529i	$-0.324932\pi$
$331$	2.08370e6	1.04536	0.522680	+	0.852529i	$0.324932\pi$
$332$	−1.00165e6	−0.498735
$333$	0	0
$334$	−305683.	−0.149936
$335$	−1.58101e6	−0.769702
$336$	0	0
$337$	2.83452e6	1.35958	0.679790	−	0.733407i	$-0.262071\pi$
$337$	2.83452e6	1.35958	0.679790	+	0.733407i	$0.262071\pi$
$338$	851742.	0.405524
$339$	0	0
$340$	2.76292e6	1.29620
$341$	263023.	0.122492
$342$	0	0
$343$	355408.	0.163114
$344$	−249043.	−0.113469
$345$	0	0
$346$	153190.	0.0687922
$347$	1.27148e6	0.566875	0.283438	−	0.958991i	$-0.408525\pi$
$347$	1.27148e6	0.566875	0.283438	+	0.958991i	$0.408525\pi$
$348$	0	0
$349$	−1.22017e6	−0.536235	−0.268118	−	0.963386i	$-0.586402\pi$
$349$	−1.22017e6	−0.536235	−0.268118	+	0.963386i	$0.586402\pi$
$350$	89195.0	0.0389198
$351$	0	0
$352$	−1.67100e6	−0.718818
$353$	2.40018e6	1.02520	0.512598	−	0.858629i	$-0.328683\pi$
$353$	2.40018e6	1.02520	0.512598	+	0.858629i	$0.328683\pi$
$354$	0	0
$355$	−3.77521e6	−1.58990
$356$	−205455.	−0.0859193
$357$	0	0
$358$	482917.	0.199143
$359$	715265.	0.292908	0.146454	−	0.989217i	$-0.453214\pi$
$359$	715265.	0.292908	0.146454	+	0.989217i	$0.453214\pi$
$360$	0	0
$361$	−1.11211e6	−0.449137
$362$	−1.02664e6	−0.411764
$363$	0	0
$364$	−5.95646e6	−2.35632
$365$	408223.	0.160385
$366$	0	0
$367$	−1.68812e6	−0.654240	−0.327120	−	0.944983i	$-0.606078\pi$
$367$	−1.68812e6	−0.654240	−0.327120	+	0.944983i	$0.606078\pi$
$368$	−2.71260e6	−1.04416
$369$	0	0
$370$	57970.8	0.0220143
$371$	5.69071e6	2.14650
$372$	0	0
$373$	−3.01889e6	−1.12351	−0.561753	−	0.827305i	$-0.689873\pi$
$373$	−3.01889e6	−1.12351	−0.561753	+	0.827305i	$0.689873\pi$
$374$	884175.	0.326858
$375$	0	0
$376$	704793.	0.257094
$377$	−4.46557e6	−1.61817
$378$	0	0
$379$	−3.61016e6	−1.29101	−0.645503	−	0.763757i	$-0.723352\pi$
$379$	−3.61016e6	−1.29101	−0.645503	+	0.763757i	$0.723352\pi$
$380$	2.11354e6	0.750846
$381$	0	0
$382$	666563.	0.233713
$383$	3.82230e6	1.33146	0.665730	−	0.746193i	$-0.268120\pi$
$383$	3.82230e6	1.33146	0.665730	+	0.746193i	$0.268120\pi$
$384$	0	0
$385$	−5.32642e6	−1.83140
$386$	101097.	0.0345358
$387$	0	0
$388$	−1.33269e6	−0.449419
$389$	2.19485e6	0.735412	0.367706	−	0.929942i	$-0.380143\pi$
$389$	2.19485e6	0.735412	0.367706	+	0.929942i	$0.380143\pi$
$390$	0	0
$391$	4.68185e6	1.54873
$392$	−1.41953e6	−0.466583
$393$	0	0
$394$	663269.	0.215253
$395$	−1.77899e6	−0.573696
$396$	0	0
$397$	−3.27312e6	−1.04228	−0.521142	−	0.853470i	$-0.674494\pi$
$397$	−3.27312e6	−1.04228	−0.521142	+	0.853470i	$0.674494\pi$
$398$	−807877.	−0.255645
$399$	0	0
$400$	344789.	0.107746
$401$	−3.16257e6	−0.982154	−0.491077	−	0.871116i	$-0.663397\pi$
$401$	−3.16257e6	−0.982154	−0.491077	+	0.871116i	$0.663397\pi$
$402$	0	0
$403$	−571269.	−0.175218
$404$	−1.15269e6	−0.351367
$405$	0	0
$406$	−986687.	−0.297074
$407$	−383907.	−0.114879
$408$	0	0
$409$	4.76838e6	1.40949	0.704746	−	0.709460i	$-0.251061\pi$
$409$	4.76838e6	1.40949	0.704746	+	0.709460i	$0.251061\pi$
$410$	−79037.7	−0.0232207
$411$	0	0
$412$	−581259.	−0.168704
$413$	−3.65499e6	−1.05441
$414$	0	0
$415$	1.94540e6	0.554484
$416$	3.62930e6	1.02823
$417$	0	0
$418$	676364.	0.189339
$419$	4.33162e6	1.20536	0.602679	−	0.797984i	$-0.294100\pi$
$419$	4.33162e6	1.20536	0.602679	+	0.797984i	$0.294100\pi$
$420$	0	0
$421$	−2.44457e6	−0.672199	−0.336099	−	0.941827i	$-0.609108\pi$
$421$	−2.44457e6	−0.672199	−0.336099	+	0.941827i	$0.609108\pi$
$422$	−829561.	−0.226760
$423$	0	0
$424$	−2.29355e6	−0.619574
$425$	−595092.	−0.159813
$426$	0	0
$427$	1.34729e6	0.357594
$428$	1.47726e6	0.389806
$429$	0	0
$430$	236141.	0.0615885
$431$	1.41794e6	0.367676	0.183838	−	0.982957i	$-0.441148\pi$
$431$	1.41794e6	0.367676	0.183838	+	0.982957i	$0.441148\pi$
$432$	0	0
$433$	4.13697e6	1.06038	0.530191	−	0.847878i	$-0.322120\pi$
$433$	4.13697e6	1.06038	0.530191	+	0.847878i	$0.322120\pi$
$434$	−126224.	−0.0321676
$435$	0	0
$436$	929373.	0.234139
$437$	3.58145e6	0.897130
$438$	0	0
$439$	1.17695e6	0.291473	0.145736	−	0.989323i	$-0.453445\pi$
$439$	1.17695e6	0.291473	0.145736	+	0.989323i	$0.453445\pi$
$440$	2.14673e6	0.528622
$441$	0	0
$442$	−1.92037e6	−0.467552
$443$	−6.07021e6	−1.46958	−0.734792	−	0.678293i	$-0.762720\pi$
$443$	−6.07021e6	−1.46958	−0.734792	+	0.678293i	$0.762720\pi$
$444$	0	0
$445$	399034.	0.0955234
$446$	−748914.	−0.178277
$447$	0	0
$448$	−4.53142e6	−1.06669
$449$	4.30002e6	1.00659	0.503297	−	0.864113i	$-0.332120\pi$
$449$	4.30002e6	1.00659	0.503297	+	0.864113i	$0.332120\pi$
$450$	0	0
$451$	523421.	0.121174
$452$	535879.	0.123373
$453$	0	0
$454$	−1.34510e6	−0.306277
$455$	1.15686e7	2.61972
$456$	0	0
$457$	6.52135e6	1.46065	0.730326	−	0.683099i	$-0.239368\pi$
$457$	6.52135e6	1.46065	0.730326	+	0.683099i	$0.239368\pi$
$458$	406475.	0.0905463
$459$	0	0
$460$	5.54954e6	1.22282
$461$	−1.17187e6	−0.256819	−0.128409	−	0.991721i	$-0.540987\pi$
$461$	−1.17187e6	−0.256819	−0.128409	+	0.991721i	$0.540987\pi$
$462$	0	0
$463$	−3.88640e6	−0.842549	−0.421274	−	0.906933i	$-0.638417\pi$
$463$	−3.88640e6	−0.842549	−0.421274	+	0.906933i	$0.638417\pi$
$464$	−3.81409e6	−0.822425
$465$	0	0
$466$	1.82945e6	0.390261
$467$	1.14887e6	0.243770	0.121885	−	0.992544i	$-0.461106\pi$
$467$	1.14887e6	0.243770	0.121885	+	0.992544i	$0.461106\pi$
$468$	0	0
$469$	−5.02460e6	−1.05480
$470$	−668279.	−0.139545
$471$	0	0
$472$	1.47308e6	0.304350
$473$	−1.56382e6	−0.321392
$474$	0	0
$475$	−455225.	−0.0925747
$476$	8.78081e6	1.77630
$477$	0	0
$478$	−402560.	−0.0805863
$479$	−4.41400e6	−0.879009	−0.439504	−	0.898240i	$-0.644846\pi$
$479$	−4.41400e6	−0.879009	−0.439504	+	0.898240i	$0.644846\pi$
$480$	0	0
$481$	833821.	0.164328
$482$	−714430.	−0.140069
$483$	0	0
$484$	−2.02450e6	−0.392829
$485$	2.58836e6	0.499655
$486$	0	0
$487$	864205.	0.165118	0.0825590	−	0.996586i	$-0.473691\pi$
$487$	864205.	0.165118	0.0825590	+	0.996586i	$0.473691\pi$
$488$	−543003.	−0.103217
$489$	0	0
$490$	1.34598e6	0.253250
$491$	4.85379e6	0.908609	0.454304	−	0.890846i	$-0.349888\pi$
$491$	4.85379e6	0.908609	0.454304	+	0.890846i	$0.349888\pi$
$492$	0	0
$493$	6.58299e6	1.21985
$494$	−1.46902e6	−0.270838
$495$	0	0
$496$	−487928.	−0.0890536
$497$	−1.19980e7	−2.17880
$498$	0	0
$499$	−852095.	−0.153192	−0.0765961	−	0.997062i	$-0.524405\pi$
$499$	−852095.	−0.153192	−0.0765961	+	0.997062i	$0.524405\pi$
$500$	4.94990e6	0.885464
$501$	0	0
$502$	−629024.	−0.111406
$503$	5.79050e6	1.02046	0.510230	−	0.860038i	$-0.329560\pi$
$503$	5.79050e6	1.02046	0.510230	+	0.860038i	$0.329560\pi$
$504$	0	0
$505$	2.23876e6	0.390643
$506$	1.77594e6	0.308355
$507$	0	0
$508$	−6.99894e6	−1.20330
$509$	1.07914e7	1.84622	0.923108	−	0.384540i	$-0.125640\pi$
$509$	1.07914e7	1.84622	0.923108	+	0.384540i	$0.125640\pi$
$510$	0	0
$511$	1.29737e6	0.219792
$512$	5.24934e6	0.884973
$513$	0	0
$514$	−1.62069e6	−0.270577
$515$	1.12892e6	0.187562
$516$	0	0
$517$	4.42562e6	0.728195
$518$	184237.	0.0301683
$519$	0	0
$520$	−4.66256e6	−0.756164
$521$	2.46455e6	0.397781	0.198890	−	0.980022i	$-0.436266\pi$
$521$	2.46455e6	0.397781	0.198890	+	0.980022i	$0.436266\pi$
$522$	0	0
$523$	−2.09488e6	−0.334892	−0.167446	−	0.985881i	$-0.553552\pi$
$523$	−2.09488e6	−0.334892	−0.167446	+	0.985881i	$0.553552\pi$
$524$	1.58851e6	0.252733
$525$	0	0
$526$	−1.83211e6	−0.288727
$527$	842145.	0.132087
$528$	0	0
$529$	2.96752e6	0.461057
$530$	2.17472e6	0.336290
$531$	0	0
$532$	6.71702e6	1.02896
$533$	−1.13684e6	−0.173333
$534$	0	0
$535$	−2.86914e6	−0.433379
$536$	2.02508e6	0.304460
$537$	0	0
$538$	714576.	0.106437
$539$	−8.91366e6	−1.32155
$540$	0	0
$541$	7.87225e6	1.15639	0.578197	−	0.815897i	$-0.303757\pi$
$541$	7.87225e6	1.15639	0.578197	+	0.815897i	$0.303757\pi$
$542$	993667.	0.145292
$543$	0	0
$544$	−5.35019e6	−0.775126
$545$	−1.80503e6	−0.260311
$546$	0	0
$547$	−1.27010e7	−1.81497	−0.907483	−	0.420088i	$-0.861999\pi$
$547$	−1.27010e7	−1.81497	−0.907483	+	0.420088i	$0.861999\pi$
$548$	3.84877e6	0.547482
$549$	0	0
$550$	−225733.	−0.0318191
$551$	5.03576e6	0.706620
$552$	0	0
$553$	−5.65381e6	−0.786192
$554$	2.74715e6	0.380285
$555$	0	0
$556$	3.02321e6	0.414745
$557$	−8.58602e6	−1.17261	−0.586306	−	0.810090i	$-0.699418\pi$
$557$	−8.58602e6	−1.17261	−0.586306	+	0.810090i	$0.699418\pi$
$558$	0	0
$559$	3.39653e6	0.459733
$560$	9.88092e6	1.33146
$561$	0	0
$562$	1.03538e6	0.138280
$563$	4.20431e6	0.559015	0.279508	−	0.960143i	$-0.409829\pi$
$563$	4.20431e6	0.559015	0.279508	+	0.960143i	$0.409829\pi$
$564$	0	0
$565$	−1.04079e6	−0.137164
$566$	−2.83128e6	−0.371486
$567$	0	0
$568$	4.83559e6	0.628895
$569$	−7.11254e6	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$569$	−7.11254e6	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$570$	0	0
$571$	1.00692e7	1.29242	0.646209	−	0.763161i	$-0.276354\pi$
$571$	1.00692e7	1.29242	0.646209	+	0.763161i	$0.276354\pi$
$572$	1.50745e7	1.92643
$573$	0	0
$574$	−251189.	−0.0318216
$575$	−1.19529e6	−0.150766
$576$	0	0
$577$	1.32728e7	1.65967	0.829834	−	0.558010i	$-0.188435\pi$
$577$	1.32728e7	1.65967	0.829834	+	0.558010i	$0.188435\pi$
$578$	1.10651e6	0.137763
$579$	0	0
$580$	7.80303e6	0.963148
$581$	6.18267e6	0.759864
$582$	0	0
$583$	−1.44019e7	−1.75489
$584$	−522884.	−0.0634415
$585$	0	0
$586$	−3.07004e6	−0.369317
$587$	−5.57554e6	−0.667870	−0.333935	−	0.942596i	$-0.608377\pi$
$587$	−5.57554e6	−0.667870	−0.333935	+	0.942596i	$0.608377\pi$
$588$	0	0
$589$	644212.	0.0765140
$590$	−1.39677e6	−0.165194
$591$	0	0
$592$	712177.	0.0835187
$593$	−1.67038e6	−0.195064	−0.0975321	−	0.995232i	$-0.531095\pi$
$593$	−1.67038e6	−0.195064	−0.0975321	+	0.995232i	$0.531095\pi$
$594$	0	0
$595$	−1.70541e7	−1.97486
$596$	1.33231e7	1.53634
$597$	0	0
$598$	−3.85722e6	−0.441085
$599$	1.54832e7	1.76316	0.881582	−	0.472032i	$-0.156479\pi$
$599$	1.54832e7	1.76316	0.881582	+	0.472032i	$0.156479\pi$
$600$	0	0
$601$	2.93414e6	0.331356	0.165678	−	0.986180i	$-0.447019\pi$
$601$	2.93414e6	0.331356	0.165678	+	0.986180i	$0.447019\pi$
$602$	750478.	0.0844008
$603$	0	0
$604$	1.20919e7	1.34866
$605$	3.93198e6	0.436740
$606$	0	0
$607$	8.41272e6	0.926755	0.463377	−	0.886161i	$-0.346637\pi$
$607$	8.41272e6	0.926755	0.463377	+	0.886161i	$0.346637\pi$
$608$	−4.09271e6	−0.449006
$609$	0	0
$610$	514870.	0.0560239
$611$	−9.61217e6	−1.04164
$612$	0	0
$613$	−7.52319e6	−0.808631	−0.404316	−	0.914619i	$-0.632490\pi$
$613$	−7.52319e6	−0.808631	−0.404316	+	0.914619i	$0.632490\pi$
$614$	3.35916e6	0.359591
$615$	0	0
$616$	6.82250e6	0.724423
$617$	−1.61778e7	−1.71083	−0.855415	−	0.517944i	$-0.826698\pi$
$617$	−1.61778e7	−1.71083	−0.855415	+	0.517944i	$0.826698\pi$
$618$	0	0
$619$	−1.25380e7	−1.31523	−0.657617	−	0.753353i	$-0.728435\pi$
$619$	−1.25380e7	−1.31523	−0.657617	+	0.753353i	$0.728435\pi$
$620$	998222.	0.104291
$621$	0	0
$622$	−2.16949e6	−0.224844
$623$	1.26817e6	0.130905
$624$	0	0
$625$	−1.08318e7	−1.10917
$626$	1.77631e6	0.181168
$627$	0	0
$628$	1.48625e7	1.50381
$629$	−1.22919e6	−0.123878
$630$	0	0
$631$	−1.20660e7	−1.20639	−0.603196	−	0.797593i	$-0.706106\pi$
$631$	−1.20660e7	−1.20639	−0.603196	+	0.797593i	$0.706106\pi$
$632$	2.27868e6	0.226929
$633$	0	0
$634$	−2.74383e6	−0.271102
$635$	1.35933e7	1.33780
$636$	0	0
$637$	1.93599e7	1.89041
$638$	2.49709e6	0.242874
$639$	0	0
$640$	−8.37990e6	−0.808703
$641$	−9.33561e6	−0.897424	−0.448712	−	0.893676i	$-0.648117\pi$
$641$	−9.33561e6	−0.897424	−0.448712	+	0.893676i	$0.648117\pi$
$642$	0	0
$643$	4.67767e6	0.446172	0.223086	−	0.974799i	$-0.428387\pi$
$643$	4.67767e6	0.446172	0.223086	+	0.974799i	$0.428387\pi$
$644$	1.76370e7	1.67575
$645$	0	0
$646$	2.16558e6	0.204170
$647$	1.12649e7	1.05796	0.528979	−	0.848635i	$-0.322575\pi$
$647$	1.12649e7	1.05796	0.528979	+	0.848635i	$0.322575\pi$
$648$	0	0
$649$	9.24997e6	0.862042
$650$	490278.	0.0455154
$651$	0	0
$652$	−7.04613e6	−0.649130
$653$	−2.09671e7	−1.92422	−0.962111	−	0.272657i	$-0.912098\pi$
$653$	−2.09671e7	−1.92422	−0.962111	+	0.272657i	$0.912098\pi$
$654$	0	0
$655$	−3.08520e6	−0.280983
$656$	−970987.	−0.0880955
$657$	0	0
$658$	−2.12385e6	−0.191232
$659$	1.64642e7	1.47682	0.738411	−	0.674351i	$-0.235577\pi$
$659$	1.64642e7	1.47682	0.738411	+	0.674351i	$0.235577\pi$
$660$	0	0
$661$	−7.79114e6	−0.693581	−0.346791	−	0.937943i	$-0.612729\pi$
$661$	−7.79114e6	−0.693581	−0.346791	+	0.937943i	$0.612729\pi$
$662$	2.53069e6	0.224437
$663$	0	0
$664$	−2.49183e6	−0.219330
$665$	−1.30458e7	−1.14398
$666$	0	0
$667$	1.32225e7	1.15079
$668$	7.68283e6	0.666163
$669$	0	0
$670$	−1.92017e6	−0.165254
$671$	−3.40969e6	−0.292353
$672$	0	0
$673$	5.20044e6	0.442591	0.221295	−	0.975207i	$-0.428972\pi$
$673$	5.20044e6	0.442591	0.221295	+	0.975207i	$0.428972\pi$
$674$	3.44258e6	0.291900
$675$	0	0
$676$	−2.14071e7	−1.80174
$677$	−2.23757e7	−1.87631	−0.938155	−	0.346215i	$-0.887467\pi$
$677$	−2.23757e7	−1.87631	−0.938155	+	0.346215i	$0.887467\pi$
$678$	0	0
$679$	8.22606e6	0.684727
$680$	6.87338e6	0.570031
$681$	0	0
$682$	319446.	0.0262988
$683$	−9.67491e6	−0.793588	−0.396794	−	0.917908i	$-0.629877\pi$
$683$	−9.67491e6	−0.793588	−0.396794	+	0.917908i	$0.629877\pi$
$684$	0	0
$685$	−7.47508e6	−0.608680
$686$	431650.	0.0350204
$687$	0	0
$688$	2.90101e6	0.233657
$689$	3.12801e7	2.51026
$690$	0	0
$691$	2.21745e7	1.76668	0.883340	−	0.468732i	$-0.155289\pi$
$691$	2.21745e7	1.76668	0.883340	+	0.468732i	$0.155289\pi$
$692$	−3.85017e6	−0.305643
$693$	0	0
$694$	1.54424e6	0.121707
$695$	−5.87168e6	−0.461105
$696$	0	0
$697$	1.67589e6	0.130666
$698$	−1.48191e6	−0.115129
$699$	0	0
$700$	−2.24177e6	−0.172920
$701$	1.14342e7	0.878839	0.439419	−	0.898282i	$-0.355184\pi$
$701$	1.14342e7	0.878839	0.439419	+	0.898282i	$0.355184\pi$
$702$	0	0
$703$	−940289.	−0.0717585
$704$	1.14680e7	0.872081
$705$	0	0
$706$	2.91506e6	0.220108
$707$	7.11500e6	0.535336
$708$	0	0
$709$	1.39598e7	1.04295	0.521474	−	0.853267i	$-0.325382\pi$
$709$	1.39598e7	1.04295	0.521474	+	0.853267i	$0.325382\pi$
$710$	−4.58506e6	−0.341349
$711$	0	0
$712$	−511115.	−0.0377849
$713$	1.69152e6	0.124610
$714$	0	0
$715$	−2.92777e7	−2.14176
$716$	−1.21373e7	−0.884790
$717$	0	0
$718$	868703.	0.0628869
$719$	4.53600e6	0.327228	0.163614	−	0.986524i	$-0.447685\pi$
$719$	4.53600e6	0.327228	0.163614	+	0.986524i	$0.447685\pi$
$720$	0	0
$721$	3.58782e6	0.257035
$722$	−1.35067e6	−0.0964290
$723$	0	0
$724$	2.58030e7	1.82946
$725$	−1.68066e6	−0.118750
$726$	0	0
$727$	−1.43228e7	−1.00506	−0.502531	−	0.864559i	$-0.667598\pi$
$727$	−1.43228e7	−1.00506	−0.502531	+	0.864559i	$0.667598\pi$
$728$	−1.48180e7	−1.03625
$729$	0	0
$730$	495794.	0.0344345
$731$	−5.00704e6	−0.346568
$732$	0	0
$733$	−1.99703e6	−0.137285	−0.0686426	−	0.997641i	$-0.521867\pi$
$733$	−1.99703e6	−0.137285	−0.0686426	+	0.997641i	$0.521867\pi$
$734$	−2.05025e6	−0.140464
$735$	0	0
$736$	−1.07463e7	−0.731247
$737$	1.27161e7	0.862356
$738$	0	0
$739$	1.51730e6	0.102202	0.0511010	−	0.998693i	$-0.483727\pi$
$739$	1.51730e6	0.102202	0.0511010	+	0.998693i	$0.483727\pi$
$740$	−1.45700e6	−0.0978093
$741$	0	0
$742$	6.91147e6	0.460851
$743$	−2.30596e7	−1.53243	−0.766215	−	0.642584i	$-0.777862\pi$
$743$	−2.30596e7	−1.53243	−0.766215	+	0.642584i	$0.777862\pi$
$744$	0	0
$745$	−2.58761e7	−1.70808
$746$	−3.66650e6	−0.241215
$747$	0	0
$748$	−2.22223e7	−1.45223
$749$	−9.11840e6	−0.593901
$750$	0	0
$751$	3.15455e6	0.204098	0.102049	−	0.994779i	$-0.467460\pi$
$751$	3.15455e6	0.204098	0.102049	+	0.994779i	$0.467460\pi$
$752$	−8.20988e6	−0.529410
$753$	0	0
$754$	−5.42351e6	−0.347418
$755$	−2.34849e7	−1.49941
$756$	0	0
$757$	−1.85638e7	−1.17741	−0.588704	−	0.808349i	$-0.700362\pi$
$757$	−1.85638e7	−1.17741	−0.588704	+	0.808349i	$0.700362\pi$
$758$	−4.38461e6	−0.277177
$759$	0	0
$760$	5.25790e6	0.330201
$761$	−1.02077e7	−0.638946	−0.319473	−	0.947595i	$-0.603506\pi$
$761$	−1.02077e7	−0.638946	−0.319473	+	0.947595i	$0.603506\pi$
$762$	0	0
$763$	−5.73656e6	−0.356730
$764$	−1.67530e7	−1.03839
$765$	0	0
$766$	4.64226e6	0.285862
$767$	−2.00903e7	−1.23310
$768$	0	0
$769$	330559.	0.0201573	0.0100787	−	0.999949i	$-0.496792\pi$
$769$	330559.	0.0201573	0.0100787	+	0.999949i	$0.496792\pi$
$770$	−6.46904e6	−0.393199
$771$	0	0
$772$	−2.54091e6	−0.153442
$773$	−6.75463e6	−0.406586	−0.203293	−	0.979118i	$-0.565164\pi$
$773$	−6.75463e6	−0.406586	−0.203293	+	0.979118i	$0.565164\pi$
$774$	0	0
$775$	−215003.	−0.0128585
$776$	−3.31538e6	−0.197642
$777$	0	0
$778$	2.66568e6	0.157892
$779$	1.28200e6	0.0756908
$780$	0	0
$781$	3.03642e7	1.78129
$782$	5.68619e6	0.332510
$783$	0	0
$784$	1.65355e7	0.960790
$785$	−2.88660e7	−1.67191
$786$	0	0
$787$	−1.22793e7	−0.706700	−0.353350	−	0.935491i	$-0.614958\pi$
$787$	−1.22793e7	−0.706700	−0.353350	+	0.935491i	$0.614958\pi$
$788$	−1.66702e7	−0.956368
$789$	0	0
$790$	−2.16062e6	−0.123172
$791$	−3.30771e6	−0.187969
$792$	0	0
$793$	7.40563e6	0.418195
$794$	−3.97527e6	−0.223777
$795$	0	0
$796$	2.03047e7	1.13583
$797$	−2.31038e6	−0.128836	−0.0644180	−	0.997923i	$-0.520519\pi$
$797$	−2.31038e6	−0.128836	−0.0644180	+	0.997923i	$0.520519\pi$
$798$	0	0
$799$	1.41699e7	0.785238
$800$	1.36592e6	0.0754573
$801$	0	0
$802$	−3.84100e6	−0.210867
$803$	−3.28336e6	−0.179692
$804$	0	0
$805$	−3.42546e7	−1.86307
$806$	−693817.	−0.0376190
$807$	0	0
$808$	−2.86759e6	−0.154521
$809$	2.17473e7	1.16825	0.584124	−	0.811665i	$-0.301438\pi$
$809$	2.17473e7	1.16825	0.584124	+	0.811665i	$0.301438\pi$
$810$	0	0
$811$	1.35287e7	0.722275	0.361138	−	0.932512i	$-0.382388\pi$
$811$	1.35287e7	0.722275	0.361138	+	0.932512i	$0.382388\pi$
$812$	2.47988e7	1.31990
$813$	0	0
$814$	−466262.	−0.0246643
$815$	1.36850e7	0.721690
$816$	0	0
$817$	−3.83022e6	−0.200756
$818$	5.79129e6	0.302616
$819$	0	0
$820$	1.98648e6	0.103169
$821$	−9.88648e6	−0.511898	−0.255949	−	0.966690i	$-0.582388\pi$
$821$	−9.88648e6	−0.511898	−0.255949	+	0.966690i	$0.582388\pi$
$822$	0	0
$823$	972714.	0.0500594	0.0250297	−	0.999687i	$-0.492032\pi$
$823$	972714.	0.0500594	0.0250297	+	0.999687i	$0.492032\pi$
$824$	−1.44601e6	−0.0741915
$825$	0	0
$826$	−4.43905e6	−0.226381
$827$	2.09980e7	1.06762	0.533808	−	0.845606i	$-0.320760\pi$
$827$	2.09980e7	1.06762	0.533808	+	0.845606i	$0.320760\pi$
$828$	0	0
$829$	2.15418e7	1.08867	0.544335	−	0.838868i	$-0.316782\pi$
$829$	2.15418e7	1.08867	0.544335	+	0.838868i	$0.316782\pi$
$830$	2.36273e6	0.119047
$831$	0	0
$832$	−2.49078e7	−1.24746
$833$	−2.85397e7	−1.42507
$834$	0	0
$835$	−1.49216e7	−0.740627
$836$	−1.69993e7	−0.841231
$837$	0	0
$838$	5.26084e6	0.258789
$839$	1.72577e7	0.846406	0.423203	−	0.906035i	$-0.360906\pi$
$839$	1.72577e7	0.846406	0.423203	+	0.906035i	$0.360906\pi$
$840$	0	0
$841$	−1.91949e6	−0.0935829
$842$	−2.96898e6	−0.144320
$843$	0	0
$844$	2.08497e7	1.00749
$845$	4.15770e7	2.00314
$846$	0	0
$847$	1.24962e7	0.598508
$848$	2.67167e7	1.27583
$849$	0	0
$850$	−722751.	−0.0343116
$851$	−2.46893e6	−0.116865
$852$	0	0
$853$	8.27076e6	0.389200	0.194600	−	0.980883i	$-0.437659\pi$
$853$	8.27076e6	0.389200	0.194600	+	0.980883i	$0.437659\pi$
$854$	1.63631e6	0.0767750
$855$	0	0
$856$	3.67502e6	0.171426
$857$	−1.66522e6	−0.0774497	−0.0387248	−	0.999250i	$-0.512330\pi$
$857$	−1.66522e6	−0.0774497	−0.0387248	+	0.999250i	$0.512330\pi$
$858$	0	0
$859$	4.64918e6	0.214978	0.107489	−	0.994206i	$-0.465719\pi$
$859$	4.64918e6	0.214978	0.107489	+	0.994206i	$0.465719\pi$
$860$	−5.93501e6	−0.273637
$861$	0	0
$862$	1.72212e6	0.0789395
$863$	3.50301e7	1.60109	0.800543	−	0.599275i	$-0.204544\pi$
$863$	3.50301e7	1.60109	0.800543	+	0.599275i	$0.204544\pi$
$864$	0	0
$865$	7.47781e6	0.339808
$866$	5.02442e6	0.227663
$867$	0	0
$868$	3.17244e6	0.142921
$869$	1.43085e7	0.642756
$870$	0	0
$871$	−2.76187e7	−1.23355
$872$	2.31203e6	0.102968
$873$	0	0
$874$	4.34974e6	0.192613
$875$	−3.05532e7	−1.34908
$876$	0	0
$877$	2.49761e7	1.09654	0.548272	−	0.836300i	$-0.315286\pi$
$877$	2.49761e7	1.09654	0.548272	+	0.836300i	$0.315286\pi$
$878$	1.42943e6	0.0625788
$879$	0	0
$880$	−2.50064e7	−1.08854
$881$	213981.	0.00928827	0.00464413	−	0.999989i	$-0.498522\pi$
$881$	213981.	0.00928827	0.00464413	+	0.999989i	$0.498522\pi$
$882$	0	0
$883$	1.91856e7	0.828082	0.414041	−	0.910258i	$-0.364117\pi$
$883$	1.91856e7	0.828082	0.414041	+	0.910258i	$0.364117\pi$
$884$	4.82654e7	2.07733
$885$	0	0
$886$	−7.37238e6	−0.315518
$887$	1.16365e7	0.496609	0.248304	−	0.968682i	$-0.420127\pi$
$887$	1.16365e7	0.496609	0.248304	+	0.968682i	$0.420127\pi$
$888$	0	0
$889$	4.32010e7	1.83332
$890$	484634.	0.0205088
$891$	0	0
$892$	1.88227e7	0.792082
$893$	1.08395e7	0.454864
$894$	0	0
$895$	2.35731e7	0.983692
$896$	−2.66321e7	−1.10825
$897$	0	0
$898$	5.22246e6	0.216115
$899$	2.37838e6	0.0981483
$900$	0	0
$901$	−4.61120e7	−1.89235
$902$	635705.	0.0260159
$903$	0	0
$904$	1.33312e6	0.0542561
$905$	−5.01146e7	−2.03396
$906$	0	0
$907$	−188989.	−0.00762814	−0.00381407	−	0.999993i	$-0.501214\pi$
$907$	−188989.	−0.00762814	−0.00381407	+	0.999993i	$0.501214\pi$
$908$	3.38068e7	1.36079
$909$	0	0
$910$	1.40503e7	0.562449
$911$	8.41219e6	0.335825	0.167913	−	0.985802i	$-0.446297\pi$
$911$	8.41219e6	0.335825	0.167913	+	0.985802i	$0.446297\pi$
$912$	0	0
$913$	−1.56470e7	−0.621232
$914$	7.92030e6	0.313600
$915$	0	0
$916$	−1.02161e7	−0.402296
$917$	−9.80507e6	−0.385059
$918$	0	0
$919$	2.13773e7	0.834958	0.417479	−	0.908687i	$-0.362914\pi$
$919$	2.13773e7	0.834958	0.417479	+	0.908687i	$0.362914\pi$
$920$	1.38057e7	0.537762
$921$	0	0
$922$	−1.42326e6	−0.0551387
$923$	−6.59491e7	−2.54803
$924$	0	0
$925$	313817.	0.0120593
$926$	−4.72011e6	−0.180894
$927$	0	0
$928$	−1.51100e7	−0.575963
$929$	1.67541e7	0.636915	0.318457	−	0.947937i	$-0.396835\pi$
$929$	1.67541e7	0.636915	0.318457	+	0.947937i	$0.396835\pi$
$930$	0	0
$931$	−2.18319e7	−0.825501
$932$	−4.59802e7	−1.73393
$933$	0	0
$934$	1.39533e6	0.0523370
$935$	4.31602e7	1.61456
$936$	0	0
$937$	−5.31715e7	−1.97847	−0.989236	−	0.146326i	$-0.953255\pi$
$937$	−5.31715e7	−1.97847	−0.989236	+	0.146326i	$0.953255\pi$
$938$	−6.10247e6	−0.226464
$939$	0	0
$940$	1.67961e7	0.619995
$941$	1.54389e7	0.568385	0.284192	−	0.958767i	$-0.408275\pi$
$941$	1.54389e7	0.568385	0.284192	+	0.958767i	$0.408275\pi$
$942$	0	0
$943$	3.36616e6	0.123269
$944$	−1.71594e7	−0.626718
$945$	0	0
$946$	−1.89929e6	−0.0690024
$947$	2.28654e7	0.828523	0.414261	−	0.910158i	$-0.364040\pi$
$947$	2.28654e7	0.828523	0.414261	+	0.910158i	$0.364040\pi$
$948$	0	0
$949$	7.13125e6	0.257040
$950$	−552880.	−0.0198757
$951$	0	0
$952$	2.18443e7	0.781169
$953$	−7.65442e6	−0.273011	−0.136505	−	0.990639i	$-0.543587\pi$
$953$	−7.65442e6	−0.273011	−0.136505	+	0.990639i	$0.543587\pi$
$954$	0	0
$955$	3.25376e7	1.15446
$956$	1.01177e7	0.358045
$957$	0	0
$958$	−5.36088e6	−0.188722
$959$	−2.37565e7	−0.834134
$960$	0	0
$961$	−2.83249e7	−0.989372
$962$	1.01269e6	0.0352809
$963$	0	0
$964$	1.79560e7	0.622325
$965$	4.93495e6	0.170594
$966$	0	0
$967$	3.63466e7	1.24996	0.624982	−	0.780639i	$-0.285106\pi$
$967$	3.63466e7	1.24996	0.624982	+	0.780639i	$0.285106\pi$
$968$	−5.03640e6	−0.172755
$969$	0	0
$970$	3.14361e6	0.107275
$971$	−5.35242e7	−1.82181	−0.910903	−	0.412620i	$-0.864614\pi$
$971$	−5.35242e7	−1.82181	−0.910903	+	0.412620i	$0.864614\pi$
$972$	0	0
$973$	−1.86608e7	−0.631898
$974$	1.04959e6	0.0354506
$975$	0	0
$976$	6.32524e6	0.212546
$977$	1.61047e6	0.0539779	0.0269889	−	0.999636i	$-0.491408\pi$
$977$	1.61047e6	0.0539779	0.0269889	+	0.999636i	$0.491408\pi$
$978$	0	0
$979$	−3.20945e6	−0.107022
$980$	−3.38291e7	−1.12519
$981$	0	0
$982$	5.89502e6	0.195077
$983$	−2.65403e7	−0.876035	−0.438018	−	0.898966i	$-0.644319\pi$
$983$	−2.65403e7	−0.876035	−0.438018	+	0.898966i	$0.644319\pi$
$984$	0	0
$985$	3.23769e7	1.06327
$986$	7.99516e6	0.261900
$987$	0	0
$988$	3.69214e7	1.20333
$989$	−1.00571e7	−0.326949
$990$	0	0
$991$	−1.78093e7	−0.576054	−0.288027	−	0.957622i	$-0.592999\pi$
$991$	−1.78093e7	−0.576054	−0.288027	+	0.957622i	$0.592999\pi$
$992$	−1.93299e6	−0.0623662
$993$	0	0
$994$	−1.45718e7	−0.467785
$995$	−3.94358e7	−1.26279
$996$	0	0
$997$	6.18024e6	0.196910	0.0984550	−	0.995142i	$-0.468610\pi$
$997$	6.18024e6	0.196910	0.0984550	+	0.995142i	$0.468610\pi$
$998$	−1.03489e6	−0.0328902
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.6.a.c.1.23		42
3.2	odd	2		729.6.a.e.1.20		42
27.2	odd	18		81.6.e.a.64.9		84
27.13	even	9		27.6.e.a.7.6	yes	84
27.14	odd	18		81.6.e.a.19.9		84
27.25	even	9		27.6.e.a.4.6	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.6	✓	84	27.25	even	9
27.6.e.a.7.6	yes	84	27.13	even	9
81.6.e.a.19.9		84	27.14	odd	18
81.6.e.a.64.9		84	27.2	odd	18
729.6.a.c.1.23		42	1.1	even	1	trivial
729.6.a.e.1.20		42	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q+1.21452 q^{2} -30.5249 q^{4} +59.2856 q^{5} +188.415 q^{7} -75.9377 q^{8} +O(q^{10})\)	\(q+1.21452 q^{2} -30.5249 q^{4} +59.2856 q^{5} +188.415 q^{7} -75.9377 q^{8} +72.0035 q^{10} -476.837 q^{11} +1035.66 q^{13} +228.834 q^{14} +884.570 q^{16} -1526.74 q^{17} -1167.90 q^{19} -1809.69 q^{20} -579.128 q^{22} -3066.57 q^{23} +389.781 q^{25} +1257.83 q^{26} -5751.36 q^{28} -4311.80 q^{29} -551.599 q^{31} +3504.33 q^{32} -1854.25 q^{34} +11170.3 q^{35} +805.111 q^{37} -1418.44 q^{38} -4502.01 q^{40} -1097.69 q^{41} +3279.57 q^{43} +14555.4 q^{44} -3724.41 q^{46} -9281.20 q^{47} +18693.3 q^{49} +473.396 q^{50} -31613.5 q^{52} +30203.0 q^{53} -28269.6 q^{55} -14307.8 q^{56} -5236.77 q^{58} -19398.6 q^{59} +7150.63 q^{61} -669.927 q^{62} -24050.2 q^{64} +61399.7 q^{65} -26667.7 q^{67} +46603.5 q^{68} +13566.5 q^{70} -63678.3 q^{71} +6885.70 q^{73} +977.822 q^{74} +35650.1 q^{76} -89843.4 q^{77} -30007.2 q^{79} +52442.3 q^{80} -1333.17 q^{82} +32814.1 q^{83} -90513.4 q^{85} +3983.10 q^{86} +36209.9 q^{88} +6730.71 q^{89} +195134. q^{91} +93606.9 q^{92} -11272.2 q^{94} -69239.7 q^{95} +43659.2 q^{97} +22703.4 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 12 q^{2} + 624 q^{4} - 150 q^{5} - 573 q^{8}+O(q^{10}) \) 42 * q - 12 * q^2 + 624 * q^4 - 150 * q^5 - 573 * q^8	\( 42 q - 12 q^{2} + 624 q^{4} - 150 q^{5} - 573 q^{8} + 3 q^{10} - 1452 q^{11} - 2256 q^{14} + 8448 q^{16} - 3465 q^{17} + 3 q^{19} - 4128 q^{20} + 96 q^{22} - 5019 q^{23} + 18750 q^{25} - 3903 q^{26} - 6 q^{28} - 13008 q^{29} - 24273 q^{32} - 35868 q^{35} + 3 q^{37} - 51801 q^{38} + 96 q^{40} - 55833 q^{41} - 110757 q^{44} + 3 q^{46} - 90129 q^{47} + 57624 q^{49} - 145362 q^{50} + 3072 q^{52} - 103203 q^{53} - 6 q^{55} - 227154 q^{56} - 192 q^{58} - 176856 q^{59} - 31851 q^{61} - 246066 q^{62} + 86019 q^{64} - 167160 q^{65} - 801 q^{67} - 374589 q^{68} + 9375 q^{70} - 279531 q^{71} + 27012 q^{73} - 413970 q^{74} + 96 q^{76} - 185190 q^{77} - 462057 q^{80} - 6 q^{82} - 295536 q^{83} - 319803 q^{86} + 3072 q^{88} - 154827 q^{89} + 91002 q^{91} - 330558 q^{92} + 96 q^{94} - 353244 q^{95} - 463410 q^{98}+O(q^{100}) \) 42 * q - 12 * q^2 + 624 * q^4 - 150 * q^5 - 573 * q^8 + 3 * q^10 - 1452 * q^11 - 2256 * q^14 + 8448 * q^16 - 3465 * q^17 + 3 * q^19 - 4128 * q^20 + 96 * q^22 - 5019 * q^23 + 18750 * q^25 - 3903 * q^26 - 6 * q^28 - 13008 * q^29 - 24273 * q^32 - 35868 * q^35 + 3 * q^37 - 51801 * q^38 + 96 * q^40 - 55833 * q^41 - 110757 * q^44 + 3 * q^46 - 90129 * q^47 + 57624 * q^49 - 145362 * q^50 + 3072 * q^52 - 103203 * q^53 - 6 * q^55 - 227154 * q^56 - 192 * q^58 - 176856 * q^59 - 31851 * q^61 - 246066 * q^62 + 86019 * q^64 - 167160 * q^65 - 801 * q^67 - 374589 * q^68 + 9375 * q^70 - 279531 * q^71 + 27012 * q^73 - 413970 * q^74 + 96 * q^76 - 185190 * q^77 - 462057 * q^80 - 6 * q^82 - 295536 * q^83 - 319803 * q^86 + 3072 * q^88 - 154827 * q^89 + 91002 * q^91 - 330558 * q^92 + 96 * q^94 - 353244 * q^95 - 463410 * q^98

Introduction

L-functions

Modular forms

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Database

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