LMFDB - Embedded newform 585.6.a.p.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,6,Mod(1,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, 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("585.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	585.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:	$93.8245345906$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4 x^{8} - 205 x^{7} + 608 x^{6} + 13727 x^{5} - 27536 x^{4} - 346839 x^{3} + 433844 x^{2} + \cdots - 3899136 $ x^9 - 4x^8 - 205x^7 + 608x^6 + 13727x^5 - 27536x^4 - 346839x^3 + 433844x^2 + 2477700x - 3899136
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-9.62674$ of defining polynomial
Character	$\chi$	$=$	585.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+10.6267 q^{2} +80.9276 q^{4} -25.0000 q^{5} +85.3847 q^{7} +519.941 q^{8} -265.669 q^{10} +269.013 q^{11} +169.000 q^{13} +907.362 q^{14} +2935.60 q^{16} -815.926 q^{17} +560.423 q^{19} -2023.19 q^{20} +2858.73 q^{22} +996.008 q^{23} +625.000 q^{25} +1795.92 q^{26} +6909.99 q^{28} +3950.82 q^{29} +4690.82 q^{31} +14557.7 q^{32} -8670.63 q^{34} -2134.62 q^{35} -7644.37 q^{37} +5955.47 q^{38} -12998.5 q^{40} +7871.47 q^{41} -1044.34 q^{43} +21770.6 q^{44} +10584.3 q^{46} +19035.2 q^{47} -9516.44 q^{49} +6641.71 q^{50} +13676.8 q^{52} +12068.9 q^{53} -6725.32 q^{55} +44395.1 q^{56} +41984.4 q^{58} -16364.9 q^{59} -24988.3 q^{61} +49848.1 q^{62} +60762.0 q^{64} -4225.00 q^{65} +21731.8 q^{67} -66031.0 q^{68} -22684.0 q^{70} +24.9983 q^{71} +60591.4 q^{73} -81234.8 q^{74} +45353.7 q^{76} +22969.6 q^{77} -62329.0 q^{79} -73390.0 q^{80} +83648.1 q^{82} -39744.2 q^{83} +20398.1 q^{85} -11098.0 q^{86} +139871. q^{88} +115144. q^{89} +14430.0 q^{91} +80604.6 q^{92} +202282. q^{94} -14010.6 q^{95} +85281.2 q^{97} -101129. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 139 q^{4} - 225 q^{5} - 153 q^{7} + 255 q^{8} - 125 q^{10} + 503 q^{11} + 1521 q^{13} + 1110 q^{14} + 763 q^{16} + 1897 q^{17} - 2412 q^{19} - 3475 q^{20} - 3776 q^{22} + 3901 q^{23} + 5625 q^{25}+ \cdots + 119225 q^{98}+O(q^{100}) $ 9 * q + 5 * q^2 + 139 * q^4 - 225 * q^5 - 153 * q^7 + 255 * q^8 - 125 * q^10 + 503 * q^11 + 1521 * q^13 + 1110 * q^14 + 763 * q^16 + 1897 * q^17 - 2412 * q^19 - 3475 * q^20 - 3776 * q^22 + 3901 * q^23 + 5625 * q^25 + 845 * q^26 - 8518 * q^28 + 7906 * q^29 - 4622 * q^31 + 4535 * q^32 - 20300 * q^34 + 3825 * q^35 - 6375 * q^37 + 4650 * q^38 - 6375 * q^40 + 13513 * q^41 - 29862 * q^43 + 34004 * q^44 + 4024 * q^46 + 46534 * q^47 - 15668 * q^49 + 3125 * q^50 + 23491 * q^52 + 49769 * q^53 - 12575 * q^55 + 94250 * q^56 + 11176 * q^58 + 54652 * q^59 + 33319 * q^61 + 110430 * q^62 + 49539 * q^64 - 38025 * q^65 + 76852 * q^67 + 127996 * q^68 - 27750 * q^70 + 108871 * q^71 - 52522 * q^73 + 71210 * q^74 + 89210 * q^76 - 72837 * q^77 - 35955 * q^79 - 19075 * q^80 + 222310 * q^82 + 9400 * q^83 - 47425 * q^85 - 28320 * q^86 + 114224 * q^88 + 73995 * q^89 - 25857 * q^91 + 132288 * q^92 + 303672 * q^94 + 60300 * q^95 + 53623 * q^97 + 119225 * q^98

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 $

$2$	10.6267	1.87856	0.939280	−	0.343151i	$-0.111494\pi$
$2$	10.6267	1.87856	0.939280	+	0.343151i	$0.111494\pi$
$3$	0	0
$3$	0	0
$4$	80.9276	2.52899
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	85.3847	0.658620	0.329310	−	0.944222i	$-0.393184\pi$
$7$	85.3847	0.658620	0.329310	+	0.944222i	$0.393184\pi$
$8$	519.941	2.87230
$9$	0	0
$10$	−265.669	−0.840118
$11$	269.013	0.670334	0.335167	−	0.942159i	$-0.391207\pi$
$11$	269.013	0.670334	0.335167	+	0.942159i	$0.391207\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	907.362	1.23726
$15$	0	0
$16$	2935.60	2.86680
$17$	−815.926	−0.684744	−0.342372	−	0.939564i	$-0.611230\pi$
$17$	−815.926	−0.684744	−0.342372	+	0.939564i	$0.611230\pi$
$18$	0	0
$19$	560.423	0.356149	0.178074	−	0.984017i	$-0.443013\pi$
$19$	560.423	0.356149	0.178074	+	0.984017i	$0.443013\pi$
$20$	−2023.19	−1.13100
$21$	0	0
$22$	2858.73	1.25926
$23$	996.008	0.392594	0.196297	−	0.980545i	$-0.437108\pi$
$23$	996.008	0.392594	0.196297	+	0.980545i	$0.437108\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	1795.92	0.521019
$27$	0	0
$28$	6909.99	1.66564
$29$	3950.82	0.872353	0.436177	−	0.899861i	$-0.356332\pi$
$29$	3950.82	0.872353	0.436177	+	0.899861i	$0.356332\pi$
$30$	0	0
$31$	4690.82	0.876687	0.438344	−	0.898807i	$-0.355565\pi$
$31$	4690.82	0.876687	0.438344	+	0.898807i	$0.355565\pi$
$32$	14557.7	2.51315
$33$	0	0
$34$	−8670.63	−1.28633
$35$	−2134.62	−0.294544
$36$	0	0
$37$	−7644.37	−0.917989	−0.458995	−	0.888439i	$-0.651790\pi$
$37$	−7644.37	−0.917989	−0.458995	+	0.888439i	$0.651790\pi$
$38$	5955.47	0.669047
$39$	0	0
$40$	−12998.5	−1.28453
$41$	7871.47	0.731301	0.365651	−	0.930752i	$-0.380847\pi$
$41$	7871.47	0.731301	0.365651	+	0.930752i	$0.380847\pi$
$42$	0	0
$43$	−1044.34	−0.0861336	−0.0430668	−	0.999072i	$-0.513713\pi$
$43$	−1044.34	−0.0861336	−0.0430668	+	0.999072i	$0.513713\pi$
$44$	21770.6	1.69527
$45$	0	0
$46$	10584.3	0.737511
$47$	19035.2	1.25694	0.628468	−	0.777835i	$-0.283682\pi$
$47$	19035.2	1.25694	0.628468	+	0.777835i	$0.283682\pi$
$48$	0	0
$49$	−9516.44	−0.566219
$50$	6641.71	0.375712
$51$	0	0
$52$	13676.8	0.701415
$53$	12068.9	0.590173	0.295087	−	0.955471i	$-0.404652\pi$
$53$	12068.9	0.590173	0.295087	+	0.955471i	$0.404652\pi$
$54$	0	0
$55$	−6725.32	−0.299782
$56$	44395.1	1.89175
$57$	0	0
$58$	41984.4	1.63877
$59$	−16364.9	−0.612045	−0.306023	−	0.952024i	$-0.598998\pi$
$59$	−16364.9	−0.612045	−0.306023	+	0.952024i	$0.598998\pi$
$60$	0	0
$61$	−24988.3	−0.859829	−0.429915	−	0.902870i	$-0.641456\pi$
$61$	−24988.3	−0.859829	−0.429915	+	0.902870i	$0.641456\pi$
$62$	49848.1	1.64691
$63$	0	0
$64$	60762.0	1.85431
$65$	−4225.00	−0.124035
$66$	0	0
$67$	21731.8	0.591437	0.295719	−	0.955275i	$-0.404441\pi$
$67$	21731.8	0.591437	0.295719	+	0.955275i	$0.404441\pi$
$68$	−66031.0	−1.73171
$69$	0	0
$70$	−22684.0	−0.553319
$71$	24.9983	0.000588526 0	0.000294263	−	1.00000i	$-0.499906\pi$
$71$	24.9983	0.000588526 0	0.000294263		1.00000i	$0.499906\pi$
$72$	0	0
$73$	60591.4	1.33077	0.665386	−	0.746499i	$-0.268267\pi$
$73$	60591.4	1.33077	0.665386	+	0.746499i	$0.268267\pi$
$74$	−81234.8	−1.72450
$75$	0	0
$76$	45353.7	0.900697
$77$	22969.6	0.441496
$78$	0	0
$79$	−62329.0	−1.12363	−0.561814	−	0.827264i	$-0.689896\pi$
$79$	−62329.0	−1.12363	−0.561814	+	0.827264i	$0.689896\pi$
$80$	−73390.0	−1.28207
$81$	0	0
$82$	83648.1	1.37379
$83$	−39744.2	−0.633255	−0.316627	−	0.948550i	$-0.602551\pi$
$83$	−39744.2	−0.633255	−0.316627	+	0.948550i	$0.602551\pi$
$84$	0	0
$85$	20398.1	0.306227
$86$	−11098.0	−0.161807
$87$	0	0
$88$	139871.	1.92540
$89$	115144.	1.54087	0.770434	−	0.637520i	$-0.220040\pi$
$89$	115144.	1.54087	0.770434	+	0.637520i	$0.220040\pi$
$90$	0	0
$91$	14430.0	0.182668
$92$	80604.6	0.992865
$93$	0	0
$94$	202282.	2.36123
$95$	−14010.6	−0.159275
$96$	0	0
$97$	85281.2	0.920288	0.460144	−	0.887844i	$-0.347798\pi$
$97$	85281.2	0.920288	0.460144	+	0.887844i	$0.347798\pi$
$98$	−101129.	−1.06368
$99$	0	0
$100$	50579.8	0.505798
$101$	126866.	1.23749	0.618745	−	0.785592i	$-0.287641\pi$
$101$	126866.	1.23749	0.618745	+	0.785592i	$0.287641\pi$
$102$	0	0
$103$	63703.2	0.591654	0.295827	−	0.955241i	$-0.404405\pi$
$103$	63703.2	0.591654	0.295827	+	0.955241i	$0.404405\pi$
$104$	87870.1	0.796632
$105$	0	0
$106$	128253.	1.10868
$107$	−699.277	−0.00590459	−0.00295230	−	0.999996i	$-0.500940\pi$
$107$	−699.277	−0.00590459	−0.00295230	+	0.999996i	$0.500940\pi$
$108$	0	0
$109$	−28609.3	−0.230643	−0.115322	−	0.993328i	$-0.536790\pi$
$109$	−28609.3	−0.230643	−0.115322	+	0.993328i	$0.536790\pi$
$110$	−71468.2	−0.563159
$111$	0	0
$112$	250655.	1.88813
$113$	−243998.	−1.79759	−0.898794	−	0.438372i	$-0.855555\pi$
$113$	−243998.	−1.79759	−0.898794	+	0.438372i	$0.855555\pi$
$114$	0	0
$115$	−24900.2	−0.175573
$116$	319731.	2.20617
$117$	0	0
$118$	−173906.	−1.14976
$119$	−69667.6	−0.450987
$120$	0	0
$121$	−88683.2	−0.550653
$122$	−265544.	−1.61524
$123$	0	0
$124$	379617.	2.21713
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−29691.2	−0.163350	−0.0816748	−	0.996659i	$-0.526027\pi$
$127$	−29691.2	−0.163350	−0.0816748	+	0.996659i	$0.526027\pi$
$128$	179855.	0.970282
$129$	0	0
$130$	−44898.0	−0.233007
$131$	−88586.3	−0.451012	−0.225506	−	0.974242i	$-0.572404\pi$
$131$	−88586.3	−0.451012	−0.225506	+	0.974242i	$0.572404\pi$
$132$	0	0
$133$	47851.5	0.234567
$134$	230938.	1.11105
$135$	0	0
$136$	−424234.	−1.96679
$137$	−236459.	−1.07635	−0.538177	−	0.842832i	$-0.680887\pi$
$137$	−236459.	−1.07635	−0.538177	+	0.842832i	$0.680887\pi$
$138$	0	0
$139$	280621.	1.23192	0.615962	−	0.787776i	$-0.288768\pi$
$139$	280621.	1.23192	0.615962	+	0.787776i	$0.288768\pi$
$140$	−172750.	−0.744899
$141$	0	0
$142$	265.651	0.00110558
$143$	45463.2	0.185917
$144$	0	0
$145$	−98770.5	−0.390128
$146$	643889.	2.49994
$147$	0	0
$148$	−618641.	−2.32158
$149$	−83880.5	−0.309525	−0.154762	−	0.987952i	$-0.549461\pi$
$149$	−83880.5	−0.309525	−0.154762	+	0.987952i	$0.549461\pi$
$150$	0	0
$151$	−529999.	−1.89162	−0.945808	−	0.324727i	$-0.894728\pi$
$151$	−529999.	−1.89162	−0.945808	+	0.324727i	$0.894728\pi$
$152$	291387.	1.02297
$153$	0	0
$154$	244092.	0.829376
$155$	−117271.	−0.392066
$156$	0	0
$157$	−82619.6	−0.267506	−0.133753	−	0.991015i	$-0.542703\pi$
$157$	−82619.6	−0.267506	−0.133753	+	0.991015i	$0.542703\pi$
$158$	−662354.	−2.11080
$159$	0	0
$160$	−363943.	−1.12392
$161$	85043.9	0.258570
$162$	0	0
$163$	−460688.	−1.35812	−0.679059	−	0.734083i	$-0.737612\pi$
$163$	−460688.	−1.35812	−0.679059	+	0.734083i	$0.737612\pi$
$164$	637020.	1.84945
$165$	0	0
$166$	−422351.	−1.18961
$167$	548177.	1.52100	0.760501	−	0.649337i	$-0.224954\pi$
$167$	548177.	1.52100	0.760501	+	0.649337i	$0.224954\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	216766.	0.575266
$171$	0	0
$172$	−84516.4	−0.217831
$173$	36316.3	0.0922543	0.0461271	−	0.998936i	$-0.485312\pi$
$173$	36316.3	0.0922543	0.0461271	+	0.998936i	$0.485312\pi$
$174$	0	0
$175$	53365.5	0.131724
$176$	789713.	1.92171
$177$	0	0
$178$	1.22360e6	2.89461
$179$	−19749.0	−0.0460694	−0.0230347	−	0.999735i	$-0.507333\pi$
$179$	−19749.0	−0.0460694	−0.0230347	+	0.999735i	$0.507333\pi$
$180$	0	0
$181$	−452389.	−1.02640	−0.513199	−	0.858270i	$-0.671540\pi$
$181$	−452389.	−1.02640	−0.513199	+	0.858270i	$0.671540\pi$
$182$	153344.	0.343154
$183$	0	0
$184$	517866.	1.12765
$185$	191109.	0.410537
$186$	0	0
$187$	−219494.	−0.459007
$188$	1.54048e6	3.17878
$189$	0	0
$190$	−148887.	−0.299207
$191$	−219095.	−0.434559	−0.217279	−	0.976109i	$-0.569718\pi$
$191$	−219095.	−0.434559	−0.217279	+	0.976109i	$0.569718\pi$
$192$	0	0
$193$	495944.	0.958384	0.479192	−	0.877710i	$-0.340930\pi$
$193$	495944.	0.958384	0.479192	+	0.877710i	$0.340930\pi$
$194$	906261.	1.72882
$195$	0	0
$196$	−770143.	−1.43196
$197$	−347570.	−0.638083	−0.319041	−	0.947741i	$-0.603361\pi$
$197$	−347570.	−0.638083	−0.319041	+	0.947741i	$0.603361\pi$
$198$	0	0
$199$	−597289.	−1.06918	−0.534591	−	0.845111i	$-0.679534\pi$
$199$	−597289.	−1.06918	−0.534591	+	0.845111i	$0.679534\pi$
$200$	324963.	0.574460
$201$	0	0
$202$	1.34817e6	2.32470
$203$	337340.	0.574550
$204$	0	0
$205$	−196787.	−0.327048
$206$	676957.	1.11146
$207$	0	0
$208$	496116.	0.795106
$209$	150761.	0.238739
$210$	0	0
$211$	−728985.	−1.12723	−0.563615	−	0.826038i	$-0.690590\pi$
$211$	−728985.	−1.12723	−0.563615	+	0.826038i	$0.690590\pi$
$212$	976711.	1.49254
$213$	0	0
$214$	−7431.04	−0.0110921
$215$	26108.6	0.0385201
$216$	0	0
$217$	400525.	0.577404
$218$	−304023.	−0.433277
$219$	0	0
$220$	−544264.	−0.758146
$221$	−137891.	−0.189914
$222$	0	0
$223$	−428566.	−0.577106	−0.288553	−	0.957464i	$-0.593174\pi$
$223$	−428566.	−0.577106	−0.288553	+	0.957464i	$0.593174\pi$
$224$	1.24301e6	1.65521
$225$	0	0
$226$	−2.59290e6	−3.37688
$227$	−1.18213e6	−1.52265	−0.761324	−	0.648371i	$-0.775450\pi$
$227$	−1.18213e6	−1.52265	−0.761324	+	0.648371i	$0.775450\pi$
$228$	0	0
$229$	−114333.	−0.144073	−0.0720364	−	0.997402i	$-0.522950\pi$
$229$	−114333.	−0.144073	−0.0720364	+	0.997402i	$0.522950\pi$
$230$	−264608.	−0.329825
$231$	0	0
$232$	2.05420e6	2.50566
$233$	1.21655e6	1.46805	0.734026	−	0.679122i	$-0.237639\pi$
$233$	1.21655e6	1.46805	0.734026	+	0.679122i	$0.237639\pi$
$234$	0	0
$235$	−475881.	−0.562119
$236$	−1.32437e6	−1.54786
$237$	0	0
$238$	−740340.	−0.847205
$239$	921510.	1.04353	0.521765	−	0.853089i	$-0.325274\pi$
$239$	921510.	1.04353	0.521765	+	0.853089i	$0.325274\pi$
$240$	0	0
$241$	494467.	0.548396	0.274198	−	0.961673i	$-0.411588\pi$
$241$	494467.	0.548396	0.274198	+	0.961673i	$0.411588\pi$
$242$	−942413.	−1.03443
$243$	0	0
$244$	−2.02224e6	−2.17450
$245$	237911.	0.253221
$246$	0	0
$247$	94711.4	0.0987779
$248$	2.43895e6	2.51811
$249$	0	0
$250$	−166043.	−0.168024
$251$	−738969.	−0.740359	−0.370179	−	0.928960i	$-0.620704\pi$
$251$	−738969.	−0.740359	−0.370179	+	0.928960i	$0.620704\pi$
$252$	0	0
$253$	267939.	0.263169
$254$	−315520.	−0.306862
$255$	0	0
$256$	−33110.4	−0.0315765
$257$	−886405.	−0.837143	−0.418571	−	0.908184i	$-0.637469\pi$
$257$	−886405.	−0.837143	−0.418571	+	0.908184i	$0.637469\pi$
$258$	0	0
$259$	−652713.	−0.604606
$260$	−341919.	−0.313682
$261$	0	0
$262$	−941384.	−0.847253
$263$	−511416.	−0.455917	−0.227958	−	0.973671i	$-0.573205\pi$
$263$	−511416.	−0.455917	−0.227958	+	0.973671i	$0.573205\pi$
$264$	0	0
$265$	−301723.	−0.263933
$266$	508506.	0.440648
$267$	0	0
$268$	1.75870e6	1.49574
$269$	913248.	0.769499	0.384749	−	0.923021i	$-0.374288\pi$
$269$	913248.	0.769499	0.384749	+	0.923021i	$0.374288\pi$
$270$	0	0
$271$	−658235.	−0.544450	−0.272225	−	0.962234i	$-0.587759\pi$
$271$	−658235.	−0.544450	−0.272225	+	0.962234i	$0.587759\pi$
$272$	−2.39523e6	−1.96302
$273$	0	0
$274$	−2.51279e6	−2.02200
$275$	168133.	0.134067
$276$	0	0
$277$	−2.19515e6	−1.71895	−0.859477	−	0.511175i	$-0.829210\pi$
$277$	−2.19515e6	−1.71895	−0.859477	+	0.511175i	$0.829210\pi$
$278$	2.98209e6	2.31424
$279$	0	0
$280$	−1.10988e6	−0.846018
$281$	−2.29020e6	−1.73024	−0.865122	−	0.501562i	$-0.832759\pi$
$281$	−2.29020e6	−1.73024	−0.865122	+	0.501562i	$0.832759\pi$
$282$	0	0
$283$	−2.15723e6	−1.60114	−0.800572	−	0.599237i	$-0.795471\pi$
$283$	−2.15723e6	−1.60114	−0.800572	+	0.599237i	$0.795471\pi$
$284$	2023.06	0.00148837
$285$	0	0
$286$	483125.	0.349257
$287$	672104.	0.481650
$288$	0	0
$289$	−754122.	−0.531125
$290$	−1.04961e6	−0.732879
$291$	0	0
$292$	4.90352e6	3.36551
$293$	−1.01479e6	−0.690570	−0.345285	−	0.938498i	$-0.612218\pi$
$293$	−1.01479e6	−0.690570	−0.345285	+	0.938498i	$0.612218\pi$
$294$	0	0
$295$	409123.	0.273715
$296$	−3.97463e6	−2.63674
$297$	0	0
$298$	−891377.	−0.581461
$299$	168325.	0.108886
$300$	0	0
$301$	−89171.1	−0.0567294
$302$	−5.63217e6	−3.55351
$303$	0	0
$304$	1.64518e6	1.02101
$305$	624707.	0.384527
$306$	0	0
$307$	2.01524e6	1.22034	0.610170	−	0.792271i	$-0.291101\pi$
$307$	2.01524e6	1.22034	0.610170	+	0.792271i	$0.291101\pi$
$308$	1.85887e6	1.11654
$309$	0	0
$310$	−1.24620e6	−0.736520
$311$	−1.60884e6	−0.943220	−0.471610	−	0.881807i	$-0.656327\pi$
$311$	−1.60884e6	−0.943220	−0.471610	+	0.881807i	$0.656327\pi$
$312$	0	0
$313$	−101542.	−0.0585848	−0.0292924	−	0.999571i	$-0.509325\pi$
$313$	−101542.	−0.0585848	−0.0292924	+	0.999571i	$0.509325\pi$
$314$	−877977.	−0.502526
$315$	0	0
$316$	−5.04414e6	−2.84164
$317$	−387514.	−0.216591	−0.108295	−	0.994119i	$-0.534539\pi$
$317$	−387514.	−0.216591	−0.108295	+	0.994119i	$0.534539\pi$
$318$	0	0
$319$	1.06282e6	0.584768
$320$	−1.51905e6	−0.829273
$321$	0	0
$322$	903740.	0.485740
$323$	−457263.	−0.243871
$324$	0	0
$325$	105625.	0.0554700
$326$	−4.89561e6	−2.55131
$327$	0	0
$328$	4.09270e6	2.10051
$329$	1.62532e6	0.827844
$330$	0	0
$331$	1.06859e6	0.536092	0.268046	−	0.963406i	$-0.413622\pi$
$331$	1.06859e6	0.536092	0.268046	+	0.963406i	$0.413622\pi$
$332$	−3.21640e6	−1.60149
$333$	0	0
$334$	5.82534e6	2.85729
$335$	−543295.	−0.264499
$336$	0	0
$337$	2.48704e6	1.19291	0.596456	−	0.802646i	$-0.296575\pi$
$337$	2.48704e6	1.19291	0.596456	+	0.802646i	$0.296575\pi$
$338$	303510.	0.144505
$339$	0	0
$340$	1.65077e6	0.774445
$341$	1.26189e6	0.587673
$342$	0	0
$343$	−2.24762e6	−1.03154
$344$	−542998.	−0.247401
$345$	0	0
$346$	385924.	0.173305
$347$	3.73571e6	1.66552	0.832758	−	0.553636i	$-0.186760\pi$
$347$	3.73571e6	1.66552	0.832758	+	0.553636i	$0.186760\pi$
$348$	0	0
$349$	2.84684e6	1.25112	0.625560	−	0.780176i	$-0.284870\pi$
$349$	2.84684e6	1.25112	0.625560	+	0.780176i	$0.284870\pi$
$350$	567101.	0.247452
$351$	0	0
$352$	3.91621e6	1.68465
$353$	3.85986e6	1.64867	0.824337	−	0.566100i	$-0.191548\pi$
$353$	3.85986e6	1.64867	0.824337	+	0.566100i	$0.191548\pi$
$354$	0	0
$355$	−624.959	−0.000263197 0
$356$	9.31831e6	3.89684
$357$	0	0
$358$	−209867.	−0.0865441
$359$	1.32848e6	0.544026	0.272013	−	0.962294i	$-0.412311\pi$
$359$	1.32848e6	0.544026	0.272013	+	0.962294i	$0.412311\pi$
$360$	0	0
$361$	−2.16203e6	−0.873158
$362$	−4.80742e6	−1.92815
$363$	0	0
$364$	1.16779e6	0.461966
$365$	−1.51478e6	−0.595139
$366$	0	0
$367$	−3.92916e6	−1.52277	−0.761385	−	0.648300i	$-0.775480\pi$
$367$	−3.92916e6	−1.52277	−0.761385	+	0.648300i	$0.775480\pi$
$368$	2.92388e6	1.12549
$369$	0	0
$370$	2.03087e6	0.771219
$371$	1.03050e6	0.388700
$372$	0	0
$373$	−2.58980e6	−0.963816	−0.481908	−	0.876222i	$-0.660056\pi$
$373$	−2.58980e6	−0.963816	−0.481908	+	0.876222i	$0.660056\pi$
$374$	−2.33251e6	−0.862273
$375$	0	0
$376$	9.89720e6	3.61030
$377$	667689.	0.241947
$378$	0	0
$379$	2.27245e6	0.812636	0.406318	−	0.913732i	$-0.366813\pi$
$379$	2.27245e6	0.812636	0.406318	+	0.913732i	$0.366813\pi$
$380$	−1.13384e6	−0.402804
$381$	0	0
$382$	−2.32826e6	−0.816345
$383$	−709295.	−0.247076	−0.123538	−	0.992340i	$-0.539424\pi$
$383$	−709295.	−0.247076	−0.123538	+	0.992340i	$0.539424\pi$
$384$	0	0
$385$	−574240.	−0.197443
$386$	5.27027e6	1.80038
$387$	0	0
$388$	6.90160e6	2.32740
$389$	3.98220e6	1.33429	0.667143	−	0.744930i	$-0.267517\pi$
$389$	3.98220e6	1.33429	0.667143	+	0.744930i	$0.267517\pi$
$390$	0	0
$391$	−812669.	−0.268826
$392$	−4.94799e6	−1.62635
$393$	0	0
$394$	−3.69354e6	−1.19868
$395$	1.55822e6	0.502501
$396$	0	0
$397$	−2.62063e6	−0.834507	−0.417253	−	0.908790i	$-0.637007\pi$
$397$	−2.62063e6	−0.834507	−0.417253	+	0.908790i	$0.637007\pi$
$398$	−6.34723e6	−2.00852
$399$	0	0
$400$	1.83475e6	0.573359
$401$	3.16139e6	0.981786	0.490893	−	0.871220i	$-0.336671\pi$
$401$	3.16139e6	0.981786	0.490893	+	0.871220i	$0.336671\pi$
$402$	0	0
$403$	792749.	0.243149
$404$	1.02670e7	3.12960
$405$	0	0
$406$	3.58482e6	1.07933
$407$	−2.05643e6	−0.615359
$408$	0	0
$409$	−2.12708e6	−0.628746	−0.314373	−	0.949299i	$-0.601794\pi$
$409$	−2.12708e6	−0.628746	−0.314373	+	0.949299i	$0.601794\pi$
$410$	−2.09120e6	−0.614379
$411$	0	0
$412$	5.15535e6	1.49629
$413$	−1.39731e6	−0.403106
$414$	0	0
$415$	993605.	0.283200
$416$	2.46026e6	0.697023
$417$	0	0
$418$	1.60210e6	0.448485
$419$	−6.07786e6	−1.69128	−0.845640	−	0.533754i	$-0.820781\pi$
$419$	−6.07786e6	−1.69128	−0.845640	+	0.533754i	$0.820781\pi$
$420$	0	0
$421$	−1.34964e6	−0.371119	−0.185560	−	0.982633i	$-0.559410\pi$
$421$	−1.34964e6	−0.371119	−0.185560	+	0.982633i	$0.559410\pi$
$422$	−7.74673e6	−2.11757
$423$	0	0
$424$	6.27514e6	1.69515
$425$	−509954.	−0.136949
$426$	0	0
$427$	−2.13362e6	−0.566301
$428$	−56590.9	−0.0149327
$429$	0	0
$430$	277450.	0.0723624
$431$	2.13931e6	0.554728	0.277364	−	0.960765i	$-0.410539\pi$
$431$	2.13931e6	0.554728	0.277364	+	0.960765i	$0.410539\pi$
$432$	0	0
$433$	2.31411e6	0.593150	0.296575	−	0.955010i	$-0.404156\pi$
$433$	2.31411e6	0.593150	0.296575	+	0.955010i	$0.404156\pi$
$434$	4.25627e6	1.08469
$435$	0	0
$436$	−2.31528e6	−0.583294
$437$	558186.	0.139822
$438$	0	0
$439$	−7.33549e6	−1.81663	−0.908317	−	0.418282i	$-0.862632\pi$
$439$	−7.33549e6	−1.81663	−0.908317	+	0.418282i	$0.862632\pi$
$440$	−3.49677e6	−0.861064
$441$	0	0
$442$	−1.46534e6	−0.356765
$443$	1.69171e6	0.409559	0.204779	−	0.978808i	$-0.434352\pi$
$443$	1.69171e6	0.409559	0.204779	+	0.978808i	$0.434352\pi$
$444$	0	0
$445$	−2.87859e6	−0.689097
$446$	−4.55426e6	−1.08413
$447$	0	0
$448$	5.18815e6	1.22129
$449$	4.04297e6	0.946423	0.473211	−	0.880949i	$-0.343095\pi$
$449$	4.04297e6	0.946423	0.473211	+	0.880949i	$0.343095\pi$
$450$	0	0
$451$	2.11753e6	0.490216
$452$	−1.97462e7	−4.54608
$453$	0	0
$454$	−1.25622e7	−2.86039
$455$	−360751.	−0.0816918
$456$	0	0
$457$	−5.37081e6	−1.20296	−0.601478	−	0.798890i	$-0.705421\pi$
$457$	−5.37081e6	−1.20296	−0.601478	+	0.798890i	$0.705421\pi$
$458$	−1.21498e6	−0.270649
$459$	0	0
$460$	−2.01512e6	−0.444023
$461$	2.85273e6	0.625184	0.312592	−	0.949888i	$-0.398803\pi$
$461$	2.85273e6	0.625184	0.312592	+	0.949888i	$0.398803\pi$
$462$	0	0
$463$	−6.00554e6	−1.30197	−0.650983	−	0.759092i	$-0.725643\pi$
$463$	−6.00554e6	−1.30197	−0.650983	+	0.759092i	$0.725643\pi$
$464$	1.15980e7	2.50086
$465$	0	0
$466$	1.29280e7	2.75782
$467$	7.18700e6	1.52495	0.762475	−	0.647018i	$-0.223984\pi$
$467$	7.18700e6	1.52495	0.762475	+	0.647018i	$0.223984\pi$
$468$	0	0
$469$	1.85556e6	0.389533
$470$	−5.05706e6	−1.05597
$471$	0	0
$472$	−8.50880e6	−1.75798
$473$	−280942.	−0.0577383
$474$	0	0
$475$	350264.	0.0712298
$476$	−5.63804e6	−1.14054
$477$	0	0
$478$	9.79265e6	1.96034
$479$	4.06312e6	0.809135	0.404568	−	0.914508i	$-0.367422\pi$
$479$	4.06312e6	0.809135	0.404568	+	0.914508i	$0.367422\pi$
$480$	0	0
$481$	−1.29190e6	−0.254604
$482$	5.25457e6	1.03020
$483$	0	0
$484$	−7.17692e6	−1.39259
$485$	−2.13203e6	−0.411565
$486$	0	0
$487$	6.45147e6	1.23264	0.616320	−	0.787496i	$-0.288623\pi$
$487$	6.45147e6	1.23264	0.616320	+	0.787496i	$0.288623\pi$
$488$	−1.29925e7	−2.46969
$489$	0	0
$490$	2.52822e6	0.475691
$491$	−5.03693e6	−0.942893	−0.471447	−	0.881895i	$-0.656268\pi$
$491$	−5.03693e6	−0.942893	−0.471447	+	0.881895i	$0.656268\pi$
$492$	0	0
$493$	−3.22358e6	−0.597339
$494$	1.00647e6	0.185560
$495$	0	0
$496$	1.37704e7	2.51328
$497$	2134.48	0.000387615 0
$498$	0	0
$499$	−4.99941e6	−0.898808	−0.449404	−	0.893329i	$-0.648364\pi$
$499$	−4.99941e6	−0.898808	−0.449404	+	0.893329i	$0.648364\pi$
$500$	−1.26449e6	−0.226200
$501$	0	0
$502$	−7.85284e6	−1.39081
$503$	89890.2	0.0158414	0.00792068	−	0.999969i	$-0.497479\pi$
$503$	89890.2	0.0158414	0.00792068	+	0.999969i	$0.497479\pi$
$504$	0	0
$505$	−3.17165e6	−0.553423
$506$	2.84732e6	0.494379
$507$	0	0
$508$	−2.40284e6	−0.413109
$509$	−2.32833e6	−0.398337	−0.199168	−	0.979965i	$-0.563824\pi$
$509$	−2.32833e6	−0.398337	−0.199168	+	0.979965i	$0.563824\pi$
$510$	0	0
$511$	5.17358e6	0.876474
$512$	−6.10722e6	−1.02960
$513$	0	0
$514$	−9.41960e6	−1.57262
$515$	−1.59258e6	−0.264596
$516$	0	0
$517$	5.12072e6	0.842567
$518$	−6.93621e6	−1.13579
$519$	0	0
$520$	−2.19675e6	−0.356265
$521$	7.76731e6	1.25365	0.626825	−	0.779160i	$-0.284354\pi$
$521$	7.76731e6	1.25365	0.626825	+	0.779160i	$0.284354\pi$
$522$	0	0
$523$	1.66847e6	0.266725	0.133363	−	0.991067i	$-0.457423\pi$
$523$	1.66847e6	0.266725	0.133363	+	0.991067i	$0.457423\pi$
$524$	−7.16908e6	−1.14060
$525$	0	0
$526$	−5.43469e6	−0.856467
$527$	−3.82736e6	−0.600307
$528$	0	0
$529$	−5.44431e6	−0.845870
$530$	−3.20634e6	−0.495815
$531$	0	0
$532$	3.87251e6	0.593217
$533$	1.33028e6	0.202826
$534$	0	0
$535$	17481.9	0.00264061
$536$	1.12993e7	1.69878
$537$	0	0
$538$	9.70485e6	1.44555
$539$	−2.56004e6	−0.379556
$540$	0	0
$541$	8.65439e6	1.27129	0.635643	−	0.771983i	$-0.280735\pi$
$541$	8.65439e6	1.27129	0.635643	+	0.771983i	$0.280735\pi$
$542$	−6.99489e6	−1.02278
$543$	0	0
$544$	−1.18780e7	−1.72087
$545$	715231.	0.103147
$546$	0	0
$547$	−8.07733e6	−1.15425	−0.577125	−	0.816656i	$-0.695825\pi$
$547$	−8.07733e6	−1.15425	−0.577125	+	0.816656i	$0.695825\pi$
$548$	−1.91361e7	−2.72209
$549$	0	0
$550$	1.78671e6	0.251852
$551$	2.21413e6	0.310688
$552$	0	0
$553$	−5.32194e6	−0.740044
$554$	−2.33273e7	−3.22916
$555$	0	0
$556$	2.27100e7	3.11552
$557$	−2.16561e6	−0.295762	−0.147881	−	0.989005i	$-0.547245\pi$
$557$	−2.16561e6	−0.295762	−0.147881	+	0.989005i	$0.547245\pi$
$558$	0	0
$559$	−176494.	−0.0238892
$560$	−6.26638e6	−0.844398
$561$	0	0
$562$	−2.43373e7	−3.25037
$563$	1.20703e7	1.60489	0.802446	−	0.596724i	$-0.203531\pi$
$563$	1.20703e7	1.60489	0.802446	+	0.596724i	$0.203531\pi$
$564$	0	0
$565$	6.09995e6	0.803906
$566$	−2.29243e7	−3.00785
$567$	0	0
$568$	12997.7	0.00169042
$569$	−5.94776e6	−0.770146	−0.385073	−	0.922886i	$-0.625824\pi$
$569$	−5.94776e6	−0.770146	−0.385073	+	0.922886i	$0.625824\pi$
$570$	0	0
$571$	−1.59497e6	−0.204721	−0.102361	−	0.994747i	$-0.532640\pi$
$571$	−1.59497e6	−0.204721	−0.102361	+	0.994747i	$0.532640\pi$
$572$	3.67923e6	0.470182
$573$	0	0
$574$	7.14227e6	0.904808
$575$	622505.	0.0785187
$576$	0	0
$577$	9.98490e6	1.24855	0.624273	−	0.781206i	$-0.285395\pi$
$577$	9.98490e6	1.24855	0.624273	+	0.781206i	$0.285395\pi$
$578$	−8.01386e6	−0.997751
$579$	0	0
$580$	−7.99327e6	−0.986630
$581$	−3.39355e6	−0.417075
$582$	0	0
$583$	3.24670e6	0.395613
$584$	3.15040e7	3.82237
$585$	0	0
$586$	−1.07839e7	−1.29728
$587$	−9.33702e6	−1.11844	−0.559221	−	0.829019i	$-0.688899\pi$
$587$	−9.33702e6	−1.11844	−0.559221	+	0.829019i	$0.688899\pi$
$588$	0	0
$589$	2.62884e6	0.312231
$590$	4.34764e6	0.514190
$591$	0	0
$592$	−2.24408e7	−2.63169
$593$	−4.30100e6	−0.502265	−0.251133	−	0.967953i	$-0.580803\pi$
$593$	−4.30100e6	−0.502265	−0.251133	+	0.967953i	$0.580803\pi$
$594$	0	0
$595$	1.74169e6	0.201687
$596$	−6.78825e6	−0.782785
$597$	0	0
$598$	1.78875e6	0.204549
$599$	1.25411e7	1.42814	0.714069	−	0.700076i	$-0.246850\pi$
$599$	1.25411e7	1.42814	0.714069	+	0.700076i	$0.246850\pi$
$600$	0	0
$601$	8.28854e6	0.936034	0.468017	−	0.883719i	$-0.344969\pi$
$601$	8.28854e6	0.936034	0.468017	+	0.883719i	$0.344969\pi$
$602$	−947599.	−0.106570
$603$	0	0
$604$	−4.28916e7	−4.78388
$605$	2.21708e6	0.246259
$606$	0	0
$607$	−1.35304e7	−1.49053	−0.745263	−	0.666771i	$-0.767676\pi$
$607$	−1.35304e7	−1.49053	−0.745263	+	0.666771i	$0.767676\pi$
$608$	8.15848e6	0.895056
$609$	0	0
$610$	6.63861e6	0.722358
$611$	3.21695e6	0.348611
$612$	0	0
$613$	−5.37532e6	−0.577767	−0.288884	−	0.957364i	$-0.593284\pi$
$613$	−5.37532e6	−0.577767	−0.288884	+	0.957364i	$0.593284\pi$
$614$	2.14154e7	2.29248
$615$	0	0
$616$	1.19428e7	1.26811
$617$	1.68362e6	0.178045	0.0890227	−	0.996030i	$-0.471626\pi$
$617$	1.68362e6	0.178045	0.0890227	+	0.996030i	$0.471626\pi$
$618$	0	0
$619$	9.62531e6	1.00969	0.504845	−	0.863210i	$-0.331550\pi$
$619$	9.62531e6	1.00969	0.504845	+	0.863210i	$0.331550\pi$
$620$	−9.49043e6	−0.991532
$621$	0	0
$622$	−1.70968e7	−1.77190
$623$	9.83152e6	1.01485
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.07906e6	−0.110055
$627$	0	0
$628$	−6.68621e6	−0.676520
$629$	6.23724e6	0.628588
$630$	0	0
$631$	−1.69384e7	−1.69355	−0.846777	−	0.531947i	$-0.821460\pi$
$631$	−1.69384e7	−1.69355	−0.846777	+	0.531947i	$0.821460\pi$
$632$	−3.24074e7	−3.22739
$633$	0	0
$634$	−4.11802e6	−0.406879
$635$	742279.	0.0730521
$636$	0	0
$637$	−1.60828e6	−0.157041
$638$	1.12943e7	1.09852
$639$	0	0
$640$	−4.49638e6	−0.433923
$641$	3.73423e6	0.358968	0.179484	−	0.983761i	$-0.442557\pi$
$641$	3.73423e6	0.358968	0.179484	+	0.983761i	$0.442557\pi$
$642$	0	0
$643$	−1.30798e7	−1.24760	−0.623798	−	0.781586i	$-0.714411\pi$
$643$	−1.30798e7	−1.24760	−0.623798	+	0.781586i	$0.714411\pi$
$644$	6.88240e6	0.653921
$645$	0	0
$646$	−4.85922e6	−0.458126
$647$	1.31074e6	0.123099	0.0615497	−	0.998104i	$-0.480396\pi$
$647$	1.31074e6	0.123099	0.0615497	+	0.998104i	$0.480396\pi$
$648$	0	0
$649$	−4.40237e6	−0.410275
$650$	1.12245e6	0.104204
$651$	0	0
$652$	−3.72824e7	−3.43467
$653$	1.53204e7	1.40600	0.703002	−	0.711188i	$-0.251843\pi$
$653$	1.53204e7	1.40600	0.703002	+	0.711188i	$0.251843\pi$
$654$	0	0
$655$	2.21466e6	0.201699
$656$	2.31075e7	2.09649
$657$	0	0
$658$	1.72718e7	1.55515
$659$	−4.82048e6	−0.432391	−0.216195	−	0.976350i	$-0.569365\pi$
$659$	−4.82048e6	−0.432391	−0.216195	+	0.976350i	$0.569365\pi$
$660$	0	0
$661$	−8.03371e6	−0.715175	−0.357588	−	0.933880i	$-0.616401\pi$
$661$	−8.03371e6	−0.715175	−0.357588	+	0.933880i	$0.616401\pi$
$662$	1.13556e7	1.00708
$663$	0	0
$664$	−2.06647e7	−1.81890
$665$	−1.19629e6	−0.104902
$666$	0	0
$667$	3.93505e6	0.342480
$668$	4.43627e7	3.84660
$669$	0	0
$670$	−5.77346e6	−0.496877
$671$	−6.72217e6	−0.576372
$672$	0	0
$673$	440906.	0.0375239	0.0187620	−	0.999824i	$-0.494028\pi$
$673$	440906.	0.0375239	0.0187620	+	0.999824i	$0.494028\pi$
$674$	2.64291e7	2.24096
$675$	0	0
$676$	2.31137e6	0.194538
$677$	−7.24025e6	−0.607131	−0.303565	−	0.952811i	$-0.598177\pi$
$677$	−7.24025e6	−0.607131	−0.303565	+	0.952811i	$0.598177\pi$
$678$	0	0
$679$	7.28171e6	0.606120
$680$	1.06058e7	0.879575
$681$	0	0
$682$	1.34098e7	1.10398
$683$	−1.43168e6	−0.117434	−0.0587170	−	0.998275i	$-0.518701\pi$
$683$	−1.43168e6	−0.117434	−0.0587170	+	0.998275i	$0.518701\pi$
$684$	0	0
$685$	5.91148e6	0.481360
$686$	−2.38849e7	−1.93782
$687$	0	0
$688$	−3.06578e6	−0.246928
$689$	2.03965e6	0.163685
$690$	0	0
$691$	5.77535e6	0.460133	0.230066	−	0.973175i	$-0.426106\pi$
$691$	5.77535e6	0.460133	0.230066	+	0.973175i	$0.426106\pi$
$692$	2.93899e6	0.233310
$693$	0	0
$694$	3.96984e7	3.12877
$695$	−7.01553e6	−0.550933
$696$	0	0
$697$	−6.42254e6	−0.500754
$698$	3.02526e7	2.35031
$699$	0	0
$700$	4.31874e6	0.333129
$701$	2.16428e7	1.66348	0.831742	−	0.555162i	$-0.187344\pi$
$701$	2.16428e7	1.66348	0.831742	+	0.555162i	$0.187344\pi$
$702$	0	0
$703$	−4.28408e6	−0.326941
$704$	1.63458e7	1.24301
$705$	0	0
$706$	4.10177e7	3.09713
$707$	1.08324e7	0.815037
$708$	0	0
$709$	4.82444e6	0.360439	0.180219	−	0.983626i	$-0.442319\pi$
$709$	4.82444e6	0.360439	0.180219	+	0.983626i	$0.442319\pi$
$710$	−6641.27	−0.000494431 0
$711$	0	0
$712$	5.98680e7	4.42583
$713$	4.67210e6	0.344182
$714$	0	0
$715$	−1.13658e6	−0.0831447
$716$	−1.59824e6	−0.116509
$717$	0	0
$718$	1.41174e7	1.02199
$719$	−6.08046e6	−0.438646	−0.219323	−	0.975652i	$-0.570385\pi$
$719$	−6.08046e6	−0.438646	−0.219323	+	0.975652i	$0.570385\pi$
$720$	0	0
$721$	5.43928e6	0.389676
$722$	−2.29753e7	−1.64028
$723$	0	0
$724$	−3.66108e7	−2.59575
$725$	2.46926e6	0.174471
$726$	0	0
$727$	1.99326e7	1.39871	0.699357	−	0.714773i	$-0.253470\pi$
$727$	1.99326e7	1.39871	0.699357	+	0.714773i	$0.253470\pi$
$728$	7.50277e6	0.524678
$729$	0	0
$730$	−1.60972e7	−1.11801
$731$	852108.	0.0589795
$732$	0	0
$733$	2.08314e7	1.43205	0.716026	−	0.698074i	$-0.245959\pi$
$733$	2.08314e7	1.43205	0.716026	+	0.698074i	$0.245959\pi$
$734$	−4.17542e7	−2.86062
$735$	0	0
$736$	1.44996e7	0.986647
$737$	5.84613e6	0.396460
$738$	0	0
$739$	2.29192e7	1.54379	0.771895	−	0.635750i	$-0.219309\pi$
$739$	2.29192e7	1.54379	0.771895	+	0.635750i	$0.219309\pi$
$740$	1.54660e7	1.03824
$741$	0	0
$742$	1.09509e7	0.730196
$743$	−230085.	−0.0152903	−0.00764517	−	0.999971i	$-0.502434\pi$
$743$	−230085.	−0.0152903	−0.00764517	+	0.999971i	$0.502434\pi$
$744$	0	0
$745$	2.09701e6	0.138424
$746$	−2.75211e7	−1.81059
$747$	0	0
$748$	−1.77632e7	−1.16082
$749$	−59707.6	−0.00388889
$750$	0	0
$751$	7.24500e6	0.468747	0.234374	−	0.972147i	$-0.424696\pi$
$751$	7.24500e6	0.468747	0.234374	+	0.972147i	$0.424696\pi$
$752$	5.58798e7	3.60338
$753$	0	0
$754$	7.09536e6	0.454513
$755$	1.32500e7	0.845956
$756$	0	0
$757$	−2.05511e7	−1.30345	−0.651725	−	0.758455i	$-0.725954\pi$
$757$	−2.05511e7	−1.30345	−0.651725	+	0.758455i	$0.725954\pi$
$758$	2.41487e7	1.52659
$759$	0	0
$760$	−7.28467e6	−0.457484
$761$	−1.07383e7	−0.672164	−0.336082	−	0.941833i	$-0.609102\pi$
$761$	−1.07383e7	−0.672164	−0.336082	+	0.941833i	$0.609102\pi$
$762$	0	0
$763$	−2.44279e6	−0.151906
$764$	−1.77308e7	−1.09899
$765$	0	0
$766$	−7.53750e6	−0.464147
$767$	−2.76567e6	−0.169751
$768$	0	0
$769$	1.49754e7	0.913194	0.456597	−	0.889674i	$-0.349068\pi$
$769$	1.49754e7	0.913194	0.456597	+	0.889674i	$0.349068\pi$
$770$	−6.10230e6	−0.370908
$771$	0	0
$772$	4.01356e7	2.42374
$773$	1.43362e7	0.862950	0.431475	−	0.902125i	$-0.357993\pi$
$773$	1.43362e7	0.862950	0.431475	+	0.902125i	$0.357993\pi$
$774$	0	0
$775$	2.93176e6	0.175337
$776$	4.43412e7	2.64334
$777$	0	0
$778$	4.23178e7	2.50654
$779$	4.41135e6	0.260452
$780$	0	0
$781$	6724.87	0.000394509 0
$782$	−8.63602e6	−0.505006
$783$	0	0
$784$	−2.79365e7	−1.62323
$785$	2.06549e6	0.119632
$786$	0	0
$787$	9.78174e6	0.562962	0.281481	−	0.959567i	$-0.409174\pi$
$787$	9.78174e6	0.562962	0.281481	+	0.959567i	$0.409174\pi$
$788$	−2.81280e7	−1.61370
$789$	0	0
$790$	1.65588e7	0.943979
$791$	−2.08337e7	−1.18393
$792$	0	0
$793$	−4.22302e6	−0.238474
$794$	−2.78488e7	−1.56767
$795$	0	0
$796$	−4.83372e7	−2.70395
$797$	−484070.	−0.0269937	−0.0134968	−	0.999909i	$-0.504296\pi$
$797$	−484070.	−0.0269937	−0.0134968	+	0.999909i	$0.504296\pi$
$798$	0	0
$799$	−1.55313e7	−0.860680
$800$	9.09858e6	0.502630
$801$	0	0
$802$	3.35952e7	1.84434
$803$	1.62999e7	0.892061
$804$	0	0
$805$	−2.12610e6	−0.115636
$806$	8.42434e6	0.456771
$807$	0	0
$808$	6.59629e7	3.55444
$809$	−3.19445e7	−1.71603	−0.858015	−	0.513625i	$-0.828302\pi$
$809$	−3.19445e7	−1.71603	−0.858015	+	0.513625i	$0.828302\pi$
$810$	0	0
$811$	1.85446e7	0.990068	0.495034	−	0.868874i	$-0.335156\pi$
$811$	1.85446e7	0.990068	0.495034	+	0.868874i	$0.335156\pi$
$812$	2.73001e7	1.45303
$813$	0	0
$814$	−2.18532e7	−1.15599
$815$	1.15172e7	0.607369
$816$	0	0
$817$	−585275.	−0.0306764
$818$	−2.26039e7	−1.18114
$819$	0	0
$820$	−1.59255e7	−0.827100
$821$	−2.77661e6	−0.143766	−0.0718830	−	0.997413i	$-0.522901\pi$
$821$	−2.77661e6	−0.143766	−0.0718830	+	0.997413i	$0.522901\pi$
$822$	0	0
$823$	2.23596e7	1.15070	0.575352	−	0.817906i	$-0.304865\pi$
$823$	2.23596e7	1.15070	0.575352	+	0.817906i	$0.304865\pi$
$824$	3.31219e7	1.69941
$825$	0	0
$826$	−1.48489e7	−0.757258
$827$	−7.33978e6	−0.373181	−0.186590	−	0.982438i	$-0.559744\pi$
$827$	−7.33978e6	−0.373181	−0.186590	+	0.982438i	$0.559744\pi$
$828$	0	0
$829$	1.14199e7	0.577131	0.288565	−	0.957460i	$-0.406822\pi$
$829$	1.14199e7	0.577131	0.288565	+	0.957460i	$0.406822\pi$
$830$	1.05588e7	0.532009
$831$	0	0
$832$	1.02688e7	0.514293
$833$	7.76471e6	0.387715
$834$	0	0
$835$	−1.37044e7	−0.680213
$836$	1.22007e7	0.603767
$837$	0	0
$838$	−6.45878e7	−3.17717
$839$	2.79323e7	1.36994	0.684971	−	0.728570i	$-0.259815\pi$
$839$	2.79323e7	1.36994	0.684971	+	0.728570i	$0.259815\pi$
$840$	0	0
$841$	−4.90216e6	−0.239000
$842$	−1.43423e7	−0.697170
$843$	0	0
$844$	−5.89950e7	−2.85075
$845$	−714025.	−0.0344010
$846$	0	0
$847$	−7.57219e6	−0.362671
$848$	3.54296e7	1.69191
$849$	0	0
$850$	−5.41915e6	−0.257267
$851$	−7.61386e6	−0.360397
$852$	0	0
$853$	2.77296e7	1.30488	0.652440	−	0.757840i	$-0.273745\pi$
$853$	2.77296e7	1.30488	0.652440	+	0.757840i	$0.273745\pi$
$854$	−2.26734e7	−1.06383
$855$	0	0
$856$	−363583.	−0.0169598
$857$	−1.74543e7	−0.811802	−0.405901	−	0.913917i	$-0.633042\pi$
$857$	−1.74543e7	−0.811802	−0.405901	+	0.913917i	$0.633042\pi$
$858$	0	0
$859$	−7.39131e6	−0.341773	−0.170887	−	0.985291i	$-0.554663\pi$
$859$	−7.39131e6	−0.341773	−0.170887	+	0.985291i	$0.554663\pi$
$860$	2.11291e6	0.0974170
$861$	0	0
$862$	2.27339e7	1.04209
$863$	−1.15963e7	−0.530020	−0.265010	−	0.964246i	$-0.585375\pi$
$863$	−1.15963e7	−0.530020	−0.265010	+	0.964246i	$0.585375\pi$
$864$	0	0
$865$	−907908.	−0.0412574
$866$	2.45915e7	1.11427
$867$	0	0
$868$	3.24135e7	1.46025
$869$	−1.67673e7	−0.753205
$870$	0	0
$871$	3.67267e6	0.164035
$872$	−1.48751e7	−0.662475
$873$	0	0
$874$	5.93169e6	0.262664
$875$	−1.33414e6	−0.0589088
$876$	0	0
$877$	3.44009e7	1.51033	0.755163	−	0.655538i	$-0.227558\pi$
$877$	3.44009e7	1.51033	0.755163	+	0.655538i	$0.227558\pi$
$878$	−7.79523e7	−3.41266
$879$	0	0
$880$	−1.97428e7	−0.859415
$881$	−2.78755e7	−1.20999	−0.604996	−	0.796228i	$-0.706825\pi$
$881$	−2.78755e7	−1.20999	−0.604996	+	0.796228i	$0.706825\pi$
$882$	0	0
$883$	−4.29067e6	−0.185193	−0.0925963	−	0.995704i	$-0.529517\pi$
$883$	−4.29067e6	−0.185193	−0.0925963	+	0.995704i	$0.529517\pi$
$884$	−1.11592e7	−0.480290
$885$	0	0
$886$	1.79774e7	0.769381
$887$	9.37833e6	0.400236	0.200118	−	0.979772i	$-0.435867\pi$
$887$	9.37833e6	0.400236	0.200118	+	0.979772i	$0.435867\pi$
$888$	0	0
$889$	−2.53517e6	−0.107585
$890$	−3.05901e7	−1.29451
$891$	0	0
$892$	−3.46829e7	−1.45950
$893$	1.06678e7	0.447657
$894$	0	0
$895$	493725.	0.0206028
$896$	1.53569e7	0.639048
$897$	0	0
$898$	4.29637e7	1.77791
$899$	1.85326e7	0.764781
$900$	0	0
$901$	−9.84736e6	−0.404118
$902$	2.25024e7	0.920900
$903$	0	0
$904$	−1.26865e8	−5.16321
$905$	1.13097e7	0.459019
$906$	0	0
$907$	3.99368e7	1.61196	0.805982	−	0.591940i	$-0.201638\pi$
$907$	3.99368e7	1.61196	0.805982	+	0.591940i	$0.201638\pi$
$908$	−9.56668e7	−3.85076
$909$	0	0
$910$	−3.83360e6	−0.153463
$911$	1.00670e7	0.401889	0.200945	−	0.979603i	$-0.435599\pi$
$911$	1.00670e7	0.401889	0.200945	+	0.979603i	$0.435599\pi$
$912$	0	0
$913$	−1.06917e7	−0.424492
$914$	−5.70742e7	−2.25982
$915$	0	0
$916$	−9.25268e6	−0.364358
$917$	−7.56392e6	−0.297046
$918$	0	0
$919$	−2.21913e7	−0.866752	−0.433376	−	0.901213i	$-0.642678\pi$
$919$	−2.21913e7	−0.866752	−0.433376	+	0.901213i	$0.642678\pi$
$920$	−1.29466e7	−0.504299
$921$	0	0
$922$	3.03152e7	1.17445
$923$	4224.72	0.000163228 0
$924$	0	0
$925$	−4.77773e6	−0.183598
$926$	−6.38193e7	−2.44582
$927$	0	0
$928$	5.75150e7	2.19236
$929$	2.89414e7	1.10022	0.550110	−	0.835092i	$-0.314586\pi$
$929$	2.89414e7	1.10022	0.550110	+	0.835092i	$0.314586\pi$
$930$	0	0
$931$	−5.33323e6	−0.201658
$932$	9.84528e7	3.71269
$933$	0	0
$934$	7.63744e7	2.86471
$935$	5.48736e6	0.205274
$936$	0	0
$937$	1.92306e7	0.715558	0.357779	−	0.933806i	$-0.383534\pi$
$937$	1.92306e7	0.715558	0.357779	+	0.933806i	$0.383534\pi$
$938$	1.97186e7	0.731761
$939$	0	0
$940$	−3.85119e7	−1.42159
$941$	−3.89954e7	−1.43562	−0.717810	−	0.696239i	$-0.754855\pi$
$941$	−3.89954e7	−1.43562	−0.717810	+	0.696239i	$0.754855\pi$
$942$	0	0
$943$	7.84005e6	0.287104
$944$	−4.80408e7	−1.75461
$945$	0	0
$946$	−2.98550e6	−0.108465
$947$	1.85599e7	0.672511	0.336256	−	0.941771i	$-0.390839\pi$
$947$	1.85599e7	0.672511	0.336256	+	0.941771i	$0.390839\pi$
$948$	0	0
$949$	1.02399e7	0.369090
$950$	3.72217e6	0.133809
$951$	0	0
$952$	−3.62231e7	−1.29537
$953$	5.22106e7	1.86220	0.931099	−	0.364765i	$-0.118851\pi$
$953$	5.22106e7	1.86220	0.931099	+	0.364765i	$0.118851\pi$
$954$	0	0
$955$	5.47737e6	0.194341
$956$	7.45756e7	2.63908
$957$	0	0
$958$	4.31778e7	1.52001
$959$	−2.01900e7	−0.708909
$960$	0	0
$961$	−6.62535e6	−0.231420
$962$	−1.37287e7	−0.478290
$963$	0	0
$964$	4.00160e7	1.38689
$965$	−1.23986e7	−0.428603
$966$	0	0
$967$	1.09907e7	0.377971	0.188986	−	0.981980i	$-0.439480\pi$
$967$	1.09907e7	0.377971	0.188986	+	0.981980i	$0.439480\pi$
$968$	−4.61100e7	−1.58164
$969$	0	0
$970$	−2.26565e7	−0.773150
$971$	3.40548e7	1.15912	0.579562	−	0.814928i	$-0.303224\pi$
$971$	3.40548e7	1.15912	0.579562	+	0.814928i	$0.303224\pi$
$972$	0	0
$973$	2.39608e7	0.811370
$974$	6.85581e7	2.31559
$975$	0	0
$976$	−7.33556e7	−2.46495
$977$	1.94494e7	0.651883	0.325942	−	0.945390i	$-0.394319\pi$
$977$	1.94494e7	0.651883	0.325942	+	0.945390i	$0.394319\pi$
$978$	0	0
$979$	3.09751e7	1.03290
$980$	1.92536e7	0.640393
$981$	0	0
$982$	−5.35262e7	−1.77128
$983$	1.75374e7	0.578872	0.289436	−	0.957197i	$-0.406532\pi$
$983$	1.75374e7	0.578872	0.289436	+	0.957197i	$0.406532\pi$
$984$	0	0
$985$	8.68926e6	0.285359
$986$	−3.42561e7	−1.12214
$987$	0	0
$988$	7.66477e6	0.249808
$989$	−1.04018e6	−0.0338155
$990$	0	0
$991$	−9.10071e6	−0.294368	−0.147184	−	0.989109i	$-0.547021\pi$
$991$	−9.10071e6	−0.294368	−0.147184	+	0.989109i	$0.547021\pi$
$992$	6.82877e7	2.20325
$993$	0	0
$994$	22682.5	0.000728158 0
$995$	1.49322e7	0.478153
$996$	0	0
$997$	5.18062e7	1.65061	0.825303	−	0.564690i	$-0.191004\pi$
$997$	5.18062e7	1.65061	0.825303	+	0.564690i	$0.191004\pi$
$998$	−5.31274e7	−1.68847
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.6.a.p.1.9	yes	9
3.2	odd	2		585.6.a.o.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.6.a.o.1.1	✓	9	3.2	odd	2
585.6.a.p.1.9	yes	9	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:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	585.a (trivial)

\(f(q)\)	\(=\)	\(q+10.6267 q^{2} +80.9276 q^{4} -25.0000 q^{5} +85.3847 q^{7} +519.941 q^{8} -265.669 q^{10} +269.013 q^{11} +169.000 q^{13} +907.362 q^{14} +2935.60 q^{16} -815.926 q^{17} +560.423 q^{19} -2023.19 q^{20} +2858.73 q^{22} +996.008 q^{23} +625.000 q^{25} +1795.92 q^{26} +6909.99 q^{28} +3950.82 q^{29} +4690.82 q^{31} +14557.7 q^{32} -8670.63 q^{34} -2134.62 q^{35} -7644.37 q^{37} +5955.47 q^{38} -12998.5 q^{40} +7871.47 q^{41} -1044.34 q^{43} +21770.6 q^{44} +10584.3 q^{46} +19035.2 q^{47} -9516.44 q^{49} +6641.71 q^{50} +13676.8 q^{52} +12068.9 q^{53} -6725.32 q^{55} +44395.1 q^{56} +41984.4 q^{58} -16364.9 q^{59} -24988.3 q^{61} +49848.1 q^{62} +60762.0 q^{64} -4225.00 q^{65} +21731.8 q^{67} -66031.0 q^{68} -22684.0 q^{70} +24.9983 q^{71} +60591.4 q^{73} -81234.8 q^{74} +45353.7 q^{76} +22969.6 q^{77} -62329.0 q^{79} -73390.0 q^{80} +83648.1 q^{82} -39744.2 q^{83} +20398.1 q^{85} -11098.0 q^{86} +139871. q^{88} +115144. q^{89} +14430.0 q^{91} +80604.6 q^{92} +202282. q^{94} -14010.6 q^{95} +85281.2 q^{97} -101129. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 139 q^{4} - 225 q^{5} - 153 q^{7} + 255 q^{8} - 125 q^{10} + 503 q^{11} + 1521 q^{13} + 1110 q^{14} + 763 q^{16} + 1897 q^{17} - 2412 q^{19} - 3475 q^{20} - 3776 q^{22} + 3901 q^{23} + 5625 q^{25}+ \cdots + 119225 q^{98}+O(q^{100}) \) 9 * q + 5 * q^2 + 139 * q^4 - 225 * q^5 - 153 * q^7 + 255 * q^8 - 125 * q^10 + 503 * q^11 + 1521 * q^13 + 1110 * q^14 + 763 * q^16 + 1897 * q^17 - 2412 * q^19 - 3475 * q^20 - 3776 * q^22 + 3901 * q^23 + 5625 * q^25 + 845 * q^26 - 8518 * q^28 + 7906 * q^29 - 4622 * q^31 + 4535 * q^32 - 20300 * q^34 + 3825 * q^35 - 6375 * q^37 + 4650 * q^38 - 6375 * q^40 + 13513 * q^41 - 29862 * q^43 + 34004 * q^44 + 4024 * q^46 + 46534 * q^47 - 15668 * q^49 + 3125 * q^50 + 23491 * q^52 + 49769 * q^53 - 12575 * q^55 + 94250 * q^56 + 11176 * q^58 + 54652 * q^59 + 33319 * q^61 + 110430 * q^62 + 49539 * q^64 - 38025 * q^65 + 76852 * q^67 + 127996 * q^68 - 27750 * q^70 + 108871 * q^71 - 52522 * q^73 + 71210 * q^74 + 89210 * q^76 - 72837 * q^77 - 35955 * q^79 - 19075 * q^80 + 222310 * q^82 + 9400 * q^83 - 47425 * q^85 - 28320 * q^86 + 114224 * q^88 + 73995 * q^89 - 25857 * q^91 + 132288 * q^92 + 303672 * q^94 + 60300 * q^95 + 53623 * q^97 + 119225 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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