LMFDB - Embedded newform 207.6.a.f.1.1

Newspace parameters

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$33.1994507013$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
Defining polynomial:	$ x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 $ x^5 - 113x^3 - 257x^2 + 1404*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.40352$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

$f(q)$	$=$	$q-9.34908 q^{2} +55.4053 q^{4} -70.5203 q^{5} -89.7132 q^{7} -218.818 q^{8} +O(q^{10})$	\(q-9.34908 q^{2} +55.4053 q^{4} -70.5203 q^{5} -89.7132 q^{7} -218.818 q^{8} +659.300 q^{10} +436.763 q^{11} -258.865 q^{13} +838.736 q^{14} +272.775 q^{16} -591.819 q^{17} +1663.69 q^{19} -3907.20 q^{20} -4083.33 q^{22} +529.000 q^{23} +1848.12 q^{25} +2420.15 q^{26} -4970.58 q^{28} +5121.48 q^{29} +1871.29 q^{31} +4451.97 q^{32} +5532.96 q^{34} +6326.60 q^{35} +884.534 q^{37} -15554.0 q^{38} +15431.1 q^{40} +18410.5 q^{41} +12334.9 q^{43} +24199.0 q^{44} -4945.66 q^{46} -23871.6 q^{47} -8758.55 q^{49} -17278.2 q^{50} -14342.5 q^{52} +14594.8 q^{53} -30800.7 q^{55} +19630.8 q^{56} -47881.1 q^{58} -47739.1 q^{59} -15351.5 q^{61} -17494.8 q^{62} -50350.6 q^{64} +18255.2 q^{65} -41174.4 q^{67} -32789.9 q^{68} -59147.9 q^{70} +45608.3 q^{71} -7898.24 q^{73} -8269.57 q^{74} +92177.1 q^{76} -39183.4 q^{77} +71435.2 q^{79} -19236.2 q^{80} -172121. q^{82} -112580. q^{83} +41735.3 q^{85} -115320. q^{86} -95571.5 q^{88} -137845. q^{89} +23223.6 q^{91} +29309.4 q^{92} +223177. q^{94} -117324. q^{95} +41679.8 q^{97} +81884.3 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8}+O(q^{10}) $ 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8	$ 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8} - 172 q^{10} - 1100 q^{11} - 978 q^{13} + 344 q^{14} + 1218 q^{16} - 2522 q^{17} + 2060 q^{19} - 7720 q^{20} - 2572 q^{22} + 2645 q^{23} + 12035 q^{25} - 9280 q^{26} + 8072 q^{28} - 1526 q^{29} - 7392 q^{31} + 5086 q^{32} - 15608 q^{34} - 6056 q^{35} - 8210 q^{37} + 14276 q^{38} - 37472 q^{40} - 21250 q^{41} - 4548 q^{43} + 4260 q^{44} - 4232 q^{46} - 536 q^{47} - 27979 q^{49} + 81872 q^{50} - 76380 q^{52} + 11482 q^{53} - 77064 q^{55} + 28624 q^{56} - 79680 q^{58} - 74676 q^{59} - 44618 q^{61} - 64880 q^{62} - 137382 q^{64} + 24388 q^{65} - 1412 q^{67} - 80196 q^{68} - 222304 q^{70} - 37912 q^{71} + 46546 q^{73} - 111604 q^{74} - 79548 q^{76} - 157008 q^{77} + 50544 q^{79} - 69424 q^{80} - 233720 q^{82} - 89588 q^{83} + 147892 q^{85} - 77428 q^{86} + 54484 q^{88} - 280410 q^{89} - 27416 q^{91} + 62422 q^{92} + 113632 q^{94} - 203120 q^{95} + 90074 q^{97} - 32976 q^{98}+O(q^{100}) $ 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8 - 172 * q^10 - 1100 * q^11 - 978 * q^13 + 344 * q^14 + 1218 * q^16 - 2522 * q^17 + 2060 * q^19 - 7720 * q^20 - 2572 * q^22 + 2645 * q^23 + 12035 * q^25 - 9280 * q^26 + 8072 * q^28 - 1526 * q^29 - 7392 * q^31 + 5086 * q^32 - 15608 * q^34 - 6056 * q^35 - 8210 * q^37 + 14276 * q^38 - 37472 * q^40 - 21250 * q^41 - 4548 * q^43 + 4260 * q^44 - 4232 * q^46 - 536 * q^47 - 27979 * q^49 + 81872 * q^50 - 76380 * q^52 + 11482 * q^53 - 77064 * q^55 + 28624 * q^56 - 79680 * q^58 - 74676 * q^59 - 44618 * q^61 - 64880 * q^62 - 137382 * q^64 + 24388 * q^65 - 1412 * q^67 - 80196 * q^68 - 222304 * q^70 - 37912 * q^71 + 46546 * q^73 - 111604 * q^74 - 79548 * q^76 - 157008 * q^77 + 50544 * q^79 - 69424 * q^80 - 233720 * q^82 - 89588 * q^83 + 147892 * q^85 - 77428 * q^86 + 54484 * q^88 - 280410 * q^89 - 27416 * q^91 + 62422 * q^92 + 113632 * q^94 - 203120 * q^95 + 90074 * q^97 - 32976 * 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 $

$2$	−9.34908	−1.65270	−0.826350	−	0.563157i	$-0.809586\pi$
$2$	−9.34908	−1.65270	−0.826350	+	0.563157i	$0.809586\pi$
$3$	0	0
$3$	0	0
$4$	55.4053	1.73141
$5$	−70.5203	−1.26151	−0.630753	−	0.775984i	$-0.717254\pi$
$5$	−70.5203	−1.26151	−0.630753	+	0.775984i	$0.717254\pi$
$6$	0	0
$7$	−89.7132	−0.692008	−0.346004	−	0.938233i	$-0.612462\pi$
$7$	−89.7132	−0.692008	−0.346004	+	0.938233i	$0.612462\pi$
$8$	−218.818	−1.20881
$9$	0	0
$10$	659.300	2.08489
$11$	436.763	1.08834	0.544170	−	0.838975i	$-0.316845\pi$
$11$	436.763	1.08834	0.544170	+	0.838975i	$0.316845\pi$
$12$	0	0
$13$	−258.865	−0.424830	−0.212415	−	0.977180i	$-0.568133\pi$
$13$	−258.865	−0.424830	−0.212415	+	0.977180i	$0.568133\pi$
$14$	838.736	1.14368
$15$	0	0
$16$	272.775	0.266382
$17$	−591.819	−0.496669	−0.248334	−	0.968674i	$-0.579883\pi$
$17$	−591.819	−0.496669	−0.248334	+	0.968674i	$0.579883\pi$
$18$	0	0
$19$	1663.69	1.05727	0.528637	−	0.848848i	$-0.322703\pi$
$19$	1663.69	1.05727	0.528637	+	0.848848i	$0.322703\pi$
$20$	−3907.20	−2.18419
$21$	0	0
$22$	−4083.33	−1.79870
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	1848.12	0.591398
$26$	2420.15	0.702116
$27$	0	0
$28$	−4970.58	−1.19815
$29$	5121.48	1.13084	0.565419	−	0.824804i	$-0.308714\pi$
$29$	5121.48	1.13084	0.565419	+	0.824804i	$0.308714\pi$
$30$	0	0
$31$	1871.29	0.349733	0.174867	−	0.984592i	$-0.444051\pi$
$31$	1871.29	0.349733	0.174867	+	0.984592i	$0.444051\pi$
$32$	4451.97	0.768559
$33$	0	0
$34$	5532.96	0.820844
$35$	6326.60	0.872972
$36$	0	0
$37$	884.534	0.106221	0.0531105	−	0.998589i	$-0.483086\pi$
$37$	884.534	0.106221	0.0531105	+	0.998589i	$0.483086\pi$
$38$	−15554.0	−1.74736
$39$	0	0
$40$	15431.1	1.52492
$41$	18410.5	1.71043	0.855216	−	0.518271i	$-0.173424\pi$
$41$	18410.5	1.71043	0.855216	+	0.518271i	$0.173424\pi$
$42$	0	0
$43$	12334.9	1.01734	0.508669	−	0.860962i	$-0.330138\pi$
$43$	12334.9	1.01734	0.508669	+	0.860962i	$0.330138\pi$
$44$	24199.0	1.88437
$45$	0	0
$46$	−4945.66	−0.344612
$47$	−23871.6	−1.57629	−0.788146	−	0.615489i	$-0.788959\pi$
$47$	−23871.6	−1.57629	−0.788146	+	0.615489i	$0.788959\pi$
$48$	0	0
$49$	−8758.55	−0.521125
$50$	−17278.2	−0.977403
$51$	0	0
$52$	−14342.5	−0.735556
$53$	14594.8	0.713690	0.356845	−	0.934164i	$-0.383853\pi$
$53$	14594.8	0.713690	0.356845	+	0.934164i	$0.383853\pi$
$54$	0	0
$55$	−30800.7	−1.37295
$56$	19630.8	0.836505
$57$	0	0
$58$	−47881.1	−1.86894
$59$	−47739.1	−1.78544	−0.892718	−	0.450616i	$-0.851205\pi$
$59$	−47739.1	−1.78544	−0.892718	+	0.450616i	$0.851205\pi$
$60$	0	0
$61$	−15351.5	−0.528233	−0.264116	−	0.964491i	$-0.585080\pi$
$61$	−15351.5	−0.528233	−0.264116	+	0.964491i	$0.585080\pi$
$62$	−17494.8	−0.578004
$63$	0	0
$64$	−50350.6	−1.53658
$65$	18255.2	0.535925
$66$	0	0
$67$	−41174.4	−1.12057	−0.560287	−	0.828299i	$-0.689309\pi$
$67$	−41174.4	−1.12057	−0.560287	+	0.828299i	$0.689309\pi$
$68$	−32789.9	−0.859939
$69$	0	0
$70$	−59147.9	−1.44276
$71$	45608.3	1.07374	0.536869	−	0.843666i	$-0.319607\pi$
$71$	45608.3	1.07374	0.536869	+	0.843666i	$0.319607\pi$
$72$	0	0
$73$	−7898.24	−0.173470	−0.0867348	−	0.996231i	$-0.527643\pi$
$73$	−7898.24	−0.173470	−0.0867348	+	0.996231i	$0.527643\pi$
$74$	−8269.57	−0.175551
$75$	0	0
$76$	92177.1	1.83058
$77$	−39183.4	−0.753140
$78$	0	0
$79$	71435.2	1.28779	0.643894	−	0.765115i	$-0.277318\pi$
$79$	71435.2	1.28779	0.643894	+	0.765115i	$0.277318\pi$
$80$	−19236.2	−0.336043
$81$	0	0
$82$	−172121.	−2.82683
$83$	−112580.	−1.79377	−0.896885	−	0.442264i	$-0.854175\pi$
$83$	−112580.	−1.79377	−0.896885	+	0.442264i	$0.854175\pi$
$84$	0	0
$85$	41735.3	0.626551
$86$	−115320.	−1.68136
$87$	0	0
$88$	−95571.5	−1.31559
$89$	−137845.	−1.84466	−0.922330	−	0.386404i	$-0.873717\pi$
$89$	−137845.	−1.84466	−0.922330	+	0.386404i	$0.873717\pi$
$90$	0	0
$91$	23223.6	0.293985
$92$	29309.4	0.361025
$93$	0	0
$94$	223177.	2.60514
$95$	−117324.	−1.33376
$96$	0	0
$97$	41679.8	0.449776	0.224888	−	0.974385i	$-0.427798\pi$
$97$	41679.8	0.449776	0.224888	+	0.974385i	$0.427798\pi$
$98$	81884.3	0.861263
$99$	0	0
$100$	102395.	1.02395
$101$	−37100.3	−0.361888	−0.180944	−	0.983493i	$-0.557915\pi$
$101$	−37100.3	−0.361888	−0.180944	+	0.983493i	$0.557915\pi$
$102$	0	0
$103$	−123323.	−1.14538	−0.572692	−	0.819771i	$-0.694101\pi$
$103$	−123323.	−1.14538	−0.572692	+	0.819771i	$0.694101\pi$
$104$	56644.2	0.513538
$105$	0	0
$106$	−136448.	−1.17951
$107$	−107455.	−0.907338	−0.453669	−	0.891170i	$-0.649885\pi$
$107$	−107455.	−0.907338	−0.453669	+	0.891170i	$0.649885\pi$
$108$	0	0
$109$	96652.7	0.779198	0.389599	−	0.920985i	$-0.372614\pi$
$109$	96652.7	0.779198	0.389599	+	0.920985i	$0.372614\pi$
$110$	287958.	2.26907
$111$	0	0
$112$	−24471.5	−0.184338
$113$	−55788.7	−0.411008	−0.205504	−	0.978656i	$-0.565883\pi$
$113$	−55788.7	−0.411008	−0.205504	+	0.978656i	$0.565883\pi$
$114$	0	0
$115$	−37305.3	−0.263042
$116$	283757.	1.95795
$117$	0	0
$118$	446317.	2.95079
$119$	53094.0	0.343699
$120$	0	0
$121$	29711.1	0.184483
$122$	143522.	0.873010
$123$	0	0
$124$	103679.	0.605533
$125$	90046.2	0.515454
$126$	0	0
$127$	167839.	0.923388	0.461694	−	0.887039i	$-0.347242\pi$
$127$	167839.	0.923388	0.461694	+	0.887039i	$0.347242\pi$
$128$	328269.	1.77094
$129$	0	0
$130$	−170670.	−0.885723
$131$	−129729.	−0.660476	−0.330238	−	0.943898i	$-0.607129\pi$
$131$	−129729.	−0.660476	−0.330238	+	0.943898i	$0.607129\pi$
$132$	0	0
$133$	−149255.	−0.731643
$134$	384943.	1.85197
$135$	0	0
$136$	129501.	0.600377
$137$	226702.	1.03194	0.515970	−	0.856607i	$-0.327432\pi$
$137$	226702.	1.03194	0.515970	+	0.856607i	$0.327432\pi$
$138$	0	0
$139$	−419175.	−1.84017	−0.920086	−	0.391716i	$-0.871882\pi$
$139$	−419175.	−1.84017	−0.920086	+	0.391716i	$0.871882\pi$
$140$	350527.	1.51148
$141$	0	0
$142$	−426396.	−1.77457
$143$	−113063.	−0.462359
$144$	0	0
$145$	−361169.	−1.42656
$146$	73841.3	0.286693
$147$	0	0
$148$	49007.8	0.183912
$149$	−503270.	−1.85710	−0.928551	−	0.371206i	$-0.878945\pi$
$149$	−503270.	−1.85710	−0.928551	+	0.371206i	$0.878945\pi$
$150$	0	0
$151$	517943.	1.84858	0.924292	−	0.381685i	$-0.124656\pi$
$151$	517943.	1.84858	0.924292	+	0.381685i	$0.124656\pi$
$152$	−364044.	−1.27804
$153$	0	0
$154$	366329.	1.24471
$155$	−131964.	−0.441190
$156$	0	0
$157$	−39747.9	−0.128696	−0.0643480	−	0.997928i	$-0.520497\pi$
$157$	−39747.9	−0.128696	−0.0643480	+	0.997928i	$0.520497\pi$
$158$	−667853.	−2.12833
$159$	0	0
$160$	−313954.	−0.969542
$161$	−47458.3	−0.144294
$162$	0	0
$163$	−157828.	−0.465279	−0.232640	−	0.972563i	$-0.574736\pi$
$163$	−157828.	−0.465279	−0.232640	+	0.972563i	$0.574736\pi$
$164$	1.02004e6	2.96147
$165$	0	0
$166$	1.05252e6	2.96456
$167$	101000.	0.280241	0.140120	−	0.990134i	$-0.455251\pi$
$167$	101000.	0.280241	0.140120	+	0.990134i	$0.455251\pi$
$168$	0	0
$169$	−304282.	−0.819520
$170$	−390187.	−1.03550
$171$	0	0
$172$	683420.	1.76144
$173$	−61568.5	−0.156402	−0.0782012	−	0.996938i	$-0.524918\pi$
$173$	−61568.5	−0.156402	−0.0782012	+	0.996938i	$0.524918\pi$
$174$	0	0
$175$	−165801.	−0.409252
$176$	119138.	0.289914
$177$	0	0
$178$	1.28872e6	3.04867
$179$	610536.	1.42423	0.712113	−	0.702065i	$-0.247738\pi$
$179$	610536.	1.42423	0.712113	+	0.702065i	$0.247738\pi$
$180$	0	0
$181$	121378.	0.275387	0.137694	−	0.990475i	$-0.456031\pi$
$181$	121378.	0.275387	0.137694	+	0.990475i	$0.456031\pi$
$182$	−217119.	−0.485870
$183$	0	0
$184$	−115755.	−0.252054
$185$	−62377.6	−0.133998
$186$	0	0
$187$	−258485.	−0.540544
$188$	−1.32261e6	−2.72921
$189$	0	0
$190$	1.09687e6	2.20430
$191$	294581.	0.584280	0.292140	−	0.956376i	$-0.405633\pi$
$191$	294581.	0.584280	0.292140	+	0.956376i	$0.405633\pi$
$192$	0	0
$193$	−8895.53	−0.0171901	−0.00859505	−	0.999963i	$-0.502736\pi$
$193$	−8895.53	−0.0171901	−0.00859505	+	0.999963i	$0.502736\pi$
$194$	−389668.	−0.743345
$195$	0	0
$196$	−485270.	−0.902283
$197$	698113.	1.28162	0.640811	−	0.767699i	$-0.278598\pi$
$197$	698113.	1.28162	0.640811	+	0.767699i	$0.278598\pi$
$198$	0	0
$199$	−533271.	−0.954586	−0.477293	−	0.878744i	$-0.658382\pi$
$199$	−533271.	−0.954586	−0.477293	+	0.878744i	$0.658382\pi$
$200$	−404401.	−0.714887
$201$	0	0
$202$	346854.	0.598092
$203$	−459464.	−0.782550
$204$	0	0
$205$	−1.29831e6	−2.15772
$206$	1.15296e6	1.89298
$207$	0	0
$208$	−70611.9	−0.113167
$209$	726638.	1.15067
$210$	0	0
$211$	−658407.	−1.01809	−0.509047	−	0.860739i	$-0.670002\pi$
$211$	−658407.	−1.01809	−0.509047	+	0.860739i	$0.670002\pi$
$212$	808630.	1.23569
$213$	0	0
$214$	1.00461e6	1.49956
$215$	−869864.	−1.28338
$216$	0	0
$217$	−167879.	−0.242018
$218$	−903614.	−1.28778
$219$	0	0
$220$	−1.70652e6	−2.37714
$221$	153201.	0.211000
$222$	0	0
$223$	79895.6	0.107587	0.0537936	−	0.998552i	$-0.482869\pi$
$223$	79895.6	0.107587	0.0537936	+	0.998552i	$0.482869\pi$
$224$	−399400.	−0.531849
$225$	0	0
$226$	521573.	0.679273
$227$	−716111.	−0.922393	−0.461196	−	0.887298i	$-0.652580\pi$
$227$	−716111.	−0.922393	−0.461196	+	0.887298i	$0.652580\pi$
$228$	0	0
$229$	−581299.	−0.732506	−0.366253	−	0.930515i	$-0.619360\pi$
$229$	−581299.	−0.732506	−0.366253	+	0.930515i	$0.619360\pi$
$230$	348770.	0.434730
$231$	0	0
$232$	−1.12067e6	−1.36697
$233$	−122734.	−0.148107	−0.0740534	−	0.997254i	$-0.523594\pi$
$233$	−122734.	−0.148107	−0.0740534	+	0.997254i	$0.523594\pi$
$234$	0	0
$235$	1.68343e6	1.98850
$236$	−2.64500e6	−3.09133
$237$	0	0
$238$	−496380.	−0.568031
$239$	−210076.	−0.237893	−0.118946	−	0.992901i	$-0.537952\pi$
$239$	−210076.	−0.237893	−0.118946	+	0.992901i	$0.537952\pi$
$240$	0	0
$241$	1.23224e6	1.36664	0.683319	−	0.730120i	$-0.260536\pi$
$241$	1.23224e6	1.36664	0.683319	+	0.730120i	$0.260536\pi$
$242$	−277772.	−0.304894
$243$	0	0
$244$	−850553.	−0.914590
$245$	617656.	0.657402
$246$	0	0
$247$	−430670.	−0.449162
$248$	−409471.	−0.422760
$249$	0	0
$250$	−841849.	−0.851891
$251$	545986.	0.547012	0.273506	−	0.961870i	$-0.411817\pi$
$251$	545986.	0.547012	0.273506	+	0.961870i	$0.411817\pi$
$252$	0	0
$253$	231048.	0.226934
$254$	−1.56914e6	−1.52608
$255$	0	0
$256$	−1.45779e6	−1.39026
$257$	−556705.	−0.525765	−0.262883	−	0.964828i	$-0.584673\pi$
$257$	−556705.	−0.525765	−0.262883	+	0.964828i	$0.584673\pi$
$258$	0	0
$259$	−79354.3	−0.0735057
$260$	1.01144e6	0.927909
$261$	0	0
$262$	1.21284e6	1.09157
$263$	−1.65392e6	−1.47443	−0.737217	−	0.675656i	$-0.763860\pi$
$263$	−1.65392e6	−1.47443	−0.737217	+	0.675656i	$0.763860\pi$
$264$	0	0
$265$	−1.02923e6	−0.900324
$266$	1.39539e6	1.20919
$267$	0	0
$268$	−2.28128e6	−1.94018
$269$	2.06545e6	1.74034	0.870168	−	0.492755i	$-0.164010\pi$
$269$	2.06545e6	1.74034	0.870168	+	0.492755i	$0.164010\pi$
$270$	0	0
$271$	708326.	0.585881	0.292941	−	0.956131i	$-0.405366\pi$
$271$	708326.	0.585881	0.292941	+	0.956131i	$0.405366\pi$
$272$	−161434.	−0.132304
$273$	0	0
$274$	−2.11946e6	−1.70549
$275$	807190.	0.643641
$276$	0	0
$277$	−17729.4	−0.0138833	−0.00694167	−	0.999976i	$-0.502210\pi$
$277$	−17729.4	−0.0138833	−0.00694167	+	0.999976i	$0.502210\pi$
$278$	3.91890e6	3.04125
$279$	0	0
$280$	−1.38437e6	−1.05526
$281$	−754660.	−0.570145	−0.285073	−	0.958506i	$-0.592018\pi$
$281$	−754660.	−0.570145	−0.285073	+	0.958506i	$0.592018\pi$
$282$	0	0
$283$	−417633.	−0.309976	−0.154988	−	0.987916i	$-0.549534\pi$
$283$	−417633.	−0.309976	−0.154988	+	0.987916i	$0.549534\pi$
$284$	2.52694e6	1.85909
$285$	0	0
$286$	1.05703e6	0.764140
$287$	−1.65166e6	−1.18363
$288$	0	0
$289$	−1.06961e6	−0.753320
$290$	3.37659e6	2.35767
$291$	0	0
$292$	−437604.	−0.300348
$293$	2.68010e6	1.82382	0.911908	−	0.410394i	$-0.134609\pi$
$293$	2.68010e6	1.82382	0.911908	+	0.410394i	$0.134609\pi$
$294$	0	0
$295$	3.36658e6	2.25234
$296$	−193552.	−0.128401
$297$	0	0
$298$	4.70511e6	3.06923
$299$	−136940.	−0.0885831
$300$	0	0
$301$	−1.10661e6	−0.704007
$302$	−4.84229e6	−3.05515
$303$	0	0
$304$	453813.	0.281639
$305$	1.08259e6	0.666369
$306$	0	0
$307$	−2.42927e6	−1.47106	−0.735528	−	0.677494i	$-0.763066\pi$
$307$	−2.42927e6	−1.47106	−0.735528	+	0.677494i	$0.763066\pi$
$308$	−2.17097e6	−1.30400
$309$	0	0
$310$	1.23374e6	0.729155
$311$	−1.71783e6	−1.00712	−0.503559	−	0.863961i	$-0.667976\pi$
$311$	−1.71783e6	−1.00712	−0.503559	+	0.863961i	$0.667976\pi$
$312$	0	0
$313$	1.56583e6	0.903406	0.451703	−	0.892168i	$-0.350817\pi$
$313$	1.56583e6	0.903406	0.451703	+	0.892168i	$0.350817\pi$
$314$	371606.	0.212696
$315$	0	0
$316$	3.95789e6	2.22970
$317$	−673750.	−0.376574	−0.188287	−	0.982114i	$-0.560294\pi$
$317$	−673750.	−0.376574	−0.188287	+	0.982114i	$0.560294\pi$
$318$	0	0
$319$	2.23688e6	1.23074
$320$	3.55074e6	1.93840
$321$	0	0
$322$	443691.	0.238474
$323$	−984603.	−0.525115
$324$	0	0
$325$	−478413.	−0.251243
$326$	1.47554e6	0.768967
$327$	0	0
$328$	−4.02854e6	−2.06759
$329$	2.14159e6	1.09081
$330$	0	0
$331$	1.69768e6	0.851699	0.425850	−	0.904794i	$-0.359975\pi$
$331$	1.69768e6	0.851699	0.425850	+	0.904794i	$0.359975\pi$
$332$	−6.23754e6	−3.10576
$333$	0	0
$334$	−944259.	−0.463153
$335$	2.90363e6	1.41361
$336$	0	0
$337$	−2.37281e6	−1.13812	−0.569061	−	0.822295i	$-0.692693\pi$
$337$	−2.37281e6	−1.13812	−0.569061	+	0.822295i	$0.692693\pi$
$338$	2.84476e6	1.35442
$339$	0	0
$340$	2.31236e6	1.08482
$341$	817310.	0.380628
$342$	0	0
$343$	2.29357e6	1.05263
$344$	−2.69910e6	−1.22977
$345$	0	0
$346$	575609.	0.258486
$347$	−3.98316e6	−1.77584	−0.887921	−	0.459996i	$-0.847851\pi$
$347$	−3.98316e6	−1.77584	−0.887921	+	0.459996i	$0.847851\pi$
$348$	0	0
$349$	−2.10463e6	−0.924935	−0.462468	−	0.886636i	$-0.653036\pi$
$349$	−2.10463e6	−0.924935	−0.462468	+	0.886636i	$0.653036\pi$
$350$	1.55008e6	0.676370
$351$	0	0
$352$	1.94446e6	0.836453
$353$	−3.41107e6	−1.45698	−0.728490	−	0.685056i	$-0.759778\pi$
$353$	−3.41107e6	−1.45698	−0.728490	+	0.685056i	$0.759778\pi$
$354$	0	0
$355$	−3.21631e6	−1.35453
$356$	−7.63734e6	−3.19387
$357$	0	0
$358$	−5.70795e6	−2.35382
$359$	1.86760e6	0.764800	0.382400	−	0.923997i	$-0.375098\pi$
$359$	1.86760e6	0.764800	0.382400	+	0.923997i	$0.375098\pi$
$360$	0	0
$361$	291759.	0.117830
$362$	−1.13477e6	−0.455132
$363$	0	0
$364$	1.28671e6	0.509011
$365$	556987.	0.218833
$366$	0	0
$367$	−437210.	−0.169444	−0.0847218	−	0.996405i	$-0.527000\pi$
$367$	−437210.	−0.169444	−0.0847218	+	0.996405i	$0.527000\pi$
$368$	144298.	0.0555445
$369$	0	0
$370$	583173.	0.221459
$371$	−1.30935e6	−0.493879
$372$	0	0
$373$	−921734.	−0.343031	−0.171516	−	0.985181i	$-0.554866\pi$
$373$	−921734.	−0.343031	−0.171516	+	0.985181i	$0.554866\pi$
$374$	2.41660e6	0.893357
$375$	0	0
$376$	5.22352e6	1.90543
$377$	−1.32577e6	−0.480414
$378$	0	0
$379$	−3.80141e6	−1.35940	−0.679699	−	0.733491i	$-0.737889\pi$
$379$	−3.80141e6	−1.35940	−0.679699	+	0.733491i	$0.737889\pi$
$380$	−6.50036e6	−2.30929
$381$	0	0
$382$	−2.75406e6	−0.965639
$383$	4.52311e6	1.57558	0.787790	−	0.615944i	$-0.211225\pi$
$383$	4.52311e6	1.57558	0.787790	+	0.615944i	$0.211225\pi$
$384$	0	0
$385$	2.76323e6	0.950090
$386$	83165.0	0.0284101
$387$	0	0
$388$	2.30928e6	0.778749
$389$	−4.48168e6	−1.50164	−0.750822	−	0.660505i	$-0.770342\pi$
$389$	−4.48168e6	−1.50164	−0.750822	+	0.660505i	$0.770342\pi$
$390$	0	0
$391$	−313072.	−0.103563
$392$	1.91652e6	0.629940
$393$	0	0
$394$	−6.52671e6	−2.11814
$395$	−5.03763e6	−1.62455
$396$	0	0
$397$	1.76656e6	0.562539	0.281269	−	0.959629i	$-0.409245\pi$
$397$	1.76656e6	0.562539	0.281269	+	0.959629i	$0.409245\pi$
$398$	4.98559e6	1.57764
$399$	0	0
$400$	504121.	0.157538
$401$	522373.	0.162226	0.0811128	−	0.996705i	$-0.474153\pi$
$401$	522373.	0.162226	0.0811128	+	0.996705i	$0.474153\pi$
$402$	0	0
$403$	−484411.	−0.148577
$404$	−2.05555e6	−0.626579
$405$	0	0
$406$	4.29557e6	1.29332
$407$	386332.	0.115604
$408$	0	0
$409$	−1.60642e6	−0.474845	−0.237422	−	0.971407i	$-0.576303\pi$
$409$	−1.60642e6	−0.474845	−0.237422	+	0.971407i	$0.576303\pi$
$410$	1.21380e7	3.56606
$411$	0	0
$412$	−6.83275e6	−1.98314
$413$	4.28283e6	1.23554
$414$	0	0
$415$	7.93919e6	2.26285
$416$	−1.15246e6	−0.326507
$417$	0	0
$418$	−6.79339e6	−1.90172
$419$	5.30658e6	1.47666	0.738329	−	0.674441i	$-0.235615\pi$
$419$	5.30658e6	1.47666	0.738329	+	0.674441i	$0.235615\pi$
$420$	0	0
$421$	−2.30575e6	−0.634025	−0.317012	−	0.948421i	$-0.602680\pi$
$421$	−2.30575e6	−0.634025	−0.317012	+	0.948421i	$0.602680\pi$
$422$	6.15549e6	1.68260
$423$	0	0
$424$	−3.19361e6	−0.862714
$425$	−1.09375e6	−0.293729
$426$	0	0
$427$	1.37723e6	0.365541
$428$	−5.95360e6	−1.57098
$429$	0	0
$430$	8.13242e6	2.12104
$431$	1.27091e6	0.329549	0.164775	−	0.986331i	$-0.447310\pi$
$431$	1.27091e6	0.329549	0.164775	+	0.986331i	$0.447310\pi$
$432$	0	0
$433$	−3.71880e6	−0.953199	−0.476599	−	0.879121i	$-0.658131\pi$
$433$	−3.71880e6	−0.953199	−0.476599	+	0.879121i	$0.658131\pi$
$434$	1.56952e6	0.399983
$435$	0	0
$436$	5.35507e6	1.34911
$437$	880091.	0.220457
$438$	0	0
$439$	−3.51069e6	−0.869423	−0.434711	−	0.900570i	$-0.643150\pi$
$439$	−3.51069e6	−0.869423	−0.434711	+	0.900570i	$0.643150\pi$
$440$	6.73974e6	1.65963
$441$	0	0
$442$	−1.43229e6	−0.348719
$443$	−2.67040e6	−0.646497	−0.323248	−	0.946314i	$-0.604775\pi$
$443$	−2.67040e6	−0.646497	−0.323248	+	0.946314i	$0.604775\pi$
$444$	0	0
$445$	9.72088e6	2.32705
$446$	−746950.	−0.177809
$447$	0	0
$448$	4.51711e6	1.06332
$449$	−808771.	−0.189326	−0.0946628	−	0.995509i	$-0.530177\pi$
$449$	−808771.	−0.189326	−0.0946628	+	0.995509i	$0.530177\pi$
$450$	0	0
$451$	8.04103e6	1.86153
$452$	−3.09099e6	−0.711625
$453$	0	0
$454$	6.69498e6	1.52444
$455$	−1.63774e6	−0.370864
$456$	0	0
$457$	−6.03865e6	−1.35254	−0.676269	−	0.736655i	$-0.736404\pi$
$457$	−6.03865e6	−1.35254	−0.676269	+	0.736655i	$0.736404\pi$
$458$	5.43461e6	1.21061
$459$	0	0
$460$	−2.06691e6	−0.455435
$461$	6.06334e6	1.32880	0.664399	−	0.747378i	$-0.268687\pi$
$461$	6.06334e6	1.32880	0.664399	+	0.747378i	$0.268687\pi$
$462$	0	0
$463$	−987185.	−0.214016	−0.107008	−	0.994258i	$-0.534127\pi$
$463$	−987185.	−0.214016	−0.107008	+	0.994258i	$0.534127\pi$
$464$	1.39701e6	0.301235
$465$	0	0
$466$	1.14745e6	0.244776
$467$	2.75869e6	0.585344	0.292672	−	0.956213i	$-0.405456\pi$
$467$	2.75869e6	0.585344	0.292672	+	0.956213i	$0.405456\pi$
$468$	0	0
$469$	3.69389e6	0.775446
$470$	−1.57385e7	−3.28639
$471$	0	0
$472$	1.04462e7	2.15825
$473$	5.38745e6	1.10721
$474$	0	0
$475$	3.07469e6	0.625270
$476$	2.94169e6	0.595085
$477$	0	0
$478$	1.96402e6	0.393165
$479$	1.43871e6	0.286507	0.143254	−	0.989686i	$-0.454244\pi$
$479$	1.43871e6	0.286507	0.143254	+	0.989686i	$0.454244\pi$
$480$	0	0
$481$	−228975.	−0.0451258
$482$	−1.15203e7	−2.25864
$483$	0	0
$484$	1.64615e6	0.319416
$485$	−2.93927e6	−0.567395
$486$	0	0
$487$	−9.54793e6	−1.82426	−0.912130	−	0.409901i	$-0.865563\pi$
$487$	−9.54793e6	−1.82426	−0.912130	+	0.409901i	$0.865563\pi$
$488$	3.35917e6	0.638532
$489$	0	0
$490$	−5.77451e6	−1.08649
$491$	−8.94131e6	−1.67378	−0.836889	−	0.547373i	$-0.815628\pi$
$491$	−8.94131e6	−1.67378	−0.836889	+	0.547373i	$0.815628\pi$
$492$	0	0
$493$	−3.03099e6	−0.561652
$494$	4.02637e6	0.742329
$495$	0	0
$496$	510441.	0.0931626
$497$	−4.09167e6	−0.743035
$498$	0	0
$499$	2.34135e6	0.420934	0.210467	−	0.977601i	$-0.432501\pi$
$499$	2.34135e6	0.420934	0.210467	+	0.977601i	$0.432501\pi$
$500$	4.98903e6	0.892465
$501$	0	0
$502$	−5.10446e6	−0.904046
$503$	−41307.7	−0.00727966	−0.00363983	−	0.999993i	$-0.501159\pi$
$503$	−41307.7	−0.00727966	−0.00363983	+	0.999993i	$0.501159\pi$
$504$	0	0
$505$	2.61633e6	0.456524
$506$	−2.16008e6	−0.375054
$507$	0	0
$508$	9.29918e6	1.59877
$509$	−3.09166e6	−0.528929	−0.264464	−	0.964395i	$-0.585195\pi$
$509$	−3.09166e6	−0.528929	−0.264464	+	0.964395i	$0.585195\pi$
$510$	0	0
$511$	708576.	0.120042
$512$	3.12440e6	0.526735
$513$	0	0
$514$	5.20467e6	0.868932
$515$	8.69678e6	1.44491
$516$	0	0
$517$	−1.04262e7	−1.71554
$518$	741890.	0.121483
$519$	0	0
$520$	−3.99457e6	−0.647831
$521$	−6.02002e6	−0.971635	−0.485818	−	0.874060i	$-0.661478\pi$
$521$	−6.02002e6	−0.971635	−0.485818	+	0.874060i	$0.661478\pi$
$522$	0	0
$523$	−6.39458e6	−1.02225	−0.511125	−	0.859506i	$-0.670771\pi$
$523$	−6.39458e6	−1.02225	−0.511125	+	0.859506i	$0.670771\pi$
$524$	−7.18764e6	−1.14356
$525$	0	0
$526$	1.54626e7	2.43679
$527$	−1.10746e6	−0.173701
$528$	0	0
$529$	279841.	0.0434783
$530$	9.62237e6	1.48796
$531$	0	0
$532$	−8.26950e6	−1.26678
$533$	−4.76583e6	−0.726642
$534$	0	0
$535$	7.57779e6	1.14461
$536$	9.00969e6	1.35456
$537$	0	0
$538$	−1.93100e7	−2.87625
$539$	−3.82541e6	−0.567161
$540$	0	0
$541$	−3.24248e6	−0.476304	−0.238152	−	0.971228i	$-0.576542\pi$
$541$	−3.24248e6	−0.476304	−0.238152	+	0.971228i	$0.576542\pi$
$542$	−6.62219e6	−0.968286
$543$	0	0
$544$	−2.63476e6	−0.381719
$545$	−6.81598e6	−0.982963
$546$	0	0
$547$	9.39508e6	1.34255	0.671277	−	0.741206i	$-0.265746\pi$
$547$	9.39508e6	1.34255	0.671277	+	0.741206i	$0.265746\pi$
$548$	1.25605e7	1.78671
$549$	0	0
$550$	−7.54648e6	−1.06375
$551$	8.52055e6	1.19561
$552$	0	0
$553$	−6.40868e6	−0.891160
$554$	165753.	0.0229450
$555$	0	0
$556$	−2.32245e7	−3.18610
$557$	−2.95696e6	−0.403839	−0.201919	−	0.979402i	$-0.564718\pi$
$557$	−2.95696e6	−0.403839	−0.201919	+	0.979402i	$0.564718\pi$
$558$	0	0
$559$	−3.19308e6	−0.432196
$560$	1.72574e6	0.232544
$561$	0	0
$562$	7.05538e6	0.942279
$563$	1.35314e7	1.79917	0.899584	−	0.436747i	$-0.143870\pi$
$563$	1.35314e7	1.79917	0.899584	+	0.436747i	$0.143870\pi$
$564$	0	0
$565$	3.93424e6	0.518489
$566$	3.90448e6	0.512297
$567$	0	0
$568$	−9.97991e6	−1.29794
$569$	−1.32776e7	−1.71924	−0.859622	−	0.510931i	$-0.829301\pi$
$569$	−1.32776e7	−1.71924	−0.859622	+	0.510931i	$0.829301\pi$
$570$	0	0
$571$	7.74960e6	0.994693	0.497346	−	0.867552i	$-0.334308\pi$
$571$	7.74960e6	0.994693	0.497346	+	0.867552i	$0.334308\pi$
$572$	−6.26427e6	−0.800535
$573$	0	0
$574$	1.54415e7	1.95619
$575$	977654.	0.123315
$576$	0	0
$577$	−4.83479e6	−0.604558	−0.302279	−	0.953220i	$-0.597747\pi$
$577$	−4.83479e6	−0.604558	−0.302279	+	0.953220i	$0.597747\pi$
$578$	9.99984e6	1.24501
$579$	0	0
$580$	−2.00106e7	−2.46997
$581$	1.00999e7	1.24130
$582$	0	0
$583$	6.37448e6	0.776736
$584$	1.72827e6	0.209691
$585$	0	0
$586$	−2.50564e7	−3.01422
$587$	−5.13380e6	−0.614955	−0.307478	−	0.951555i	$-0.599485\pi$
$587$	−5.13380e6	−0.614955	−0.307478	+	0.951555i	$0.599485\pi$
$588$	0	0
$589$	3.11324e6	0.369764
$590$	−3.14744e7	−3.72244
$591$	0	0
$592$	241279.	0.0282953
$593$	−1.29134e7	−1.50801	−0.754003	−	0.656871i	$-0.771879\pi$
$593$	−1.29134e7	−1.50801	−0.754003	+	0.656871i	$0.771879\pi$
$594$	0	0
$595$	−3.74421e6	−0.433578
$596$	−2.78838e7	−3.21541
$597$	0	0
$598$	1.28026e6	0.146401
$599$	5.79980e6	0.660460	0.330230	−	0.943901i	$-0.392874\pi$
$599$	5.79980e6	0.660460	0.330230	+	0.943901i	$0.392874\pi$
$600$	0	0
$601$	1.06993e7	1.20829	0.604144	−	0.796875i	$-0.293515\pi$
$601$	1.06993e7	1.20829	0.604144	+	0.796875i	$0.293515\pi$
$602$	1.03457e7	1.16351
$603$	0	0
$604$	2.86968e7	3.20067
$605$	−2.09524e6	−0.232726
$606$	0	0
$607$	−1.48509e7	−1.63600	−0.817998	−	0.575221i	$-0.804916\pi$
$607$	−1.48509e7	−1.63600	−0.817998	+	0.575221i	$0.804916\pi$
$608$	7.40669e6	0.812578
$609$	0	0
$610$	−1.01212e7	−1.10131
$611$	6.17951e6	0.669655
$612$	0	0
$613$	−7.64415e6	−0.821633	−0.410816	−	0.911718i	$-0.634756\pi$
$613$	−7.64415e6	−0.821633	−0.410816	+	0.911718i	$0.634756\pi$
$614$	2.27114e7	2.43121
$615$	0	0
$616$	8.57402e6	0.910401
$617$	8.02025e6	0.848155	0.424077	−	0.905626i	$-0.360598\pi$
$617$	8.02025e6	0.848155	0.424077	+	0.905626i	$0.360598\pi$
$618$	0	0
$619$	9.46459e6	0.992831	0.496416	−	0.868085i	$-0.334649\pi$
$619$	9.46459e6	0.992831	0.496416	+	0.868085i	$0.334649\pi$
$620$	−7.31150e6	−0.763883
$621$	0	0
$622$	1.60602e7	1.66446
$623$	1.23665e7	1.27652
$624$	0	0
$625$	−1.21255e7	−1.24165
$626$	−1.46390e7	−1.49306
$627$	0	0
$628$	−2.20224e6	−0.222826
$629$	−523484.	−0.0527566
$630$	0	0
$631$	8.76525e6	0.876377	0.438189	−	0.898883i	$-0.355620\pi$
$631$	8.76525e6	0.876377	0.438189	+	0.898883i	$0.355620\pi$
$632$	−1.56313e7	−1.55669
$633$	0	0
$634$	6.29894e6	0.622364
$635$	−1.18361e7	−1.16486
$636$	0	0
$637$	2.26728e6	0.221389
$638$	−2.09127e7	−2.03404
$639$	0	0
$640$	−2.31496e7	−2.23406
$641$	8.04677e6	0.773529	0.386764	−	0.922179i	$-0.373593\pi$
$641$	8.04677e6	0.773529	0.386764	+	0.922179i	$0.373593\pi$
$642$	0	0
$643$	1.81157e7	1.72794	0.863970	−	0.503543i	$-0.167970\pi$
$643$	1.81157e7	1.72794	0.863970	+	0.503543i	$0.167970\pi$
$644$	−2.62944e6	−0.249832
$645$	0	0
$646$	9.20513e6	0.867858
$647$	−6.09047e6	−0.571992	−0.285996	−	0.958231i	$-0.592324\pi$
$647$	−6.09047e6	−0.571992	−0.285996	+	0.958231i	$0.592324\pi$
$648$	0	0
$649$	−2.08507e7	−1.94316
$650$	4.47272e6	0.415230
$651$	0	0
$652$	−8.74448e6	−0.805591
$653$	−1.73408e7	−1.59142	−0.795712	−	0.605675i	$-0.792903\pi$
$653$	−1.73408e7	−1.59142	−0.795712	+	0.605675i	$0.792903\pi$
$654$	0	0
$655$	9.14850e6	0.833195
$656$	5.02193e6	0.455629
$657$	0	0
$658$	−2.00219e7	−1.80277
$659$	−1.27618e7	−1.14471	−0.572357	−	0.820004i	$-0.693971\pi$
$659$	−1.27618e7	−1.14471	−0.572357	+	0.820004i	$0.693971\pi$
$660$	0	0
$661$	−6.44195e6	−0.573474	−0.286737	−	0.958009i	$-0.592571\pi$
$661$	−6.44195e6	−0.573474	−0.286737	+	0.958009i	$0.592571\pi$
$662$	−1.58718e7	−1.40760
$663$	0	0
$664$	2.46345e7	2.16832
$665$	1.05255e7	0.922972
$666$	0	0
$667$	2.70926e6	0.235796
$668$	5.59594e6	0.485213
$669$	0	0
$670$	−2.71463e7	−2.33627
$671$	−6.70496e6	−0.574897
$672$	0	0
$673$	8.64864e6	0.736055	0.368028	−	0.929815i	$-0.380033\pi$
$673$	8.64864e6	0.736055	0.368028	+	0.929815i	$0.380033\pi$
$674$	2.21836e7	1.88097
$675$	0	0
$676$	−1.68588e7	−1.41893
$677$	1.22020e7	1.02320	0.511600	−	0.859223i	$-0.329053\pi$
$677$	1.22020e7	1.02320	0.511600	+	0.859223i	$0.329053\pi$
$678$	0	0
$679$	−3.73923e6	−0.311249
$680$	−9.13242e6	−0.757380
$681$	0	0
$682$	−7.64110e6	−0.629064
$683$	1.10784e7	0.908708	0.454354	−	0.890821i	$-0.349870\pi$
$683$	1.10784e7	0.908708	0.454354	+	0.890821i	$0.349870\pi$
$684$	0	0
$685$	−1.59871e7	−1.30180
$686$	−2.14427e7	−1.73968
$687$	0	0
$688$	3.36466e6	0.271001
$689$	−3.77809e6	−0.303196
$690$	0	0
$691$	−1.24814e7	−0.994415	−0.497207	−	0.867632i	$-0.665641\pi$
$691$	−1.24814e7	−0.994415	−0.497207	+	0.867632i	$0.665641\pi$
$692$	−3.41122e6	−0.270797
$693$	0	0
$694$	3.72389e7	2.93493
$695$	2.95604e7	2.32139
$696$	0	0
$697$	−1.08957e7	−0.849518
$698$	1.96763e7	1.52864
$699$	0	0
$700$	−9.18622e6	−0.708585
$701$	4.71570e6	0.362452	0.181226	−	0.983441i	$-0.441993\pi$
$701$	4.71570e6	0.362452	0.181226	+	0.983441i	$0.441993\pi$
$702$	0	0
$703$	1.47159e6	0.112305
$704$	−2.19913e7	−1.67232
$705$	0	0
$706$	3.18903e7	2.40795
$707$	3.32839e6	0.250430
$708$	0	0
$709$	1.05958e7	0.791620	0.395810	−	0.918332i	$-0.370464\pi$
$709$	1.05958e7	0.791620	0.395810	+	0.918332i	$0.370464\pi$
$710$	3.00696e7	2.23863
$711$	0	0
$712$	3.01629e7	2.22984
$713$	989912.	0.0729244
$714$	0	0
$715$	7.97322e6	0.583268
$716$	3.38269e7	2.46593
$717$	0	0
$718$	−1.74603e7	−1.26398
$719$	−2.95783e6	−0.213379	−0.106689	−	0.994292i	$-0.534025\pi$
$719$	−2.95783e6	−0.213379	−0.106689	+	0.994292i	$0.534025\pi$
$720$	0	0
$721$	1.10637e7	0.792615
$722$	−2.72768e6	−0.194738
$723$	0	0
$724$	6.72499e6	0.476810
$725$	9.46510e6	0.668776
$726$	0	0
$727$	−1.30768e7	−0.917625	−0.458813	−	0.888533i	$-0.651725\pi$
$727$	−1.30768e7	−0.917625	−0.458813	+	0.888533i	$0.651725\pi$
$728$	−5.08173e6	−0.355372
$729$	0	0
$730$	−5.20731e6	−0.361665
$731$	−7.30005e6	−0.505280
$732$	0	0
$733$	5.06819e6	0.348412	0.174206	−	0.984709i	$-0.444264\pi$
$733$	5.06819e6	0.348412	0.174206	+	0.984709i	$0.444264\pi$
$734$	4.08751e6	0.280039
$735$	0	0
$736$	2.35509e6	0.160256
$737$	−1.79835e7	−1.21956
$738$	0	0
$739$	5.39832e6	0.363620	0.181810	−	0.983334i	$-0.441804\pi$
$739$	5.39832e6	0.363620	0.181810	+	0.983334i	$0.441804\pi$
$740$	−3.45605e6	−0.232007
$741$	0	0
$742$	1.22412e7	0.816233
$743$	−3.71690e6	−0.247007	−0.123503	−	0.992344i	$-0.539413\pi$
$743$	−3.71690e6	−0.247007	−0.123503	+	0.992344i	$0.539413\pi$
$744$	0	0
$745$	3.54908e7	2.34274
$746$	8.61736e6	0.566927
$747$	0	0
$748$	−1.43214e7	−0.935906
$749$	9.64017e6	0.627885
$750$	0	0
$751$	1.00711e7	0.651594	0.325797	−	0.945440i	$-0.394367\pi$
$751$	1.00711e7	0.651594	0.325797	+	0.945440i	$0.394367\pi$
$752$	−6.51157e6	−0.419896
$753$	0	0
$754$	1.23947e7	0.793979
$755$	−3.65255e7	−2.33200
$756$	0	0
$757$	−8.49996e6	−0.539110	−0.269555	−	0.962985i	$-0.586877\pi$
$757$	−8.49996e6	−0.539110	−0.269555	+	0.962985i	$0.586877\pi$
$758$	3.55397e7	2.24667
$759$	0	0
$760$	2.56725e7	1.61226
$761$	−2.62780e7	−1.64487	−0.822434	−	0.568861i	$-0.807384\pi$
$761$	−2.62780e7	−1.64487	−0.822434	+	0.568861i	$0.807384\pi$
$762$	0	0
$763$	−8.67102e6	−0.539211
$764$	1.63213e7	1.01163
$765$	0	0
$766$	−4.22869e7	−2.60396
$767$	1.23580e7	0.758506
$768$	0	0
$769$	899889.	0.0548748	0.0274374	−	0.999624i	$-0.491265\pi$
$769$	899889.	0.0548748	0.0274374	+	0.999624i	$0.491265\pi$
$770$	−2.58336e7	−1.57021
$771$	0	0
$772$	−492859.	−0.0297632
$773$	1.33758e7	0.805140	0.402570	−	0.915389i	$-0.368117\pi$
$773$	1.33758e7	0.805140	0.402570	+	0.915389i	$0.368117\pi$
$774$	0	0
$775$	3.45836e6	0.206831
$776$	−9.12028e6	−0.543693
$777$	0	0
$778$	4.18996e7	2.48177
$779$	3.06293e7	1.80840
$780$	0	0
$781$	1.99200e7	1.16859
$782$	2.92694e6	0.171158
$783$	0	0
$784$	−2.38911e6	−0.138818
$785$	2.80304e6	0.162351
$786$	0	0
$787$	1.67690e7	0.965098	0.482549	−	0.875869i	$-0.339711\pi$
$787$	1.67690e7	0.965098	0.482549	+	0.875869i	$0.339711\pi$
$788$	3.86791e7	2.21902
$789$	0	0
$790$	4.70972e7	2.68490
$791$	5.00498e6	0.284421
$792$	0	0
$793$	3.97396e6	0.224409
$794$	−1.65157e7	−0.929707
$795$	0	0
$796$	−2.95460e7	−1.65278
$797$	−2.78280e7	−1.55180	−0.775902	−	0.630854i	$-0.782705\pi$
$797$	−2.78280e7	−1.55180	−0.775902	+	0.630854i	$0.782705\pi$
$798$	0	0
$799$	1.41277e7	0.782894
$800$	8.22776e6	0.454524
$801$	0	0
$802$	−4.88370e6	−0.268110
$803$	−3.44966e6	−0.188794
$804$	0	0
$805$	3.34677e6	0.182027
$806$	4.52880e6	0.245553
$807$	0	0
$808$	8.11821e6	0.437454
$809$	1.98606e7	1.06689	0.533447	−	0.845834i	$-0.320897\pi$
$809$	1.98606e7	1.06689	0.533447	+	0.845834i	$0.320897\pi$
$810$	0	0
$811$	1.95245e7	1.04238	0.521192	−	0.853439i	$-0.325488\pi$
$811$	1.95245e7	1.04238	0.521192	+	0.853439i	$0.325488\pi$
$812$	−2.54568e7	−1.35492
$813$	0	0
$814$	−3.61185e6	−0.191059
$815$	1.11301e7	0.586953
$816$	0	0
$817$	2.05215e7	1.07561
$818$	1.50186e7	0.784775
$819$	0	0
$820$	−7.19335e7	−3.73591
$821$	2.39444e7	1.23978	0.619891	−	0.784688i	$-0.287177\pi$
$821$	2.39444e7	1.23978	0.619891	+	0.784688i	$0.287177\pi$
$822$	0	0
$823$	−2.78310e7	−1.43229	−0.716143	−	0.697953i	$-0.754094\pi$
$823$	−2.78310e7	−1.43229	−0.716143	+	0.697953i	$0.754094\pi$
$824$	2.69853e7	1.38455
$825$	0	0
$826$	−4.00405e7	−2.04197
$827$	1.94204e7	0.987405	0.493703	−	0.869631i	$-0.335643\pi$
$827$	1.94204e7	0.987405	0.493703	+	0.869631i	$0.335643\pi$
$828$	0	0
$829$	2.66039e7	1.34449	0.672247	−	0.740327i	$-0.265329\pi$
$829$	2.66039e7	1.34449	0.672247	+	0.740327i	$0.265329\pi$
$830$	−7.42241e7	−3.73981
$831$	0	0
$832$	1.30340e7	0.652784
$833$	5.18348e6	0.258826
$834$	0	0
$835$	−7.12257e6	−0.353525
$836$	4.02596e7	1.99229
$837$	0	0
$838$	−4.96117e7	−2.44047
$839$	1.41085e7	0.691954	0.345977	−	0.938243i	$-0.387547\pi$
$839$	1.41085e7	0.691954	0.345977	+	0.938243i	$0.387547\pi$
$840$	0	0
$841$	5.71843e6	0.278796
$842$	2.15566e7	1.04785
$843$	0	0
$844$	−3.64792e7	−1.76274
$845$	2.14581e7	1.03383
$846$	0	0
$847$	−2.66548e6	−0.127664
$848$	3.98111e6	0.190114
$849$	0	0
$850$	1.02256e7	0.485445
$851$	467918.	0.0221486
$852$	0	0
$853$	−1.87882e7	−0.884122	−0.442061	−	0.896985i	$-0.645752\pi$
$853$	−1.87882e7	−0.884122	−0.442061	+	0.896985i	$0.645752\pi$
$854$	−1.28758e7	−0.604130
$855$	0	0
$856$	2.35131e7	1.09680
$857$	−3.67369e7	−1.70864	−0.854320	−	0.519748i	$-0.826026\pi$
$857$	−3.67369e7	−1.70864	−0.854320	+	0.519748i	$0.826026\pi$
$858$	0	0
$859$	2.03228e6	0.0939723	0.0469861	−	0.998896i	$-0.485038\pi$
$859$	2.03228e6	0.0939723	0.0469861	+	0.998896i	$0.485038\pi$
$860$	−4.81950e7	−2.22206
$861$	0	0
$862$	−1.18818e7	−0.544646
$863$	1.21735e7	0.556400	0.278200	−	0.960523i	$-0.410262\pi$
$863$	1.21735e7	0.556400	0.278200	+	0.960523i	$0.410262\pi$
$864$	0	0
$865$	4.34183e6	0.197303
$866$	3.47674e7	1.57535
$867$	0	0
$868$	−9.30140e6	−0.419034
$869$	3.12003e7	1.40155
$870$	0	0
$871$	1.06586e7	0.476053
$872$	−2.11493e7	−0.941901
$873$	0	0
$874$	−8.22804e6	−0.364349
$875$	−8.07833e6	−0.356698
$876$	0	0
$877$	4.13453e7	1.81521	0.907605	−	0.419825i	$-0.137909\pi$
$877$	4.13453e7	1.81521	0.907605	+	0.419825i	$0.137909\pi$
$878$	3.28217e7	1.43689
$879$	0	0
$880$	−8.40166e6	−0.365728
$881$	−1.92417e7	−0.835227	−0.417613	−	0.908625i	$-0.637133\pi$
$881$	−1.92417e7	−0.835227	−0.417613	+	0.908625i	$0.637133\pi$
$882$	0	0
$883$	4.59010e6	0.198117	0.0990583	−	0.995082i	$-0.468417\pi$
$883$	4.59010e6	0.198117	0.0990583	+	0.995082i	$0.468417\pi$
$884$	8.48815e6	0.365328
$885$	0	0
$886$	2.49657e7	1.06846
$887$	1.34822e7	0.575374	0.287687	−	0.957724i	$-0.407114\pi$
$887$	1.34822e7	0.575374	0.287687	+	0.957724i	$0.407114\pi$
$888$	0	0
$889$	−1.50574e7	−0.638992
$890$	−9.08813e7	−3.84591
$891$	0	0
$892$	4.42663e6	0.186278
$893$	−3.97149e7	−1.66657
$894$	0	0
$895$	−4.30552e7	−1.79667
$896$	−2.94500e7	−1.22551
$897$	0	0
$898$	7.56126e6	0.312898
$899$	9.58377e6	0.395492
$900$	0	0
$901$	−8.63750e6	−0.354467
$902$	−7.51762e7	−3.07655
$903$	0	0
$904$	1.22076e7	0.496830
$905$	−8.55962e6	−0.347403
$906$	0	0
$907$	5.13039e6	0.207077	0.103539	−	0.994625i	$-0.466983\pi$
$907$	5.13039e6	0.207077	0.103539	+	0.994625i	$0.466983\pi$
$908$	−3.96763e7	−1.59704
$909$	0	0
$910$	1.53113e7	0.612927
$911$	−3.88471e7	−1.55082	−0.775411	−	0.631456i	$-0.782457\pi$
$911$	−3.88471e7	−1.55082	−0.775411	+	0.631456i	$0.782457\pi$
$912$	0	0
$913$	−4.91709e7	−1.95223
$914$	5.64558e7	2.23534
$915$	0	0
$916$	−3.22071e7	−1.26827
$917$	1.16384e7	0.457055
$918$	0	0
$919$	2.43482e7	0.950993	0.475496	−	0.879718i	$-0.342268\pi$
$919$	2.43482e7	0.950993	0.475496	+	0.879718i	$0.342268\pi$
$920$	8.16305e6	0.317968
$921$	0	0
$922$	−5.66866e7	−2.19610
$923$	−1.18064e7	−0.456156
$924$	0	0
$925$	1.63472e6	0.0628188
$926$	9.22927e6	0.353704
$927$	0	0
$928$	2.28007e7	0.869116
$929$	−2.17027e7	−0.825040	−0.412520	−	0.910948i	$-0.635351\pi$
$929$	−2.17027e7	−0.825040	−0.412520	+	0.910948i	$0.635351\pi$
$930$	0	0
$931$	−1.45715e7	−0.550972
$932$	−6.80010e6	−0.256434
$933$	0	0
$934$	−2.57912e7	−0.967397
$935$	1.82284e7	0.681900
$936$	0	0
$937$	−2.01218e6	−0.0748718	−0.0374359	−	0.999299i	$-0.511919\pi$
$937$	−2.01218e6	−0.0748718	−0.0374359	+	0.999299i	$0.511919\pi$
$938$	−3.45344e7	−1.28158
$939$	0	0
$940$	9.32710e7	3.44292
$941$	1.17136e7	0.431237	0.215618	−	0.976478i	$-0.430823\pi$
$941$	1.17136e7	0.431237	0.215618	+	0.976478i	$0.430823\pi$
$942$	0	0
$943$	9.73916e6	0.356650
$944$	−1.30220e7	−0.475608
$945$	0	0
$946$	−5.03676e7	−1.82989
$947$	−571455.	−0.0207065	−0.0103533	−	0.999946i	$-0.503296\pi$
$947$	−571455.	−0.0207065	−0.0103533	+	0.999946i	$0.503296\pi$
$948$	0	0
$949$	2.04458e6	0.0736950
$950$	−2.87455e7	−1.03338
$951$	0	0
$952$	−1.16179e7	−0.415466
$953$	−2.03678e7	−0.726461	−0.363231	−	0.931699i	$-0.618326\pi$
$953$	−2.03678e7	−0.726461	−0.363231	+	0.931699i	$0.618326\pi$
$954$	0	0
$955$	−2.07739e7	−0.737073
$956$	−1.16393e7	−0.411891
$957$	0	0
$958$	−1.34506e7	−0.473510
$959$	−2.03382e7	−0.714110
$960$	0	0
$961$	−2.51274e7	−0.877687
$962$	2.14070e6	0.0745794
$963$	0	0
$964$	6.82727e7	2.36622
$965$	627316.	0.0216854
$966$	0	0
$967$	−3.93183e7	−1.35216	−0.676081	−	0.736827i	$-0.736323\pi$
$967$	−3.93183e7	−1.35216	−0.676081	+	0.736827i	$0.736323\pi$
$968$	−6.50132e6	−0.223004
$969$	0	0
$970$	2.74795e7	0.937734
$971$	−4.86930e7	−1.65736	−0.828682	−	0.559719i	$-0.810909\pi$
$971$	−4.86930e7	−1.65736	−0.828682	+	0.559719i	$0.810909\pi$
$972$	0	0
$973$	3.76055e7	1.27341
$974$	8.92643e7	3.01495
$975$	0	0
$976$	−4.18750e6	−0.140712
$977$	1.11758e7	0.374577	0.187288	−	0.982305i	$-0.440030\pi$
$977$	1.11758e7	0.374577	0.187288	+	0.982305i	$0.440030\pi$
$978$	0	0
$979$	−6.02056e7	−2.00762
$980$	3.42214e7	1.13824
$981$	0	0
$982$	8.35930e7	2.76625
$983$	−2.09439e7	−0.691310	−0.345655	−	0.938362i	$-0.612343\pi$
$983$	−2.09439e7	−0.691310	−0.345655	+	0.938362i	$0.612343\pi$
$984$	0	0
$985$	−4.92311e7	−1.61677
$986$	2.83370e7	0.928242
$987$	0	0
$988$	−2.38614e7	−0.777685
$989$	6.52518e6	0.212130
$990$	0	0
$991$	−1.95125e7	−0.631144	−0.315572	−	0.948902i	$-0.602196\pi$
$991$	−1.95125e7	−0.631144	−0.315572	+	0.948902i	$0.602196\pi$
$992$	8.33092e6	0.268790
$993$	0	0
$994$	3.82533e7	1.22801
$995$	3.76064e7	1.20422
$996$	0	0
$997$	1.32696e7	0.422786	0.211393	−	0.977401i	$-0.432200\pi$
$997$	1.32696e7	0.422786	0.211393	+	0.977401i	$0.432200\pi$
$998$	−2.18894e7	−0.695678
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.f.1.1		5
3.2	odd	2		69.6.a.e.1.5	✓	5
12.11	even	2		1104.6.a.r.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.e.1.5	✓	5	3.2	odd	2
207.6.a.f.1.1		5	1.1	even	1	trivial
1104.6.a.r.1.4		5	12.11	even	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-9.34908 q^{2} +55.4053 q^{4} -70.5203 q^{5} -89.7132 q^{7} -218.818 q^{8} +O(q^{10})\)	\(q-9.34908 q^{2} +55.4053 q^{4} -70.5203 q^{5} -89.7132 q^{7} -218.818 q^{8} +659.300 q^{10} +436.763 q^{11} -258.865 q^{13} +838.736 q^{14} +272.775 q^{16} -591.819 q^{17} +1663.69 q^{19} -3907.20 q^{20} -4083.33 q^{22} +529.000 q^{23} +1848.12 q^{25} +2420.15 q^{26} -4970.58 q^{28} +5121.48 q^{29} +1871.29 q^{31} +4451.97 q^{32} +5532.96 q^{34} +6326.60 q^{35} +884.534 q^{37} -15554.0 q^{38} +15431.1 q^{40} +18410.5 q^{41} +12334.9 q^{43} +24199.0 q^{44} -4945.66 q^{46} -23871.6 q^{47} -8758.55 q^{49} -17278.2 q^{50} -14342.5 q^{52} +14594.8 q^{53} -30800.7 q^{55} +19630.8 q^{56} -47881.1 q^{58} -47739.1 q^{59} -15351.5 q^{61} -17494.8 q^{62} -50350.6 q^{64} +18255.2 q^{65} -41174.4 q^{67} -32789.9 q^{68} -59147.9 q^{70} +45608.3 q^{71} -7898.24 q^{73} -8269.57 q^{74} +92177.1 q^{76} -39183.4 q^{77} +71435.2 q^{79} -19236.2 q^{80} -172121. q^{82} -112580. q^{83} +41735.3 q^{85} -115320. q^{86} -95571.5 q^{88} -137845. q^{89} +23223.6 q^{91} +29309.4 q^{92} +223177. q^{94} -117324. q^{95} +41679.8 q^{97} +81884.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8}+O(q^{10}) \) 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8	\( 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8} - 172 q^{10} - 1100 q^{11} - 978 q^{13} + 344 q^{14} + 1218 q^{16} - 2522 q^{17} + 2060 q^{19} - 7720 q^{20} - 2572 q^{22} + 2645 q^{23} + 12035 q^{25} - 9280 q^{26} + 8072 q^{28} - 1526 q^{29} - 7392 q^{31} + 5086 q^{32} - 15608 q^{34} - 6056 q^{35} - 8210 q^{37} + 14276 q^{38} - 37472 q^{40} - 21250 q^{41} - 4548 q^{43} + 4260 q^{44} - 4232 q^{46} - 536 q^{47} - 27979 q^{49} + 81872 q^{50} - 76380 q^{52} + 11482 q^{53} - 77064 q^{55} + 28624 q^{56} - 79680 q^{58} - 74676 q^{59} - 44618 q^{61} - 64880 q^{62} - 137382 q^{64} + 24388 q^{65} - 1412 q^{67} - 80196 q^{68} - 222304 q^{70} - 37912 q^{71} + 46546 q^{73} - 111604 q^{74} - 79548 q^{76} - 157008 q^{77} + 50544 q^{79} - 69424 q^{80} - 233720 q^{82} - 89588 q^{83} + 147892 q^{85} - 77428 q^{86} + 54484 q^{88} - 280410 q^{89} - 27416 q^{91} + 62422 q^{92} + 113632 q^{94} - 203120 q^{95} + 90074 q^{97} - 32976 q^{98}+O(q^{100}) \) 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8 - 172 * q^10 - 1100 * q^11 - 978 * q^13 + 344 * q^14 + 1218 * q^16 - 2522 * q^17 + 2060 * q^19 - 7720 * q^20 - 2572 * q^22 + 2645 * q^23 + 12035 * q^25 - 9280 * q^26 + 8072 * q^28 - 1526 * q^29 - 7392 * q^31 + 5086 * q^32 - 15608 * q^34 - 6056 * q^35 - 8210 * q^37 + 14276 * q^38 - 37472 * q^40 - 21250 * q^41 - 4548 * q^43 + 4260 * q^44 - 4232 * q^46 - 536 * q^47 - 27979 * q^49 + 81872 * q^50 - 76380 * q^52 + 11482 * q^53 - 77064 * q^55 + 28624 * q^56 - 79680 * q^58 - 74676 * q^59 - 44618 * q^61 - 64880 * q^62 - 137382 * q^64 + 24388 * q^65 - 1412 * q^67 - 80196 * q^68 - 222304 * q^70 - 37912 * q^71 + 46546 * q^73 - 111604 * q^74 - 79548 * q^76 - 157008 * q^77 + 50544 * q^79 - 69424 * q^80 - 233720 * q^82 - 89588 * q^83 + 147892 * q^85 - 77428 * q^86 + 54484 * q^88 - 280410 * q^89 - 27416 * q^91 + 62422 * q^92 + 113632 * q^94 - 203120 * q^95 + 90074 * q^97 - 32976 * q^98

Introduction

L-functions

Modular forms

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Database

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