LMFDB - Embedded newform 207.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("207.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.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:	$33.1994507013$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.7925.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 12 $ x^3 - x^2 - 13*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.65736$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.31473 q^{2} -3.75366 q^{4} +23.8768 q^{5} -11.7843 q^{7} +190.021 q^{8} +O(q^{10})$	\(q-5.31473 q^{2} -3.75366 q^{4} +23.8768 q^{5} -11.7843 q^{7} +190.021 q^{8} -126.899 q^{10} -280.887 q^{11} -835.171 q^{13} +62.6305 q^{14} -889.793 q^{16} +1801.86 q^{17} +2357.32 q^{19} -89.6256 q^{20} +1492.84 q^{22} +529.000 q^{23} -2554.90 q^{25} +4438.71 q^{26} +44.2344 q^{28} +1206.86 q^{29} -3394.53 q^{31} -1351.66 q^{32} -9576.38 q^{34} -281.372 q^{35} +8351.42 q^{37} -12528.5 q^{38} +4537.10 q^{40} +10701.7 q^{41} -803.535 q^{43} +1054.36 q^{44} -2811.49 q^{46} +4896.03 q^{47} -16668.1 q^{49} +13578.6 q^{50} +3134.95 q^{52} -39358.6 q^{53} -6706.69 q^{55} -2239.27 q^{56} -6414.13 q^{58} -39514.6 q^{59} -15309.3 q^{61} +18041.0 q^{62} +35657.1 q^{64} -19941.2 q^{65} -60783.9 q^{67} -6763.56 q^{68} +1495.42 q^{70} -39125.6 q^{71} -5872.67 q^{73} -44385.5 q^{74} -8848.60 q^{76} +3310.07 q^{77} -63656.1 q^{79} -21245.4 q^{80} -56876.4 q^{82} +63639.8 q^{83} +43022.6 q^{85} +4270.57 q^{86} -53374.5 q^{88} -36262.8 q^{89} +9841.93 q^{91} -1985.69 q^{92} -26021.1 q^{94} +56285.4 q^{95} -64187.5 q^{97} +88586.6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8}+O(q^{10}) $ 3 * q + 4 * q^2 + 16 * q^4 + 58 * q^5 - 282 * q^7 + 360 * q^8	$ 3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8} - 156 q^{10} - 136 q^{11} - 1116 q^{13} - 2016 q^{14} + 128 q^{16} + 896 q^{17} + 1654 q^{19} - 1504 q^{20} + 1352 q^{22} + 1587 q^{23} - 7347 q^{25} - 1998 q^{26} - 10264 q^{28} + 844 q^{29} - 3020 q^{31} - 6656 q^{32} - 11212 q^{34} - 1072 q^{35} + 8938 q^{37} - 10728 q^{38} + 3440 q^{40} + 12792 q^{41} - 16730 q^{43} - 5112 q^{44} + 2116 q^{46} - 22500 q^{47} + 2887 q^{49} - 15156 q^{50} - 47412 q^{52} - 17108 q^{53} - 436 q^{55} - 42640 q^{56} - 55678 q^{58} - 54176 q^{59} - 71324 q^{61} + 72710 q^{62} - 49984 q^{64} - 846 q^{65} - 62960 q^{67} + 8352 q^{68} + 9224 q^{70} - 98400 q^{71} - 81772 q^{73} + 59044 q^{74} + 31488 q^{76} + 304 q^{77} + 58224 q^{79} + 17568 q^{80} + 61926 q^{82} - 9892 q^{83} + 15536 q^{85} - 191140 q^{86} - 58400 q^{88} - 27542 q^{89} + 151974 q^{91} + 8464 q^{92} - 146990 q^{94} + 20644 q^{95} - 273672 q^{97} + 401276 q^{98}+O(q^{100}) $ 3 * q + 4 * q^2 + 16 * q^4 + 58 * q^5 - 282 * q^7 + 360 * q^8 - 156 * q^10 - 136 * q^11 - 1116 * q^13 - 2016 * q^14 + 128 * q^16 + 896 * q^17 + 1654 * q^19 - 1504 * q^20 + 1352 * q^22 + 1587 * q^23 - 7347 * q^25 - 1998 * q^26 - 10264 * q^28 + 844 * q^29 - 3020 * q^31 - 6656 * q^32 - 11212 * q^34 - 1072 * q^35 + 8938 * q^37 - 10728 * q^38 + 3440 * q^40 + 12792 * q^41 - 16730 * q^43 - 5112 * q^44 + 2116 * q^46 - 22500 * q^47 + 2887 * q^49 - 15156 * q^50 - 47412 * q^52 - 17108 * q^53 - 436 * q^55 - 42640 * q^56 - 55678 * q^58 - 54176 * q^59 - 71324 * q^61 + 72710 * q^62 - 49984 * q^64 - 846 * q^65 - 62960 * q^67 + 8352 * q^68 + 9224 * q^70 - 98400 * q^71 - 81772 * q^73 + 59044 * q^74 + 31488 * q^76 + 304 * q^77 + 58224 * q^79 + 17568 * q^80 + 61926 * q^82 - 9892 * q^83 + 15536 * q^85 - 191140 * q^86 - 58400 * q^88 - 27542 * q^89 + 151974 * q^91 + 8464 * q^92 - 146990 * q^94 + 20644 * q^95 - 273672 * q^97 + 401276 * 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 $

$2$	−5.31473	−0.939520	−0.469760	−	0.882794i	$-0.655660\pi$
$2$	−5.31473	−0.939520	−0.469760	+	0.882794i	$0.655660\pi$
$3$	0	0
$3$	0	0
$4$	−3.75366	−0.117302
$5$	23.8768	0.427122	0.213561	−	0.976930i	$-0.431494\pi$
$5$	23.8768	0.427122	0.213561	+	0.976930i	$0.431494\pi$
$6$	0	0
$7$	−11.7843	−0.0908991	−0.0454496	−	0.998967i	$-0.514472\pi$
$7$	−11.7843	−0.0908991	−0.0454496	+	0.998967i	$0.514472\pi$
$8$	190.021	1.04973
$9$	0	0
$10$	−126.899	−0.401289
$11$	−280.887	−0.699923	−0.349961	−	0.936764i	$-0.613805\pi$
$11$	−280.887	−0.699923	−0.349961	+	0.936764i	$0.613805\pi$
$12$	0	0
$13$	−835.171	−1.37062	−0.685310	−	0.728251i	$-0.740333\pi$
$13$	−835.171	−1.37062	−0.685310	+	0.728251i	$0.740333\pi$
$14$	62.6305	0.0854015
$15$	0	0
$16$	−889.793	−0.868938
$17$	1801.86	1.51216	0.756080	−	0.654479i	$-0.227112\pi$
$17$	1801.86	1.51216	0.756080	+	0.654479i	$0.227112\pi$
$18$	0	0
$19$	2357.32	1.49808	0.749041	−	0.662524i	$-0.230515\pi$
$19$	2357.32	1.49808	0.749041	+	0.662524i	$0.230515\pi$
$20$	−89.6256	−0.0501022
$21$	0	0
$22$	1492.84	0.657592
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−2554.90	−0.817567
$26$	4438.71	1.28773
$27$	0	0
$28$	44.2344	0.0106626
$29$	1206.86	0.266478	0.133239	−	0.991084i	$-0.457462\pi$
$29$	1206.86	0.266478	0.133239	+	0.991084i	$0.457462\pi$
$30$	0	0
$31$	−3394.53	−0.634418	−0.317209	−	0.948356i	$-0.602746\pi$
$31$	−3394.53	−0.634418	−0.317209	+	0.948356i	$0.602746\pi$
$32$	−1351.66	−0.233343
$33$	0	0
$34$	−9576.38	−1.42071
$35$	−281.372	−0.0388250
$36$	0	0
$37$	8351.42	1.00290	0.501448	−	0.865188i	$-0.332801\pi$
$37$	8351.42	1.00290	0.501448	+	0.865188i	$0.332801\pi$
$38$	−12528.5	−1.40748
$39$	0	0
$40$	4537.10	0.448362
$41$	10701.7	0.994240	0.497120	−	0.867682i	$-0.334391\pi$
$41$	10701.7	0.994240	0.497120	+	0.867682i	$0.334391\pi$
$42$	0	0
$43$	−803.535	−0.0662726	−0.0331363	−	0.999451i	$-0.510550\pi$
$43$	−803.535	−0.0662726	−0.0331363	+	0.999451i	$0.510550\pi$
$44$	1054.36	0.0821023
$45$	0	0
$46$	−2811.49	−0.195903
$47$	4896.03	0.323295	0.161648	−	0.986849i	$-0.448319\pi$
$47$	4896.03	0.323295	0.161648	+	0.986849i	$0.448319\pi$
$48$	0	0
$49$	−16668.1	−0.991737
$50$	13578.6	0.768121
$51$	0	0
$52$	3134.95	0.160776
$53$	−39358.6	−1.92464	−0.962322	−	0.271913i	$-0.912344\pi$
$53$	−39358.6	−1.92464	−0.962322	+	0.271913i	$0.912344\pi$
$54$	0	0
$55$	−6706.69	−0.298952
$56$	−2239.27	−0.0954193
$57$	0	0
$58$	−6414.13	−0.250362
$59$	−39514.6	−1.47784	−0.738921	−	0.673792i	$-0.764664\pi$
$59$	−39514.6	−1.47784	−0.738921	+	0.673792i	$0.764664\pi$
$60$	0	0
$61$	−15309.3	−0.526783	−0.263392	−	0.964689i	$-0.584841\pi$
$61$	−15309.3	−0.526783	−0.263392	+	0.964689i	$0.584841\pi$
$62$	18041.0	0.596048
$63$	0	0
$64$	35657.1	1.08817
$65$	−19941.2	−0.585422
$66$	0	0
$67$	−60783.9	−1.65425	−0.827126	−	0.562016i	$-0.810026\pi$
$67$	−60783.9	−1.65425	−0.827126	+	0.562016i	$0.810026\pi$
$68$	−6763.56	−0.177379
$69$	0	0
$70$	1495.42	0.0364769
$71$	−39125.6	−0.921118	−0.460559	−	0.887629i	$-0.652351\pi$
$71$	−39125.6	−0.921118	−0.460559	+	0.887629i	$0.652351\pi$
$72$	0	0
$73$	−5872.67	−0.128982	−0.0644909	−	0.997918i	$-0.520542\pi$
$73$	−5872.67	−0.128982	−0.0644909	+	0.997918i	$0.520542\pi$
$74$	−44385.5	−0.942241
$75$	0	0
$76$	−8848.60	−0.175728
$77$	3310.07	0.0636224
$78$	0	0
$79$	−63656.1	−1.14755	−0.573776	−	0.819012i	$-0.694522\pi$
$79$	−63656.1	−1.14755	−0.573776	+	0.819012i	$0.694522\pi$
$80$	−21245.4	−0.371142
$81$	0	0
$82$	−56876.4	−0.934109
$83$	63639.8	1.01399	0.506995	−	0.861949i	$-0.330756\pi$
$83$	63639.8	1.01399	0.506995	+	0.861949i	$0.330756\pi$
$84$	0	0
$85$	43022.6	0.645877
$86$	4270.57	0.0622644
$87$	0	0
$88$	−53374.5	−0.734728
$89$	−36262.8	−0.485273	−0.242637	−	0.970117i	$-0.578012\pi$
$89$	−36262.8	−0.485273	−0.242637	+	0.970117i	$0.578012\pi$
$90$	0	0
$91$	9841.93	0.124588
$92$	−1985.69	−0.0244591
$93$	0	0
$94$	−26021.1	−0.303742
$95$	56285.4	0.639863
$96$	0	0
$97$	−64187.5	−0.692662	−0.346331	−	0.938112i	$-0.612573\pi$
$97$	−64187.5	−0.692662	−0.346331	+	0.938112i	$0.612573\pi$
$98$	88586.6	0.931757
$99$	0	0
$100$	9590.22	0.0959022
$101$	108356.	1.05693	0.528467	−	0.848954i	$-0.322767\pi$
$101$	108356.	1.05693	0.528467	+	0.848954i	$0.322767\pi$
$102$	0	0
$103$	−88102.3	−0.818265	−0.409133	−	0.912475i	$-0.634169\pi$
$103$	−88102.3	−0.818265	−0.409133	+	0.912475i	$0.634169\pi$
$104$	−158700.	−1.43878
$105$	0	0
$106$	209180.	1.80824
$107$	16813.2	0.141968	0.0709842	−	0.997477i	$-0.477386\pi$
$107$	16813.2	0.141968	0.0709842	+	0.997477i	$0.477386\pi$
$108$	0	0
$109$	−192169.	−1.54923	−0.774616	−	0.632432i	$-0.782057\pi$
$109$	−192169.	−1.54923	−0.774616	+	0.632432i	$0.782057\pi$
$110$	35644.3	0.280872
$111$	0	0
$112$	10485.6	0.0789857
$113$	−24975.1	−0.183997	−0.0919985	−	0.995759i	$-0.529326\pi$
$113$	−24975.1	−0.183997	−0.0919985	+	0.995759i	$0.529326\pi$
$114$	0	0
$115$	12630.8	0.0890610
$116$	−4530.14	−0.0312584
$117$	0	0
$118$	210010.	1.38846
$119$	−21233.7	−0.137454
$120$	0	0
$121$	−82153.4	−0.510108
$122$	81365.0	0.494923
$123$	0	0
$124$	12741.9	0.0744185
$125$	−135618.	−0.776322
$126$	0	0
$127$	215290.	1.18445	0.592223	−	0.805774i	$-0.298251\pi$
$127$	215290.	1.18445	0.592223	+	0.805774i	$0.298251\pi$
$128$	−146255.	−0.789013
$129$	0	0
$130$	105982.	0.550015
$131$	−34626.7	−0.176292	−0.0881461	−	0.996108i	$-0.528094\pi$
$131$	−34626.7	−0.176292	−0.0881461	+	0.996108i	$0.528094\pi$
$132$	0	0
$133$	−27779.5	−0.136174
$134$	323050.	1.55420
$135$	0	0
$136$	342391.	1.58736
$137$	−172235.	−0.784007	−0.392003	−	0.919964i	$-0.628218\pi$
$137$	−172235.	−0.784007	−0.392003	+	0.919964i	$0.628218\pi$
$138$	0	0
$139$	153367.	0.673278	0.336639	−	0.941634i	$-0.390710\pi$
$139$	153367.	0.673278	0.336639	+	0.941634i	$0.390710\pi$
$140$	1056.18	0.00455425
$141$	0	0
$142$	207942.	0.865409
$143$	234589.	0.959328
$144$	0	0
$145$	28816.0	0.113819
$146$	31211.7	0.121181
$147$	0	0
$148$	−31348.4	−0.117642
$149$	28131.3	0.103807	0.0519033	−	0.998652i	$-0.483471\pi$
$149$	28131.3	0.103807	0.0519033	+	0.998652i	$0.483471\pi$
$150$	0	0
$151$	160593.	0.573172	0.286586	−	0.958055i	$-0.407480\pi$
$151$	160593.	0.573172	0.286586	+	0.958055i	$0.407480\pi$
$152$	447941.	1.57258
$153$	0	0
$154$	−17592.1	−0.0597745
$155$	−81050.6	−0.270974
$156$	0	0
$157$	242015.	0.783599	0.391800	−	0.920051i	$-0.371853\pi$
$157$	242015.	0.783599	0.391800	+	0.920051i	$0.371853\pi$
$158$	338315.	1.07815
$159$	0	0
$160$	−32273.5	−0.0996657
$161$	−6233.91	−0.0189538
$162$	0	0
$163$	−136177.	−0.401453	−0.200727	−	0.979647i	$-0.564330\pi$
$163$	−136177.	−0.401453	−0.200727	+	0.979647i	$0.564330\pi$
$164$	−40170.4	−0.116626
$165$	0	0
$166$	−338228.	−0.952664
$167$	9384.75	0.0260394	0.0130197	−	0.999915i	$-0.495856\pi$
$167$	9384.75	0.0260394	0.0130197	+	0.999915i	$0.495856\pi$
$168$	0	0
$169$	326218.	0.878600
$170$	−228654.	−0.606814
$171$	0	0
$172$	3016.20	0.00777390
$173$	398962.	1.01348	0.506741	−	0.862098i	$-0.330850\pi$
$173$	398962.	1.01348	0.506741	+	0.862098i	$0.330850\pi$
$174$	0	0
$175$	30107.7	0.0743161
$176$	249931.	0.608190
$177$	0	0
$178$	192727.	0.455924
$179$	−456085.	−1.06393	−0.531965	−	0.846766i	$-0.678546\pi$
$179$	−456085.	−1.06393	−0.531965	+	0.846766i	$0.678546\pi$
$180$	0	0
$181$	−120816.	−0.274111	−0.137056	−	0.990563i	$-0.543764\pi$
$181$	−120816.	−0.274111	−0.137056	+	0.990563i	$0.543764\pi$
$182$	−52307.2	−0.117053
$183$	0	0
$184$	100521.	0.218883
$185$	199405.	0.428359
$186$	0	0
$187$	−506118.	−1.05840
$188$	−18378.0	−0.0379232
$189$	0	0
$190$	−299142.	−0.601164
$191$	−416377.	−0.825854	−0.412927	−	0.910764i	$-0.635494\pi$
$191$	−416377.	−0.825854	−0.412927	+	0.910764i	$0.635494\pi$
$192$	0	0
$193$	−381178.	−0.736605	−0.368302	−	0.929706i	$-0.620061\pi$
$193$	−381178.	−0.736605	−0.368302	+	0.929706i	$0.620061\pi$
$194$	341139.	0.650770
$195$	0	0
$196$	62566.5	0.116333
$197$	−617955.	−1.13447	−0.567233	−	0.823558i	$-0.691986\pi$
$197$	−617955.	−1.13447	−0.567233	+	0.823558i	$0.691986\pi$
$198$	0	0
$199$	455121.	0.814693	0.407346	−	0.913274i	$-0.366454\pi$
$199$	455121.	0.814693	0.407346	+	0.913274i	$0.366454\pi$
$200$	−485484.	−0.858223
$201$	0	0
$202$	−575881.	−0.993011
$203$	−14222.0	−0.0242226
$204$	0	0
$205$	255522.	0.424662
$206$	468240.	0.768776
$207$	0	0
$208$	743129.	1.19098
$209$	−662142.	−1.04854
$210$	0	0
$211$	−870625.	−1.34625	−0.673124	−	0.739530i	$-0.735048\pi$
$211$	−870625.	−1.34625	−0.673124	+	0.739530i	$0.735048\pi$
$212$	147739.	0.225764
$213$	0	0
$214$	−89357.8	−0.133382
$215$	−19185.9	−0.0283065
$216$	0	0
$217$	40002.3	0.0576680
$218$	1.02132e6	1.45553
$219$	0	0
$220$	25174.7	0.0350677
$221$	−1.50486e6	−2.07260
$222$	0	0
$223$	−135558.	−0.182541	−0.0912707	−	0.995826i	$-0.529093\pi$
$223$	−135558.	−0.182541	−0.0912707	+	0.995826i	$0.529093\pi$
$224$	15928.5	0.0212106
$225$	0	0
$226$	132736.	0.172869
$227$	443206.	0.570875	0.285438	−	0.958397i	$-0.407861\pi$
$227$	443206.	0.570875	0.285438	+	0.958397i	$0.407861\pi$
$228$	0	0
$229$	689970.	0.869444	0.434722	−	0.900565i	$-0.356847\pi$
$229$	689970.	0.869444	0.434722	+	0.900565i	$0.356847\pi$
$230$	−67129.5	−0.0836746
$231$	0	0
$232$	229329.	0.279730
$233$	−487389.	−0.588147	−0.294074	−	0.955783i	$-0.595011\pi$
$233$	−487389.	−0.588147	−0.294074	+	0.955783i	$0.595011\pi$
$234$	0	0
$235$	116902.	0.138086
$236$	148325.	0.173354
$237$	0	0
$238$	112851.	0.129141
$239$	1.12530e6	1.27430	0.637150	−	0.770740i	$-0.280113\pi$
$239$	1.12530e6	1.27430	0.637150	+	0.770740i	$0.280113\pi$
$240$	0	0
$241$	−1.48985e6	−1.65235	−0.826173	−	0.563416i	$-0.809487\pi$
$241$	−1.48985e6	−1.65235	−0.826173	+	0.563416i	$0.809487\pi$
$242$	436623.	0.479257
$243$	0	0
$244$	57466.1	0.0617927
$245$	−397982.	−0.423593
$246$	0	0
$247$	−1.96877e6	−2.05330
$248$	−645032.	−0.665966
$249$	0	0
$250$	720773.	0.729371
$251$	−63456.5	−0.0635758	−0.0317879	−	0.999495i	$-0.510120\pi$
$251$	−63456.5	−0.0635758	−0.0317879	+	0.999495i	$0.510120\pi$
$252$	0	0
$253$	−148589.	−0.145944
$254$	−1.14421e6	−1.11281
$255$	0	0
$256$	−363724.	−0.346874
$257$	−1.08874e6	−1.02823	−0.514116	−	0.857720i	$-0.671880\pi$
$257$	−1.08874e6	−1.02823	−0.514116	+	0.857720i	$0.671880\pi$
$258$	0	0
$259$	−98415.8	−0.0911624
$260$	74852.7	0.0686711
$261$	0	0
$262$	184032.	0.165630
$263$	542790.	0.483885	0.241943	−	0.970291i	$-0.422215\pi$
$263$	542790.	0.483885	0.241943	+	0.970291i	$0.422215\pi$
$264$	0	0
$265$	−939760.	−0.822057
$266$	147640.	0.127938
$267$	0	0
$268$	228162.	0.194047
$269$	1.66169e6	1.40013	0.700065	−	0.714079i	$-0.253154\pi$
$269$	1.66169e6	1.40013	0.700065	+	0.714079i	$0.253154\pi$
$270$	0	0
$271$	621380.	0.513966	0.256983	−	0.966416i	$-0.417272\pi$
$271$	621380.	0.513966	0.256983	+	0.966416i	$0.417272\pi$
$272$	−1.60328e6	−1.31397
$273$	0	0
$274$	915382.	0.736590
$275$	717638.	0.572234
$276$	0	0
$277$	266442.	0.208643	0.104321	−	0.994544i	$-0.466733\pi$
$277$	266442.	0.208643	0.104321	+	0.994544i	$0.466733\pi$
$278$	−815103.	−0.632558
$279$	0	0
$280$	−53466.7	−0.0407557
$281$	−472574.	−0.357029	−0.178515	−	0.983937i	$-0.557129\pi$
$281$	−472574.	−0.357029	−0.178515	+	0.983937i	$0.557129\pi$
$282$	0	0
$283$	914756.	0.678952	0.339476	−	0.940615i	$-0.389750\pi$
$283$	914756.	0.678952	0.339476	+	0.940615i	$0.389750\pi$
$284$	146864.	0.108049
$285$	0	0
$286$	−1.24678e6	−0.901308
$287$	−126112.	−0.0903755
$288$	0	0
$289$	1.82683e6	1.28663
$290$	−153149.	−0.106935
$291$	0	0
$292$	22044.0	0.0151298
$293$	906414.	0.616819	0.308409	−	0.951254i	$-0.400203\pi$
$293$	906414.	0.616819	0.308409	+	0.951254i	$0.400203\pi$
$294$	0	0
$295$	−943484.	−0.631218
$296$	1.58695e6	1.05277
$297$	0	0
$298$	−149510.	−0.0975283
$299$	−441806.	−0.285794
$300$	0	0
$301$	9469.12	0.00602412
$302$	−853509.	−0.538506
$303$	0	0
$304$	−2.09753e6	−1.30174
$305$	−365539.	−0.225001
$306$	0	0
$307$	102614.	0.0621387	0.0310694	−	0.999517i	$-0.490109\pi$
$307$	102614.	0.0621387	0.0310694	+	0.999517i	$0.490109\pi$
$308$	−12424.9	−0.00746303
$309$	0	0
$310$	430762.	0.254585
$311$	−1.21303e6	−0.711168	−0.355584	−	0.934644i	$-0.615718\pi$
$311$	−1.21303e6	−0.711168	−0.355584	+	0.934644i	$0.615718\pi$
$312$	0	0
$313$	−3.04190e6	−1.75503	−0.877514	−	0.479551i	$-0.840800\pi$
$313$	−3.04190e6	−1.75503	−0.877514	+	0.479551i	$0.840800\pi$
$314$	−1.28625e6	−0.736207
$315$	0	0
$316$	238944.	0.134610
$317$	2.10095e6	1.17427	0.587133	−	0.809490i	$-0.300256\pi$
$317$	2.10095e6	1.17427	0.587133	+	0.809490i	$0.300256\pi$
$318$	0	0
$319$	−338991.	−0.186514
$320$	851379.	0.464780
$321$	0	0
$322$	33131.5	0.0178075
$323$	4.24756e6	2.26534
$324$	0	0
$325$	2.13378e6	1.12057
$326$	723744.	0.377173
$327$	0	0
$328$	2.03354e6	1.04368
$329$	−57696.4	−0.0293873
$330$	0	0
$331$	−2.91623e6	−1.46303	−0.731513	−	0.681827i	$-0.761186\pi$
$331$	−2.91623e6	−1.46303	−0.731513	+	0.681827i	$0.761186\pi$
$332$	−238882.	−0.118943
$333$	0	0
$334$	−49877.4	−0.0244646
$335$	−1.45133e6	−0.706567
$336$	0	0
$337$	−2.93330e6	−1.40696	−0.703480	−	0.710715i	$-0.748372\pi$
$337$	−2.93330e6	−1.40696	−0.703480	+	0.710715i	$0.748372\pi$
$338$	−1.73376e6	−0.825462
$339$	0	0
$340$	−161492.	−0.0757626
$341$	953480.	0.444044
$342$	0	0
$343$	394482.	0.181047
$344$	−152689.	−0.0695682
$345$	0	0
$346$	−2.12038e6	−0.952188
$347$	1.48233e6	0.660877	0.330438	−	0.943828i	$-0.392803\pi$
$347$	1.48233e6	0.660877	0.330438	+	0.943828i	$0.392803\pi$
$348$	0	0
$349$	3.14342e6	1.38146	0.690731	−	0.723112i	$-0.257289\pi$
$349$	3.14342e6	1.38146	0.690731	+	0.723112i	$0.257289\pi$
$350$	−160014.	−0.0698215
$351$	0	0
$352$	379665.	0.163322
$353$	−2.99556e6	−1.27950	−0.639751	−	0.768582i	$-0.720963\pi$
$353$	−2.99556e6	−1.27950	−0.639751	+	0.768582i	$0.720963\pi$
$354$	0	0
$355$	−934196.	−0.393430
$356$	136118.	0.0569235
$357$	0	0
$358$	2.42397e6	0.999584
$359$	2.23362e6	0.914687	0.457344	−	0.889290i	$-0.348801\pi$
$359$	2.23362e6	0.914687	0.457344	+	0.889290i	$0.348801\pi$
$360$	0	0
$361$	3.08088e6	1.24425
$362$	642102.	0.257533
$363$	0	0
$364$	−36943.3	−0.0146144
$365$	−140221.	−0.0550910
$366$	0	0
$367$	−923000.	−0.357715	−0.178857	−	0.983875i	$-0.557240\pi$
$367$	−923000.	−0.357715	−0.178857	+	0.983875i	$0.557240\pi$
$368$	−470700.	−0.181186
$369$	0	0
$370$	−1.05979e6	−0.402452
$371$	463815.	0.174948
$372$	0	0
$373$	−2.31000e6	−0.859688	−0.429844	−	0.902903i	$-0.641431\pi$
$373$	−2.31000e6	−0.859688	−0.429844	+	0.902903i	$0.641431\pi$
$374$	2.68988e6	0.994384
$375$	0	0
$376$	930348.	0.339372
$377$	−1.00793e6	−0.365240
$378$	0	0
$379$	−1.89297e6	−0.676934	−0.338467	−	0.940978i	$-0.609908\pi$
$379$	−1.89297e6	−0.676934	−0.338467	+	0.940978i	$0.609908\pi$
$380$	−211277.	−0.0750572
$381$	0	0
$382$	2.21293e6	0.775907
$383$	−2.56252e6	−0.892629	−0.446315	−	0.894876i	$-0.647264\pi$
$383$	−2.56252e6	−0.892629	−0.446315	+	0.894876i	$0.647264\pi$
$384$	0	0
$385$	79033.9	0.0271745
$386$	2.02586e6	0.692055
$387$	0	0
$388$	240938.	0.0812506
$389$	−5.33851e6	−1.78873	−0.894367	−	0.447334i	$-0.852373\pi$
$389$	−5.33851e6	−1.78873	−0.894367	+	0.447334i	$0.852373\pi$
$390$	0	0
$391$	953182.	0.315307
$392$	−3.16729e6	−1.04105
$393$	0	0
$394$	3.28426e6	1.06585
$395$	−1.51991e6	−0.490145
$396$	0	0
$397$	2.26002e6	0.719676	0.359838	−	0.933015i	$-0.382832\pi$
$397$	2.26002e6	0.719676	0.359838	+	0.933015i	$0.382832\pi$
$398$	−2.41884e6	−0.765420
$399$	0	0
$400$	2.27333e6	0.710415
$401$	5.78253e6	1.79580	0.897899	−	0.440202i	$-0.145093\pi$
$401$	5.78253e6	1.79580	0.897899	+	0.440202i	$0.145093\pi$
$402$	0	0
$403$	2.83501e6	0.869546
$404$	−406730.	−0.123980
$405$	0	0
$406$	75586.2	0.0227577
$407$	−2.34581e6	−0.701950
$408$	0	0
$409$	−1.80868e6	−0.534629	−0.267315	−	0.963609i	$-0.586136\pi$
$409$	−1.80868e6	−0.534629	−0.267315	+	0.963609i	$0.586136\pi$
$410$	−1.35803e6	−0.398978
$411$	0	0
$412$	330706.	0.0959841
$413$	465653.	0.134335
$414$	0	0
$415$	1.51952e6	0.433097
$416$	1.12887e6	0.319824
$417$	0	0
$418$	3.51911e6	0.985126
$419$	2.44304e6	0.679822	0.339911	−	0.940458i	$-0.389603\pi$
$419$	2.44304e6	0.679822	0.339911	+	0.940458i	$0.389603\pi$
$420$	0	0
$421$	−66516.0	−0.0182903	−0.00914515	−	0.999958i	$-0.502911\pi$
$421$	−66516.0	−0.0182903	−0.00914515	+	0.999958i	$0.502911\pi$
$422$	4.62713e6	1.26483
$423$	0	0
$424$	−7.47897e6	−2.02035
$425$	−4.60356e6	−1.23629
$426$	0	0
$427$	180410.	0.0478841
$428$	−63111.2	−0.0166532
$429$	0	0
$430$	101968.	0.0265945
$431$	583815.	0.151385	0.0756924	−	0.997131i	$-0.475883\pi$
$431$	583815.	0.151385	0.0756924	+	0.997131i	$0.475883\pi$
$432$	0	0
$433$	2.28085e6	0.584624	0.292312	−	0.956323i	$-0.405575\pi$
$433$	2.28085e6	0.584624	0.292312	+	0.956323i	$0.405575\pi$
$434$	−212601.	−0.0541803
$435$	0	0
$436$	721337.	0.181728
$437$	1.24702e6	0.312372
$438$	0	0
$439$	6.19515e6	1.53423	0.767116	−	0.641509i	$-0.221691\pi$
$439$	6.19515e6	1.53423	0.767116	+	0.641509i	$0.221691\pi$
$440$	−1.27441e6	−0.313818
$441$	0	0
$442$	7.99791e6	1.94725
$443$	1.01675e6	0.246153	0.123076	−	0.992397i	$-0.460724\pi$
$443$	1.01675e6	0.246153	0.123076	+	0.992397i	$0.460724\pi$
$444$	0	0
$445$	−865841.	−0.207271
$446$	720451.	0.171501
$447$	0	0
$448$	−420195.	−0.0989135
$449$	3.04664e6	0.713191	0.356595	−	0.934259i	$-0.383938\pi$
$449$	3.04664e6	0.713191	0.356595	+	0.934259i	$0.383938\pi$
$450$	0	0
$451$	−3.00596e6	−0.695891
$452$	93748.0	0.0215832
$453$	0	0
$454$	−2.35552e6	−0.536349
$455$	234994.	0.0532143
$456$	0	0
$457$	4.24293e6	0.950333	0.475166	−	0.879896i	$-0.342388\pi$
$457$	4.24293e6	0.950333	0.475166	+	0.879896i	$0.342388\pi$
$458$	−3.66700e6	−0.816860
$459$	0	0
$460$	−47411.9	−0.0104470
$461$	−1.08445e6	−0.237662	−0.118831	−	0.992915i	$-0.537915\pi$
$461$	−1.08445e6	−0.237662	−0.118831	+	0.992915i	$0.537915\pi$
$462$	0	0
$463$	932916.	0.202251	0.101125	−	0.994874i	$-0.467756\pi$
$463$	932916.	0.202251	0.101125	+	0.994874i	$0.467756\pi$
$464$	−1.07386e6	−0.231553
$465$	0	0
$466$	2.59034e6	0.552576
$467$	252067.	0.0534839	0.0267420	−	0.999642i	$-0.491487\pi$
$467$	252067.	0.0534839	0.0267420	+	0.999642i	$0.491487\pi$
$468$	0	0
$469$	716298.	0.150370
$470$	−621301.	−0.129735
$471$	0	0
$472$	−7.50861e6	−1.55133
$473$	225703.	0.0463857
$474$	0	0
$475$	−6.02272e6	−1.22478
$476$	79704.0	0.0161236
$477$	0	0
$478$	−5.98064e6	−1.19723
$479$	638652.	0.127182	0.0635909	−	0.997976i	$-0.479745\pi$
$479$	638652.	0.127182	0.0635909	+	0.997976i	$0.479745\pi$
$480$	0	0
$481$	−6.97487e6	−1.37459
$482$	7.91817e6	1.55241
$483$	0	0
$484$	308376.	0.0598367
$485$	−1.53259e6	−0.295851
$486$	0	0
$487$	−6.40830e6	−1.22439	−0.612196	−	0.790706i	$-0.709714\pi$
$487$	−6.40830e6	−1.22439	−0.612196	+	0.790706i	$0.709714\pi$
$488$	−2.90910e6	−0.552979
$489$	0	0
$490$	2.11517e6	0.397974
$491$	8.07458e6	1.51153	0.755764	−	0.654844i	$-0.227266\pi$
$491$	8.07458e6	1.51153	0.755764	+	0.654844i	$0.227266\pi$
$492$	0	0
$493$	2.17459e6	0.402958
$494$	1.04635e7	1.92912
$495$	0	0
$496$	3.02043e6	0.551270
$497$	461069.	0.0837288
$498$	0	0
$499$	7.25360e6	1.30407	0.652037	−	0.758187i	$-0.273915\pi$
$499$	7.25360e6	1.30407	0.652037	+	0.758187i	$0.273915\pi$
$500$	509064.	0.0910641
$501$	0	0
$502$	337254.	0.0597308
$503$	4.02697e6	0.709673	0.354836	−	0.934928i	$-0.384537\pi$
$503$	4.02697e6	0.709673	0.354836	+	0.934928i	$0.384537\pi$
$504$	0	0
$505$	2.58719e6	0.451440
$506$	789712.	0.137117
$507$	0	0
$508$	−808127.	−0.138938
$509$	98493.6	0.0168505	0.00842526	−	0.999965i	$-0.497318\pi$
$509$	98493.6	0.0168505	0.00842526	+	0.999965i	$0.497318\pi$
$510$	0	0
$511$	69205.5	0.0117243
$512$	6.61324e6	1.11491
$513$	0	0
$514$	5.78636e6	0.966045
$515$	−2.10360e6	−0.349499
$516$	0	0
$517$	−1.37523e6	−0.226282
$518$	523053.	0.0856489
$519$	0	0
$520$	−3.78925e6	−0.614533
$521$	7.76729e6	1.25365	0.626824	−	0.779161i	$-0.284355\pi$
$521$	7.76729e6	1.25365	0.626824	+	0.779161i	$0.284355\pi$
$522$	0	0
$523$	1.05753e7	1.69058	0.845292	−	0.534305i	$-0.179427\pi$
$523$	1.05753e7	1.69058	0.845292	+	0.534305i	$0.179427\pi$
$524$	129977.	0.0206794
$525$	0	0
$526$	−2.88478e6	−0.454620
$527$	−6.11646e6	−0.959342
$528$	0	0
$529$	279841.	0.0434783
$530$	4.99457e6	0.772339
$531$	0	0
$532$	104275.	0.0159735
$533$	−8.93771e6	−1.36273
$534$	0	0
$535$	401447.	0.0606378
$536$	−1.15502e7	−1.73651
$537$	0	0
$538$	−8.83141e6	−1.31545
$539$	4.68186e6	0.694140
$540$	0	0
$541$	−2.72554e6	−0.400369	−0.200184	−	0.979758i	$-0.564154\pi$
$541$	−2.72554e6	−0.400369	−0.200184	+	0.979758i	$0.564154\pi$
$542$	−3.30247e6	−0.482881
$543$	0	0
$544$	−2.43551e6	−0.352852
$545$	−4.58838e6	−0.661711
$546$	0	0
$547$	−1.12681e7	−1.61020	−0.805101	−	0.593137i	$-0.797889\pi$
$547$	−1.12681e7	−1.61020	−0.805101	+	0.593137i	$0.797889\pi$
$548$	646512.	0.0919655
$549$	0	0
$550$	−3.81405e6	−0.537625
$551$	2.84496e6	0.399206
$552$	0	0
$553$	750145.	0.104311
$554$	−1.41607e6	−0.196024
$555$	0	0
$556$	−575687.	−0.0789768
$557$	−1.42068e7	−1.94025	−0.970125	−	0.242607i	$-0.921998\pi$
$557$	−1.42068e7	−1.94025	−0.970125	+	0.242607i	$0.921998\pi$
$558$	0	0
$559$	671090.	0.0908345
$560$	250363.	0.0337365
$561$	0	0
$562$	2.51160e6	0.335436
$563$	−9.93624e6	−1.32115	−0.660574	−	0.750761i	$-0.729687\pi$
$563$	−9.93624e6	−1.32115	−0.660574	+	0.750761i	$0.729687\pi$
$564$	0	0
$565$	−596326.	−0.0785892
$566$	−4.86168e6	−0.637889
$567$	0	0
$568$	−7.43469e6	−0.966923
$569$	2.43903e6	0.315818	0.157909	−	0.987454i	$-0.449525\pi$
$569$	2.43903e6	0.315818	0.157909	+	0.987454i	$0.449525\pi$
$570$	0	0
$571$	1.54691e7	1.98552	0.992759	−	0.120124i	$-0.0383292\pi$
$571$	1.54691e7	1.98552	0.992759	+	0.120124i	$0.0383292\pi$
$572$	−880567.	−0.112531
$573$	0	0
$574$	670250.	0.0849096
$575$	−1.35154e6	−0.170475
$576$	0	0
$577$	−1.30892e7	−1.63672	−0.818360	−	0.574706i	$-0.805116\pi$
$577$	−1.30892e7	−1.63672	−0.818360	+	0.574706i	$0.805116\pi$
$578$	−9.70910e6	−1.20881
$579$	0	0
$580$	−108165.	−0.0133511
$581$	−749952.	−0.0921708
$582$	0	0
$583$	1.10553e7	1.34710
$584$	−1.11593e6	−0.135396
$585$	0	0
$586$	−4.81734e6	−0.579514
$587$	−2.37887e6	−0.284954	−0.142477	−	0.989798i	$-0.545507\pi$
$587$	−2.37887e6	−0.284954	−0.142477	+	0.989798i	$0.545507\pi$
$588$	0	0
$589$	−8.00201e6	−0.950410
$590$	5.01436e6	0.593042
$591$	0	0
$592$	−7.43103e6	−0.871455
$593$	3.47325e6	0.405601	0.202800	−	0.979220i	$-0.434996\pi$
$593$	3.47325e6	0.405601	0.202800	+	0.979220i	$0.434996\pi$
$594$	0	0
$595$	−506993.	−0.0587096
$596$	−105596.	−0.0121767
$597$	0	0
$598$	2.34808e6	0.268509
$599$	−5.31390e6	−0.605127	−0.302564	−	0.953129i	$-0.597842\pi$
$599$	−5.31390e6	−0.605127	−0.302564	+	0.953129i	$0.597842\pi$
$600$	0	0
$601$	−1.35246e6	−0.152734	−0.0763672	−	0.997080i	$-0.524332\pi$
$601$	−1.35246e6	−0.152734	−0.0763672	+	0.997080i	$0.524332\pi$
$602$	−50325.8	−0.00565978
$603$	0	0
$604$	−602813.	−0.0672341
$605$	−1.96156e6	−0.217878
$606$	0	0
$607$	−384207.	−0.0423247	−0.0211623	−	0.999776i	$-0.506737\pi$
$607$	−384207.	−0.0423247	−0.0211623	+	0.999776i	$0.506737\pi$
$608$	−3.18631e6	−0.349566
$609$	0	0
$610$	1.94274e6	0.211393
$611$	−4.08902e6	−0.443115
$612$	0	0
$613$	−1.67749e7	−1.80306	−0.901530	−	0.432718i	$-0.857555\pi$
$613$	−1.67749e7	−1.80306	−0.901530	+	0.432718i	$0.857555\pi$
$614$	−545368.	−0.0583806
$615$	0	0
$616$	628982.	0.0667862
$617$	−1.27567e6	−0.134905	−0.0674523	−	0.997723i	$-0.521487\pi$
$617$	−1.27567e6	−0.134905	−0.0674523	+	0.997723i	$0.521487\pi$
$618$	0	0
$619$	1.46795e7	1.53988	0.769938	−	0.638118i	$-0.220287\pi$
$619$	1.46795e7	1.53988	0.769938	+	0.638118i	$0.220287\pi$
$620$	304237.	0.0317857
$621$	0	0
$622$	6.44695e6	0.668157
$623$	427333.	0.0441109
$624$	0	0
$625$	4.74593e6	0.485983
$626$	1.61669e7	1.64888
$627$	0	0
$628$	−908444.	−0.0919177
$629$	1.50481e7	1.51654
$630$	0	0
$631$	−1.65066e7	−1.65038	−0.825190	−	0.564856i	$-0.808932\pi$
$631$	−1.65066e7	−1.65038	−0.825190	+	0.564856i	$0.808932\pi$
$632$	−1.20960e7	−1.20462
$633$	0	0
$634$	−1.11660e7	−1.10325
$635$	5.14045e6	0.505902
$636$	0	0
$637$	1.39207e7	1.35930
$638$	1.80165e6	0.175234
$639$	0	0
$640$	−3.49209e6	−0.337005
$641$	−1.87653e6	−0.180389	−0.0901947	−	0.995924i	$-0.528749\pi$
$641$	−1.87653e6	−0.180389	−0.0901947	+	0.995924i	$0.528749\pi$
$642$	0	0
$643$	1.72531e6	0.164566	0.0822830	−	0.996609i	$-0.473779\pi$
$643$	1.72531e6	0.164566	0.0822830	+	0.996609i	$0.473779\pi$
$644$	23400.0	0.00222331
$645$	0	0
$646$	−2.25746e7	−2.12833
$647$	−1.18002e7	−1.10822	−0.554112	−	0.832442i	$-0.686942\pi$
$647$	−1.18002e7	−1.10822	−0.554112	+	0.832442i	$0.686942\pi$
$648$	0	0
$649$	1.10992e7	1.03438
$650$	−1.13404e7	−1.05280
$651$	0	0
$652$	511163.	0.0470912
$653$	−1.00372e7	−0.921147	−0.460573	−	0.887622i	$-0.652356\pi$
$653$	−1.00372e7	−0.921147	−0.460573	+	0.887622i	$0.652356\pi$
$654$	0	0
$655$	−826776.	−0.0752982
$656$	−9.52226e6	−0.863933
$657$	0	0
$658$	306641.	0.0276099
$659$	−1.02223e7	−0.916929	−0.458464	−	0.888713i	$-0.651600\pi$
$659$	−1.02223e7	−0.916929	−0.458464	+	0.888713i	$0.651600\pi$
$660$	0	0
$661$	−1.87007e6	−0.166477	−0.0832386	−	0.996530i	$-0.526526\pi$
$661$	−1.87007e6	−0.166477	−0.0832386	+	0.996530i	$0.526526\pi$
$662$	1.54990e7	1.37454
$663$	0	0
$664$	1.20929e7	1.06441
$665$	−663286.	−0.0581630
$666$	0	0
$667$	638429.	0.0555646
$668$	−35227.2	−0.00305448
$669$	0	0
$670$	7.71341e6	0.663834
$671$	4.30020e6	0.368708
$672$	0	0
$673$	−2.14057e6	−0.182176	−0.0910881	−	0.995843i	$-0.529034\pi$
$673$	−2.14057e6	−0.182176	−0.0910881	+	0.995843i	$0.529034\pi$
$674$	1.55897e7	1.32187
$675$	0	0
$676$	−1.22451e6	−0.103061
$677$	−1.42893e7	−1.19822	−0.599112	−	0.800665i	$-0.704480\pi$
$677$	−1.42893e7	−1.19822	−0.599112	+	0.800665i	$0.704480\pi$
$678$	0	0
$679$	756407.	0.0629624
$680$	8.17520e6	0.677995
$681$	0	0
$682$	−5.06749e6	−0.417188
$683$	7.29980e6	0.598769	0.299384	−	0.954133i	$-0.403219\pi$
$683$	7.29980e6	0.598769	0.299384	+	0.954133i	$0.403219\pi$
$684$	0	0
$685$	−4.11242e6	−0.334866
$686$	−2.09656e6	−0.170097
$687$	0	0
$688$	714980.	0.0575868
$689$	3.28712e7	2.63796
$690$	0	0
$691$	3.77617e6	0.300855	0.150427	−	0.988621i	$-0.451935\pi$
$691$	3.77617e6	0.300855	0.150427	+	0.988621i	$0.451935\pi$
$692$	−1.49757e6	−0.118884
$693$	0	0
$694$	−7.87817e6	−0.620907
$695$	3.66191e6	0.287572
$696$	0	0
$697$	1.92828e7	1.50345
$698$	−1.67064e7	−1.29791
$699$	0	0
$700$	−113014.	−0.00871743
$701$	1.27732e7	0.981761	0.490880	−	0.871227i	$-0.336675\pi$
$701$	1.27732e7	0.981761	0.490880	+	0.871227i	$0.336675\pi$
$702$	0	0
$703$	1.96870e7	1.50242
$704$	−1.00156e7	−0.761634
$705$	0	0
$706$	1.59206e7	1.20212
$707$	−1.27690e6	−0.0960744
$708$	0	0
$709$	−2.21287e7	−1.65326	−0.826628	−	0.562748i	$-0.809744\pi$
$709$	−2.21287e7	−1.65326	−0.826628	+	0.562748i	$0.809744\pi$
$710$	4.96500e6	0.369635
$711$	0	0
$712$	−6.89069e6	−0.509405
$713$	−1.79571e6	−0.132285
$714$	0	0
$715$	5.60124e6	0.409750
$716$	1.71199e6	0.124801
$717$	0	0
$718$	−1.18711e7	−0.859367
$719$	6.86509e6	0.495249	0.247625	−	0.968856i	$-0.420350\pi$
$719$	6.86509e6	0.495249	0.247625	+	0.968856i	$0.420350\pi$
$720$	0	0
$721$	1.03823e6	0.0743796
$722$	−1.63740e7	−1.16900
$723$	0	0
$724$	453501.	0.0321538
$725$	−3.08340e6	−0.217864
$726$	0	0
$727$	4.75733e6	0.333831	0.166916	−	0.985971i	$-0.446619\pi$
$727$	4.75733e6	0.333831	0.166916	+	0.985971i	$0.446619\pi$
$728$	1.87017e6	0.130784
$729$	0	0
$730$	745236.	0.0517591
$731$	−1.44786e6	−0.100215
$732$	0	0
$733$	2.41232e7	1.65835	0.829173	−	0.558993i	$-0.188812\pi$
$733$	2.41232e7	1.65835	0.829173	+	0.558993i	$0.188812\pi$
$734$	4.90550e6	0.336080
$735$	0	0
$736$	−715031.	−0.0486553
$737$	1.70734e7	1.15785
$738$	0	0
$739$	−1.01851e7	−0.686050	−0.343025	−	0.939326i	$-0.611452\pi$
$739$	−1.01851e7	−0.686050	−0.343025	+	0.939326i	$0.611452\pi$
$740$	−748501.	−0.0502473
$741$	0	0
$742$	−2.46505e6	−0.164368
$743$	1.55515e7	1.03348	0.516738	−	0.856144i	$-0.327146\pi$
$743$	1.55515e7	1.03348	0.516738	+	0.856144i	$0.327146\pi$
$744$	0	0
$745$	671687.	0.0443380
$746$	1.22770e7	0.807694
$747$	0	0
$748$	1.89980e6	0.124152
$749$	−198133.	−0.0129048
$750$	0	0
$751$	−1.03320e7	−0.668473	−0.334237	−	0.942489i	$-0.608478\pi$
$751$	−1.03320e7	−0.668473	−0.334237	+	0.942489i	$0.608478\pi$
$752$	−4.35645e6	−0.280924
$753$	0	0
$754$	5.35690e6	0.343151
$755$	3.83446e6	0.244814
$756$	0	0
$757$	−1.61676e7	−1.02543	−0.512714	−	0.858559i	$-0.671360\pi$
$757$	−1.61676e7	−1.02543	−0.512714	+	0.858559i	$0.671360\pi$
$758$	1.00606e7	0.635993
$759$	0	0
$760$	1.06954e7	0.671682
$761$	−8.40313e6	−0.525993	−0.262996	−	0.964797i	$-0.584711\pi$
$761$	−8.40313e6	−0.525993	−0.262996	+	0.964797i	$0.584711\pi$
$762$	0	0
$763$	2.26458e6	0.140824
$764$	1.56294e6	0.0968743
$765$	0	0
$766$	1.36191e7	0.838643
$767$	3.30015e7	2.02556
$768$	0	0
$769$	−1.74997e7	−1.06713	−0.533563	−	0.845760i	$-0.679147\pi$
$769$	−1.74997e7	−1.06713	−0.533563	+	0.845760i	$0.679147\pi$
$770$	−420044.	−0.0255310
$771$	0	0
$772$	1.43081e6	0.0864052
$773$	2.11605e7	1.27373	0.636863	−	0.770977i	$-0.280232\pi$
$773$	2.11605e7	1.27373	0.636863	+	0.770977i	$0.280232\pi$
$774$	0	0
$775$	8.67268e6	0.518679
$776$	−1.21970e7	−0.727106
$777$	0	0
$778$	2.83727e7	1.68055
$779$	2.52273e7	1.48945
$780$	0	0
$781$	1.09899e7	0.644712
$782$	−5.06590e6	−0.296237
$783$	0	0
$784$	1.48312e7	0.861759
$785$	5.77856e6	0.334692
$786$	0	0
$787$	−1.19600e7	−0.688325	−0.344163	−	0.938910i	$-0.611837\pi$
$787$	−1.19600e7	−0.688325	−0.344163	+	0.938910i	$0.611837\pi$
$788$	2.31959e6	0.133075
$789$	0	0
$790$	8.07789e6	0.460501
$791$	294315.	0.0167252
$792$	0	0
$793$	1.27859e7	0.722020
$794$	−1.20114e7	−0.676150
$795$	0	0
$796$	−1.70837e6	−0.0955650
$797$	−1.32264e7	−0.737559	−0.368779	−	0.929517i	$-0.620224\pi$
$797$	−1.32264e7	−0.737559	−0.368779	+	0.929517i	$0.620224\pi$
$798$	0	0
$799$	8.82194e6	0.488874
$800$	3.45336e6	0.190773
$801$	0	0
$802$	−3.07326e7	−1.68719
$803$	1.64956e6	0.0902773
$804$	0	0
$805$	−148846.	−0.00809557
$806$	−1.50673e7	−0.816956
$807$	0	0
$808$	2.05898e7	1.10949
$809$	−2.81536e6	−0.151239	−0.0756194	−	0.997137i	$-0.524093\pi$
$809$	−2.81536e6	−0.151239	−0.0756194	+	0.997137i	$0.524093\pi$
$810$	0	0
$811$	2.03162e7	1.08465	0.542325	−	0.840169i	$-0.317544\pi$
$811$	2.03162e7	1.08465	0.542325	+	0.840169i	$0.317544\pi$
$812$	53384.7	0.00284136
$813$	0	0
$814$	1.24673e7	0.659496
$815$	−3.25148e6	−0.171469
$816$	0	0
$817$	−1.89419e6	−0.0992817
$818$	9.61263e6	0.502295
$819$	0	0
$820$	−959142.	−0.0498136
$821$	−2.06237e7	−1.06785	−0.533924	−	0.845532i	$-0.679283\pi$
$821$	−2.06237e7	−1.06785	−0.533924	+	0.845532i	$0.679283\pi$
$822$	0	0
$823$	3.05683e7	1.57316	0.786578	−	0.617491i	$-0.211851\pi$
$823$	3.05683e7	1.57316	0.786578	+	0.617491i	$0.211851\pi$
$824$	−1.67413e7	−0.858955
$825$	0	0
$826$	−2.47482e6	−0.126210
$827$	−3.23053e7	−1.64252	−0.821258	−	0.570556i	$-0.806728\pi$
$827$	−3.23053e7	−1.64252	−0.821258	+	0.570556i	$0.806728\pi$
$828$	0	0
$829$	1.30561e7	0.659823	0.329912	−	0.944012i	$-0.392981\pi$
$829$	1.30561e7	0.659823	0.329912	+	0.944012i	$0.392981\pi$
$830$	−8.07582e6	−0.406904
$831$	0	0
$832$	−2.97798e7	−1.49147
$833$	−3.00336e7	−1.49967
$834$	0	0
$835$	224078.	0.0111220
$836$	2.48546e6	0.122996
$837$	0	0
$838$	−1.29841e7	−0.638707
$839$	6.50241e6	0.318911	0.159455	−	0.987205i	$-0.449026\pi$
$839$	6.50241e6	0.318911	0.159455	+	0.987205i	$0.449026\pi$
$840$	0	0
$841$	−1.90546e7	−0.928989
$842$	353514.	0.0171841
$843$	0	0
$844$	3.26803e6	0.157917
$845$	7.78905e6	0.375269
$846$	0	0
$847$	968123.	0.0463684
$848$	3.50210e7	1.67240
$849$	0	0
$850$	2.44667e7	1.16152
$851$	4.41790e6	0.209118
$852$	0	0
$853$	−3.18228e7	−1.49750	−0.748749	−	0.662854i	$-0.769345\pi$
$853$	−3.18228e7	−1.49750	−0.748749	+	0.662854i	$0.769345\pi$
$854$	−958832.	−0.0449881
$855$	0	0
$856$	3.19487e6	0.149028
$857$	1.82430e7	0.848487	0.424243	−	0.905548i	$-0.360540\pi$
$857$	1.82430e7	0.848487	0.424243	+	0.905548i	$0.360540\pi$
$858$	0	0
$859$	2.66125e7	1.23056	0.615281	−	0.788308i	$-0.289043\pi$
$859$	2.66125e7	1.23056	0.615281	+	0.788308i	$0.289043\pi$
$860$	72017.3	0.00332040
$861$	0	0
$862$	−3.10282e6	−0.142229
$863$	−4.88600e6	−0.223320	−0.111660	−	0.993746i	$-0.535617\pi$
$863$	−4.88600e6	−0.223320	−0.111660	+	0.993746i	$0.535617\pi$
$864$	0	0
$865$	9.52595e6	0.432881
$866$	−1.21221e7	−0.549266
$867$	0	0
$868$	−150155.	−0.00676457
$869$	1.78802e7	0.803198
$870$	0	0
$871$	5.07650e7	2.26735
$872$	−3.65161e7	−1.62627
$873$	0	0
$874$	−6.62760e6	−0.293479
$875$	1.59817e6	0.0705670
$876$	0	0
$877$	−2.36006e7	−1.03615	−0.518077	−	0.855334i	$-0.673352\pi$
$877$	−2.36006e7	−1.03615	−0.518077	+	0.855334i	$0.673352\pi$
$878$	−3.29256e7	−1.44144
$879$	0	0
$880$	5.96757e6	0.259771
$881$	−7.11920e6	−0.309023	−0.154512	−	0.987991i	$-0.549380\pi$
$881$	−7.11920e6	−0.309023	−0.154512	+	0.987991i	$0.549380\pi$
$882$	0	0
$883$	−3.69545e7	−1.59502	−0.797509	−	0.603307i	$-0.793849\pi$
$883$	−3.69545e7	−1.59502	−0.797509	+	0.603307i	$0.793849\pi$
$884$	5.64873e6	0.243120
$885$	0	0
$886$	−5.40374e6	−0.231265
$887$	−2.40116e6	−0.102474	−0.0512368	−	0.998687i	$-0.516316\pi$
$887$	−2.40116e6	−0.102474	−0.0512368	+	0.998687i	$0.516316\pi$
$888$	0	0
$889$	−2.53705e6	−0.107665
$890$	4.60171e6	0.194735
$891$	0	0
$892$	508837.	0.0214125
$893$	1.15415e7	0.484323
$894$	0	0
$895$	−1.08899e7	−0.454428
$896$	1.72351e6	0.0717206
$897$	0	0
$898$	−1.61921e7	−0.670057
$899$	−4.09672e6	−0.169059
$900$	0	0
$901$	−7.09186e7	−2.91037
$902$	1.59758e7	0.653804
$903$	0	0
$904$	−4.74579e6	−0.193147
$905$	−2.88469e6	−0.117079
$906$	0	0
$907$	1.34643e7	0.543458	0.271729	−	0.962374i	$-0.412405\pi$
$907$	1.34643e7	0.543458	0.271729	+	0.962374i	$0.412405\pi$
$908$	−1.66365e6	−0.0669648
$909$	0	0
$910$	−1.24893e6	−0.0499959
$911$	−3.07348e7	−1.22697	−0.613485	−	0.789707i	$-0.710233\pi$
$911$	−3.07348e7	−1.22697	−0.613485	+	0.789707i	$0.710233\pi$
$912$	0	0
$913$	−1.78756e7	−0.709715
$914$	−2.25500e7	−0.892857
$915$	0	0
$916$	−2.58991e6	−0.101987
$917$	408052.	0.0160248
$918$	0	0
$919$	2.52416e7	0.985888	0.492944	−	0.870061i	$-0.335921\pi$
$919$	2.52416e7	0.985888	0.492944	+	0.870061i	$0.335921\pi$
$920$	2.40013e6	0.0934898
$921$	0	0
$922$	5.76358e6	0.223288
$923$	3.26766e7	1.26250
$924$	0	0
$925$	−2.13370e7	−0.819935
$926$	−4.95820e6	−0.190019
$927$	0	0
$928$	−1.63127e6	−0.0621807
$929$	−1.43072e7	−0.543897	−0.271948	−	0.962312i	$-0.587668\pi$
$929$	−1.43072e7	−0.543897	−0.271948	+	0.962312i	$0.587668\pi$
$930$	0	0
$931$	−3.92922e7	−1.48570
$932$	1.82949e6	0.0689908
$933$	0	0
$934$	−1.33967e6	−0.0502492
$935$	−1.20845e7	−0.452064
$936$	0	0
$937$	−1.37068e7	−0.510020	−0.255010	−	0.966938i	$-0.582079\pi$
$937$	−1.37068e7	−0.510020	−0.255010	+	0.966938i	$0.582079\pi$
$938$	−3.80693e6	−0.141276
$939$	0	0
$940$	−438809.	−0.0161978
$941$	−7.51582e6	−0.276696	−0.138348	−	0.990384i	$-0.544179\pi$
$941$	−7.51582e6	−0.276696	−0.138348	+	0.990384i	$0.544179\pi$
$942$	0	0
$943$	5.66118e6	0.207313
$944$	3.51598e7	1.28415
$945$	0	0
$946$	−1.19955e6	−0.0435803
$947$	−1.59343e7	−0.577375	−0.288688	−	0.957423i	$-0.593219\pi$
$947$	−1.59343e7	−0.577375	−0.288688	+	0.957423i	$0.593219\pi$
$948$	0	0
$949$	4.90469e6	0.176785
$950$	3.20091e7	1.15071
$951$	0	0
$952$	−4.03484e6	−0.144289
$953$	−2.00561e7	−0.715343	−0.357672	−	0.933847i	$-0.616429\pi$
$953$	−2.00561e7	−0.715343	−0.357672	+	0.933847i	$0.616429\pi$
$954$	0	0
$955$	−9.94177e6	−0.352740
$956$	−4.22398e6	−0.149478
$957$	0	0
$958$	−3.39426e6	−0.119490
$959$	2.02967e6	0.0712655
$960$	0	0
$961$	−1.71063e7	−0.597514
$962$	3.70695e7	1.29145
$963$	0	0
$964$	5.59241e6	0.193823
$965$	−9.10132e6	−0.314620
$966$	0	0
$967$	1.66228e7	0.571661	0.285831	−	0.958280i	$-0.407731\pi$
$967$	1.66228e7	0.571661	0.285831	+	0.958280i	$0.407731\pi$
$968$	−1.56109e7	−0.535475
$969$	0	0
$970$	8.14533e6	0.277958
$971$	5.74792e7	1.95642	0.978212	−	0.207609i	$-0.0665680\pi$
$971$	5.74792e7	1.95642	0.978212	+	0.207609i	$0.0665680\pi$
$972$	0	0
$973$	−1.80732e6	−0.0612004
$974$	3.40584e7	1.15034
$975$	0	0
$976$	1.36221e7	0.457742
$977$	−2.65537e7	−0.889996	−0.444998	−	0.895532i	$-0.646796\pi$
$977$	−2.65537e7	−0.889996	−0.444998	+	0.895532i	$0.646796\pi$
$978$	0	0
$979$	1.01858e7	0.339654
$980$	1.49389e6	0.0496882
$981$	0	0
$982$	−4.29142e7	−1.42011
$983$	−2.70907e6	−0.0894205	−0.0447102	−	0.999000i	$-0.514236\pi$
$983$	−2.70907e6	−0.0894205	−0.0447102	+	0.999000i	$0.514236\pi$
$984$	0	0
$985$	−1.47548e7	−0.484555
$986$	−1.15573e7	−0.378587
$987$	0	0
$988$	7.39010e6	0.240856
$989$	−425070.	−0.0138188
$990$	0	0
$991$	2.13175e7	0.689530	0.344765	−	0.938689i	$-0.387959\pi$
$991$	2.13175e7	0.689530	0.344765	+	0.938689i	$0.387959\pi$
$992$	4.58827e6	0.148037
$993$	0	0
$994$	−2.45046e6	−0.0786649
$995$	1.08668e7	0.347973
$996$	0	0
$997$	4.42548e7	1.41001	0.705005	−	0.709202i	$-0.250945\pi$
$997$	4.42548e7	1.41001	0.705005	+	0.709202i	$0.250945\pi$
$998$	−3.85509e7	−1.22520
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.b.1.1		3
3.2	odd	2		23.6.a.a.1.3	✓	3
12.11	even	2		368.6.a.e.1.3		3
15.14	odd	2		575.6.a.b.1.1		3
69.68	even	2		529.6.a.a.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.6.a.a.1.3	✓	3	3.2	odd	2
207.6.a.b.1.1		3	1.1	even	1	trivial
368.6.a.e.1.3		3	12.11	even	2
529.6.a.a.1.3		3	69.68	even	2
575.6.a.b.1.1		3	15.14	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-5.31473 q^{2} -3.75366 q^{4} +23.8768 q^{5} -11.7843 q^{7} +190.021 q^{8} +O(q^{10})\)	\(q-5.31473 q^{2} -3.75366 q^{4} +23.8768 q^{5} -11.7843 q^{7} +190.021 q^{8} -126.899 q^{10} -280.887 q^{11} -835.171 q^{13} +62.6305 q^{14} -889.793 q^{16} +1801.86 q^{17} +2357.32 q^{19} -89.6256 q^{20} +1492.84 q^{22} +529.000 q^{23} -2554.90 q^{25} +4438.71 q^{26} +44.2344 q^{28} +1206.86 q^{29} -3394.53 q^{31} -1351.66 q^{32} -9576.38 q^{34} -281.372 q^{35} +8351.42 q^{37} -12528.5 q^{38} +4537.10 q^{40} +10701.7 q^{41} -803.535 q^{43} +1054.36 q^{44} -2811.49 q^{46} +4896.03 q^{47} -16668.1 q^{49} +13578.6 q^{50} +3134.95 q^{52} -39358.6 q^{53} -6706.69 q^{55} -2239.27 q^{56} -6414.13 q^{58} -39514.6 q^{59} -15309.3 q^{61} +18041.0 q^{62} +35657.1 q^{64} -19941.2 q^{65} -60783.9 q^{67} -6763.56 q^{68} +1495.42 q^{70} -39125.6 q^{71} -5872.67 q^{73} -44385.5 q^{74} -8848.60 q^{76} +3310.07 q^{77} -63656.1 q^{79} -21245.4 q^{80} -56876.4 q^{82} +63639.8 q^{83} +43022.6 q^{85} +4270.57 q^{86} -53374.5 q^{88} -36262.8 q^{89} +9841.93 q^{91} -1985.69 q^{92} -26021.1 q^{94} +56285.4 q^{95} -64187.5 q^{97} +88586.6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8}+O(q^{10}) \) 3 * q + 4 * q^2 + 16 * q^4 + 58 * q^5 - 282 * q^7 + 360 * q^8	\( 3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8} - 156 q^{10} - 136 q^{11} - 1116 q^{13} - 2016 q^{14} + 128 q^{16} + 896 q^{17} + 1654 q^{19} - 1504 q^{20} + 1352 q^{22} + 1587 q^{23} - 7347 q^{25} - 1998 q^{26} - 10264 q^{28} + 844 q^{29} - 3020 q^{31} - 6656 q^{32} - 11212 q^{34} - 1072 q^{35} + 8938 q^{37} - 10728 q^{38} + 3440 q^{40} + 12792 q^{41} - 16730 q^{43} - 5112 q^{44} + 2116 q^{46} - 22500 q^{47} + 2887 q^{49} - 15156 q^{50} - 47412 q^{52} - 17108 q^{53} - 436 q^{55} - 42640 q^{56} - 55678 q^{58} - 54176 q^{59} - 71324 q^{61} + 72710 q^{62} - 49984 q^{64} - 846 q^{65} - 62960 q^{67} + 8352 q^{68} + 9224 q^{70} - 98400 q^{71} - 81772 q^{73} + 59044 q^{74} + 31488 q^{76} + 304 q^{77} + 58224 q^{79} + 17568 q^{80} + 61926 q^{82} - 9892 q^{83} + 15536 q^{85} - 191140 q^{86} - 58400 q^{88} - 27542 q^{89} + 151974 q^{91} + 8464 q^{92} - 146990 q^{94} + 20644 q^{95} - 273672 q^{97} + 401276 q^{98}+O(q^{100}) \) 3 * q + 4 * q^2 + 16 * q^4 + 58 * q^5 - 282 * q^7 + 360 * q^8 - 156 * q^10 - 136 * q^11 - 1116 * q^13 - 2016 * q^14 + 128 * q^16 + 896 * q^17 + 1654 * q^19 - 1504 * q^20 + 1352 * q^22 + 1587 * q^23 - 7347 * q^25 - 1998 * q^26 - 10264 * q^28 + 844 * q^29 - 3020 * q^31 - 6656 * q^32 - 11212 * q^34 - 1072 * q^35 + 8938 * q^37 - 10728 * q^38 + 3440 * q^40 + 12792 * q^41 - 16730 * q^43 - 5112 * q^44 + 2116 * q^46 - 22500 * q^47 + 2887 * q^49 - 15156 * q^50 - 47412 * q^52 - 17108 * q^53 - 436 * q^55 - 42640 * q^56 - 55678 * q^58 - 54176 * q^59 - 71324 * q^61 + 72710 * q^62 - 49984 * q^64 - 846 * q^65 - 62960 * q^67 + 8352 * q^68 + 9224 * q^70 - 98400 * q^71 - 81772 * q^73 + 59044 * q^74 + 31488 * q^76 + 304 * q^77 + 58224 * q^79 + 17568 * q^80 + 61926 * q^82 - 9892 * q^83 + 15536 * q^85 - 191140 * q^86 - 58400 * q^88 - 27542 * q^89 + 151974 * q^91 + 8464 * q^92 - 146990 * q^94 + 20644 * q^95 - 273672 * q^97 + 401276 * q^98

Introduction

L-functions

Modular forms

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Database

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