LMFDB - Embedded newform 207.6.a.g.1.3

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:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 149x^{4} + 215x^{3} + 6182x^{2} - 4625x - 79150 $ x^6 - 2x^5 - 149x^4 + 215x^3 + 6182x^2 - 4625*x - 79150
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.81709$ 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)

$f(q)$	$=$	$q-5.81709 q^{2} +1.83858 q^{4} -83.3054 q^{5} +84.5782 q^{7} +175.452 q^{8} +O(q^{10})$	\(q-5.81709 q^{2} +1.83858 q^{4} -83.3054 q^{5} +84.5782 q^{7} +175.452 q^{8} +484.596 q^{10} -137.706 q^{11} -802.180 q^{13} -492.000 q^{14} -1079.45 q^{16} -1069.97 q^{17} -2610.09 q^{19} -153.164 q^{20} +801.050 q^{22} -529.000 q^{23} +3814.80 q^{25} +4666.35 q^{26} +155.504 q^{28} -5992.34 q^{29} +6084.35 q^{31} +664.828 q^{32} +6224.11 q^{34} -7045.83 q^{35} +1803.78 q^{37} +15183.1 q^{38} -14616.1 q^{40} -159.280 q^{41} +184.420 q^{43} -253.184 q^{44} +3077.24 q^{46} +8877.78 q^{47} -9653.52 q^{49} -22191.0 q^{50} -1474.87 q^{52} -17482.4 q^{53} +11471.7 q^{55} +14839.4 q^{56} +34858.0 q^{58} -443.279 q^{59} +50100.9 q^{61} -35393.2 q^{62} +30675.2 q^{64} +66825.9 q^{65} -32934.8 q^{67} -1967.22 q^{68} +40986.2 q^{70} +63590.3 q^{71} -44619.8 q^{73} -10492.8 q^{74} -4798.86 q^{76} -11647.0 q^{77} +89229.0 q^{79} +89924.4 q^{80} +926.547 q^{82} +86848.6 q^{83} +89134.2 q^{85} -1072.79 q^{86} -24160.8 q^{88} -34347.9 q^{89} -67846.9 q^{91} -972.609 q^{92} -51642.9 q^{94} +217435. q^{95} +125384. q^{97} +56155.4 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8}+O(q^{10}) $ 6 * q - 4 * q^2 + 112 * q^4 - 42 * q^5 + 300 * q^7 - 393 * q^8	$ 6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8} - 10 q^{10} + 58 q^{11} + 792 q^{13} + 2984 q^{14} - 1904 q^{16} + 400 q^{17} + 2738 q^{19} + 7124 q^{20} - 1972 q^{22} - 3174 q^{23} + 9966 q^{25} - 4511 q^{26} + 9570 q^{28} - 11244 q^{29} + 13748 q^{31} - 6600 q^{32} - 16226 q^{34} + 4296 q^{35} + 25426 q^{37} + 8028 q^{38} + 10230 q^{40} + 14268 q^{41} - 18082 q^{43} + 51146 q^{44} + 2116 q^{46} + 23084 q^{47} + 37422 q^{49} + 67436 q^{50} + 36807 q^{52} - 17522 q^{53} + 47576 q^{55} - 44946 q^{56} + 141001 q^{58} + 36392 q^{59} + 27062 q^{61} - 48971 q^{62} + 89451 q^{64} - 7108 q^{65} + 37138 q^{67} - 17260 q^{68} + 248380 q^{70} + 158556 q^{71} + 112228 q^{73} + 66878 q^{74} + 157816 q^{76} + 89760 q^{77} + 36844 q^{79} + 158530 q^{80} + 150039 q^{82} + 76350 q^{83} - 102132 q^{85} + 100578 q^{86} - 219028 q^{88} - 16100 q^{89} - 250592 q^{91} - 59248 q^{92} + 12887 q^{94} + 190096 q^{95} + 259432 q^{97} + 325816 q^{98}+O(q^{100}) $ 6 * q - 4 * q^2 + 112 * q^4 - 42 * q^5 + 300 * q^7 - 393 * q^8 - 10 * q^10 + 58 * q^11 + 792 * q^13 + 2984 * q^14 - 1904 * q^16 + 400 * q^17 + 2738 * q^19 + 7124 * q^20 - 1972 * q^22 - 3174 * q^23 + 9966 * q^25 - 4511 * q^26 + 9570 * q^28 - 11244 * q^29 + 13748 * q^31 - 6600 * q^32 - 16226 * q^34 + 4296 * q^35 + 25426 * q^37 + 8028 * q^38 + 10230 * q^40 + 14268 * q^41 - 18082 * q^43 + 51146 * q^44 + 2116 * q^46 + 23084 * q^47 + 37422 * q^49 + 67436 * q^50 + 36807 * q^52 - 17522 * q^53 + 47576 * q^55 - 44946 * q^56 + 141001 * q^58 + 36392 * q^59 + 27062 * q^61 - 48971 * q^62 + 89451 * q^64 - 7108 * q^65 + 37138 * q^67 - 17260 * q^68 + 248380 * q^70 + 158556 * q^71 + 112228 * q^73 + 66878 * q^74 + 157816 * q^76 + 89760 * q^77 + 36844 * q^79 + 158530 * q^80 + 150039 * q^82 + 76350 * q^83 - 102132 * q^85 + 100578 * q^86 - 219028 * q^88 - 16100 * q^89 - 250592 * q^91 - 59248 * q^92 + 12887 * q^94 + 190096 * q^95 + 259432 * q^97 + 325816 * 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 $

$2$	−5.81709	−1.02833	−0.514163	−	0.857692i	$-0.671897\pi$
$2$	−5.81709	−1.02833	−0.514163	+	0.857692i	$0.671897\pi$
$3$	0	0
$3$	0	0
$4$	1.83858	0.0574556
$5$	−83.3054	−1.49021	−0.745106	−	0.666946i	$-0.767601\pi$
$5$	−83.3054	−1.49021	−0.745106	+	0.666946i	$0.767601\pi$
$6$	0	0
$7$	84.5782	0.652399	0.326200	−	0.945301i	$-0.394232\pi$
$7$	84.5782	0.652399	0.326200	+	0.945301i	$0.394232\pi$
$8$	175.452	0.969243
$9$	0	0
$10$	484.596	1.53243
$11$	−137.706	−0.343140	−0.171570	−	0.985172i	$-0.554884\pi$
$11$	−137.706	−0.343140	−0.171570	+	0.985172i	$0.554884\pi$
$12$	0	0
$13$	−802.180	−1.31648	−0.658238	−	0.752809i	$-0.728698\pi$
$13$	−802.180	−1.31648	−0.658238	+	0.752809i	$0.728698\pi$
$14$	−492.000	−0.670880
$15$	0	0
$16$	−1079.45	−1.05415
$17$	−1069.97	−0.897943	−0.448972	−	0.893546i	$-0.648210\pi$
$17$	−1069.97	−0.897943	−0.448972	+	0.893546i	$0.648210\pi$
$18$	0	0
$19$	−2610.09	−1.65872	−0.829358	−	0.558718i	$-0.811293\pi$
$19$	−2610.09	−1.65872	−0.829358	+	0.558718i	$0.811293\pi$
$20$	−153.164	−0.0856211
$21$	0	0
$22$	801.050	0.352860
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	3814.80	1.22073
$26$	4666.35	1.35377
$27$	0	0
$28$	155.504	0.0374840
$29$	−5992.34	−1.32313	−0.661564	−	0.749889i	$-0.730107\pi$
$29$	−5992.34	−1.32313	−0.661564	+	0.749889i	$0.730107\pi$
$30$	0	0
$31$	6084.35	1.13713	0.568565	−	0.822638i	$-0.307499\pi$
$31$	6084.35	1.13713	0.568565	+	0.822638i	$0.307499\pi$
$32$	664.828	0.114772
$33$	0	0
$34$	6224.11	0.923379
$35$	−7045.83	−0.972214
$36$	0	0
$37$	1803.78	0.216610	0.108305	−	0.994118i	$-0.465458\pi$
$37$	1803.78	0.216610	0.108305	+	0.994118i	$0.465458\pi$
$38$	15183.1	1.70570
$39$	0	0
$40$	−14616.1	−1.44438
$41$	−159.280	−0.0147980	−0.00739898	−	0.999973i	$-0.502355\pi$
$41$	−159.280	−0.0147980	−0.00739898	+	0.999973i	$0.502355\pi$
$42$	0	0
$43$	184.420	0.0152103	0.00760513	−	0.999971i	$-0.497579\pi$
$43$	184.420	0.0152103	0.00760513	+	0.999971i	$0.497579\pi$
$44$	−253.184	−0.0197153
$45$	0	0
$46$	3077.24	0.214421
$47$	8877.78	0.586218	0.293109	−	0.956079i	$-0.405310\pi$
$47$	8877.78	0.586218	0.293109	+	0.956079i	$0.405310\pi$
$48$	0	0
$49$	−9653.52	−0.574375
$50$	−22191.0	−1.25531
$51$	0	0
$52$	−1474.87	−0.0756390
$53$	−17482.4	−0.854894	−0.427447	−	0.904040i	$-0.640587\pi$
$53$	−17482.4	−0.854894	−0.427447	+	0.904040i	$0.640587\pi$
$54$	0	0
$55$	11471.7	0.511352
$56$	14839.4	0.632334
$57$	0	0
$58$	34858.0	1.36061
$59$	−443.279	−0.0165786	−0.00828929	−	0.999966i	$-0.502639\pi$
$59$	−443.279	−0.0165786	−0.00828929	+	0.999966i	$0.502639\pi$
$60$	0	0
$61$	50100.9	1.72394	0.861968	−	0.506963i	$-0.169232\pi$
$61$	50100.9	1.72394	0.861968	+	0.506963i	$0.169232\pi$
$62$	−35393.2	−1.16934
$63$	0	0
$64$	30675.2	0.936132
$65$	66825.9	1.96183
$66$	0	0
$67$	−32934.8	−0.896331	−0.448166	−	0.893950i	$-0.647923\pi$
$67$	−32934.8	−0.896331	−0.448166	+	0.893950i	$0.647923\pi$
$68$	−1967.22	−0.0515919
$69$	0	0
$70$	40986.2	0.999754
$71$	63590.3	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$71$	63590.3	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$72$	0	0
$73$	−44619.8	−0.979988	−0.489994	−	0.871726i	$-0.663001\pi$
$73$	−44619.8	−0.979988	−0.489994	+	0.871726i	$0.663001\pi$
$74$	−10492.8	−0.222746
$75$	0	0
$76$	−4798.86	−0.0953025
$77$	−11647.0	−0.223865
$78$	0	0
$79$	89229.0	1.60856	0.804282	−	0.594247i	$-0.202550\pi$
$79$	89229.0	1.60856	0.804282	+	0.594247i	$0.202550\pi$
$80$	89924.4	1.57091
$81$	0	0
$82$	926.547	0.0152171
$83$	86848.6	1.38378	0.691891	−	0.722002i	$-0.256778\pi$
$83$	86848.6	1.38378	0.691891	+	0.722002i	$0.256778\pi$
$84$	0	0
$85$	89134.2	1.33813
$86$	−1072.79	−0.0156411
$87$	0	0
$88$	−24160.8	−0.332587
$89$	−34347.9	−0.459648	−0.229824	−	0.973232i	$-0.573815\pi$
$89$	−34347.9	−0.459648	−0.229824	+	0.973232i	$0.573815\pi$
$90$	0	0
$91$	−67846.9	−0.858869
$92$	−972.609	−0.0119803
$93$	0	0
$94$	−51642.9	−0.602824
$95$	217435.	2.47184
$96$	0	0
$97$	125384.	1.35305	0.676524	−	0.736421i	$-0.263486\pi$
$97$	125384.	1.35305	0.676524	+	0.736421i	$0.263486\pi$
$98$	56155.4	0.590645
$99$	0	0
$100$	7013.80	0.0701380
$101$	−200673.	−1.95743	−0.978714	−	0.205229i	$-0.934206\pi$
$101$	−200673.	−1.95743	−0.978714	+	0.205229i	$0.934206\pi$
$102$	0	0
$103$	−47183.6	−0.438226	−0.219113	−	0.975699i	$-0.570316\pi$
$103$	−47183.6	−0.438226	−0.219113	+	0.975699i	$0.570316\pi$
$104$	−140744.	−1.27599
$105$	0	0
$106$	101697.	0.879110
$107$	−15670.6	−0.132320	−0.0661600	−	0.997809i	$-0.521075\pi$
$107$	−15670.6	−0.132320	−0.0661600	+	0.997809i	$0.521075\pi$
$108$	0	0
$109$	−122599.	−0.988370	−0.494185	−	0.869357i	$-0.664533\pi$
$109$	−122599.	−0.988370	−0.494185	+	0.869357i	$0.664533\pi$
$110$	−66731.8	−0.525837
$111$	0	0
$112$	−91298.3	−0.687730
$113$	−34542.6	−0.254483	−0.127242	−	0.991872i	$-0.540612\pi$
$113$	−34542.6	−0.254483	−0.127242	+	0.991872i	$0.540612\pi$
$114$	0	0
$115$	44068.6	0.310731
$116$	−11017.4	−0.0760211
$117$	0	0
$118$	2578.60	0.0170482
$119$	−90496.1	−0.585818
$120$	0	0
$121$	−142088.	−0.882255
$122$	−291442.	−1.77277
$123$	0	0
$124$	11186.6	0.0653345
$125$	−57463.7	−0.328942
$126$	0	0
$127$	141955.	0.780985	0.390492	−	0.920606i	$-0.372305\pi$
$127$	141955.	0.780985	0.390492	+	0.920606i	$0.372305\pi$
$128$	−199715.	−1.07742
$129$	0	0
$130$	−388733.	−2.01740
$131$	267589.	1.36235	0.681176	−	0.732120i	$-0.261469\pi$
$131$	267589.	1.36235	0.681176	+	0.732120i	$0.261469\pi$
$132$	0	0
$133$	−220757.	−1.08214
$134$	191585.	0.921722
$135$	0	0
$136$	−187728.	−0.870326
$137$	−54729.6	−0.249127	−0.124563	−	0.992212i	$-0.539753\pi$
$137$	−54729.6	−0.249127	−0.124563	+	0.992212i	$0.539753\pi$
$138$	0	0
$139$	291566.	1.27997	0.639986	−	0.768387i	$-0.278940\pi$
$139$	291566.	1.27997	0.639986	+	0.768387i	$0.278940\pi$
$140$	−12954.3	−0.0558591
$141$	0	0
$142$	−369910.	−1.53949
$143$	110465.	0.451736
$144$	0	0
$145$	499195.	1.97174
$146$	259558.	1.00775
$147$	0	0
$148$	3316.39	0.0124455
$149$	311383.	1.14903	0.574513	−	0.818496i	$-0.305192\pi$
$149$	311383.	1.14903	0.574513	+	0.818496i	$0.305192\pi$
$150$	0	0
$151$	−145636.	−0.519789	−0.259895	−	0.965637i	$-0.583688\pi$
$151$	−145636.	−0.519789	−0.259895	+	0.965637i	$0.583688\pi$
$152$	−457945.	−1.60770
$153$	0	0
$154$	67751.4	0.230206
$155$	−506860.	−1.69457
$156$	0	0
$157$	33225.9	0.107579	0.0537895	−	0.998552i	$-0.482870\pi$
$157$	33225.9	0.107579	0.0537895	+	0.998552i	$0.482870\pi$
$158$	−519054.	−1.65413
$159$	0	0
$160$	−55383.8	−0.171034
$161$	−44741.9	−0.136035
$162$	0	0
$163$	−228521.	−0.673685	−0.336842	−	0.941561i	$-0.609359\pi$
$163$	−228521.	−0.673685	−0.336842	+	0.941561i	$0.609359\pi$
$164$	−292.849	−0.000850226 0
$165$	0	0
$166$	−505207.	−1.42298
$167$	596374.	1.65473	0.827366	−	0.561663i	$-0.189838\pi$
$167$	596374.	1.65473	0.827366	+	0.561663i	$0.189838\pi$
$168$	0	0
$169$	272199.	0.733112
$170$	−518502.	−1.37603
$171$	0	0
$172$	339.071	0.000873914 0
$173$	−164750.	−0.418513	−0.209257	−	0.977861i	$-0.567104\pi$
$173$	−164750.	−0.418513	−0.209257	+	0.977861i	$0.567104\pi$
$174$	0	0
$175$	322649.	0.796407
$176$	148648.	0.361723
$177$	0	0
$178$	199805.	0.472668
$179$	444881.	1.03780	0.518898	−	0.854836i	$-0.326343\pi$
$179$	444881.	1.03780	0.518898	+	0.854836i	$0.326343\pi$
$180$	0	0
$181$	−793333.	−1.79995	−0.899973	−	0.435947i	$-0.856414\pi$
$181$	−793333.	−1.79995	−0.899973	+	0.435947i	$0.856414\pi$
$182$	394672.	0.883198
$183$	0	0
$184$	−92814.0	−0.202101
$185$	−150265.	−0.322796
$186$	0	0
$187$	147341.	0.308121
$188$	16322.5	0.0336815
$189$	0	0
$190$	−1.26484e6	−2.54186
$191$	819938.	1.62629	0.813144	−	0.582062i	$-0.197754\pi$
$191$	819938.	1.62629	0.813144	+	0.582062i	$0.197754\pi$
$192$	0	0
$193$	188411.	0.364093	0.182047	−	0.983290i	$-0.441728\pi$
$193$	188411.	0.364093	0.182047	+	0.983290i	$0.441728\pi$
$194$	−729371.	−1.39137
$195$	0	0
$196$	−17748.8	−0.0330011
$197$	−696734.	−1.27909	−0.639545	−	0.768753i	$-0.720877\pi$
$197$	−696734.	−1.27909	−0.639545	+	0.768753i	$0.720877\pi$
$198$	0	0
$199$	−342889.	−0.613792	−0.306896	−	0.951743i	$-0.599290\pi$
$199$	−342889.	−0.613792	−0.306896	+	0.951743i	$0.599290\pi$
$200$	669313.	1.18319
$201$	0	0
$202$	1.16733e6	2.01288
$203$	−506822.	−0.863208
$204$	0	0
$205$	13268.9	0.0220521
$206$	274472.	0.450640
$207$	0	0
$208$	865916.	1.38777
$209$	359426.	0.569172
$210$	0	0
$211$	−410466.	−0.634704	−0.317352	−	0.948308i	$-0.602794\pi$
$211$	−410466.	−0.634704	−0.317352	+	0.948308i	$0.602794\pi$
$212$	−32142.9	−0.0491185
$213$	0	0
$214$	91157.2	0.136068
$215$	−15363.2	−0.0226665
$216$	0	0
$217$	514604.	0.741863
$218$	713168.	1.01637
$219$	0	0
$220$	21091.6	0.0293801
$221$	858307.	1.18212
$222$	0	0
$223$	342896.	0.461742	0.230871	−	0.972984i	$-0.425842\pi$
$223$	342896.	0.461742	0.230871	+	0.972984i	$0.425842\pi$
$224$	56230.0	0.0748769
$225$	0	0
$226$	200938.	0.261692
$227$	−98200.3	−0.126488	−0.0632438	−	0.997998i	$-0.520145\pi$
$227$	−98200.3	−0.126488	−0.0632438	+	0.997998i	$0.520145\pi$
$228$	0	0
$229$	355410.	0.447859	0.223929	−	0.974605i	$-0.428111\pi$
$229$	355410.	0.447859	0.223929	+	0.974605i	$0.428111\pi$
$230$	−256351.	−0.319533
$231$	0	0
$232$	−1.05137e6	−1.28243
$233$	−1.14019e6	−1.37590	−0.687949	−	0.725759i	$-0.741489\pi$
$233$	−1.14019e6	−1.37590	−0.687949	+	0.725759i	$0.741489\pi$
$234$	0	0
$235$	−739567.	−0.873590
$236$	−815.004	−0.000952533 0
$237$	0	0
$238$	526424.	0.602412
$239$	−790634.	−0.895325	−0.447663	−	0.894203i	$-0.647743\pi$
$239$	−790634.	−0.895325	−0.447663	+	0.894203i	$0.647743\pi$
$240$	0	0
$241$	573570.	0.636127	0.318064	−	0.948069i	$-0.396967\pi$
$241$	573570.	0.636127	0.318064	+	0.948069i	$0.396967\pi$
$242$	826539.	0.907246
$243$	0	0
$244$	92114.5	0.0990498
$245$	804191.	0.855941
$246$	0	0
$247$	2.09376e6	2.18366
$248$	1.06751e6	1.10216
$249$	0	0
$250$	334272.	0.338259
$251$	187911.	0.188264	0.0941320	−	0.995560i	$-0.469992\pi$
$251$	187911.	0.188264	0.0941320	+	0.995560i	$0.469992\pi$
$252$	0	0
$253$	72846.6	0.0715497
$254$	−825768.	−0.803107
$255$	0	0
$256$	180154.	0.171809
$257$	−290953.	−0.274783	−0.137391	−	0.990517i	$-0.543872\pi$
$257$	−290953.	−0.274783	−0.137391	+	0.990517i	$0.543872\pi$
$258$	0	0
$259$	152561.	0.141316
$260$	122865.	0.112718
$261$	0	0
$262$	−1.55659e6	−1.40094
$263$	2759.71	0.00246022	0.00123011	−	0.999999i	$-0.499608\pi$
$263$	2759.71	0.00246022	0.00123011	+	0.999999i	$0.499608\pi$
$264$	0	0
$265$	1.45638e6	1.27397
$266$	1.28416e6	1.11280
$267$	0	0
$268$	−60553.3	−0.0514993
$269$	42939.6	0.0361807	0.0180904	−	0.999836i	$-0.494241\pi$
$269$	42939.6	0.0361807	0.0180904	+	0.999836i	$0.494241\pi$
$270$	0	0
$271$	649292.	0.537053	0.268526	−	0.963272i	$-0.413463\pi$
$271$	649292.	0.537053	0.268526	+	0.963272i	$0.413463\pi$
$272$	1.15498e6	0.946571
$273$	0	0
$274$	318367.	0.256184
$275$	−525321.	−0.418883
$276$	0	0
$277$	−2.27898e6	−1.78460	−0.892300	−	0.451444i	$-0.850909\pi$
$277$	−2.27898e6	−1.78460	−0.892300	+	0.451444i	$0.850909\pi$
$278$	−1.69607e6	−1.31623
$279$	0	0
$280$	−1.23620e6	−0.942312
$281$	574307.	0.433888	0.216944	−	0.976184i	$-0.430391\pi$
$281$	574307.	0.433888	0.216944	+	0.976184i	$0.430391\pi$
$282$	0	0
$283$	1.76769e6	1.31202	0.656008	−	0.754754i	$-0.272244\pi$
$283$	1.76769e6	1.31202	0.656008	+	0.754754i	$0.272244\pi$
$284$	116916.	0.0860156
$285$	0	0
$286$	−642586.	−0.464533
$287$	−13471.6	−0.00965418
$288$	0	0
$289$	−275023.	−0.193698
$290$	−2.90386e6	−2.02759
$291$	0	0
$292$	−82037.1	−0.0563058
$293$	2.02147e6	1.37562	0.687809	−	0.725892i	$-0.258573\pi$
$293$	2.02147e6	1.37562	0.687809	+	0.725892i	$0.258573\pi$
$294$	0	0
$295$	36927.6	0.0247056
$296$	316476.	0.209948
$297$	0	0
$298$	−1.81135e6	−1.18157
$299$	424353.	0.274504
$300$	0	0
$301$	15597.9	0.00992316
$302$	847180.	0.534513
$303$	0	0
$304$	2.81747e6	1.74854
$305$	−4.17368e6	−2.56903
$306$	0	0
$307$	1.66465e6	1.00804	0.504020	−	0.863692i	$-0.331854\pi$
$307$	1.66465e6	1.00804	0.504020	+	0.863692i	$0.331854\pi$
$308$	−21413.8	−0.0128623
$309$	0	0
$310$	2.94845e6	1.74257
$311$	−1.93216e6	−1.13277	−0.566385	−	0.824141i	$-0.691659\pi$
$311$	−1.93216e6	−1.13277	−0.566385	+	0.824141i	$0.691659\pi$
$312$	0	0
$313$	−1.27561e6	−0.735966	−0.367983	−	0.929833i	$-0.619951\pi$
$313$	−1.27561e6	−0.735966	−0.367983	+	0.929833i	$0.619951\pi$
$314$	−193278.	−0.110626
$315$	0	0
$316$	164055.	0.0924211
$317$	−2.15141e6	−1.20247	−0.601236	−	0.799071i	$-0.705325\pi$
$317$	−2.15141e6	−1.20247	−0.601236	+	0.799071i	$0.705325\pi$
$318$	0	0
$319$	825183.	0.454019
$320$	−2.55541e6	−1.39504
$321$	0	0
$322$	260268.	0.139888
$323$	2.79272e6	1.48943
$324$	0	0
$325$	−3.06015e6	−1.60707
$326$	1.32933e6	0.692768
$327$	0	0
$328$	−27946.0	−0.0143428
$329$	750867.	0.382449
$330$	0	0
$331$	1.93177e6	0.969139	0.484570	−	0.874753i	$-0.338976\pi$
$331$	1.93177e6	0.969139	0.484570	+	0.874753i	$0.338976\pi$
$332$	159678.	0.0795060
$333$	0	0
$334$	−3.46916e6	−1.70161
$335$	2.74365e6	1.33572
$336$	0	0
$337$	852949.	0.409118	0.204559	−	0.978854i	$-0.434424\pi$
$337$	852949.	0.409118	0.204559	+	0.978854i	$0.434424\pi$
$338$	−1.58341e6	−0.753878
$339$	0	0
$340$	163880.	0.0768829
$341$	−837853.	−0.390195
$342$	0	0
$343$	−2.23798e6	−1.02712
$344$	32356.8	0.0147424
$345$	0	0
$346$	958364.	0.430368
$347$	874549.	0.389907	0.194953	−	0.980813i	$-0.437544\pi$
$347$	874549.	0.389907	0.194953	+	0.980813i	$0.437544\pi$
$348$	0	0
$349$	−1.29496e6	−0.569106	−0.284553	−	0.958660i	$-0.591845\pi$
$349$	−1.29496e6	−0.569106	−0.284553	+	0.958660i	$0.591845\pi$
$350$	−1.87688e6	−0.818966
$351$	0	0
$352$	−91551.0	−0.0393828
$353$	3.09448e6	1.32176	0.660878	−	0.750494i	$-0.270184\pi$
$353$	3.09448e6	1.32176	0.660878	+	0.750494i	$0.270184\pi$
$354$	0	0
$355$	−5.29741e6	−2.23097
$356$	−63151.3	−0.0264093
$357$	0	0
$358$	−2.58792e6	−1.06719
$359$	3.92693e6	1.60812	0.804058	−	0.594550i	$-0.202670\pi$
$359$	3.92693e6	1.60812	0.804058	+	0.594550i	$0.202670\pi$
$360$	0	0
$361$	4.33648e6	1.75134
$362$	4.61489e6	1.85093
$363$	0	0
$364$	−124742.	−0.0493468
$365$	3.71707e6	1.46039
$366$	0	0
$367$	2.54422e6	0.986027	0.493014	−	0.870022i	$-0.335895\pi$
$367$	2.54422e6	0.986027	0.493014	+	0.870022i	$0.335895\pi$
$368$	571031.	0.219806
$369$	0	0
$370$	874104.	0.331939
$371$	−1.47863e6	−0.557732
$372$	0	0
$373$	4.34785e6	1.61809	0.809045	−	0.587747i	$-0.199985\pi$
$373$	4.34785e6	1.61809	0.809045	+	0.587747i	$0.199985\pi$
$374$	−857099.	−0.316849
$375$	0	0
$376$	1.55762e6	0.568188
$377$	4.80694e6	1.74187
$378$	0	0
$379$	−719733.	−0.257379	−0.128690	−	0.991685i	$-0.541077\pi$
$379$	−719733.	−0.257379	−0.128690	+	0.991685i	$0.541077\pi$
$380$	399771.	0.142021
$381$	0	0
$382$	−4.76966e6	−1.67236
$383$	−5.36052e6	−1.86728	−0.933641	−	0.358211i	$-0.883387\pi$
$383$	−5.36052e6	−1.86728	−0.933641	+	0.358211i	$0.883387\pi$
$384$	0	0
$385$	970254.	0.333606
$386$	−1.09600e6	−0.374407
$387$	0	0
$388$	230529.	0.0777402
$389$	697098.	0.233571	0.116786	−	0.993157i	$-0.462741\pi$
$389$	697098.	0.233571	0.116786	+	0.993157i	$0.462741\pi$
$390$	0	0
$391$	566014.	0.187234
$392$	−1.69373e6	−0.556709
$393$	0	0
$394$	4.05297e6	1.31532
$395$	−7.43326e6	−2.39710
$396$	0	0
$397$	2.36778e6	0.753991	0.376995	−	0.926215i	$-0.376957\pi$
$397$	2.36778e6	0.753991	0.376995	+	0.926215i	$0.376957\pi$
$398$	1.99462e6	0.631179
$399$	0	0
$400$	−4.11790e6	−1.28684
$401$	−2.43186e6	−0.755227	−0.377613	−	0.925963i	$-0.623255\pi$
$401$	−2.43186e6	−0.755227	−0.377613	+	0.925963i	$0.623255\pi$
$402$	0	0
$403$	−4.88074e6	−1.49701
$404$	−368954.	−0.112465
$405$	0	0
$406$	2.94823e6	0.887659
$407$	−248392.	−0.0743278
$408$	0	0
$409$	214496.	0.0634032	0.0317016	−	0.999497i	$-0.489907\pi$
$409$	214496.	0.0634032	0.0317016	+	0.999497i	$0.489907\pi$
$410$	−77186.4	−0.0226768
$411$	0	0
$412$	−86750.9	−0.0251786
$413$	−37491.8	−0.0108159
$414$	0	0
$415$	−7.23496e6	−2.06213
$416$	−533312.	−0.151094
$417$	0	0
$418$	−2.09081e6	−0.585295
$419$	−2.77435e6	−0.772016	−0.386008	−	0.922496i	$-0.626146\pi$
$419$	−2.77435e6	−0.772016	−0.386008	+	0.922496i	$0.626146\pi$
$420$	0	0
$421$	−3.74289e6	−1.02920	−0.514602	−	0.857429i	$-0.672060\pi$
$421$	−3.74289e6	−1.02920	−0.514602	+	0.857429i	$0.672060\pi$
$422$	2.38772e6	0.652683
$423$	0	0
$424$	−3.06733e6	−0.828601
$425$	−4.08171e6	−1.09615
$426$	0	0
$427$	4.23745e6	1.12469
$428$	−28811.6	−0.00760253
$429$	0	0
$430$	89369.0	0.0233086
$431$	−4.66860e6	−1.21058	−0.605290	−	0.796005i	$-0.706943\pi$
$431$	−4.66860e6	−1.21058	−0.605290	+	0.796005i	$0.706943\pi$
$432$	0	0
$433$	822830.	0.210907	0.105453	−	0.994424i	$-0.466371\pi$
$433$	822830.	0.210907	0.105453	+	0.994424i	$0.466371\pi$
$434$	−2.99350e6	−0.762877
$435$	0	0
$436$	−225407.	−0.0567874
$437$	1.38074e6	0.345866
$438$	0	0
$439$	−2.34164e6	−0.579908	−0.289954	−	0.957041i	$-0.593640\pi$
$439$	−2.34164e6	−0.579908	−0.289954	+	0.957041i	$0.593640\pi$
$440$	2.01273e6	0.495625
$441$	0	0
$442$	−4.99286e6	−1.21561
$443$	1.66238e6	0.402458	0.201229	−	0.979544i	$-0.435506\pi$
$443$	1.66238e6	0.402458	0.201229	+	0.979544i	$0.435506\pi$
$444$	0	0
$445$	2.86137e6	0.684973
$446$	−1.99466e6	−0.474822
$447$	0	0
$448$	2.59445e6	0.610732
$449$	−1.56953e6	−0.367411	−0.183706	−	0.982981i	$-0.558809\pi$
$449$	−1.56953e6	−0.367411	−0.183706	+	0.982981i	$0.558809\pi$
$450$	0	0
$451$	21933.9	0.00507778
$452$	−63509.4	−0.0146215
$453$	0	0
$454$	571240.	0.130071
$455$	5.65202e6	1.27990
$456$	0	0
$457$	−43940.4	−0.00984177	−0.00492088	−	0.999988i	$-0.501566\pi$
$457$	−43940.4	−0.00984177	−0.00492088	+	0.999988i	$0.501566\pi$
$458$	−2.06745e6	−0.460545
$459$	0	0
$460$	81023.6	0.0178532
$461$	4.35291e6	0.953953	0.476976	−	0.878916i	$-0.341733\pi$
$461$	4.35291e6	0.953953	0.476976	+	0.878916i	$0.341733\pi$
$462$	0	0
$463$	−2.47496e6	−0.536558	−0.268279	−	0.963341i	$-0.586455\pi$
$463$	−2.47496e6	−0.536558	−0.268279	+	0.963341i	$0.586455\pi$
$464$	6.46846e6	1.39478
$465$	0	0
$466$	6.63257e6	1.41487
$467$	2.98547e6	0.633462	0.316731	−	0.948515i	$-0.397415\pi$
$467$	2.98547e6	0.633462	0.316731	+	0.948515i	$0.397415\pi$
$468$	0	0
$469$	−2.78557e6	−0.584766
$470$	4.30213e6	0.898336
$471$	0	0
$472$	−77774.2	−0.0160687
$473$	−25395.8	−0.00521925
$474$	0	0
$475$	−9.95697e6	−2.02485
$476$	−166384.	−0.0336585
$477$	0	0
$478$	4.59919e6	0.920687
$479$	−1.85203e6	−0.368816	−0.184408	−	0.982850i	$-0.559037\pi$
$479$	−1.85203e6	−0.368816	−0.184408	+	0.982850i	$0.559037\pi$
$480$	0	0
$481$	−1.44696e6	−0.285163
$482$	−3.33651e6	−0.654147
$483$	0	0
$484$	−261240.	−0.0506905
$485$	−1.04452e7	−2.01633
$486$	0	0
$487$	1.24473e6	0.237823	0.118912	−	0.992905i	$-0.462059\pi$
$487$	1.24473e6	0.237823	0.118912	+	0.992905i	$0.462059\pi$
$488$	8.79029e6	1.67091
$489$	0	0
$490$	−4.67805e6	−0.880187
$491$	−4.68986e6	−0.877922	−0.438961	−	0.898506i	$-0.644653\pi$
$491$	−4.68986e6	−0.877922	−0.438961	+	0.898506i	$0.644653\pi$
$492$	0	0
$493$	6.41162e6	1.18809
$494$	−1.21796e7	−2.24552
$495$	0	0
$496$	−6.56778e6	−1.19871
$497$	5.37835e6	0.976693
$498$	0	0
$499$	−8.68030e6	−1.56057	−0.780285	−	0.625424i	$-0.784926\pi$
$499$	−8.68030e6	−1.56057	−0.780285	+	0.625424i	$0.784926\pi$
$500$	−105652.	−0.0188995
$501$	0	0
$502$	−1.09309e6	−0.193597
$503$	−2.92667e6	−0.515767	−0.257884	−	0.966176i	$-0.583025\pi$
$503$	−2.92667e6	−0.515767	−0.257884	+	0.966176i	$0.583025\pi$
$504$	0	0
$505$	1.67172e7	2.91698
$506$	−423756.	−0.0735765
$507$	0	0
$508$	260996.	0.0448719
$509$	−1.67515e6	−0.286588	−0.143294	−	0.989680i	$-0.545769\pi$
$509$	−1.67515e6	−0.286588	−0.143294	+	0.989680i	$0.545769\pi$
$510$	0	0
$511$	−3.77387e6	−0.639344
$512$	5.34290e6	0.900745
$513$	0	0
$514$	1.69250e6	0.282567
$515$	3.93065e6	0.653050
$516$	0	0
$517$	−1.22253e6	−0.201155
$518$	−887459.	−0.145320
$519$	0	0
$520$	1.17247e7	1.90149
$521$	−2.88782e6	−0.466096	−0.233048	−	0.972465i	$-0.574870\pi$
$521$	−2.88782e6	−0.466096	−0.233048	+	0.972465i	$0.574870\pi$
$522$	0	0
$523$	8.07823e6	1.29140	0.645702	−	0.763589i	$-0.276565\pi$
$523$	8.07823e6	1.29140	0.645702	+	0.763589i	$0.276565\pi$
$524$	491983.	0.0782747
$525$	0	0
$526$	−16053.5	−0.00252991
$527$	−6.51007e6	−1.02108
$528$	0	0
$529$	279841.	0.0434783
$530$	−8.47191e6	−1.31006
$531$	0	0
$532$	−405879.	−0.0621753
$533$	127771.	0.0194812
$534$	0	0
$535$	1.30544e6	0.197185
$536$	−5.77848e6	−0.868763
$537$	0	0
$538$	−249784.	−0.0372056
$539$	1.32935e6	0.197091
$540$	0	0
$541$	−7.62056e6	−1.11942	−0.559711	−	0.828688i	$-0.689088\pi$
$541$	−7.62056e6	−1.11942	−0.559711	+	0.828688i	$0.689088\pi$
$542$	−3.77699e6	−0.552265
$543$	0	0
$544$	−711346.	−0.103058
$545$	1.02131e7	1.47288
$546$	0	0
$547$	−8.99192e6	−1.28494	−0.642472	−	0.766309i	$-0.722091\pi$
$547$	−8.99192e6	−1.28494	−0.642472	+	0.766309i	$0.722091\pi$
$548$	−100625.	−0.0143137
$549$	0	0
$550$	3.05584e6	0.430749
$551$	1.56406e7	2.19469
$552$	0	0
$553$	7.54684e6	1.04943
$554$	1.32570e7	1.83515
$555$	0	0
$556$	536068.	0.0735416
$557$	4.76720e6	0.651067	0.325534	−	0.945530i	$-0.394456\pi$
$557$	4.76720e6	0.651067	0.325534	+	0.945530i	$0.394456\pi$
$558$	0	0
$559$	−147938.	−0.0200240
$560$	7.60565e6	1.02486
$561$	0	0
$562$	−3.34080e6	−0.446179
$563$	1.68560e6	0.224122	0.112061	−	0.993701i	$-0.464255\pi$
$563$	1.68560e6	0.224122	0.112061	+	0.993701i	$0.464255\pi$
$564$	0	0
$565$	2.87759e6	0.379234
$566$	−1.02828e7	−1.34918
$567$	0	0
$568$	1.11570e7	1.45103
$569$	−7.52765e6	−0.974717	−0.487358	−	0.873202i	$-0.662039\pi$
$569$	−7.52765e6	−0.974717	−0.487358	+	0.873202i	$0.662039\pi$
$570$	0	0
$571$	1.01640e6	0.130459	0.0652294	−	0.997870i	$-0.479222\pi$
$571$	1.01640e6	0.130459	0.0652294	+	0.997870i	$0.479222\pi$
$572$	203099.	0.0259548
$573$	0	0
$574$	78365.8	0.00992765
$575$	−2.01803e6	−0.254541
$576$	0	0
$577$	1.13287e7	1.41657	0.708286	−	0.705925i	$-0.249469\pi$
$577$	1.13287e7	1.41657	0.708286	+	0.705925i	$0.249469\pi$
$578$	1.59983e6	0.199184
$579$	0	0
$580$	917809.	0.113288
$581$	7.34550e6	0.902779
$582$	0	0
$583$	2.40744e6	0.293349
$584$	−7.82863e6	−0.949847
$585$	0	0
$586$	−1.17591e7	−1.41458
$587$	8.72506e6	1.04514	0.522569	−	0.852597i	$-0.324974\pi$
$587$	8.72506e6	1.04514	0.522569	+	0.852597i	$0.324974\pi$
$588$	0	0
$589$	−1.58807e7	−1.88617
$590$	−214811.	−0.0254055
$591$	0	0
$592$	−1.94710e6	−0.228341
$593$	−5.90768e6	−0.689890	−0.344945	−	0.938623i	$-0.612102\pi$
$593$	−5.90768e6	−0.689890	−0.344945	+	0.938623i	$0.612102\pi$
$594$	0	0
$595$	7.53882e6	0.872993
$596$	572503.	0.0660180
$597$	0	0
$598$	−2.46850e6	−0.282280
$599$	2.18195e6	0.248472	0.124236	−	0.992253i	$-0.460352\pi$
$599$	2.18195e6	0.248472	0.124236	+	0.992253i	$0.460352\pi$
$600$	0	0
$601$	1.49332e7	1.68642	0.843210	−	0.537585i	$-0.180663\pi$
$601$	1.49332e7	1.68642	0.843210	+	0.537585i	$0.180663\pi$
$602$	−90734.5	−0.0102043
$603$	0	0
$604$	−267764.	−0.0298648
$605$	1.18367e7	1.31475
$606$	0	0
$607$	8.49835e6	0.936188	0.468094	−	0.883679i	$-0.344941\pi$
$607$	8.49835e6	0.936188	0.468094	+	0.883679i	$0.344941\pi$
$608$	−1.73526e6	−0.190373
$609$	0	0
$610$	2.42787e7	2.64180
$611$	−7.12157e6	−0.771743
$612$	0	0
$613$	−1.75648e7	−1.88795	−0.943977	−	0.330010i	$-0.892948\pi$
$613$	−1.75648e7	−1.88795	−0.943977	+	0.330010i	$0.892948\pi$
$614$	−9.68344e6	−1.03659
$615$	0	0
$616$	−2.04348e6	−0.216979
$617$	5.02553e6	0.531459	0.265729	−	0.964048i	$-0.414387\pi$
$617$	5.02553e6	0.531459	0.265729	+	0.964048i	$0.414387\pi$
$618$	0	0
$619$	−1.44445e7	−1.51522	−0.757611	−	0.652706i	$-0.773634\pi$
$619$	−1.44445e7	−1.51522	−0.757611	+	0.652706i	$0.773634\pi$
$620$	−931901.	−0.0973623
$621$	0	0
$622$	1.12396e7	1.16486
$623$	−2.90508e6	−0.299874
$624$	0	0
$625$	−7.13420e6	−0.730542
$626$	7.42035e6	0.756813
$627$	0	0
$628$	61088.5	0.00618102
$629$	−1.92999e6	−0.194504
$630$	0	0
$631$	6.28116e6	0.628010	0.314005	−	0.949421i	$-0.398329\pi$
$631$	6.28116e6	0.628010	0.314005	+	0.949421i	$0.398329\pi$
$632$	1.56554e7	1.55909
$633$	0	0
$634$	1.25150e7	1.23653
$635$	−1.18257e7	−1.16383
$636$	0	0
$637$	7.74386e6	0.756151
$638$	−4.80017e6	−0.466879
$639$	0	0
$640$	1.66373e7	1.60559
$641$	1.14060e7	1.09645	0.548226	−	0.836330i	$-0.315303\pi$
$641$	1.14060e7	1.09645	0.548226	+	0.836330i	$0.315303\pi$
$642$	0	0
$643$	1.26116e7	1.20293	0.601467	−	0.798898i	$-0.294583\pi$
$643$	1.26116e7	1.20293	0.601467	+	0.798898i	$0.294583\pi$
$644$	−82261.5	−0.00781596
$645$	0	0
$646$	−1.62455e7	−1.53162
$647$	1.78780e7	1.67903	0.839514	−	0.543339i	$-0.182840\pi$
$647$	1.78780e7	1.67903	0.839514	+	0.543339i	$0.182840\pi$
$648$	0	0
$649$	61042.3	0.00568878
$650$	1.78012e7	1.65259
$651$	0	0
$652$	−420154.	−0.0387070
$653$	−1.49962e7	−1.37625	−0.688125	−	0.725592i	$-0.741566\pi$
$653$	−1.49962e7	−1.37625	−0.688125	+	0.725592i	$0.741566\pi$
$654$	0	0
$655$	−2.22916e7	−2.03019
$656$	171936.	0.0155993
$657$	0	0
$658$	−4.36786e6	−0.393282
$659$	−4.42725e6	−0.397119	−0.198560	−	0.980089i	$-0.563626\pi$
$659$	−4.42725e6	−0.397119	−0.198560	+	0.980089i	$0.563626\pi$
$660$	0	0
$661$	−1.18881e7	−1.05830	−0.529148	−	0.848529i	$-0.677488\pi$
$661$	−1.18881e7	−1.05830	−0.529148	+	0.848529i	$0.677488\pi$
$662$	−1.12373e7	−0.996592
$663$	0	0
$664$	1.52377e7	1.34122
$665$	1.83903e7	1.61263
$666$	0	0
$667$	3.16995e6	0.275891
$668$	1.09648e6	0.0950736
$669$	0	0
$670$	−1.59601e7	−1.37356
$671$	−6.89921e6	−0.591552
$672$	0	0
$673$	1.21976e7	1.03810	0.519048	−	0.854745i	$-0.326286\pi$
$673$	1.21976e7	1.03810	0.519048	+	0.854745i	$0.326286\pi$
$674$	−4.96169e6	−0.420707
$675$	0	0
$676$	500460.	0.0421214
$677$	−6.53028e6	−0.547595	−0.273798	−	0.961787i	$-0.588280\pi$
$677$	−6.53028e6	−0.547595	−0.273798	+	0.961787i	$0.588280\pi$
$678$	0	0
$679$	1.06048e7	0.882727
$680$	1.56388e7	1.29697
$681$	0	0
$682$	4.87387e6	0.401248
$683$	894173.	0.0733449	0.0366725	−	0.999327i	$-0.488324\pi$
$683$	894173.	0.0733449	0.0366725	+	0.999327i	$0.488324\pi$
$684$	0	0
$685$	4.55927e6	0.371252
$686$	1.30186e7	1.05622
$687$	0	0
$688$	−199073.	−0.0160340
$689$	1.40241e7	1.12545
$690$	0	0
$691$	1.50110e7	1.19595	0.597976	−	0.801514i	$-0.295972\pi$
$691$	1.50110e7	1.19595	0.597976	+	0.801514i	$0.295972\pi$
$692$	−302905.	−0.0240459
$693$	0	0
$694$	−5.08734e6	−0.400951
$695$	−2.42891e7	−1.90743
$696$	0	0
$697$	170425.	0.0132877
$698$	7.53291e6	0.585227
$699$	0	0
$700$	593215.	0.0457580
$701$	1.26434e6	0.0971780	0.0485890	−	0.998819i	$-0.484528\pi$
$701$	1.26434e6	0.0971780	0.0485890	+	0.998819i	$0.484528\pi$
$702$	0	0
$703$	−4.70803e6	−0.359295
$704$	−4.22416e6	−0.321225
$705$	0	0
$706$	−1.80009e7	−1.35920
$707$	−1.69726e7	−1.27702
$708$	0	0
$709$	−1.87275e7	−1.39915	−0.699576	−	0.714558i	$-0.746628\pi$
$709$	−1.87275e7	−1.39915	−0.699576	+	0.714558i	$0.746628\pi$
$710$	3.08156e7	2.29416
$711$	0	0
$712$	−6.02640e6	−0.445511
$713$	−3.21862e6	−0.237108
$714$	0	0
$715$	−9.20235e6	−0.673184
$716$	817950.	0.0596271
$717$	0	0
$718$	−2.28433e7	−1.65367
$719$	−1.38371e7	−0.998211	−0.499105	−	0.866541i	$-0.666338\pi$
$719$	−1.38371e7	−0.998211	−0.499105	+	0.866541i	$0.666338\pi$
$720$	0	0
$721$	−3.99071e6	−0.285898
$722$	−2.52257e7	−1.80095
$723$	0	0
$724$	−1.45861e6	−0.103417
$725$	−2.28596e7	−1.61519
$726$	0	0
$727$	−2.89283e6	−0.202996	−0.101498	−	0.994836i	$-0.532364\pi$
$727$	−2.89283e6	−0.202996	−0.101498	+	0.994836i	$0.532364\pi$
$728$	−1.19039e7	−0.832453
$729$	0	0
$730$	−2.16226e7	−1.50176
$731$	−197324.	−0.0136579
$732$	0	0
$733$	1.54902e7	1.06487	0.532436	−	0.846471i	$-0.321277\pi$
$733$	1.54902e7	1.06487	0.532436	+	0.846471i	$0.321277\pi$
$734$	−1.47999e7	−1.01396
$735$	0	0
$736$	−351694.	−0.0239315
$737$	4.53533e6	0.307568
$738$	0	0
$739$	−1.45940e7	−0.983025	−0.491513	−	0.870870i	$-0.663556\pi$
$739$	−1.45940e7	−0.983025	−0.491513	+	0.870870i	$0.663556\pi$
$740$	−276274.	−0.0185464
$741$	0	0
$742$	8.60135e6	0.573531
$743$	−1.63620e7	−1.08734	−0.543669	−	0.839300i	$-0.682965\pi$
$743$	−1.63620e7	−1.08734	−0.543669	+	0.839300i	$0.682965\pi$
$744$	0	0
$745$	−2.59399e7	−1.71229
$746$	−2.52918e7	−1.66392
$747$	0	0
$748$	270899.	0.0177033
$749$	−1.32539e6	−0.0863255
$750$	0	0
$751$	−1.92035e7	−1.24245	−0.621226	−	0.783632i	$-0.713365\pi$
$751$	−1.92035e7	−1.24245	−0.621226	+	0.783632i	$0.713365\pi$
$752$	−9.58315e6	−0.617965
$753$	0	0
$754$	−2.79624e7	−1.79121
$755$	1.21323e7	0.774596
$756$	0	0
$757$	1.02273e7	0.648668	0.324334	−	0.945943i	$-0.394860\pi$
$757$	1.02273e7	0.648668	0.324334	+	0.945943i	$0.394860\pi$
$758$	4.18676e6	0.264670
$759$	0	0
$760$	3.81493e7	2.39581
$761$	−2.20154e7	−1.37805	−0.689024	−	0.724738i	$-0.741961\pi$
$761$	−2.20154e7	−1.37805	−0.689024	+	0.724738i	$0.741961\pi$
$762$	0	0
$763$	−1.03692e7	−0.644812
$764$	1.50752e6	0.0934394
$765$	0	0
$766$	3.11826e7	1.92017
$767$	355590.	0.0218253
$768$	0	0
$769$	−1.70212e7	−1.03795	−0.518974	−	0.854790i	$-0.673686\pi$
$769$	−1.70212e7	−1.03795	−0.518974	+	0.854790i	$0.673686\pi$
$770$	−5.64406e6	−0.343056
$771$	0	0
$772$	346408.	0.0209192
$773$	−1.01099e7	−0.608554	−0.304277	−	0.952584i	$-0.598415\pi$
$773$	−1.01099e7	−0.608554	−0.304277	+	0.952584i	$0.598415\pi$
$774$	0	0
$775$	2.32106e7	1.38813
$776$	2.19989e7	1.31143
$777$	0	0
$778$	−4.05508e6	−0.240188
$779$	415736.	0.0245456
$780$	0	0
$781$	−8.75677e6	−0.513708
$782$	−3.29255e6	−0.192538
$783$	0	0
$784$	1.04205e7	0.605480
$785$	−2.76790e6	−0.160316
$786$	0	0
$787$	−3.91665e6	−0.225412	−0.112706	−	0.993628i	$-0.535952\pi$
$787$	−3.91665e6	−0.225412	−0.112706	+	0.993628i	$0.535952\pi$
$788$	−1.28100e6	−0.0734909
$789$	0	0
$790$	4.32400e7	2.46501
$791$	−2.92156e6	−0.166025
$792$	0	0
$793$	−4.01899e7	−2.26952
$794$	−1.37736e7	−0.775349
$795$	0	0
$796$	−630430.	−0.0352658
$797$	1.68096e7	0.937374	0.468687	−	0.883364i	$-0.344727\pi$
$797$	1.68096e7	0.937374	0.468687	+	0.883364i	$0.344727\pi$
$798$	0	0
$799$	−9.49895e6	−0.526391
$800$	2.53618e6	0.140106
$801$	0	0
$802$	1.41463e7	0.776620
$803$	6.14443e6	0.336274
$804$	0	0
$805$	3.72724e6	0.202721
$806$	2.83917e7	1.53941
$807$	0	0
$808$	−3.52085e7	−1.89722
$809$	−1.06849e7	−0.573981	−0.286990	−	0.957933i	$-0.592655\pi$
$809$	−1.06849e7	−0.573981	−0.286990	+	0.957933i	$0.592655\pi$
$810$	0	0
$811$	1.17019e7	0.624748	0.312374	−	0.949959i	$-0.398876\pi$
$811$	1.17019e7	0.624748	0.312374	+	0.949959i	$0.398876\pi$
$812$	−931832.	−0.0495961
$813$	0	0
$814$	1.44492e6	0.0764332
$815$	1.90370e7	1.00393
$816$	0	0
$817$	−481353.	−0.0252295
$818$	−1.24774e6	−0.0651992
$819$	0	0
$820$	24395.9	0.00126702
$821$	6.83785e6	0.354048	0.177024	−	0.984207i	$-0.443353\pi$
$821$	6.83785e6	0.354048	0.177024	+	0.984207i	$0.443353\pi$
$822$	0	0
$823$	894041.	0.0460106	0.0230053	−	0.999735i	$-0.492677\pi$
$823$	894041.	0.0460106	0.0230053	+	0.999735i	$0.492677\pi$
$824$	−8.27845e6	−0.424748
$825$	0	0
$826$	218093.	0.0111222
$827$	1.05054e7	0.534131	0.267066	−	0.963678i	$-0.413946\pi$
$827$	1.05054e7	0.534131	0.267066	+	0.963678i	$0.413946\pi$
$828$	0	0
$829$	−7.05600e6	−0.356593	−0.178296	−	0.983977i	$-0.557059\pi$
$829$	−7.05600e6	−0.356593	−0.178296	+	0.983977i	$0.557059\pi$
$830$	4.20865e7	2.12054
$831$	0	0
$832$	−2.46070e7	−1.23240
$833$	1.03290e7	0.515756
$834$	0	0
$835$	−4.96812e7	−2.46590
$836$	660833.	0.0327021
$837$	0	0
$838$	1.61387e7	0.793884
$839$	−8.25855e6	−0.405041	−0.202521	−	0.979278i	$-0.564913\pi$
$839$	−8.25855e6	−0.405041	−0.202521	+	0.979278i	$0.564913\pi$
$840$	0	0
$841$	1.53970e7	0.750667
$842$	2.17727e7	1.05836
$843$	0	0
$844$	−754675.	−0.0364673
$845$	−2.26757e7	−1.09249
$846$	0	0
$847$	−1.20176e7	−0.575582
$848$	1.88715e7	0.901190
$849$	0	0
$850$	2.37437e7	1.12720
$851$	−954200.	−0.0451664
$852$	0	0
$853$	−1.92869e6	−0.0907590	−0.0453795	−	0.998970i	$-0.514450\pi$
$853$	−1.92869e6	−0.0907590	−0.0453795	+	0.998970i	$0.514450\pi$
$854$	−2.46496e7	−1.15655
$855$	0	0
$856$	−2.74943e6	−0.128250
$857$	1.54723e7	0.719620	0.359810	−	0.933026i	$-0.382842\pi$
$857$	1.54723e7	0.719620	0.359810	+	0.933026i	$0.382842\pi$
$858$	0	0
$859$	−7.58697e6	−0.350821	−0.175410	−	0.984495i	$-0.556125\pi$
$859$	−7.58697e6	−0.350821	−0.175410	+	0.984495i	$0.556125\pi$
$860$	−28246.4	−0.00130232
$861$	0	0
$862$	2.71577e7	1.24487
$863$	−2.10765e7	−0.963321	−0.481661	−	0.876358i	$-0.659966\pi$
$863$	−2.10765e7	−0.963321	−0.481661	+	0.876358i	$0.659966\pi$
$864$	0	0
$865$	1.37245e7	0.623674
$866$	−4.78648e6	−0.216881
$867$	0	0
$868$	946140.	0.0426242
$869$	−1.22874e7	−0.551964
$870$	0	0
$871$	2.64197e7	1.18000
$872$	−2.15102e7	−0.957971
$873$	0	0
$874$	−8.03189e6	−0.355663
$875$	−4.86018e6	−0.214601
$876$	0	0
$877$	3.91056e7	1.71688	0.858441	−	0.512913i	$-0.171434\pi$
$877$	3.91056e7	1.71688	0.858441	+	0.512913i	$0.171434\pi$
$878$	1.36216e7	0.596335
$879$	0	0
$880$	−1.23832e7	−0.539044
$881$	5.95068e6	0.258301	0.129151	−	0.991625i	$-0.458775\pi$
$881$	5.95068e6	0.258301	0.129151	+	0.991625i	$0.458775\pi$
$882$	0	0
$883$	3.10273e7	1.33919	0.669596	−	0.742726i	$-0.266467\pi$
$883$	3.10273e7	1.33919	0.669596	+	0.742726i	$0.266467\pi$
$884$	1.57807e6	0.0679195
$885$	0	0
$886$	−9.67022e6	−0.413859
$887$	6.33333e6	0.270286	0.135143	−	0.990826i	$-0.456851\pi$
$887$	6.33333e6	0.270286	0.135143	+	0.990826i	$0.456851\pi$
$888$	0	0
$889$	1.20063e7	0.509514
$890$	−1.66448e7	−0.704376
$891$	0	0
$892$	630441.	0.0265297
$893$	−2.31718e7	−0.972369
$894$	0	0
$895$	−3.70610e7	−1.54654
$896$	−1.68915e7	−0.702909
$897$	0	0
$898$	9.13008e6	0.377819
$899$	−3.64595e7	−1.50457
$900$	0	0
$901$	1.87057e7	0.767647
$902$	−127591.	−0.00522162
$903$	0	0
$904$	−6.06057e6	−0.246656
$905$	6.60890e7	2.68230
$906$	0	0
$907$	6.19943e6	0.250227	0.125113	−	0.992142i	$-0.460071\pi$
$907$	6.19943e6	0.250227	0.125113	+	0.992142i	$0.460071\pi$
$908$	−180549.	−0.00726742
$909$	0	0
$910$	−3.28783e7	−1.31615
$911$	3.51643e7	1.40380	0.701902	−	0.712274i	$-0.252334\pi$
$911$	3.51643e7	1.40380	0.701902	+	0.712274i	$0.252334\pi$
$912$	0	0
$913$	−1.19596e7	−0.474832
$914$	255605.	0.0101206
$915$	0	0
$916$	653450.	0.0257320
$917$	2.26322e7	0.888797
$918$	0	0
$919$	3.98124e7	1.55500	0.777499	−	0.628885i	$-0.216488\pi$
$919$	3.98124e7	1.55500	0.777499	+	0.628885i	$0.216488\pi$
$920$	7.73191e6	0.301174
$921$	0	0
$922$	−2.53213e7	−0.980975
$923$	−5.10108e7	−1.97087
$924$	0	0
$925$	6.88105e6	0.264424
$926$	1.43971e7	0.551756
$927$	0	0
$928$	−3.98388e6	−0.151857
$929$	3.07640e6	0.116951	0.0584754	−	0.998289i	$-0.481376\pi$
$929$	3.07640e6	0.116951	0.0584754	+	0.998289i	$0.481376\pi$
$930$	0	0
$931$	2.51966e7	0.952725
$932$	−2.09632e6	−0.0790531
$933$	0	0
$934$	−1.73668e7	−0.651406
$935$	−1.22743e7	−0.459165
$936$	0	0
$937$	2.36250e7	0.879067	0.439534	−	0.898226i	$-0.355144\pi$
$937$	2.36250e7	0.879067	0.439534	+	0.898226i	$0.355144\pi$
$938$	1.62039e7	0.601331
$939$	0	0
$940$	−1.35975e6	−0.0501927
$941$	4.14719e6	0.152679	0.0763397	−	0.997082i	$-0.475677\pi$
$941$	4.14719e6	0.152679	0.0763397	+	0.997082i	$0.475677\pi$
$942$	0	0
$943$	84259.2	0.00308559
$944$	478500.	0.0174764
$945$	0	0
$946$	147730.	0.00536710
$947$	3.91993e7	1.42038	0.710188	−	0.704012i	$-0.248610\pi$
$947$	3.91993e7	1.42038	0.710188	+	0.704012i	$0.248610\pi$
$948$	0	0
$949$	3.57931e7	1.29013
$950$	5.79206e7	2.08221
$951$	0	0
$952$	−1.58777e7	−0.567800
$953$	9.84683e6	0.351208	0.175604	−	0.984461i	$-0.443812\pi$
$953$	9.84683e6	0.351208	0.175604	+	0.984461i	$0.443812\pi$
$954$	0	0
$955$	−6.83053e7	−2.42352
$956$	−1.45364e6	−0.0514414
$957$	0	0
$958$	1.07735e7	0.379264
$959$	−4.62893e6	−0.162530
$960$	0	0
$961$	8.39018e6	0.293064
$962$	8.41708e6	0.293240
$963$	0	0
$964$	1.05455e6	0.0365491
$965$	−1.56956e7	−0.542576
$966$	0	0
$967$	−2.14979e7	−0.739315	−0.369658	−	0.929168i	$-0.620525\pi$
$967$	−2.14979e7	−0.739315	−0.369658	+	0.929168i	$0.620525\pi$
$968$	−2.49296e7	−0.855120
$969$	0	0
$970$	6.07606e7	2.07344
$971$	3.13999e7	1.06876	0.534379	−	0.845245i	$-0.320545\pi$
$971$	3.13999e7	1.06876	0.534379	+	0.845245i	$0.320545\pi$
$972$	0	0
$973$	2.46602e7	0.835053
$974$	−7.24073e6	−0.244560
$975$	0	0
$976$	−5.40816e7	−1.81729
$977$	−2.96815e7	−0.994831	−0.497416	−	0.867512i	$-0.665718\pi$
$977$	−2.96815e7	−0.994831	−0.497416	+	0.867512i	$0.665718\pi$
$978$	0	0
$979$	4.72992e6	0.157724
$980$	1.47857e6	0.0491786
$981$	0	0
$982$	2.72813e7	0.902790
$983$	−282449.	−0.00932301	−0.00466151	−	0.999989i	$-0.501484\pi$
$983$	−282449.	−0.00932301	−0.00466151	+	0.999989i	$0.501484\pi$
$984$	0	0
$985$	5.80417e7	1.90612
$986$	−3.72970e7	−1.22175
$987$	0	0
$988$	3.84955e6	0.125464
$989$	−97558.1	−0.00317156
$990$	0	0
$991$	−1.15923e7	−0.374959	−0.187480	−	0.982268i	$-0.560032\pi$
$991$	−1.15923e7	−0.374959	−0.187480	+	0.982268i	$0.560032\pi$
$992$	4.04505e6	0.130510
$993$	0	0
$994$	−3.12864e7	−1.00436
$995$	2.85646e7	0.914681
$996$	0	0
$997$	5.68352e7	1.81084	0.905420	−	0.424518i	$-0.139556\pi$
$997$	5.68352e7	1.81084	0.905420	+	0.424518i	$0.139556\pi$
$998$	5.04941e7	1.60478
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.g.1.3		6
3.2	odd	2		23.6.a.b.1.4	✓	6
12.11	even	2		368.6.a.h.1.3		6
15.14	odd	2		575.6.a.c.1.3		6
69.68	even	2		529.6.a.c.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.6.a.b.1.4	✓	6	3.2	odd	2
207.6.a.g.1.3		6	1.1	even	1	trivial
368.6.a.h.1.3		6	12.11	even	2
529.6.a.c.1.4		6	69.68	even	2
575.6.a.c.1.3		6	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)

\(f(q)\)	\(=\)	\(q-5.81709 q^{2} +1.83858 q^{4} -83.3054 q^{5} +84.5782 q^{7} +175.452 q^{8} +O(q^{10})\)	\(q-5.81709 q^{2} +1.83858 q^{4} -83.3054 q^{5} +84.5782 q^{7} +175.452 q^{8} +484.596 q^{10} -137.706 q^{11} -802.180 q^{13} -492.000 q^{14} -1079.45 q^{16} -1069.97 q^{17} -2610.09 q^{19} -153.164 q^{20} +801.050 q^{22} -529.000 q^{23} +3814.80 q^{25} +4666.35 q^{26} +155.504 q^{28} -5992.34 q^{29} +6084.35 q^{31} +664.828 q^{32} +6224.11 q^{34} -7045.83 q^{35} +1803.78 q^{37} +15183.1 q^{38} -14616.1 q^{40} -159.280 q^{41} +184.420 q^{43} -253.184 q^{44} +3077.24 q^{46} +8877.78 q^{47} -9653.52 q^{49} -22191.0 q^{50} -1474.87 q^{52} -17482.4 q^{53} +11471.7 q^{55} +14839.4 q^{56} +34858.0 q^{58} -443.279 q^{59} +50100.9 q^{61} -35393.2 q^{62} +30675.2 q^{64} +66825.9 q^{65} -32934.8 q^{67} -1967.22 q^{68} +40986.2 q^{70} +63590.3 q^{71} -44619.8 q^{73} -10492.8 q^{74} -4798.86 q^{76} -11647.0 q^{77} +89229.0 q^{79} +89924.4 q^{80} +926.547 q^{82} +86848.6 q^{83} +89134.2 q^{85} -1072.79 q^{86} -24160.8 q^{88} -34347.9 q^{89} -67846.9 q^{91} -972.609 q^{92} -51642.9 q^{94} +217435. q^{95} +125384. q^{97} +56155.4 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8}+O(q^{10}) \) 6 * q - 4 * q^2 + 112 * q^4 - 42 * q^5 + 300 * q^7 - 393 * q^8	\( 6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8} - 10 q^{10} + 58 q^{11} + 792 q^{13} + 2984 q^{14} - 1904 q^{16} + 400 q^{17} + 2738 q^{19} + 7124 q^{20} - 1972 q^{22} - 3174 q^{23} + 9966 q^{25} - 4511 q^{26} + 9570 q^{28} - 11244 q^{29} + 13748 q^{31} - 6600 q^{32} - 16226 q^{34} + 4296 q^{35} + 25426 q^{37} + 8028 q^{38} + 10230 q^{40} + 14268 q^{41} - 18082 q^{43} + 51146 q^{44} + 2116 q^{46} + 23084 q^{47} + 37422 q^{49} + 67436 q^{50} + 36807 q^{52} - 17522 q^{53} + 47576 q^{55} - 44946 q^{56} + 141001 q^{58} + 36392 q^{59} + 27062 q^{61} - 48971 q^{62} + 89451 q^{64} - 7108 q^{65} + 37138 q^{67} - 17260 q^{68} + 248380 q^{70} + 158556 q^{71} + 112228 q^{73} + 66878 q^{74} + 157816 q^{76} + 89760 q^{77} + 36844 q^{79} + 158530 q^{80} + 150039 q^{82} + 76350 q^{83} - 102132 q^{85} + 100578 q^{86} - 219028 q^{88} - 16100 q^{89} - 250592 q^{91} - 59248 q^{92} + 12887 q^{94} + 190096 q^{95} + 259432 q^{97} + 325816 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 + 112 * q^4 - 42 * q^5 + 300 * q^7 - 393 * q^8 - 10 * q^10 + 58 * q^11 + 792 * q^13 + 2984 * q^14 - 1904 * q^16 + 400 * q^17 + 2738 * q^19 + 7124 * q^20 - 1972 * q^22 - 3174 * q^23 + 9966 * q^25 - 4511 * q^26 + 9570 * q^28 - 11244 * q^29 + 13748 * q^31 - 6600 * q^32 - 16226 * q^34 + 4296 * q^35 + 25426 * q^37 + 8028 * q^38 + 10230 * q^40 + 14268 * q^41 - 18082 * q^43 + 51146 * q^44 + 2116 * q^46 + 23084 * q^47 + 37422 * q^49 + 67436 * q^50 + 36807 * q^52 - 17522 * q^53 + 47576 * q^55 - 44946 * q^56 + 141001 * q^58 + 36392 * q^59 + 27062 * q^61 - 48971 * q^62 + 89451 * q^64 - 7108 * q^65 + 37138 * q^67 - 17260 * q^68 + 248380 * q^70 + 158556 * q^71 + 112228 * q^73 + 66878 * q^74 + 157816 * q^76 + 89760 * q^77 + 36844 * q^79 + 158530 * q^80 + 150039 * q^82 + 76350 * q^83 - 102132 * q^85 + 100578 * q^86 - 219028 * q^88 - 16100 * q^89 - 250592 * q^91 - 59248 * q^92 + 12887 * q^94 + 190096 * q^95 + 259432 * q^97 + 325816 * q^98

Introduction

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Database

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