LMFDB - Embedded newform 207.6.a.d.1.4

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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 39x^{2} - 30x + 20 $ x^4 - 39x^2 - 30x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-5.76108$ 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)

$f(q)$	$=$	$q+4.76108 q^{2} -9.33211 q^{4} +97.1410 q^{5} -211.220 q^{7} -196.786 q^{8} +O(q^{10})$	\(q+4.76108 q^{2} -9.33211 q^{4} +97.1410 q^{5} -211.220 q^{7} -196.786 q^{8} +462.496 q^{10} +633.540 q^{11} -59.3594 q^{13} -1005.63 q^{14} -638.284 q^{16} +1809.15 q^{17} +1986.94 q^{19} -906.530 q^{20} +3016.33 q^{22} -529.000 q^{23} +6311.37 q^{25} -282.615 q^{26} +1971.12 q^{28} +2631.22 q^{29} -4916.81 q^{31} +3258.21 q^{32} +8613.53 q^{34} -20518.1 q^{35} -4871.13 q^{37} +9459.96 q^{38} -19115.9 q^{40} +13838.6 q^{41} -1272.49 q^{43} -5912.26 q^{44} -2518.61 q^{46} +17922.1 q^{47} +27806.7 q^{49} +30048.9 q^{50} +553.948 q^{52} +19517.0 q^{53} +61542.6 q^{55} +41565.0 q^{56} +12527.5 q^{58} +17569.0 q^{59} -20453.8 q^{61} -23409.3 q^{62} +35937.7 q^{64} -5766.23 q^{65} -11300.7 q^{67} -16883.2 q^{68} -97688.2 q^{70} -33263.2 q^{71} +22571.9 q^{73} -23191.9 q^{74} -18542.3 q^{76} -133816. q^{77} -9902.06 q^{79} -62003.6 q^{80} +65886.5 q^{82} +24218.1 q^{83} +175743. q^{85} -6058.43 q^{86} -124671. q^{88} -40463.9 q^{89} +12537.9 q^{91} +4936.69 q^{92} +85328.5 q^{94} +193013. q^{95} -18327.4 q^{97} +132390. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8	$ 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8} + 642 q^{10} - 32 q^{11} + 1364 q^{13} - 2754 q^{14} + 18 q^{16} - 278 q^{17} + 2862 q^{19} - 3830 q^{20} + 3176 q^{22} - 2116 q^{23} + 5944 q^{25} - 6996 q^{26} + 4738 q^{28} + 5180 q^{29} - 1788 q^{31} + 7352 q^{32} + 15818 q^{34} - 11768 q^{35} + 3348 q^{37} - 1050 q^{38} - 13462 q^{40} + 17664 q^{41} + 25398 q^{43} + 16848 q^{44} + 2116 q^{46} + 26040 q^{47} + 55720 q^{49} + 35256 q^{50} - 2752 q^{52} + 32006 q^{53} + 34904 q^{55} + 68542 q^{56} - 40804 q^{58} + 61136 q^{59} + 35844 q^{61} + 47524 q^{62} - 35142 q^{64} + 48036 q^{65} + 73458 q^{67} - 17910 q^{68} - 59104 q^{70} - 24432 q^{71} + 122512 q^{73} + 20828 q^{74} + 56834 q^{76} - 159496 q^{77} + 90170 q^{79} + 36546 q^{80} + 84144 q^{82} - 28592 q^{83} + 355124 q^{85} - 103778 q^{86} - 150776 q^{88} + 27926 q^{89} + 334180 q^{91} + 24334 q^{92} + 113632 q^{94} - 113392 q^{95} + 16580 q^{97} - 49688 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8 + 642 * q^10 - 32 * q^11 + 1364 * q^13 - 2754 * q^14 + 18 * q^16 - 278 * q^17 + 2862 * q^19 - 3830 * q^20 + 3176 * q^22 - 2116 * q^23 + 5944 * q^25 - 6996 * q^26 + 4738 * q^28 + 5180 * q^29 - 1788 * q^31 + 7352 * q^32 + 15818 * q^34 - 11768 * q^35 + 3348 * q^37 - 1050 * q^38 - 13462 * q^40 + 17664 * q^41 + 25398 * q^43 + 16848 * q^44 + 2116 * q^46 + 26040 * q^47 + 55720 * q^49 + 35256 * q^50 - 2752 * q^52 + 32006 * q^53 + 34904 * q^55 + 68542 * q^56 - 40804 * q^58 + 61136 * q^59 + 35844 * q^61 + 47524 * q^62 - 35142 * q^64 + 48036 * q^65 + 73458 * q^67 - 17910 * q^68 - 59104 * q^70 - 24432 * q^71 + 122512 * q^73 + 20828 * q^74 + 56834 * q^76 - 159496 * q^77 + 90170 * q^79 + 36546 * q^80 + 84144 * q^82 - 28592 * q^83 + 355124 * q^85 - 103778 * q^86 - 150776 * q^88 + 27926 * q^89 + 334180 * q^91 + 24334 * q^92 + 113632 * q^94 - 113392 * q^95 + 16580 * q^97 - 49688 * 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 $

$2$	4.76108	0.841648	0.420824	−	0.907142i	$-0.361741\pi$
$2$	4.76108	0.841648	0.420824	+	0.907142i	$0.361741\pi$
$3$	0	0
$3$	0	0
$4$	−9.33211	−0.291628
$5$	97.1410	1.73771	0.868855	−	0.495066i	$-0.164856\pi$
$5$	97.1410	1.73771	0.868855	+	0.495066i	$0.164856\pi$
$6$	0	0
$7$	−211.220	−1.62926	−0.814628	−	0.579985i	$-0.803059\pi$
$7$	−211.220	−1.62926	−0.814628	+	0.579985i	$0.803059\pi$
$8$	−196.786	−1.08710
$9$	0	0
$10$	462.496	1.46254
$11$	633.540	1.57867	0.789336	−	0.613961i	$-0.210425\pi$
$11$	633.540	1.57867	0.789336	+	0.613961i	$0.210425\pi$
$12$	0	0
$13$	−59.3594	−0.0974162	−0.0487081	−	0.998813i	$-0.515510\pi$
$13$	−59.3594	−0.0974162	−0.0487081	+	0.998813i	$0.515510\pi$
$14$	−1005.63	−1.37126
$15$	0	0
$16$	−638.284	−0.623325
$17$	1809.15	1.51828	0.759142	−	0.650925i	$-0.225619\pi$
$17$	1809.15	1.51828	0.759142	+	0.650925i	$0.225619\pi$
$18$	0	0
$19$	1986.94	1.26270	0.631350	−	0.775498i	$-0.282501\pi$
$19$	1986.94	1.26270	0.631350	+	0.775498i	$0.282501\pi$
$20$	−906.530	−0.506766
$21$	0	0
$22$	3016.33	1.32869
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	6311.37	2.01964
$26$	−282.615	−0.0819902
$27$	0	0
$28$	1971.12	0.475137
$29$	2631.22	0.580982	0.290491	−	0.956878i	$-0.406181\pi$
$29$	2631.22	0.580982	0.290491	+	0.956878i	$0.406181\pi$
$30$	0	0
$31$	−4916.81	−0.918923	−0.459462	−	0.888198i	$-0.651958\pi$
$31$	−4916.81	−0.918923	−0.459462	+	0.888198i	$0.651958\pi$
$32$	3258.21	0.562477
$33$	0	0
$34$	8613.53	1.27786
$35$	−20518.1	−2.83117
$36$	0	0
$37$	−4871.13	−0.584959	−0.292480	−	0.956272i	$-0.594480\pi$
$37$	−4871.13	−0.584959	−0.292480	+	0.956272i	$0.594480\pi$
$38$	9459.96	1.06275
$39$	0	0
$40$	−19115.9	−1.88906
$41$	13838.6	1.28567	0.642837	−	0.766003i	$-0.277757\pi$
$41$	13838.6	1.28567	0.642837	+	0.766003i	$0.277757\pi$
$42$	0	0
$43$	−1272.49	−0.104950	−0.0524751	−	0.998622i	$-0.516711\pi$
$43$	−1272.49	−0.104950	−0.0524751	+	0.998622i	$0.516711\pi$
$44$	−5912.26	−0.460386
$45$	0	0
$46$	−2518.61	−0.175496
$47$	17922.1	1.18343	0.591717	−	0.806146i	$-0.298450\pi$
$47$	17922.1	1.18343	0.591717	+	0.806146i	$0.298450\pi$
$48$	0	0
$49$	27806.7	1.65447
$50$	30048.9	1.69982
$51$	0	0
$52$	553.948	0.0284093
$53$	19517.0	0.954383	0.477192	−	0.878799i	$-0.341655\pi$
$53$	19517.0	0.954383	0.477192	+	0.878799i	$0.341655\pi$
$54$	0	0
$55$	61542.6	2.74328
$56$	41565.0	1.77116
$57$	0	0
$58$	12527.5	0.488982
$59$	17569.0	0.657078	0.328539	−	0.944490i	$-0.393444\pi$
$59$	17569.0	0.657078	0.328539	+	0.944490i	$0.393444\pi$
$60$	0	0
$61$	−20453.8	−0.703801	−0.351901	−	0.936037i	$-0.614464\pi$
$61$	−20453.8	−0.703801	−0.351901	+	0.936037i	$0.614464\pi$
$62$	−23409.3	−0.773410
$63$	0	0
$64$	35937.7	1.09673
$65$	−5766.23	−0.169281
$66$	0	0
$67$	−11300.7	−0.307552	−0.153776	−	0.988106i	$-0.549143\pi$
$67$	−11300.7	−0.307552	−0.153776	+	0.988106i	$0.549143\pi$
$68$	−16883.2	−0.442775
$69$	0	0
$70$	−97688.2	−2.38285
$71$	−33263.2	−0.783102	−0.391551	−	0.920156i	$-0.628061\pi$
$71$	−33263.2	−0.783102	−0.391551	+	0.920156i	$0.628061\pi$
$72$	0	0
$73$	22571.9	0.495749	0.247874	−	0.968792i	$-0.420268\pi$
$73$	22571.9	0.495749	0.247874	+	0.968792i	$0.420268\pi$
$74$	−23191.9	−0.492330
$75$	0	0
$76$	−18542.3	−0.368239
$77$	−133816.	−2.57206
$78$	0	0
$79$	−9902.06	−0.178508	−0.0892540	−	0.996009i	$-0.528448\pi$
$79$	−9902.06	−0.178508	−0.0892540	+	0.996009i	$0.528448\pi$
$80$	−62003.6	−1.08316
$81$	0	0
$82$	65886.5	1.08209
$83$	24218.1	0.385873	0.192937	−	0.981211i	$-0.438199\pi$
$83$	24218.1	0.385873	0.192937	+	0.981211i	$0.438199\pi$
$84$	0	0
$85$	175743.	2.63834
$86$	−6058.43	−0.0883311
$87$	0	0
$88$	−124671.	−1.71617
$89$	−40463.9	−0.541493	−0.270746	−	0.962651i	$-0.587270\pi$
$89$	−40463.9	−0.541493	−0.270746	+	0.962651i	$0.587270\pi$
$90$	0	0
$91$	12537.9	0.158716
$92$	4936.69	0.0608087
$93$	0	0
$94$	85328.5	0.996035
$95$	193013.	2.19421
$96$	0	0
$97$	−18327.4	−0.197776	−0.0988878	−	0.995099i	$-0.531528\pi$
$97$	−18327.4	−0.197776	−0.0988878	+	0.995099i	$0.531528\pi$
$98$	132390.	1.39248
$99$	0	0
$100$	−58898.4	−0.588984
$101$	−135506.	−1.32177	−0.660886	−	0.750486i	$-0.729819\pi$
$101$	−135506.	−1.32177	−0.660886	+	0.750486i	$0.729819\pi$
$102$	0	0
$103$	13708.4	0.127319	0.0636595	−	0.997972i	$-0.479723\pi$
$103$	13708.4	0.127319	0.0636595	+	0.997972i	$0.479723\pi$
$104$	11681.1	0.105901
$105$	0	0
$106$	92921.9	0.803255
$107$	−193324.	−1.63240	−0.816200	−	0.577770i	$-0.803923\pi$
$107$	−193324.	−1.63240	−0.816200	+	0.577770i	$0.803923\pi$
$108$	0	0
$109$	−116451.	−0.938804	−0.469402	−	0.882984i	$-0.655531\pi$
$109$	−116451.	−0.938804	−0.469402	+	0.882984i	$0.655531\pi$
$110$	293010.	2.30887
$111$	0	0
$112$	134818.	1.01555
$113$	179696.	1.32386	0.661929	−	0.749567i	$-0.269738\pi$
$113$	179696.	1.32386	0.661929	+	0.749567i	$0.269738\pi$
$114$	0	0
$115$	−51387.6	−0.362338
$116$	−24554.9	−0.169431
$117$	0	0
$118$	83647.4	0.553028
$119$	−382129.	−2.47367
$120$	0	0
$121$	240321.	1.49221
$122$	−97382.3	−0.592353
$123$	0	0
$124$	45884.2	0.267984
$125$	309527.	1.77183
$126$	0	0
$127$	−246258.	−1.35482	−0.677408	−	0.735607i	$-0.736897\pi$
$127$	−246258.	−1.35482	−0.677408	+	0.735607i	$0.736897\pi$
$128$	66839.5	0.360586
$129$	0	0
$130$	−27453.5	−0.142475
$131$	−68004.2	−0.346224	−0.173112	−	0.984902i	$-0.555382\pi$
$131$	−68004.2	−0.346224	−0.173112	+	0.984902i	$0.555382\pi$
$132$	0	0
$133$	−419680.	−2.05726
$134$	−53803.6	−0.258851
$135$	0	0
$136$	−356015.	−1.65052
$137$	378451.	1.72269	0.861346	−	0.508018i	$-0.169622\pi$
$137$	378451.	1.72269	0.861346	+	0.508018i	$0.169622\pi$
$138$	0	0
$139$	−112126.	−0.492231	−0.246115	−	0.969241i	$-0.579154\pi$
$139$	−112126.	−0.492231	−0.246115	+	0.969241i	$0.579154\pi$
$140$	191477.	0.825651
$141$	0	0
$142$	−158369.	−0.659096
$143$	−37606.5	−0.153788
$144$	0	0
$145$	255599.	1.00958
$146$	107467.	0.417246
$147$	0	0
$148$	45458.0	0.170591
$149$	−290306.	−1.07125	−0.535625	−	0.844456i	$-0.679924\pi$
$149$	−290306.	−1.07125	−0.535625	+	0.844456i	$0.679924\pi$
$150$	0	0
$151$	125549.	0.448095	0.224048	−	0.974578i	$-0.428073\pi$
$151$	125549.	0.448095	0.224048	+	0.974578i	$0.428073\pi$
$152$	−391000.	−1.37268
$153$	0	0
$154$	−637109.	−2.16477
$155$	−477624.	−1.59682
$156$	0	0
$157$	−165073.	−0.534473	−0.267237	−	0.963631i	$-0.586111\pi$
$157$	−165073.	−0.534473	−0.267237	+	0.963631i	$0.586111\pi$
$158$	−47144.5	−0.150241
$159$	0	0
$160$	316506.	0.977422
$161$	111735.	0.339723
$162$	0	0
$163$	330262.	0.973620	0.486810	−	0.873508i	$-0.338160\pi$
$163$	330262.	0.973620	0.486810	+	0.873508i	$0.338160\pi$
$164$	−129143.	−0.374939
$165$	0	0
$166$	115304.	0.324769
$167$	21121.5	0.0586048	0.0293024	−	0.999571i	$-0.490671\pi$
$167$	21121.5	0.0586048	0.0293024	+	0.999571i	$0.490671\pi$
$168$	0	0
$169$	−367769.	−0.990510
$170$	836726.	2.22055
$171$	0	0
$172$	11875.0	0.0306065
$173$	519214.	1.31896	0.659480	−	0.751722i	$-0.270777\pi$
$173$	519214.	1.31896	0.659480	+	0.751722i	$0.270777\pi$
$174$	0	0
$175$	−1.33308e6	−3.29050
$176$	−404378.	−0.984025
$177$	0	0
$178$	−192652.	−0.455746
$179$	−420339.	−0.980544	−0.490272	−	0.871570i	$-0.663102\pi$
$179$	−420339.	−0.980544	−0.490272	+	0.871570i	$0.663102\pi$
$180$	0	0
$181$	79336.5	0.180002	0.0900008	−	0.995942i	$-0.471313\pi$
$181$	79336.5	0.180002	0.0900008	+	0.995942i	$0.471313\pi$
$182$	59693.8	0.133583
$183$	0	0
$184$	104100.	0.226675
$185$	−473187.	−1.01649
$186$	0	0
$187$	1.14617e6	2.39687
$188$	−167251.	−0.345123
$189$	0	0
$190$	918950.	1.84675
$191$	−409572.	−0.812357	−0.406179	−	0.913794i	$-0.633139\pi$
$191$	−409572.	−0.812357	−0.406179	+	0.913794i	$0.633139\pi$
$192$	0	0
$193$	424832.	0.820963	0.410482	−	0.911869i	$-0.365361\pi$
$193$	424832.	0.820963	0.410482	+	0.911869i	$0.365361\pi$
$194$	−87258.5	−0.166457
$195$	0	0
$196$	−259495.	−0.482491
$197$	716052.	1.31456	0.657278	−	0.753648i	$-0.271708\pi$
$197$	716052.	1.31456	0.657278	+	0.753648i	$0.271708\pi$
$198$	0	0
$199$	−636722.	−1.13977	−0.569885	−	0.821724i	$-0.693012\pi$
$199$	−636722.	−1.13977	−0.569885	+	0.821724i	$0.693012\pi$
$200$	−1.24199e6	−2.19554
$201$	0	0
$202$	−645157.	−1.11247
$203$	−555766.	−0.946568
$204$	0	0
$205$	1.34429e6	2.23413
$206$	65266.7	0.107158
$207$	0	0
$208$	37888.2	0.0607219
$209$	1.25880e6	1.99339
$210$	0	0
$211$	694318.	1.07362	0.536812	−	0.843702i	$-0.319628\pi$
$211$	694318.	1.07362	0.536812	+	0.843702i	$0.319628\pi$
$212$	−182134.	−0.278325
$213$	0	0
$214$	−920432.	−1.37391
$215$	−123611.	−0.182373
$216$	0	0
$217$	1.03853e6	1.49716
$218$	−554430.	−0.790143
$219$	0	0
$220$	−574323.	−0.800017
$221$	−107390.	−0.147906
$222$	0	0
$223$	−555721.	−0.748333	−0.374166	−	0.927362i	$-0.622071\pi$
$223$	−555721.	−0.748333	−0.374166	+	0.927362i	$0.622071\pi$
$224$	−688198.	−0.916418
$225$	0	0
$226$	855545.	1.11422
$227$	513115.	0.660922	0.330461	−	0.943820i	$-0.392796\pi$
$227$	513115.	0.660922	0.330461	+	0.943820i	$0.392796\pi$
$228$	0	0
$229$	−1.29876e6	−1.63659	−0.818296	−	0.574798i	$-0.805081\pi$
$229$	−1.29876e6	−1.63659	−0.818296	+	0.574798i	$0.805081\pi$
$230$	−244660.	−0.304961
$231$	0	0
$232$	−517786.	−0.631583
$233$	522889.	0.630986	0.315493	−	0.948928i	$-0.397830\pi$
$233$	522889.	0.630986	0.315493	+	0.948928i	$0.397830\pi$
$234$	0	0
$235$	1.74097e6	2.05646
$236$	−163956.	−0.191623
$237$	0	0
$238$	−1.81935e6	−2.08196
$239$	−1.02363e6	−1.15917	−0.579587	−	0.814910i	$-0.696786\pi$
$239$	−1.02363e6	−1.15917	−0.579587	+	0.814910i	$0.696786\pi$
$240$	0	0
$241$	−1.66479e6	−1.84636	−0.923180	−	0.384367i	$-0.874420\pi$
$241$	−1.66479e6	−1.84636	−0.923180	+	0.384367i	$0.874420\pi$
$242$	1.14419e6	1.25591
$243$	0	0
$244$	190877.	0.205248
$245$	2.70117e6	2.87499
$246$	0	0
$247$	−117943.	−0.123007
$248$	967557.	0.998958
$249$	0	0
$250$	1.47368e6	1.49126
$251$	−1.05735e6	−1.05934	−0.529671	−	0.848203i	$-0.677685\pi$
$251$	−1.05735e6	−1.05934	−0.529671	+	0.848203i	$0.677685\pi$
$252$	0	0
$253$	−335142.	−0.329176
$254$	−1.17245e6	−1.14028
$255$	0	0
$256$	−831778.	−0.793246
$257$	1.06720e6	1.00789	0.503943	−	0.863737i	$-0.331882\pi$
$257$	1.06720e6	1.00789	0.503943	+	0.863737i	$0.331882\pi$
$258$	0	0
$259$	1.02888e6	0.953048
$260$	53811.1	0.0493672
$261$	0	0
$262$	−323773.	−0.291399
$263$	1.72834e6	1.54077	0.770387	−	0.637577i	$-0.220063\pi$
$263$	1.72834e6	1.54077	0.770387	+	0.637577i	$0.220063\pi$
$264$	0	0
$265$	1.89590e6	1.65844
$266$	−1.99813e6	−1.73149
$267$	0	0
$268$	105459.	0.0896909
$269$	−447676.	−0.377210	−0.188605	−	0.982053i	$-0.560397\pi$
$269$	−447676.	−0.377210	−0.188605	+	0.982053i	$0.560397\pi$
$270$	0	0
$271$	−1.72413e6	−1.42609	−0.713047	−	0.701117i	$-0.752685\pi$
$271$	−1.72413e6	−1.42609	−0.713047	+	0.701117i	$0.752685\pi$
$272$	−1.15475e6	−0.946384
$273$	0	0
$274$	1.80183e6	1.44990
$275$	3.99850e6	3.18835
$276$	0	0
$277$	−114905.	−0.0899784	−0.0449892	−	0.998987i	$-0.514325\pi$
$277$	−114905.	−0.0899784	−0.0449892	+	0.998987i	$0.514325\pi$
$278$	−533840.	−0.414285
$279$	0	0
$280$	4.03766e6	3.07776
$281$	2.02575e6	1.53046	0.765228	−	0.643760i	$-0.222626\pi$
$281$	2.02575e6	1.53046	0.765228	+	0.643760i	$0.222626\pi$
$282$	0	0
$283$	−1.29230e6	−0.959177	−0.479589	−	0.877493i	$-0.659214\pi$
$283$	−1.29230e6	−0.959177	−0.479589	+	0.877493i	$0.659214\pi$
$284$	310416.	0.228375
$285$	0	0
$286$	−179048.	−0.129436
$287$	−2.92297e6	−2.09469
$288$	0	0
$289$	1.85318e6	1.30519
$290$	1.21693e6	0.849710
$291$	0	0
$292$	−210644.	−0.144574
$293$	838415.	0.570545	0.285272	−	0.958446i	$-0.407916\pi$
$293$	838415.	0.570545	0.285272	+	0.958446i	$0.407916\pi$
$294$	0	0
$295$	1.70667e6	1.14181
$296$	958569.	0.635908
$297$	0	0
$298$	−1.38217e6	−0.901616
$299$	31401.1	0.0203127
$300$	0	0
$301$	268775.	0.170991
$302$	597748.	0.377139
$303$	0	0
$304$	−1.26823e6	−0.787071
$305$	−1.98690e6	−1.22300
$306$	0	0
$307$	1.15370e6	0.698632	0.349316	−	0.937005i	$-0.386414\pi$
$307$	1.15370e6	0.698632	0.349316	+	0.937005i	$0.386414\pi$
$308$	1.24878e6	0.750086
$309$	0	0
$310$	−2.27401e6	−1.34396
$311$	−1.46590e6	−0.859414	−0.429707	−	0.902968i	$-0.641383\pi$
$311$	−1.46590e6	−0.859414	−0.429707	+	0.902968i	$0.641383\pi$
$312$	0	0
$313$	2.94425e6	1.69869	0.849343	−	0.527841i	$-0.176998\pi$
$313$	2.94425e6	1.69869	0.849343	+	0.527841i	$0.176998\pi$
$314$	−785924.	−0.449838
$315$	0	0
$316$	92407.1	0.0520580
$317$	561175.	0.313653	0.156827	−	0.987626i	$-0.449874\pi$
$317$	561175.	0.313653	0.156827	+	0.987626i	$0.449874\pi$
$318$	0	0
$319$	1.66698e6	0.917180
$320$	3.49102e6	1.90580
$321$	0	0
$322$	531980.	0.285927
$323$	3.59467e6	1.91714
$324$	0	0
$325$	−374639.	−0.196745
$326$	1.57240e6	0.819445
$327$	0	0
$328$	−2.72323e6	−1.39765
$329$	−3.78550e6	−1.92812
$330$	0	0
$331$	3.55649e6	1.78423	0.892117	−	0.451805i	$-0.149220\pi$
$331$	3.55649e6	1.78423	0.892117	+	0.451805i	$0.149220\pi$
$332$	−226006.	−0.112532
$333$	0	0
$334$	100561.	0.0493246
$335$	−1.09776e6	−0.534436
$336$	0	0
$337$	−1.16442e6	−0.558516	−0.279258	−	0.960216i	$-0.590088\pi$
$337$	−1.16442e6	−0.558516	−0.279258	+	0.960216i	$0.590088\pi$
$338$	−1.75098e6	−0.833661
$339$	0	0
$340$	−1.64005e6	−0.769415
$341$	−3.11499e6	−1.45068
$342$	0	0
$343$	−2.32335e6	−1.06630
$344$	250408.	0.114091
$345$	0	0
$346$	2.47202e6	1.11010
$347$	−2.09076e6	−0.932139	−0.466069	−	0.884748i	$-0.654330\pi$
$347$	−2.09076e6	−0.932139	−0.466069	+	0.884748i	$0.654330\pi$
$348$	0	0
$349$	2.28656e6	1.00489	0.502445	−	0.864609i	$-0.332434\pi$
$349$	2.28656e6	1.00489	0.502445	+	0.864609i	$0.332434\pi$
$350$	−6.34692e6	−2.76945
$351$	0	0
$352$	2.06421e6	0.887966
$353$	−1.42523e6	−0.608764	−0.304382	−	0.952550i	$-0.598450\pi$
$353$	−1.42523e6	−0.608764	−0.304382	+	0.952550i	$0.598450\pi$
$354$	0	0
$355$	−3.23122e6	−1.36080
$356$	377613.	0.157915
$357$	0	0
$358$	−2.00127e6	−0.825273
$359$	1.73659e6	0.711152	0.355576	−	0.934647i	$-0.384285\pi$
$359$	1.73659e6	0.711152	0.355576	+	0.934647i	$0.384285\pi$
$360$	0	0
$361$	1.47182e6	0.594409
$362$	377727.	0.151498
$363$	0	0
$364$	−117005.	−0.0462860
$365$	2.19266e6	0.861467
$366$	0	0
$367$	1.37269e6	0.531994	0.265997	−	0.963974i	$-0.414299\pi$
$367$	1.37269e6	0.531994	0.265997	+	0.963974i	$0.414299\pi$
$368$	337652.	0.129972
$369$	0	0
$370$	−2.25288e6	−0.855527
$371$	−4.12237e6	−1.55493
$372$	0	0
$373$	−539618.	−0.200824	−0.100412	−	0.994946i	$-0.532016\pi$
$373$	−539618.	−0.200824	−0.100412	+	0.994946i	$0.532016\pi$
$374$	5.45701e6	2.01732
$375$	0	0
$376$	−3.52681e6	−1.28651
$377$	−156188.	−0.0565970
$378$	0	0
$379$	2.91919e6	1.04391	0.521956	−	0.852972i	$-0.325203\pi$
$379$	2.91919e6	1.04391	0.521956	+	0.852972i	$0.325203\pi$
$380$	−1.80122e6	−0.639893
$381$	0	0
$382$	−1.95001e6	−0.683719
$383$	2.78248e6	0.969247	0.484623	−	0.874723i	$-0.338957\pi$
$383$	2.78248e6	0.969247	0.484623	+	0.874723i	$0.338957\pi$
$384$	0	0
$385$	−1.29990e7	−4.46950
$386$	2.02266e6	0.690962
$387$	0	0
$388$	171034.	0.0576770
$389$	−1.77472e6	−0.594642	−0.297321	−	0.954778i	$-0.596093\pi$
$389$	−1.77472e6	−0.594642	−0.297321	+	0.954778i	$0.596093\pi$
$390$	0	0
$391$	−957042.	−0.316584
$392$	−5.47196e6	−1.79857
$393$	0	0
$394$	3.40918e6	1.10639
$395$	−961896.	−0.310195
$396$	0	0
$397$	−470654.	−0.149874	−0.0749369	−	0.997188i	$-0.523876\pi$
$397$	−470654.	−0.149874	−0.0749369	+	0.997188i	$0.523876\pi$
$398$	−3.03149e6	−0.959285
$399$	0	0
$400$	−4.02845e6	−1.25889
$401$	−2.32453e6	−0.721894	−0.360947	−	0.932586i	$-0.617547\pi$
$401$	−2.32453e6	−0.721894	−0.360947	+	0.932586i	$0.617547\pi$
$402$	0	0
$403$	291859.	0.0895180
$404$	1.26456e6	0.385466
$405$	0	0
$406$	−2.64605e6	−0.796677
$407$	−3.08606e6	−0.923459
$408$	0	0
$409$	−3.54081e6	−1.04663	−0.523317	−	0.852138i	$-0.675305\pi$
$409$	−3.54081e6	−1.04663	−0.523317	+	0.852138i	$0.675305\pi$
$410$	6.40027e6	1.88035
$411$	0	0
$412$	−127928.	−0.0371298
$413$	−3.71091e6	−1.07055
$414$	0	0
$415$	2.35257e6	0.670536
$416$	−193406.	−0.0547943
$417$	0	0
$418$	5.99326e6	1.67773
$419$	−1.84820e6	−0.514297	−0.257149	−	0.966372i	$-0.582783\pi$
$419$	−1.84820e6	−0.514297	−0.257149	+	0.966372i	$0.582783\pi$
$420$	0	0
$421$	1.97062e6	0.541874	0.270937	−	0.962597i	$-0.412667\pi$
$421$	1.97062e6	0.541874	0.270937	+	0.962597i	$0.412667\pi$
$422$	3.30571e6	0.903614
$423$	0	0
$424$	−3.84066e6	−1.03751
$425$	1.14182e7	3.06638
$426$	0	0
$427$	4.32025e6	1.14667
$428$	1.80412e6	0.476054
$429$	0	0
$430$	−588522.	−0.153494
$431$	3.45702e6	0.896415	0.448208	−	0.893929i	$-0.352063\pi$
$431$	3.45702e6	0.896415	0.448208	+	0.893929i	$0.352063\pi$
$432$	0	0
$433$	−4.90614e6	−1.25753	−0.628767	−	0.777594i	$-0.716440\pi$
$433$	−4.90614e6	−1.25753	−0.628767	+	0.777594i	$0.716440\pi$
$434$	4.94451e6	1.26008
$435$	0	0
$436$	1.08673e6	0.273782
$437$	−1.05109e6	−0.263291
$438$	0	0
$439$	400905.	0.0992842	0.0496421	−	0.998767i	$-0.484192\pi$
$439$	400905.	0.0992842	0.0496421	+	0.998767i	$0.484192\pi$
$440$	−1.21107e7	−2.98221
$441$	0	0
$442$	−511294.	−0.124484
$443$	5.31050e6	1.28566	0.642829	−	0.766009i	$-0.277760\pi$
$443$	5.31050e6	1.28566	0.642829	+	0.766009i	$0.277760\pi$
$444$	0	0
$445$	−3.93070e6	−0.940958
$446$	−2.64583e6	−0.629833
$447$	0	0
$448$	−7.59075e6	−1.78686
$449$	501557.	0.117410	0.0587049	−	0.998275i	$-0.481303\pi$
$449$	501557.	0.117410	0.0587049	+	0.998275i	$0.481303\pi$
$450$	0	0
$451$	8.76727e6	2.02966
$452$	−1.67694e6	−0.386075
$453$	0	0
$454$	2.44298e6	0.556264
$455$	1.21794e6	0.275802
$456$	0	0
$457$	−5.15796e6	−1.15528	−0.577641	−	0.816291i	$-0.696027\pi$
$457$	−5.15796e6	−1.15528	−0.577641	+	0.816291i	$0.696027\pi$
$458$	−6.18350e6	−1.37743
$459$	0	0
$460$	479554.	0.105668
$461$	3.35669e6	0.735628	0.367814	−	0.929899i	$-0.380106\pi$
$461$	3.35669e6	0.735628	0.367814	+	0.929899i	$0.380106\pi$
$462$	0	0
$463$	1.83888e6	0.398658	0.199329	−	0.979933i	$-0.436124\pi$
$463$	1.83888e6	0.398658	0.199329	+	0.979933i	$0.436124\pi$
$464$	−1.67947e6	−0.362140
$465$	0	0
$466$	2.48952e6	0.531068
$467$	−4.47719e6	−0.949977	−0.474988	−	0.879992i	$-0.657548\pi$
$467$	−4.47719e6	−0.949977	−0.474988	+	0.879992i	$0.657548\pi$
$468$	0	0
$469$	2.38693e6	0.501081
$470$	8.28889e6	1.73082
$471$	0	0
$472$	−3.45732e6	−0.714307
$473$	−806173.	−0.165682
$474$	0	0
$475$	1.25403e7	2.55019
$476$	3.56607e6	0.721393
$477$	0	0
$478$	−4.87359e6	−0.975617
$479$	1.13426e6	0.225878	0.112939	−	0.993602i	$-0.463974\pi$
$479$	1.13426e6	0.225878	0.112939	+	0.993602i	$0.463974\pi$
$480$	0	0
$481$	289148.	0.0569845
$482$	−7.92620e6	−1.55399
$483$	0	0
$484$	−2.24270e6	−0.435170
$485$	−1.78035e6	−0.343677
$486$	0	0
$487$	−4.84753e6	−0.926185	−0.463093	−	0.886310i	$-0.653260\pi$
$487$	−4.84753e6	−0.926185	−0.463093	+	0.886310i	$0.653260\pi$
$488$	4.02502e6	0.765100
$489$	0	0
$490$	1.28605e7	2.41973
$491$	−5.80829e6	−1.08729	−0.543644	−	0.839316i	$-0.682956\pi$
$491$	−5.80829e6	−1.08729	−0.543644	+	0.839316i	$0.682956\pi$
$492$	0	0
$493$	4.76029e6	0.882096
$494$	−561538.	−0.103529
$495$	0	0
$496$	3.13832e6	0.572787
$497$	7.02584e6	1.27587
$498$	0	0
$499$	−3.67933e6	−0.661481	−0.330741	−	0.943722i	$-0.607299\pi$
$499$	−3.67933e6	−0.661481	−0.330741	+	0.943722i	$0.607299\pi$
$500$	−2.88854e6	−0.516717
$501$	0	0
$502$	−5.03415e6	−0.891593
$503$	−8.28397e6	−1.45988	−0.729942	−	0.683509i	$-0.760453\pi$
$503$	−8.28397e6	−1.45988	−0.729942	+	0.683509i	$0.760453\pi$
$504$	0	0
$505$	−1.31632e7	−2.29686
$506$	−1.59564e6	−0.277050
$507$	0	0
$508$	2.29810e6	0.395103
$509$	1.50326e6	0.257182	0.128591	−	0.991698i	$-0.458955\pi$
$509$	1.50326e6	0.257182	0.128591	+	0.991698i	$0.458955\pi$
$510$	0	0
$511$	−4.76763e6	−0.807701
$512$	−6.09903e6	−1.02822
$513$	0	0
$514$	5.08100e6	0.848285
$515$	1.33165e6	0.221243
$516$	0	0
$517$	1.13544e7	1.86825
$518$	4.89858e6	0.802131
$519$	0	0
$520$	1.13471e6	0.184025
$521$	7.04086e6	1.13640	0.568200	−	0.822891i	$-0.307640\pi$
$521$	7.04086e6	1.13640	0.568200	+	0.822891i	$0.307640\pi$
$522$	0	0
$523$	−1.68119e6	−0.268759	−0.134380	−	0.990930i	$-0.542904\pi$
$523$	−1.68119e6	−0.268759	−0.134380	+	0.990930i	$0.542904\pi$
$524$	634622.	0.100969
$525$	0	0
$526$	8.22875e6	1.29679
$527$	−8.89527e6	−1.39519
$528$	0	0
$529$	279841.	0.0434783
$530$	9.02652e6	1.39582
$531$	0	0
$532$	3.91650e6	0.599955
$533$	−821448.	−0.125245
$534$	0	0
$535$	−1.87797e7	−2.83664
$536$	2.22382e6	0.334339
$537$	0	0
$538$	−2.13142e6	−0.317478
$539$	1.76166e7	2.61187
$540$	0	0
$541$	−2.09033e6	−0.307059	−0.153529	−	0.988144i	$-0.549064\pi$
$541$	−2.09033e6	−0.307059	−0.153529	+	0.988144i	$0.549064\pi$
$542$	−8.20874e6	−1.20027
$543$	0	0
$544$	5.89461e6	0.854000
$545$	−1.13121e7	−1.63137
$546$	0	0
$547$	−9.24304e6	−1.32083	−0.660414	−	0.750901i	$-0.729619\pi$
$547$	−9.24304e6	−1.32083	−0.660414	+	0.750901i	$0.729619\pi$
$548$	−3.53174e6	−0.502386
$549$	0	0
$550$	1.90372e7	2.68347
$551$	5.22807e6	0.733605
$552$	0	0
$553$	2.09151e6	0.290835
$554$	−547071.	−0.0757302
$555$	0	0
$556$	1.04637e6	0.143548
$557$	1.67077e6	0.228181	0.114091	−	0.993470i	$-0.463605\pi$
$557$	1.67077e6	0.228181	0.114091	+	0.993470i	$0.463605\pi$
$558$	0	0
$559$	75534.3	0.0102238
$560$	1.30964e7	1.76474
$561$	0	0
$562$	9.64477e6	1.28810
$563$	1.12609e7	1.49728	0.748639	−	0.662978i	$-0.230708\pi$
$563$	1.12609e7	1.49728	0.748639	+	0.662978i	$0.230708\pi$
$564$	0	0
$565$	1.74558e7	2.30048
$566$	−6.15277e6	−0.807290
$567$	0	0
$568$	6.54572e6	0.851307
$569$	−1.05116e7	−1.36110	−0.680549	−	0.732703i	$-0.738259\pi$
$569$	−1.05116e7	−1.36110	−0.680549	+	0.732703i	$0.738259\pi$
$570$	0	0
$571$	1.48085e7	1.90073	0.950363	−	0.311142i	$-0.100711\pi$
$571$	1.48085e7	1.90073	0.950363	+	0.311142i	$0.100711\pi$
$572$	350948.	0.0448490
$573$	0	0
$574$	−1.39165e7	−1.76299
$575$	−3.33871e6	−0.421124
$576$	0	0
$577$	−3.06552e6	−0.383323	−0.191661	−	0.981461i	$-0.561387\pi$
$577$	−3.06552e6	−0.383323	−0.191661	+	0.981461i	$0.561387\pi$
$578$	8.82315e6	1.09851
$579$	0	0
$580$	−2.38528e6	−0.294422
$581$	−5.11533e6	−0.628686
$582$	0	0
$583$	1.23648e7	1.50666
$584$	−4.44183e6	−0.538927
$585$	0	0
$586$	3.99176e6	0.480198
$587$	−4.49808e6	−0.538805	−0.269403	−	0.963028i	$-0.586826\pi$
$587$	−4.49808e6	−0.538805	−0.269403	+	0.963028i	$0.586826\pi$
$588$	0	0
$589$	−9.76939e6	−1.16032
$590$	8.12559e6	0.961003
$591$	0	0
$592$	3.10917e6	0.364620
$593$	−2.42205e6	−0.282844	−0.141422	−	0.989949i	$-0.545167\pi$
$593$	−2.42205e6	−0.282844	−0.141422	+	0.989949i	$0.545167\pi$
$594$	0	0
$595$	−3.71204e7	−4.29853
$596$	2.70917e6	0.312407
$597$	0	0
$598$	149503.	0.0170961
$599$	−2.86365e6	−0.326102	−0.163051	−	0.986618i	$-0.552134\pi$
$599$	−2.86365e6	−0.326102	−0.163051	+	0.986618i	$0.552134\pi$
$600$	0	0
$601$	3.17881e6	0.358987	0.179493	−	0.983759i	$-0.442554\pi$
$601$	3.17881e6	0.358987	0.179493	+	0.983759i	$0.442554\pi$
$602$	1.27966e6	0.143914
$603$	0	0
$604$	−1.17164e6	−0.130677
$605$	2.33450e7	2.59302
$606$	0	0
$607$	−1.62063e7	−1.78531	−0.892653	−	0.450745i	$-0.851158\pi$
$607$	−1.62063e7	−1.78531	−0.892653	+	0.450745i	$0.851158\pi$
$608$	6.47386e6	0.710239
$609$	0	0
$610$	−9.45981e6	−1.02934
$611$	−1.06384e6	−0.115286
$612$	0	0
$613$	5.85294e6	0.629105	0.314552	−	0.949240i	$-0.398146\pi$
$613$	5.85294e6	0.629105	0.314552	+	0.949240i	$0.398146\pi$
$614$	5.49288e6	0.588002
$615$	0	0
$616$	2.63330e7	2.79608
$617$	−6.26326e6	−0.662350	−0.331175	−	0.943569i	$-0.607445\pi$
$617$	−6.26326e6	−0.662350	−0.331175	+	0.943569i	$0.607445\pi$
$618$	0	0
$619$	−4.36108e6	−0.457475	−0.228737	−	0.973488i	$-0.573460\pi$
$619$	−4.36108e6	−0.457475	−0.228737	+	0.973488i	$0.573460\pi$
$620$	4.45724e6	0.465679
$621$	0	0
$622$	−6.97925e6	−0.723324
$623$	8.54677e6	0.882230
$624$	0	0
$625$	1.03447e7	1.05930
$626$	1.40178e7	1.42970
$627$	0	0
$628$	1.54048e6	0.155868
$629$	−8.81263e6	−0.888135
$630$	0	0
$631$	−1.48998e7	−1.48973	−0.744866	−	0.667214i	$-0.767487\pi$
$631$	−1.48998e7	−1.48973	−0.744866	+	0.667214i	$0.767487\pi$
$632$	1.94858e6	0.194056
$633$	0	0
$634$	2.67180e6	0.263986
$635$	−2.39217e7	−2.35428
$636$	0	0
$637$	−1.65059e6	−0.161172
$638$	7.93664e6	0.771943
$639$	0	0
$640$	6.49286e6	0.626594
$641$	−826180.	−0.0794199	−0.0397100	−	0.999211i	$-0.512643\pi$
$641$	−826180.	−0.0794199	−0.0397100	+	0.999211i	$0.512643\pi$
$642$	0	0
$643$	−1.37326e7	−1.30986	−0.654929	−	0.755691i	$-0.727301\pi$
$643$	−1.37326e7	−1.30986	−0.654929	+	0.755691i	$0.727301\pi$
$644$	−1.04272e6	−0.0990729
$645$	0	0
$646$	1.71145e7	1.61355
$647$	−2.70016e6	−0.253588	−0.126794	−	0.991929i	$-0.540469\pi$
$647$	−2.70016e6	−0.253588	−0.126794	+	0.991929i	$0.540469\pi$
$648$	0	0
$649$	1.11306e7	1.03731
$650$	−1.78369e6	−0.165590
$651$	0	0
$652$	−3.08204e6	−0.283935
$653$	1.01591e7	0.932339	0.466169	−	0.884696i	$-0.345634\pi$
$653$	1.01591e7	0.932339	0.466169	+	0.884696i	$0.345634\pi$
$654$	0	0
$655$	−6.60599e6	−0.601637
$656$	−8.83293e6	−0.801392
$657$	0	0
$658$	−1.80230e7	−1.62279
$659$	−1.53744e7	−1.37906	−0.689531	−	0.724256i	$-0.742183\pi$
$659$	−1.53744e7	−1.37906	−0.689531	+	0.724256i	$0.742183\pi$
$660$	0	0
$661$	−1.47641e7	−1.31433	−0.657163	−	0.753748i	$-0.728244\pi$
$661$	−1.47641e7	−1.31433	−0.657163	+	0.753748i	$0.728244\pi$
$662$	1.69327e7	1.50170
$663$	0	0
$664$	−4.76577e6	−0.419481
$665$	−4.07681e7	−3.57492
$666$	0	0
$667$	−1.39192e6	−0.121143
$668$	−197108.	−0.0170908
$669$	0	0
$670$	−5.22653e6	−0.449807
$671$	−1.29583e7	−1.11107
$672$	0	0
$673$	1.79584e7	1.52837	0.764187	−	0.644995i	$-0.223141\pi$
$673$	1.79584e7	1.52837	0.764187	+	0.644995i	$0.223141\pi$
$674$	−5.54391e6	−0.470074
$675$	0	0
$676$	3.43206e6	0.288861
$677$	−7.66782e6	−0.642984	−0.321492	−	0.946912i	$-0.604184\pi$
$677$	−7.66782e6	−0.642984	−0.321492	+	0.946912i	$0.604184\pi$
$678$	0	0
$679$	3.87112e6	0.322227
$680$	−3.45837e7	−2.86813
$681$	0	0
$682$	−1.48307e7	−1.22096
$683$	3.48360e6	0.285744	0.142872	−	0.989741i	$-0.454366\pi$
$683$	3.48360e6	0.285744	0.142872	+	0.989741i	$0.454366\pi$
$684$	0	0
$685$	3.67631e7	2.99354
$686$	−1.10617e7	−0.897451
$687$	0	0
$688$	812211.	0.0654180
$689$	−1.15852e6	−0.0929724
$690$	0	0
$691$	−1.24991e7	−0.995829	−0.497914	−	0.867226i	$-0.665901\pi$
$691$	−1.24991e7	−0.995829	−0.497914	+	0.867226i	$0.665901\pi$
$692$	−4.84537e6	−0.384646
$693$	0	0
$694$	−9.95428e6	−0.784533
$695$	−1.08920e7	−0.855355
$696$	0	0
$697$	2.50361e7	1.95202
$698$	1.08865e7	0.845765
$699$	0	0
$700$	1.24405e7	0.959604
$701$	2.20409e7	1.69408	0.847042	−	0.531525i	$-0.178381\pi$
$701$	2.20409e7	1.69408	0.847042	+	0.531525i	$0.178381\pi$
$702$	0	0
$703$	−9.67863e6	−0.738628
$704$	2.27680e7	1.73138
$705$	0	0
$706$	−6.78564e6	−0.512365
$707$	2.86216e7	2.15350
$708$	0	0
$709$	4.99549e6	0.373218	0.186609	−	0.982434i	$-0.440250\pi$
$709$	4.99549e6	0.373218	0.186609	+	0.982434i	$0.440250\pi$
$710$	−1.53841e7	−1.14532
$711$	0	0
$712$	7.96271e6	0.588655
$713$	2.60099e6	0.191609
$714$	0	0
$715$	−3.65313e6	−0.267239
$716$	3.92265e6	0.285954
$717$	0	0
$718$	8.26806e6	0.598539
$719$	−2.06128e6	−0.148701	−0.0743507	−	0.997232i	$-0.523688\pi$
$719$	−2.06128e6	−0.148701	−0.0743507	+	0.997232i	$0.523688\pi$
$720$	0	0
$721$	−2.89548e6	−0.207435
$722$	7.00744e6	0.500284
$723$	0	0
$724$	−740377.	−0.0524936
$725$	1.66066e7	1.17337
$726$	0	0
$727$	1.74867e7	1.22708	0.613539	−	0.789665i	$-0.289746\pi$
$727$	1.74867e7	1.22708	0.613539	+	0.789665i	$0.289746\pi$
$728$	−2.46727e6	−0.172539
$729$	0	0
$730$	1.04394e7	0.725052
$731$	−2.30213e6	−0.159344
$732$	0	0
$733$	8.68392e6	0.596975	0.298487	−	0.954414i	$-0.403518\pi$
$733$	8.68392e6	0.596975	0.298487	+	0.954414i	$0.403518\pi$
$734$	6.53549e6	0.447752
$735$	0	0
$736$	−1.72359e6	−0.117285
$737$	−7.15944e6	−0.485524
$738$	0	0
$739$	1.15780e7	0.779871	0.389936	−	0.920842i	$-0.372497\pi$
$739$	1.15780e7	0.779871	0.389936	+	0.920842i	$0.372497\pi$
$740$	4.41583e6	0.296437
$741$	0	0
$742$	−1.96269e7	−1.30871
$743$	6.91849e6	0.459768	0.229884	−	0.973218i	$-0.426165\pi$
$743$	6.91849e6	0.459768	0.229884	+	0.973218i	$0.426165\pi$
$744$	0	0
$745$	−2.82006e7	−1.86152
$746$	−2.56917e6	−0.169023
$747$	0	0
$748$	−1.06962e7	−0.698997
$749$	4.08338e7	2.65960
$750$	0	0
$751$	3.10339e6	0.200787	0.100394	−	0.994948i	$-0.467990\pi$
$751$	3.10339e6	0.200787	0.100394	+	0.994948i	$0.467990\pi$
$752$	−1.14394e7	−0.737663
$753$	0	0
$754$	−743623.	−0.0476348
$755$	1.21959e7	0.778660
$756$	0	0
$757$	−9.03023e6	−0.572742	−0.286371	−	0.958119i	$-0.592449\pi$
$757$	−9.03023e6	−0.572742	−0.286371	+	0.958119i	$0.592449\pi$
$758$	1.38985e7	0.878606
$759$	0	0
$760$	−3.79821e7	−2.38531
$761$	−9.70856e6	−0.607706	−0.303853	−	0.952719i	$-0.598273\pi$
$761$	−9.70856e6	−0.607706	−0.303853	+	0.952719i	$0.598273\pi$
$762$	0	0
$763$	2.45966e7	1.52955
$764$	3.82217e6	0.236906
$765$	0	0
$766$	1.32476e7	0.815765
$767$	−1.04288e6	−0.0640100
$768$	0	0
$769$	−1.31499e7	−0.801874	−0.400937	−	0.916106i	$-0.631315\pi$
$769$	−1.31499e7	−0.801874	−0.400937	+	0.916106i	$0.631315\pi$
$770$	−6.18893e7	−3.76174
$771$	0	0
$772$	−3.96458e6	−0.239416
$773$	1.58606e7	0.954706	0.477353	−	0.878712i	$-0.341596\pi$
$773$	1.58606e7	0.954706	0.477353	+	0.878712i	$0.341596\pi$
$774$	0	0
$775$	−3.10318e7	−1.85589
$776$	3.60658e6	0.215001
$777$	0	0
$778$	−8.44958e6	−0.500479
$779$	2.74963e7	1.62342
$780$	0	0
$781$	−2.10736e7	−1.23626
$782$	−4.55656e6	−0.266453
$783$	0	0
$784$	−1.77486e7	−1.03127
$785$	−1.60353e7	−0.928760
$786$	0	0
$787$	3.25773e7	1.87490	0.937449	−	0.348122i	$-0.113180\pi$
$787$	3.25773e7	1.87490	0.937449	+	0.348122i	$0.113180\pi$
$788$	−6.68228e6	−0.383362
$789$	0	0
$790$	−4.57966e6	−0.261075
$791$	−3.79552e7	−2.15690
$792$	0	0
$793$	1.21413e6	0.0685616
$794$	−2.24082e6	−0.126141
$795$	0	0
$796$	5.94196e6	0.332389
$797$	−2.89765e7	−1.61585	−0.807923	−	0.589288i	$-0.799408\pi$
$797$	−2.89765e7	−1.61585	−0.807923	+	0.589288i	$0.799408\pi$
$798$	0	0
$799$	3.24238e7	1.79679
$800$	2.05638e7	1.13600
$801$	0	0
$802$	−1.10673e7	−0.607581
$803$	1.43002e7	0.782625
$804$	0	0
$805$	1.08541e7	0.590340
$806$	1.38956e6	0.0753427
$807$	0	0
$808$	2.66657e7	1.43689
$809$	1.73592e7	0.932521	0.466261	−	0.884647i	$-0.345601\pi$
$809$	1.73592e7	0.932521	0.466261	+	0.884647i	$0.345601\pi$
$810$	0	0
$811$	−2.90389e7	−1.55034	−0.775172	−	0.631750i	$-0.782337\pi$
$811$	−2.90389e7	−1.55034	−0.775172	+	0.631750i	$0.782337\pi$
$812$	5.18647e6	0.276046
$813$	0	0
$814$	−1.46930e7	−0.777228
$815$	3.20820e7	1.69187
$816$	0	0
$817$	−2.52836e6	−0.132521
$818$	−1.68581e7	−0.880897
$819$	0	0
$820$	−1.25451e7	−0.651536
$821$	−2.46510e7	−1.27637	−0.638186	−	0.769882i	$-0.720315\pi$
$821$	−2.46510e7	−1.27637	−0.638186	+	0.769882i	$0.720315\pi$
$822$	0	0
$823$	−2.84702e7	−1.46518	−0.732589	−	0.680671i	$-0.761688\pi$
$823$	−2.84702e7	−1.46518	−0.732589	+	0.680671i	$0.761688\pi$
$824$	−2.69761e6	−0.138408
$825$	0	0
$826$	−1.76680e7	−0.901024
$827$	−2.38791e7	−1.21410	−0.607050	−	0.794664i	$-0.707647\pi$
$827$	−2.38791e7	−1.21410	−0.607050	+	0.794664i	$0.707647\pi$
$828$	0	0
$829$	−9.49660e6	−0.479934	−0.239967	−	0.970781i	$-0.577137\pi$
$829$	−9.49660e6	−0.479934	−0.239967	+	0.970781i	$0.577137\pi$
$830$	1.12008e7	0.564355
$831$	0	0
$832$	−2.13324e6	−0.106839
$833$	5.03066e7	2.51196
$834$	0	0
$835$	2.05176e6	0.101838
$836$	−1.17473e7	−0.581329
$837$	0	0
$838$	−8.79944e6	−0.432857
$839$	2.25952e7	1.10818	0.554091	−	0.832456i	$-0.313066\pi$
$839$	2.25952e7	1.10818	0.554091	+	0.832456i	$0.313066\pi$
$840$	0	0
$841$	−1.35878e7	−0.662460
$842$	9.38229e6	0.456067
$843$	0	0
$844$	−6.47945e6	−0.313099
$845$	−3.57255e7	−1.72122
$846$	0	0
$847$	−5.07606e7	−2.43119
$848$	−1.24574e7	−0.594890
$849$	0	0
$850$	5.43631e7	2.58082
$851$	2.57683e6	0.121972
$852$	0	0
$853$	−687989.	−0.0323749	−0.0161875	−	0.999869i	$-0.505153\pi$
$853$	−687989.	−0.0323749	−0.0161875	+	0.999869i	$0.505153\pi$
$854$	2.05691e7	0.965094
$855$	0	0
$856$	3.80434e7	1.77458
$857$	−1.52420e7	−0.708906	−0.354453	−	0.935074i	$-0.615333\pi$
$857$	−1.52420e7	−0.708906	−0.354453	+	0.935074i	$0.615333\pi$
$858$	0	0
$859$	−2.56393e7	−1.18556	−0.592780	−	0.805364i	$-0.701970\pi$
$859$	−2.56393e7	−1.18556	−0.592780	+	0.805364i	$0.701970\pi$
$860$	1.15355e6	0.0531852
$861$	0	0
$862$	1.64592e7	0.754466
$863$	−4.25337e7	−1.94405	−0.972023	−	0.234888i	$-0.924528\pi$
$863$	−4.25337e7	−1.94405	−0.972023	+	0.234888i	$0.924528\pi$
$864$	0	0
$865$	5.04370e7	2.29197
$866$	−2.33585e7	−1.05840
$867$	0	0
$868$	−9.69164e6	−0.436614
$869$	−6.27335e6	−0.281806
$870$	0	0
$871$	670803.	0.0299605
$872$	2.29158e7	1.02057
$873$	0	0
$874$	−5.00432e6	−0.221598
$875$	−6.53781e7	−2.88677
$876$	0	0
$877$	−2.00396e7	−0.879811	−0.439906	−	0.898044i	$-0.644988\pi$
$877$	−2.00396e7	−0.879811	−0.439906	+	0.898044i	$0.644988\pi$
$878$	1.90874e6	0.0835624
$879$	0	0
$880$	−3.92817e7	−1.70995
$881$	−1.60786e7	−0.697925	−0.348962	−	0.937137i	$-0.613466\pi$
$881$	−1.60786e7	−0.697925	−0.348962	+	0.937137i	$0.613466\pi$
$882$	0	0
$883$	1.82621e7	0.788222	0.394111	−	0.919063i	$-0.371053\pi$
$883$	1.82621e7	0.788222	0.394111	+	0.919063i	$0.371053\pi$
$884$	1.00218e6	0.0431334
$885$	0	0
$886$	2.52837e7	1.08207
$887$	2.37902e7	1.01529	0.507645	−	0.861567i	$-0.330516\pi$
$887$	2.37902e7	1.01529	0.507645	+	0.861567i	$0.330516\pi$
$888$	0	0
$889$	5.20145e7	2.20734
$890$	−1.87144e7	−0.791955
$891$	0	0
$892$	5.18605e6	0.218235
$893$	3.56100e7	1.49432
$894$	0	0
$895$	−4.08321e7	−1.70390
$896$	−1.41178e7	−0.587486
$897$	0	0
$898$	2.38795e6	0.0988178
$899$	−1.29372e7	−0.533878
$900$	0	0
$901$	3.53092e7	1.44903
$902$	4.17417e7	1.70826
$903$	0	0
$904$	−3.53615e7	−1.43916
$905$	7.70682e6	0.312791
$906$	0	0
$907$	7.37144e6	0.297532	0.148766	−	0.988872i	$-0.452470\pi$
$907$	7.37144e6	0.297532	0.148766	+	0.988872i	$0.452470\pi$
$908$	−4.78845e6	−0.192744
$909$	0	0
$910$	5.79871e6	0.232128
$911$	4.34077e7	1.73289	0.866445	−	0.499273i	$-0.166399\pi$
$911$	4.34077e7	1.73289	0.866445	+	0.499273i	$0.166399\pi$
$912$	0	0
$913$	1.53431e7	0.609167
$914$	−2.45575e7	−0.972341
$915$	0	0
$916$	1.21202e7	0.477276
$917$	1.43638e7	0.564087
$918$	0	0
$919$	1.26700e6	0.0494865	0.0247433	−	0.999694i	$-0.492123\pi$
$919$	1.26700e6	0.0494865	0.0247433	+	0.999694i	$0.492123\pi$
$920$	1.01123e7	0.393896
$921$	0	0
$922$	1.59815e7	0.619140
$923$	1.97448e6	0.0762868
$924$	0	0
$925$	−3.07435e7	−1.18141
$926$	8.75506e6	0.335530
$927$	0	0
$928$	8.57308e6	0.326789
$929$	−632559.	−0.0240470	−0.0120235	−	0.999928i	$-0.503827\pi$
$929$	−632559.	−0.0240470	−0.0120235	+	0.999928i	$0.503827\pi$
$930$	0	0
$931$	5.52502e7	2.08910
$932$	−4.87966e6	−0.184013
$933$	0	0
$934$	−2.13163e7	−0.799546
$935$	1.11340e8	4.16507
$936$	0	0
$937$	−3.21230e6	−0.119527	−0.0597637	−	0.998213i	$-0.519035\pi$
$937$	−3.21230e6	−0.119527	−0.0597637	+	0.998213i	$0.519035\pi$
$938$	1.13644e7	0.421734
$939$	0	0
$940$	−1.62469e7	−0.599724
$941$	2.08925e7	0.769161	0.384580	−	0.923091i	$-0.374346\pi$
$941$	2.08925e7	0.769161	0.384580	+	0.923091i	$0.374346\pi$
$942$	0	0
$943$	−7.32059e6	−0.268082
$944$	−1.12140e7	−0.409573
$945$	0	0
$946$	−3.83825e6	−0.139446
$947$	−4.64335e7	−1.68250	−0.841252	−	0.540643i	$-0.818181\pi$
$947$	−4.64335e7	−1.68250	−0.841252	+	0.540643i	$0.818181\pi$
$948$	0	0
$949$	−1.33986e6	−0.0482939
$950$	5.97053e7	2.14637
$951$	0	0
$952$	7.51974e7	2.68912
$953$	−3.54643e7	−1.26491	−0.632454	−	0.774598i	$-0.717952\pi$
$953$	−3.54643e7	−1.26491	−0.632454	+	0.774598i	$0.717952\pi$
$954$	0	0
$955$	−3.97862e7	−1.41164
$956$	9.55264e6	0.338048
$957$	0	0
$958$	5.40030e6	0.190110
$959$	−7.99362e7	−2.80671
$960$	0	0
$961$	−4.45412e6	−0.155580
$962$	1.37666e6	0.0479609
$963$	0	0
$964$	1.55360e7	0.538451
$965$	4.12686e7	1.42660
$966$	0	0
$967$	4.27394e7	1.46981	0.734906	−	0.678169i	$-0.237226\pi$
$967$	4.27394e7	1.46981	0.734906	+	0.678169i	$0.237226\pi$
$968$	−4.72918e7	−1.62217
$969$	0	0
$970$	−8.47637e6	−0.289255
$971$	−1.05226e7	−0.358157	−0.179079	−	0.983835i	$-0.557312\pi$
$971$	−1.05226e7	−0.358157	−0.179079	+	0.983835i	$0.557312\pi$
$972$	0	0
$973$	2.36832e7	0.801970
$974$	−2.30795e7	−0.779522
$975$	0	0
$976$	1.30554e7	0.438696
$977$	−5.68345e7	−1.90492	−0.952458	−	0.304670i	$-0.901454\pi$
$977$	−5.68345e7	−1.90492	−0.952458	+	0.304670i	$0.901454\pi$
$978$	0	0
$979$	−2.56355e7	−0.854840
$980$	−2.52076e7	−0.838430
$981$	0	0
$982$	−2.76537e7	−0.915114
$983$	−1.82745e7	−0.603201	−0.301600	−	0.953434i	$-0.597521\pi$
$983$	−1.82745e7	−0.603201	−0.301600	+	0.953434i	$0.597521\pi$
$984$	0	0
$985$	6.95580e7	2.28432
$986$	2.26641e7	0.742414
$987$	0	0
$988$	1.10066e6	0.0358724
$989$	673147.	0.0218836
$990$	0	0
$991$	−1.33296e7	−0.431155	−0.215577	−	0.976487i	$-0.569163\pi$
$991$	−1.33296e7	−0.431155	−0.215577	+	0.976487i	$0.569163\pi$
$992$	−1.60200e7	−0.516873
$993$	0	0
$994$	3.34506e7	1.07384
$995$	−6.18518e7	−1.98059
$996$	0	0
$997$	−1.11514e7	−0.355296	−0.177648	−	0.984094i	$-0.556849\pi$
$997$	−1.11514e7	−0.355296	−0.177648	+	0.984094i	$0.556849\pi$
$998$	−1.75176e7	−0.556735
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.d.1.4		4
3.2	odd	2		69.6.a.c.1.1	✓	4
12.11	even	2		1104.6.a.n.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.c.1.1	✓	4	3.2	odd	2
207.6.a.d.1.4		4	1.1	even	1	trivial
1104.6.a.n.1.1		4	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)

\(f(q)\)	\(=\)	\(q+4.76108 q^{2} -9.33211 q^{4} +97.1410 q^{5} -211.220 q^{7} -196.786 q^{8} +O(q^{10})\)	\(q+4.76108 q^{2} -9.33211 q^{4} +97.1410 q^{5} -211.220 q^{7} -196.786 q^{8} +462.496 q^{10} +633.540 q^{11} -59.3594 q^{13} -1005.63 q^{14} -638.284 q^{16} +1809.15 q^{17} +1986.94 q^{19} -906.530 q^{20} +3016.33 q^{22} -529.000 q^{23} +6311.37 q^{25} -282.615 q^{26} +1971.12 q^{28} +2631.22 q^{29} -4916.81 q^{31} +3258.21 q^{32} +8613.53 q^{34} -20518.1 q^{35} -4871.13 q^{37} +9459.96 q^{38} -19115.9 q^{40} +13838.6 q^{41} -1272.49 q^{43} -5912.26 q^{44} -2518.61 q^{46} +17922.1 q^{47} +27806.7 q^{49} +30048.9 q^{50} +553.948 q^{52} +19517.0 q^{53} +61542.6 q^{55} +41565.0 q^{56} +12527.5 q^{58} +17569.0 q^{59} -20453.8 q^{61} -23409.3 q^{62} +35937.7 q^{64} -5766.23 q^{65} -11300.7 q^{67} -16883.2 q^{68} -97688.2 q^{70} -33263.2 q^{71} +22571.9 q^{73} -23191.9 q^{74} -18542.3 q^{76} -133816. q^{77} -9902.06 q^{79} -62003.6 q^{80} +65886.5 q^{82} +24218.1 q^{83} +175743. q^{85} -6058.43 q^{86} -124671. q^{88} -40463.9 q^{89} +12537.9 q^{91} +4936.69 q^{92} +85328.5 q^{94} +193013. q^{95} -18327.4 q^{97} +132390. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8	\( 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8} + 642 q^{10} - 32 q^{11} + 1364 q^{13} - 2754 q^{14} + 18 q^{16} - 278 q^{17} + 2862 q^{19} - 3830 q^{20} + 3176 q^{22} - 2116 q^{23} + 5944 q^{25} - 6996 q^{26} + 4738 q^{28} + 5180 q^{29} - 1788 q^{31} + 7352 q^{32} + 15818 q^{34} - 11768 q^{35} + 3348 q^{37} - 1050 q^{38} - 13462 q^{40} + 17664 q^{41} + 25398 q^{43} + 16848 q^{44} + 2116 q^{46} + 26040 q^{47} + 55720 q^{49} + 35256 q^{50} - 2752 q^{52} + 32006 q^{53} + 34904 q^{55} + 68542 q^{56} - 40804 q^{58} + 61136 q^{59} + 35844 q^{61} + 47524 q^{62} - 35142 q^{64} + 48036 q^{65} + 73458 q^{67} - 17910 q^{68} - 59104 q^{70} - 24432 q^{71} + 122512 q^{73} + 20828 q^{74} + 56834 q^{76} - 159496 q^{77} + 90170 q^{79} + 36546 q^{80} + 84144 q^{82} - 28592 q^{83} + 355124 q^{85} - 103778 q^{86} - 150776 q^{88} + 27926 q^{89} + 334180 q^{91} + 24334 q^{92} + 113632 q^{94} - 113392 q^{95} + 16580 q^{97} - 49688 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8 + 642 * q^10 - 32 * q^11 + 1364 * q^13 - 2754 * q^14 + 18 * q^16 - 278 * q^17 + 2862 * q^19 - 3830 * q^20 + 3176 * q^22 - 2116 * q^23 + 5944 * q^25 - 6996 * q^26 + 4738 * q^28 + 5180 * q^29 - 1788 * q^31 + 7352 * q^32 + 15818 * q^34 - 11768 * q^35 + 3348 * q^37 - 1050 * q^38 - 13462 * q^40 + 17664 * q^41 + 25398 * q^43 + 16848 * q^44 + 2116 * q^46 + 26040 * q^47 + 55720 * q^49 + 35256 * q^50 - 2752 * q^52 + 32006 * q^53 + 34904 * q^55 + 68542 * q^56 - 40804 * q^58 + 61136 * q^59 + 35844 * q^61 + 47524 * q^62 - 35142 * q^64 + 48036 * q^65 + 73458 * q^67 - 17910 * q^68 - 59104 * q^70 - 24432 * q^71 + 122512 * q^73 + 20828 * q^74 + 56834 * q^76 - 159496 * q^77 + 90170 * q^79 + 36546 * q^80 + 84144 * q^82 - 28592 * q^83 + 355124 * q^85 - 103778 * q^86 - 150776 * q^88 + 27926 * q^89 + 334180 * q^91 + 24334 * q^92 + 113632 * q^94 - 113392 * q^95 + 16580 * q^97 - 49688 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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