LMFDB - Embedded newform 207.6.a.h.1.8

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 251 x^{8} + 296 x^{7} + 21169 x^{6} - 9042 x^{5} - 685949 x^{4} - 270764 x^{3} + \cdots + 5446216 $ x^10 - 2x^9 - 251x^8 + 296x^7 + 21169x^6 - 9042x^5 - 685949x^4 - 270764x^3 + 6230110x^2 + 11573432*x + 5446216
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$7.31180$ 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)

$f(q)$	$=$	$q+6.31180 q^{2} +7.83883 q^{4} -40.1381 q^{5} +201.144 q^{7} -152.500 q^{8} +O(q^{10})$	\(q+6.31180 q^{2} +7.83883 q^{4} -40.1381 q^{5} +201.144 q^{7} -152.500 q^{8} -253.344 q^{10} -542.420 q^{11} +734.379 q^{13} +1269.58 q^{14} -1213.40 q^{16} -1643.16 q^{17} -950.789 q^{19} -314.636 q^{20} -3423.65 q^{22} -529.000 q^{23} -1513.93 q^{25} +4635.26 q^{26} +1576.74 q^{28} -466.749 q^{29} +4353.69 q^{31} -2778.69 q^{32} -10371.3 q^{34} -8073.56 q^{35} -15249.6 q^{37} -6001.19 q^{38} +6121.09 q^{40} -10434.3 q^{41} -21013.4 q^{43} -4251.94 q^{44} -3338.94 q^{46} -19689.6 q^{47} +23652.1 q^{49} -9555.62 q^{50} +5756.68 q^{52} +14525.0 q^{53} +21771.7 q^{55} -30674.6 q^{56} -2946.03 q^{58} +43707.9 q^{59} +25350.4 q^{61} +27479.6 q^{62} +21290.1 q^{64} -29476.6 q^{65} -9538.72 q^{67} -12880.5 q^{68} -50958.7 q^{70} -65369.7 q^{71} +17281.7 q^{73} -96252.6 q^{74} -7453.07 q^{76} -109105. q^{77} +88640.9 q^{79} +48703.4 q^{80} -65859.3 q^{82} +52866.9 q^{83} +65953.4 q^{85} -132632. q^{86} +82719.3 q^{88} -7534.11 q^{89} +147716. q^{91} -4146.74 q^{92} -124277. q^{94} +38162.9 q^{95} +21902.5 q^{97} +149287. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8}+O(q^{10}) $ 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8	$ 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8} - 250 q^{10} - 460 q^{11} + 464 q^{13} - 3676 q^{14} + 4612 q^{16} - 4756 q^{17} - 1780 q^{19} - 10314 q^{20} - 4214 q^{22} - 5290 q^{23} + 1330 q^{25} + 5152 q^{26} + 7072 q^{28} - 4048 q^{29} + 2816 q^{31} - 27436 q^{32} + 420 q^{34} - 9452 q^{35} + 2872 q^{37} - 31038 q^{38} + 2618 q^{40} - 34056 q^{41} + 7316 q^{43} - 33562 q^{44} + 4232 q^{46} - 49300 q^{47} + 45118 q^{49} - 44764 q^{50} - 25120 q^{52} - 86676 q^{53} - 2120 q^{55} - 290684 q^{56} - 87408 q^{58} - 67100 q^{59} - 40432 q^{61} - 230992 q^{62} + 136776 q^{64} - 184000 q^{65} - 50108 q^{67} - 270592 q^{68} + 117456 q^{70} - 238584 q^{71} - 13804 q^{73} - 150074 q^{74} - 197622 q^{76} - 116248 q^{77} - 9228 q^{79} - 313010 q^{80} - 68604 q^{82} - 155300 q^{83} + 80444 q^{85} + 80914 q^{86} - 237738 q^{88} - 213732 q^{89} - 264352 q^{91} - 101568 q^{92} + 140280 q^{94} + 123612 q^{95} + 42516 q^{97} + 151388 q^{98}+O(q^{100}) $ 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8 - 250 * q^10 - 460 * q^11 + 464 * q^13 - 3676 * q^14 + 4612 * q^16 - 4756 * q^17 - 1780 * q^19 - 10314 * q^20 - 4214 * q^22 - 5290 * q^23 + 1330 * q^25 + 5152 * q^26 + 7072 * q^28 - 4048 * q^29 + 2816 * q^31 - 27436 * q^32 + 420 * q^34 - 9452 * q^35 + 2872 * q^37 - 31038 * q^38 + 2618 * q^40 - 34056 * q^41 + 7316 * q^43 - 33562 * q^44 + 4232 * q^46 - 49300 * q^47 + 45118 * q^49 - 44764 * q^50 - 25120 * q^52 - 86676 * q^53 - 2120 * q^55 - 290684 * q^56 - 87408 * q^58 - 67100 * q^59 - 40432 * q^61 - 230992 * q^62 + 136776 * q^64 - 184000 * q^65 - 50108 * q^67 - 270592 * q^68 + 117456 * q^70 - 238584 * q^71 - 13804 * q^73 - 150074 * q^74 - 197622 * q^76 - 116248 * q^77 - 9228 * q^79 - 313010 * q^80 - 68604 * q^82 - 155300 * q^83 + 80444 * q^85 + 80914 * q^86 - 237738 * q^88 - 213732 * q^89 - 264352 * q^91 - 101568 * q^92 + 140280 * q^94 + 123612 * q^95 + 42516 * q^97 + 151388 * 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 $

$2$	6.31180	1.11578	0.557890	−	0.829915i	$-0.311611\pi$
$2$	6.31180	1.11578	0.557890	+	0.829915i	$0.311611\pi$
$3$	0	0
$3$	0	0
$4$	7.83883	0.244963
$5$	−40.1381	−0.718013	−0.359006	−	0.933335i	$-0.616884\pi$
$5$	−40.1381	−0.718013	−0.359006	+	0.933335i	$0.616884\pi$
$6$	0	0
$7$	201.144	1.55154	0.775770	−	0.631016i	$-0.217362\pi$
$7$	201.144	1.55154	0.775770	+	0.631016i	$0.217362\pi$
$8$	−152.500	−0.842454
$9$	0	0
$10$	−253.344	−0.801144
$11$	−542.420	−1.35162	−0.675809	−	0.737077i	$-0.736205\pi$
$11$	−542.420	−1.35162	−0.675809	+	0.737077i	$0.736205\pi$
$12$	0	0
$13$	734.379	1.20521	0.602604	−	0.798040i	$-0.294130\pi$
$13$	734.379	1.20521	0.602604	+	0.798040i	$0.294130\pi$
$14$	1269.58	1.73118
$15$	0	0
$16$	−1213.40	−1.18496
$17$	−1643.16	−1.37898	−0.689490	−	0.724296i	$-0.742165\pi$
$17$	−1643.16	−1.37898	−0.689490	+	0.724296i	$0.742165\pi$
$18$	0	0
$19$	−950.789	−0.604227	−0.302113	−	0.953272i	$-0.597692\pi$
$19$	−950.789	−0.604227	−0.302113	+	0.953272i	$0.597692\pi$
$20$	−314.636	−0.175887
$21$	0	0
$22$	−3423.65	−1.50811
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−1513.93	−0.484457
$26$	4635.26	1.34475
$27$	0	0
$28$	1576.74	0.380070
$29$	−466.749	−0.103060	−0.0515298	−	0.998671i	$-0.516410\pi$
$29$	−466.749	−0.103060	−0.0515298	+	0.998671i	$0.516410\pi$
$30$	0	0
$31$	4353.69	0.813679	0.406839	−	0.913500i	$-0.366631\pi$
$31$	4353.69	0.813679	0.406839	+	0.913500i	$0.366631\pi$
$32$	−2778.69	−0.479696
$33$	0	0
$34$	−10371.3	−1.53864
$35$	−8073.56	−1.11403
$36$	0	0
$37$	−15249.6	−1.83128	−0.915641	−	0.401998i	$-0.868316\pi$
$37$	−15249.6	−1.83128	−0.915641	+	0.401998i	$0.868316\pi$
$38$	−6001.19	−0.674184
$39$	0	0
$40$	6121.09	0.604893
$41$	−10434.3	−0.969403	−0.484702	−	0.874680i	$-0.661072\pi$
$41$	−10434.3	−0.969403	−0.484702	+	0.874680i	$0.661072\pi$
$42$	0	0
$43$	−21013.4	−1.73311	−0.866554	−	0.499084i	$-0.833670\pi$
$43$	−21013.4	−1.73311	−0.866554	+	0.499084i	$0.833670\pi$
$44$	−4251.94	−0.331097
$45$	0	0
$46$	−3338.94	−0.232656
$47$	−19689.6	−1.30015	−0.650075	−	0.759870i	$-0.725262\pi$
$47$	−19689.6	−1.30015	−0.650075	+	0.759870i	$0.725262\pi$
$48$	0	0
$49$	23652.1	1.40727
$50$	−9555.62	−0.540548
$51$	0	0
$52$	5756.68	0.295232
$53$	14525.0	0.710277	0.355138	−	0.934814i	$-0.384434\pi$
$53$	14525.0	0.710277	0.355138	+	0.934814i	$0.384434\pi$
$54$	0	0
$55$	21771.7	0.970479
$56$	−30674.6	−1.30710
$57$	0	0
$58$	−2946.03	−0.114992
$59$	43707.9	1.63467	0.817335	−	0.576163i	$-0.195450\pi$
$59$	43707.9	1.63467	0.817335	+	0.576163i	$0.195450\pi$
$60$	0	0
$61$	25350.4	0.872288	0.436144	−	0.899877i	$-0.356344\pi$
$61$	25350.4	0.872288	0.436144	+	0.899877i	$0.356344\pi$
$62$	27479.6	0.907886
$63$	0	0
$64$	21290.1	0.649722
$65$	−29476.6	−0.865355
$66$	0	0
$67$	−9538.72	−0.259599	−0.129799	−	0.991540i	$-0.541433\pi$
$67$	−9538.72	−0.259599	−0.129799	+	0.991540i	$0.541433\pi$
$68$	−12880.5	−0.337800
$69$	0	0
$70$	−50958.7	−1.24301
$71$	−65369.7	−1.53897	−0.769485	−	0.638664i	$-0.779487\pi$
$71$	−65369.7	−1.53897	−0.769485	+	0.638664i	$0.779487\pi$
$72$	0	0
$73$	17281.7	0.379559	0.189780	−	0.981827i	$-0.439223\pi$
$73$	17281.7	0.379559	0.189780	+	0.981827i	$0.439223\pi$
$74$	−96252.6	−2.04331
$75$	0	0
$76$	−7453.07	−0.148013
$77$	−109105.	−2.09709
$78$	0	0
$79$	88640.9	1.59796	0.798981	−	0.601356i	$-0.205373\pi$
$79$	88640.9	1.59796	0.798981	+	0.601356i	$0.205373\pi$
$80$	48703.4	0.850814
$81$	0	0
$82$	−65859.3	−1.08164
$83$	52866.9	0.842343	0.421171	−	0.906981i	$-0.361619\pi$
$83$	52866.9	0.842343	0.421171	+	0.906981i	$0.361619\pi$
$84$	0	0
$85$	65953.4	0.990125
$86$	−132632.	−1.93377
$87$	0	0
$88$	82719.3	1.13868
$89$	−7534.11	−0.100822	−0.0504112	−	0.998729i	$-0.516053\pi$
$89$	−7534.11	−0.100822	−0.0504112	+	0.998729i	$0.516053\pi$
$90$	0	0
$91$	147716.	1.86993
$92$	−4146.74	−0.0510784
$93$	0	0
$94$	−124277.	−1.45068
$95$	38162.9	0.433843
$96$	0	0
$97$	21902.5	0.236355	0.118177	−	0.992992i	$-0.462295\pi$
$97$	21902.5	0.236355	0.118177	+	0.992992i	$0.462295\pi$
$98$	149287.	1.57021
$99$	0	0
$100$	−11867.4	−0.118674
$101$	169082.	1.64928	0.824639	−	0.565659i	$-0.191378\pi$
$101$	169082.	1.64928	0.824639	+	0.565659i	$0.191378\pi$
$102$	0	0
$103$	−153452.	−1.42521	−0.712606	−	0.701564i	$-0.752485\pi$
$103$	−153452.	−1.42521	−0.712606	+	0.701564i	$0.752485\pi$
$104$	−111993.	−1.01533
$105$	0	0
$106$	91679.1	0.792512
$107$	9404.93	0.0794138	0.0397069	−	0.999211i	$-0.487358\pi$
$107$	9404.93	0.0794138	0.0397069	+	0.999211i	$0.487358\pi$
$108$	0	0
$109$	−203472.	−1.64036	−0.820178	−	0.572108i	$-0.806126\pi$
$109$	−203472.	−1.64036	−0.820178	+	0.572108i	$0.806126\pi$
$110$	137419.	1.08284
$111$	0	0
$112$	−244068.	−1.83851
$113$	−86777.4	−0.639309	−0.319654	−	0.947534i	$-0.603567\pi$
$113$	−86777.4	−0.639309	−0.319654	+	0.947534i	$0.603567\pi$
$114$	0	0
$115$	21233.1	0.149716
$116$	−3658.77	−0.0252458
$117$	0	0
$118$	275876.	1.82393
$119$	−330512.	−2.13954
$120$	0	0
$121$	133168.	0.826870
$122$	160007.	0.973281
$123$	0	0
$124$	34127.8	0.199322
$125$	186198.	1.06586
$126$	0	0
$127$	214305.	1.17903	0.589514	−	0.807758i	$-0.299319\pi$
$127$	214305.	1.17903	0.589514	+	0.807758i	$0.299319\pi$
$128$	223297.	1.20464
$129$	0	0
$130$	−186051.	−0.965545
$131$	38454.5	0.195780	0.0978902	−	0.995197i	$-0.468791\pi$
$131$	38454.5	0.195780	0.0978902	+	0.995197i	$0.468791\pi$
$132$	0	0
$133$	−191246.	−0.937481
$134$	−60206.5	−0.289655
$135$	0	0
$136$	250583.	1.16173
$137$	253995.	1.15617	0.578087	−	0.815975i	$-0.303800\pi$
$137$	253995.	1.15617	0.578087	+	0.815975i	$0.303800\pi$
$138$	0	0
$139$	−185560.	−0.814605	−0.407303	−	0.913293i	$-0.633531\pi$
$139$	−185560.	−0.814605	−0.407303	+	0.913293i	$0.633531\pi$
$140$	−63287.3	−0.272895
$141$	0	0
$142$	−412600.	−1.71715
$143$	−398342.	−1.62898
$144$	0	0
$145$	18734.4	0.0739981
$146$	109079.	0.423504
$147$	0	0
$148$	−119539.	−0.448597
$149$	241946.	0.892796	0.446398	−	0.894834i	$-0.352707\pi$
$149$	241946.	0.892796	0.446398	+	0.894834i	$0.352707\pi$
$150$	0	0
$151$	−357679.	−1.27659	−0.638294	−	0.769793i	$-0.720360\pi$
$151$	−357679.	−1.27659	−0.638294	+	0.769793i	$0.720360\pi$
$152$	144996.	0.509033
$153$	0	0
$154$	−688647.	−2.33989
$155$	−174749.	−0.584232
$156$	0	0
$157$	−168853.	−0.546712	−0.273356	−	0.961913i	$-0.588134\pi$
$157$	−168853.	−0.546712	−0.273356	+	0.961913i	$0.588134\pi$
$158$	559484.	1.78297
$159$	0	0
$160$	111532.	0.344428
$161$	−106405.	−0.323518
$162$	0	0
$163$	490132.	1.44492	0.722460	−	0.691412i	$-0.243011\pi$
$163$	490132.	1.44492	0.722460	+	0.691412i	$0.243011\pi$
$164$	−81792.9	−0.237468
$165$	0	0
$166$	333686.	0.939869
$167$	136272.	0.378108	0.189054	−	0.981967i	$-0.439458\pi$
$167$	136272.	0.378108	0.189054	+	0.981967i	$0.439458\pi$
$168$	0	0
$169$	168020.	0.452526
$170$	416285.	1.10476
$171$	0	0
$172$	−164721.	−0.424548
$173$	−121132.	−0.307711	−0.153855	−	0.988093i	$-0.549169\pi$
$173$	−121132.	−0.307711	−0.153855	+	0.988093i	$0.549169\pi$
$174$	0	0
$175$	−304518.	−0.751655
$176$	658170.	1.60161
$177$	0	0
$178$	−47553.8	−0.112496
$179$	−569859.	−1.32934	−0.664669	−	0.747138i	$-0.731427\pi$
$179$	−569859.	−1.32934	−0.664669	+	0.747138i	$0.731427\pi$
$180$	0	0
$181$	710204.	1.61134	0.805669	−	0.592365i	$-0.201806\pi$
$181$	710204.	1.61134	0.805669	+	0.592365i	$0.201806\pi$
$182$	932356.	2.08643
$183$	0	0
$184$	80672.8	0.175664
$185$	612092.	1.31488
$186$	0	0
$187$	891283.	1.86385
$188$	−154344.	−0.318489
$189$	0	0
$190$	240877.	0.484073
$191$	−153974.	−0.305397	−0.152698	−	0.988273i	$-0.548796\pi$
$191$	−153974.	−0.305397	−0.152698	+	0.988273i	$0.548796\pi$
$192$	0	0
$193$	276457.	0.534238	0.267119	−	0.963664i	$-0.413928\pi$
$193$	276457.	0.534238	0.267119	+	0.963664i	$0.413928\pi$
$194$	138244.	0.263720
$195$	0	0
$196$	185404.	0.344731
$197$	−621691.	−1.14132	−0.570662	−	0.821185i	$-0.693313\pi$
$197$	−621691.	−1.14132	−0.570662	+	0.821185i	$0.693313\pi$
$198$	0	0
$199$	−444054.	−0.794883	−0.397442	−	0.917627i	$-0.630102\pi$
$199$	−444054.	−0.794883	−0.397442	+	0.917627i	$0.630102\pi$
$200$	230875.	0.408133
$201$	0	0
$202$	1.06721e6	1.84023
$203$	−93883.9	−0.159901
$204$	0	0
$205$	418814.	0.696044
$206$	−968558.	−1.59022
$207$	0	0
$208$	−891092.	−1.42812
$209$	515727.	0.816683
$210$	0	0
$211$	−711330.	−1.09993	−0.549965	−	0.835188i	$-0.685359\pi$
$211$	−711330.	−1.09993	−0.549965	+	0.835188i	$0.685359\pi$
$212$	113859.	0.173992
$213$	0	0
$214$	59362.0	0.0886083
$215$	843439.	1.24439
$216$	0	0
$217$	875720.	1.26245
$218$	−1.28427e6	−1.83028
$219$	0	0
$220$	170665.	0.237732
$221$	−1.20670e6	−1.66196
$222$	0	0
$223$	−29103.7	−0.0391910	−0.0195955	−	0.999808i	$-0.506238\pi$
$223$	−29103.7	−0.0391910	−0.0195955	+	0.999808i	$0.506238\pi$
$224$	−558919.	−0.744267
$225$	0	0
$226$	−547722.	−0.713327
$227$	−85117.5	−0.109636	−0.0548181	−	0.998496i	$-0.517458\pi$
$227$	−85117.5	−0.109636	−0.0548181	+	0.998496i	$0.517458\pi$
$228$	0	0
$229$	550138.	0.693239	0.346620	−	0.938006i	$-0.387329\pi$
$229$	550138.	0.693239	0.346620	+	0.938006i	$0.387329\pi$
$230$	134019.	0.167050
$231$	0	0
$232$	71179.4	0.0868229
$233$	329291.	0.397366	0.198683	−	0.980064i	$-0.436334\pi$
$233$	329291.	0.397366	0.198683	+	0.980064i	$0.436334\pi$
$234$	0	0
$235$	790306.	0.933524
$236$	342619.	0.400435
$237$	0	0
$238$	−2.08613e6	−2.38725
$239$	−1.02340e6	−1.15891	−0.579457	−	0.815003i	$-0.696735\pi$
$239$	−1.02340e6	−1.15891	−0.579457	+	0.815003i	$0.696735\pi$
$240$	0	0
$241$	−88522.0	−0.0981768	−0.0490884	−	0.998794i	$-0.515632\pi$
$241$	−88522.0	−0.0981768	−0.0490884	+	0.998794i	$0.515632\pi$
$242$	840531.	0.922604
$243$	0	0
$244$	198717.	0.213679
$245$	−949349.	−1.01044
$246$	0	0
$247$	−698239.	−0.728219
$248$	−663940.	−0.685487
$249$	0	0
$250$	1.17524e6	1.18926
$251$	365086.	0.365772	0.182886	−	0.983134i	$-0.441456\pi$
$251$	365086.	0.365772	0.182886	+	0.983134i	$0.441456\pi$
$252$	0	0
$253$	286940.	0.281832
$254$	1.35265e6	1.31553
$255$	0	0
$256$	728123.	0.694393
$257$	414152.	0.391135	0.195568	−	0.980690i	$-0.437345\pi$
$257$	414152.	0.391135	0.195568	+	0.980690i	$0.437345\pi$
$258$	0	0
$259$	−3.06738e6	−2.84130
$260$	−231062.	−0.211980
$261$	0	0
$262$	242717.	0.218448
$263$	−2.06704e6	−1.84272	−0.921358	−	0.388715i	$-0.872919\pi$
$263$	−2.06704e6	−1.84272	−0.921358	+	0.388715i	$0.872919\pi$
$264$	0	0
$265$	−583008.	−0.509988
$266$	−1.20711e6	−1.04602
$267$	0	0
$268$	−74772.4	−0.0635923
$269$	−1.49730e6	−1.26162	−0.630809	−	0.775938i	$-0.717277\pi$
$269$	−1.49730e6	−1.26162	−0.630809	+	0.775938i	$0.717277\pi$
$270$	0	0
$271$	82305.7	0.0680780	0.0340390	−	0.999421i	$-0.489163\pi$
$271$	82305.7	0.0680780	0.0340390	+	0.999421i	$0.489163\pi$
$272$	1.99380e6	1.63403
$273$	0	0
$274$	1.60316e6	1.29003
$275$	821185.	0.654801
$276$	0	0
$277$	521084.	0.408045	0.204022	−	0.978966i	$-0.434598\pi$
$277$	521084.	0.408045	0.204022	+	0.978966i	$0.434598\pi$
$278$	−1.17122e6	−0.908920
$279$	0	0
$280$	1.23122e6	0.938515
$281$	−175623.	−0.132683	−0.0663417	−	0.997797i	$-0.521133\pi$
$281$	−175623.	−0.132683	−0.0663417	+	0.997797i	$0.521133\pi$
$282$	0	0
$283$	−1.94588e6	−1.44427	−0.722137	−	0.691750i	$-0.756840\pi$
$283$	−1.94588e6	−1.44427	−0.722137	+	0.691750i	$0.756840\pi$
$284$	−512422.	−0.376992
$285$	0	0
$286$	−2.51425e6	−1.81758
$287$	−2.09880e6	−1.50407
$288$	0	0
$289$	1.28012e6	0.901583
$290$	118248.	0.0825655
$291$	0	0
$292$	135468.	0.0929781
$293$	−641956.	−0.436854	−0.218427	−	0.975853i	$-0.570093\pi$
$293$	−641956.	−0.436854	−0.218427	+	0.975853i	$0.570093\pi$
$294$	0	0
$295$	−1.75435e6	−1.17371
$296$	2.32558e6	1.54277
$297$	0	0
$298$	1.52711e6	0.996163
$299$	−388487.	−0.251303
$300$	0	0
$301$	−4.22673e6	−2.68898
$302$	−2.25760e6	−1.42439
$303$	0	0
$304$	1.15368e6	0.715982
$305$	−1.01752e6	−0.626314
$306$	0	0
$307$	999557.	0.605288	0.302644	−	0.953104i	$-0.402131\pi$
$307$	999557.	0.605288	0.302644	+	0.953104i	$0.402131\pi$
$308$	−855253.	−0.513710
$309$	0	0
$310$	−1.10298e6	−0.651874
$311$	2.06253e6	1.20920	0.604601	−	0.796528i	$-0.293333\pi$
$311$	2.06253e6	1.20920	0.604601	+	0.796528i	$0.293333\pi$
$312$	0	0
$313$	−1.72440e6	−0.994895	−0.497447	−	0.867494i	$-0.665729\pi$
$313$	−1.72440e6	−0.994895	−0.497447	+	0.867494i	$0.665729\pi$
$314$	−1.06576e6	−0.610010
$315$	0	0
$316$	694841.	0.391443
$317$	−1.11274e6	−0.621936	−0.310968	−	0.950420i	$-0.600653\pi$
$317$	−1.11274e6	−0.621936	−0.310968	+	0.950420i	$0.600653\pi$
$318$	0	0
$319$	253174.	0.139297
$320$	−854544.	−0.466509
$321$	0	0
$322$	−671609.	−0.360975
$323$	1.56230e6	0.833216
$324$	0	0
$325$	−1.11180e6	−0.583872
$326$	3.09362e6	1.61221
$327$	0	0
$328$	1.59124e6	0.816678
$329$	−3.96046e6	−2.01723
$330$	0	0
$331$	1.15759e6	0.580746	0.290373	−	0.956913i	$-0.406221\pi$
$331$	1.15759e6	0.580746	0.290373	+	0.956913i	$0.406221\pi$
$332$	414415.	0.206343
$333$	0	0
$334$	860122.	0.421885
$335$	382866.	0.186395
$336$	0	0
$337$	1.43245e6	0.687074	0.343537	−	0.939139i	$-0.388375\pi$
$337$	1.43245e6	0.687074	0.343537	+	0.939139i	$0.388375\pi$
$338$	1.06051e6	0.504920
$339$	0	0
$340$	516998.	0.242544
$341$	−2.36153e6	−1.09978
$342$	0	0
$343$	1.37684e6	0.631901
$344$	3.20456e6	1.46006
$345$	0	0
$346$	−764560.	−0.343337
$347$	1.74657e6	0.778688	0.389344	−	0.921092i	$-0.372702\pi$
$347$	1.74657e6	0.778688	0.389344	+	0.921092i	$0.372702\pi$
$348$	0	0
$349$	176582.	0.0776040	0.0388020	−	0.999247i	$-0.487646\pi$
$349$	176582.	0.0776040	0.0388020	+	0.999247i	$0.487646\pi$
$350$	−1.92206e6	−0.838681
$351$	0	0
$352$	1.50722e6	0.648365
$353$	280945.	0.120001	0.0600004	−	0.998198i	$-0.480890\pi$
$353$	280945.	0.120001	0.0600004	+	0.998198i	$0.480890\pi$
$354$	0	0
$355$	2.62382e6	1.10500
$356$	−59058.6	−0.0246978
$357$	0	0
$358$	−3.59684e6	−1.48325
$359$	3.26247e6	1.33601	0.668006	−	0.744156i	$-0.267148\pi$
$359$	3.26247e6	1.33601	0.668006	+	0.744156i	$0.267148\pi$
$360$	0	0
$361$	−1.57210e6	−0.634910
$362$	4.48267e6	1.79790
$363$	0	0
$364$	1.15792e6	0.458064
$365$	−693655.	−0.272528
$366$	0	0
$367$	−2.12325e6	−0.822878	−0.411439	−	0.911437i	$-0.634974\pi$
$367$	−2.12325e6	−0.822878	−0.411439	+	0.911437i	$0.634974\pi$
$368$	641886.	0.247080
$369$	0	0
$370$	3.86340e6	1.46712
$371$	2.92163e6	1.10202
$372$	0	0
$373$	−4.67779e6	−1.74088	−0.870441	−	0.492274i	$-0.836166\pi$
$373$	−4.67779e6	−1.74088	−0.870441	+	0.492274i	$0.836166\pi$
$374$	5.62560e6	2.07965
$375$	0	0
$376$	3.00268e6	1.09532
$377$	−342771.	−0.124208
$378$	0	0
$379$	1.45398e6	0.519949	0.259975	−	0.965615i	$-0.416286\pi$
$379$	1.45398e6	0.519949	0.259975	+	0.965615i	$0.416286\pi$
$380$	299152.	0.106276
$381$	0	0
$382$	−971854.	−0.340755
$383$	−3.67397e6	−1.27979	−0.639895	−	0.768462i	$-0.721022\pi$
$383$	−3.67397e6	−1.27979	−0.639895	+	0.768462i	$0.721022\pi$
$384$	0	0
$385$	4.37926e6	1.50574
$386$	1.74494e6	0.596092
$387$	0	0
$388$	171690.	0.0578983
$389$	−5.16131e6	−1.72936	−0.864682	−	0.502320i	$-0.832480\pi$
$389$	−5.16131e6	−1.72936	−0.864682	+	0.502320i	$0.832480\pi$
$390$	0	0
$391$	869232.	0.287537
$392$	−3.60695e6	−1.18556
$393$	0	0
$394$	−3.92399e6	−1.27347
$395$	−3.55788e6	−1.14736
$396$	0	0
$397$	2.87104e6	0.914246	0.457123	−	0.889404i	$-0.348880\pi$
$397$	2.87104e6	0.914246	0.457123	+	0.889404i	$0.348880\pi$
$398$	−2.80278e6	−0.886914
$399$	0	0
$400$	1.83700e6	0.574061
$401$	−2.00344e6	−0.622178	−0.311089	−	0.950381i	$-0.600694\pi$
$401$	−2.00344e6	−0.622178	−0.311089	+	0.950381i	$0.600694\pi$
$402$	0	0
$403$	3.19726e6	0.980652
$404$	1.32541e6	0.404013
$405$	0	0
$406$	−592576.	−0.178414
$407$	8.27170e6	2.47519
$408$	0	0
$409$	−4.04541e6	−1.19579	−0.597894	−	0.801575i	$-0.703996\pi$
$409$	−4.04541e6	−1.19579	−0.597894	+	0.801575i	$0.703996\pi$
$410$	2.64347e6	0.776631
$411$	0	0
$412$	−1.20288e6	−0.349125
$413$	8.79160e6	2.53626
$414$	0	0
$415$	−2.12198e6	−0.604813
$416$	−2.04062e6	−0.578133
$417$	0	0
$418$	3.25516e6	0.911238
$419$	3.04893e6	0.848422	0.424211	−	0.905563i	$-0.360551\pi$
$419$	3.04893e6	0.848422	0.424211	+	0.905563i	$0.360551\pi$
$420$	0	0
$421$	3.06960e6	0.844067	0.422033	−	0.906580i	$-0.361317\pi$
$421$	3.06960e6	0.844067	0.422033	+	0.906580i	$0.361317\pi$
$422$	−4.48977e6	−1.22728
$423$	0	0
$424$	−2.21508e6	−0.598375
$425$	2.48763e6	0.668057
$426$	0	0
$427$	5.09908e6	1.35339
$428$	73723.6	0.0194535
$429$	0	0
$430$	5.32362e6	1.38847
$431$	−4.88623e6	−1.26701	−0.633506	−	0.773738i	$-0.718385\pi$
$431$	−4.88623e6	−1.26701	−0.633506	+	0.773738i	$0.718385\pi$
$432$	0	0
$433$	986825.	0.252942	0.126471	−	0.991970i	$-0.459635\pi$
$433$	986825.	0.252942	0.126471	+	0.991970i	$0.459635\pi$
$434$	5.52737e6	1.40862
$435$	0	0
$436$	−1.59498e6	−0.401828
$437$	502967.	0.125990
$438$	0	0
$439$	−1.42580e6	−0.353099	−0.176550	−	0.984292i	$-0.556494\pi$
$439$	−1.42580e6	−0.353099	−0.176550	+	0.984292i	$0.556494\pi$
$440$	−3.32020e6	−0.817584
$441$	0	0
$442$	−7.61647e6	−1.85438
$443$	7.26093e6	1.75786	0.878928	−	0.476955i	$-0.158260\pi$
$443$	7.26093e6	1.75786	0.878928	+	0.476955i	$0.158260\pi$
$444$	0	0
$445$	302405.	0.0723918
$446$	−183697.	−0.0437285
$447$	0	0
$448$	4.28238e6	1.00807
$449$	−1.09789e6	−0.257007	−0.128503	−	0.991709i	$-0.541017\pi$
$449$	−1.09789e6	−0.257007	−0.128503	+	0.991709i	$0.541017\pi$
$450$	0	0
$451$	5.65978e6	1.31026
$452$	−680233.	−0.156607
$453$	0	0
$454$	−537245.	−0.122330
$455$	−5.92906e6	−1.34263
$456$	0	0
$457$	−7.15913e6	−1.60350	−0.801752	−	0.597658i	$-0.796098\pi$
$457$	−7.15913e6	−1.60350	−0.801752	+	0.597658i	$0.796098\pi$
$458$	3.47236e6	0.773502
$459$	0	0
$460$	166443.	0.0366750
$461$	4.66760e6	1.02292	0.511460	−	0.859307i	$-0.329105\pi$
$461$	4.66760e6	1.02292	0.511460	+	0.859307i	$0.329105\pi$
$462$	0	0
$463$	253136.	0.0548783	0.0274392	−	0.999623i	$-0.491265\pi$
$463$	253136.	0.0548783	0.0274392	+	0.999623i	$0.491265\pi$
$464$	566351.	0.122121
$465$	0	0
$466$	2.07842e6	0.443372
$467$	4.53338e6	0.961899	0.480950	−	0.876748i	$-0.340292\pi$
$467$	4.53338e6	0.961899	0.480950	+	0.876748i	$0.340292\pi$
$468$	0	0
$469$	−1.91866e6	−0.402778
$470$	4.98825e6	1.04161
$471$	0	0
$472$	−6.66548e6	−1.37713
$473$	1.13981e7	2.34250
$474$	0	0
$475$	1.43943e6	0.292722
$476$	−2.59083e6	−0.524109
$477$	0	0
$478$	−6.45950e6	−1.29309
$479$	−1.75739e6	−0.349969	−0.174984	−	0.984571i	$-0.555988\pi$
$479$	−1.75739e6	−0.349969	−0.174984	+	0.984571i	$0.555988\pi$
$480$	0	0
$481$	−1.11990e7	−2.20707
$482$	−558733.	−0.109544
$483$	0	0
$484$	1.04388e6	0.202553
$485$	−879127.	−0.169706
$486$	0	0
$487$	1.16937e6	0.223423	0.111712	−	0.993741i	$-0.464367\pi$
$487$	1.16937e6	0.223423	0.111712	+	0.993741i	$0.464367\pi$
$488$	−3.86594e6	−0.734862
$489$	0	0
$490$	−5.99210e6	−1.12743
$491$	8.67448e6	1.62383	0.811914	−	0.583777i	$-0.198426\pi$
$491$	8.67448e6	1.62383	0.811914	+	0.583777i	$0.198426\pi$
$492$	0	0
$493$	766943.	0.142117
$494$	−4.40715e6	−0.812532
$495$	0	0
$496$	−5.28274e6	−0.964174
$497$	−1.31487e7	−2.38777
$498$	0	0
$499$	−308107.	−0.0553924	−0.0276962	−	0.999616i	$-0.508817\pi$
$499$	−308107.	−0.0553924	−0.0276962	+	0.999616i	$0.508817\pi$
$500$	1.45957e6	0.261097
$501$	0	0
$502$	2.30435e6	0.408121
$503$	−961070.	−0.169369	−0.0846847	−	0.996408i	$-0.526988\pi$
$503$	−961070.	−0.169369	−0.0846847	+	0.996408i	$0.526988\pi$
$504$	0	0
$505$	−6.78664e6	−1.18420
$506$	1.81111e6	0.314462
$507$	0	0
$508$	1.67990e6	0.288819
$509$	−1.90478e6	−0.325874	−0.162937	−	0.986636i	$-0.552097\pi$
$509$	−1.90478e6	−0.325874	−0.162937	+	0.986636i	$0.552097\pi$
$510$	0	0
$511$	3.47612e6	0.588901
$512$	−2.54973e6	−0.429853
$513$	0	0
$514$	2.61404e6	0.436420
$515$	6.15928e6	1.02332
$516$	0	0
$517$	1.06801e7	1.75730
$518$	−1.93607e7	−3.17027
$519$	0	0
$520$	4.49520e6	0.729022
$521$	−9.33725e6	−1.50704	−0.753520	−	0.657425i	$-0.771646\pi$
$521$	−9.33725e6	−1.50704	−0.753520	+	0.657425i	$0.771646\pi$
$522$	0	0
$523$	−8.18714e6	−1.30881	−0.654407	−	0.756142i	$-0.727082\pi$
$523$	−8.18714e6	−1.30881	−0.654407	+	0.756142i	$0.727082\pi$
$524$	301438.	0.0479590
$525$	0	0
$526$	−1.30467e7	−2.05606
$527$	−7.15381e6	−1.12205
$528$	0	0
$529$	279841.	0.0434783
$530$	−3.67983e6	−0.569034
$531$	0	0
$532$	−1.49914e6	−0.229649
$533$	−7.66275e6	−1.16833
$534$	0	0
$535$	−377496.	−0.0570201
$536$	1.45466e6	0.218700
$537$	0	0
$538$	−9.45066e6	−1.40769
$539$	−1.28293e7	−1.90210
$540$	0	0
$541$	1.18424e7	1.73959	0.869793	−	0.493416i	$-0.164252\pi$
$541$	1.18424e7	1.73959	0.869793	+	0.493416i	$0.164252\pi$
$542$	519497.	0.0759600
$543$	0	0
$544$	4.56584e6	0.661490
$545$	8.16699e6	1.17780
$546$	0	0
$547$	1.07959e6	0.154273	0.0771367	−	0.997021i	$-0.475422\pi$
$547$	1.07959e6	0.154273	0.0771367	+	0.997021i	$0.475422\pi$
$548$	1.99102e6	0.283220
$549$	0	0
$550$	5.18316e6	0.730614
$551$	443779.	0.0622713
$552$	0	0
$553$	1.78296e7	2.47930
$554$	3.28898e6	0.455288
$555$	0	0
$556$	−1.45457e6	−0.199549
$557$	−3.27821e6	−0.447712	−0.223856	−	0.974622i	$-0.571865\pi$
$557$	−3.27821e6	−0.447712	−0.223856	+	0.974622i	$0.571865\pi$
$558$	0	0
$559$	−1.54318e7	−2.08875
$560$	9.79642e6	1.32007
$561$	0	0
$562$	−1.10850e6	−0.148045
$563$	3.89243e6	0.517547	0.258774	−	0.965938i	$-0.416682\pi$
$563$	3.89243e6	0.517547	0.258774	+	0.965938i	$0.416682\pi$
$564$	0	0
$565$	3.48308e6	0.459032
$566$	−1.22820e7	−1.61149
$567$	0	0
$568$	9.96891e6	1.29651
$569$	1.00848e7	1.30583	0.652916	−	0.757430i	$-0.273545\pi$
$569$	1.00848e7	1.30583	0.652916	+	0.757430i	$0.273545\pi$
$570$	0	0
$571$	1.84357e6	0.236630	0.118315	−	0.992976i	$-0.462251\pi$
$571$	1.84357e6	0.236630	0.118315	+	0.992976i	$0.462251\pi$
$572$	−3.12253e6	−0.399041
$573$	0	0
$574$	−1.32472e7	−1.67821
$575$	800869.	0.101016
$576$	0	0
$577$	6.09702e6	0.762391	0.381196	−	0.924494i	$-0.375512\pi$
$577$	6.09702e6	0.762391	0.381196	+	0.924494i	$0.375512\pi$
$578$	8.07986e6	1.00597
$579$	0	0
$580$	146856.	0.0181268
$581$	1.06339e7	1.30693
$582$	0	0
$583$	−7.87867e6	−0.960022
$584$	−2.63547e6	−0.319761
$585$	0	0
$586$	−4.05190e6	−0.487433
$587$	7.72100e6	0.924864	0.462432	−	0.886655i	$-0.346977\pi$
$587$	7.72100e6	0.924864	0.462432	+	0.886655i	$0.346977\pi$
$588$	0	0
$589$	−4.13944e6	−0.491647
$590$	−1.10731e7	−1.30961
$591$	0	0
$592$	1.85038e7	2.16999
$593$	−7.53507e6	−0.879935	−0.439967	−	0.898014i	$-0.645010\pi$
$593$	−7.53507e6	−0.879935	−0.439967	+	0.898014i	$0.645010\pi$
$594$	0	0
$595$	1.32662e7	1.53622
$596$	1.89657e6	0.218702
$597$	0	0
$598$	−2.45205e6	−0.280399
$599$	9.33044e6	1.06252	0.531258	−	0.847210i	$-0.321720\pi$
$599$	9.33044e6	1.06252	0.531258	+	0.847210i	$0.321720\pi$
$600$	0	0
$601$	5.54845e6	0.626593	0.313296	−	0.949655i	$-0.398567\pi$
$601$	5.54845e6	0.626593	0.313296	+	0.949655i	$0.398567\pi$
$602$	−2.66783e7	−3.00031
$603$	0	0
$604$	−2.80378e6	−0.312717
$605$	−5.34512e6	−0.593703
$606$	0	0
$607$	1.04524e7	1.15145	0.575726	−	0.817643i	$-0.304720\pi$
$607$	1.04524e7	1.15145	0.575726	+	0.817643i	$0.304720\pi$
$608$	2.64195e6	0.289845
$609$	0	0
$610$	−6.42236e6	−0.698828
$611$	−1.44597e7	−1.56695
$612$	0	0
$613$	3.96320e6	0.425986	0.212993	−	0.977054i	$-0.431679\pi$
$613$	3.96320e6	0.425986	0.212993	+	0.977054i	$0.431679\pi$
$614$	6.30901e6	0.675367
$615$	0	0
$616$	1.66385e7	1.76670
$617$	−1.10859e7	−1.17235	−0.586174	−	0.810185i	$-0.699367\pi$
$617$	−1.10859e7	−1.17235	−0.586174	+	0.810185i	$0.699367\pi$
$618$	0	0
$619$	−6.76471e6	−0.709615	−0.354807	−	0.934939i	$-0.615454\pi$
$619$	−6.76471e6	−0.709615	−0.354807	+	0.934939i	$0.615454\pi$
$620$	−1.36983e6	−0.143116
$621$	0	0
$622$	1.30183e7	1.34920
$623$	−1.51544e6	−0.156430
$624$	0	0
$625$	−2.74261e6	−0.280843
$626$	−1.08841e7	−1.11008
$627$	0	0
$628$	−1.32361e6	−0.133925
$629$	2.50576e7	2.52530
$630$	0	0
$631$	7.21131e6	0.721009	0.360505	−	0.932757i	$-0.382604\pi$
$631$	7.21131e6	0.721009	0.360505	+	0.932757i	$0.382604\pi$
$632$	−1.35178e7	−1.34621
$633$	0	0
$634$	−7.02340e6	−0.693944
$635$	−8.60182e6	−0.846557
$636$	0	0
$637$	1.73696e7	1.69606
$638$	1.59798e6	0.155425
$639$	0	0
$640$	−8.96273e6	−0.864948
$641$	1.09748e7	1.05500	0.527498	−	0.849556i	$-0.323130\pi$
$641$	1.09748e7	1.05500	0.527498	+	0.849556i	$0.323130\pi$
$642$	0	0
$643$	1.35577e6	0.129318	0.0646592	−	0.997907i	$-0.479404\pi$
$643$	1.35577e6	0.129318	0.0646592	+	0.997907i	$0.479404\pi$
$644$	−834094.	−0.0792502
$645$	0	0
$646$	9.86091e6	0.929685
$647$	−6.31351e6	−0.592939	−0.296470	−	0.955042i	$-0.595809\pi$
$647$	−6.31351e6	−0.592939	−0.296470	+	0.955042i	$0.595809\pi$
$648$	0	0
$649$	−2.37080e7	−2.20945
$650$	−7.01745e6	−0.651472
$651$	0	0
$652$	3.84206e6	0.353953
$653$	−7.72686e6	−0.709121	−0.354560	−	0.935033i	$-0.615369\pi$
$653$	−7.72686e6	−0.709121	−0.354560	+	0.935033i	$0.615369\pi$
$654$	0	0
$655$	−1.54349e6	−0.140573
$656$	1.26610e7	1.14870
$657$	0	0
$658$	−2.49976e7	−2.25079
$659$	1.28976e7	1.15690	0.578451	−	0.815717i	$-0.303657\pi$
$659$	1.28976e7	1.15690	0.578451	+	0.815717i	$0.303657\pi$
$660$	0	0
$661$	−9.60790e6	−0.855313	−0.427656	−	0.903941i	$-0.640661\pi$
$661$	−9.60790e6	−0.855313	−0.427656	+	0.903941i	$0.640661\pi$
$662$	7.30651e6	0.647985
$663$	0	0
$664$	−8.06223e6	−0.709635
$665$	7.67625e6	0.673124
$666$	0	0
$667$	246910.	0.0214894
$668$	1.06821e6	0.0926226
$669$	0	0
$670$	2.41658e6	0.207976
$671$	−1.37505e7	−1.17900
$672$	0	0
$673$	−9.95632e6	−0.847347	−0.423673	−	0.905815i	$-0.639260\pi$
$673$	−9.95632e6	−0.847347	−0.423673	+	0.905815i	$0.639260\pi$
$674$	9.04132e6	0.766623
$675$	0	0
$676$	1.31708e6	0.110852
$677$	−1.61699e7	−1.35593	−0.677963	−	0.735096i	$-0.737137\pi$
$677$	−1.61699e7	−1.35593	−0.677963	+	0.735096i	$0.737137\pi$
$678$	0	0
$679$	4.40557e6	0.366714
$680$	−1.00579e7	−0.834135
$681$	0	0
$682$	−1.49055e7	−1.22711
$683$	−2.03557e6	−0.166968	−0.0834841	−	0.996509i	$-0.526605\pi$
$683$	−2.03557e6	−0.166968	−0.0834841	+	0.996509i	$0.526605\pi$
$684$	0	0
$685$	−1.01949e7	−0.830148
$686$	8.69036e6	0.705062
$687$	0	0
$688$	2.54976e7	2.05366
$689$	1.06669e7	0.856031
$690$	0	0
$691$	−2.05390e7	−1.63638	−0.818192	−	0.574945i	$-0.805023\pi$
$691$	−2.05390e7	−1.63638	−0.818192	+	0.574945i	$0.805023\pi$
$692$	−949532.	−0.0753779
$693$	0	0
$694$	1.10240e7	0.868844
$695$	7.44803e6	0.584897
$696$	0	0
$697$	1.71453e7	1.33679
$698$	1.11455e6	0.0865889
$699$	0	0
$700$	−2.38707e6	−0.184128
$701$	−6.82344e6	−0.524455	−0.262227	−	0.965006i	$-0.584457\pi$
$701$	−6.82344e6	−0.524455	−0.262227	+	0.965006i	$0.584457\pi$
$702$	0	0
$703$	1.44992e7	1.10651
$704$	−1.15482e7	−0.878175
$705$	0	0
$706$	1.77327e6	0.133894
$707$	3.40099e7	2.55892
$708$	0	0
$709$	3.54740e6	0.265030	0.132515	−	0.991181i	$-0.457695\pi$
$709$	3.54740e6	0.265030	0.132515	+	0.991181i	$0.457695\pi$
$710$	1.65610e7	1.23294
$711$	0	0
$712$	1.14896e6	0.0849383
$713$	−2.30310e6	−0.169664
$714$	0	0
$715$	1.59887e7	1.16963
$716$	−4.46703e6	−0.325639
$717$	0	0
$718$	2.05921e7	1.49069
$719$	2.10838e7	1.52099	0.760495	−	0.649344i	$-0.224957\pi$
$719$	2.10838e7	1.52099	0.760495	+	0.649344i	$0.224957\pi$
$720$	0	0
$721$	−3.08660e7	−2.21127
$722$	−9.92278e6	−0.708420
$723$	0	0
$724$	5.56717e6	0.394719
$725$	706625.	0.0499280
$726$	0	0
$727$	7.98925e6	0.560622	0.280311	−	0.959909i	$-0.409562\pi$
$727$	7.98925e6	0.560622	0.280311	+	0.959909i	$0.409562\pi$
$728$	−2.25268e7	−1.57533
$729$	0	0
$730$	−4.37821e6	−0.304081
$731$	3.45284e7	2.38992
$732$	0	0
$733$	−4.87246e6	−0.334956	−0.167478	−	0.985876i	$-0.553562\pi$
$733$	−4.87246e6	−0.334956	−0.167478	+	0.985876i	$0.553562\pi$
$734$	−1.34015e7	−0.918151
$735$	0	0
$736$	1.46993e6	0.100023
$737$	5.17399e6	0.350879
$738$	0	0
$739$	−4.76528e6	−0.320980	−0.160490	−	0.987037i	$-0.551307\pi$
$739$	−4.76528e6	−0.320980	−0.160490	+	0.987037i	$0.551307\pi$
$740$	4.79808e6	0.322098
$741$	0	0
$742$	1.84407e7	1.22961
$743$	2.46720e6	0.163958	0.0819788	−	0.996634i	$-0.473876\pi$
$743$	2.46720e6	0.163958	0.0819788	+	0.996634i	$0.473876\pi$
$744$	0	0
$745$	−9.71125e6	−0.641039
$746$	−2.95253e7	−1.94244
$747$	0	0
$748$	6.98662e6	0.456576
$749$	1.89175e6	0.123214
$750$	0	0
$751$	−1.39895e7	−0.905114	−0.452557	−	0.891735i	$-0.649488\pi$
$751$	−1.39895e7	−0.905114	−0.452557	+	0.891735i	$0.649488\pi$
$752$	2.38913e7	1.54062
$753$	0	0
$754$	−2.16350e6	−0.138589
$755$	1.43566e7	0.916607
$756$	0	0
$757$	4.81014e6	0.305083	0.152542	−	0.988297i	$-0.451254\pi$
$757$	4.81014e6	0.305083	0.152542	+	0.988297i	$0.451254\pi$
$758$	9.17724e6	0.580149
$759$	0	0
$760$	−5.81986e6	−0.365492
$761$	−2.98173e7	−1.86641	−0.933205	−	0.359344i	$-0.883001\pi$
$761$	−2.98173e7	−1.86641	−0.933205	+	0.359344i	$0.883001\pi$
$762$	0	0
$763$	−4.09272e7	−2.54508
$764$	−1.20698e6	−0.0748110
$765$	0	0
$766$	−2.31894e7	−1.42796
$767$	3.20982e7	1.97012
$768$	0	0
$769$	−3.25572e7	−1.98532	−0.992661	−	0.120932i	$-0.961412\pi$
$769$	−3.25572e7	−1.98532	−0.992661	+	0.120932i	$0.961412\pi$
$770$	2.76410e7	1.68007
$771$	0	0
$772$	2.16710e6	0.130869
$773$	2.57319e7	1.54890	0.774449	−	0.632637i	$-0.218027\pi$
$773$	2.57319e7	1.54890	0.774449	+	0.632637i	$0.218027\pi$
$774$	0	0
$775$	−6.59118e6	−0.394193
$776$	−3.34015e6	−0.199118
$777$	0	0
$778$	−3.25772e7	−1.92959
$779$	9.92083e6	0.585739
$780$	0	0
$781$	3.54578e7	2.08010
$782$	5.48642e6	0.320828
$783$	0	0
$784$	−2.86993e7	−1.66756
$785$	6.77743e6	0.392547
$786$	0	0
$787$	2.26438e6	0.130321	0.0651603	−	0.997875i	$-0.479244\pi$
$787$	2.26438e6	0.130321	0.0651603	+	0.997875i	$0.479244\pi$
$788$	−4.87333e6	−0.279583
$789$	0	0
$790$	−2.24566e7	−1.28020
$791$	−1.74548e7	−0.991912
$792$	0	0
$793$	1.86168e7	1.05129
$794$	1.81214e7	1.02010
$795$	0	0
$796$	−3.48087e6	−0.194717
$797$	−3.31745e7	−1.84995	−0.924973	−	0.380034i	$-0.875912\pi$
$797$	−3.31745e7	−1.84995	−0.924973	+	0.380034i	$0.875912\pi$
$798$	0	0
$799$	3.23532e7	1.79288
$800$	4.20675e6	0.232392
$801$	0	0
$802$	−1.26453e7	−0.694213
$803$	−9.37394e6	−0.513019
$804$	0	0
$805$	4.27091e6	0.232290
$806$	2.01805e7	1.09419
$807$	0	0
$808$	−2.57851e7	−1.38944
$809$	−1.71782e7	−0.922798	−0.461399	−	0.887193i	$-0.652652\pi$
$809$	−1.71782e7	−0.922798	−0.461399	+	0.887193i	$0.652652\pi$
$810$	0	0
$811$	−1.33026e7	−0.710208	−0.355104	−	0.934827i	$-0.615555\pi$
$811$	−1.33026e7	−0.710208	−0.355104	+	0.934827i	$0.615555\pi$
$812$	−735940.	−0.0391699
$813$	0	0
$814$	5.22093e7	2.76177
$815$	−1.96730e7	−1.03747
$816$	0	0
$817$	1.99793e7	1.04719
$818$	−2.55338e7	−1.33424
$819$	0	0
$820$	3.28301e6	0.170505
$821$	3.68832e7	1.90972	0.954862	−	0.297051i	$-0.0960032\pi$
$821$	3.68832e7	1.90972	0.954862	+	0.297051i	$0.0960032\pi$
$822$	0	0
$823$	−1.30396e7	−0.671065	−0.335532	−	0.942029i	$-0.608916\pi$
$823$	−1.30396e7	−0.671065	−0.335532	+	0.942029i	$0.608916\pi$
$824$	2.34015e7	1.20068
$825$	0	0
$826$	5.54908e7	2.82990
$827$	2.88478e7	1.46672	0.733362	−	0.679838i	$-0.237950\pi$
$827$	2.88478e7	1.46672	0.733362	+	0.679838i	$0.237950\pi$
$828$	0	0
$829$	−2.67180e7	−1.35026	−0.675130	−	0.737699i	$-0.735912\pi$
$829$	−2.67180e7	−1.35026	−0.675130	+	0.737699i	$0.735912\pi$
$830$	−1.33935e7	−0.674838
$831$	0	0
$832$	1.56350e7	0.783050
$833$	−3.88641e7	−1.94060
$834$	0	0
$835$	−5.46970e6	−0.271486
$836$	4.04269e6	0.200058
$837$	0	0
$838$	1.92442e7	0.946652
$839$	−1.66490e7	−0.816553	−0.408277	−	0.912858i	$-0.633870\pi$
$839$	−1.66490e7	−0.816553	−0.408277	+	0.912858i	$0.633870\pi$
$840$	0	0
$841$	−2.02933e7	−0.989379
$842$	1.93747e7	0.941792
$843$	0	0
$844$	−5.57600e6	−0.269443
$845$	−6.74401e6	−0.324920
$846$	0	0
$847$	2.67860e7	1.28292
$848$	−1.76246e7	−0.841647
$849$	0	0
$850$	1.57014e7	0.745404
$851$	8.06705e6	0.381849
$852$	0	0
$853$	6.34090e6	0.298386	0.149193	−	0.988808i	$-0.452332\pi$
$853$	6.34090e6	0.298386	0.149193	+	0.988808i	$0.452332\pi$
$854$	3.21844e7	1.51008
$855$	0	0
$856$	−1.43426e6	−0.0669025
$857$	−1.73493e7	−0.806921	−0.403460	−	0.914997i	$-0.632193\pi$
$857$	−1.73493e7	−0.806921	−0.403460	+	0.914997i	$0.632193\pi$
$858$	0	0
$859$	−121607.	−0.00562311	−0.00281156	−	0.999996i	$-0.500895\pi$
$859$	−121607.	−0.00562311	−0.00281156	+	0.999996i	$0.500895\pi$
$860$	6.61158e6	0.304831
$861$	0	0
$862$	−3.08409e7	−1.41371
$863$	−3.49627e6	−0.159800	−0.0799001	−	0.996803i	$-0.525460\pi$
$863$	−3.49627e6	−0.159800	−0.0799001	+	0.996803i	$0.525460\pi$
$864$	0	0
$865$	4.86200e6	0.220940
$866$	6.22864e6	0.282227
$867$	0	0
$868$	6.86462e6	0.309255
$869$	−4.80806e7	−2.15983
$870$	0	0
$871$	−7.00504e6	−0.312871
$872$	3.10296e7	1.38193
$873$	0	0
$874$	3.17463e6	0.140577
$875$	3.74527e7	1.65372
$876$	0	0
$877$	3.96872e6	0.174241	0.0871207	−	0.996198i	$-0.472233\pi$
$877$	3.96872e6	0.174241	0.0871207	+	0.996198i	$0.472233\pi$
$878$	−8.99935e6	−0.393981
$879$	0	0
$880$	−2.64177e7	−1.14998
$881$	−1.86303e7	−0.808688	−0.404344	−	0.914607i	$-0.632500\pi$
$881$	−1.86303e7	−0.808688	−0.404344	+	0.914607i	$0.632500\pi$
$882$	0	0
$883$	7.60829e6	0.328386	0.164193	−	0.986428i	$-0.447498\pi$
$883$	7.60829e6	0.328386	0.164193	+	0.986428i	$0.447498\pi$
$884$	−9.45914e6	−0.407119
$885$	0	0
$886$	4.58296e7	1.96138
$887$	3.68363e7	1.57205	0.786027	−	0.618192i	$-0.212135\pi$
$887$	3.68363e7	1.57205	0.786027	+	0.618192i	$0.212135\pi$
$888$	0	0
$889$	4.31063e7	1.82931
$890$	1.90872e6	0.0807733
$891$	0	0
$892$	−228139.	−0.00960036
$893$	1.87207e7	0.785585
$894$	0	0
$895$	2.28731e7	0.954481
$896$	4.49149e7	1.86905
$897$	0	0
$898$	−6.92969e6	−0.286763
$899$	−2.03208e6	−0.0838574
$900$	0	0
$901$	−2.38670e7	−0.979457
$902$	3.57234e7	1.46196
$903$	0	0
$904$	1.32336e7	0.538588
$905$	−2.85063e7	−1.15696
$906$	0	0
$907$	−1.63522e7	−0.660023	−0.330011	−	0.943977i	$-0.607053\pi$
$907$	−1.63522e7	−0.660023	−0.330011	+	0.943977i	$0.607053\pi$
$908$	−667222.	−0.0268569
$909$	0	0
$910$	−3.74230e7	−1.49808
$911$	−2.09450e7	−0.836149	−0.418074	−	0.908413i	$-0.637295\pi$
$911$	−2.09450e7	−0.836149	−0.418074	+	0.908413i	$0.637295\pi$
$912$	0	0
$913$	−2.86761e7	−1.13853
$914$	−4.51870e7	−1.78916
$915$	0	0
$916$	4.31244e6	0.169818
$917$	7.73491e6	0.303761
$918$	0	0
$919$	−2.30439e7	−0.900051	−0.450026	−	0.893016i	$-0.648585\pi$
$919$	−2.30439e7	−0.900051	−0.450026	+	0.893016i	$0.648585\pi$
$920$	−3.23805e6	−0.126129
$921$	0	0
$922$	2.94610e7	1.14135
$923$	−4.80061e7	−1.85478
$924$	0	0
$925$	2.30869e7	0.887178
$926$	1.59774e6	0.0612321
$927$	0	0
$928$	1.29695e6	0.0494372
$929$	−2.12962e7	−0.809586	−0.404793	−	0.914408i	$-0.632656\pi$
$929$	−2.12962e7	−0.809586	−0.404793	+	0.914408i	$0.632656\pi$
$930$	0	0
$931$	−2.24881e7	−0.850312
$932$	2.58126e6	0.0973400
$933$	0	0
$934$	2.86138e7	1.07327
$935$	−3.57744e7	−1.33827
$936$	0	0
$937$	1.31590e7	0.489638	0.244819	−	0.969569i	$-0.421271\pi$
$937$	1.31590e7	0.489638	0.244819	+	0.969569i	$0.421271\pi$
$938$	−1.21102e7	−0.449411
$939$	0	0
$940$	6.19507e6	0.228679
$941$	−1.96126e7	−0.722040	−0.361020	−	0.932558i	$-0.617571\pi$
$941$	−1.96126e7	−0.722040	−0.361020	+	0.932558i	$0.617571\pi$
$942$	0	0
$943$	5.51975e6	0.202135
$944$	−5.30350e7	−1.93701
$945$	0	0
$946$	7.19425e7	2.61371
$947$	−7.75068e6	−0.280844	−0.140422	−	0.990092i	$-0.544846\pi$
$947$	−7.75068e6	−0.280844	−0.140422	+	0.990092i	$0.544846\pi$
$948$	0	0
$949$	1.26913e7	0.457448
$950$	9.08538e6	0.326613
$951$	0	0
$952$	5.04033e7	1.80246
$953$	−3.72852e6	−0.132986	−0.0664928	−	0.997787i	$-0.521181\pi$
$953$	−3.72852e6	−0.132986	−0.0664928	+	0.997787i	$0.521181\pi$
$954$	0	0
$955$	6.18024e6	0.219279
$956$	−8.02227e6	−0.283892
$957$	0	0
$958$	−1.10923e7	−0.390488
$959$	5.10896e7	1.79385
$960$	0	0
$961$	−9.67455e6	−0.337927
$962$	−7.06859e7	−2.46261
$963$	0	0
$964$	−693909.	−0.0240497
$965$	−1.10965e7	−0.383590
$966$	0	0
$967$	−2.61409e7	−0.898989	−0.449495	−	0.893283i	$-0.648396\pi$
$967$	−2.61409e7	−0.898989	−0.449495	+	0.893283i	$0.648396\pi$
$968$	−2.03082e7	−0.696600
$969$	0	0
$970$	−5.54887e6	−0.189354
$971$	−7.33965e6	−0.249820	−0.124910	−	0.992168i	$-0.539864\pi$
$971$	−7.33965e6	−0.249820	−0.124910	+	0.992168i	$0.539864\pi$
$972$	0	0
$973$	−3.73243e7	−1.26389
$974$	7.38081e6	0.249291
$975$	0	0
$976$	−3.07600e7	−1.03362
$977$	2.41710e7	0.810136	0.405068	−	0.914287i	$-0.367248\pi$
$977$	2.41710e7	0.810136	0.405068	+	0.914287i	$0.367248\pi$
$978$	0	0
$979$	4.08665e6	0.136273
$980$	−7.44179e6	−0.247521
$981$	0	0
$982$	5.47516e7	1.81183
$983$	4.64683e7	1.53381	0.766907	−	0.641758i	$-0.221795\pi$
$983$	4.64683e7	1.53381	0.766907	+	0.641758i	$0.221795\pi$
$984$	0	0
$985$	2.49535e7	0.819485
$986$	4.84079e6	0.158571
$987$	0	0
$988$	−5.47338e6	−0.178387
$989$	1.11161e7	0.361378
$990$	0	0
$991$	−4.04999e6	−0.130999	−0.0654997	−	0.997853i	$-0.520864\pi$
$991$	−4.04999e6	−0.130999	−0.0654997	+	0.997853i	$0.520864\pi$
$992$	−1.20976e7	−0.390318
$993$	0	0
$994$	−8.29922e7	−2.66423
$995$	1.78235e7	0.570737
$996$	0	0
$997$	−4.74444e7	−1.51163	−0.755817	−	0.654782i	$-0.772760\pi$
$997$	−4.74444e7	−1.51163	−0.755817	+	0.654782i	$0.772760\pi$
$998$	−1.94471e6	−0.0618057
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.h.1.8	✓	10
3.2	odd	2		207.6.a.i.1.3	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.a.h.1.8	✓	10	1.1	even	1	trivial
207.6.a.i.1.3	yes	10	3.2	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)

\(f(q)\)	\(=\)	\(q+6.31180 q^{2} +7.83883 q^{4} -40.1381 q^{5} +201.144 q^{7} -152.500 q^{8} +O(q^{10})\)	\(q+6.31180 q^{2} +7.83883 q^{4} -40.1381 q^{5} +201.144 q^{7} -152.500 q^{8} -253.344 q^{10} -542.420 q^{11} +734.379 q^{13} +1269.58 q^{14} -1213.40 q^{16} -1643.16 q^{17} -950.789 q^{19} -314.636 q^{20} -3423.65 q^{22} -529.000 q^{23} -1513.93 q^{25} +4635.26 q^{26} +1576.74 q^{28} -466.749 q^{29} +4353.69 q^{31} -2778.69 q^{32} -10371.3 q^{34} -8073.56 q^{35} -15249.6 q^{37} -6001.19 q^{38} +6121.09 q^{40} -10434.3 q^{41} -21013.4 q^{43} -4251.94 q^{44} -3338.94 q^{46} -19689.6 q^{47} +23652.1 q^{49} -9555.62 q^{50} +5756.68 q^{52} +14525.0 q^{53} +21771.7 q^{55} -30674.6 q^{56} -2946.03 q^{58} +43707.9 q^{59} +25350.4 q^{61} +27479.6 q^{62} +21290.1 q^{64} -29476.6 q^{65} -9538.72 q^{67} -12880.5 q^{68} -50958.7 q^{70} -65369.7 q^{71} +17281.7 q^{73} -96252.6 q^{74} -7453.07 q^{76} -109105. q^{77} +88640.9 q^{79} +48703.4 q^{80} -65859.3 q^{82} +52866.9 q^{83} +65953.4 q^{85} -132632. q^{86} +82719.3 q^{88} -7534.11 q^{89} +147716. q^{91} -4146.74 q^{92} -124277. q^{94} +38162.9 q^{95} +21902.5 q^{97} +149287. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8}+O(q^{10}) \) 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8	\( 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8} - 250 q^{10} - 460 q^{11} + 464 q^{13} - 3676 q^{14} + 4612 q^{16} - 4756 q^{17} - 1780 q^{19} - 10314 q^{20} - 4214 q^{22} - 5290 q^{23} + 1330 q^{25} + 5152 q^{26} + 7072 q^{28} - 4048 q^{29} + 2816 q^{31} - 27436 q^{32} + 420 q^{34} - 9452 q^{35} + 2872 q^{37} - 31038 q^{38} + 2618 q^{40} - 34056 q^{41} + 7316 q^{43} - 33562 q^{44} + 4232 q^{46} - 49300 q^{47} + 45118 q^{49} - 44764 q^{50} - 25120 q^{52} - 86676 q^{53} - 2120 q^{55} - 290684 q^{56} - 87408 q^{58} - 67100 q^{59} - 40432 q^{61} - 230992 q^{62} + 136776 q^{64} - 184000 q^{65} - 50108 q^{67} - 270592 q^{68} + 117456 q^{70} - 238584 q^{71} - 13804 q^{73} - 150074 q^{74} - 197622 q^{76} - 116248 q^{77} - 9228 q^{79} - 313010 q^{80} - 68604 q^{82} - 155300 q^{83} + 80444 q^{85} + 80914 q^{86} - 237738 q^{88} - 213732 q^{89} - 264352 q^{91} - 101568 q^{92} + 140280 q^{94} + 123612 q^{95} + 42516 q^{97} + 151388 q^{98}+O(q^{100}) \) 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8 - 250 * q^10 - 460 * q^11 + 464 * q^13 - 3676 * q^14 + 4612 * q^16 - 4756 * q^17 - 1780 * q^19 - 10314 * q^20 - 4214 * q^22 - 5290 * q^23 + 1330 * q^25 + 5152 * q^26 + 7072 * q^28 - 4048 * q^29 + 2816 * q^31 - 27436 * q^32 + 420 * q^34 - 9452 * q^35 + 2872 * q^37 - 31038 * q^38 + 2618 * q^40 - 34056 * q^41 + 7316 * q^43 - 33562 * q^44 + 4232 * q^46 - 49300 * q^47 + 45118 * q^49 - 44764 * q^50 - 25120 * q^52 - 86676 * q^53 - 2120 * q^55 - 290684 * q^56 - 87408 * q^58 - 67100 * q^59 - 40432 * q^61 - 230992 * q^62 + 136776 * q^64 - 184000 * q^65 - 50108 * q^67 - 270592 * q^68 + 117456 * q^70 - 238584 * q^71 - 13804 * q^73 - 150074 * q^74 - 197622 * q^76 - 116248 * q^77 - 9228 * q^79 - 313010 * q^80 - 68604 * q^82 - 155300 * q^83 + 80444 * q^85 + 80914 * q^86 - 237738 * q^88 - 213732 * q^89 - 264352 * q^91 - 101568 * q^92 + 140280 * q^94 + 123612 * q^95 + 42516 * q^97 + 151388 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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