LMFDB - Embedded newform 207.6.a.g.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:	$6$
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			1.4
Root			$5.64307$ 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)

$f(q)$	$=$	$q+4.64307 q^{2} -10.4419 q^{4} -9.65155 q^{5} +199.091 q^{7} -197.061 q^{8} +O(q^{10})$	\(q+4.64307 q^{2} -10.4419 q^{4} -9.65155 q^{5} +199.091 q^{7} -197.061 q^{8} -44.8128 q^{10} -201.247 q^{11} -120.060 q^{13} +924.394 q^{14} -580.825 q^{16} +2038.91 q^{17} +724.684 q^{19} +100.781 q^{20} -934.403 q^{22} -529.000 q^{23} -3031.85 q^{25} -557.445 q^{26} -2078.89 q^{28} +6362.52 q^{29} +8295.97 q^{31} +3609.13 q^{32} +9466.82 q^{34} -1921.54 q^{35} +2046.78 q^{37} +3364.76 q^{38} +1901.94 q^{40} +4185.63 q^{41} +14841.9 q^{43} +2101.40 q^{44} -2456.18 q^{46} +13555.1 q^{47} +22830.3 q^{49} -14077.1 q^{50} +1253.65 q^{52} +22422.9 q^{53} +1942.34 q^{55} -39233.0 q^{56} +29541.6 q^{58} -29012.4 q^{59} -53380.7 q^{61} +38518.7 q^{62} +35343.8 q^{64} +1158.76 q^{65} -19518.7 q^{67} -21290.2 q^{68} -8921.83 q^{70} +23162.8 q^{71} +71902.0 q^{73} +9503.33 q^{74} -7567.10 q^{76} -40066.5 q^{77} -18852.4 q^{79} +5605.86 q^{80} +19434.2 q^{82} +18897.0 q^{83} -19678.7 q^{85} +68911.7 q^{86} +39657.9 q^{88} +13591.1 q^{89} -23902.8 q^{91} +5523.78 q^{92} +62937.1 q^{94} -6994.33 q^{95} -73921.2 q^{97} +106003. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.64307	0.820786	0.410393	−	0.911909i	$-0.365391\pi$
$2$	4.64307	0.820786	0.410393	+	0.911909i	$0.365391\pi$
$3$	0	0
$3$	0	0
$4$	−10.4419	−0.326310
$5$	−9.65155	−0.172652	−0.0863261	−	0.996267i	$-0.527513\pi$
$5$	−9.65155	−0.172652	−0.0863261	+	0.996267i	$0.527513\pi$
$6$	0	0
$7$	199.091	1.53570	0.767851	−	0.640629i	$-0.221326\pi$
$7$	199.091	1.53570	0.767851	+	0.640629i	$0.221326\pi$
$8$	−197.061	−1.08862
$9$	0	0
$10$	−44.8128	−0.141711
$11$	−201.247	−0.501473	−0.250736	−	0.968055i	$-0.580673\pi$
$11$	−201.247	−0.501473	−0.250736	+	0.968055i	$0.580673\pi$
$12$	0	0
$13$	−120.060	−0.197033	−0.0985165	−	0.995135i	$-0.531410\pi$
$13$	−120.060	−0.197033	−0.0985165	+	0.995135i	$0.531410\pi$
$14$	924.394	1.26048
$15$	0	0
$16$	−580.825	−0.567212
$17$	2038.91	1.71111	0.855553	−	0.517716i	$-0.173217\pi$
$17$	2038.91	1.71111	0.855553	+	0.517716i	$0.173217\pi$
$18$	0	0
$19$	724.684	0.460537	0.230269	−	0.973127i	$-0.426039\pi$
$19$	724.684	0.460537	0.230269	+	0.973127i	$0.426039\pi$
$20$	100.781	0.0563381
$21$	0	0
$22$	−934.403	−0.411602
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−3031.85	−0.970191
$26$	−557.445	−0.161722
$27$	0	0
$28$	−2078.89	−0.501115
$29$	6362.52	1.40486	0.702431	−	0.711752i	$-0.252098\pi$
$29$	6362.52	1.40486	0.702431	+	0.711752i	$0.252098\pi$
$30$	0	0
$31$	8295.97	1.55047	0.775234	−	0.631674i	$-0.217632\pi$
$31$	8295.97	1.55047	0.775234	+	0.631674i	$0.217632\pi$
$32$	3609.13	0.623057
$33$	0	0
$34$	9466.82	1.40445
$35$	−1921.54	−0.265142
$36$	0	0
$37$	2046.78	0.245791	0.122896	−	0.992420i	$-0.460782\pi$
$37$	2046.78	0.245791	0.122896	+	0.992420i	$0.460782\pi$
$38$	3364.76	0.378003
$39$	0	0
$40$	1901.94	0.187952
$41$	4185.63	0.388867	0.194434	−	0.980916i	$-0.437713\pi$
$41$	4185.63	0.388867	0.194434	+	0.980916i	$0.437713\pi$
$42$	0	0
$43$	14841.9	1.22410	0.612050	−	0.790819i	$-0.290345\pi$
$43$	14841.9	1.22410	0.612050	+	0.790819i	$0.290345\pi$
$44$	2101.40	0.163636
$45$	0	0
$46$	−2456.18	−0.171146
$47$	13555.1	0.895070	0.447535	−	0.894266i	$-0.352302\pi$
$47$	13555.1	0.895070	0.447535	+	0.894266i	$0.352302\pi$
$48$	0	0
$49$	22830.3	1.35838
$50$	−14077.1	−0.796320
$51$	0	0
$52$	1253.65	0.0642938
$53$	22422.9	1.09648	0.548242	−	0.836319i	$-0.315297\pi$
$53$	22422.9	1.09648	0.548242	+	0.836319i	$0.315297\pi$
$54$	0	0
$55$	1942.34	0.0865804
$56$	−39233.0	−1.67179
$57$	0	0
$58$	29541.6	1.15309
$59$	−29012.4	−1.08506	−0.542530	−	0.840036i	$-0.682534\pi$
$59$	−29012.4	−1.08506	−0.542530	+	0.840036i	$0.682534\pi$
$60$	0	0
$61$	−53380.7	−1.83679	−0.918395	−	0.395665i	$-0.870514\pi$
$61$	−53380.7	−1.83679	−0.918395	+	0.395665i	$0.870514\pi$
$62$	38518.7	1.27260
$63$	0	0
$64$	35343.8	1.07861
$65$	1158.76	0.0340182
$66$	0	0
$67$	−19518.7	−0.531206	−0.265603	−	0.964082i	$-0.585571\pi$
$67$	−19518.7	−0.531206	−0.265603	+	0.964082i	$0.585571\pi$
$68$	−21290.2	−0.558351
$69$	0	0
$70$	−8921.83	−0.217625
$71$	23162.8	0.545311	0.272656	−	0.962112i	$-0.412098\pi$
$71$	23162.8	0.545311	0.272656	+	0.962112i	$0.412098\pi$
$72$	0	0
$73$	71902.0	1.57919	0.789594	−	0.613630i	$-0.210291\pi$
$73$	71902.0	1.57919	0.789594	+	0.613630i	$0.210291\pi$
$74$	9503.33	0.201742
$75$	0	0
$76$	−7567.10	−0.150278
$77$	−40066.5	−0.770113
$78$	0	0
$79$	−18852.4	−0.339859	−0.169930	−	0.985456i	$-0.554354\pi$
$79$	−18852.4	−0.339859	−0.169930	+	0.985456i	$0.554354\pi$
$80$	5605.86	0.0979303
$81$	0	0
$82$	19434.2	0.319177
$83$	18897.0	0.301090	0.150545	−	0.988603i	$-0.451897\pi$
$83$	18897.0	0.301090	0.150545	+	0.988603i	$0.451897\pi$
$84$	0	0
$85$	−19678.7	−0.295426
$86$	68911.7	1.00472
$87$	0	0
$88$	39657.9	0.545912
$89$	13591.1	0.181878	0.0909390	−	0.995856i	$-0.471013\pi$
$89$	13591.1	0.181878	0.0909390	+	0.995856i	$0.471013\pi$
$90$	0	0
$91$	−23902.8	−0.302584
$92$	5523.78	0.0680404
$93$	0	0
$94$	62937.1	0.734661
$95$	−6994.33	−0.0795128
$96$	0	0
$97$	−73921.2	−0.797701	−0.398850	−	0.917016i	$-0.630591\pi$
$97$	−73921.2	−0.797701	−0.398850	+	0.917016i	$0.630591\pi$
$98$	106003.	1.11494
$99$	0	0
$100$	31658.3	0.316583
$101$	−156101.	−1.52266	−0.761329	−	0.648366i	$-0.775453\pi$
$101$	−156101.	−1.52266	−0.761329	+	0.648366i	$0.775453\pi$
$102$	0	0
$103$	−151059.	−1.40298	−0.701492	−	0.712677i	$-0.747482\pi$
$103$	−151059.	−1.40298	−0.701492	+	0.712677i	$0.747482\pi$
$104$	23659.1	0.214493
$105$	0	0
$106$	104111.	0.899980
$107$	−13258.7	−0.111955	−0.0559773	−	0.998432i	$-0.517827\pi$
$107$	−13258.7	−0.111955	−0.0559773	+	0.998432i	$0.517827\pi$
$108$	0	0
$109$	205843.	1.65947	0.829736	−	0.558156i	$-0.188491\pi$
$109$	205843.	1.65947	0.829736	+	0.558156i	$0.188491\pi$
$110$	9018.44	0.0710640
$111$	0	0
$112$	−115637.	−0.871068
$113$	−147946.	−1.08995	−0.544975	−	0.838453i	$-0.683461\pi$
$113$	−147946.	−1.08995	−0.544975	+	0.838453i	$0.683461\pi$
$114$	0	0
$115$	5105.67	0.0360005
$116$	−66436.9	−0.458421
$117$	0	0
$118$	−134707.	−0.890603
$119$	405930.	2.62775
$120$	0	0
$121$	−120551.	−0.748525
$122$	−247850.	−1.50761
$123$	0	0
$124$	−86625.8	−0.505933
$125$	59423.1	0.340158
$126$	0	0
$127$	152790.	0.840591	0.420295	−	0.907387i	$-0.361926\pi$
$127$	152790.	0.840591	0.420295	+	0.907387i	$0.361926\pi$
$128$	48611.6	0.262250
$129$	0	0
$130$	5380.21	0.0279216
$131$	−50676.8	−0.258007	−0.129003	−	0.991644i	$-0.541178\pi$
$131$	−50676.8	−0.258007	−0.129003	+	0.991644i	$0.541178\pi$
$132$	0	0
$133$	144278.	0.707248
$134$	−90626.5	−0.436007
$135$	0	0
$136$	−401790.	−1.86274
$137$	−261052.	−1.18830	−0.594148	−	0.804355i	$-0.702511\pi$
$137$	−261052.	−1.18830	−0.594148	+	0.804355i	$0.702511\pi$
$138$	0	0
$139$	166128.	0.729299	0.364649	−	0.931145i	$-0.381189\pi$
$139$	166128.	0.729299	0.364649	+	0.931145i	$0.381189\pi$
$140$	20064.5	0.0865186
$141$	0	0
$142$	107546.	0.447584
$143$	24161.6	0.0988067
$144$	0	0
$145$	−61408.1	−0.242553
$146$	333846.	1.29618
$147$	0	0
$148$	−21372.3	−0.0802041
$149$	392547.	1.44852	0.724262	−	0.689525i	$-0.242181\pi$
$149$	392547.	1.44852	0.724262	+	0.689525i	$0.242181\pi$
$150$	0	0
$151$	176679.	0.630584	0.315292	−	0.948995i	$-0.397898\pi$
$151$	176679.	0.630584	0.315292	+	0.948995i	$0.397898\pi$
$152$	−142807.	−0.501349
$153$	0	0
$154$	−186031.	−0.632098
$155$	−80068.9	−0.267692
$156$	0	0
$157$	−193221.	−0.625613	−0.312807	−	0.949817i	$-0.601269\pi$
$157$	−193221.	−0.625613	−0.312807	+	0.949817i	$0.601269\pi$
$158$	−87533.0	−0.278952
$159$	0	0
$160$	−34833.7	−0.107572
$161$	−105319.	−0.320216
$162$	0	0
$163$	500963.	1.47685	0.738426	−	0.674335i	$-0.235570\pi$
$163$	500963.	1.47685	0.738426	+	0.674335i	$0.235570\pi$
$164$	−43706.1	−0.126891
$165$	0	0
$166$	87739.9	0.247131
$167$	−45897.2	−0.127349	−0.0636745	−	0.997971i	$-0.520282\pi$
$167$	−45897.2	−0.127349	−0.0636745	+	0.997971i	$0.520282\pi$
$168$	0	0
$169$	−356879.	−0.961178
$170$	−91369.5	−0.242482
$171$	0	0
$172$	−154977.	−0.399436
$173$	−457792.	−1.16293	−0.581464	−	0.813572i	$-0.697520\pi$
$173$	−457792.	−1.16293	−0.581464	+	0.813572i	$0.697520\pi$
$174$	0	0
$175$	−603614.	−1.48992
$176$	116889.	0.284441
$177$	0	0
$178$	63104.5	0.149283
$179$	−486479.	−1.13483	−0.567416	−	0.823431i	$-0.692057\pi$
$179$	−486479.	−1.13483	−0.567416	+	0.823431i	$0.692057\pi$
$180$	0	0
$181$	−553152.	−1.25501	−0.627506	−	0.778612i	$-0.715924\pi$
$181$	−553152.	−1.25501	−0.627506	+	0.778612i	$0.715924\pi$
$182$	−110982.	−0.248357
$183$	0	0
$184$	104245.	0.226992
$185$	−19754.6	−0.0424364
$186$	0	0
$187$	−410325.	−0.858073
$188$	−141541.	−0.292070
$189$	0	0
$190$	−32475.1	−0.0652630
$191$	255237.	0.506245	0.253122	−	0.967434i	$-0.418542\pi$
$191$	255237.	0.506245	0.253122	+	0.967434i	$0.418542\pi$
$192$	0	0
$193$	−43528.2	−0.0841158	−0.0420579	−	0.999115i	$-0.513391\pi$
$193$	−43528.2	−0.0841158	−0.0420579	+	0.999115i	$0.513391\pi$
$194$	−343221.	−0.654742
$195$	0	0
$196$	−238392.	−0.443253
$197$	69882.8	0.128294	0.0641468	−	0.997940i	$-0.479567\pi$
$197$	69882.8	0.128294	0.0641468	+	0.997940i	$0.479567\pi$
$198$	0	0
$199$	298205.	0.533804	0.266902	−	0.963724i	$-0.414000\pi$
$199$	298205.	0.533804	0.266902	+	0.963724i	$0.414000\pi$
$200$	597458.	1.05617
$201$	0	0
$202$	−724788.	−1.24978
$203$	1.26672e6	2.15745
$204$	0	0
$205$	−40397.8	−0.0671388
$206$	−701376.	−1.15155
$207$	0	0
$208$	69733.6	0.111759
$209$	−145840.	−0.230947
$210$	0	0
$211$	256504.	0.396632	0.198316	−	0.980138i	$-0.436453\pi$
$211$	256504.	0.396632	0.198316	+	0.980138i	$0.436453\pi$
$212$	−234138.	−0.357794
$213$	0	0
$214$	−61561.1	−0.0918908
$215$	−143247.	−0.211344
$216$	0	0
$217$	1.65165e6	2.38106
$218$	955743.	1.36207
$219$	0	0
$220$	−20281.8	−0.0282521
$221$	−244792.	−0.337144
$222$	0	0
$223$	493700.	0.664815	0.332407	−	0.943136i	$-0.392139\pi$
$223$	493700.	0.664815	0.332407	+	0.943136i	$0.392139\pi$
$224$	718547.	0.956830
$225$	0	0
$226$	−686922.	−0.894615
$227$	−342384.	−0.441011	−0.220506	−	0.975386i	$-0.570771\pi$
$227$	−342384.	−0.441011	−0.220506	+	0.975386i	$0.570771\pi$
$228$	0	0
$229$	−277393.	−0.349547	−0.174774	−	0.984609i	$-0.555919\pi$
$229$	−277393.	−0.349547	−0.174774	+	0.984609i	$0.555919\pi$
$230$	23706.0	0.0295487
$231$	0	0
$232$	−1.25380e6	−1.52936
$233$	−838954.	−1.01239	−0.506196	−	0.862419i	$-0.668949\pi$
$233$	−838954.	−1.01239	−0.506196	+	0.862419i	$0.668949\pi$
$234$	0	0
$235$	−130827.	−0.154536
$236$	302946.	0.354066
$237$	0	0
$238$	1.88476e6	2.15682
$239$	−376526.	−0.426384	−0.213192	−	0.977010i	$-0.568386\pi$
$239$	−376526.	−0.426384	−0.213192	+	0.977010i	$0.568386\pi$
$240$	0	0
$241$	1.48161e6	1.64320	0.821599	−	0.570065i	$-0.193082\pi$
$241$	1.48161e6	1.64320	0.821599	+	0.570065i	$0.193082\pi$
$242$	−559725.	−0.614379
$243$	0	0
$244$	557397.	0.599363
$245$	−220348.	−0.234527
$246$	0	0
$247$	−87005.4	−0.0907410
$248$	−1.63481e6	−1.68787
$249$	0	0
$250$	275906.	0.279197
$251$	−671128.	−0.672389	−0.336195	−	0.941792i	$-0.609140\pi$
$251$	−671128.	−0.672389	−0.336195	+	0.941792i	$0.609140\pi$
$252$	0	0
$253$	106460.	0.104564
$254$	709413.	0.689945
$255$	0	0
$256$	−905296.	−0.863358
$257$	1.11616e6	1.05413	0.527064	−	0.849826i	$-0.323293\pi$
$257$	1.11616e6	1.05413	0.527064	+	0.849826i	$0.323293\pi$
$258$	0	0
$259$	407495.	0.377462
$260$	−12099.7	−0.0111005
$261$	0	0
$262$	−235296.	−0.211768
$263$	607272.	0.541369	0.270685	−	0.962668i	$-0.412750\pi$
$263$	607272.	0.541369	0.270685	+	0.962668i	$0.412750\pi$
$264$	0	0
$265$	−216416.	−0.189310
$266$	669894.	0.580499
$267$	0	0
$268$	203812.	0.173338
$269$	1.09087e6	0.919162	0.459581	−	0.888136i	$-0.348000\pi$
$269$	1.09087e6	0.919162	0.459581	+	0.888136i	$0.348000\pi$
$270$	0	0
$271$	2.03452e6	1.68283	0.841414	−	0.540391i	$-0.181724\pi$
$271$	2.03452e6	1.68283	0.841414	+	0.540391i	$0.181724\pi$
$272$	−1.18425e6	−0.970559
$273$	0	0
$274$	−1.21208e6	−0.975338
$275$	610150.	0.486525
$276$	0	0
$277$	−488077.	−0.382198	−0.191099	−	0.981571i	$-0.561205\pi$
$277$	−488077.	−0.382198	−0.191099	+	0.981571i	$0.561205\pi$
$278$	771343.	0.598598
$279$	0	0
$280$	378660.	0.288638
$281$	1.12318e6	0.848559	0.424279	−	0.905531i	$-0.360527\pi$
$281$	1.12318e6	0.848559	0.424279	+	0.905531i	$0.360527\pi$
$282$	0	0
$283$	−553602.	−0.410895	−0.205448	−	0.978668i	$-0.565865\pi$
$283$	−553602.	−0.410895	−0.205448	+	0.978668i	$0.565865\pi$
$284$	−241864.	−0.177941
$285$	0	0
$286$	112184.	0.0810992
$287$	833323.	0.597184
$288$	0	0
$289$	2.73732e6	1.92788
$290$	−285122.	−0.199084
$291$	0	0
$292$	−750795.	−0.515305
$293$	−388164.	−0.264147	−0.132074	−	0.991240i	$-0.542164\pi$
$293$	−388164.	−0.264147	−0.132074	+	0.991240i	$0.542164\pi$
$294$	0	0
$295$	280015.	0.187338
$296$	−403339.	−0.267572
$297$	0	0
$298$	1.82262e6	1.18893
$299$	63511.6	0.0410842
$300$	0	0
$301$	2.95488e6	1.87985
$302$	820333.	0.517574
$303$	0	0
$304$	−420915.	−0.261222
$305$	515206.	0.317126
$306$	0	0
$307$	−1.30078e6	−0.787693	−0.393847	−	0.919176i	$-0.628856\pi$
$307$	−1.30078e6	−0.787693	−0.393847	+	0.919176i	$0.628856\pi$
$308$	418371.	0.251296
$309$	0	0
$310$	−371766.	−0.219718
$311$	1.32327e6	0.775797	0.387899	−	0.921702i	$-0.373201\pi$
$311$	1.32327e6	0.775797	0.387899	+	0.921702i	$0.373201\pi$
$312$	0	0
$313$	1.42368e6	0.821392	0.410696	−	0.911772i	$-0.365286\pi$
$313$	1.42368e6	0.821392	0.410696	+	0.911772i	$0.365286\pi$
$314$	−897140.	−0.513495
$315$	0	0
$316$	196855.	0.110899
$317$	1.64810e6	0.921159	0.460580	−	0.887618i	$-0.347642\pi$
$317$	1.64810e6	0.921159	0.460580	+	0.887618i	$0.347642\pi$
$318$	0	0
$319$	−1.28044e6	−0.704501
$320$	−341123.	−0.186224
$321$	0	0
$322$	−489004.	−0.262829
$323$	1.47757e6	0.788028
$324$	0	0
$325$	364003.	0.191160
$326$	2.32601e6	1.21218
$327$	0	0
$328$	−824824.	−0.423328
$329$	2.69869e6	1.37456
$330$	0	0
$331$	−697552.	−0.349950	−0.174975	−	0.984573i	$-0.555985\pi$
$331$	−697552.	−0.349950	−0.174975	+	0.984573i	$0.555985\pi$
$332$	−197321.	−0.0982488
$333$	0	0
$334$	−213104.	−0.104526
$335$	188385.	0.0917139
$336$	0	0
$337$	740122.	0.355000	0.177500	−	0.984121i	$-0.443199\pi$
$337$	740122.	0.355000	0.177500	+	0.984121i	$0.443199\pi$
$338$	−1.65701e6	−0.788922
$339$	0	0
$340$	205483.	0.0964005
$341$	−1.66954e6	−0.777518
$342$	0	0
$343$	1.19918e6	0.550363
$344$	−2.92475e6	−1.33258
$345$	0	0
$346$	−2.12556e6	−0.954516
$347$	−3.73356e6	−1.66456	−0.832279	−	0.554356i	$-0.812964\pi$
$347$	−3.73356e6	−1.66456	−0.832279	+	0.554356i	$0.812964\pi$
$348$	0	0
$349$	−2.03172e6	−0.892894	−0.446447	−	0.894810i	$-0.647311\pi$
$349$	−2.03172e6	−0.892894	−0.446447	+	0.894810i	$0.647311\pi$
$350$	−2.80262e6	−1.22291
$351$	0	0
$352$	−726327.	−0.312446
$353$	131264.	0.0560671	0.0280335	−	0.999607i	$-0.491075\pi$
$353$	131264.	0.0560671	0.0280335	+	0.999607i	$0.491075\pi$
$354$	0	0
$355$	−223557.	−0.0941492
$356$	−141917.	−0.0593486
$357$	0	0
$358$	−2.25876e6	−0.931455
$359$	680478.	0.278662	0.139331	−	0.990246i	$-0.455505\pi$
$359$	680478.	0.278662	0.139331	+	0.990246i	$0.455505\pi$
$360$	0	0
$361$	−1.95093e6	−0.787905
$362$	−2.56832e6	−1.03010
$363$	0	0
$364$	249591.	0.0987361
$365$	−693966.	−0.272650
$366$	0	0
$367$	−1.69577e6	−0.657207	−0.328604	−	0.944468i	$-0.606578\pi$
$367$	−1.69577e6	−0.657207	−0.328604	+	0.944468i	$0.606578\pi$
$368$	307256.	0.118272
$369$	0	0
$370$	−91721.8	−0.0348312
$371$	4.46421e6	1.68387
$372$	0	0
$373$	−2.54955e6	−0.948836	−0.474418	−	0.880300i	$-0.657341\pi$
$373$	−2.54955e6	−0.948836	−0.474418	+	0.880300i	$0.657341\pi$
$374$	−1.90517e6	−0.704294
$375$	0	0
$376$	−2.67117e6	−0.974388
$377$	−763882.	−0.276804
$378$	0	0
$379$	−160439.	−0.0573737	−0.0286869	−	0.999588i	$-0.509133\pi$
$379$	−160439.	−0.0573737	−0.0286869	+	0.999588i	$0.509133\pi$
$380$	73034.2	0.0259458
$381$	0	0
$382$	1.18508e6	0.415519
$383$	−1.26183e6	−0.439547	−0.219773	−	0.975551i	$-0.570532\pi$
$383$	−1.26183e6	−0.439547	−0.219773	+	0.975551i	$0.570532\pi$
$384$	0	0
$385$	386704.	0.132962
$386$	−202104.	−0.0690410
$387$	0	0
$388$	771880.	0.260298
$389$	2.04518e6	0.685263	0.342631	−	0.939470i	$-0.388682\pi$
$389$	2.04518e6	0.685263	0.342631	+	0.939470i	$0.388682\pi$
$390$	0	0
$391$	−1.07859e6	−0.356790
$392$	−4.49895e6	−1.47875
$393$	0	0
$394$	324471.	0.105302
$395$	181955.	0.0586774
$396$	0	0
$397$	780767.	0.248625	0.124313	−	0.992243i	$-0.460327\pi$
$397$	780767.	0.248625	0.124313	+	0.992243i	$0.460327\pi$
$398$	1.38459e6	0.438139
$399$	0	0
$400$	1.76097e6	0.550304
$401$	689038.	0.213984	0.106992	−	0.994260i	$-0.465878\pi$
$401$	689038.	0.213984	0.106992	+	0.994260i	$0.465878\pi$
$402$	0	0
$403$	−996011.	−0.305493
$404$	1.63000e6	0.496859
$405$	0	0
$406$	5.88147e6	1.77080
$407$	−411908.	−0.123258
$408$	0	0
$409$	6.20401e6	1.83385	0.916926	−	0.399058i	$-0.130663\pi$
$409$	6.20401e6	1.83385	0.916926	+	0.399058i	$0.130663\pi$
$410$	−187570.	−0.0551066
$411$	0	0
$412$	1.57734e6	0.457808
$413$	−5.77612e6	−1.66633
$414$	0	0
$415$	−182385.	−0.0519839
$416$	−433312.	−0.122763
$417$	0	0
$418$	−677147.	−0.189558
$419$	678639.	0.188844	0.0944222	−	0.995532i	$-0.469900\pi$
$419$	678639.	0.188844	0.0944222	+	0.995532i	$0.469900\pi$
$420$	0	0
$421$	−6381.94	−0.00175488	−0.000877440	−	1.00000i	$-0.500279\pi$
$421$	−6381.94	−0.00175488	−0.000877440		1.00000i	$0.500279\pi$
$422$	1.19097e6	0.325550
$423$	0	0
$424$	−4.41868e6	−1.19365
$425$	−6.18168e6	−1.66010
$426$	0	0
$427$	−1.06276e7	−2.82076
$428$	138447.	0.0365319
$429$	0	0
$430$	−665105.	−0.173468
$431$	−4.55329e6	−1.18068	−0.590340	−	0.807155i	$-0.701006\pi$
$431$	−4.55329e6	−1.18068	−0.590340	+	0.807155i	$0.701006\pi$
$432$	0	0
$433$	−2.17265e6	−0.556892	−0.278446	−	0.960452i	$-0.589819\pi$
$433$	−2.17265e6	−0.556892	−0.278446	+	0.960452i	$0.589819\pi$
$434$	7.66874e6	1.95434
$435$	0	0
$436$	−2.14940e6	−0.541502
$437$	−383358.	−0.0960287
$438$	0	0
$439$	−5.48294e6	−1.35785	−0.678926	−	0.734207i	$-0.737554\pi$
$439$	−5.48294e6	−1.35785	−0.678926	+	0.734207i	$0.737554\pi$
$440$	−382760.	−0.0942529
$441$	0	0
$442$	−1.13658e6	−0.276723
$443$	−4.75651e6	−1.15154	−0.575770	−	0.817612i	$-0.695298\pi$
$443$	−4.75651e6	−1.15154	−0.575770	+	0.817612i	$0.695298\pi$
$444$	0	0
$445$	−131175.	−0.0314016
$446$	2.29228e6	0.545671
$447$	0	0
$448$	7.03665e6	1.65642
$449$	798537.	0.186930	0.0934650	−	0.995623i	$-0.470206\pi$
$449$	798537.	0.186930	0.0934650	+	0.995623i	$0.470206\pi$
$450$	0	0
$451$	−842346.	−0.195006
$452$	1.54484e6	0.355661
$453$	0	0
$454$	−1.58971e6	−0.361976
$455$	230699.	0.0522418
$456$	0	0
$457$	3.54849e6	0.794792	0.397396	−	0.917647i	$-0.369914\pi$
$457$	3.54849e6	0.794792	0.397396	+	0.917647i	$0.369914\pi$
$458$	−1.28795e6	−0.286904
$459$	0	0
$460$	−53313.0	−0.0117473
$461$	3.09208e6	0.677638	0.338819	−	0.940852i	$-0.389973\pi$
$461$	3.09208e6	0.677638	0.338819	+	0.940852i	$0.389973\pi$
$462$	0	0
$463$	3.94502e6	0.855257	0.427629	−	0.903955i	$-0.359349\pi$
$463$	3.94502e6	0.855257	0.427629	+	0.903955i	$0.359349\pi$
$464$	−3.69551e6	−0.796855
$465$	0	0
$466$	−3.89532e6	−0.830957
$467$	2.04479e6	0.433867	0.216933	−	0.976186i	$-0.430395\pi$
$467$	2.04479e6	0.433867	0.216933	+	0.976186i	$0.430395\pi$
$468$	0	0
$469$	−3.88599e6	−0.815774
$470$	−607440.	−0.126841
$471$	0	0
$472$	5.71721e6	1.18122
$473$	−2.98688e6	−0.613853
$474$	0	0
$475$	−2.19713e6	−0.446809
$476$	−4.23869e6	−0.857460
$477$	0	0
$478$	−1.74824e6	−0.349970
$479$	5.42589e6	1.08052	0.540259	−	0.841498i	$-0.318326\pi$
$479$	5.42589e6	1.08052	0.540259	+	0.841498i	$0.318326\pi$
$480$	0	0
$481$	−245736.	−0.0484290
$482$	6.87920e6	1.34871
$483$	0	0
$484$	1.25878e6	0.244251
$485$	713455.	0.137725
$486$	0	0
$487$	2.30401e6	0.440212	0.220106	−	0.975476i	$-0.429360\pi$
$487$	2.30401e6	0.440212	0.220106	+	0.975476i	$0.429360\pi$
$488$	1.05192e7	1.99956
$489$	0	0
$490$	−1.02309e6	−0.192497
$491$	−875983.	−0.163980	−0.0819902	−	0.996633i	$-0.526128\pi$
$491$	−875983.	−0.163980	−0.0819902	+	0.996633i	$0.526128\pi$
$492$	0	0
$493$	1.29726e7	2.40387
$494$	−403972.	−0.0744790
$495$	0	0
$496$	−4.81850e6	−0.879444
$497$	4.61150e6	0.837436
$498$	0	0
$499$	−9.99352e6	−1.79667	−0.898333	−	0.439316i	$-0.855221\pi$
$499$	−9.99352e6	−1.79667	−0.898333	+	0.439316i	$0.855221\pi$
$500$	−620492.	−0.110997
$501$	0	0
$502$	−3.11609e6	−0.551888
$503$	−8.68084e6	−1.52983	−0.764913	−	0.644134i	$-0.777218\pi$
$503$	−8.68084e6	−1.52983	−0.764913	+	0.644134i	$0.777218\pi$
$504$	0	0
$505$	1.50662e6	0.262890
$506$	494299.	0.0858250
$507$	0	0
$508$	−1.59542e6	−0.274293
$509$	5.24126e6	0.896688	0.448344	−	0.893861i	$-0.352014\pi$
$509$	5.24126e6	0.896688	0.448344	+	0.893861i	$0.352014\pi$
$510$	0	0
$511$	1.43150e7	2.42516
$512$	−5.75892e6	−0.970882
$513$	0	0
$514$	5.18240e6	0.865214
$515$	1.45795e6	0.242228
$516$	0	0
$517$	−2.72791e6	−0.448853
$518$	1.89203e6	0.309816
$519$	0	0
$520$	−228347.	−0.0370328
$521$	−2.67105e6	−0.431109	−0.215555	−	0.976492i	$-0.569156\pi$
$521$	−2.67105e6	−0.431109	−0.215555	+	0.976492i	$0.569156\pi$
$522$	0	0
$523$	−3.01162e6	−0.481444	−0.240722	−	0.970594i	$-0.577384\pi$
$523$	−3.01162e6	−0.481444	−0.240722	+	0.970594i	$0.577384\pi$
$524$	529163.	0.0841902
$525$	0	0
$526$	2.81960e6	0.444348
$527$	1.69148e7	2.65301
$528$	0	0
$529$	279841.	0.0434783
$530$	−1.00483e6	−0.155383
$531$	0	0
$532$	−1.50654e6	−0.230782
$533$	−502526.	−0.0766197
$534$	0	0
$535$	127967.	0.0193292
$536$	3.84636e6	0.578280
$537$	0	0
$538$	5.06498e6	0.754436
$539$	−4.59452e6	−0.681190
$540$	0	0
$541$	−763243.	−0.112116	−0.0560582	−	0.998428i	$-0.517853\pi$
$541$	−763243.	−0.112116	−0.0560582	+	0.998428i	$0.517853\pi$
$542$	9.44643e6	1.38124
$543$	0	0
$544$	7.35872e6	1.06612
$545$	−1.98670e6	−0.286511
$546$	0	0
$547$	−3.34872e6	−0.478532	−0.239266	−	0.970954i	$-0.576907\pi$
$547$	−3.34872e6	−0.478532	−0.239266	+	0.970954i	$0.576907\pi$
$548$	2.72588e6	0.387753
$549$	0	0
$550$	2.83297e6	0.399333
$551$	4.61082e6	0.646992
$552$	0	0
$553$	−3.75334e6	−0.521922
$554$	−2.26617e6	−0.313703
$555$	0	0
$556$	−1.73469e6	−0.237978
$557$	−5.49716e6	−0.750758	−0.375379	−	0.926871i	$-0.622487\pi$
$557$	−5.49716e6	−0.750758	−0.375379	+	0.926871i	$0.622487\pi$
$558$	0	0
$559$	−1.78191e6	−0.241188
$560$	1.11608e6	0.150392
$561$	0	0
$562$	5.21498e6	0.696485
$563$	−1.18868e7	−1.58050	−0.790251	−	0.612783i	$-0.790050\pi$
$563$	−1.18868e7	−1.58050	−0.790251	+	0.612783i	$0.790050\pi$
$564$	0	0
$565$	1.42791e6	0.188182
$566$	−2.57041e6	−0.337257
$567$	0	0
$568$	−4.56447e6	−0.593635
$569$	549705.	0.0711786	0.0355893	−	0.999367i	$-0.488669\pi$
$569$	549705.	0.0711786	0.0355893	+	0.999367i	$0.488669\pi$
$570$	0	0
$571$	6.55322e6	0.841133	0.420566	−	0.907262i	$-0.361831\pi$
$571$	6.55322e6	0.841133	0.420566	+	0.907262i	$0.361831\pi$
$572$	−252294.	−0.0322416
$573$	0	0
$574$	3.86917e6	0.490161
$575$	1.60385e6	0.202299
$576$	0	0
$577$	857385.	0.107210	0.0536052	−	0.998562i	$-0.482929\pi$
$577$	857385.	0.107210	0.0536052	+	0.998562i	$0.482929\pi$
$578$	1.27095e7	1.58238
$579$	0	0
$580$	641219.	0.0791473
$581$	3.76222e6	0.462385
$582$	0	0
$583$	−4.51254e6	−0.549857
$584$	−1.41691e7	−1.71913
$585$	0	0
$586$	−1.80227e6	−0.216809
$587$	−8.04515e6	−0.963694	−0.481847	−	0.876255i	$-0.660034\pi$
$587$	−8.04515e6	−0.963694	−0.481847	+	0.876255i	$0.660034\pi$
$588$	0	0
$589$	6.01196e6	0.714049
$590$	1.30013e6	0.153765
$591$	0	0
$592$	−1.18882e6	−0.139416
$593$	−7.69882e6	−0.899057	−0.449529	−	0.893266i	$-0.648408\pi$
$593$	−7.69882e6	−0.899057	−0.449529	+	0.893266i	$0.648408\pi$
$594$	0	0
$595$	−3.91785e6	−0.453686
$596$	−4.09894e6	−0.472668
$597$	0	0
$598$	294889.	0.0337214
$599$	−1.62753e7	−1.85337	−0.926687	−	0.375835i	$-0.877356\pi$
$599$	−1.62753e7	−1.85337	−0.926687	+	0.375835i	$0.877356\pi$
$600$	0	0
$601$	−1.45214e7	−1.63992	−0.819959	−	0.572423i	$-0.806004\pi$
$601$	−1.45214e7	−1.63992	−0.819959	+	0.572423i	$0.806004\pi$
$602$	1.37197e7	1.54296
$603$	0	0
$604$	−1.84487e6	−0.205766
$605$	1.16350e6	0.129234
$606$	0	0
$607$	1.12242e7	1.23647	0.618235	−	0.785993i	$-0.287848\pi$
$607$	1.12242e7	1.23647	0.618235	+	0.785993i	$0.287848\pi$
$608$	2.61548e6	0.286941
$609$	0	0
$610$	2.39214e6	0.260292
$611$	−1.62742e6	−0.176358
$612$	0	0
$613$	4.12814e6	0.443715	0.221857	−	0.975079i	$-0.428788\pi$
$613$	4.12814e6	0.443715	0.221857	+	0.975079i	$0.428788\pi$
$614$	−6.03960e6	−0.646528
$615$	0	0
$616$	7.89553e6	0.838358
$617$	−1.51382e7	−1.60088	−0.800442	−	0.599410i	$-0.795402\pi$
$617$	−1.51382e7	−1.60088	−0.800442	+	0.599410i	$0.795402\pi$
$618$	0	0
$619$	−3.19410e6	−0.335060	−0.167530	−	0.985867i	$-0.553579\pi$
$619$	−3.19410e6	−0.335060	−0.167530	+	0.985867i	$0.553579\pi$
$620$	836074.	0.0873505
$621$	0	0
$622$	6.14404e6	0.636764
$623$	2.70587e6	0.279310
$624$	0	0
$625$	8.90100e6	0.911462
$626$	6.61023e6	0.674187
$627$	0	0
$628$	2.01760e6	0.204144
$629$	4.17321e6	0.420575
$630$	0	0
$631$	1.08417e7	1.08398	0.541991	−	0.840384i	$-0.317671\pi$
$631$	1.08417e7	1.08398	0.541991	+	0.840384i	$0.317671\pi$
$632$	3.71507e6	0.369976
$633$	0	0
$634$	7.65223e6	0.756075
$635$	−1.47466e6	−0.145130
$636$	0	0
$637$	−2.74100e6	−0.267645
$638$	−5.94515e6	−0.578244
$639$	0	0
$640$	−469177.	−0.0452780
$641$	7.28037e6	0.699856	0.349928	−	0.936777i	$-0.386206\pi$
$641$	7.28037e6	0.699856	0.349928	+	0.936777i	$0.386206\pi$
$642$	0	0
$643$	−4.07419e6	−0.388610	−0.194305	−	0.980941i	$-0.562245\pi$
$643$	−4.07419e6	−0.388610	−0.194305	+	0.980941i	$0.562245\pi$
$644$	1.09973e6	0.104490
$645$	0	0
$646$	6.86046e6	0.646803
$647$	−1.11052e7	−1.04295	−0.521476	−	0.853266i	$-0.674618\pi$
$647$	−1.11052e7	−1.04295	−0.521476	+	0.853266i	$0.674618\pi$
$648$	0	0
$649$	5.83866e6	0.544129
$650$	1.69009e6	0.156901
$651$	0	0
$652$	−5.23102e6	−0.481911
$653$	7.07241e6	0.649059	0.324530	−	0.945875i	$-0.394794\pi$
$653$	7.07241e6	0.649059	0.324530	+	0.945875i	$0.394794\pi$
$654$	0	0
$655$	489110.	0.0445454
$656$	−2.43112e6	−0.220570
$657$	0	0
$658$	1.25302e7	1.12822
$659$	2.03717e7	1.82732	0.913660	−	0.406480i	$-0.133244\pi$
$659$	2.03717e7	1.82732	0.913660	+	0.406480i	$0.133244\pi$
$660$	0	0
$661$	−2.45288e6	−0.218360	−0.109180	−	0.994022i	$-0.534822\pi$
$661$	−2.45288e6	−0.218360	−0.109180	+	0.994022i	$0.534822\pi$
$662$	−3.23878e6	−0.287234
$663$	0	0
$664$	−3.72385e6	−0.327772
$665$	−1.39251e6	−0.122108
$666$	0	0
$667$	−3.36577e6	−0.292934
$668$	479255.	0.0415552
$669$	0	0
$670$	874686.	0.0752775
$671$	1.07427e7	0.921100
$672$	0	0
$673$	−1.89339e7	−1.61140	−0.805699	−	0.592326i	$-0.798210\pi$
$673$	−1.89339e7	−1.61140	−0.805699	+	0.592326i	$0.798210\pi$
$674$	3.43644e6	0.291379
$675$	0	0
$676$	3.72650e6	0.313642
$677$	9.47745e6	0.794730	0.397365	−	0.917661i	$-0.369925\pi$
$677$	9.47745e6	0.794730	0.397365	+	0.917661i	$0.369925\pi$
$678$	0	0
$679$	−1.47171e7	−1.22503
$680$	3.87790e6	0.321606
$681$	0	0
$682$	−7.75178e6	−0.638176
$683$	−9.49319e6	−0.778683	−0.389341	−	0.921094i	$-0.627297\pi$
$683$	−9.49319e6	−0.778683	−0.389341	+	0.921094i	$0.627297\pi$
$684$	0	0
$685$	2.51955e6	0.205162
$686$	5.56788e6	0.451731
$687$	0	0
$688$	−8.62052e6	−0.694324
$689$	−2.69209e6	−0.216044
$690$	0	0
$691$	−1.71375e7	−1.36538	−0.682689	−	0.730709i	$-0.739190\pi$
$691$	−1.71375e7	−1.36538	−0.682689	+	0.730709i	$0.739190\pi$
$692$	4.78023e6	0.379475
$693$	0	0
$694$	−1.73352e7	−1.36625
$695$	−1.60339e6	−0.125915
$696$	0	0
$697$	8.53415e6	0.665393
$698$	−9.43341e6	−0.732875
$699$	0	0
$700$	6.30289e6	0.486177
$701$	1.57431e7	1.21003	0.605013	−	0.796216i	$-0.293168\pi$
$701$	1.57431e7	1.21003	0.605013	+	0.796216i	$0.293168\pi$
$702$	0	0
$703$	1.48327e6	0.113196
$704$	−7.11284e6	−0.540893
$705$	0	0
$706$	609466.	0.0460191
$707$	−3.10783e7	−2.33835
$708$	0	0
$709$	−1.11880e7	−0.835867	−0.417933	−	0.908478i	$-0.637245\pi$
$709$	−1.11880e7	−0.835867	−0.417933	+	0.908478i	$0.637245\pi$
$710$	−1.03799e6	−0.0772764
$711$	0	0
$712$	−2.67828e6	−0.197996
$713$	−4.38857e6	−0.323295
$714$	0	0
$715$	−233197.	−0.0170592
$716$	5.07978e6	0.370307
$717$	0	0
$718$	3.15951e6	0.228722
$719$	5.34846e6	0.385840	0.192920	−	0.981215i	$-0.438204\pi$
$719$	5.34846e6	0.385840	0.192920	+	0.981215i	$0.438204\pi$
$720$	0	0
$721$	−3.00745e7	−2.15457
$722$	−9.05831e6	−0.646702
$723$	0	0
$724$	5.77597e6	0.409523
$725$	−1.92902e7	−1.36299
$726$	0	0
$727$	5.48333e6	0.384776	0.192388	−	0.981319i	$-0.438377\pi$
$727$	5.48333e6	0.384776	0.192388	+	0.981319i	$0.438377\pi$
$728$	4.71031e6	0.329398
$729$	0	0
$730$	−3.22213e6	−0.223788
$731$	3.02613e7	2.09456
$732$	0	0
$733$	−9.62220e6	−0.661477	−0.330738	−	0.943722i	$-0.607298\pi$
$733$	−9.62220e6	−0.661477	−0.330738	+	0.943722i	$0.607298\pi$
$734$	−7.87359e6	−0.539427
$735$	0	0
$736$	−1.90923e6	−0.129916
$737$	3.92807e6	0.266385
$738$	0	0
$739$	−5.22946e6	−0.352246	−0.176123	−	0.984368i	$-0.556356\pi$
$739$	−5.22946e6	−0.352246	−0.176123	+	0.984368i	$0.556356\pi$
$740$	206276.	0.0138474
$741$	0	0
$742$	2.07276e7	1.38210
$743$	2.25365e7	1.49766	0.748832	−	0.662760i	$-0.230615\pi$
$743$	2.25365e7	1.49766	0.748832	+	0.662760i	$0.230615\pi$
$744$	0	0
$745$	−3.78868e6	−0.250091
$746$	−1.18377e7	−0.778791
$747$	0	0
$748$	4.28458e6	0.279998
$749$	−2.63969e6	−0.171929
$750$	0	0
$751$	−8.05173e6	−0.520942	−0.260471	−	0.965482i	$-0.583878\pi$
$751$	−8.05173e6	−0.520942	−0.260471	+	0.965482i	$0.583878\pi$
$752$	−7.87312e6	−0.507694
$753$	0	0
$754$	−3.54675e6	−0.227197
$755$	−1.70523e6	−0.108872
$756$	0	0
$757$	−1.30095e7	−0.825128	−0.412564	−	0.910929i	$-0.635367\pi$
$757$	−1.30095e7	−0.825128	−0.412564	+	0.910929i	$0.635367\pi$
$758$	−744931.	−0.0470916
$759$	0	0
$760$	1.37831e6	0.0865590
$761$	−1.81782e7	−1.13786	−0.568930	−	0.822386i	$-0.692643\pi$
$761$	−1.81782e7	−1.13786	−0.568930	+	0.822386i	$0.692643\pi$
$762$	0	0
$763$	4.09815e7	2.54845
$764$	−2.66517e6	−0.165193
$765$	0	0
$766$	−5.85878e6	−0.360774
$767$	3.48322e6	0.213793
$768$	0	0
$769$	1.93965e7	1.18279	0.591394	−	0.806383i	$-0.298578\pi$
$769$	1.93965e7	1.18279	0.591394	+	0.806383i	$0.298578\pi$
$770$	1.79549e6	0.109133
$771$	0	0
$772$	454518.	0.0274478
$773$	3.23823e7	1.94921	0.974607	−	0.223923i	$-0.0718866\pi$
$773$	3.23823e7	1.94921	0.974607	+	0.223923i	$0.0718866\pi$
$774$	0	0
$775$	−2.51521e7	−1.50425
$776$	1.45670e7	0.868390
$777$	0	0
$778$	9.49590e6	0.562454
$779$	3.03326e6	0.179088
$780$	0	0
$781$	−4.66144e6	−0.273459
$782$	−5.00795e6	−0.292848
$783$	0	0
$784$	−1.32604e7	−0.770489
$785$	1.86489e6	0.108014
$786$	0	0
$787$	−1.11105e7	−0.639436	−0.319718	−	0.947513i	$-0.603588\pi$
$787$	−1.11105e7	−0.639436	−0.319718	+	0.947513i	$0.603588\pi$
$788$	−729711.	−0.0418635
$789$	0	0
$790$	844829.	0.0481616
$791$	−2.94547e7	−1.67384
$792$	0	0
$793$	6.40887e6	0.361908
$794$	3.62515e6	0.204068
$795$	0	0
$796$	−3.11383e6	−0.174186
$797$	−1.56224e7	−0.871170	−0.435585	−	0.900148i	$-0.643458\pi$
$797$	−1.56224e7	−0.871170	−0.435585	+	0.900148i	$0.643458\pi$
$798$	0	0
$799$	2.76376e7	1.53156
$800$	−1.09423e7	−0.604485
$801$	0	0
$802$	3.19925e6	0.175636
$803$	−1.44701e7	−0.791920
$804$	0	0
$805$	1.01649e6	0.0552860
$806$	−4.62455e6	−0.250745
$807$	0	0
$808$	3.07614e7	1.65759
$809$	1.86288e7	1.00072	0.500362	−	0.865817i	$-0.333200\pi$
$809$	1.86288e7	1.00072	0.500362	+	0.865817i	$0.333200\pi$
$810$	0	0
$811$	−1.25574e7	−0.670418	−0.335209	−	0.942144i	$-0.608807\pi$
$811$	−1.25574e7	−0.670418	−0.335209	+	0.942144i	$0.608807\pi$
$812$	−1.32270e7	−0.703998
$813$	0	0
$814$	−1.91252e6	−0.101168
$815$	−4.83507e6	−0.254982
$816$	0	0
$817$	1.07557e7	0.563744
$818$	2.88056e7	1.50520
$819$	0	0
$820$	421831.	0.0219081
$821$	1.98406e7	1.02730	0.513650	−	0.858000i	$-0.328293\pi$
$821$	1.98406e7	1.02730	0.513650	+	0.858000i	$0.328293\pi$
$822$	0	0
$823$	−4.61707e6	−0.237611	−0.118805	−	0.992918i	$-0.537906\pi$
$823$	−4.61707e6	−0.237611	−0.118805	+	0.992918i	$0.537906\pi$
$824$	2.97677e7	1.52731
$825$	0	0
$826$	−2.68189e7	−1.36770
$827$	−3.37591e7	−1.71644	−0.858218	−	0.513286i	$-0.828428\pi$
$827$	−3.37591e7	−1.71644	−0.858218	+	0.513286i	$0.828428\pi$
$828$	0	0
$829$	−8.87084e6	−0.448310	−0.224155	−	0.974554i	$-0.571962\pi$
$829$	−8.87084e6	−0.448310	−0.224155	+	0.974554i	$0.571962\pi$
$830$	−846826.	−0.0426677
$831$	0	0
$832$	−4.24337e6	−0.212521
$833$	4.65490e7	2.32433
$834$	0	0
$835$	442979.	0.0219871
$836$	1.52285e6	0.0753603
$837$	0	0
$838$	3.15097e6	0.155001
$839$	−2.35103e7	−1.15307	−0.576533	−	0.817074i	$-0.695595\pi$
$839$	−2.35103e7	−1.15307	−0.576533	+	0.817074i	$0.695595\pi$
$840$	0	0
$841$	1.99705e7	0.973639
$842$	−29631.8	−0.00144038
$843$	0	0
$844$	−2.67840e6	−0.129425
$845$	3.44443e6	0.165949
$846$	0	0
$847$	−2.40006e7	−1.14951
$848$	−1.30238e7	−0.621939
$849$	0	0
$850$	−2.87020e7	−1.36259
$851$	−1.08275e6	−0.0512510
$852$	0	0
$853$	1.33579e7	0.628588	0.314294	−	0.949326i	$-0.398232\pi$
$853$	1.33579e7	0.628588	0.314294	+	0.949326i	$0.398232\pi$
$854$	−4.93448e7	−2.31524
$855$	0	0
$856$	2.61277e6	0.121876
$857$	−8.10728e6	−0.377071	−0.188536	−	0.982066i	$-0.560374\pi$
$857$	−8.10728e6	−0.377071	−0.188536	+	0.982066i	$0.560374\pi$
$858$	0	0
$859$	−2.66270e6	−0.123123	−0.0615615	−	0.998103i	$-0.519608\pi$
$859$	−2.66270e6	−0.123123	−0.0615615	+	0.998103i	$0.519608\pi$
$860$	1.49577e6	0.0689635
$861$	0	0
$862$	−2.11412e7	−0.969086
$863$	−2.79756e7	−1.27865	−0.639327	−	0.768935i	$-0.720787\pi$
$863$	−2.79756e7	−1.27865	−0.639327	+	0.768935i	$0.720787\pi$
$864$	0	0
$865$	4.41840e6	0.200782
$866$	−1.00878e7	−0.457089
$867$	0	0
$868$	−1.72464e7	−0.776963
$869$	3.79399e6	0.170430
$870$	0	0
$871$	2.34340e6	0.104665
$872$	−4.05636e7	−1.80653
$873$	0	0
$874$	−1.77996e6	−0.0788190
$875$	1.18306e7	0.522381
$876$	0	0
$877$	−7.33435e6	−0.322005	−0.161003	−	0.986954i	$-0.551473\pi$
$877$	−7.33435e6	−0.322005	−0.161003	+	0.986954i	$0.551473\pi$
$878$	−2.54577e7	−1.11451
$879$	0	0
$880$	−1.12816e6	−0.0491094
$881$	2.07072e7	0.898838	0.449419	−	0.893321i	$-0.351631\pi$
$881$	2.07072e7	0.898838	0.449419	+	0.893321i	$0.351631\pi$
$882$	0	0
$883$	−61880.5	−0.00267087	−0.00133543	−	0.999999i	$-0.500425\pi$
$883$	−61880.5	−0.00267087	−0.00133543	+	0.999999i	$0.500425\pi$
$884$	2.55609e6	0.110014
$885$	0	0
$886$	−2.20848e7	−0.945168
$887$	−9.17720e6	−0.391653	−0.195826	−	0.980639i	$-0.562739\pi$
$887$	−9.17720e6	−0.391653	−0.195826	+	0.980639i	$0.562739\pi$
$888$	0	0
$889$	3.04191e7	1.29090
$890$	−609056.	−0.0257740
$891$	0	0
$892$	−5.15517e6	−0.216936
$893$	9.82314e6	0.412213
$894$	0	0
$895$	4.69528e6	0.195931
$896$	9.67814e6	0.402737
$897$	0	0
$898$	3.70766e6	0.153430
$899$	5.27832e7	2.17819
$900$	0	0
$901$	4.57184e7	1.87620
$902$	−3.91107e6	−0.160059
$903$	0	0
$904$	2.91543e7	1.18654
$905$	5.33877e6	0.216681
$906$	0	0
$907$	−3.66605e7	−1.47972	−0.739861	−	0.672759i	$-0.765109\pi$
$907$	−3.66605e7	−1.47972	−0.739861	+	0.672759i	$0.765109\pi$
$908$	3.57515e6	0.143906
$909$	0	0
$910$	1.07115e6	0.0428793
$911$	−2.52554e7	−1.00823	−0.504113	−	0.863638i	$-0.668180\pi$
$911$	−2.52554e7	−1.00823	−0.504113	+	0.863638i	$0.668180\pi$
$912$	0	0
$913$	−3.80295e6	−0.150989
$914$	1.64759e7	0.652354
$915$	0	0
$916$	2.89651e6	0.114061
$917$	−1.00893e7	−0.396221
$918$	0	0
$919$	−1.63983e7	−0.640485	−0.320243	−	0.947336i	$-0.603764\pi$
$919$	−1.63983e7	−0.640485	−0.320243	+	0.947336i	$0.603764\pi$
$920$	−1.00613e6	−0.0391907
$921$	0	0
$922$	1.43567e7	0.556196
$923$	−2.78092e6	−0.107444
$924$	0	0
$925$	−6.20552e6	−0.238464
$926$	1.83170e7	0.701983
$927$	0	0
$928$	2.29632e7	0.875310
$929$	2.13650e7	0.812200	0.406100	−	0.913829i	$-0.366888\pi$
$929$	2.13650e7	0.812200	0.406100	+	0.913829i	$0.366888\pi$
$930$	0	0
$931$	1.65447e7	0.625584
$932$	8.76029e6	0.330353
$933$	0	0
$934$	9.49409e6	0.356112
$935$	3.96027e6	0.148148
$936$	0	0
$937$	1.62645e6	0.0605191	0.0302595	−	0.999542i	$-0.490367\pi$
$937$	1.62645e6	0.0605191	0.0302595	+	0.999542i	$0.490367\pi$
$938$	−1.80429e7	−0.669576
$939$	0	0
$940$	1.36609e6	0.0504266
$941$	−4.47294e7	−1.64672	−0.823359	−	0.567520i	$-0.807903\pi$
$941$	−4.47294e7	−1.64672	−0.823359	+	0.567520i	$0.807903\pi$
$942$	0	0
$943$	−2.21420e6	−0.0810845
$944$	1.68511e7	0.615459
$945$	0	0
$946$	−1.38683e7	−0.503842
$947$	4.97502e7	1.80269	0.901343	−	0.433107i	$-0.142583\pi$
$947$	4.97502e7	1.80269	0.901343	+	0.433107i	$0.142583\pi$
$948$	0	0
$949$	−8.63253e6	−0.311152
$950$	−1.02014e7	−0.366735
$951$	0	0
$952$	−7.99928e7	−2.86061
$953$	3.67502e7	1.31077	0.655386	−	0.755294i	$-0.272506\pi$
$953$	3.67502e7	1.31077	0.655386	+	0.755294i	$0.272506\pi$
$954$	0	0
$955$	−2.46343e6	−0.0874042
$956$	3.93166e6	0.139133
$957$	0	0
$958$	2.51928e7	0.886875
$959$	−5.19731e7	−1.82487
$960$	0	0
$961$	4.01939e7	1.40395
$962$	−1.14097e6	−0.0397498
$963$	0	0
$964$	−1.54708e7	−0.536192
$965$	420115.	0.0145228
$966$	0	0
$967$	1.25853e7	0.432809	0.216404	−	0.976304i	$-0.430567\pi$
$967$	1.25853e7	0.432809	0.216404	+	0.976304i	$0.430567\pi$
$968$	2.37558e7	0.814857
$969$	0	0
$970$	3.31262e6	0.113043
$971$	−5.09098e7	−1.73282	−0.866410	−	0.499334i	$-0.833578\pi$
$971$	−5.09098e7	−1.73282	−0.866410	+	0.499334i	$0.833578\pi$
$972$	0	0
$973$	3.30746e7	1.11999
$974$	1.06977e7	0.361320
$975$	0	0
$976$	3.10048e7	1.04185
$977$	3.11350e7	1.04355	0.521773	−	0.853084i	$-0.325271\pi$
$977$	3.11350e7	1.04355	0.521773	+	0.853084i	$0.325271\pi$
$978$	0	0
$979$	−2.73517e6	−0.0912069
$980$	2.30085e6	0.0765286
$981$	0	0
$982$	−4.06725e6	−0.134593
$983$	−4.74669e6	−0.156678	−0.0783388	−	0.996927i	$-0.524962\pi$
$983$	−4.74669e6	−0.156678	−0.0783388	+	0.996927i	$0.524962\pi$
$984$	0	0
$985$	−674477.	−0.0221502
$986$	6.02328e7	1.97306
$987$	0	0
$988$	908503.	0.0296097
$989$	−7.85134e6	−0.255243
$990$	0	0
$991$	4.46146e7	1.44309	0.721545	−	0.692368i	$-0.243432\pi$
$991$	4.46146e7	1.44309	0.721545	+	0.692368i	$0.243432\pi$
$992$	2.99413e7	0.966031
$993$	0	0
$994$	2.14115e7	0.687356
$995$	−2.87814e6	−0.0921625
$996$	0	0
$997$	4.86185e7	1.54904	0.774522	−	0.632546i	$-0.217990\pi$
$997$	4.86185e7	1.54904	0.774522	+	0.632546i	$0.217990\pi$
$998$	−4.64006e7	−1.47468
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.g.1.4		6
3.2	odd	2		23.6.a.b.1.3	✓	6
12.11	even	2		368.6.a.h.1.2		6
15.14	odd	2		575.6.a.c.1.4		6
69.68	even	2		529.6.a.c.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.6.a.b.1.3	✓	6	3.2	odd	2
207.6.a.g.1.4		6	1.1	even	1	trivial
368.6.a.h.1.2		6	12.11	even	2
529.6.a.c.1.3		6	69.68	even	2
575.6.a.c.1.4		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)

\(f(q)\)	\(=\)	\(q+4.64307 q^{2} -10.4419 q^{4} -9.65155 q^{5} +199.091 q^{7} -197.061 q^{8} +O(q^{10})\)	\(q+4.64307 q^{2} -10.4419 q^{4} -9.65155 q^{5} +199.091 q^{7} -197.061 q^{8} -44.8128 q^{10} -201.247 q^{11} -120.060 q^{13} +924.394 q^{14} -580.825 q^{16} +2038.91 q^{17} +724.684 q^{19} +100.781 q^{20} -934.403 q^{22} -529.000 q^{23} -3031.85 q^{25} -557.445 q^{26} -2078.89 q^{28} +6362.52 q^{29} +8295.97 q^{31} +3609.13 q^{32} +9466.82 q^{34} -1921.54 q^{35} +2046.78 q^{37} +3364.76 q^{38} +1901.94 q^{40} +4185.63 q^{41} +14841.9 q^{43} +2101.40 q^{44} -2456.18 q^{46} +13555.1 q^{47} +22830.3 q^{49} -14077.1 q^{50} +1253.65 q^{52} +22422.9 q^{53} +1942.34 q^{55} -39233.0 q^{56} +29541.6 q^{58} -29012.4 q^{59} -53380.7 q^{61} +38518.7 q^{62} +35343.8 q^{64} +1158.76 q^{65} -19518.7 q^{67} -21290.2 q^{68} -8921.83 q^{70} +23162.8 q^{71} +71902.0 q^{73} +9503.33 q^{74} -7567.10 q^{76} -40066.5 q^{77} -18852.4 q^{79} +5605.86 q^{80} +19434.2 q^{82} +18897.0 q^{83} -19678.7 q^{85} +68911.7 q^{86} +39657.9 q^{88} +13591.1 q^{89} -23902.8 q^{91} +5523.78 q^{92} +62937.1 q^{94} -6994.33 q^{95} -73921.2 q^{97} +106003. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8}+O(q^{10}) \) 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

L-functions

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Database

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