LMFDB - Embedded newform 207.6.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("207.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.2
Root			$0.917748$ 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)

$f(q)$	$=$	$q+0.164504 q^{2} -31.9729 q^{4} +37.9865 q^{5} -43.8366 q^{7} -10.5238 q^{8} +O(q^{10})$	\(q+0.164504 q^{2} -31.9729 q^{4} +37.9865 q^{5} -43.8366 q^{7} -10.5238 q^{8} +6.24894 q^{10} +163.213 q^{11} +430.356 q^{13} -7.21131 q^{14} +1021.40 q^{16} -740.415 q^{17} -916.578 q^{19} -1214.54 q^{20} +26.8493 q^{22} +529.000 q^{23} -1682.03 q^{25} +70.7955 q^{26} +1401.59 q^{28} +5112.96 q^{29} -5702.60 q^{31} +504.787 q^{32} -121.802 q^{34} -1665.20 q^{35} -10913.1 q^{37} -150.781 q^{38} -399.763 q^{40} -11092.6 q^{41} +5528.76 q^{43} -5218.41 q^{44} +87.0228 q^{46} -14435.2 q^{47} -14885.4 q^{49} -276.701 q^{50} -13759.8 q^{52} -6852.79 q^{53} +6199.90 q^{55} +461.329 q^{56} +841.104 q^{58} -4155.17 q^{59} -21911.4 q^{61} -938.102 q^{62} -32601.9 q^{64} +16347.7 q^{65} -15164.2 q^{67} +23673.3 q^{68} -273.932 q^{70} -14031.4 q^{71} -23310.1 q^{73} -1795.25 q^{74} +29305.7 q^{76} -7154.72 q^{77} +64276.8 q^{79} +38799.5 q^{80} -1824.78 q^{82} -114349. q^{83} -28125.8 q^{85} +909.504 q^{86} -1717.63 q^{88} +63420.7 q^{89} -18865.4 q^{91} -16913.7 q^{92} -2374.66 q^{94} -34817.6 q^{95} -91987.6 q^{97} -2448.70 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8}+O(q^{10}) $ 3 * q + 4 * q^2 + 16 * q^4 + 58 * q^5 - 282 * q^7 + 360 * q^8	$ 3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8} - 156 q^{10} - 136 q^{11} - 1116 q^{13} - 2016 q^{14} + 128 q^{16} + 896 q^{17} + 1654 q^{19} - 1504 q^{20} + 1352 q^{22} + 1587 q^{23} - 7347 q^{25} - 1998 q^{26} - 10264 q^{28} + 844 q^{29} - 3020 q^{31} - 6656 q^{32} - 11212 q^{34} - 1072 q^{35} + 8938 q^{37} - 10728 q^{38} + 3440 q^{40} + 12792 q^{41} - 16730 q^{43} - 5112 q^{44} + 2116 q^{46} - 22500 q^{47} + 2887 q^{49} - 15156 q^{50} - 47412 q^{52} - 17108 q^{53} - 436 q^{55} - 42640 q^{56} - 55678 q^{58} - 54176 q^{59} - 71324 q^{61} + 72710 q^{62} - 49984 q^{64} - 846 q^{65} - 62960 q^{67} + 8352 q^{68} + 9224 q^{70} - 98400 q^{71} - 81772 q^{73} + 59044 q^{74} + 31488 q^{76} + 304 q^{77} + 58224 q^{79} + 17568 q^{80} + 61926 q^{82} - 9892 q^{83} + 15536 q^{85} - 191140 q^{86} - 58400 q^{88} - 27542 q^{89} + 151974 q^{91} + 8464 q^{92} - 146990 q^{94} + 20644 q^{95} - 273672 q^{97} + 401276 q^{98}+O(q^{100}) $ 3 * q + 4 * q^2 + 16 * q^4 + 58 * q^5 - 282 * q^7 + 360 * q^8 - 156 * q^10 - 136 * q^11 - 1116 * q^13 - 2016 * q^14 + 128 * q^16 + 896 * q^17 + 1654 * q^19 - 1504 * q^20 + 1352 * q^22 + 1587 * q^23 - 7347 * q^25 - 1998 * q^26 - 10264 * q^28 + 844 * q^29 - 3020 * q^31 - 6656 * q^32 - 11212 * q^34 - 1072 * q^35 + 8938 * q^37 - 10728 * q^38 + 3440 * q^40 + 12792 * q^41 - 16730 * q^43 - 5112 * q^44 + 2116 * q^46 - 22500 * q^47 + 2887 * q^49 - 15156 * q^50 - 47412 * q^52 - 17108 * q^53 - 436 * q^55 - 42640 * q^56 - 55678 * q^58 - 54176 * q^59 - 71324 * q^61 + 72710 * q^62 - 49984 * q^64 - 846 * q^65 - 62960 * q^67 + 8352 * q^68 + 9224 * q^70 - 98400 * q^71 - 81772 * q^73 + 59044 * q^74 + 31488 * q^76 + 304 * q^77 + 58224 * q^79 + 17568 * q^80 + 61926 * q^82 - 9892 * q^83 + 15536 * q^85 - 191140 * q^86 - 58400 * q^88 - 27542 * q^89 + 151974 * q^91 + 8464 * q^92 - 146990 * q^94 + 20644 * q^95 - 273672 * q^97 + 401276 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.164504	0.0290805	0.0145403	−	0.999894i	$-0.495372\pi$
$2$	0.164504	0.0290805	0.0145403	+	0.999894i	$0.495372\pi$
$3$	0	0
$3$	0	0
$4$	−31.9729	−0.999154
$5$	37.9865	0.679523	0.339761	−	0.940512i	$-0.389654\pi$
$5$	37.9865	0.679523	0.339761	+	0.940512i	$0.389654\pi$
$6$	0	0
$7$	−43.8366	−0.338136	−0.169068	−	0.985604i	$-0.554076\pi$
$7$	−43.8366	−0.338136	−0.169068	+	0.985604i	$0.554076\pi$
$8$	−10.5238	−0.0581365
$9$	0	0
$10$	6.24894	0.0197609
$11$	163.213	0.406700	0.203350	−	0.979106i	$-0.434817\pi$
$11$	163.213	0.406700	0.203350	+	0.979106i	$0.434817\pi$
$12$	0	0
$13$	430.356	0.706269	0.353134	−	0.935573i	$-0.385116\pi$
$13$	430.356	0.706269	0.353134	+	0.935573i	$0.385116\pi$
$14$	−7.21131	−0.00983318
$15$	0	0
$16$	1021.40	0.997464
$17$	−740.415	−0.621374	−0.310687	−	0.950512i	$-0.600559\pi$
$17$	−740.415	−0.621374	−0.310687	+	0.950512i	$0.600559\pi$
$18$	0	0
$19$	−916.578	−0.582486	−0.291243	−	0.956649i	$-0.594069\pi$
$19$	−916.578	−0.582486	−0.291243	+	0.956649i	$0.594069\pi$
$20$	−1214.54	−0.678948
$21$	0	0
$22$	26.8493	0.0118270
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−1682.03	−0.538249
$26$	70.7955	0.0205387
$27$	0	0
$28$	1401.59	0.337850
$29$	5112.96	1.12896	0.564478	−	0.825448i	$-0.309077\pi$
$29$	5112.96	1.12896	0.564478	+	0.825448i	$0.309077\pi$
$30$	0	0
$31$	−5702.60	−1.06578	−0.532891	−	0.846184i	$-0.678895\pi$
$31$	−5702.60	−1.06578	−0.532891	+	0.846184i	$0.678895\pi$
$32$	504.787	0.0871432
$33$	0	0
$34$	−121.802	−0.0180699
$35$	−1665.20	−0.229771
$36$	0	0
$37$	−10913.1	−1.31052	−0.655261	−	0.755402i	$-0.727441\pi$
$37$	−10913.1	−1.31052	−0.655261	+	0.755402i	$0.727441\pi$
$38$	−150.781	−0.0169390
$39$	0	0
$40$	−399.763	−0.0395050
$41$	−11092.6	−1.03056	−0.515282	−	0.857021i	$-0.672313\pi$
$41$	−11092.6	−1.03056	−0.515282	+	0.857021i	$0.672313\pi$
$42$	0	0
$43$	5528.76	0.455991	0.227996	−	0.973662i	$-0.426783\pi$
$43$	5528.76	0.455991	0.227996	+	0.973662i	$0.426783\pi$
$44$	−5218.41	−0.406356
$45$	0	0
$46$	87.0228	0.00606371
$47$	−14435.2	−0.953189	−0.476594	−	0.879123i	$-0.658129\pi$
$47$	−14435.2	−0.953189	−0.476594	+	0.879123i	$0.658129\pi$
$48$	0	0
$49$	−14885.4	−0.885664
$50$	−276.701	−0.0156526
$51$	0	0
$52$	−13759.8	−0.705671
$53$	−6852.79	−0.335102	−0.167551	−	0.985863i	$-0.553586\pi$
$53$	−6852.79	−0.335102	−0.167551	+	0.985863i	$0.553586\pi$
$54$	0	0
$55$	6199.90	0.276362
$56$	461.329	0.0196580
$57$	0	0
$58$	841.104	0.0328307
$59$	−4155.17	−0.155403	−0.0777014	−	0.996977i	$-0.524758\pi$
$59$	−4155.17	−0.155403	−0.0777014	+	0.996977i	$0.524758\pi$
$60$	0	0
$61$	−21911.4	−0.753955	−0.376978	−	0.926222i	$-0.623037\pi$
$61$	−21911.4	−0.753955	−0.376978	+	0.926222i	$0.623037\pi$
$62$	−938.102	−0.0309935
$63$	0	0
$64$	−32601.9	−0.994930
$65$	16347.7	0.479925
$66$	0	0
$67$	−15164.2	−0.412699	−0.206349	−	0.978478i	$-0.566158\pi$
$67$	−15164.2	−0.412699	−0.206349	+	0.978478i	$0.566158\pi$
$68$	23673.3	0.620849
$69$	0	0
$70$	−273.932	−0.00668187
$71$	−14031.4	−0.330334	−0.165167	−	0.986266i	$-0.552816\pi$
$71$	−14031.4	−0.330334	−0.165167	+	0.986266i	$0.552816\pi$
$72$	0	0
$73$	−23310.1	−0.511962	−0.255981	−	0.966682i	$-0.582398\pi$
$73$	−23310.1	−0.511962	−0.255981	+	0.966682i	$0.582398\pi$
$74$	−1795.25	−0.0381107
$75$	0	0
$76$	29305.7	0.581994
$77$	−7154.72	−0.137520
$78$	0	0
$79$	64276.8	1.15874	0.579371	−	0.815064i	$-0.303298\pi$
$79$	64276.8	1.15874	0.579371	+	0.815064i	$0.303298\pi$
$80$	38799.5	0.677799
$81$	0	0
$82$	−1824.78	−0.0299693
$83$	−114349.	−1.82195	−0.910974	−	0.412465i	$-0.864668\pi$
$83$	−114349.	−1.82195	−0.910974	+	0.412465i	$0.864668\pi$
$84$	0	0
$85$	−28125.8	−0.422238
$86$	909.504	0.0132605
$87$	0	0
$88$	−1717.63	−0.0236441
$89$	63420.7	0.848703	0.424351	−	0.905498i	$-0.360502\pi$
$89$	63420.7	0.848703	0.424351	+	0.905498i	$0.360502\pi$
$90$	0	0
$91$	−18865.4	−0.238815
$92$	−16913.7	−0.208338
$93$	0	0
$94$	−2374.66	−0.0277192
$95$	−34817.6	−0.395813
$96$	0	0
$97$	−91987.6	−0.992658	−0.496329	−	0.868134i	$-0.665319\pi$
$97$	−91987.6	−0.992658	−0.496329	+	0.868134i	$0.665319\pi$
$98$	−2448.70	−0.0257556
$99$	0	0
$100$	53779.4	0.537794
$101$	−122133.	−1.19132	−0.595660	−	0.803236i	$-0.703110\pi$
$101$	−122133.	−1.19132	−0.595660	+	0.803236i	$0.703110\pi$
$102$	0	0
$103$	−145517.	−1.35151	−0.675756	−	0.737125i	$-0.736183\pi$
$103$	−145517.	−1.35151	−0.675756	+	0.737125i	$0.736183\pi$
$104$	−4528.99	−0.0410600
$105$	0	0
$106$	−1127.31	−0.00974495
$107$	129578.	1.09414	0.547068	−	0.837088i	$-0.315744\pi$
$107$	129578.	1.09414	0.547068	+	0.837088i	$0.315744\pi$
$108$	0	0
$109$	−10434.1	−0.0841178	−0.0420589	−	0.999115i	$-0.513392\pi$
$109$	−10434.1	−0.0841178	−0.0420589	+	0.999115i	$0.513392\pi$
$110$	1019.91	0.00803674
$111$	0	0
$112$	−44774.8	−0.337279
$113$	253268.	1.86589	0.932943	−	0.360025i	$-0.117232\pi$
$113$	253268.	1.86589	0.932943	+	0.360025i	$0.117232\pi$
$114$	0	0
$115$	20094.8	0.141690
$116$	−163476.	−1.12800
$117$	0	0
$118$	−683.543	−0.00451919
$119$	32457.3	0.210109
$120$	0	0
$121$	−134412.	−0.834595
$122$	−3604.52	−0.0219254
$123$	0	0
$124$	182329.	1.06488
$125$	−182602.	−1.04527
$126$	0	0
$127$	235260.	1.29431	0.647156	−	0.762358i	$-0.275958\pi$
$127$	235260.	1.29431	0.647156	+	0.762358i	$0.275958\pi$
$128$	−21516.3	−0.116076
$129$	0	0
$130$	2689.27	0.0139565
$131$	−62921.5	−0.320347	−0.160174	−	0.987089i	$-0.551205\pi$
$131$	−62921.5	−0.320347	−0.160174	+	0.987089i	$0.551205\pi$
$132$	0	0
$133$	40179.7	0.196960
$134$	−2494.58	−0.0120015
$135$	0	0
$136$	7792.00	0.0361245
$137$	134562.	0.612521	0.306260	−	0.951948i	$-0.400922\pi$
$137$	134562.	0.612521	0.306260	+	0.951948i	$0.400922\pi$
$138$	0	0
$139$	287261.	1.26107	0.630535	−	0.776161i	$-0.282835\pi$
$139$	287261.	1.26107	0.630535	+	0.776161i	$0.282835\pi$
$140$	53241.3	0.229577
$141$	0	0
$142$	−2308.22	−0.00960629
$143$	70239.9	0.287239
$144$	0	0
$145$	194223.	0.767152
$146$	−3834.61	−0.0148881
$147$	0	0
$148$	348924.	1.30941
$149$	418105.	1.54284	0.771418	−	0.636329i	$-0.219548\pi$
$149$	418105.	1.54284	0.771418	+	0.636329i	$0.219548\pi$
$150$	0	0
$151$	486983.	1.73809	0.869043	−	0.494736i	$-0.164735\pi$
$151$	486983.	1.73809	0.869043	+	0.494736i	$0.164735\pi$
$152$	9645.91	0.0338637
$153$	0	0
$154$	−1176.98	−0.00399915
$155$	−216622.	−0.724223
$156$	0	0
$157$	−439288.	−1.42233	−0.711165	−	0.703026i	$-0.751832\pi$
$157$	−439288.	−1.42233	−0.711165	+	0.703026i	$0.751832\pi$
$158$	10573.8	0.0336968
$159$	0	0
$160$	19175.1	0.0592158
$161$	−23189.6	−0.0705063
$162$	0	0
$163$	−125475.	−0.369905	−0.184952	−	0.982747i	$-0.559213\pi$
$163$	−125475.	−0.369905	−0.184952	+	0.982747i	$0.559213\pi$
$164$	354664.	1.02969
$165$	0	0
$166$	−18810.8	−0.0529832
$167$	626996.	1.73970	0.869848	−	0.493319i	$-0.164217\pi$
$167$	626996.	1.73970	0.869848	+	0.493319i	$0.164217\pi$
$168$	0	0
$169$	−186086.	−0.501185
$170$	−4626.81	−0.0122789
$171$	0	0
$172$	−176771.	−0.455605
$173$	−502364.	−1.27615	−0.638077	−	0.769972i	$-0.720270\pi$
$173$	−502364.	−1.27615	−0.638077	+	0.769972i	$0.720270\pi$
$174$	0	0
$175$	73734.4	0.182002
$176$	166706.	0.405668
$177$	0	0
$178$	10433.0	0.0246807
$179$	131681.	0.307178	0.153589	−	0.988135i	$-0.450917\pi$
$179$	131681.	0.307178	0.153589	+	0.988135i	$0.450917\pi$
$180$	0	0
$181$	343209.	0.778686	0.389343	−	0.921093i	$-0.372702\pi$
$181$	343209.	0.778686	0.389343	+	0.921093i	$0.372702\pi$
$182$	−3103.43	−0.00694487
$183$	0	0
$184$	−5567.10	−0.0121223
$185$	−414551.	−0.890529
$186$	0	0
$187$	−120846.	−0.252713
$188$	461536.	0.952383
$189$	0	0
$190$	−5727.64	−0.0115104
$191$	−831093.	−1.64841	−0.824207	−	0.566289i	$-0.808379\pi$
$191$	−831093.	−1.64841	−0.824207	+	0.566289i	$0.808379\pi$
$192$	0	0
$193$	498764.	0.963832	0.481916	−	0.876217i	$-0.339941\pi$
$193$	498764.	0.963832	0.481916	+	0.876217i	$0.339941\pi$
$194$	−15132.3	−0.0288670
$195$	0	0
$196$	475928.	0.884915
$197$	−415236.	−0.762306	−0.381153	−	0.924512i	$-0.624473\pi$
$197$	−415236.	−0.762306	−0.381153	+	0.924512i	$0.624473\pi$
$198$	0	0
$199$	−969194.	−1.73491	−0.867457	−	0.497512i	$-0.834247\pi$
$199$	−969194.	−1.73491	−0.867457	+	0.497512i	$0.834247\pi$
$200$	17701.4	0.0312919
$201$	0	0
$202$	−20091.4	−0.0346442
$203$	−224135.	−0.381741
$204$	0	0
$205$	−421370.	−0.700291
$206$	−23938.1	−0.0393027
$207$	0	0
$208$	439567.	0.704477
$209$	−149598.	−0.236897
$210$	0	0
$211$	372669.	0.576259	0.288129	−	0.957592i	$-0.406967\pi$
$211$	372669.	0.576259	0.288129	+	0.957592i	$0.406967\pi$
$212$	219104.	0.334819
$213$	0	0
$214$	21316.1	0.0318181
$215$	210018.	0.309856
$216$	0	0
$217$	249983.	0.360380
$218$	−1716.45	−0.00244619
$219$	0	0
$220$	−198229.	−0.276128
$221$	−318643.	−0.438857
$222$	0	0
$223$	−668196.	−0.899791	−0.449895	−	0.893081i	$-0.648539\pi$
$223$	−668196.	−0.899791	−0.449895	+	0.893081i	$0.648539\pi$
$224$	−22128.2	−0.0294663
$225$	0	0
$226$	41663.7	0.0542609
$227$	1.01295e6	1.30473	0.652367	−	0.757903i	$-0.273776\pi$
$227$	1.01295e6	1.30473	0.652367	+	0.757903i	$0.273776\pi$
$228$	0	0
$229$	−342003.	−0.430964	−0.215482	−	0.976508i	$-0.569132\pi$
$229$	−342003.	−0.430964	−0.215482	+	0.976508i	$0.569132\pi$
$230$	3305.69	0.00412043
$231$	0	0
$232$	−53807.9	−0.0656335
$233$	792950.	0.956877	0.478438	−	0.878121i	$-0.341203\pi$
$233$	792950.	0.956877	0.478438	+	0.878121i	$0.341203\pi$
$234$	0	0
$235$	−548343.	−0.647713
$236$	132853.	0.155271
$237$	0	0
$238$	5339.36	0.00611008
$239$	−520763.	−0.589719	−0.294859	−	0.955541i	$-0.595273\pi$
$239$	−520763.	−0.589719	−0.294859	+	0.955541i	$0.595273\pi$
$240$	0	0
$241$	−867128.	−0.961702	−0.480851	−	0.876802i	$-0.659672\pi$
$241$	−867128.	−0.961702	−0.480851	+	0.876802i	$0.659672\pi$
$242$	−22111.4	−0.0242705
$243$	0	0
$244$	700572.	0.753317
$245$	−565442.	−0.601829
$246$	0	0
$247$	−394455.	−0.411392
$248$	60013.1	0.0619608
$249$	0	0
$250$	−30038.8	−0.0303971
$251$	590873.	0.591983	0.295992	−	0.955191i	$-0.404350\pi$
$251$	590873.	0.591983	0.295992	+	0.955191i	$0.404350\pi$
$252$	0	0
$253$	86339.8	0.0848027
$254$	38701.3	0.0376393
$255$	0	0
$256$	1.03972e6	0.991554
$257$	448650.	0.423716	0.211858	−	0.977300i	$-0.432049\pi$
$257$	448650.	0.423716	0.211858	+	0.977300i	$0.432049\pi$
$258$	0	0
$259$	478394.	0.443135
$260$	−522685.	−0.479520
$261$	0	0
$262$	−10350.9	−0.00931586
$263$	−434704.	−0.387529	−0.193765	−	0.981048i	$-0.562070\pi$
$263$	−434704.	−0.387529	−0.193765	+	0.981048i	$0.562070\pi$
$264$	0	0
$265$	−260313.	−0.227710
$266$	6609.73	0.00572769
$267$	0	0
$268$	484845.	0.412350
$269$	−873326.	−0.735861	−0.367930	−	0.929853i	$-0.619933\pi$
$269$	−873326.	−0.735861	−0.367930	+	0.929853i	$0.619933\pi$
$270$	0	0
$271$	−1.16572e6	−0.964208	−0.482104	−	0.876114i	$-0.660127\pi$
$271$	−1.16572e6	−0.964208	−0.482104	+	0.876114i	$0.660127\pi$
$272$	−756262.	−0.619798
$273$	0	0
$274$	22136.0	0.0178124
$275$	−274529.	−0.218906
$276$	0	0
$277$	−604025.	−0.472994	−0.236497	−	0.971632i	$-0.575999\pi$
$277$	−604025.	−0.472994	−0.236497	+	0.971632i	$0.575999\pi$
$278$	47255.6	0.0366726
$279$	0	0
$280$	17524.2	0.0133581
$281$	−1.72154e6	−1.30062	−0.650310	−	0.759669i	$-0.725361\pi$
$281$	−1.72154e6	−1.30062	−0.650310	+	0.759669i	$0.725361\pi$
$282$	0	0
$283$	−2.01557e6	−1.49600	−0.748000	−	0.663699i	$-0.768986\pi$
$283$	−2.01557e6	−1.49600	−0.748000	+	0.663699i	$0.768986\pi$
$284$	448624.	0.330055
$285$	0	0
$286$	11554.8	0.00835306
$287$	486263.	0.348471
$288$	0	0
$289$	−871642.	−0.613894
$290$	31950.6	0.0223092
$291$	0	0
$292$	745293.	0.511529
$293$	−1.45451e6	−0.989798	−0.494899	−	0.868951i	$-0.664795\pi$
$293$	−1.45451e6	−0.989798	−0.494899	+	0.868951i	$0.664795\pi$
$294$	0	0
$295$	−157840.	−0.105600
$296$	114848.	0.0761891
$297$	0	0
$298$	68780.1	0.0448665
$299$	227659.	0.147267
$300$	0	0
$301$	−242362.	−0.154187
$302$	80110.8	0.0505445
$303$	0	0
$304$	−936196.	−0.581009
$305$	−832337.	−0.512329
$306$	0	0
$307$	272463.	0.164991	0.0824957	−	0.996591i	$-0.473711\pi$
$307$	272463.	0.164991	0.0824957	+	0.996591i	$0.473711\pi$
$308$	228757.	0.137404
$309$	0	0
$310$	−35635.2	−0.0210608
$311$	2.33041e6	1.36625	0.683126	−	0.730301i	$-0.260620\pi$
$311$	2.33041e6	1.36625	0.683126	+	0.730301i	$0.260620\pi$
$312$	0	0
$313$	−1.67227e6	−0.964818	−0.482409	−	0.875946i	$-0.660238\pi$
$313$	−1.67227e6	−0.964818	−0.482409	+	0.875946i	$0.660238\pi$
$314$	−72264.7	−0.0413621
$315$	0	0
$316$	−2.05512e6	−1.15776
$317$	−1.18782e6	−0.663901	−0.331950	−	0.943297i	$-0.607707\pi$
$317$	−1.18782e6	−0.663901	−0.331950	+	0.943297i	$0.607707\pi$
$318$	0	0
$319$	834503.	0.459146
$320$	−1.23843e6	−0.676077
$321$	0	0
$322$	−3814.78	−0.00205036
$323$	678649.	0.361942
$324$	0	0
$325$	−723872.	−0.380148
$326$	−20641.3	−0.0107570
$327$	0	0
$328$	116737.	0.0599133
$329$	632791.	0.322308
$330$	0	0
$331$	−2.09261e6	−1.04983	−0.524915	−	0.851155i	$-0.675903\pi$
$331$	−2.09261e6	−1.04983	−0.524915	+	0.851155i	$0.675903\pi$
$332$	3.65606e6	1.82041
$333$	0	0
$334$	103144.	0.0505913
$335$	−576035.	−0.280438
$336$	0	0
$337$	2.13994e6	1.02642	0.513211	−	0.858262i	$-0.328456\pi$
$337$	2.13994e6	1.02642	0.513211	+	0.858262i	$0.328456\pi$
$338$	−30612.0	−0.0145747
$339$	0	0
$340$	899263.	0.421881
$341$	−930740.	−0.433453
$342$	0	0
$343$	1.38929e6	0.637611
$344$	−58183.6	−0.0265097
$345$	0	0
$346$	−82641.0	−0.0371112
$347$	−3.78048e6	−1.68548	−0.842739	−	0.538322i	$-0.819059\pi$
$347$	−3.78048e6	−1.68548	−0.842739	+	0.538322i	$0.819059\pi$
$348$	0	0
$349$	3.46473e6	1.52267	0.761334	−	0.648360i	$-0.224545\pi$
$349$	3.46473e6	1.52267	0.761334	+	0.648360i	$0.224545\pi$
$350$	12129.6	0.00529270
$351$	0	0
$352$	82388.0	0.0354411
$353$	2.39834e6	1.02441	0.512206	−	0.858863i	$-0.328829\pi$
$353$	2.39834e6	1.02441	0.512206	+	0.858863i	$0.328829\pi$
$354$	0	0
$355$	−533002.	−0.224470
$356$	−2.02774e6	−0.847985
$357$	0	0
$358$	21662.0	0.00893289
$359$	2.04171e6	0.836098	0.418049	−	0.908424i	$-0.362714\pi$
$359$	2.04171e6	0.836098	0.418049	+	0.908424i	$0.362714\pi$
$360$	0	0
$361$	−1.63598e6	−0.660710
$362$	56459.3	0.0226446
$363$	0	0
$364$	603181.	0.238613
$365$	−885469.	−0.347889
$366$	0	0
$367$	2.41363e6	0.935417	0.467709	−	0.883883i	$-0.345080\pi$
$367$	2.41363e6	0.935417	0.467709	+	0.883883i	$0.345080\pi$
$368$	540322.	0.207986
$369$	0	0
$370$	−68195.4	−0.0258971
$371$	300403.	0.113310
$372$	0	0
$373$	1.83073e6	0.681321	0.340660	−	0.940186i	$-0.389349\pi$
$373$	1.83073e6	0.681321	0.340660	+	0.940186i	$0.389349\pi$
$374$	−19879.6	−0.00734901
$375$	0	0
$376$	151914.	0.0554150
$377$	2.20039e6	0.797347
$378$	0	0
$379$	5.24532e6	1.87575	0.937874	−	0.346977i	$-0.112792\pi$
$379$	5.24532e6	1.87575	0.937874	+	0.346977i	$0.112792\pi$
$380$	1.11322e6	0.395478
$381$	0	0
$382$	−136718.	−0.0479367
$383$	754501.	0.262823	0.131411	−	0.991328i	$-0.458049\pi$
$383$	754501.	0.262823	0.131411	+	0.991328i	$0.458049\pi$
$384$	0	0
$385$	−271782.	−0.0934479
$386$	82048.7	0.0280287
$387$	0	0
$388$	2.94111e6	0.991819
$389$	−2.05221e6	−0.687618	−0.343809	−	0.939040i	$-0.611717\pi$
$389$	−2.05221e6	−0.687618	−0.343809	+	0.939040i	$0.611717\pi$
$390$	0	0
$391$	−391680.	−0.129565
$392$	156651.	0.0514894
$393$	0	0
$394$	−68308.0	−0.0221682
$395$	2.44165e6	0.787391
$396$	0	0
$397$	3.99917e6	1.27348	0.636742	−	0.771077i	$-0.280281\pi$
$397$	3.99917e6	1.27348	0.636742	+	0.771077i	$0.280281\pi$
$398$	−159437.	−0.0504522
$399$	0	0
$400$	−1.71803e6	−0.536884
$401$	5.10734e6	1.58611	0.793056	−	0.609149i	$-0.208489\pi$
$401$	5.10734e6	1.58611	0.793056	+	0.609149i	$0.208489\pi$
$402$	0	0
$403$	−2.45415e6	−0.752729
$404$	3.90494e6	1.19031
$405$	0	0
$406$	−36871.1	−0.0111012
$407$	−1.78116e6	−0.532989
$408$	0	0
$409$	−3.06823e6	−0.906942	−0.453471	−	0.891271i	$-0.649814\pi$
$409$	−3.06823e6	−0.906942	−0.453471	+	0.891271i	$0.649814\pi$
$410$	−69317.1	−0.0203648
$411$	0	0
$412$	4.65260e6	1.35037
$413$	182149.	0.0525473
$414$	0	0
$415$	−4.34370e6	−1.23805
$416$	217239.	0.0615465
$417$	0	0
$418$	−24609.5	−0.00688908
$419$	−3.33601e6	−0.928308	−0.464154	−	0.885754i	$-0.653642\pi$
$419$	−3.33601e6	−0.928308	−0.464154	+	0.885754i	$0.653642\pi$
$420$	0	0
$421$	−2.75080e6	−0.756404	−0.378202	−	0.925723i	$-0.623458\pi$
$421$	−2.75080e6	−0.756404	−0.378202	+	0.925723i	$0.623458\pi$
$422$	61305.7	0.0167579
$423$	0	0
$424$	72117.5	0.0194817
$425$	1.24540e6	0.334454
$426$	0	0
$427$	960521.	0.254940
$428$	−4.14299e6	−1.09321
$429$	0	0
$430$	34548.8	0.00901078
$431$	−926380.	−0.240213	−0.120106	−	0.992761i	$-0.538324\pi$
$431$	−926380.	−0.240213	−0.120106	+	0.992761i	$0.538324\pi$
$432$	0	0
$433$	680820.	0.174507	0.0872535	−	0.996186i	$-0.472191\pi$
$433$	680820.	0.174507	0.0872535	+	0.996186i	$0.472191\pi$
$434$	41123.2	0.0104800
$435$	0	0
$436$	333608.	0.0840467
$437$	−484870.	−0.121457
$438$	0	0
$439$	2.17406e6	0.538406	0.269203	−	0.963083i	$-0.413240\pi$
$439$	2.17406e6	0.538406	0.269203	+	0.963083i	$0.413240\pi$
$440$	−65246.6	−0.0160667
$441$	0	0
$442$	−52418.1	−0.0127622
$443$	−4.17517e6	−1.01080	−0.505400	−	0.862885i	$-0.668655\pi$
$443$	−4.17517e6	−1.01080	−0.505400	+	0.862885i	$0.668655\pi$
$444$	0	0
$445$	2.40913e6	0.576713
$446$	−109921.	−0.0261664
$447$	0	0
$448$	1.42915e6	0.336422
$449$	4.68946e6	1.09776	0.548880	−	0.835901i	$-0.315055\pi$
$449$	4.68946e6	1.09776	0.548880	+	0.835901i	$0.315055\pi$
$450$	0	0
$451$	−1.81046e6	−0.419130
$452$	−8.09774e6	−1.86431
$453$	0	0
$454$	166634.	0.0379424
$455$	−716629.	−0.162280
$456$	0	0
$457$	−1.68983e6	−0.378487	−0.189244	−	0.981930i	$-0.560604\pi$
$457$	−1.68983e6	−0.378487	−0.189244	+	0.981930i	$0.560604\pi$
$458$	−56260.9	−0.0125326
$459$	0	0
$460$	−642491.	−0.141570
$461$	4.28765e6	0.939652	0.469826	−	0.882759i	$-0.344317\pi$
$461$	4.28765e6	0.939652	0.469826	+	0.882759i	$0.344317\pi$
$462$	0	0
$463$	5.26738e6	1.14194	0.570969	−	0.820972i	$-0.306568\pi$
$463$	5.26738e6	1.14194	0.570969	+	0.820972i	$0.306568\pi$
$464$	5.22239e6	1.12609
$465$	0	0
$466$	130444.	0.0278265
$467$	3.66305e6	0.777231	0.388616	−	0.921400i	$-0.372953\pi$
$467$	3.66305e6	0.777231	0.388616	+	0.921400i	$0.372953\pi$
$468$	0	0
$469$	664748.	0.139548
$470$	−90204.8	−0.0188358
$471$	0	0
$472$	43728.3	0.00903457
$473$	902366.	0.185451
$474$	0	0
$475$	1.54171e6	0.313523
$476$	−1.03776e6	−0.209931
$477$	0	0
$478$	−85667.7	−0.0171493
$479$	−7.03353e6	−1.40067	−0.700333	−	0.713816i	$-0.746965\pi$
$479$	−7.03353e6	−1.40067	−0.700333	+	0.713816i	$0.746965\pi$
$480$	0	0
$481$	−4.69653e6	−0.925581
$482$	−142646.	−0.0279668
$483$	0	0
$484$	4.29756e6	0.833890
$485$	−3.49428e6	−0.674534
$486$	0	0
$487$	5.05170e6	0.965195	0.482598	−	0.875842i	$-0.339693\pi$
$487$	5.05170e6	0.965195	0.482598	+	0.875842i	$0.339693\pi$
$488$	230592.	0.0438323
$489$	0	0
$490$	−93017.6	−0.0175015
$491$	3.90292e6	0.730610	0.365305	−	0.930888i	$-0.380965\pi$
$491$	3.90292e6	0.730610	0.365305	+	0.930888i	$0.380965\pi$
$492$	0	0
$493$	−3.78571e6	−0.701505
$494$	−64889.6	−0.0119635
$495$	0	0
$496$	−5.82465e6	−1.06308
$497$	615087.	0.111698
$498$	0	0
$499$	1.13818e6	0.204625	0.102312	−	0.994752i	$-0.467376\pi$
$499$	1.13818e6	0.204625	0.102312	+	0.994752i	$0.467376\pi$
$500$	5.83832e6	1.04439
$501$	0	0
$502$	97201.1	0.0172152
$503$	7.48030e6	1.31825	0.659127	−	0.752032i	$-0.270926\pi$
$503$	7.48030e6	1.31825	0.659127	+	0.752032i	$0.270926\pi$
$504$	0	0
$505$	−4.63939e6	−0.809529
$506$	14203.3	0.00246611
$507$	0	0
$508$	−7.52195e6	−1.29322
$509$	−3.03026e6	−0.518424	−0.259212	−	0.965821i	$-0.583463\pi$
$509$	−3.03026e6	−0.518424	−0.259212	+	0.965821i	$0.583463\pi$
$510$	0	0
$511$	1.02184e6	0.173113
$512$	859561.	0.144911
$513$	0	0
$514$	73804.8	0.0123219
$515$	−5.52767e6	−0.918383
$516$	0	0
$517$	−2.35602e6	−0.387661
$518$	78697.8	0.0128866
$519$	0	0
$520$	−172040.	−0.0279012
$521$	1.90771e6	0.307905	0.153953	−	0.988078i	$-0.450800\pi$
$521$	1.90771e6	0.307905	0.153953	+	0.988078i	$0.450800\pi$
$522$	0	0
$523$	2.40656e6	0.384718	0.192359	−	0.981325i	$-0.438386\pi$
$523$	2.40656e6	0.384718	0.192359	+	0.981325i	$0.438386\pi$
$524$	2.01178e6	0.320076
$525$	0	0
$526$	−71510.7	−0.0112696
$527$	4.22229e6	0.662250
$528$	0	0
$529$	279841.	0.0434783
$530$	−42822.6	−0.00662192
$531$	0	0
$532$	−1.28466e6	−0.196793
$533$	−4.77378e6	−0.727855
$534$	0	0
$535$	4.92221e6	0.743490
$536$	159586.	0.0239928
$537$	0	0
$538$	−143666.	−0.0213992
$539$	−2.42949e6	−0.360199
$540$	0	0
$541$	−711055.	−0.104450	−0.0522252	−	0.998635i	$-0.516631\pi$
$541$	−711055.	−0.104450	−0.0522252	+	0.998635i	$0.516631\pi$
$542$	−191766.	−0.0280397
$543$	0	0
$544$	−373752.	−0.0541485
$545$	−396354.	−0.0571600
$546$	0	0
$547$	7.21528e6	1.03106	0.515531	−	0.856871i	$-0.327595\pi$
$547$	7.21528e6	1.03106	0.515531	+	0.856871i	$0.327595\pi$
$548$	−4.30234e6	−0.612003
$549$	0	0
$550$	−45161.2	−0.00636589
$551$	−4.68643e6	−0.657602
$552$	0	0
$553$	−2.81768e6	−0.391813
$554$	−99364.7	−0.0137549
$555$	0	0
$556$	−9.18457e6	−1.26000
$557$	920675.	0.125739	0.0628693	−	0.998022i	$-0.479975\pi$
$557$	920675.	0.125739	0.0628693	+	0.998022i	$0.479975\pi$
$558$	0	0
$559$	2.37934e6	0.322052
$560$	−1.70084e6	−0.229189
$561$	0	0
$562$	−283200.	−0.0378227
$563$	7.13949e6	0.949284	0.474642	−	0.880179i	$-0.342578\pi$
$563$	7.13949e6	0.949284	0.474642	+	0.880179i	$0.342578\pi$
$564$	0	0
$565$	9.62077e6	1.26791
$566$	−331570.	−0.0435045
$567$	0	0
$568$	147663.	0.0192045
$569$	−1.32562e7	−1.71648	−0.858239	−	0.513250i	$-0.828441\pi$
$569$	−1.32562e7	−1.71648	−0.858239	+	0.513250i	$0.828441\pi$
$570$	0	0
$571$	−8.74927e6	−1.12300	−0.561502	−	0.827475i	$-0.689776\pi$
$571$	−8.74927e6	−1.12300	−0.561502	+	0.827475i	$0.689776\pi$
$572$	−2.24578e6	−0.286996
$573$	0	0
$574$	79992.4	0.0101337
$575$	−889793.	−0.112233
$576$	0	0
$577$	2.28815e6	0.286118	0.143059	−	0.989714i	$-0.454306\pi$
$577$	2.28815e6	0.286118	0.143059	+	0.989714i	$0.454306\pi$
$578$	−143389.	−0.0178524
$579$	0	0
$580$	−6.20989e6	−0.766503
$581$	5.01266e6	0.616067
$582$	0	0
$583$	−1.11847e6	−0.136286
$584$	245312.	0.0297636
$585$	0	0
$586$	−239272.	−0.0287838
$587$	−1.48384e7	−1.77743	−0.888714	−	0.458463i	$-0.848400\pi$
$587$	−1.48384e7	−1.77743	−0.888714	+	0.458463i	$0.848400\pi$
$588$	0	0
$589$	5.22688e6	0.620804
$590$	−25965.4	−0.00307090
$591$	0	0
$592$	−1.11467e7	−1.30720
$593$	7.67836e6	0.896668	0.448334	−	0.893866i	$-0.352018\pi$
$593$	7.67836e6	0.896668	0.448334	+	0.893866i	$0.352018\pi$
$594$	0	0
$595$	1.23294e6	0.142774
$596$	−1.33680e7	−1.54153
$597$	0	0
$598$	37450.8	0.00428261
$599$	599539.	0.0682733	0.0341366	−	0.999417i	$-0.489132\pi$
$599$	599539.	0.0682733	0.0341366	+	0.999417i	$0.489132\pi$
$600$	0	0
$601$	−6.92017e6	−0.781503	−0.390751	−	0.920496i	$-0.627785\pi$
$601$	−6.92017e6	−0.781503	−0.390751	+	0.920496i	$0.627785\pi$
$602$	−39869.6	−0.00448384
$603$	0	0
$604$	−1.55703e7	−1.73662
$605$	−5.10585e6	−0.567126
$606$	0	0
$607$	−6.65094e6	−0.732675	−0.366337	−	0.930482i	$-0.619388\pi$
$607$	−6.65094e6	−0.732675	−0.366337	+	0.930482i	$0.619388\pi$
$608$	−462677.	−0.0507597
$609$	0	0
$610$	−136923.	−0.0148988
$611$	−6.21229e6	−0.673207
$612$	0	0
$613$	−1.07456e7	−1.15499	−0.577497	−	0.816393i	$-0.695970\pi$
$613$	−1.07456e7	−1.15499	−0.577497	+	0.816393i	$0.695970\pi$
$614$	44821.3	0.00479804
$615$	0	0
$616$	75295.0	0.00799492
$617$	1.66965e7	1.76568	0.882841	−	0.469673i	$-0.155628\pi$
$617$	1.66965e7	1.76568	0.882841	+	0.469673i	$0.155628\pi$
$618$	0	0
$619$	−1.06977e7	−1.12218	−0.561090	−	0.827755i	$-0.689618\pi$
$619$	−1.06977e7	−1.12218	−0.561090	+	0.827755i	$0.689618\pi$
$620$	6.92603e6	0.723611
$621$	0	0
$622$	383362.	0.0397313
$623$	−2.78015e6	−0.286977
$624$	0	0
$625$	−1.68007e6	−0.172039
$626$	−275095.	−0.0280574
$627$	0	0
$628$	1.40453e7	1.42113
$629$	8.08024e6	0.814325
$630$	0	0
$631$	1.31871e7	1.31849	0.659243	−	0.751930i	$-0.270877\pi$
$631$	1.31871e7	1.31849	0.659243	+	0.751930i	$0.270877\pi$
$632$	−676438.	−0.0673651
$633$	0	0
$634$	−195402.	−0.0193066
$635$	8.93670e6	0.879514
$636$	0	0
$637$	−6.40601e6	−0.625517
$638$	137279.	0.0133522
$639$	0	0
$640$	−817330.	−0.0788765
$641$	849150.	0.0816280	0.0408140	−	0.999167i	$-0.487005\pi$
$641$	849150.	0.0816280	0.0408140	+	0.999167i	$0.487005\pi$
$642$	0	0
$643$	−2.31164e6	−0.220491	−0.110246	−	0.993904i	$-0.535164\pi$
$643$	−2.31164e6	−0.220491	−0.110246	+	0.993904i	$0.535164\pi$
$644$	741439.	0.0704467
$645$	0	0
$646$	111641.	0.0105255
$647$	1.51827e7	1.42590	0.712949	−	0.701216i	$-0.247359\pi$
$647$	1.51827e7	1.42590	0.712949	+	0.701216i	$0.247359\pi$
$648$	0	0
$649$	−678179.	−0.0632023
$650$	−119080.	−0.0110549
$651$	0	0
$652$	4.01182e6	0.369592
$653$	−1.57201e7	−1.44269	−0.721343	−	0.692578i	$-0.756475\pi$
$653$	−1.57201e7	−1.44269	−0.721343	+	0.692578i	$0.756475\pi$
$654$	0	0
$655$	−2.39016e6	−0.217683
$656$	−1.13300e7	−1.02795
$657$	0	0
$658$	104097.	0.00937288
$659$	1.71369e7	1.53716	0.768580	−	0.639754i	$-0.220964\pi$
$659$	1.71369e7	1.53716	0.768580	+	0.639754i	$0.220964\pi$
$660$	0	0
$661$	1.52598e7	1.35845	0.679227	−	0.733928i	$-0.262315\pi$
$661$	1.52598e7	1.35845	0.679227	+	0.733928i	$0.262315\pi$
$662$	−344244.	−0.0305296
$663$	0	0
$664$	1.20338e6	0.105922
$665$	1.52628e6	0.133839
$666$	0	0
$667$	2.70476e6	0.235404
$668$	−2.00469e7	−1.73823
$669$	0	0
$670$	−94760.3	−0.00815529
$671$	−3.57623e6	−0.306633
$672$	0	0
$673$	−1.35466e7	−1.15290	−0.576451	−	0.817132i	$-0.695563\pi$
$673$	−1.35466e7	−1.15290	−0.576451	+	0.817132i	$0.695563\pi$
$674$	352029.	0.0298489
$675$	0	0
$676$	5.94973e6	0.500761
$677$	−1.13711e7	−0.953520	−0.476760	−	0.879034i	$-0.658189\pi$
$677$	−1.13711e7	−0.953520	−0.476760	+	0.879034i	$0.658189\pi$
$678$	0	0
$679$	4.03242e6	0.335654
$680$	295991.	0.0245474
$681$	0	0
$682$	−153111.	−0.0126050
$683$	498823.	0.0409161	0.0204581	−	0.999791i	$-0.493488\pi$
$683$	498823.	0.0409161	0.0204581	+	0.999791i	$0.493488\pi$
$684$	0	0
$685$	5.11153e6	0.416222
$686$	228543.	0.0185421
$687$	0	0
$688$	5.64709e6	0.454834
$689$	−2.94914e6	−0.236672
$690$	0	0
$691$	1.03020e7	0.820783	0.410392	−	0.911909i	$-0.365392\pi$
$691$	1.03020e7	0.820783	0.410392	+	0.911909i	$0.365392\pi$
$692$	1.60621e7	1.27508
$693$	0	0
$694$	−621905.	−0.0490146
$695$	1.09120e7	0.856926
$696$	0	0
$697$	8.21315e6	0.640366
$698$	569962.	0.0442800
$699$	0	0
$700$	−2.35751e6	−0.181848
$701$	3.29623e6	0.253351	0.126675	−	0.991944i	$-0.459569\pi$
$701$	3.29623e6	0.253351	0.126675	+	0.991944i	$0.459569\pi$
$702$	0	0
$703$	1.00027e7	0.763361
$704$	−5.32105e6	−0.404637
$705$	0	0
$706$	394537.	0.0297904
$707$	5.35388e6	0.402829
$708$	0	0
$709$	1.35819e7	1.01472	0.507359	−	0.861735i	$-0.330622\pi$
$709$	1.35819e7	1.01472	0.507359	+	0.861735i	$0.330622\pi$
$710$	−87681.0	−0.00652769
$711$	0	0
$712$	−667428.	−0.0493406
$713$	−3.01667e6	−0.222231
$714$	0	0
$715$	2.66816e6	0.195185
$716$	−4.21022e6	−0.306918
$717$	0	0
$718$	335870.	0.0243142
$719$	8.30096e6	0.598834	0.299417	−	0.954122i	$-0.403208\pi$
$719$	8.30096e6	0.598834	0.299417	+	0.954122i	$0.403208\pi$
$720$	0	0
$721$	6.37896e6	0.456995
$722$	−269126.	−0.0192138
$723$	0	0
$724$	−1.09734e7	−0.778027
$725$	−8.60014e6	−0.607660
$726$	0	0
$727$	−5.01347e6	−0.351806	−0.175903	−	0.984408i	$-0.556284\pi$
$727$	−5.01347e6	−0.351806	−0.175903	+	0.984408i	$0.556284\pi$
$728$	198536.	0.0138839
$729$	0	0
$730$	−145663.	−0.0101168
$731$	−4.09358e6	−0.283341
$732$	0	0
$733$	−1.48950e7	−1.02395	−0.511977	−	0.858999i	$-0.671087\pi$
$733$	−1.48950e7	−1.02395	−0.511977	+	0.858999i	$0.671087\pi$
$734$	397052.	0.0272024
$735$	0	0
$736$	267033.	0.0181706
$737$	−2.47500e6	−0.167844
$738$	0	0
$739$	8.39705e6	0.565608	0.282804	−	0.959178i	$-0.408735\pi$
$739$	8.39705e6	0.565608	0.282804	+	0.959178i	$0.408735\pi$
$740$	1.32544e7	0.889776
$741$	0	0
$742$	49417.6	0.00329512
$743$	1.54362e7	1.02582	0.512908	−	0.858444i	$-0.328568\pi$
$743$	1.54362e7	1.02582	0.512908	+	0.858444i	$0.328568\pi$
$744$	0	0
$745$	1.58823e7	1.04839
$746$	301163.	0.0198132
$747$	0	0
$748$	3.86379e6	0.252499
$749$	−5.68025e6	−0.369967
$750$	0	0
$751$	−5.26274e6	−0.340496	−0.170248	−	0.985401i	$-0.554457\pi$
$751$	−5.26274e6	−0.340496	−0.170248	+	0.985401i	$0.554457\pi$
$752$	−1.47442e7	−0.950771
$753$	0	0
$754$	361974.	0.0231873
$755$	1.84988e7	1.18107
$756$	0	0
$757$	−2.58188e7	−1.63756	−0.818779	−	0.574109i	$-0.805348\pi$
$757$	−2.58188e7	−1.63756	−0.818779	+	0.574109i	$0.805348\pi$
$758$	862878.	0.0545477
$759$	0	0
$760$	366414.	0.0230111
$761$	−4.74356e6	−0.296922	−0.148461	−	0.988918i	$-0.547432\pi$
$761$	−4.74356e6	−0.296922	−0.148461	+	0.988918i	$0.547432\pi$
$762$	0	0
$763$	457395.	0.0284433
$764$	2.65725e7	1.64702
$765$	0	0
$766$	124119.	0.00764302
$767$	−1.78820e6	−0.109756
$768$	0	0
$769$	2.16055e7	1.31749	0.658746	−	0.752365i	$-0.271087\pi$
$769$	2.16055e7	1.31749	0.658746	+	0.752365i	$0.271087\pi$
$770$	−44709.4	−0.00271751
$771$	0	0
$772$	−1.59469e7	−0.963017
$773$	−4.60087e6	−0.276944	−0.138472	−	0.990366i	$-0.544219\pi$
$773$	−4.60087e6	−0.276944	−0.138472	+	0.990366i	$0.544219\pi$
$774$	0	0
$775$	9.59193e6	0.573656
$776$	968061.	0.0577096
$777$	0	0
$778$	−337597.	−0.0199963
$779$	1.01673e7	0.600289
$780$	0	0
$781$	−2.29010e6	−0.134347
$782$	−64433.0	−0.00376783
$783$	0	0
$784$	−1.52039e7	−0.883417
$785$	−1.66870e7	−0.966505
$786$	0	0
$787$	3.20780e7	1.84616	0.923082	−	0.384603i	$-0.125662\pi$
$787$	3.20780e7	1.84616	0.923082	+	0.384603i	$0.125662\pi$
$788$	1.32763e7	0.761661
$789$	0	0
$790$	401662.	0.0228977
$791$	−1.11024e7	−0.630924
$792$	0	0
$793$	−9.42971e6	−0.532495
$794$	657881.	0.0370336
$795$	0	0
$796$	3.09880e7	1.73345
$797$	−950693.	−0.0530145	−0.0265072	−	0.999649i	$-0.508439\pi$
$797$	−950693.	−0.0530145	−0.0265072	+	0.999649i	$0.508439\pi$
$798$	0	0
$799$	1.06881e7	0.592287
$800$	−849067.	−0.0469047
$801$	0	0
$802$	840179.	0.0461250
$803$	−3.80452e6	−0.208215
$804$	0	0
$805$	−880890.	−0.0479106
$806$	−403718.	−0.0218897
$807$	0	0
$808$	1.28530e6	0.0692592
$809$	−1.89226e7	−1.01650	−0.508252	−	0.861208i	$-0.669708\pi$
$809$	−1.89226e7	−1.01650	−0.508252	+	0.861208i	$0.669708\pi$
$810$	0	0
$811$	4.78517e6	0.255473	0.127737	−	0.991808i	$-0.459229\pi$
$811$	4.78517e6	0.255473	0.127737	+	0.991808i	$0.459229\pi$
$812$	7.16625e6	0.381419
$813$	0	0
$814$	−293009.	−0.0154996
$815$	−4.76637e6	−0.251359
$816$	0	0
$817$	−5.06754e6	−0.265608
$818$	−504737.	−0.0263744
$819$	0	0
$820$	1.34724e7	0.699699
$821$	−5.51231e6	−0.285414	−0.142707	−	0.989765i	$-0.545581\pi$
$821$	−5.51231e6	−0.285414	−0.142707	+	0.989765i	$0.545581\pi$
$822$	0	0
$823$	−1.44617e7	−0.744249	−0.372125	−	0.928183i	$-0.621371\pi$
$823$	−1.44617e7	−0.744249	−0.372125	+	0.928183i	$0.621371\pi$
$824$	1.53139e6	0.0785721
$825$	0	0
$826$	29964.2	0.00152810
$827$	2.78469e7	1.41584	0.707919	−	0.706294i	$-0.249634\pi$
$827$	2.78469e7	1.41584	0.707919	+	0.706294i	$0.249634\pi$
$828$	0	0
$829$	1.89279e6	0.0956569	0.0478285	−	0.998856i	$-0.484770\pi$
$829$	1.89279e6	0.0956569	0.0478285	+	0.998856i	$0.484770\pi$
$830$	−714557.	−0.0360033
$831$	0	0
$832$	−1.40304e7	−0.702687
$833$	1.10213e7	0.550329
$834$	0	0
$835$	2.38174e7	1.18216
$836$	4.78308e6	0.236697
$837$	0	0
$838$	−548788.	−0.0269957
$839$	−8.81372e6	−0.432269	−0.216135	−	0.976364i	$-0.569345\pi$
$839$	−8.81372e6	−0.432269	−0.216135	+	0.976364i	$0.569345\pi$
$840$	0	0
$841$	5.63121e6	0.274544
$842$	−452518.	−0.0219966
$843$	0	0
$844$	−1.19153e7	−0.575771
$845$	−7.06876e6	−0.340566
$846$	0	0
$847$	5.89219e6	0.282207
$848$	−6.99946e6	−0.334253
$849$	0	0
$850$	204874.	0.00972610
$851$	−5.77304e6	−0.273263
$852$	0	0
$853$	4.21641e6	0.198413	0.0992064	−	0.995067i	$-0.468370\pi$
$853$	4.21641e6	0.198413	0.0992064	+	0.995067i	$0.468370\pi$
$854$	158010.	0.00741378
$855$	0	0
$856$	−1.36365e6	−0.0636092
$857$	−1.36923e7	−0.636829	−0.318415	−	0.947952i	$-0.603150\pi$
$857$	−1.36923e7	−0.636829	−0.318415	+	0.947952i	$0.603150\pi$
$858$	0	0
$859$	1.31118e7	0.606291	0.303145	−	0.952944i	$-0.401963\pi$
$859$	1.31118e7	0.606291	0.303145	+	0.952944i	$0.401963\pi$
$860$	−6.71489e6	−0.309594
$861$	0	0
$862$	−152393.	−0.00698551
$863$	−5.67126e6	−0.259210	−0.129605	−	0.991566i	$-0.541371\pi$
$863$	−5.67126e6	−0.259210	−0.129605	+	0.991566i	$0.541371\pi$
$864$	0	0
$865$	−1.90830e7	−0.867176
$866$	111998.	0.00507475
$867$	0	0
$868$	−7.99268e6	−0.360075
$869$	1.04908e7	0.471260
$870$	0	0
$871$	−6.52602e6	−0.291476
$872$	109806.	0.00489031
$873$	0	0
$874$	−79763.2	−0.00353203
$875$	8.00465e6	0.353445
$876$	0	0
$877$	−2.51970e7	−1.10624	−0.553120	−	0.833101i	$-0.686563\pi$
$877$	−2.51970e7	−1.10624	−0.553120	+	0.833101i	$0.686563\pi$
$878$	357642.	0.0156571
$879$	0	0
$880$	6.33259e6	0.275661
$881$	2.99532e7	1.30018	0.650090	−	0.759857i	$-0.274731\pi$
$881$	2.99532e7	1.30018	0.650090	+	0.759857i	$0.274731\pi$
$882$	0	0
$883$	−2.98802e7	−1.28968	−0.644840	−	0.764318i	$-0.723076\pi$
$883$	−2.98802e7	−1.28968	−0.644840	+	0.764318i	$0.723076\pi$
$884$	1.01879e7	0.438486
$885$	0	0
$886$	−686834.	−0.0293946
$887$	7.65547e6	0.326710	0.163355	−	0.986567i	$-0.447768\pi$
$887$	7.65547e6	0.326710	0.163355	+	0.986567i	$0.447768\pi$
$888$	0	0
$889$	−1.03130e7	−0.437654
$890$	396312.	0.0167711
$891$	0	0
$892$	2.13642e7	0.899030
$893$	1.32310e7	0.555219
$894$	0	0
$895$	5.00208e6	0.208734
$896$	943203.	0.0392496
$897$	0	0
$898$	771436.	0.0319234
$899$	−2.91572e7	−1.20322
$900$	0	0
$901$	5.07391e6	0.208224
$902$	−297829.	−0.0121885
$903$	0	0
$904$	−2.66535e6	−0.108476
$905$	1.30373e7	0.529134
$906$	0	0
$907$	−3.50928e7	−1.41644	−0.708222	−	0.705990i	$-0.750502\pi$
$907$	−3.50928e7	−1.41644	−0.708222	+	0.705990i	$0.750502\pi$
$908$	−3.23869e7	−1.30363
$909$	0	0
$910$	−117888.	−0.00471919
$911$	3.03942e7	1.21337	0.606687	−	0.794941i	$-0.292498\pi$
$911$	3.03942e7	1.21337	0.606687	+	0.794941i	$0.292498\pi$
$912$	0	0
$913$	−1.86632e7	−0.740985
$914$	−277983.	−0.0110066
$915$	0	0
$916$	1.09348e7	0.430599
$917$	2.75826e6	0.108321
$918$	0	0
$919$	3.59276e7	1.40326	0.701632	−	0.712540i	$-0.252455\pi$
$919$	3.59276e7	1.40326	0.701632	+	0.712540i	$0.252455\pi$
$920$	−211475.	−0.00823737
$921$	0	0
$922$	705337.	0.0273256
$923$	−6.03848e6	−0.233305
$924$	0	0
$925$	1.83562e7	0.705387
$926$	866507.	0.0332081
$927$	0	0
$928$	2.58096e6	0.0983809
$929$	9.41890e6	0.358064	0.179032	−	0.983843i	$-0.442703\pi$
$929$	9.41890e6	0.358064	0.179032	+	0.983843i	$0.442703\pi$
$930$	0	0
$931$	1.36436e7	0.515887
$932$	−2.53529e7	−0.956068
$933$	0	0
$934$	602587.	0.0226023
$935$	−4.59050e6	−0.171724
$936$	0	0
$937$	−2.13837e7	−0.795673	−0.397836	−	0.917456i	$-0.630239\pi$
$937$	−2.13837e7	−0.795673	−0.397836	+	0.917456i	$0.630239\pi$
$938$	109354.	0.00405814
$939$	0	0
$940$	1.75321e7	0.647165
$941$	−4.31706e7	−1.58933	−0.794666	−	0.607047i	$-0.792354\pi$
$941$	−4.31706e7	−1.58933	−0.794666	+	0.607047i	$0.792354\pi$
$942$	0	0
$943$	−5.86800e6	−0.214887
$944$	−4.24410e6	−0.155009
$945$	0	0
$946$	148443.	0.00539302
$947$	−1.75109e7	−0.634504	−0.317252	−	0.948341i	$-0.602760\pi$
$947$	−1.75109e7	−0.634504	−0.317252	+	0.948341i	$0.602760\pi$
$948$	0	0
$949$	−1.00317e7	−0.361582
$950$	253618.	0.00911740
$951$	0	0
$952$	−341575.	−0.0122150
$953$	4.05106e6	0.144489	0.0722447	−	0.997387i	$-0.476984\pi$
$953$	4.05106e6	0.144489	0.0722447	+	0.997387i	$0.476984\pi$
$954$	0	0
$955$	−3.15703e7	−1.12013
$956$	1.66503e7	0.589220
$957$	0	0
$958$	−1.15705e6	−0.0407321
$959$	−5.89874e6	−0.207116
$960$	0	0
$961$	3.89048e6	0.135892
$962$	−772599.	−0.0269164
$963$	0	0
$964$	2.77246e7	0.960889
$965$	1.89463e7	0.654946
$966$	0	0
$967$	−2.54319e7	−0.874608	−0.437304	−	0.899314i	$-0.644067\pi$
$967$	−2.54319e7	−0.874608	−0.437304	+	0.899314i	$0.644067\pi$
$968$	1.41453e6	0.0485204
$969$	0	0
$970$	−574824.	−0.0196158
$971$	4.42829e6	0.150726	0.0753630	−	0.997156i	$-0.475988\pi$
$971$	4.42829e6	0.150726	0.0753630	+	0.997156i	$0.475988\pi$
$972$	0	0
$973$	−1.25925e7	−0.426414
$974$	831026.	0.0280684
$975$	0	0
$976$	−2.23804e7	−0.752043
$977$	−5.60569e7	−1.87885	−0.939427	−	0.342750i	$-0.888642\pi$
$977$	−5.60569e7	−1.87885	−0.939427	+	0.342750i	$0.888642\pi$
$978$	0	0
$979$	1.03511e7	0.345167
$980$	1.80788e7	0.601320
$981$	0	0
$982$	642046.	0.0212465
$983$	−3.41997e7	−1.12886	−0.564428	−	0.825482i	$-0.690903\pi$
$983$	−3.41997e7	−1.12886	−0.564428	+	0.825482i	$0.690903\pi$
$984$	0	0
$985$	−1.57733e7	−0.518004
$986$	−622766.	−0.0204001
$987$	0	0
$988$	1.26119e7	0.411044
$989$	2.92471e6	0.0950807
$990$	0	0
$991$	−904509.	−0.0292569	−0.0146285	−	0.999893i	$-0.504657\pi$
$991$	−904509.	−0.0292569	−0.0146285	+	0.999893i	$0.504657\pi$
$992$	−2.87860e6	−0.0928757
$993$	0	0
$994$	101184.	0.00324824
$995$	−3.68163e7	−1.17891
$996$	0	0
$997$	−4.00447e7	−1.27587	−0.637936	−	0.770089i	$-0.720212\pi$
$997$	−4.00447e7	−1.27587	−0.637936	+	0.770089i	$0.720212\pi$
$998$	187235.	0.00595060
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.b.1.2		3
3.2	odd	2		23.6.a.a.1.2	✓	3
12.11	even	2		368.6.a.e.1.2		3
15.14	odd	2		575.6.a.b.1.2		3
69.68	even	2		529.6.a.a.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.6.a.a.1.2	✓	3	3.2	odd	2
207.6.a.b.1.2		3	1.1	even	1	trivial
368.6.a.e.1.2		3	12.11	even	2
529.6.a.a.1.2		3	69.68	even	2
575.6.a.b.1.2		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+0.164504 q^{2} -31.9729 q^{4} +37.9865 q^{5} -43.8366 q^{7} -10.5238 q^{8} +O(q^{10})\)	\(q+0.164504 q^{2} -31.9729 q^{4} +37.9865 q^{5} -43.8366 q^{7} -10.5238 q^{8} +6.24894 q^{10} +163.213 q^{11} +430.356 q^{13} -7.21131 q^{14} +1021.40 q^{16} -740.415 q^{17} -916.578 q^{19} -1214.54 q^{20} +26.8493 q^{22} +529.000 q^{23} -1682.03 q^{25} +70.7955 q^{26} +1401.59 q^{28} +5112.96 q^{29} -5702.60 q^{31} +504.787 q^{32} -121.802 q^{34} -1665.20 q^{35} -10913.1 q^{37} -150.781 q^{38} -399.763 q^{40} -11092.6 q^{41} +5528.76 q^{43} -5218.41 q^{44} +87.0228 q^{46} -14435.2 q^{47} -14885.4 q^{49} -276.701 q^{50} -13759.8 q^{52} -6852.79 q^{53} +6199.90 q^{55} +461.329 q^{56} +841.104 q^{58} -4155.17 q^{59} -21911.4 q^{61} -938.102 q^{62} -32601.9 q^{64} +16347.7 q^{65} -15164.2 q^{67} +23673.3 q^{68} -273.932 q^{70} -14031.4 q^{71} -23310.1 q^{73} -1795.25 q^{74} +29305.7 q^{76} -7154.72 q^{77} +64276.8 q^{79} +38799.5 q^{80} -1824.78 q^{82} -114349. q^{83} -28125.8 q^{85} +909.504 q^{86} -1717.63 q^{88} +63420.7 q^{89} -18865.4 q^{91} -16913.7 q^{92} -2374.66 q^{94} -34817.6 q^{95} -91987.6 q^{97} -2448.70 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8}+O(q^{10}) \) 3 * q + 4 * q^2 + 16 * q^4 + 58 * q^5 - 282 * q^7 + 360 * q^8	\( 3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8} - 156 q^{10} - 136 q^{11} - 1116 q^{13} - 2016 q^{14} + 128 q^{16} + 896 q^{17} + 1654 q^{19} - 1504 q^{20} + 1352 q^{22} + 1587 q^{23} - 7347 q^{25} - 1998 q^{26} - 10264 q^{28} + 844 q^{29} - 3020 q^{31} - 6656 q^{32} - 11212 q^{34} - 1072 q^{35} + 8938 q^{37} - 10728 q^{38} + 3440 q^{40} + 12792 q^{41} - 16730 q^{43} - 5112 q^{44} + 2116 q^{46} - 22500 q^{47} + 2887 q^{49} - 15156 q^{50} - 47412 q^{52} - 17108 q^{53} - 436 q^{55} - 42640 q^{56} - 55678 q^{58} - 54176 q^{59} - 71324 q^{61} + 72710 q^{62} - 49984 q^{64} - 846 q^{65} - 62960 q^{67} + 8352 q^{68} + 9224 q^{70} - 98400 q^{71} - 81772 q^{73} + 59044 q^{74} + 31488 q^{76} + 304 q^{77} + 58224 q^{79} + 17568 q^{80} + 61926 q^{82} - 9892 q^{83} + 15536 q^{85} - 191140 q^{86} - 58400 q^{88} - 27542 q^{89} + 151974 q^{91} + 8464 q^{92} - 146990 q^{94} + 20644 q^{95} - 273672 q^{97} + 401276 q^{98}+O(q^{100}) \) 3 * q + 4 * q^2 + 16 * q^4 + 58 * q^5 - 282 * q^7 + 360 * q^8 - 156 * q^10 - 136 * q^11 - 1116 * q^13 - 2016 * q^14 + 128 * q^16 + 896 * q^17 + 1654 * q^19 - 1504 * q^20 + 1352 * q^22 + 1587 * q^23 - 7347 * q^25 - 1998 * q^26 - 10264 * q^28 + 844 * q^29 - 3020 * q^31 - 6656 * q^32 - 11212 * q^34 - 1072 * q^35 + 8938 * q^37 - 10728 * q^38 + 3440 * q^40 + 12792 * q^41 - 16730 * q^43 - 5112 * q^44 + 2116 * q^46 - 22500 * q^47 + 2887 * q^49 - 15156 * q^50 - 47412 * q^52 - 17108 * q^53 - 436 * q^55 - 42640 * q^56 - 55678 * q^58 - 54176 * q^59 - 71324 * q^61 + 72710 * q^62 - 49984 * q^64 - 846 * q^65 - 62960 * q^67 + 8352 * q^68 + 9224 * q^70 - 98400 * q^71 - 81772 * q^73 + 59044 * q^74 + 31488 * q^76 + 304 * q^77 + 58224 * q^79 + 17568 * q^80 + 61926 * q^82 - 9892 * q^83 + 15536 * q^85 - 191140 * q^86 - 58400 * q^88 - 27542 * q^89 + 151974 * q^91 + 8464 * q^92 - 146990 * q^94 + 20644 * q^95 - 273672 * q^97 + 401276 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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