LMFDB - Embedded newform 27.6.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [27,6,Mod(1,27)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("27.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	27.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.33036313495$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	27.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+10.6847 q^{2} +82.1619 q^{4} -25.8466 q^{5} -115.324 q^{7} +535.963 q^{8} -276.162 q^{10} +63.0909 q^{11} -797.295 q^{13} -1232.20 q^{14} +3097.40 q^{16} +1147.02 q^{17} -752.057 q^{19} -2123.61 q^{20} +674.105 q^{22} -265.386 q^{23} -2456.95 q^{25} -8518.83 q^{26} -9475.23 q^{28} +7183.76 q^{29} +2696.39 q^{31} +15943.8 q^{32} +12255.5 q^{34} +2980.73 q^{35} +11181.4 q^{37} -8035.47 q^{38} -13852.8 q^{40} -9945.87 q^{41} +9965.85 q^{43} +5183.67 q^{44} -2835.56 q^{46} +20250.0 q^{47} -3507.41 q^{49} -26251.7 q^{50} -65507.3 q^{52} +15738.2 q^{53} -1630.69 q^{55} -61809.3 q^{56} +76756.0 q^{58} -40290.0 q^{59} -10702.5 q^{61} +28810.0 q^{62} +71237.8 q^{64} +20607.4 q^{65} -54262.5 q^{67} +94241.6 q^{68} +31848.1 q^{70} -51165.1 q^{71} +7578.37 q^{73} +119470. q^{74} -61790.4 q^{76} -7275.89 q^{77} -26589.2 q^{79} -80057.2 q^{80} -106268. q^{82} -91718.4 q^{83} -29646.6 q^{85} +106482. q^{86} +33814.4 q^{88} +45378.2 q^{89} +91947.2 q^{91} -21804.7 q^{92} +216364. q^{94} +19438.1 q^{95} +376.873 q^{97} -37475.5 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 9 q^{2} + 53 q^{4} + 72 q^{5} - 8 q^{7} + 639 q^{8} - 441 q^{10} + 522 q^{11} - 704 q^{13} - 1413 q^{14} + 3857 q^{16} + 216 q^{17} - 2840 q^{19} - 4977 q^{20} - 99 q^{22} - 36 q^{23} + 3992 q^{25}+ \cdots - 28566 q^{98}+O(q^{100}) $ 2 * q + 9 * q^2 + 53 * q^4 + 72 * q^5 - 8 * q^7 + 639 * q^8 - 441 * q^10 + 522 * q^11 - 704 * q^13 - 1413 * q^14 + 3857 * q^16 + 216 * q^17 - 2840 * q^19 - 4977 * q^20 - 99 * q^22 - 36 * q^23 + 3992 * q^25 - 8676 * q^26 - 12605 * q^28 + 12240 * q^29 - 1064 * q^31 + 11367 * q^32 + 13824 * q^34 + 13482 * q^35 + 9004 * q^37 - 4518 * q^38 - 3771 * q^40 + 5688 * q^41 + 784 * q^43 - 8199 * q^44 - 3222 * q^46 + 1116 * q^47 - 8796 * q^49 - 37116 * q^50 - 68228 * q^52 + 4536 * q^53 + 43272 * q^55 - 50751 * q^56 + 68238 * q^58 - 67320 * q^59 - 49904 * q^61 + 35145 * q^62 + 54641 * q^64 + 29736 * q^65 - 42176 * q^67 + 121392 * q^68 + 14157 * q^70 - 43848 * q^71 + 47218 * q^73 + 123138 * q^74 - 902 * q^76 + 41976 * q^77 - 49616 * q^79 - 5733 * q^80 - 132606 * q^82 - 102294 * q^83 - 120744 * q^85 + 121950 * q^86 + 81099 * q^88 - 35856 * q^89 + 101960 * q^91 - 28494 * q^92 + 248598 * q^94 - 184860 * q^95 + 169966 * q^97 - 28566 * q^98

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 $

$2$	10.6847	1.88880	0.944399	−	0.328801i	$-0.106644\pi$
$2$	10.6847	1.88880	0.944399	+	0.328801i	$0.106644\pi$
$3$	0	0
$3$	0	0
$4$	82.1619	2.56756
$5$	−25.8466	−0.462358	−0.231179	−	0.972911i	$-0.574258\pi$
$5$	−25.8466	−0.462358	−0.231179	+	0.972911i	$0.574258\pi$
$6$	0	0
$7$	−115.324	−0.889558	−0.444779	−	0.895640i	$-0.646718\pi$
$7$	−115.324	−0.889558	−0.444779	+	0.895640i	$0.646718\pi$
$8$	535.963	2.96081
$9$	0	0
$10$	−276.162	−0.873301
$11$	63.0909	0.157212	0.0786059	−	0.996906i	$-0.474953\pi$
$11$	63.0909	0.157212	0.0786059	+	0.996906i	$0.474953\pi$
$12$	0	0
$13$	−797.295	−1.30846	−0.654231	−	0.756295i	$-0.727007\pi$
$13$	−797.295	−1.30846	−0.654231	+	0.756295i	$0.727007\pi$
$14$	−1232.20	−1.68020
$15$	0	0
$16$	3097.40	3.02481
$17$	1147.02	0.962608	0.481304	−	0.876554i	$-0.340163\pi$
$17$	1147.02	0.962608	0.481304	+	0.876554i	$0.340163\pi$
$18$	0	0
$19$	−752.057	−0.477933	−0.238966	−	0.971028i	$-0.576809\pi$
$19$	−752.057	−0.477933	−0.238966	+	0.971028i	$0.576809\pi$
$20$	−2123.61	−1.18713
$21$	0	0
$22$	674.105	0.296941
$23$	−265.386	−0.104607	−0.0523033	−	0.998631i	$-0.516656\pi$
$23$	−265.386	−0.104607	−0.0523033	+	0.998631i	$0.516656\pi$
$24$	0	0
$25$	−2456.95	−0.786225
$26$	−8518.83	−2.47142
$27$	0	0
$28$	−9475.23	−2.28399
$29$	7183.76	1.58620	0.793098	−	0.609094i	$-0.208467\pi$
$29$	7183.76	1.58620	0.793098	+	0.609094i	$0.208467\pi$
$30$	0	0
$31$	2696.39	0.503940	0.251970	−	0.967735i	$-0.418922\pi$
$31$	2696.39	0.503940	0.251970	+	0.967735i	$0.418922\pi$
$32$	15943.8	2.75244
$33$	0	0
$34$	12255.5	1.81817
$35$	2980.73	0.411294
$36$	0	0
$37$	11181.4	1.34274	0.671372	−	0.741121i	$-0.265705\pi$
$37$	11181.4	1.34274	0.671372	+	0.741121i	$0.265705\pi$
$38$	−8035.47	−0.902719
$39$	0	0
$40$	−13852.8	−1.36895
$41$	−9945.87	−0.924024	−0.462012	−	0.886874i	$-0.652872\pi$
$41$	−9945.87	−0.924024	−0.462012	+	0.886874i	$0.652872\pi$
$42$	0	0
$43$	9965.85	0.821946	0.410973	−	0.911648i	$-0.365189\pi$
$43$	9965.85	0.821946	0.410973	+	0.911648i	$0.365189\pi$
$44$	5183.67	0.403651
$45$	0	0
$46$	−2835.56	−0.197581
$47$	20250.0	1.33715	0.668574	−	0.743646i	$-0.266905\pi$
$47$	20250.0	1.33715	0.668574	+	0.743646i	$0.266905\pi$
$48$	0	0
$49$	−3507.41	−0.208687
$50$	−26251.7	−1.48502
$51$	0	0
$52$	−65507.3	−3.35955
$53$	15738.2	0.769600	0.384800	−	0.923000i	$-0.374270\pi$
$53$	15738.2	0.769600	0.384800	+	0.923000i	$0.374270\pi$
$54$	0	0
$55$	−1630.69	−0.0726881
$56$	−61809.3	−2.63381
$57$	0	0
$58$	76756.0	2.99601
$59$	−40290.0	−1.50684	−0.753419	−	0.657540i	$-0.771597\pi$
$59$	−40290.0	−1.50684	−0.753419	+	0.657540i	$0.771597\pi$
$60$	0	0
$61$	−10702.5	−0.368267	−0.184133	−	0.982901i	$-0.558948\pi$
$61$	−10702.5	−0.368267	−0.184133	+	0.982901i	$0.558948\pi$
$62$	28810.0	0.951841
$63$	0	0
$64$	71237.8	2.17400
$65$	20607.4	0.604977
$66$	0	0
$67$	−54262.5	−1.47677	−0.738385	−	0.674379i	$-0.764411\pi$
$67$	−54262.5	−1.47677	−0.738385	+	0.674379i	$0.764411\pi$
$68$	94241.6	2.47156
$69$	0	0
$70$	31848.1	0.776851
$71$	−51165.1	−1.20456	−0.602279	−	0.798286i	$-0.705740\pi$
$71$	−51165.1	−1.20456	−0.602279	+	0.798286i	$0.705740\pi$
$72$	0	0
$73$	7578.37	0.166444	0.0832220	−	0.996531i	$-0.473479\pi$
$73$	7578.37	0.166444	0.0832220	+	0.996531i	$0.473479\pi$
$74$	119470.	2.53617
$75$	0	0
$76$	−61790.4	−1.22712
$77$	−7275.89	−0.139849
$78$	0	0
$79$	−26589.2	−0.479333	−0.239666	−	0.970855i	$-0.577038\pi$
$79$	−26589.2	−0.479333	−0.239666	+	0.970855i	$0.577038\pi$
$80$	−80057.2	−1.39854
$81$	0	0
$82$	−106268.	−1.74530
$83$	−91718.4	−1.46137	−0.730686	−	0.682713i	$-0.760800\pi$
$83$	−91718.4	−1.46137	−0.730686	+	0.682713i	$0.760800\pi$
$84$	0	0
$85$	−29646.6	−0.445069
$86$	106482.	1.55249
$87$	0	0
$88$	33814.4	0.465474
$89$	45378.2	0.607256	0.303628	−	0.952791i	$-0.401802\pi$
$89$	45378.2	0.607256	0.303628	+	0.952791i	$0.401802\pi$
$90$	0	0
$91$	91947.2	1.16395
$92$	−21804.7	−0.268584
$93$	0	0
$94$	216364.	2.52560
$95$	19438.1	0.220976
$96$	0	0
$97$	376.873	0.00406692	0.00203346	−	0.999998i	$-0.499353\pi$
$97$	376.873	0.00406692	0.00203346	+	0.999998i	$0.499353\pi$
$98$	−37475.5	−0.394168
$99$	0	0
$100$	−201868.	−2.01868
$101$	1626.35	0.0158639	0.00793195	−	0.999969i	$-0.497475\pi$
$101$	1626.35	0.0158639	0.00793195	+	0.999969i	$0.497475\pi$
$102$	0	0
$103$	59635.7	0.553876	0.276938	−	0.960888i	$-0.410680\pi$
$103$	59635.7	0.553876	0.276938	+	0.960888i	$0.410680\pi$
$104$	−427321.	−3.87410
$105$	0	0
$106$	168157.	1.45362
$107$	−113386.	−0.957412	−0.478706	−	0.877975i	$-0.658894\pi$
$107$	−113386.	−0.957412	−0.478706	+	0.877975i	$0.658894\pi$
$108$	0	0
$109$	167511.	1.35044	0.675221	−	0.737615i	$-0.264048\pi$
$109$	167511.	1.35044	0.675221	+	0.737615i	$0.264048\pi$
$110$	−17423.3	−0.137293
$111$	0	0
$112$	−357204.	−2.69074
$113$	181432.	1.33665	0.668326	−	0.743868i	$-0.267011\pi$
$113$	181432.	1.33665	0.668326	+	0.743868i	$0.267011\pi$
$114$	0	0
$115$	6859.33	0.0483657
$116$	590232.	4.07265
$117$	0	0
$118$	−430484.	−2.84611
$119$	−132279.	−0.856296
$120$	0	0
$121$	−157071.	−0.975284
$122$	−114353.	−0.695582
$123$	0	0
$124$	221541.	1.29390
$125$	144274.	0.825875
$126$	0	0
$127$	65774.9	0.361868	0.180934	−	0.983495i	$-0.442088\pi$
$127$	65774.9	0.361868	0.180934	+	0.983495i	$0.442088\pi$
$128$	250948.	1.35381
$129$	0	0
$130$	220183.	1.14268
$131$	88139.0	0.448735	0.224367	−	0.974505i	$-0.427968\pi$
$131$	88139.0	0.448735	0.224367	+	0.974505i	$0.427968\pi$
$132$	0	0
$133$	86730.1	0.425149
$134$	−579776.	−2.78932
$135$	0	0
$136$	614762.	2.85010
$137$	45228.8	0.205880	0.102940	−	0.994688i	$-0.467175\pi$
$137$	45228.8	0.205880	0.102940	+	0.994688i	$0.467175\pi$
$138$	0	0
$139$	−71637.5	−0.314488	−0.157244	−	0.987560i	$-0.550261\pi$
$139$	−71637.5	−0.314488	−0.157244	+	0.987560i	$0.550261\pi$
$140$	244902.	1.05602
$141$	0	0
$142$	−546681.	−2.27517
$143$	−50302.1	−0.205706
$144$	0	0
$145$	−185676.	−0.733390
$146$	80972.2	0.314379
$147$	0	0
$148$	918688.	3.44757
$149$	461330.	1.70234	0.851169	−	0.524891i	$-0.175894\pi$
$149$	461330.	1.70234	0.851169	+	0.524891i	$0.175894\pi$
$150$	0	0
$151$	−364935.	−1.30249	−0.651243	−	0.758869i	$-0.725752\pi$
$151$	−364935.	−1.30249	−0.651243	+	0.758869i	$0.725752\pi$
$152$	−403075.	−1.41507
$153$	0	0
$154$	−77740.4	−0.264147
$155$	−69692.5	−0.233001
$156$	0	0
$157$	−270764.	−0.876681	−0.438341	−	0.898809i	$-0.644434\pi$
$157$	−270764.	−0.876681	−0.438341	+	0.898809i	$0.644434\pi$
$158$	−284096.	−0.905363
$159$	0	0
$160$	−412094.	−1.27261
$161$	30605.4	0.0930536
$162$	0	0
$163$	−119086.	−0.351069	−0.175535	−	0.984473i	$-0.556165\pi$
$163$	−119086.	−0.351069	−0.175535	+	0.984473i	$0.556165\pi$
$164$	−817172.	−2.37249
$165$	0	0
$166$	−979979.	−2.76024
$167$	−402672.	−1.11728	−0.558638	−	0.829412i	$-0.688676\pi$
$167$	−402672.	−1.11728	−0.558638	+	0.829412i	$0.688676\pi$
$168$	0	0
$169$	264387.	0.712071
$170$	−316764.	−0.840647
$171$	0	0
$172$	818814.	2.11040
$173$	467534.	1.18768	0.593839	−	0.804584i	$-0.297612\pi$
$173$	467534.	1.18768	0.593839	+	0.804584i	$0.297612\pi$
$174$	0	0
$175$	283345.	0.699393
$176$	195418.	0.475535
$177$	0	0
$178$	484850.	1.14698
$179$	49932.9	0.116481	0.0582404	−	0.998303i	$-0.481451\pi$
$179$	49932.9	0.116481	0.0582404	+	0.998303i	$0.481451\pi$
$180$	0	0
$181$	82498.4	0.187176	0.0935878	−	0.995611i	$-0.470166\pi$
$181$	82498.4	0.187176	0.0935878	+	0.995611i	$0.470166\pi$
$182$	982424.	2.19847
$183$	0	0
$184$	−142237.	−0.309720
$185$	−289002.	−0.620828
$186$	0	0
$187$	72366.7	0.151333
$188$	1.66378e6	3.43321
$189$	0	0
$190$	207689.	0.417379
$191$	−344744.	−0.683775	−0.341887	−	0.939741i	$-0.611066\pi$
$191$	−344744.	−0.683775	−0.341887	+	0.939741i	$0.611066\pi$
$192$	0	0
$193$	591965.	1.14394	0.571970	−	0.820275i	$-0.306179\pi$
$193$	591965.	1.14394	0.571970	+	0.820275i	$0.306179\pi$
$194$	4026.75	0.00768158
$195$	0	0
$196$	−288175.	−0.535817
$197$	236171.	0.433572	0.216786	−	0.976219i	$-0.430443\pi$
$197$	236171.	0.433572	0.216786	+	0.976219i	$0.430443\pi$
$198$	0	0
$199$	36776.0	0.0658313	0.0329156	−	0.999458i	$-0.489521\pi$
$199$	36776.0	0.0658313	0.0329156	+	0.999458i	$0.489521\pi$
$200$	−1.31684e6	−2.32786
$201$	0	0
$202$	17377.0	0.0299637
$203$	−828459.	−1.41101
$204$	0	0
$205$	257067.	0.427230
$206$	637187.	1.04616
$207$	0	0
$208$	−2.46954e6	−3.95784
$209$	−47448.0	−0.0751367
$210$	0	0
$211$	526111.	0.813525	0.406763	−	0.913534i	$-0.366658\pi$
$211$	526111.	0.813525	0.406763	+	0.913534i	$0.366658\pi$
$212$	1.29308e6	1.97599
$213$	0	0
$214$	−1.21149e6	−1.80836
$215$	−257583.	−0.380033
$216$	0	0
$217$	−310958.	−0.448284
$218$	1.78979e6	2.55071
$219$	0	0
$220$	−133980.	−0.186631
$221$	−914516.	−1.25954
$222$	0	0
$223$	172656.	0.232498	0.116249	−	0.993220i	$-0.462913\pi$
$223$	172656.	0.232498	0.116249	+	0.993220i	$0.462913\pi$
$224$	−1.83871e6	−2.44846
$225$	0	0
$226$	1.93854e6	2.52467
$227$	1.13692e6	1.46442	0.732208	−	0.681081i	$-0.238490\pi$
$227$	1.13692e6	1.46442	0.732208	+	0.681081i	$0.238490\pi$
$228$	0	0
$229$	−783946.	−0.987865	−0.493933	−	0.869500i	$-0.664441\pi$
$229$	−783946.	−0.987865	−0.493933	+	0.869500i	$0.664441\pi$
$230$	73289.6	0.0913530
$231$	0	0
$232$	3.85023e6	4.69642
$233$	−852850.	−1.02916	−0.514580	−	0.857442i	$-0.672052\pi$
$233$	−852850.	−1.02916	−0.514580	+	0.857442i	$0.672052\pi$
$234$	0	0
$235$	−523392.	−0.618241
$236$	−3.31030e6	−3.86890
$237$	0	0
$238$	−1.41336e6	−1.61737
$239$	625959.	0.708845	0.354423	−	0.935085i	$-0.384677\pi$
$239$	625959.	0.708845	0.354423	+	0.935085i	$0.384677\pi$
$240$	0	0
$241$	−114154.	−0.126604	−0.0633021	−	0.997994i	$-0.520163\pi$
$241$	−114154.	−0.126604	−0.0633021	+	0.997994i	$0.520163\pi$
$242$	−1.67825e6	−1.84212
$243$	0	0
$244$	−879342.	−0.945547
$245$	90654.5	0.0964882
$246$	0	0
$247$	599612.	0.625356
$248$	1.44517e6	1.49207
$249$	0	0
$250$	1.54152e6	1.55991
$251$	572829.	0.573906	0.286953	−	0.957945i	$-0.407358\pi$
$251$	572829.	0.573906	0.286953	+	0.957945i	$0.407358\pi$
$252$	0	0
$253$	−16743.5	−0.0164454
$254$	702782.	0.683496
$255$	0	0
$256$	401685.	0.383077
$257$	−286863.	−0.270920	−0.135460	−	0.990783i	$-0.543251\pi$
$257$	−286863.	−0.270920	−0.135460	+	0.990783i	$0.543251\pi$
$258$	0	0
$259$	−1.28949e6	−1.19445
$260$	1.69314e6	1.55332
$261$	0	0
$262$	941735.	0.847569
$263$	−296534.	−0.264353	−0.132177	−	0.991226i	$-0.542197\pi$
$263$	−296534.	−0.264353	−0.132177	+	0.991226i	$0.542197\pi$
$264$	0	0
$265$	−406778.	−0.355830
$266$	926681.	0.803020
$267$	0	0
$268$	−4.45831e6	−3.79170
$269$	−60270.2	−0.0507835	−0.0253917	−	0.999678i	$-0.508083\pi$
$269$	−60270.2	−0.0507835	−0.0253917	+	0.999678i	$0.508083\pi$
$270$	0	0
$271$	−2.18183e6	−1.80467	−0.902335	−	0.431036i	$-0.858148\pi$
$271$	−2.18183e6	−1.80467	−0.902335	+	0.431036i	$0.858148\pi$
$272$	3.55279e6	2.91170
$273$	0	0
$274$	483254.	0.388866
$275$	−155012.	−0.123604
$276$	0	0
$277$	−39559.4	−0.0309778	−0.0154889	−	0.999880i	$-0.504930\pi$
$277$	−39559.4	−0.0309778	−0.0154889	+	0.999880i	$0.504930\pi$
$278$	−765422.	−0.594004
$279$	0	0
$280$	1.59756e6	1.21776
$281$	−955918.	−0.722196	−0.361098	−	0.932528i	$-0.617598\pi$
$281$	−955918.	−0.722196	−0.361098	+	0.932528i	$0.617598\pi$
$282$	0	0
$283$	2.38449e6	1.76982	0.884911	−	0.465759i	$-0.154219\pi$
$283$	2.38449e6	1.76982	0.884911	+	0.465759i	$0.154219\pi$
$284$	−4.20382e6	−3.09277
$285$	0	0
$286$	−537461.	−0.388536
$287$	1.14700e6	0.821973
$288$	0	0
$289$	−104196.	−0.0733849
$290$	−1.98388e6	−1.38523
$291$	0	0
$292$	622653.	0.427355
$293$	2.31380e6	1.57455	0.787276	−	0.616600i	$-0.211490\pi$
$293$	2.31380e6	1.57455	0.787276	+	0.616600i	$0.211490\pi$
$294$	0	0
$295$	1.04136e6	0.696699
$296$	5.99283e6	3.97560
$297$	0	0
$298$	4.92915e6	3.21537
$299$	211591.	0.136874
$300$	0	0
$301$	−1.14930e6	−0.731168
$302$	−3.89921e6	−2.46013
$303$	0	0
$304$	−2.32942e6	−1.44565
$305$	276624.	0.170271
$306$	0	0
$307$	175518.	0.106286	0.0531430	−	0.998587i	$-0.483076\pi$
$307$	175518.	0.106286	0.0531430	+	0.998587i	$0.483076\pi$
$308$	−597801.	−0.359071
$309$	0	0
$310$	−744641.	−0.440091
$311$	3.31716e6	1.94476	0.972379	−	0.233408i	$-0.0749877\pi$
$311$	3.31716e6	1.94476	0.972379	+	0.233408i	$0.0749877\pi$
$312$	0	0
$313$	3.15765e6	1.82181	0.910904	−	0.412617i	$-0.135385\pi$
$313$	3.15765e6	1.82181	0.910904	+	0.412617i	$0.135385\pi$
$314$	−2.89302e6	−1.65587
$315$	0	0
$316$	−2.18462e6	−1.23072
$317$	−1.25137e6	−0.699420	−0.349710	−	0.936858i	$-0.613720\pi$
$317$	−1.25137e6	−0.699420	−0.349710	+	0.936858i	$0.613720\pi$
$318$	0	0
$319$	453230.	0.249369
$320$	−1.84125e6	−1.00517
$321$	0	0
$322$	327008.	0.175759
$323$	−862626.	−0.460062
$324$	0	0
$325$	1.95892e6	1.02875
$326$	−1.27240e6	−0.663100
$327$	0	0
$328$	−5.33062e6	−2.73586
$329$	−2.33530e6	−1.18947
$330$	0	0
$331$	−2.47582e6	−1.24208	−0.621039	−	0.783780i	$-0.713289\pi$
$331$	−2.47582e6	−1.24208	−0.621039	+	0.783780i	$0.713289\pi$
$332$	−7.53576e6	−3.75216
$333$	0	0
$334$	−4.30241e6	−2.11031
$335$	1.40250e6	0.682796
$336$	0	0
$337$	−1.03952e6	−0.498607	−0.249304	−	0.968425i	$-0.580202\pi$
$337$	−1.03952e6	−0.498607	−0.249304	+	0.968425i	$0.580202\pi$
$338$	2.82488e6	1.34496
$339$	0	0
$340$	−2.43582e6	−1.14274
$341$	170118.	0.0792253
$342$	0	0
$343$	2.34274e6	1.07520
$344$	5.34133e6	2.43362
$345$	0	0
$346$	4.99545e6	2.24328
$347$	3.47924e6	1.55118	0.775588	−	0.631240i	$-0.217454\pi$
$347$	3.47924e6	1.55118	0.775588	+	0.631240i	$0.217454\pi$
$348$	0	0
$349$	2.20547e6	0.969252	0.484626	−	0.874721i	$-0.338956\pi$
$349$	2.20547e6	0.969252	0.484626	+	0.874721i	$0.338956\pi$
$350$	3.02745e6	1.32101
$351$	0	0
$352$	1.00591e6	0.432716
$353$	−3.67997e6	−1.57184	−0.785920	−	0.618329i	$-0.787810\pi$
$353$	−3.67997e6	−1.57184	−0.785920	+	0.618329i	$0.787810\pi$
$354$	0	0
$355$	1.32244e6	0.556937
$356$	3.72836e6	1.55917
$357$	0	0
$358$	533516.	0.220009
$359$	1.30796e6	0.535620	0.267810	−	0.963472i	$-0.413700\pi$
$359$	1.30796e6	0.535620	0.267810	+	0.963472i	$0.413700\pi$
$360$	0	0
$361$	−1.91051e6	−0.771580
$362$	881467.	0.353537
$363$	0	0
$364$	7.55456e6	2.98852
$365$	−195875.	−0.0769567
$366$	0	0
$367$	−4.37650e6	−1.69614	−0.848069	−	0.529885i	$-0.822235\pi$
$367$	−4.37650e6	−1.69614	−0.848069	+	0.529885i	$0.822235\pi$
$368$	−822008.	−0.316414
$369$	0	0
$370$	−3.08789e6	−1.17262
$371$	−1.81499e6	−0.684603
$372$	0	0
$373$	−3.02935e6	−1.12740	−0.563699	−	0.825980i	$-0.690622\pi$
$373$	−3.02935e6	−1.12740	−0.563699	+	0.825980i	$0.690622\pi$
$374$	773214.	0.285838
$375$	0	0
$376$	1.08532e7	3.95903
$377$	−5.72758e6	−2.07548
$378$	0	0
$379$	833764.	0.298157	0.149078	−	0.988825i	$-0.452369\pi$
$379$	833764.	0.298157	0.149078	+	0.988825i	$0.452369\pi$
$380$	1.59707e6	0.567369
$381$	0	0
$382$	−3.68347e6	−1.29151
$383$	−1.63521e6	−0.569609	−0.284804	−	0.958586i	$-0.591929\pi$
$383$	−1.63521e6	−0.569609	−0.284804	+	0.958586i	$0.591929\pi$
$384$	0	0
$385$	188057.	0.0646603
$386$	6.32495e6	2.16067
$387$	0	0
$388$	30964.6	0.0104421
$389$	4.48524e6	1.50284	0.751418	−	0.659827i	$-0.229370\pi$
$389$	4.48524e6	1.50284	0.751418	+	0.659827i	$0.229370\pi$
$390$	0	0
$391$	−304404.	−0.100695
$392$	−1.87984e6	−0.617883
$393$	0	0
$394$	2.52341e6	0.818930
$395$	687240.	0.221623
$396$	0	0
$397$	−2.08717e6	−0.664632	−0.332316	−	0.943168i	$-0.607830\pi$
$397$	−2.08717e6	−0.664632	−0.332316	+	0.943168i	$0.607830\pi$
$398$	392939.	0.124342
$399$	0	0
$400$	−7.61017e6	−2.37818
$401$	−4.02831e6	−1.25101	−0.625506	−	0.780219i	$-0.715108\pi$
$401$	−4.02831e6	−1.25101	−0.625506	+	0.780219i	$0.715108\pi$
$402$	0	0
$403$	−2.14982e6	−0.659386
$404$	133624.	0.0407315
$405$	0	0
$406$	−8.85180e6	−2.66512
$407$	705447.	0.211095
$408$	0	0
$409$	−4.31168e6	−1.27450	−0.637248	−	0.770658i	$-0.719927\pi$
$409$	−4.31168e6	−1.27450	−0.637248	+	0.770658i	$0.719927\pi$
$410$	2.74667e6	0.806951
$411$	0	0
$412$	4.89978e6	1.42211
$413$	4.64639e6	1.34042
$414$	0	0
$415$	2.37061e6	0.675677
$416$	−1.27120e7	−3.60146
$417$	0	0
$418$	−506965.	−0.141918
$419$	−4.33250e6	−1.20560	−0.602800	−	0.797892i	$-0.705948\pi$
$419$	−4.33250e6	−1.20560	−0.602800	+	0.797892i	$0.705948\pi$
$420$	0	0
$421$	249525.	0.0686135	0.0343067	−	0.999411i	$-0.489078\pi$
$421$	249525.	0.0686135	0.0343067	+	0.999411i	$0.489078\pi$
$422$	5.62131e6	1.53659
$423$	0	0
$424$	8.43509e6	2.27864
$425$	−2.81818e6	−0.756827
$426$	0	0
$427$	1.23426e6	0.327594
$428$	−9.31599e6	−2.45821
$429$	0	0
$430$	−2.75219e6	−0.717806
$431$	−4.16691e6	−1.08049	−0.540245	−	0.841508i	$-0.681668\pi$
$431$	−4.16691e6	−1.08049	−0.540245	+	0.841508i	$0.681668\pi$
$432$	0	0
$433$	6.75137e6	1.73050	0.865251	−	0.501340i	$-0.167159\pi$
$433$	6.75137e6	1.73050	0.865251	+	0.501340i	$0.167159\pi$
$434$	−3.32248e6	−0.846717
$435$	0	0
$436$	1.37630e7	3.46734
$437$	199586.	0.0499949
$438$	0	0
$439$	−343561.	−0.0850831	−0.0425415	−	0.999095i	$-0.513545\pi$
$439$	−343561.	−0.0850831	−0.0425415	+	0.999095i	$0.513545\pi$
$440$	−873987.	−0.215215
$441$	0	0
$442$	−9.77129e6	−2.37901
$443$	5.17293e6	1.25235	0.626177	−	0.779681i	$-0.284619\pi$
$443$	5.17293e6	1.25235	0.626177	+	0.779681i	$0.284619\pi$
$444$	0	0
$445$	−1.17287e6	−0.280770
$446$	1.84477e6	0.439141
$447$	0	0
$448$	−8.21541e6	−1.93390
$449$	913932.	0.213943	0.106972	−	0.994262i	$-0.465885\pi$
$449$	913932.	0.213943	0.106972	+	0.994262i	$0.465885\pi$
$450$	0	0
$451$	−627494.	−0.145268
$452$	1.49068e7	3.43194
$453$	0	0
$454$	1.21476e7	2.76599
$455$	−2.37652e6	−0.538162
$456$	0	0
$457$	1.96268e6	0.439602	0.219801	−	0.975545i	$-0.429459\pi$
$457$	1.96268e6	0.439602	0.219801	+	0.975545i	$0.429459\pi$
$458$	−8.37620e6	−1.86588
$459$	0	0
$460$	563576.	0.124182
$461$	2.23347e6	0.489472	0.244736	−	0.969590i	$-0.421299\pi$
$461$	2.23347e6	0.489472	0.244736	+	0.969590i	$0.421299\pi$
$462$	0	0
$463$	6.27635e6	1.36068	0.680338	−	0.732898i	$-0.261833\pi$
$463$	6.27635e6	1.36068	0.680338	+	0.732898i	$0.261833\pi$
$464$	2.22510e7	4.79793
$465$	0	0
$466$	−9.11241e6	−1.94388
$467$	4.69184e6	0.995522	0.497761	−	0.867314i	$-0.334156\pi$
$467$	4.69184e6	0.995522	0.497761	+	0.867314i	$0.334156\pi$
$468$	0	0
$469$	6.25776e6	1.31367
$470$	−5.59227e6	−1.16773
$471$	0	0
$472$	−2.15939e7	−4.46146
$473$	628755.	0.129220
$474$	0	0
$475$	1.84777e6	0.375763
$476$	−1.08683e7	−2.19859
$477$	0	0
$478$	6.68816e6	1.33887
$479$	−2.04271e6	−0.406788	−0.203394	−	0.979097i	$-0.565197\pi$
$479$	−2.04271e6	−0.406788	−0.203394	+	0.979097i	$0.565197\pi$
$480$	0	0
$481$	−8.91490e6	−1.75693
$482$	−1.21969e6	−0.239130
$483$	0	0
$484$	−1.29052e7	−2.50410
$485$	−9740.87	−0.00188037
$486$	0	0
$487$	−6.45535e6	−1.23338	−0.616690	−	0.787206i	$-0.711527\pi$
$487$	−6.45535e6	−1.23338	−0.616690	+	0.787206i	$0.711527\pi$
$488$	−5.73617e6	−1.09037
$489$	0	0
$490$	968613.	0.182247
$491$	−5.68549e6	−1.06430	−0.532150	−	0.846650i	$-0.678616\pi$
$491$	−5.68549e6	−1.06430	−0.532150	+	0.846650i	$0.678616\pi$
$492$	0	0
$493$	8.23994e6	1.52689
$494$	6.40664e6	1.18117
$495$	0	0
$496$	8.35180e6	1.52432
$497$	5.90055e6	1.07152
$498$	0	0
$499$	202061.	0.0363272	0.0181636	−	0.999835i	$-0.494218\pi$
$499$	202061.	0.0363272	0.0181636	+	0.999835i	$0.494218\pi$
$500$	1.18539e7	2.12048
$501$	0	0
$502$	6.12048e6	1.08399
$503$	−1.48616e6	−0.261906	−0.130953	−	0.991389i	$-0.541804\pi$
$503$	−1.48616e6	−0.261906	−0.130953	+	0.991389i	$0.541804\pi$
$504$	0	0
$505$	−42035.5	−0.00733480
$506$	−178898.	−0.0310620
$507$	0	0
$508$	5.40419e6	0.929119
$509$	8.86109e6	1.51598	0.757989	−	0.652268i	$-0.226182\pi$
$509$	8.86109e6	1.51598	0.757989	+	0.652268i	$0.226182\pi$
$510$	0	0
$511$	−873966.	−0.148062
$512$	−3.73847e6	−0.630258
$513$	0	0
$514$	−3.06503e6	−0.511714
$515$	−1.54138e6	−0.256089
$516$	0	0
$517$	1.27759e6	0.210215
$518$	−1.37777e7	−2.25607
$519$	0	0
$520$	1.10448e7	1.79122
$521$	885212.	0.142874	0.0714370	−	0.997445i	$-0.477242\pi$
$521$	885212.	0.142874	0.0714370	+	0.997445i	$0.477242\pi$
$522$	0	0
$523$	3.19649e6	0.510998	0.255499	−	0.966809i	$-0.417760\pi$
$523$	3.19649e6	0.510998	0.255499	+	0.966809i	$0.417760\pi$
$524$	7.24167e6	1.15215
$525$	0	0
$526$	−3.16836e6	−0.499310
$527$	3.09282e6	0.485097
$528$	0	0
$529$	−6.36591e6	−0.989057
$530$	−4.34629e6	−0.672092
$531$	0	0
$532$	7.12591e6	1.09159
$533$	7.92980e6	1.20905
$534$	0	0
$535$	2.93063e6	0.442667
$536$	−2.90827e7	−4.37243
$537$	0	0
$538$	−643967.	−0.0959197
$539$	−221286.	−0.0328081
$540$	0	0
$541$	−6.95533e6	−1.02170	−0.510851	−	0.859669i	$-0.670670\pi$
$541$	−6.95533e6	−1.02170	−0.510851	+	0.859669i	$0.670670\pi$
$542$	−2.33121e7	−3.40866
$543$	0	0
$544$	1.82880e7	2.64952
$545$	−4.32958e6	−0.624388
$546$	0	0
$547$	−9.48749e6	−1.35576	−0.677880	−	0.735173i	$-0.737101\pi$
$547$	−9.48749e6	−1.35576	−0.677880	+	0.735173i	$0.737101\pi$
$548$	3.71609e6	0.528609
$549$	0	0
$550$	−1.65625e6	−0.233463
$551$	−5.40260e6	−0.758095
$552$	0	0
$553$	3.06637e6	0.426394
$554$	−422678.	−0.0585107
$555$	0	0
$556$	−5.88588e6	−0.807466
$557$	8.41675e6	1.14949	0.574747	−	0.818331i	$-0.305101\pi$
$557$	8.41675e6	1.14949	0.574747	+	0.818331i	$0.305101\pi$
$558$	0	0
$559$	−7.94573e6	−1.07548
$560$	9.23251e6	1.24408
$561$	0	0
$562$	−1.02137e7	−1.36408
$563$	7.81759e6	1.03945	0.519723	−	0.854335i	$-0.326035\pi$
$563$	7.81759e6	1.03945	0.519723	+	0.854335i	$0.326035\pi$
$564$	0	0
$565$	−4.68941e6	−0.618012
$566$	2.54775e7	3.34284
$567$	0	0
$568$	−2.74226e7	−3.56646
$569$	−9.22271e6	−1.19420	−0.597101	−	0.802166i	$-0.703681\pi$
$569$	−9.22271e6	−1.19420	−0.597101	+	0.802166i	$0.703681\pi$
$570$	0	0
$571$	−4.37099e6	−0.561035	−0.280518	−	0.959849i	$-0.590506\pi$
$571$	−4.37099e6	−0.561035	−0.280518	+	0.959849i	$0.590506\pi$
$572$	−4.13292e6	−0.528161
$573$	0	0
$574$	1.22553e7	1.55254
$575$	652042.	0.0822443
$576$	0	0
$577$	4.98397e6	0.623212	0.311606	−	0.950211i	$-0.399133\pi$
$577$	4.98397e6	0.623212	0.311606	+	0.950211i	$0.399133\pi$
$578$	−1.11330e6	−0.138609
$579$	0	0
$580$	−1.52555e7	−1.88302
$581$	1.05773e7	1.29998
$582$	0	0
$583$	992937.	0.120990
$584$	4.06172e6	0.492809
$585$	0	0
$586$	2.47222e7	2.97401
$587$	−9.17700e6	−1.09927	−0.549636	−	0.835404i	$-0.685234\pi$
$587$	−9.17700e6	−1.09927	−0.549636	+	0.835404i	$0.685234\pi$
$588$	0	0
$589$	−2.02784e6	−0.240849
$590$	1.11266e7	1.31592
$591$	0	0
$592$	3.46334e7	4.06154
$593$	−1.25421e7	−1.46465	−0.732327	−	0.680953i	$-0.761566\pi$
$593$	−1.25421e7	−1.46465	−0.732327	+	0.680953i	$0.761566\pi$
$594$	0	0
$595$	3.41896e6	0.395915
$596$	3.79038e7	4.37086
$597$	0	0
$598$	2.26078e6	0.258527
$599$	−8.28426e6	−0.943380	−0.471690	−	0.881765i	$-0.656356\pi$
$599$	−8.28426e6	−0.943380	−0.471690	+	0.881765i	$0.656356\pi$
$600$	0	0
$601$	8.27223e6	0.934193	0.467096	−	0.884206i	$-0.345300\pi$
$601$	8.27223e6	0.934193	0.467096	+	0.884206i	$0.345300\pi$
$602$	−1.22799e7	−1.38103
$603$	0	0
$604$	−2.99838e7	−3.34421
$605$	4.05974e6	0.450930
$606$	0	0
$607$	6.18488e6	0.681333	0.340666	−	0.940184i	$-0.389347\pi$
$607$	6.18488e6	0.681333	0.340666	+	0.940184i	$0.389347\pi$
$608$	−1.19907e7	−1.31548
$609$	0	0
$610$	2.95564e6	0.321608
$611$	−1.61452e7	−1.74961
$612$	0	0
$613$	−1.25753e7	−1.35166	−0.675829	−	0.737059i	$-0.736214\pi$
$613$	−1.25753e7	−1.35166	−0.675829	+	0.737059i	$0.736214\pi$
$614$	1.87535e6	0.200753
$615$	0	0
$616$	−3.89961e6	−0.414066
$617$	6.68148e6	0.706577	0.353289	−	0.935514i	$-0.385063\pi$
$617$	6.68148e6	0.706577	0.353289	+	0.935514i	$0.385063\pi$
$618$	0	0
$619$	1.04346e7	1.09458	0.547292	−	0.836942i	$-0.315659\pi$
$619$	1.04346e7	1.09458	0.547292	+	0.836942i	$0.315659\pi$
$620$	−5.72607e6	−0.598243
$621$	0	0
$622$	3.54427e7	3.67326
$623$	−5.23318e6	−0.540189
$624$	0	0
$625$	3.94898e6	0.404376
$626$	3.37384e7	3.44103
$627$	0	0
$628$	−2.22465e7	−2.25093
$629$	1.28254e7	1.29254
$630$	0	0
$631$	4.36703e6	0.436629	0.218315	−	0.975878i	$-0.429944\pi$
$631$	4.36703e6	0.436629	0.218315	+	0.975878i	$0.429944\pi$
$632$	−1.42508e7	−1.41921
$633$	0	0
$634$	−1.33705e7	−1.32106
$635$	−1.70006e6	−0.167313
$636$	0	0
$637$	2.79644e6	0.273059
$638$	4.84261e6	0.471007
$639$	0	0
$640$	−6.48615e6	−0.625946
$641$	−9.99391e6	−0.960705	−0.480353	−	0.877075i	$-0.659491\pi$
$641$	−9.99391e6	−0.960705	−0.480353	+	0.877075i	$0.659491\pi$
$642$	0	0
$643$	1.03306e7	0.985364	0.492682	−	0.870209i	$-0.336017\pi$
$643$	1.03306e7	0.985364	0.492682	+	0.870209i	$0.336017\pi$
$644$	2.51460e6	0.238921
$645$	0	0
$646$	−9.21687e6	−0.868964
$647$	−404393.	−0.0379790	−0.0189895	−	0.999820i	$-0.506045\pi$
$647$	−404393.	−0.0379790	−0.0189895	+	0.999820i	$0.506045\pi$
$648$	0	0
$649$	−2.54193e6	−0.236893
$650$	2.09304e7	1.94309
$651$	0	0
$652$	−9.78437e6	−0.901392
$653$	8.98734e6	0.824799	0.412399	−	0.911003i	$-0.364691\pi$
$653$	8.98734e6	0.824799	0.412399	+	0.911003i	$0.364691\pi$
$654$	0	0
$655$	−2.27809e6	−0.207476
$656$	−3.08064e7	−2.79499
$657$	0	0
$658$	−2.49519e7	−2.24667
$659$	−2.91006e6	−0.261028	−0.130514	−	0.991446i	$-0.541663\pi$
$659$	−2.91006e6	−0.261028	−0.130514	+	0.991446i	$0.541663\pi$
$660$	0	0
$661$	1.38558e7	1.23347	0.616733	−	0.787173i	$-0.288456\pi$
$661$	1.38558e7	1.23347	0.616733	+	0.787173i	$0.288456\pi$
$662$	−2.64533e7	−2.34603
$663$	0	0
$664$	−4.91577e7	−4.32684
$665$	−2.24168e6	−0.196571
$666$	0	0
$667$	−1.90647e6	−0.165927
$668$	−3.30843e7	−2.86867
$669$	0	0
$670$	1.49852e7	1.28966
$671$	−675234.	−0.0578959
$672$	0	0
$673$	8.27036e6	0.703861	0.351930	−	0.936026i	$-0.385525\pi$
$673$	8.27036e6	0.703861	0.351930	+	0.936026i	$0.385525\pi$
$674$	−1.11069e7	−0.941769
$675$	0	0
$676$	2.17225e7	1.82829
$677$	1.34494e6	0.112780	0.0563899	−	0.998409i	$-0.482041\pi$
$677$	1.34494e6	0.112780	0.0563899	+	0.998409i	$0.482041\pi$
$678$	0	0
$679$	−43462.4	−0.00361776
$680$	−1.58895e7	−1.31776
$681$	0	0
$682$	1.81765e6	0.149641
$683$	−8.12622e6	−0.666556	−0.333278	−	0.942829i	$-0.608155\pi$
$683$	−8.12622e6	−0.666556	−0.333278	+	0.942829i	$0.608155\pi$
$684$	0	0
$685$	−1.16901e6	−0.0951901
$686$	2.50313e7	2.03083
$687$	0	0
$688$	3.08682e7	2.48623
$689$	−1.25480e7	−1.00699
$690$	0	0
$691$	−1.43559e7	−1.14376	−0.571881	−	0.820337i	$-0.693786\pi$
$691$	−1.43559e7	−1.14376	−0.571881	+	0.820337i	$0.693786\pi$
$692$	3.84135e7	3.04943
$693$	0	0
$694$	3.71745e7	2.92986
$695$	1.85159e6	0.145406
$696$	0	0
$697$	−1.14081e7	−0.889473
$698$	2.35646e7	1.83072
$699$	0	0
$700$	2.32802e7	1.79573
$701$	2.70976e6	0.208274	0.104137	−	0.994563i	$-0.466792\pi$
$701$	2.70976e6	0.208274	0.104137	+	0.994563i	$0.466792\pi$
$702$	0	0
$703$	−8.40907e6	−0.641741
$704$	4.49446e6	0.341779
$705$	0	0
$706$	−3.93193e7	−2.96889
$707$	−187557.	−0.0141119
$708$	0	0
$709$	−1.76568e6	−0.131916	−0.0659578	−	0.997822i	$-0.521010\pi$
$709$	−1.76568e6	−0.131916	−0.0659578	+	0.997822i	$0.521010\pi$
$710$	1.41298e7	1.05194
$711$	0	0
$712$	2.43210e7	1.79797
$713$	−715586.	−0.0527154
$714$	0	0
$715$	1.30014e6	0.0951096
$716$	4.10258e6	0.299071
$717$	0	0
$718$	1.39751e7	1.01168
$719$	2.06759e7	1.49157	0.745783	−	0.666189i	$-0.232076\pi$
$719$	2.06759e7	1.49157	0.745783	+	0.666189i	$0.232076\pi$
$720$	0	0
$721$	−6.87741e6	−0.492705
$722$	−2.04131e7	−1.45736
$723$	0	0
$724$	6.77823e6	0.480585
$725$	−1.76502e7	−1.24711
$726$	0	0
$727$	1.77199e7	1.24344	0.621720	−	0.783240i	$-0.286434\pi$
$727$	1.77199e7	1.24344	0.621720	+	0.783240i	$0.286434\pi$
$728$	4.92803e7	3.44623
$729$	0	0
$730$	−2.09286e6	−0.145356
$731$	1.14311e7	0.791212
$732$	0	0
$733$	−5.13488e6	−0.352996	−0.176498	−	0.984301i	$-0.556477\pi$
$733$	−5.13488e6	−0.352996	−0.176498	+	0.984301i	$0.556477\pi$
$734$	−4.67614e7	−3.20366
$735$	0	0
$736$	−4.23128e6	−0.287924
$737$	−3.42347e6	−0.232166
$738$	0	0
$739$	−8.41879e6	−0.567072	−0.283536	−	0.958962i	$-0.591508\pi$
$739$	−8.41879e6	−0.567072	−0.283536	+	0.958962i	$0.591508\pi$
$740$	−2.37449e7	−1.59401
$741$	0	0
$742$	−1.93925e7	−1.29308
$743$	3.63942e6	0.241858	0.120929	−	0.992661i	$-0.461413\pi$
$743$	3.63942e6	0.241858	0.120929	+	0.992661i	$0.461413\pi$
$744$	0	0
$745$	−1.19238e7	−0.787090
$746$	−3.23676e7	−2.12943
$747$	0	0
$748$	5.94579e6	0.388558
$749$	1.30761e7	0.851673
$750$	0	0
$751$	−1.91843e7	−1.24121	−0.620605	−	0.784123i	$-0.713113\pi$
$751$	−1.91843e7	−1.24121	−0.620605	+	0.784123i	$0.713113\pi$
$752$	6.27222e7	4.04461
$753$	0	0
$754$	−6.11972e7	−3.92016
$755$	9.43233e6	0.602215
$756$	0	0
$757$	213745.	0.0135567	0.00677837	−	0.999977i	$-0.497842\pi$
$757$	213745.	0.0135567	0.00677837	+	0.999977i	$0.497842\pi$
$758$	8.90848e6	0.563158
$759$	0	0
$760$	1.04181e7	0.654267
$761$	−1.67486e7	−1.04838	−0.524189	−	0.851602i	$-0.675631\pi$
$761$	−1.67486e7	−1.04838	−0.524189	+	0.851602i	$0.675631\pi$
$762$	0	0
$763$	−1.93180e7	−1.20130
$764$	−2.83248e7	−1.75563
$765$	0	0
$766$	−1.74717e7	−1.07588
$767$	3.21230e7	1.97164
$768$	0	0
$769$	1.56362e7	0.953486	0.476743	−	0.879043i	$-0.341817\pi$
$769$	1.56362e7	0.953486	0.476743	+	0.879043i	$0.341817\pi$
$770$	2.00932e6	0.122130
$771$	0	0
$772$	4.86370e7	2.93713
$773$	−7.70701e6	−0.463913	−0.231957	−	0.972726i	$-0.574513\pi$
$773$	−7.70701e6	−0.463913	−0.231957	+	0.972726i	$0.574513\pi$
$774$	0	0
$775$	−6.62491e6	−0.396210
$776$	201990.	0.0120413
$777$	0	0
$778$	4.79232e7	2.83855
$779$	7.47986e6	0.441621
$780$	0	0
$781$	−3.22805e6	−0.189371
$782$	−3.25245e6	−0.190193
$783$	0	0
$784$	−1.08639e7	−0.631239
$785$	6.99832e6	0.405340
$786$	0	0
$787$	1.41953e7	0.816974	0.408487	−	0.912764i	$-0.366057\pi$
$787$	1.41953e7	0.816974	0.408487	+	0.912764i	$0.366057\pi$
$788$	1.94043e7	1.11322
$789$	0	0
$790$	7.34292e6	0.418602
$791$	−2.09235e7	−1.18903
$792$	0	0
$793$	8.53309e6	0.481863
$794$	−2.23007e7	−1.25536
$795$	0	0
$796$	3.02159e6	0.169026
$797$	1.72207e6	0.0960297	0.0480149	−	0.998847i	$-0.484711\pi$
$797$	1.72207e6	0.0960297	0.0480149	+	0.998847i	$0.484711\pi$
$798$	0	0
$799$	2.32272e7	1.28715
$800$	−3.91733e7	−2.16404
$801$	0	0
$802$	−4.30411e7	−2.36291
$803$	478126.	0.0261670
$804$	0	0
$805$	−791044.	−0.0430240
$806$	−2.29701e7	−1.24545
$807$	0	0
$808$	871663.	0.0469699
$809$	2.11981e7	1.13874	0.569372	−	0.822080i	$-0.307187\pi$
$809$	2.11981e7	1.13874	0.569372	+	0.822080i	$0.307187\pi$
$810$	0	0
$811$	2.62522e7	1.40157	0.700784	−	0.713374i	$-0.252834\pi$
$811$	2.62522e7	1.40157	0.700784	+	0.713374i	$0.252834\pi$
$812$	−6.80678e7	−3.62286
$813$	0	0
$814$	7.53746e6	0.398716
$815$	3.07798e6	0.162320
$816$	0	0
$817$	−7.49489e6	−0.392835
$818$	−4.60689e7	−2.40727
$819$	0	0
$820$	2.11211e7	1.09694
$821$	−1.03058e7	−0.533608	−0.266804	−	0.963751i	$-0.585968\pi$
$821$	−1.03058e7	−0.533608	−0.266804	+	0.963751i	$0.585968\pi$
$822$	0	0
$823$	−3.59771e7	−1.85151	−0.925756	−	0.378121i	$-0.876570\pi$
$823$	−3.59771e7	−1.85151	−0.925756	+	0.378121i	$0.876570\pi$
$824$	3.19625e7	1.63992
$825$	0	0
$826$	4.96451e7	2.53178
$827$	2.68608e7	1.36570	0.682850	−	0.730558i	$-0.260740\pi$
$827$	2.68608e7	1.36570	0.682850	+	0.730558i	$0.260740\pi$
$828$	0	0
$829$	1.79726e7	0.908290	0.454145	−	0.890928i	$-0.349945\pi$
$829$	1.79726e7	0.908290	0.454145	+	0.890928i	$0.349945\pi$
$830$	2.53291e7	1.27622
$831$	0	0
$832$	−5.67975e7	−2.84460
$833$	−4.02308e6	−0.200884
$834$	0	0
$835$	1.04077e7	0.516581
$836$	−3.89842e6	−0.192918
$837$	0	0
$838$	−4.62912e7	−2.27714
$839$	6.90706e6	0.338757	0.169378	−	0.985551i	$-0.445824\pi$
$839$	6.90706e6	0.338757	0.169378	+	0.985551i	$0.445824\pi$
$840$	0	0
$841$	3.10953e7	1.51602
$842$	2.66609e6	0.129597
$843$	0	0
$844$	4.32263e7	2.08877
$845$	−6.83350e6	−0.329232
$846$	0	0
$847$	1.81140e7	0.867572
$848$	4.87475e7	2.32789
$849$	0	0
$850$	−3.01113e7	−1.42949
$851$	−2.96740e6	−0.140460
$852$	0	0
$853$	−2.22992e7	−1.04934	−0.524671	−	0.851305i	$-0.675812\pi$
$853$	−2.22992e7	−1.04934	−0.524671	+	0.851305i	$0.675812\pi$
$854$	1.31876e7	0.618760
$855$	0	0
$856$	−6.07705e7	−2.83471
$857$	1.43986e6	0.0669683	0.0334842	−	0.999439i	$-0.489340\pi$
$857$	1.43986e6	0.0669683	0.0334842	+	0.999439i	$0.489340\pi$
$858$	0	0
$859$	−3.21807e7	−1.48803	−0.744017	−	0.668161i	$-0.767082\pi$
$859$	−3.21807e7	−1.48803	−0.744017	+	0.668161i	$0.767082\pi$
$860$	−2.11635e7	−0.975758
$861$	0	0
$862$	−4.45220e7	−2.04083
$863$	1.47425e6	0.0673818	0.0336909	−	0.999432i	$-0.489274\pi$
$863$	1.47425e6	0.0673818	0.0336909	+	0.999432i	$0.489274\pi$
$864$	0	0
$865$	−1.20842e7	−0.549132
$866$	7.21360e7	3.26857
$867$	0	0
$868$	−2.55489e7	−1.15099
$869$	−1.67754e6	−0.0753568
$870$	0	0
$871$	4.32632e7	1.93230
$872$	8.97795e7	3.99840
$873$	0	0
$874$	2.13250e6	0.0944303
$875$	−1.66383e7	−0.734663
$876$	0	0
$877$	7.33628e6	0.322090	0.161045	−	0.986947i	$-0.448514\pi$
$877$	7.33628e6	0.322090	0.161045	+	0.986947i	$0.448514\pi$
$878$	−3.67084e6	−0.160705
$879$	0	0
$880$	−5.05088e6	−0.219867
$881$	−3.95062e7	−1.71485	−0.857424	−	0.514610i	$-0.827937\pi$
$881$	−3.95062e7	−1.71485	−0.857424	+	0.514610i	$0.827937\pi$
$882$	0	0
$883$	2.54183e7	1.09709	0.548547	−	0.836120i	$-0.315181\pi$
$883$	2.54183e7	1.09709	0.548547	+	0.836120i	$0.315181\pi$
$884$	−7.51384e7	−3.23393
$885$	0	0
$886$	5.52710e7	2.36544
$887$	−6.65524e6	−0.284024	−0.142012	−	0.989865i	$-0.545357\pi$
$887$	−6.65524e6	−0.284024	−0.142012	+	0.989865i	$0.545357\pi$
$888$	0	0
$889$	−7.58541e6	−0.321903
$890$	−1.25317e7	−0.530317
$891$	0	0
$892$	1.41857e7	0.596952
$893$	−1.52291e7	−0.639067
$894$	0	0
$895$	−1.29060e6	−0.0538558
$896$	−2.89403e7	−1.20429
$897$	0	0
$898$	9.76505e6	0.404095
$899$	1.93702e7	0.799348
$900$	0	0
$901$	1.80521e7	0.740823
$902$	−6.70456e6	−0.274381
$903$	0	0
$904$	9.72410e7	3.95757
$905$	−2.13230e6	−0.0865421
$906$	0	0
$907$	−2.02162e7	−0.815984	−0.407992	−	0.912986i	$-0.633771\pi$
$907$	−2.02162e7	−0.815984	−0.407992	+	0.912986i	$0.633771\pi$
$908$	9.34114e7	3.75998
$909$	0	0
$910$	−2.53923e7	−1.01648
$911$	2.75967e6	0.110170	0.0550848	−	0.998482i	$-0.482457\pi$
$911$	2.75967e6	0.110170	0.0550848	+	0.998482i	$0.482457\pi$
$912$	0	0
$913$	−5.78660e6	−0.229745
$914$	2.09706e7	0.830320
$915$	0	0
$916$	−6.44105e7	−2.53640
$917$	−1.01645e7	−0.399175
$918$	0	0
$919$	−2.47471e7	−0.966575	−0.483288	−	0.875462i	$-0.660557\pi$
$919$	−2.47471e7	−0.966575	−0.483288	+	0.875462i	$0.660557\pi$
$920$	3.67635e6	0.143201
$921$	0	0
$922$	2.38638e7	0.924513
$923$	4.07937e7	1.57612
$924$	0	0
$925$	−2.74723e7	−1.05570
$926$	6.70607e7	2.57004
$927$	0	0
$928$	1.14537e8	4.36591
$929$	−4.64223e7	−1.76477	−0.882383	−	0.470531i	$-0.844062\pi$
$929$	−4.64223e7	−1.76477	−0.882383	+	0.470531i	$0.844062\pi$
$930$	0	0
$931$	2.63777e6	0.0997385
$932$	−7.00718e7	−2.64243
$933$	0	0
$934$	5.01307e7	1.88034
$935$	−1.87043e6	−0.0699702
$936$	0	0
$937$	−1.80567e7	−0.671877	−0.335939	−	0.941884i	$-0.609054\pi$
$937$	−1.80567e7	−0.671877	−0.335939	+	0.941884i	$0.609054\pi$
$938$	6.68620e7	2.48126
$939$	0	0
$940$	−4.30029e7	−1.58737
$941$	−9.37905e6	−0.345291	−0.172645	−	0.984984i	$-0.555231\pi$
$941$	−9.37905e6	−0.345291	−0.172645	+	0.984984i	$0.555231\pi$
$942$	0	0
$943$	2.63950e6	0.0966590
$944$	−1.24794e8	−4.55789
$945$	0	0
$946$	6.71803e6	0.244070
$947$	−4.35017e6	−0.157627	−0.0788137	−	0.996889i	$-0.525113\pi$
$947$	−4.35017e6	−0.157627	−0.0788137	+	0.996889i	$0.525113\pi$
$948$	0	0
$949$	−6.04220e6	−0.217786
$950$	1.97428e7	0.709740
$951$	0	0
$952$	−7.08967e7	−2.53532
$953$	−1.60322e7	−0.571822	−0.285911	−	0.958256i	$-0.592296\pi$
$953$	−1.60322e7	−0.571822	−0.285911	+	0.958256i	$0.592296\pi$
$954$	0	0
$955$	8.91045e6	0.316148
$956$	5.14300e7	1.82000
$957$	0	0
$958$	−2.18257e7	−0.768340
$959$	−5.21596e6	−0.183142
$960$	0	0
$961$	−2.13586e7	−0.746045
$962$	−9.52527e7	−3.31848
$963$	0	0
$964$	−9.37910e6	−0.325064
$965$	−1.53003e7	−0.528909
$966$	0	0
$967$	−4.05998e7	−1.39623	−0.698116	−	0.715984i	$-0.745978\pi$
$967$	−4.05998e7	−1.39623	−0.698116	+	0.715984i	$0.745978\pi$
$968$	−8.41840e7	−2.88763
$969$	0	0
$970$	−104078.	−0.00355164
$971$	1.54267e6	0.0525080	0.0262540	−	0.999655i	$-0.491642\pi$
$971$	1.54267e6	0.0525080	0.0262540	+	0.999655i	$0.491642\pi$
$972$	0	0
$973$	8.26152e6	0.279755
$974$	−6.89732e7	−2.32961
$975$	0	0
$976$	−3.31501e7	−1.11394
$977$	5.14968e7	1.72601	0.863006	−	0.505193i	$-0.168579\pi$
$977$	5.14968e7	1.72601	0.863006	+	0.505193i	$0.168579\pi$
$978$	0	0
$979$	2.86295e6	0.0954678
$980$	7.44835e6	0.247739
$981$	0	0
$982$	−6.07476e7	−2.01025
$983$	−2.71081e7	−0.894777	−0.447389	−	0.894340i	$-0.647646\pi$
$983$	−2.71081e7	−0.894777	−0.447389	+	0.894340i	$0.647646\pi$
$984$	0	0
$985$	−6.10421e6	−0.200465
$986$	8.80409e7	2.88398
$987$	0	0
$988$	4.92652e7	1.60564
$989$	−2.64480e6	−0.0859809
$990$	0	0
$991$	−3.04379e7	−0.984534	−0.492267	−	0.870444i	$-0.663832\pi$
$991$	−3.04379e7	−0.984534	−0.492267	+	0.870444i	$0.663832\pi$
$992$	4.29909e7	1.38707
$993$	0	0
$994$	6.30454e7	2.02389
$995$	−950535.	−0.0304376
$996$	0	0
$997$	3.27678e7	1.04402	0.522010	−	0.852939i	$-0.325182\pi$
$997$	3.27678e7	1.04402	0.522010	+	0.852939i	$0.325182\pi$
$998$	2.15896e6	0.0686147
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.6.a.d.1.2	yes	2
3.2	odd	2		27.6.a.b.1.1	✓	2
4.3	odd	2		432.6.a.v.1.1		2
5.4	even	2		675.6.a.f.1.1		2
9.2	odd	6		81.6.c.h.28.2		4
9.4	even	3		81.6.c.d.55.1		4
9.5	odd	6		81.6.c.h.55.2		4
9.7	even	3		81.6.c.d.28.1		4
12.11	even	2		432.6.a.k.1.2		2
15.14	odd	2		675.6.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.a.b.1.1	✓	2	3.2	odd	2
27.6.a.d.1.2	yes	2	1.1	even	1	trivial
81.6.c.d.28.1		4	9.7	even	3
81.6.c.d.55.1		4	9.4	even	3
81.6.c.h.28.2		4	9.2	odd	6
81.6.c.h.55.2		4	9.5	odd	6
432.6.a.k.1.2		2	12.11	even	2
432.6.a.v.1.1		2	4.3	odd	2
675.6.a.f.1.1		2	5.4	even	2
675.6.a.n.1.2		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q+10.6847 q^{2} +82.1619 q^{4} -25.8466 q^{5} -115.324 q^{7} +535.963 q^{8} -276.162 q^{10} +63.0909 q^{11} -797.295 q^{13} -1232.20 q^{14} +3097.40 q^{16} +1147.02 q^{17} -752.057 q^{19} -2123.61 q^{20} +674.105 q^{22} -265.386 q^{23} -2456.95 q^{25} -8518.83 q^{26} -9475.23 q^{28} +7183.76 q^{29} +2696.39 q^{31} +15943.8 q^{32} +12255.5 q^{34} +2980.73 q^{35} +11181.4 q^{37} -8035.47 q^{38} -13852.8 q^{40} -9945.87 q^{41} +9965.85 q^{43} +5183.67 q^{44} -2835.56 q^{46} +20250.0 q^{47} -3507.41 q^{49} -26251.7 q^{50} -65507.3 q^{52} +15738.2 q^{53} -1630.69 q^{55} -61809.3 q^{56} +76756.0 q^{58} -40290.0 q^{59} -10702.5 q^{61} +28810.0 q^{62} +71237.8 q^{64} +20607.4 q^{65} -54262.5 q^{67} +94241.6 q^{68} +31848.1 q^{70} -51165.1 q^{71} +7578.37 q^{73} +119470. q^{74} -61790.4 q^{76} -7275.89 q^{77} -26589.2 q^{79} -80057.2 q^{80} -106268. q^{82} -91718.4 q^{83} -29646.6 q^{85} +106482. q^{86} +33814.4 q^{88} +45378.2 q^{89} +91947.2 q^{91} -21804.7 q^{92} +216364. q^{94} +19438.1 q^{95} +376.873 q^{97} -37475.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{2} + 53 q^{4} + 72 q^{5} - 8 q^{7} + 639 q^{8} - 441 q^{10} + 522 q^{11} - 704 q^{13} - 1413 q^{14} + 3857 q^{16} + 216 q^{17} - 2840 q^{19} - 4977 q^{20} - 99 q^{22} - 36 q^{23} + 3992 q^{25}+ \cdots - 28566 q^{98}+O(q^{100}) \) 2 * q + 9 * q^2 + 53 * q^4 + 72 * q^5 - 8 * q^7 + 639 * q^8 - 441 * q^10 + 522 * q^11 - 704 * q^13 - 1413 * q^14 + 3857 * q^16 + 216 * q^17 - 2840 * q^19 - 4977 * q^20 - 99 * q^22 - 36 * q^23 + 3992 * q^25 - 8676 * q^26 - 12605 * q^28 + 12240 * q^29 - 1064 * q^31 + 11367 * q^32 + 13824 * q^34 + 13482 * q^35 + 9004 * q^37 - 4518 * q^38 - 3771 * q^40 + 5688 * q^41 + 784 * q^43 - 8199 * q^44 - 3222 * q^46 + 1116 * q^47 - 8796 * q^49 - 37116 * q^50 - 68228 * q^52 + 4536 * q^53 + 43272 * q^55 - 50751 * q^56 + 68238 * q^58 - 67320 * q^59 - 49904 * q^61 + 35145 * q^62 + 54641 * q^64 + 29736 * q^65 - 42176 * q^67 + 121392 * q^68 + 14157 * q^70 - 43848 * q^71 + 47218 * q^73 + 123138 * q^74 - 902 * q^76 + 41976 * q^77 - 49616 * q^79 - 5733 * q^80 - 132606 * q^82 - 102294 * q^83 - 120744 * q^85 + 121950 * q^86 + 81099 * q^88 - 35856 * q^89 + 101960 * q^91 - 28494 * q^92 + 248598 * q^94 - 184860 * q^95 + 169966 * q^97 - 28566 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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