LMFDB - Embedded newform 27.6.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,6,Mod(1,27)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	27.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:	$4.33036313495$
Analytic rank:	$1$
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			$-1.56155$ 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)

$f(q)$	$=$	$q+1.68466 q^{2} -29.1619 q^{4} -97.8466 q^{5} +107.324 q^{7} -103.037 q^{8} +O(q^{10})$	\(q+1.68466 q^{2} -29.1619 q^{4} -97.8466 q^{5} +107.324 q^{7} -103.037 q^{8} -164.838 q^{10} -458.909 q^{11} +93.2954 q^{13} +180.804 q^{14} +759.600 q^{16} +931.023 q^{17} -2087.94 q^{19} +2853.39 q^{20} -773.105 q^{22} -229.386 q^{23} +6448.95 q^{25} +157.171 q^{26} -3129.77 q^{28} -5056.24 q^{29} -3760.39 q^{31} +4576.85 q^{32} +1568.46 q^{34} -10501.3 q^{35} -2177.43 q^{37} -3517.47 q^{38} +10081.8 q^{40} -15633.9 q^{41} -9181.85 q^{43} +13382.7 q^{44} -386.438 q^{46} +19134.0 q^{47} -5288.59 q^{49} +10864.3 q^{50} -2720.67 q^{52} +11202.2 q^{53} +44902.7 q^{55} -11058.3 q^{56} -8518.04 q^{58} +27030.0 q^{59} -39201.5 q^{61} -6334.98 q^{62} -16596.8 q^{64} -9128.64 q^{65} +12086.5 q^{67} -27150.4 q^{68} -17691.1 q^{70} -7317.07 q^{71} +39639.6 q^{73} -3668.23 q^{74} +60888.4 q^{76} -49251.9 q^{77} -23026.8 q^{79} -74324.2 q^{80} -26337.7 q^{82} +10575.6 q^{83} -91097.4 q^{85} -15468.3 q^{86} +47284.6 q^{88} +81234.2 q^{89} +10012.8 q^{91} +6689.35 q^{92} +32234.2 q^{94} +204298. q^{95} +169589. q^{97} -8909.47 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{2} + 53 q^{4} - 72 q^{5} - 8 q^{7} - 639 q^{8}+O(q^{10}) $ 2 * q - 9 * q^2 + 53 * q^4 - 72 * q^5 - 8 * q^7 - 639 * q^8	$ 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}+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

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 $

$2$	1.68466	0.297808	0.148904	−	0.988852i	$-0.452425\pi$
$2$	1.68466	0.297808	0.148904	+	0.988852i	$0.452425\pi$
$3$	0	0
$3$	0	0
$4$	−29.1619	−0.911310
$5$	−97.8466	−1.75033	−0.875166	−	0.483822i	$-0.839248\pi$
$5$	−97.8466	−1.75033	−0.875166	+	0.483822i	$0.839248\pi$
$6$	0	0
$7$	107.324	0.827849	0.413925	−	0.910311i	$-0.364158\pi$
$7$	107.324	0.827849	0.413925	+	0.910311i	$0.364158\pi$
$8$	−103.037	−0.569204
$9$	0	0
$10$	−164.838	−0.521264
$11$	−458.909	−1.14352	−0.571762	−	0.820420i	$-0.693740\pi$
$11$	−458.909	−1.14352	−0.571762	+	0.820420i	$0.693740\pi$
$12$	0	0
$13$	93.2954	0.153109	0.0765547	−	0.997065i	$-0.475608\pi$
$13$	93.2954	0.153109	0.0765547	+	0.997065i	$0.475608\pi$
$14$	180.804	0.246540
$15$	0	0
$16$	759.600	0.741796
$17$	931.023	0.781336	0.390668	−	0.920532i	$-0.372244\pi$
$17$	931.023	0.781336	0.390668	+	0.920532i	$0.372244\pi$
$18$	0	0
$19$	−2087.94	−1.32689	−0.663445	−	0.748225i	$-0.730906\pi$
$19$	−2087.94	−1.32689	−0.663445	+	0.748225i	$0.730906\pi$
$20$	2853.39	1.59510
$21$	0	0
$22$	−773.105	−0.340551
$23$	−229.386	−0.0904166	−0.0452083	−	0.998978i	$-0.514395\pi$
$23$	−229.386	−0.0904166	−0.0452083	+	0.998978i	$0.514395\pi$
$24$	0	0
$25$	6448.95	2.06367
$26$	157.171	0.0455973
$27$	0	0
$28$	−3129.77	−0.754427
$29$	−5056.24	−1.11643	−0.558216	−	0.829695i	$-0.688514\pi$
$29$	−5056.24	−1.11643	−0.558216	+	0.829695i	$0.688514\pi$
$30$	0	0
$31$	−3760.39	−0.702795	−0.351398	−	0.936226i	$-0.614293\pi$
$31$	−3760.39	−0.702795	−0.351398	+	0.936226i	$0.614293\pi$
$32$	4576.85	0.790117
$33$	0	0
$34$	1568.46	0.232688
$35$	−10501.3	−1.44901
$36$	0	0
$37$	−2177.43	−0.261481	−0.130740	−	0.991417i	$-0.541735\pi$
$37$	−2177.43	−0.261481	−0.130740	+	0.991417i	$0.541735\pi$
$38$	−3517.47	−0.395159
$39$	0	0
$40$	10081.8	0.996297
$41$	−15633.9	−1.45247	−0.726235	−	0.687447i	$-0.758731\pi$
$41$	−15633.9	−1.45247	−0.726235	+	0.687447i	$0.758731\pi$
$42$	0	0
$43$	−9181.85	−0.757285	−0.378642	−	0.925543i	$-0.623609\pi$
$43$	−9181.85	−0.757285	−0.378642	+	0.925543i	$0.623609\pi$
$44$	13382.7	1.04210
$45$	0	0
$46$	−386.438	−0.0269268
$47$	19134.0	1.26346	0.631728	−	0.775190i	$-0.282346\pi$
$47$	19134.0	1.26346	0.631728	+	0.775190i	$0.282346\pi$
$48$	0	0
$49$	−5288.59	−0.314666
$50$	10864.3	0.614577
$51$	0	0
$52$	−2720.67	−0.139530
$53$	11202.2	0.547789	0.273894	−	0.961760i	$-0.411688\pi$
$53$	11202.2	0.547789	0.273894	+	0.961760i	$0.411688\pi$
$54$	0	0
$55$	44902.7	2.00155
$56$	−11058.3	−0.471215
$57$	0	0
$58$	−8518.04	−0.332483
$59$	27030.0	1.01092	0.505460	−	0.862850i	$-0.331323\pi$
$59$	27030.0	1.01092	0.505460	+	0.862850i	$0.331323\pi$
$60$	0	0
$61$	−39201.5	−1.34889	−0.674447	−	0.738324i	$-0.735618\pi$
$61$	−39201.5	−1.34889	−0.674447	+	0.738324i	$0.735618\pi$
$62$	−6334.98	−0.209298
$63$	0	0
$64$	−16596.8	−0.506493
$65$	−9128.64	−0.267992
$66$	0	0
$67$	12086.5	0.328938	0.164469	−	0.986382i	$-0.447409\pi$
$67$	12086.5	0.328938	0.164469	+	0.986382i	$0.447409\pi$
$68$	−27150.4	−0.712040
$69$	0	0
$70$	−17691.1	−0.431528
$71$	−7317.07	−0.172263	−0.0861313	−	0.996284i	$-0.527450\pi$
$71$	−7317.07	−0.172263	−0.0861313	+	0.996284i	$0.527450\pi$
$72$	0	0
$73$	39639.6	0.870608	0.435304	−	0.900284i	$-0.356641\pi$
$73$	39639.6	0.870608	0.435304	+	0.900284i	$0.356641\pi$
$74$	−3668.23	−0.0778712
$75$	0	0
$76$	60888.4	1.20921
$77$	−49251.9	−0.946664
$78$	0	0
$79$	−23026.8	−0.415113	−0.207556	−	0.978223i	$-0.566551\pi$
$79$	−23026.8	−0.415113	−0.207556	+	0.978223i	$0.566551\pi$
$80$	−74324.2	−1.29839
$81$	0	0
$82$	−26337.7	−0.432557
$83$	10575.6	0.168504	0.0842522	−	0.996444i	$-0.473150\pi$
$83$	10575.6	0.168504	0.0842522	+	0.996444i	$0.473150\pi$
$84$	0	0
$85$	−91097.4	−1.36760
$86$	−15468.3	−0.225526
$87$	0	0
$88$	47284.6	0.650898
$89$	81234.2	1.08709	0.543543	−	0.839381i	$-0.317083\pi$
$89$	81234.2	1.08709	0.543543	+	0.839381i	$0.317083\pi$
$90$	0	0
$91$	10012.8	0.126751
$92$	6689.35	0.0823975
$93$	0	0
$94$	32234.2	0.376268
$95$	204298.	2.32250
$96$	0	0
$97$	169589.	1.83007	0.915037	−	0.403370i	$-0.132161\pi$
$97$	169589.	1.83007	0.915037	+	0.403370i	$0.132161\pi$
$98$	−8909.47	−0.0937102
$99$	0	0
$100$	−188064.	−1.88064
$101$	−34157.7	−0.333184	−0.166592	−	0.986026i	$-0.553276\pi$
$101$	−34157.7	−0.333184	−0.166592	+	0.986026i	$0.553276\pi$
$102$	0	0
$103$	−105124.	−0.976354	−0.488177	−	0.872745i	$-0.662338\pi$
$103$	−105124.	−0.976354	−0.488177	+	0.872745i	$0.662338\pi$
$104$	−9612.87	−0.0871505
$105$	0	0
$106$	18871.9	0.163136
$107$	51044.3	0.431011	0.215505	−	0.976503i	$-0.430860\pi$
$107$	51044.3	0.431011	0.215505	+	0.976503i	$0.430860\pi$
$108$	0	0
$109$	−6154.60	−0.0496174	−0.0248087	−	0.999692i	$-0.507898\pi$
$109$	−6154.60	−0.0496174	−0.0248087	+	0.999692i	$0.507898\pi$
$110$	75645.7	0.596077
$111$	0	0
$112$	81523.2	0.614095
$113$	−216512.	−1.59509	−0.797545	−	0.603259i	$-0.793868\pi$
$113$	−216512.	−1.59509	−0.797545	+	0.603259i	$0.793868\pi$
$114$	0	0
$115$	22444.7	0.158259
$116$	147450.	1.01742
$117$	0	0
$118$	45536.4	0.301060
$119$	99920.9	0.646828
$120$	0	0
$121$	49546.5	0.307645
$122$	−66041.1	−0.401712
$123$	0	0
$124$	109660.	0.640465
$125$	−325238.	−1.86177
$126$	0	0
$127$	−101879.	−0.560499	−0.280249	−	0.959927i	$-0.590417\pi$
$127$	−101879.	−0.560499	−0.280249	+	0.959927i	$0.590417\pi$
$128$	−174419.	−0.940955
$129$	0	0
$130$	−15378.6	−0.0798104
$131$	−57463.0	−0.292557	−0.146278	−	0.989243i	$-0.546730\pi$
$131$	−57463.0	−0.292557	−0.146278	+	0.989243i	$0.546730\pi$
$132$	0	0
$133$	−224086.	−1.09846
$134$	20361.6	0.0979604
$135$	0	0
$136$	−95929.7	−0.444740
$137$	−287123.	−1.30697	−0.653487	−	0.756938i	$-0.726695\pi$
$137$	−287123.	−1.30697	−0.653487	+	0.756938i	$0.726695\pi$
$138$	0	0
$139$	−76090.5	−0.334036	−0.167018	−	0.985954i	$-0.553414\pi$
$139$	−76090.5	−0.334036	−0.167018	+	0.985954i	$0.553414\pi$
$140$	306237.	1.32050
$141$	0	0
$142$	−12326.8	−0.0513012
$143$	−42814.1	−0.175084
$144$	0	0
$145$	494736.	1.95413
$146$	66779.2	0.259274
$147$	0	0
$148$	63498.1	0.238290
$149$	96649.9	0.356645	0.178322	−	0.983972i	$-0.442933\pi$
$149$	96649.9	0.356645	0.178322	+	0.983972i	$0.442933\pi$
$150$	0	0
$151$	16015.1	0.0571594	0.0285797	−	0.999592i	$-0.490902\pi$
$151$	16015.1	0.0571594	0.0285797	+	0.999592i	$0.490902\pi$
$152$	215135.	0.755271
$153$	0	0
$154$	−82972.6	−0.281925
$155$	367941.	1.23013
$156$	0	0
$157$	−144300.	−0.467216	−0.233608	−	0.972331i	$-0.575053\pi$
$157$	−144300.	−0.467216	−0.233608	+	0.972331i	$0.575053\pi$
$158$	−38792.3	−0.123624
$159$	0	0
$160$	−447829.	−1.38297
$161$	−24618.6	−0.0748513
$162$	0	0
$163$	388550.	1.14546	0.572728	−	0.819746i	$-0.305885\pi$
$163$	388550.	1.14546	0.572728	+	0.819746i	$0.305885\pi$
$164$	455914.	1.32365
$165$	0	0
$166$	17816.3	0.0501820
$167$	−14916.1	−0.0413869	−0.0206935	−	0.999786i	$-0.506587\pi$
$167$	−14916.1	−0.0413869	−0.0206935	+	0.999786i	$0.506587\pi$
$168$	0	0
$169$	−362589.	−0.976558
$170$	−153468.	−0.407282
$171$	0	0
$172$	267760.	0.690121
$173$	−336346.	−0.854418	−0.427209	−	0.904153i	$-0.640503\pi$
$173$	−336346.	−0.854418	−0.427209	+	0.904153i	$0.640503\pi$
$174$	0	0
$175$	692127.	1.70840
$176$	−348587.	−0.848261
$177$	0	0
$178$	136852.	0.323743
$179$	729631.	1.70204	0.851022	−	0.525130i	$-0.175983\pi$
$179$	729631.	1.70204	0.851022	+	0.525130i	$0.175983\pi$
$180$	0	0
$181$	−612162.	−1.38890	−0.694449	−	0.719542i	$-0.744352\pi$
$181$	−612162.	−1.38890	−0.694449	+	0.719542i	$0.744352\pi$
$182$	16868.2	0.0377476
$183$	0	0
$184$	23635.3	0.0514655
$185$	213054.	0.457679
$186$	0	0
$187$	−427255.	−0.893476
$188$	−557983.	−1.15140
$189$	0	0
$190$	344173.	0.691659
$191$	−699128.	−1.38667	−0.693335	−	0.720615i	$-0.743859\pi$
$191$	−699128.	−1.38667	−0.693335	+	0.720615i	$0.743859\pi$
$192$	0	0
$193$	−777763.	−1.50298	−0.751492	−	0.659743i	$-0.770665\pi$
$193$	−777763.	−1.50298	−0.751492	+	0.659743i	$0.770665\pi$
$194$	285700.	0.545011
$195$	0	0
$196$	154225.	0.286758
$197$	−582685.	−1.06972	−0.534858	−	0.844942i	$-0.679635\pi$
$197$	−582685.	−1.06972	−0.534858	+	0.844942i	$0.679635\pi$
$198$	0	0
$199$	−291184.	−0.521237	−0.260618	−	0.965442i	$-0.583926\pi$
$199$	−291184.	−0.521237	−0.260618	+	0.965442i	$0.583926\pi$
$200$	−664481.	−1.17465
$201$	0	0
$202$	−57544.0	−0.0992251
$203$	−542655.	−0.924238
$204$	0	0
$205$	1.52972e6	2.54230
$206$	−177097.	−0.290766
$207$	0	0
$208$	70867.2	0.113576
$209$	958176.	1.51733
$210$	0	0
$211$	432153.	0.668239	0.334119	−	0.942531i	$-0.391561\pi$
$211$	432153.	0.668239	0.334119	+	0.942531i	$0.391561\pi$
$212$	−326677.	−0.499205
$213$	0	0
$214$	85992.2	0.128359
$215$	898413.	1.32550
$216$	0	0
$217$	−403580.	−0.581808
$218$	−10368.4	−0.0147765
$219$	0	0
$220$	−1.30945e6	−1.82403
$221$	86860.1	0.119630
$222$	0	0
$223$	−85615.7	−0.115290	−0.0576449	−	0.998337i	$-0.518359\pi$
$223$	−85615.7	−0.115290	−0.0576449	+	0.998337i	$0.518359\pi$
$224$	491205.	0.654098
$225$	0	0
$226$	−364748.	−0.475031
$227$	1.24413e6	1.60251	0.801253	−	0.598326i	$-0.204167\pi$
$227$	1.24413e6	1.60251	0.801253	+	0.598326i	$0.204167\pi$
$228$	0	0
$229$	1.21276e6	1.52822	0.764109	−	0.645087i	$-0.223179\pi$
$229$	1.21276e6	1.52822	0.764109	+	0.645087i	$0.223179\pi$
$230$	37811.6	0.0471309
$231$	0	0
$232$	520979.	0.635478
$233$	−406594.	−0.490649	−0.245325	−	0.969441i	$-0.578895\pi$
$233$	−406594.	−0.490649	−0.245325	+	0.969441i	$0.578895\pi$
$234$	0	0
$235$	−1.87219e6	−2.21147
$236$	−788248.	−0.921262
$237$	0	0
$238$	168333.	0.192631
$239$	1.07636e6	1.21888	0.609440	−	0.792832i	$-0.291394\pi$
$239$	1.07636e6	1.21888	0.609440	+	0.792832i	$0.291394\pi$
$240$	0	0
$241$	578726.	0.641845	0.320923	−	0.947105i	$-0.396007\pi$
$241$	578726.	0.641845	0.320923	+	0.947105i	$0.396007\pi$
$242$	83469.0	0.0916192
$243$	0	0
$244$	1.14319e6	1.22926
$245$	517471.	0.550770
$246$	0	0
$247$	−194796.	−0.203159
$248$	387459.	0.400034
$249$	0	0
$250$	−547914.	−0.554450
$251$	−794883.	−0.796377	−0.398189	−	0.917304i	$-0.630361\pi$
$251$	−794883.	−0.796377	−0.398189	+	0.917304i	$0.630361\pi$
$252$	0	0
$253$	105267.	0.103393
$254$	−171631.	−0.166921
$255$	0	0
$256$	237260.	0.226269
$257$	781833.	0.738382	0.369191	−	0.929353i	$-0.379635\pi$
$257$	781833.	0.738382	0.369191	+	0.929353i	$0.379635\pi$
$258$	0	0
$259$	−233690.	−0.216467
$260$	266209.	0.244224
$261$	0	0
$262$	−96805.6	−0.0871259
$263$	−1.98763e6	−1.77193	−0.885966	−	0.463750i	$-0.846503\pi$
$263$	−1.98763e6	−1.77193	−0.885966	+	0.463750i	$0.846503\pi$
$264$	0	0
$265$	−1.09610e6	−0.958813
$266$	−377509.	−0.327132
$267$	0	0
$268$	−352466.	−0.299764
$269$	175602.	0.147961	0.0739807	−	0.997260i	$-0.476430\pi$
$269$	175602.	0.147961	0.0739807	+	0.997260i	$0.476430\pi$
$270$	0	0
$271$	817902.	0.676516	0.338258	−	0.941053i	$-0.390162\pi$
$271$	817902.	0.676516	0.338258	+	0.941053i	$0.390162\pi$
$272$	707204.	0.579592
$273$	0	0
$274$	−483704.	−0.389228
$275$	−2.95948e6	−2.35985
$276$	0	0
$277$	664007.	0.519964	0.259982	−	0.965613i	$-0.416283\pi$
$277$	664007.	0.519964	0.259982	+	0.965613i	$0.416283\pi$
$278$	−128186.	−0.0994787
$279$	0	0
$280$	1.08202e6	0.824783
$281$	−194230.	−0.146741	−0.0733704	−	0.997305i	$-0.523376\pi$
$281$	−194230.	−0.146741	−0.0733704	+	0.997305i	$0.523376\pi$
$282$	0	0
$283$	2.42145e6	1.79726	0.898628	−	0.438712i	$-0.144565\pi$
$283$	2.42145e6	1.79726	0.898628	+	0.438712i	$0.144565\pi$
$284$	213380.	0.156985
$285$	0	0
$286$	−72127.1	−0.0521415
$287$	−1.67789e6	−1.20243
$288$	0	0
$289$	−553054.	−0.389514
$290$	833461.	0.581956
$291$	0	0
$292$	−1.15597e6	−0.793394
$293$	−1.07682e6	−0.732782	−0.366391	−	0.930461i	$-0.619407\pi$
$293$	−1.07682e6	−0.732782	−0.366391	+	0.930461i	$0.619407\pi$
$294$	0	0
$295$	−2.64480e6	−1.76945
$296$	224356.	0.148836
$297$	0	0
$298$	162822.	0.106212
$299$	−21400.7	−0.0138436
$300$	0	0
$301$	−985432.	−0.626917
$302$	26980.0	0.0170225
$303$	0	0
$304$	−1.58600e6	−0.984282
$305$	3.83573e6	2.36101
$306$	0	0
$307$	2.10721e6	1.27603	0.638016	−	0.770023i	$-0.279755\pi$
$307$	2.10721e6	1.27603	0.638016	+	0.770023i	$0.279755\pi$
$308$	1.43628e6	0.862705
$309$	0	0
$310$	619856.	0.366342
$311$	2.24972e6	1.31895	0.659475	−	0.751726i	$-0.270779\pi$
$311$	2.24972e6	1.31895	0.659475	+	0.751726i	$0.270779\pi$
$312$	0	0
$313$	−793013.	−0.457530	−0.228765	−	0.973482i	$-0.573469\pi$
$313$	−793013.	−0.457530	−0.228765	+	0.973482i	$0.573469\pi$
$314$	−243096.	−0.139141
$315$	0	0
$316$	671506.	0.378297
$317$	−1.91960e6	−1.07291	−0.536455	−	0.843929i	$-0.680237\pi$
$317$	−1.91960e6	−1.07291	−0.536455	+	0.843929i	$0.680237\pi$
$318$	0	0
$319$	2.32035e6	1.27667
$320$	1.62394e6	0.886531
$321$	0	0
$322$	−41474.0	−0.0222913
$323$	−1.94392e6	−1.03675
$324$	0	0
$325$	601658.	0.315967
$326$	654575.	0.341126
$327$	0	0
$328$	1.61087e6	0.826752
$329$	2.05353e6	1.04595
$330$	0	0
$331$	−1.36748e6	−0.686041	−0.343021	−	0.939328i	$-0.611450\pi$
$331$	−1.36748e6	−0.686041	−0.343021	+	0.939328i	$0.611450\pi$
$332$	−308406.	−0.153560
$333$	0	0
$334$	−25128.5	−0.0123254
$335$	−1.18262e6	−0.575751
$336$	0	0
$337$	−3.24284e6	−1.55543	−0.777716	−	0.628616i	$-0.783622\pi$
$337$	−3.24284e6	−1.55543	−0.777716	+	0.628616i	$0.783622\pi$
$338$	−610839.	−0.290827
$339$	0	0
$340$	2.65658e6	1.24631
$341$	1.72568e6	0.803663
$342$	0	0
$343$	−2.37138e6	−1.08834
$344$	946070.	0.431050
$345$	0	0
$346$	−566627.	−0.254453
$347$	−789261.	−0.351882	−0.175941	−	0.984401i	$-0.556297\pi$
$347$	−789261.	−0.351882	−0.175941	+	0.984401i	$0.556297\pi$
$348$	0	0
$349$	822378.	0.361417	0.180708	−	0.983537i	$-0.442161\pi$
$349$	822378.	0.361417	0.180708	+	0.983537i	$0.442161\pi$
$350$	1.16600e6	0.508777
$351$	0	0
$352$	−2.10036e6	−0.903517
$353$	2.61607e6	1.11741	0.558704	−	0.829367i	$-0.311299\pi$
$353$	2.61607e6	1.11741	0.558704	+	0.829367i	$0.311299\pi$
$354$	0	0
$355$	715950.	0.301517
$356$	−2.36894e6	−0.990672
$357$	0	0
$358$	1.22918e6	0.506883
$359$	−4.57405e6	−1.87312	−0.936558	−	0.350513i	$-0.886007\pi$
$359$	−4.57405e6	−1.87312	−0.936558	+	0.350513i	$0.886007\pi$
$360$	0	0
$361$	1.88341e6	0.760635
$362$	−1.03128e6	−0.413625
$363$	0	0
$364$	−291993.	−0.115510
$365$	−3.87860e6	−1.52385
$366$	0	0
$367$	3.13809e6	1.21619	0.608093	−	0.793866i	$-0.291935\pi$
$367$	3.13809e6	1.21619	0.608093	+	0.793866i	$0.291935\pi$
$368$	−174242.	−0.0670707
$369$	0	0
$370$	358924.	0.136301
$371$	1.20226e6	0.453486
$372$	0	0
$373$	−2.34983e6	−0.874508	−0.437254	−	0.899338i	$-0.644049\pi$
$373$	−2.34983e6	−0.874508	−0.437254	+	0.899338i	$0.644049\pi$
$374$	−719778.	−0.266085
$375$	0	0
$376$	−1.97150e6	−0.719164
$377$	−471724.	−0.170936
$378$	0	0
$379$	−2.27351e6	−0.813015	−0.406507	−	0.913648i	$-0.633253\pi$
$379$	−2.27351e6	−0.813015	−0.406507	+	0.913648i	$0.633253\pi$
$380$	−5.95773e6	−2.11652
$381$	0	0
$382$	−1.17779e6	−0.412962
$383$	3.69257e6	1.28627	0.643135	−	0.765753i	$-0.277633\pi$
$383$	3.69257e6	1.28627	0.643135	+	0.765753i	$0.277633\pi$
$384$	0	0
$385$	4.81913e6	1.65698
$386$	−1.31027e6	−0.447601
$387$	0	0
$388$	−4.94555e6	−1.66777
$389$	311757.	0.104458	0.0522291	−	0.998635i	$-0.483367\pi$
$389$	311757.	0.104458	0.0522291	+	0.998635i	$0.483367\pi$
$390$	0	0
$391$	−213564.	−0.0706457
$392$	544920.	0.179109
$393$	0	0
$394$	−981625.	−0.318570
$395$	2.25310e6	0.726586
$396$	0	0
$397$	1.85904e6	0.591987	0.295994	−	0.955190i	$-0.404349\pi$
$397$	1.85904e6	0.591987	0.295994	+	0.955190i	$0.404349\pi$
$398$	−490546.	−0.155229
$399$	0	0
$400$	4.89862e6	1.53082
$401$	2.94129e6	0.913434	0.456717	−	0.889612i	$-0.349025\pi$
$401$	2.94129e6	0.913434	0.456717	+	0.889612i	$0.349025\pi$
$402$	0	0
$403$	−350827.	−0.107605
$404$	996103.	0.303634
$405$	0	0
$406$	−914188.	−0.275246
$407$	999243.	0.299010
$408$	0	0
$409$	1.17435e6	0.347129	0.173565	−	0.984822i	$-0.444471\pi$
$409$	1.17435e6	0.347129	0.173565	+	0.984822i	$0.444471\pi$
$410$	2.57706e6	0.757120
$411$	0	0
$412$	3.06561e6	0.889761
$413$	2.90097e6	0.836889
$414$	0	0
$415$	−1.03479e6	−0.294939
$416$	426999.	0.120974
$417$	0	0
$418$	1.61420e6	0.451873
$419$	−3.95198e6	−1.09971	−0.549856	−	0.835259i	$-0.685318\pi$
$419$	−3.95198e6	−1.09971	−0.549856	+	0.835259i	$0.685318\pi$
$420$	0	0
$421$	−1.37669e6	−0.378558	−0.189279	−	0.981923i	$-0.560615\pi$
$421$	−1.37669e6	−0.378558	−0.189279	+	0.981923i	$0.560615\pi$
$422$	728031.	0.199007
$423$	0	0
$424$	−1.15424e6	−0.311804
$425$	6.00412e6	1.61242
$426$	0	0
$427$	−4.20725e6	−1.11668
$428$	−1.48855e6	−0.392784
$429$	0	0
$430$	1.51352e6	0.394745
$431$	3.21219e6	0.832930	0.416465	−	0.909152i	$-0.363269\pi$
$431$	3.21219e6	0.832930	0.416465	+	0.909152i	$0.363269\pi$
$432$	0	0
$433$	1.87271e6	0.480010	0.240005	−	0.970772i	$-0.422851\pi$
$433$	1.87271e6	0.480010	0.240005	+	0.970772i	$0.422851\pi$
$434$	−679894.	−0.173267
$435$	0	0
$436$	179480.	0.0452168
$437$	478946.	0.119973
$438$	0	0
$439$	−6.21723e6	−1.53970	−0.769849	−	0.638226i	$-0.779669\pi$
$439$	−6.21723e6	−1.53970	−0.769849	+	0.638226i	$0.779669\pi$
$440$	−4.62664e6	−1.13929
$441$	0	0
$442$	146330.	0.0356268
$443$	−68456.6	−0.0165732	−0.00828660	−	0.999966i	$-0.502638\pi$
$443$	−68456.6	−0.0165732	−0.00828660	+	0.999966i	$0.502638\pi$
$444$	0	0
$445$	−7.94849e6	−1.90276
$446$	−144233.	−0.0343343
$447$	0	0
$448$	−1.78123e6	−0.419300
$449$	−1.15988e6	−0.271518	−0.135759	−	0.990742i	$-0.543347\pi$
$449$	−1.15988e6	−0.271518	−0.135759	+	0.990742i	$0.543347\pi$
$450$	0	0
$451$	7.17453e6	1.66093
$452$	6.31390e6	1.45362
$453$	0	0
$454$	2.09593e6	0.477240
$455$	−979720.	−0.221857
$456$	0	0
$457$	7.46475e6	1.67196	0.835978	−	0.548762i	$-0.184901\pi$
$457$	7.46475e6	1.67196	0.835978	+	0.548762i	$0.184901\pi$
$458$	2.04308e6	0.455116
$459$	0	0
$460$	−654530.	−0.144223
$461$	−7.21041e6	−1.58018	−0.790092	−	0.612988i	$-0.789967\pi$
$461$	−7.21041e6	−1.58018	−0.790092	+	0.612988i	$0.789967\pi$
$462$	0	0
$463$	7.90903e6	1.71463	0.857316	−	0.514791i	$-0.172131\pi$
$463$	7.90903e6	1.71463	0.857316	+	0.514791i	$0.172131\pi$
$464$	−3.84072e6	−0.828166
$465$	0	0
$466$	−684972.	−0.146119
$467$	3.55919e6	0.755194	0.377597	−	0.925970i	$-0.376750\pi$
$467$	3.55919e6	0.755194	0.377597	+	0.925970i	$0.376750\pi$
$468$	0	0
$469$	1.29717e6	0.272311
$470$	−3.15400e6	−0.658594
$471$	0	0
$472$	−2.78509e6	−0.575420
$473$	4.21363e6	0.865972
$474$	0	0
$475$	−1.34650e7	−2.73826
$476$	−2.91389e6	−0.589461
$477$	0	0
$478$	1.81329e6	0.362993
$479$	−9.67727e6	−1.92714	−0.963571	−	0.267451i	$-0.913819\pi$
$479$	−9.67727e6	−1.92714	−0.963571	+	0.267451i	$0.913819\pi$
$480$	0	0
$481$	−203144.	−0.0400352
$482$	974955.	0.191147
$483$	0	0
$484$	−1.44487e6	−0.280360
$485$	−1.65937e7	−3.20324
$486$	0	0
$487$	3.41418e6	0.652325	0.326162	−	0.945314i	$-0.394244\pi$
$487$	3.41418e6	0.652325	0.326162	+	0.945314i	$0.394244\pi$
$488$	4.03920e6	0.767796
$489$	0	0
$490$	871761.	0.164024
$491$	−1.07488e6	−0.201214	−0.100607	−	0.994926i	$-0.532078\pi$
$491$	−1.07488e6	−0.201214	−0.100607	+	0.994926i	$0.532078\pi$
$492$	0	0
$493$	−4.70747e6	−0.872309
$494$	−328164.	−0.0605025
$495$	0	0
$496$	−2.85639e6	−0.521331
$497$	−785296.	−0.142607
$498$	0	0
$499$	9.37381e6	1.68525	0.842626	−	0.538499i	$-0.181009\pi$
$499$	9.37381e6	1.68525	0.842626	+	0.538499i	$0.181009\pi$
$500$	9.48455e6	1.69665
$501$	0	0
$502$	−1.33911e6	−0.237168
$503$	5.30887e6	0.935583	0.467792	−	0.883839i	$-0.345050\pi$
$503$	5.30887e6	0.935583	0.467792	+	0.883839i	$0.345050\pi$
$504$	0	0
$505$	3.34221e6	0.583184
$506$	177340.	0.0307914
$507$	0	0
$508$	2.97098e6	0.510788
$509$	−7.65996e6	−1.31048	−0.655242	−	0.755419i	$-0.727433\pi$
$509$	−7.65996e6	−1.31048	−0.655242	+	0.755419i	$0.727433\pi$
$510$	0	0
$511$	4.25428e6	0.720732
$512$	5.98111e6	1.00834
$513$	0	0
$514$	1.31712e6	0.219896
$515$	1.02860e7	1.70894
$516$	0	0
$517$	−8.78074e6	−1.44479
$518$	−393688.	−0.0644656
$519$	0	0
$520$	940587.	0.152542
$521$	7.15029e6	1.15406	0.577031	−	0.816722i	$-0.304211\pi$
$521$	7.15029e6	1.15406	0.577031	+	0.816722i	$0.304211\pi$
$522$	0	0
$523$	−1.70087e6	−0.271904	−0.135952	−	0.990715i	$-0.543409\pi$
$523$	−1.70087e6	−0.271904	−0.135952	+	0.990715i	$0.543409\pi$
$524$	1.67573e6	0.266610
$525$	0	0
$526$	−3.34848e6	−0.527696
$527$	−3.50101e6	−0.549119
$528$	0	0
$529$	−6.38372e6	−0.991825
$530$	−1.84655e6	−0.285542
$531$	0	0
$532$	6.53478e6	1.00104
$533$	−1.45857e6	−0.222387
$534$	0	0
$535$	−4.99451e6	−0.754412
$536$	−1.24536e6	−0.187233
$537$	0	0
$538$	295829.	0.0440641
$539$	2.42698e6	0.359828
$540$	0	0
$541$	−8.33931e6	−1.22500	−0.612501	−	0.790470i	$-0.709836\pi$
$541$	−8.33931e6	−1.22500	−0.612501	+	0.790470i	$0.709836\pi$
$542$	1.37789e6	0.201472
$543$	0	0
$544$	4.26115e6	0.617347
$545$	602207.	0.0868469
$546$	0	0
$547$	9.93719e6	1.42002	0.710011	−	0.704190i	$-0.248690\pi$
$547$	9.93719e6	1.42002	0.710011	+	0.704190i	$0.248690\pi$
$548$	8.37306e6	1.19106
$549$	0	0
$550$	−4.98572e6	−0.702783
$551$	1.05571e7	1.48138
$552$	0	0
$553$	−2.47133e6	−0.343651
$554$	1.11863e6	0.154850
$555$	0	0
$556$	2.21894e6	0.304410
$557$	−1.21963e7	−1.66568	−0.832840	−	0.553513i	$-0.813287\pi$
$557$	−1.21963e7	−1.66568	−0.832840	+	0.553513i	$0.813287\pi$
$558$	0	0
$559$	−856625.	−0.115947
$560$	−7.97676e6	−1.07487
$561$	0	0
$562$	−327212.	−0.0437007
$563$	9.03635e6	1.20150	0.600748	−	0.799439i	$-0.294870\pi$
$563$	9.03635e6	1.20150	0.600748	+	0.799439i	$0.294870\pi$
$564$	0	0
$565$	2.11849e7	2.79194
$566$	4.07932e6	0.535238
$567$	0	0
$568$	753928.	0.0980526
$569$	−7.15357e6	−0.926280	−0.463140	−	0.886285i	$-0.653277\pi$
$569$	−7.15357e6	−0.926280	−0.463140	+	0.886285i	$0.653277\pi$
$570$	0	0
$571$	9.49907e6	1.21924	0.609622	−	0.792692i	$-0.291321\pi$
$571$	9.49907e6	1.21924	0.609622	+	0.792692i	$0.291321\pi$
$572$	1.24854e6	0.159556
$573$	0	0
$574$	−2.82667e6	−0.358092
$575$	−1.47930e6	−0.186590
$576$	0	0
$577$	−7.05504e6	−0.882186	−0.441093	−	0.897461i	$-0.645409\pi$
$577$	−7.05504e6	−0.882186	−0.441093	+	0.897461i	$0.645409\pi$
$578$	−931707.	−0.116000
$579$	0	0
$580$	−1.44274e7	−1.78082
$581$	1.13502e6	0.139496
$582$	0	0
$583$	−5.14078e6	−0.626409
$584$	−4.08435e6	−0.495553
$585$	0	0
$586$	−1.81408e6	−0.218229
$587$	−7.27024e6	−0.870870	−0.435435	−	0.900220i	$-0.643405\pi$
$587$	−7.27024e6	−0.870870	−0.435435	+	0.900220i	$0.643405\pi$
$588$	0	0
$589$	7.85148e6	0.932531
$590$	−4.45558e6	−0.526956
$591$	0	0
$592$	−1.65398e6	−0.193966
$593$	−848335.	−0.0990674	−0.0495337	−	0.998772i	$-0.515774\pi$
$593$	−848335.	−0.0990674	−0.0495337	+	0.998772i	$0.515774\pi$
$594$	0	0
$595$	−9.77692e6	−1.13217
$596$	−2.81850e6	−0.325014
$597$	0	0
$598$	−36052.9	−0.00412275
$599$	−6.05683e6	−0.689729	−0.344864	−	0.938653i	$-0.612075\pi$
$599$	−6.05683e6	−0.689729	−0.344864	+	0.938653i	$0.612075\pi$
$600$	0	0
$601$	3.57347e6	0.403556	0.201778	−	0.979431i	$-0.435328\pi$
$601$	3.57347e6	0.403556	0.201778	+	0.979431i	$0.435328\pi$
$602$	−1.66012e6	−0.186701
$603$	0	0
$604$	−467031.	−0.0520899
$605$	−4.84796e6	−0.538481
$606$	0	0
$607$	−8.20173e6	−0.903511	−0.451756	−	0.892142i	$-0.649202\pi$
$607$	−8.20173e6	−0.903511	−0.451756	+	0.892142i	$0.649202\pi$
$608$	−9.55620e6	−1.04840
$609$	0	0
$610$	6.46189e6	0.703129
$611$	1.78511e6	0.193447
$612$	0	0
$613$	4.96223e6	0.533366	0.266683	−	0.963784i	$-0.414072\pi$
$613$	4.96223e6	0.533366	0.266683	+	0.963784i	$0.414072\pi$
$614$	3.54993e6	0.380013
$615$	0	0
$616$	5.07476e6	0.538845
$617$	−2.24371e6	−0.237276	−0.118638	−	0.992938i	$-0.537853\pi$
$617$	−2.24371e6	−0.237276	−0.118638	+	0.992938i	$0.537853\pi$
$618$	0	0
$619$	−5.63523e6	−0.591132	−0.295566	−	0.955322i	$-0.595508\pi$
$619$	−5.63523e6	−0.591132	−0.295566	+	0.955322i	$0.595508\pi$
$620$	−1.07299e7	−1.12103
$621$	0	0
$622$	3.79002e6	0.392794
$623$	8.71836e6	0.899943
$624$	0	0
$625$	1.16704e7	1.19505
$626$	−1.33596e6	−0.136256
$627$	0	0
$628$	4.20807e6	0.425778
$629$	−2.02724e6	−0.204305
$630$	0	0
$631$	−1.46326e7	−1.46301	−0.731507	−	0.681834i	$-0.761183\pi$
$631$	−1.46326e7	−1.46301	−0.731507	+	0.681834i	$0.761183\pi$
$632$	2.37261e6	0.236284
$633$	0	0
$634$	−3.23387e6	−0.319521
$635$	9.96850e6	0.981060
$636$	0	0
$637$	−493401.	−0.0481783
$638$	3.90900e6	0.380202
$639$	0	0
$640$	1.70663e7	1.64698
$641$	1.70204e7	1.63616	0.818079	−	0.575106i	$-0.195039\pi$
$641$	1.70204e7	1.63616	0.818079	+	0.575106i	$0.195039\pi$
$642$	0	0
$643$	−4.18161e6	−0.398856	−0.199428	−	0.979912i	$-0.563908\pi$
$643$	−4.18161e6	−0.398856	−0.199428	+	0.979912i	$0.563908\pi$
$644$	717927.	0.0682127
$645$	0	0
$646$	−3.27485e6	−0.308752
$647$	1.43350e7	1.34629	0.673143	−	0.739512i	$-0.264944\pi$
$647$	1.43350e7	1.34629	0.673143	+	0.739512i	$0.264944\pi$
$648$	0	0
$649$	−1.24043e7	−1.15601
$650$	1.01359e6	0.0940975
$651$	0	0
$652$	−1.13309e7	−1.04387
$653$	3.13971e6	0.288142	0.144071	−	0.989567i	$-0.453981\pi$
$653$	3.13971e6	0.288142	0.144071	+	0.989567i	$0.453981\pi$
$654$	0	0
$655$	5.62256e6	0.512072
$656$	−1.18755e7	−1.07744
$657$	0	0
$658$	3.45950e6	0.311493
$659$	−7.88430e6	−0.707212	−0.353606	−	0.935394i	$-0.615045\pi$
$659$	−7.88430e6	−0.707212	−0.353606	+	0.935394i	$0.615045\pi$
$660$	0	0
$661$	2.96118e6	0.263609	0.131805	−	0.991276i	$-0.457923\pi$
$661$	2.96118e6	0.263609	0.131805	+	0.991276i	$0.457923\pi$
$662$	−2.30373e6	−0.204309
$663$	0	0
$664$	−1.08968e6	−0.0959134
$665$	2.19261e7	1.92268
$666$	0	0
$667$	1.15983e6	0.100944
$668$	434981.	0.0377163
$669$	0	0
$670$	−1.99232e6	−0.171463
$671$	1.79899e7	1.54249
$672$	0	0
$673$	−1.01292e7	−0.862064	−0.431032	−	0.902337i	$-0.641850\pi$
$673$	−1.01292e7	−0.862064	−0.431032	+	0.902337i	$0.641850\pi$
$674$	−5.46308e6	−0.463221
$675$	0	0
$676$	1.05738e7	0.889947
$677$	−1.68631e7	−1.41406	−0.707028	−	0.707185i	$-0.749965\pi$
$677$	−1.68631e7	−1.41406	−0.707028	+	0.707185i	$0.749965\pi$
$678$	0	0
$679$	1.82010e7	1.51503
$680$	9.38640e6	0.778443
$681$	0	0
$682$	2.90718e6	0.239337
$683$	−1.55041e7	−1.27173	−0.635866	−	0.771800i	$-0.719357\pi$
$683$	−1.55041e7	−1.27173	−0.635866	+	0.771800i	$0.719357\pi$
$684$	0	0
$685$	2.80940e7	2.28764
$686$	−3.99497e6	−0.324118
$687$	0	0
$688$	−6.97453e6	−0.561751
$689$	1.04511e6	0.0838716
$690$	0	0
$691$	−9.70614e6	−0.773306	−0.386653	−	0.922225i	$-0.626369\pi$
$691$	−9.70614e6	−0.773306	−0.386653	+	0.922225i	$0.626369\pi$
$692$	9.80848e6	0.778640
$693$	0	0
$694$	−1.32964e6	−0.104793
$695$	7.44519e6	0.584674
$696$	0	0
$697$	−1.45555e7	−1.13487
$698$	1.38543e6	0.107633
$699$	0	0
$700$	−2.01837e7	−1.55689
$701$	7.36305e6	0.565930	0.282965	−	0.959130i	$-0.408682\pi$
$701$	7.36305e6	0.565930	0.282965	+	0.959130i	$0.408682\pi$
$702$	0	0
$703$	4.54635e6	0.346956
$704$	7.61640e6	0.579186
$705$	0	0
$706$	4.40718e6	0.332773
$707$	−3.66593e6	−0.275826
$708$	0	0
$709$	−2.14370e7	−1.60158	−0.800791	−	0.598943i	$-0.795587\pi$
$709$	−2.14370e7	−1.60158	−0.800791	+	0.598943i	$0.795587\pi$
$710$	1.20613e6	0.0897943
$711$	0	0
$712$	−8.37012e6	−0.618774
$713$	862582.	0.0635443
$714$	0	0
$715$	4.18921e6	0.306456
$716$	−2.12774e7	−1.55109
$717$	0	0
$718$	−7.70571e6	−0.557830
$719$	−1.13057e7	−0.815596	−0.407798	−	0.913072i	$-0.633703\pi$
$719$	−1.13057e7	−0.815596	−0.407798	+	0.913072i	$0.633703\pi$
$720$	0	0
$721$	−1.12823e7	−0.808274
$722$	3.17290e6	0.226523
$723$	0	0
$724$	1.78518e7	1.26572
$725$	−3.26075e7	−2.30394
$726$	0	0
$727$	2.26201e7	1.58730	0.793650	−	0.608374i	$-0.208178\pi$
$727$	2.26201e7	1.58730	0.793650	+	0.608374i	$0.208178\pi$
$728$	−1.03169e6	−0.0721475
$729$	0	0
$730$	−6.53412e6	−0.453816
$731$	−8.54851e6	−0.591694
$732$	0	0
$733$	1.53799e7	1.05729	0.528644	−	0.848844i	$-0.322701\pi$
$733$	1.53799e7	1.05729	0.528644	+	0.848844i	$0.322701\pi$
$734$	5.28661e6	0.362190
$735$	0	0
$736$	−1.04987e6	−0.0714397
$737$	−5.54661e6	−0.376148
$738$	0	0
$739$	2.25871e7	1.52142	0.760712	−	0.649090i	$-0.224850\pi$
$739$	2.25871e7	1.52142	0.760712	+	0.649090i	$0.224850\pi$
$740$	−6.21307e6	−0.417087
$741$	0	0
$742$	2.02540e6	0.135052
$743$	−2.01343e7	−1.33802	−0.669011	−	0.743252i	$-0.733282\pi$
$743$	−2.01343e7	−1.33802	−0.669011	+	0.743252i	$0.733282\pi$
$744$	0	0
$745$	−9.45687e6	−0.624247
$746$	−3.95866e6	−0.260436
$747$	0	0
$748$	1.24596e7	0.814234
$749$	5.47827e6	0.356812
$750$	0	0
$751$	−9.23303e6	−0.597372	−0.298686	−	0.954352i	$-0.596548\pi$
$751$	−9.23303e6	−0.597372	−0.298686	+	0.954352i	$0.596548\pi$
$752$	1.45341e7	0.937227
$753$	0	0
$754$	−794694.	−0.0509063
$755$	−1.56702e6	−0.100048
$756$	0	0
$757$	−7.14432e6	−0.453128	−0.226564	−	0.973996i	$-0.572749\pi$
$757$	−7.14432e6	−0.453128	−0.226564	+	0.973996i	$0.572749\pi$
$758$	−3.83008e6	−0.242123
$759$	0	0
$760$	−2.10503e7	−1.32198
$761$	1.12509e6	0.0704247	0.0352123	−	0.999380i	$-0.488789\pi$
$761$	1.12509e6	0.0704247	0.0352123	+	0.999380i	$0.488789\pi$
$762$	0	0
$763$	−660536.	−0.0410757
$764$	2.03879e7	1.26369
$765$	0	0
$766$	6.22073e6	0.383062
$767$	2.52178e6	0.154781
$768$	0	0
$769$	4.58392e6	0.279526	0.139763	−	0.990185i	$-0.455366\pi$
$769$	4.58392e6	0.279526	0.139763	+	0.990185i	$0.455366\pi$
$770$	8.11859e6	0.493462
$771$	0	0
$772$	2.26811e7	1.36968
$773$	−1.30448e7	−0.785215	−0.392608	−	0.919706i	$-0.628427\pi$
$773$	−1.30448e7	−0.785215	−0.392608	+	0.919706i	$0.628427\pi$
$774$	0	0
$775$	−2.42506e7	−1.45033
$776$	−1.74739e7	−1.04169
$777$	0	0
$778$	525204.	0.0311085
$779$	3.26426e7	1.92727
$780$	0	0
$781$	3.35787e6	0.196986
$782$	−359782.	−0.0210389
$783$	0	0
$784$	−4.01721e6	−0.233418
$785$	1.41193e7	0.817783
$786$	0	0
$787$	−3.23648e7	−1.86267	−0.931335	−	0.364164i	$-0.881355\pi$
$787$	−3.23648e7	−1.86267	−0.931335	+	0.364164i	$0.881355\pi$
$788$	1.69922e7	0.974843
$789$	0	0
$790$	3.79570e6	0.216383
$791$	−2.32369e7	−1.32049
$792$	0	0
$793$	−3.65732e6	−0.206528
$794$	3.13185e6	0.176299
$795$	0	0
$796$	8.49149e6	0.475008
$797$	1.70921e7	0.953126	0.476563	−	0.879140i	$-0.341882\pi$
$797$	1.70921e7	0.953126	0.476563	+	0.879140i	$0.341882\pi$
$798$	0	0
$799$	1.78141e7	0.987184
$800$	2.95159e7	1.63054
$801$	0	0
$802$	4.95507e6	0.272028
$803$	−1.81910e7	−0.995560
$804$	0	0
$805$	2.40885e6	0.131015
$806$	−591024.	−0.0320455
$807$	0	0
$808$	3.51950e6	0.189650
$809$	2.24306e7	1.20495	0.602476	−	0.798137i	$-0.294181\pi$
$809$	2.24306e7	1.20495	0.602476	+	0.798137i	$0.294181\pi$
$810$	0	0
$811$	1.87966e7	1.00352	0.501762	−	0.865006i	$-0.332685\pi$
$811$	1.87966e7	1.00352	0.501762	+	0.865006i	$0.332685\pi$
$812$	1.58249e7	0.842267
$813$	0	0
$814$	1.68338e6	0.0890475
$815$	−3.80183e7	−2.00493
$816$	0	0
$817$	1.91712e7	1.00483
$818$	1.97839e6	0.103378
$819$	0	0
$820$	−4.46096e7	−2.31683
$821$	2.46114e7	1.27432	0.637159	−	0.770732i	$-0.280109\pi$
$821$	2.46114e7	1.27432	0.637159	+	0.770732i	$0.280109\pi$
$822$	0	0
$823$	1.13362e7	0.583402	0.291701	−	0.956510i	$-0.405779\pi$
$823$	1.13362e7	0.583402	0.291701	+	0.956510i	$0.405779\pi$
$824$	1.08316e7	0.555745
$825$	0	0
$826$	4.88714e6	0.249233
$827$	−2.27950e7	−1.15898	−0.579489	−	0.814980i	$-0.696748\pi$
$827$	−2.27950e7	−1.15898	−0.579489	+	0.814980i	$0.696748\pi$
$828$	0	0
$829$	−3.05094e7	−1.54187	−0.770934	−	0.636915i	$-0.780210\pi$
$829$	−3.05094e7	−1.54187	−0.770934	+	0.636915i	$0.780210\pi$
$830$	−1.74327e6	−0.0878353
$831$	0	0
$832$	−1.54840e6	−0.0775488
$833$	−4.92380e6	−0.245860
$834$	0	0
$835$	1.45949e6	0.0724409
$836$	−2.79423e7	−1.38276
$837$	0	0
$838$	−6.65773e6	−0.327504
$839$	1.76393e7	0.865121	0.432560	−	0.901605i	$-0.357610\pi$
$839$	1.76393e7	0.865121	0.432560	+	0.901605i	$0.357610\pi$
$840$	0	0
$841$	5.05440e6	0.246422
$842$	−2.31926e6	−0.112738
$843$	0	0
$844$	−1.26024e7	−0.608973
$845$	3.54781e7	1.70930
$846$	0	0
$847$	5.31752e6	0.254684
$848$	8.50918e6	0.406348
$849$	0	0
$850$	1.01149e7	0.480191
$851$	499473.	0.0236422
$852$	0	0
$853$	−1.01233e6	−0.0476374	−0.0238187	−	0.999716i	$-0.507582\pi$
$853$	−1.01233e6	−0.0476374	−0.0238187	+	0.999716i	$0.507582\pi$
$854$	−7.08778e6	−0.332557
$855$	0	0
$856$	−5.25945e6	−0.245333
$857$	−2.20613e7	−1.02607	−0.513037	−	0.858366i	$-0.671480\pi$
$857$	−2.20613e7	−1.02607	−0.513037	+	0.858366i	$0.671480\pi$
$858$	0	0
$859$	−1.24809e7	−0.577114	−0.288557	−	0.957463i	$-0.593175\pi$
$859$	−1.24809e7	−0.577114	−0.288557	+	0.957463i	$0.593175\pi$
$860$	−2.61994e7	−1.20794
$861$	0	0
$862$	5.41145e6	0.248054
$863$	2.55835e6	0.116932	0.0584659	−	0.998289i	$-0.481379\pi$
$863$	2.55835e6	0.116932	0.0584659	+	0.998289i	$0.481379\pi$
$864$	0	0
$865$	3.29103e7	1.49552
$866$	3.15487e6	0.142951
$867$	0	0
$868$	1.17692e7	0.530208
$869$	1.05672e7	0.474691
$870$	0	0
$871$	1.12762e6	0.0503635
$872$	634152.	0.0282424
$873$	0	0
$874$	806860.	0.0357289
$875$	−3.49057e7	−1.54126
$876$	0	0
$877$	1.45109e7	0.637081	0.318541	−	0.947909i	$-0.396807\pi$
$877$	1.45109e7	0.637081	0.318541	+	0.947909i	$0.396807\pi$
$878$	−1.04739e7	−0.458535
$879$	0	0
$880$	3.41081e7	1.48474
$881$	−1.13133e7	−0.491075	−0.245538	−	0.969387i	$-0.578964\pi$
$881$	−1.13133e7	−0.491075	−0.245538	+	0.969387i	$0.578964\pi$
$882$	0	0
$883$	−1.59581e7	−0.688780	−0.344390	−	0.938827i	$-0.611914\pi$
$883$	−1.59581e7	−0.688780	−0.344390	+	0.938827i	$0.611914\pi$
$884$	−2.53301e6	−0.109020
$885$	0	0
$886$	−115326.	−0.00493564
$887$	−2.78503e7	−1.18856	−0.594280	−	0.804258i	$-0.702563\pi$
$887$	−2.78503e7	−1.18856	−0.594280	+	0.804258i	$0.702563\pi$
$888$	0	0
$889$	−1.09340e7	−0.464008
$890$	−1.33905e7	−0.566658
$891$	0	0
$892$	2.49672e6	0.105065
$893$	−3.99506e7	−1.67647
$894$	0	0
$895$	−7.13919e7	−2.97914
$896$	−1.87193e7	−0.778969
$897$	0	0
$898$	−1.95401e6	−0.0808603
$899$	1.90134e7	0.784624
$900$	0	0
$901$	1.04295e7	0.428007
$902$	1.20866e7	0.494639
$903$	0	0
$904$	2.23087e7	0.907932
$905$	5.98980e7	2.43103
$906$	0	0
$907$	−3.72380e6	−0.150303	−0.0751516	−	0.997172i	$-0.523944\pi$
$907$	−3.72380e6	−0.150303	−0.0751516	+	0.997172i	$0.523944\pi$
$908$	−3.62811e7	−1.46038
$909$	0	0
$910$	−1.65049e6	−0.0660710
$911$	3.95736e6	0.157983	0.0789913	−	0.996875i	$-0.474830\pi$
$911$	3.95736e6	0.157983	0.0789913	+	0.996875i	$0.474830\pi$
$912$	0	0
$913$	−4.85326e6	−0.192689
$914$	1.25756e7	0.497923
$915$	0	0
$916$	−3.53664e7	−1.39268
$917$	−6.16716e6	−0.242193
$918$	0	0
$919$	1.17607e7	0.459349	0.229674	−	0.973268i	$-0.426234\pi$
$919$	1.17607e7	0.459349	0.229674	+	0.973268i	$0.426234\pi$
$920$	−2.31263e6	−0.0900817
$921$	0	0
$922$	−1.21471e7	−0.470592
$923$	−682649.	−0.0263750
$924$	0	0
$925$	−1.40422e7	−0.539609
$926$	1.33240e7	0.510632
$927$	0	0
$928$	−2.31416e7	−0.882113
$929$	2.56239e7	0.974107	0.487053	−	0.873372i	$-0.338072\pi$
$929$	2.56239e7	0.974107	0.487053	+	0.873372i	$0.338072\pi$
$930$	0	0
$931$	1.10423e7	0.417527
$932$	1.18571e7	0.447134
$933$	0	0
$934$	5.99602e6	0.224903
$935$	4.18054e7	1.56388
$936$	0	0
$937$	2.78817e7	1.03746	0.518729	−	0.854939i	$-0.326405\pi$
$937$	2.78817e7	1.03746	0.518729	+	0.854939i	$0.326405\pi$
$938$	2.18529e6	0.0810965
$939$	0	0
$940$	5.45967e7	2.01533
$941$	1.09841e7	0.404379	0.202190	−	0.979346i	$-0.435194\pi$
$941$	1.09841e7	0.404379	0.202190	+	0.979346i	$0.435194\pi$
$942$	0	0
$943$	3.58620e6	0.131327
$944$	2.05320e7	0.749897
$945$	0	0
$946$	7.09854e6	0.257894
$947$	1.81979e7	0.659395	0.329697	−	0.944087i	$-0.393053\pi$
$947$	1.81979e7	0.659395	0.329697	+	0.944087i	$0.393053\pi$
$948$	0	0
$949$	3.69820e6	0.133298
$950$	−2.26840e7	−0.815475
$951$	0	0
$952$	−1.02955e7	−0.368177
$953$	−1.52991e7	−0.545674	−0.272837	−	0.962060i	$-0.587962\pi$
$953$	−1.52991e7	−0.545674	−0.272837	+	0.962060i	$0.587962\pi$
$954$	0	0
$955$	6.84073e7	2.42713
$956$	−3.13886e7	−1.11078
$957$	0	0
$958$	−1.63029e7	−0.573919
$959$	−3.08152e7	−1.08198
$960$	0	0
$961$	−1.44886e7	−0.506079
$962$	−342229.	−0.0119228
$963$	0	0
$964$	−1.68768e7	−0.584920
$965$	7.61015e7	2.63072
$966$	0	0
$967$	−8.61312e6	−0.296206	−0.148103	−	0.988972i	$-0.547317\pi$
$967$	−8.61312e6	−0.296206	−0.148103	+	0.988972i	$0.547317\pi$
$968$	−5.10512e6	−0.175113
$969$	0	0
$970$	−2.79547e7	−0.953951
$971$	2.02056e7	0.687738	0.343869	−	0.939018i	$-0.388262\pi$
$971$	2.02056e7	0.687738	0.343869	+	0.939018i	$0.388262\pi$
$972$	0	0
$973$	−8.16632e6	−0.276531
$974$	5.75173e6	0.194268
$975$	0	0
$976$	−2.97774e7	−1.00060
$977$	1.42828e7	0.478714	0.239357	−	0.970932i	$-0.423063\pi$
$977$	1.42828e7	0.478714	0.239357	+	0.970932i	$0.423063\pi$
$978$	0	0
$979$	−3.72791e7	−1.24311
$980$	−1.50904e7	−0.501922
$981$	0	0
$982$	−1.81081e6	−0.0599231
$983$	−267302.	−0.00882305	−0.00441152	−	0.999990i	$-0.501404\pi$
$983$	−267302.	−0.00882305	−0.00441152	+	0.999990i	$0.501404\pi$
$984$	0	0
$985$	5.70137e7	1.87236
$986$	−7.93048e6	−0.259781
$987$	0	0
$988$	5.68061e6	0.185141
$989$	2.10619e6	0.0684711
$990$	0	0
$991$	−1.00930e7	−0.326465	−0.163233	−	0.986588i	$-0.552192\pi$
$991$	−1.00930e7	−0.326465	−0.163233	+	0.986588i	$0.552192\pi$
$992$	−1.72107e7	−0.555291
$993$	0	0
$994$	−1.32295e6	−0.0424697
$995$	2.84914e7	0.912338
$996$	0	0
$997$	−4.94364e7	−1.57510	−0.787552	−	0.616248i	$-0.788652\pi$
$997$	−4.94364e7	−1.57510	−0.787552	+	0.616248i	$0.788652\pi$
$998$	1.57917e7	0.501882
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.68466 q^{2} -29.1619 q^{4} -97.8466 q^{5} +107.324 q^{7} -103.037 q^{8} +O(q^{10})\)	\(q+1.68466 q^{2} -29.1619 q^{4} -97.8466 q^{5} +107.324 q^{7} -103.037 q^{8} -164.838 q^{10} -458.909 q^{11} +93.2954 q^{13} +180.804 q^{14} +759.600 q^{16} +931.023 q^{17} -2087.94 q^{19} +2853.39 q^{20} -773.105 q^{22} -229.386 q^{23} +6448.95 q^{25} +157.171 q^{26} -3129.77 q^{28} -5056.24 q^{29} -3760.39 q^{31} +4576.85 q^{32} +1568.46 q^{34} -10501.3 q^{35} -2177.43 q^{37} -3517.47 q^{38} +10081.8 q^{40} -15633.9 q^{41} -9181.85 q^{43} +13382.7 q^{44} -386.438 q^{46} +19134.0 q^{47} -5288.59 q^{49} +10864.3 q^{50} -2720.67 q^{52} +11202.2 q^{53} +44902.7 q^{55} -11058.3 q^{56} -8518.04 q^{58} +27030.0 q^{59} -39201.5 q^{61} -6334.98 q^{62} -16596.8 q^{64} -9128.64 q^{65} +12086.5 q^{67} -27150.4 q^{68} -17691.1 q^{70} -7317.07 q^{71} +39639.6 q^{73} -3668.23 q^{74} +60888.4 q^{76} -49251.9 q^{77} -23026.8 q^{79} -74324.2 q^{80} -26337.7 q^{82} +10575.6 q^{83} -91097.4 q^{85} -15468.3 q^{86} +47284.6 q^{88} +81234.2 q^{89} +10012.8 q^{91} +6689.35 q^{92} +32234.2 q^{94} +204298. q^{95} +169589. q^{97} -8909.47 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{2} + 53 q^{4} - 72 q^{5} - 8 q^{7} - 639 q^{8}+O(q^{10}) \) 2 * q - 9 * q^2 + 53 * q^4 - 72 * q^5 - 8 * q^7 - 639 * q^8	\( 2 q - 9 q^{2} + 53 q^{4} - 72 q^{5} - 8 q^{7} - 639 q^{8} - 441 q^{10} - 522 q^{11} - 704 q^{13} + 1413 q^{14} + 3857 q^{16} - 216 q^{17} - 2840 q^{19} + 4977 q^{20} - 99 q^{22} + 36 q^{23} + 3992 q^{25} + 8676 q^{26} - 12605 q^{28} - 12240 q^{29} - 1064 q^{31} - 11367 q^{32} + 13824 q^{34} - 13482 q^{35} + 9004 q^{37} + 4518 q^{38} - 3771 q^{40} - 5688 q^{41} + 784 q^{43} + 8199 q^{44} - 3222 q^{46} - 1116 q^{47} - 8796 q^{49} + 37116 q^{50} - 68228 q^{52} - 4536 q^{53} + 43272 q^{55} + 50751 q^{56} + 68238 q^{58} + 67320 q^{59} - 49904 q^{61} - 35145 q^{62} + 54641 q^{64} - 29736 q^{65} - 42176 q^{67} - 121392 q^{68} + 14157 q^{70} + 43848 q^{71} + 47218 q^{73} - 123138 q^{74} - 902 q^{76} - 41976 q^{77} - 49616 q^{79} + 5733 q^{80} - 132606 q^{82} + 102294 q^{83} - 120744 q^{85} - 121950 q^{86} + 81099 q^{88} + 35856 q^{89} + 101960 q^{91} + 28494 q^{92} + 248598 q^{94} + 184860 q^{95} + 169966 q^{97} + 28566 q^{98}+O(q^{100}) \) 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

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