LMFDB - Embedded newform 6633.2.a.w.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6633,2,Mod(1,6633)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6633, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6633.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6633 = 3^{2} \cdot 11 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6633.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:	$52.9647716607$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - x^{16} - 26 x^{15} + 25 x^{14} + 272 x^{13} - 244 x^{12} - 1472 x^{11} + 1186 x^{10} + 4406 x^{9} + \cdots - 2 $ x^17 - x^16 - 26x^15 + 25x^14 + 272x^13 - 244x^12 - 1472x^11 + 1186x^10 + 4406x^9 - 3042x^8 - 7180x^7 + 4055x^6 + 5665x^5 - 2655x^4 - 1418x^3 + 810x^2 - 76*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 737)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-1.19652$ of defining polynomial
Character	$\chi$	$=$	6633.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.19652 q^{2} -0.568334 q^{4} +0.744253 q^{5} +4.07489 q^{7} -3.07307 q^{8} +O(q^{10})$	\(q+1.19652 q^{2} -0.568334 q^{4} +0.744253 q^{5} +4.07489 q^{7} -3.07307 q^{8} +0.890515 q^{10} -1.00000 q^{11} -1.12199 q^{13} +4.87570 q^{14} -2.54033 q^{16} -5.26387 q^{17} -1.09401 q^{19} -0.422984 q^{20} -1.19652 q^{22} +0.724154 q^{23} -4.44609 q^{25} -1.34249 q^{26} -2.31590 q^{28} -0.375376 q^{29} +9.00211 q^{31} +3.10658 q^{32} -6.29834 q^{34} +3.03275 q^{35} +11.2851 q^{37} -1.30900 q^{38} -2.28714 q^{40} -6.79208 q^{41} +11.2636 q^{43} +0.568334 q^{44} +0.866467 q^{46} +10.2318 q^{47} +9.60475 q^{49} -5.31984 q^{50} +0.637664 q^{52} +6.24245 q^{53} -0.744253 q^{55} -12.5224 q^{56} -0.449145 q^{58} -11.2509 q^{59} +9.85237 q^{61} +10.7712 q^{62} +8.79775 q^{64} -0.835044 q^{65} -1.00000 q^{67} +2.99163 q^{68} +3.62875 q^{70} -0.951027 q^{71} -2.36454 q^{73} +13.5029 q^{74} +0.621760 q^{76} -4.07489 q^{77} -3.24116 q^{79} -1.89065 q^{80} -8.12688 q^{82} +4.53797 q^{83} -3.91765 q^{85} +13.4771 q^{86} +3.07307 q^{88} +5.18671 q^{89} -4.57199 q^{91} -0.411561 q^{92} +12.2426 q^{94} -0.814216 q^{95} +18.5902 q^{97} +11.4923 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * 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 $

$2$	1.19652	0.846069	0.423035	−	0.906114i	$-0.360965\pi$
$2$	1.19652	0.846069	0.423035	+	0.906114i	$0.360965\pi$
$3$	0	0
$3$	0	0
$4$	−0.568334	−0.284167
$5$	0.744253	0.332840	0.166420	−	0.986055i	$-0.446779\pi$
$5$	0.744253	0.332840	0.166420	+	0.986055i	$0.446779\pi$
$6$	0	0
$7$	4.07489	1.54016	0.770082	−	0.637944i	$-0.220215\pi$
$7$	4.07489	1.54016	0.770082	+	0.637944i	$0.220215\pi$
$8$	−3.07307	−1.08649
$9$	0	0
$10$	0.890515	0.281606
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.12199	−0.311184	−0.155592	−	0.987821i	$-0.549728\pi$
$13$	−1.12199	−0.311184	−0.155592	+	0.987821i	$0.549728\pi$
$14$	4.87570	1.30309
$15$	0	0
$16$	−2.54033	−0.635082
$17$	−5.26387	−1.27668	−0.638338	−	0.769756i	$-0.720378\pi$
$17$	−5.26387	−1.27668	−0.638338	+	0.769756i	$0.720378\pi$
$18$	0	0
$19$	−1.09401	−0.250982	−0.125491	−	0.992095i	$-0.540051\pi$
$19$	−1.09401	−0.250982	−0.125491	+	0.992095i	$0.540051\pi$
$20$	−0.422984	−0.0945821
$21$	0	0
$22$	−1.19652	−0.255099
$23$	0.724154	0.150997	0.0754983	−	0.997146i	$-0.475945\pi$
$23$	0.724154	0.150997	0.0754983	+	0.997146i	$0.475945\pi$
$24$	0	0
$25$	−4.44609	−0.889218
$26$	−1.34249	−0.263283
$27$	0	0
$28$	−2.31590	−0.437664
$29$	−0.375376	−0.0697055	−0.0348528	−	0.999392i	$-0.511096\pi$
$29$	−0.375376	−0.0697055	−0.0348528	+	0.999392i	$0.511096\pi$
$30$	0	0
$31$	9.00211	1.61683	0.808413	−	0.588615i	$-0.200327\pi$
$31$	9.00211	1.61683	0.808413	+	0.588615i	$0.200327\pi$
$32$	3.10658	0.549170
$33$	0	0
$34$	−6.29834	−1.08016
$35$	3.03275	0.512628
$36$	0	0
$37$	11.2851	1.85526	0.927630	−	0.373502i	$-0.121843\pi$
$37$	11.2851	1.85526	0.927630	+	0.373502i	$0.121843\pi$
$38$	−1.30900	−0.212348
$39$	0	0
$40$	−2.28714	−0.361629
$41$	−6.79208	−1.06074	−0.530372	−	0.847765i	$-0.677948\pi$
$41$	−6.79208	−1.06074	−0.530372	+	0.847765i	$0.677948\pi$
$42$	0	0
$43$	11.2636	1.71768	0.858838	−	0.512247i	$-0.171187\pi$
$43$	11.2636	1.71768	0.858838	+	0.512247i	$0.171187\pi$
$44$	0.568334	0.0856795
$45$	0	0
$46$	0.866467	0.127754
$47$	10.2318	1.49247	0.746234	−	0.665684i	$-0.231860\pi$
$47$	10.2318	1.49247	0.746234	+	0.665684i	$0.231860\pi$
$48$	0	0
$49$	9.60475	1.37211
$50$	−5.31984	−0.752340
$51$	0	0
$52$	0.637664	0.0884281
$53$	6.24245	0.857466	0.428733	−	0.903431i	$-0.358960\pi$
$53$	6.24245	0.857466	0.428733	+	0.903431i	$0.358960\pi$
$54$	0	0
$55$	−0.744253	−0.100355
$56$	−12.5224	−1.67338
$57$	0	0
$58$	−0.449145	−0.0589757
$59$	−11.2509	−1.46474	−0.732372	−	0.680904i	$-0.761587\pi$
$59$	−11.2509	−1.46474	−0.732372	+	0.680904i	$0.761587\pi$
$60$	0	0
$61$	9.85237	1.26147	0.630733	−	0.776000i	$-0.282754\pi$
$61$	9.85237	1.26147	0.630733	+	0.776000i	$0.282754\pi$
$62$	10.7712	1.36795
$63$	0	0
$64$	8.79775	1.09972
$65$	−0.835044	−0.103574
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	2.99163	0.362789
$69$	0	0
$70$	3.62875	0.433719
$71$	−0.951027	−0.112866	−0.0564330	−	0.998406i	$-0.517973\pi$
$71$	−0.951027	−0.112866	−0.0564330	+	0.998406i	$0.517973\pi$
$72$	0	0
$73$	−2.36454	−0.276748	−0.138374	−	0.990380i	$-0.544188\pi$
$73$	−2.36454	−0.276748	−0.138374	+	0.990380i	$0.544188\pi$
$74$	13.5029	1.56968
$75$	0	0
$76$	0.621760	0.0713208
$77$	−4.07489	−0.464377
$78$	0	0
$79$	−3.24116	−0.364659	−0.182329	−	0.983238i	$-0.558364\pi$
$79$	−3.24116	−0.364659	−0.182329	+	0.983238i	$0.558364\pi$
$80$	−1.89065	−0.211381
$81$	0	0
$82$	−8.12688	−0.897463
$83$	4.53797	0.498107	0.249053	−	0.968490i	$-0.419881\pi$
$83$	4.53797	0.498107	0.249053	+	0.968490i	$0.419881\pi$
$84$	0	0
$85$	−3.91765	−0.424929
$86$	13.4771	1.45327
$87$	0	0
$88$	3.07307	0.327590
$89$	5.18671	0.549790	0.274895	−	0.961474i	$-0.411357\pi$
$89$	5.18671	0.549790	0.274895	+	0.961474i	$0.411357\pi$
$90$	0	0
$91$	−4.57199	−0.479274
$92$	−0.411561	−0.0429082
$93$	0	0
$94$	12.2426	1.26273
$95$	−0.814216	−0.0835368
$96$	0	0
$97$	18.5902	1.88755	0.943774	−	0.330590i	$-0.107248\pi$
$97$	18.5902	1.88755	0.943774	+	0.330590i	$0.107248\pi$
$98$	11.4923	1.16090
$99$	0	0
$100$	2.52686	0.252686
$101$	8.94178	0.889740	0.444870	−	0.895595i	$-0.353250\pi$
$101$	8.94178	0.889740	0.444870	+	0.895595i	$0.353250\pi$
$102$	0	0
$103$	−3.32967	−0.328082	−0.164041	−	0.986454i	$-0.552453\pi$
$103$	−3.32967	−0.328082	−0.164041	+	0.986454i	$0.552453\pi$
$104$	3.44795	0.338099
$105$	0	0
$106$	7.46923	0.725476
$107$	3.44807	0.333338	0.166669	−	0.986013i	$-0.446699\pi$
$107$	3.44807	0.333338	0.166669	+	0.986013i	$0.446699\pi$
$108$	0	0
$109$	18.5179	1.77369	0.886845	−	0.462067i	$-0.152892\pi$
$109$	18.5179	1.77369	0.886845	+	0.462067i	$0.152892\pi$
$110$	−0.890515	−0.0849073
$111$	0	0
$112$	−10.3516	−0.978132
$113$	2.67935	0.252052	0.126026	−	0.992027i	$-0.459778\pi$
$113$	2.67935	0.252052	0.126026	+	0.992027i	$0.459778\pi$
$114$	0	0
$115$	0.538954	0.0502577
$116$	0.213339	0.0198080
$117$	0	0
$118$	−13.4620	−1.23928
$119$	−21.4497	−1.96629
$120$	0	0
$121$	1.00000	0.0909091
$122$	11.7886	1.06729
$123$	0	0
$124$	−5.11620	−0.459448
$125$	−7.03028	−0.628807
$126$	0	0
$127$	−8.65780	−0.768256	−0.384128	−	0.923280i	$-0.625498\pi$
$127$	−8.65780	−0.768256	−0.384128	+	0.923280i	$0.625498\pi$
$128$	4.31355	0.381268
$129$	0	0
$130$	−0.999149	−0.0876311
$131$	17.3709	1.51771	0.758853	−	0.651262i	$-0.225760\pi$
$131$	17.3709	1.51771	0.758853	+	0.651262i	$0.225760\pi$
$132$	0	0
$133$	−4.45795	−0.386554
$134$	−1.19652	−0.103364
$135$	0	0
$136$	16.1762	1.38710
$137$	−21.4281	−1.83072	−0.915361	−	0.402634i	$-0.868095\pi$
$137$	−21.4281	−1.83072	−0.915361	+	0.402634i	$0.868095\pi$
$138$	0	0
$139$	−17.0333	−1.44474	−0.722371	−	0.691506i	$-0.756948\pi$
$139$	−17.0333	−1.44474	−0.722371	+	0.691506i	$0.756948\pi$
$140$	−1.72361	−0.145672
$141$	0	0
$142$	−1.13792	−0.0954925
$143$	1.12199	0.0938255
$144$	0	0
$145$	−0.279374	−0.0232008
$146$	−2.82922	−0.234148
$147$	0	0
$148$	−6.41370	−0.527203
$149$	20.6676	1.69316	0.846579	−	0.532263i	$-0.178658\pi$
$149$	20.6676	1.69316	0.846579	+	0.532263i	$0.178658\pi$
$150$	0	0
$151$	2.71367	0.220835	0.110418	−	0.993885i	$-0.464781\pi$
$151$	2.71367	0.220835	0.110418	+	0.993885i	$0.464781\pi$
$152$	3.36195	0.272691
$153$	0	0
$154$	−4.87570	−0.392895
$155$	6.69984	0.538144
$156$	0	0
$157$	−19.7580	−1.57686	−0.788428	−	0.615127i	$-0.789105\pi$
$157$	−19.7580	−1.57686	−0.788428	+	0.615127i	$0.789105\pi$
$158$	−3.87812	−0.308527
$159$	0	0
$160$	2.31208	0.182786
$161$	2.95085	0.232560
$162$	0	0
$163$	1.47742	0.115720	0.0578601	−	0.998325i	$-0.481572\pi$
$163$	1.47742	0.115720	0.0578601	+	0.998325i	$0.481572\pi$
$164$	3.86017	0.301428
$165$	0	0
$166$	5.42978	0.421433
$167$	−0.728612	−0.0563817	−0.0281909	−	0.999603i	$-0.508975\pi$
$167$	−0.728612	−0.0563817	−0.0281909	+	0.999603i	$0.508975\pi$
$168$	0	0
$169$	−11.7411	−0.903165
$170$	−4.68756	−0.359519
$171$	0	0
$172$	−6.40146	−0.488107
$173$	11.0768	0.842152	0.421076	−	0.907025i	$-0.361652\pi$
$173$	11.0768	0.842152	0.421076	+	0.907025i	$0.361652\pi$
$174$	0	0
$175$	−18.1173	−1.36954
$176$	2.54033	0.191485
$177$	0	0
$178$	6.20602	0.465161
$179$	−6.12132	−0.457529	−0.228765	−	0.973482i	$-0.573469\pi$
$179$	−6.12132	−0.457529	−0.228765	+	0.973482i	$0.573469\pi$
$180$	0	0
$181$	16.1112	1.19754	0.598769	−	0.800922i	$-0.295657\pi$
$181$	16.1112	1.19754	0.598769	+	0.800922i	$0.295657\pi$
$182$	−5.47049	−0.405499
$183$	0	0
$184$	−2.22538	−0.164057
$185$	8.39897	0.617504
$186$	0	0
$187$	5.26387	0.384932
$188$	−5.81510	−0.424110
$189$	0	0
$190$	−0.974228	−0.0706780
$191$	3.72993	0.269889	0.134944	−	0.990853i	$-0.456914\pi$
$191$	3.72993	0.269889	0.134944	+	0.990853i	$0.456914\pi$
$192$	0	0
$193$	−7.25204	−0.522013	−0.261007	−	0.965337i	$-0.584054\pi$
$193$	−7.25204	−0.522013	−0.261007	+	0.965337i	$0.584054\pi$
$194$	22.2436	1.59700
$195$	0	0
$196$	−5.45870	−0.389907
$197$	5.14820	0.366794	0.183397	−	0.983039i	$-0.441291\pi$
$197$	5.14820	0.366794	0.183397	+	0.983039i	$0.441291\pi$
$198$	0	0
$199$	−3.42377	−0.242704	−0.121352	−	0.992610i	$-0.538723\pi$
$199$	−3.42377	−0.242704	−0.121352	+	0.992610i	$0.538723\pi$
$200$	13.6631	0.966130
$201$	0	0
$202$	10.6990	0.752782
$203$	−1.52962	−0.107358
$204$	0	0
$205$	−5.05502	−0.353058
$206$	−3.98403	−0.277580
$207$	0	0
$208$	2.85022	0.197627
$209$	1.09401	0.0756739
$210$	0	0
$211$	−18.3075	−1.26034	−0.630172	−	0.776456i	$-0.717015\pi$
$211$	−18.3075	−1.26034	−0.630172	+	0.776456i	$0.717015\pi$
$212$	−3.54779	−0.243663
$213$	0	0
$214$	4.12570	0.282027
$215$	8.38293	0.571711
$216$	0	0
$217$	36.6826	2.49018
$218$	22.1570	1.50066
$219$	0	0
$220$	0.422984	0.0285176
$221$	5.90601	0.397281
$222$	0	0
$223$	−13.8312	−0.926205	−0.463102	−	0.886305i	$-0.653264\pi$
$223$	−13.8312	−0.926205	−0.463102	+	0.886305i	$0.653264\pi$
$224$	12.6590	0.845813
$225$	0	0
$226$	3.20590	0.213253
$227$	4.02566	0.267193	0.133596	−	0.991036i	$-0.457347\pi$
$227$	4.02566	0.267193	0.133596	+	0.991036i	$0.457347\pi$
$228$	0	0
$229$	−28.7740	−1.90144	−0.950719	−	0.310055i	$-0.899653\pi$
$229$	−28.7740	−1.90144	−0.950719	+	0.310055i	$0.899653\pi$
$230$	0.644870	0.0425215
$231$	0	0
$232$	1.15356	0.0757346
$233$	11.0675	0.725054	0.362527	−	0.931973i	$-0.381914\pi$
$233$	11.0675	0.725054	0.362527	+	0.931973i	$0.381914\pi$
$234$	0	0
$235$	7.61508	0.496753
$236$	6.39427	0.416232
$237$	0	0
$238$	−25.6651	−1.66362
$239$	−23.8339	−1.54169	−0.770844	−	0.637024i	$-0.780165\pi$
$239$	−23.8339	−1.54169	−0.770844	+	0.637024i	$0.780165\pi$
$240$	0	0
$241$	13.4590	0.866971	0.433485	−	0.901161i	$-0.357284\pi$
$241$	13.4590	0.866971	0.433485	+	0.901161i	$0.357284\pi$
$242$	1.19652	0.0769154
$243$	0	0
$244$	−5.59943	−0.358467
$245$	7.14836	0.456692
$246$	0	0
$247$	1.22746	0.0781016
$248$	−27.6641	−1.75667
$249$	0	0
$250$	−8.41189	−0.532014
$251$	27.7320	1.75043	0.875216	−	0.483733i	$-0.160719\pi$
$251$	27.7320	1.75043	0.875216	+	0.483733i	$0.160719\pi$
$252$	0	0
$253$	−0.724154	−0.0455272
$254$	−10.3593	−0.649998
$255$	0	0
$256$	−12.4342	−0.777140
$257$	15.7445	0.982114	0.491057	−	0.871127i	$-0.336611\pi$
$257$	15.7445	0.982114	0.491057	+	0.871127i	$0.336611\pi$
$258$	0	0
$259$	45.9856	2.85740
$260$	0.474583	0.0294324
$261$	0	0
$262$	20.7847	1.28408
$263$	−14.1490	−0.872466	−0.436233	−	0.899834i	$-0.643688\pi$
$263$	−14.1490	−0.872466	−0.436233	+	0.899834i	$0.643688\pi$
$264$	0	0
$265$	4.64596	0.285399
$266$	−5.33404	−0.327051
$267$	0	0
$268$	0.568334	0.0347165
$269$	−3.66802	−0.223643	−0.111821	−	0.993728i	$-0.535668\pi$
$269$	−3.66802	−0.223643	−0.111821	+	0.993728i	$0.535668\pi$
$270$	0	0
$271$	32.4689	1.97235	0.986174	−	0.165712i	$-0.0529924\pi$
$271$	32.4689	1.97235	0.986174	+	0.165712i	$0.0529924\pi$
$272$	13.3720	0.810794
$273$	0	0
$274$	−25.6391	−1.54892
$275$	4.44609	0.268109
$276$	0	0
$277$	−8.27566	−0.497236	−0.248618	−	0.968602i	$-0.579976\pi$
$277$	−8.27566	−0.497236	−0.248618	+	0.968602i	$0.579976\pi$
$278$	−20.3807	−1.22235
$279$	0	0
$280$	−9.31985	−0.556968
$281$	17.0978	1.01997	0.509986	−	0.860183i	$-0.329651\pi$
$281$	17.0978	1.01997	0.509986	+	0.860183i	$0.329651\pi$
$282$	0	0
$283$	−14.2677	−0.848130	−0.424065	−	0.905632i	$-0.639397\pi$
$283$	−14.2677	−0.848130	−0.424065	+	0.905632i	$0.639397\pi$
$284$	0.540500	0.0320728
$285$	0	0
$286$	1.34249	0.0793828
$287$	−27.6770	−1.63372
$288$	0	0
$289$	10.7083	0.629901
$290$	−0.334278	−0.0196295
$291$	0	0
$292$	1.34385	0.0786427
$293$	20.3642	1.18969	0.594843	−	0.803842i	$-0.297214\pi$
$293$	20.3642	1.18969	0.594843	+	0.803842i	$0.297214\pi$
$294$	0	0
$295$	−8.37353	−0.487526
$296$	−34.6799	−2.01573
$297$	0	0
$298$	24.7293	1.43253
$299$	−0.812493	−0.0469877
$300$	0	0
$301$	45.8978	2.64550
$302$	3.24697	0.186842
$303$	0	0
$304$	2.77913	0.159394
$305$	7.33265	0.419866
$306$	0	0
$307$	8.76598	0.500301	0.250151	−	0.968207i	$-0.419520\pi$
$307$	8.76598	0.500301	0.250151	+	0.968207i	$0.419520\pi$
$308$	2.31590	0.131961
$309$	0	0
$310$	8.01651	0.455307
$311$	−17.4388	−0.988862	−0.494431	−	0.869217i	$-0.664624\pi$
$311$	−17.4388	−0.988862	−0.494431	+	0.869217i	$0.664624\pi$
$312$	0	0
$313$	19.5878	1.10717	0.553584	−	0.832793i	$-0.313260\pi$
$313$	19.5878	1.10717	0.553584	+	0.832793i	$0.313260\pi$
$314$	−23.6408	−1.33413
$315$	0	0
$316$	1.84206	0.103624
$317$	−15.1231	−0.849398	−0.424699	−	0.905335i	$-0.639620\pi$
$317$	−15.1231	−0.849398	−0.424699	+	0.905335i	$0.639620\pi$
$318$	0	0
$319$	0.375376	0.0210170
$320$	6.54775	0.366030
$321$	0	0
$322$	3.53076	0.196762
$323$	5.75870	0.320423
$324$	0	0
$325$	4.98846	0.276710
$326$	1.76776	0.0979072
$327$	0	0
$328$	20.8725	1.15249
$329$	41.6937	2.29865
$330$	0	0
$331$	32.4727	1.78486	0.892431	−	0.451183i	$-0.148998\pi$
$331$	32.4727	1.78486	0.892431	+	0.451183i	$0.148998\pi$
$332$	−2.57908	−0.141545
$333$	0	0
$334$	−0.871801	−0.0477028
$335$	−0.744253	−0.0406629
$336$	0	0
$337$	7.46901	0.406863	0.203431	−	0.979089i	$-0.434791\pi$
$337$	7.46901	0.406863	0.203431	+	0.979089i	$0.434791\pi$
$338$	−14.0485	−0.764140
$339$	0	0
$340$	2.22653	0.120751
$341$	−9.00211	−0.487492
$342$	0	0
$343$	10.6141	0.573107
$344$	−34.6137	−1.86625
$345$	0	0
$346$	13.2536	0.712519
$347$	−8.28225	−0.444615	−0.222307	−	0.974977i	$-0.571359\pi$
$347$	−8.28225	−0.444615	−0.222307	+	0.974977i	$0.571359\pi$
$348$	0	0
$349$	25.1876	1.34826	0.674131	−	0.738612i	$-0.264518\pi$
$349$	25.1876	1.34826	0.674131	+	0.738612i	$0.264518\pi$
$350$	−21.6778	−1.15873
$351$	0	0
$352$	−3.10658	−0.165581
$353$	0.456526	0.0242984	0.0121492	−	0.999926i	$-0.496133\pi$
$353$	0.456526	0.0242984	0.0121492	+	0.999926i	$0.496133\pi$
$354$	0	0
$355$	−0.707804	−0.0375663
$356$	−2.94778	−0.156232
$357$	0	0
$358$	−7.32430	−0.387101
$359$	5.29880	0.279660	0.139830	−	0.990176i	$-0.455344\pi$
$359$	5.29880	0.279660	0.139830	+	0.990176i	$0.455344\pi$
$360$	0	0
$361$	−17.8032	−0.937008
$362$	19.2774	1.01320
$363$	0	0
$364$	2.59841	0.136194
$365$	−1.75981	−0.0921129
$366$	0	0
$367$	17.8911	0.933907	0.466954	−	0.884282i	$-0.345351\pi$
$367$	17.8911	0.933907	0.466954	+	0.884282i	$0.345351\pi$
$368$	−1.83959	−0.0958953
$369$	0	0
$370$	10.0496	0.522451
$371$	25.4373	1.32064
$372$	0	0
$373$	−18.9707	−0.982267	−0.491133	−	0.871084i	$-0.663417\pi$
$373$	−18.9707	−0.982267	−0.491133	+	0.871084i	$0.663417\pi$
$374$	6.29834	0.325679
$375$	0	0
$376$	−31.4432	−1.62156
$377$	0.421168	0.0216912
$378$	0	0
$379$	−12.6482	−0.649692	−0.324846	−	0.945767i	$-0.605313\pi$
$379$	−12.6482	−0.649692	−0.324846	+	0.945767i	$0.605313\pi$
$380$	0.462747	0.0237384
$381$	0	0
$382$	4.46295	0.228344
$383$	−11.2321	−0.573931	−0.286966	−	0.957941i	$-0.592647\pi$
$383$	−11.2321	−0.573931	−0.286966	+	0.957941i	$0.592647\pi$
$384$	0	0
$385$	−3.03275	−0.154563
$386$	−8.67723	−0.441659
$387$	0	0
$388$	−10.5654	−0.536379
$389$	31.2055	1.58218	0.791090	−	0.611700i	$-0.209514\pi$
$389$	31.2055	1.58218	0.791090	+	0.611700i	$0.209514\pi$
$390$	0	0
$391$	−3.81185	−0.192774
$392$	−29.5161	−1.49079
$393$	0	0
$394$	6.15994	0.310333
$395$	−2.41224	−0.121373
$396$	0	0
$397$	−20.6539	−1.03659	−0.518295	−	0.855202i	$-0.673433\pi$
$397$	−20.6539	−1.03659	−0.518295	+	0.855202i	$0.673433\pi$
$398$	−4.09661	−0.205345
$399$	0	0
$400$	11.2945	0.564726
$401$	31.6619	1.58112	0.790560	−	0.612385i	$-0.209790\pi$
$401$	31.6619	1.58112	0.790560	+	0.612385i	$0.209790\pi$
$402$	0	0
$403$	−10.1003	−0.503130
$404$	−5.08191	−0.252835
$405$	0	0
$406$	−1.83022	−0.0908323
$407$	−11.2851	−0.559382
$408$	0	0
$409$	2.37165	0.117271	0.0586353	−	0.998279i	$-0.481325\pi$
$409$	2.37165	0.117271	0.0586353	+	0.998279i	$0.481325\pi$
$410$	−6.04845	−0.298712
$411$	0	0
$412$	1.89236	0.0932301
$413$	−45.8463	−2.25595
$414$	0	0
$415$	3.37740	0.165790
$416$	−3.48555	−0.170893
$417$	0	0
$418$	1.30900	0.0640254
$419$	−15.3932	−0.752008	−0.376004	−	0.926618i	$-0.622702\pi$
$419$	−15.3932	−0.752008	−0.376004	+	0.926618i	$0.622702\pi$
$420$	0	0
$421$	−10.4301	−0.508331	−0.254166	−	0.967161i	$-0.581801\pi$
$421$	−10.4301	−0.508331	−0.254166	+	0.967161i	$0.581801\pi$
$422$	−21.9054	−1.06634
$423$	0	0
$424$	−19.1835	−0.931632
$425$	23.4036	1.13524
$426$	0	0
$427$	40.1473	1.94287
$428$	−1.95965	−0.0947235
$429$	0	0
$430$	10.0304	0.483707
$431$	−8.80477	−0.424111	−0.212055	−	0.977258i	$-0.568016\pi$
$431$	−8.80477	−0.424111	−0.212055	+	0.977258i	$0.568016\pi$
$432$	0	0
$433$	−25.9896	−1.24898	−0.624489	−	0.781033i	$-0.714693\pi$
$433$	−25.9896	−1.24898	−0.624489	+	0.781033i	$0.714693\pi$
$434$	43.8916	2.10686
$435$	0	0
$436$	−10.5243	−0.504024
$437$	−0.792228	−0.0378974
$438$	0	0
$439$	10.6115	0.506460	0.253230	−	0.967406i	$-0.418507\pi$
$439$	10.6115	0.506460	0.253230	+	0.967406i	$0.418507\pi$
$440$	2.28714	0.109035
$441$	0	0
$442$	7.06667	0.336127
$443$	−14.9726	−0.711368	−0.355684	−	0.934606i	$-0.615752\pi$
$443$	−14.9726	−0.711368	−0.355684	+	0.934606i	$0.615752\pi$
$444$	0	0
$445$	3.86022	0.182992
$446$	−16.5493	−0.783634
$447$	0	0
$448$	35.8499	1.69375
$449$	35.3977	1.67052	0.835260	−	0.549855i	$-0.185317\pi$
$449$	35.3977	1.67052	0.835260	+	0.549855i	$0.185317\pi$
$450$	0	0
$451$	6.79208	0.319827
$452$	−1.52276	−0.0716248
$453$	0	0
$454$	4.81680	0.226063
$455$	−3.40271	−0.159522
$456$	0	0
$457$	40.8890	1.91271	0.956354	−	0.292209i	$-0.0943904\pi$
$457$	40.8890	1.91271	0.956354	+	0.292209i	$0.0943904\pi$
$458$	−34.4287	−1.60875
$459$	0	0
$460$	−0.306305	−0.0142816
$461$	−21.2804	−0.991128	−0.495564	−	0.868571i	$-0.665039\pi$
$461$	−21.2804	−0.991128	−0.495564	+	0.868571i	$0.665039\pi$
$462$	0	0
$463$	20.1486	0.936385	0.468193	−	0.883626i	$-0.344905\pi$
$463$	20.1486	0.936385	0.468193	+	0.883626i	$0.344905\pi$
$464$	0.953578	0.0442687
$465$	0	0
$466$	13.2425	0.613446
$467$	−1.35142	−0.0625364	−0.0312682	−	0.999511i	$-0.509955\pi$
$467$	−1.35142	−0.0625364	−0.0312682	+	0.999511i	$0.509955\pi$
$468$	0	0
$469$	−4.07489	−0.188161
$470$	9.11161	0.420287
$471$	0	0
$472$	34.5749	1.59144
$473$	−11.2636	−0.517899
$474$	0	0
$475$	4.86404	0.223178
$476$	12.1906	0.558755
$477$	0	0
$478$	−28.5178	−1.30437
$479$	−16.6752	−0.761908	−0.380954	−	0.924594i	$-0.624404\pi$
$479$	−16.6752	−0.761908	−0.380954	+	0.924594i	$0.624404\pi$
$480$	0	0
$481$	−12.6618	−0.577327
$482$	16.1040	0.733517
$483$	0	0
$484$	−0.568334	−0.0258333
$485$	13.8358	0.628252
$486$	0	0
$487$	17.0062	0.770626	0.385313	−	0.922786i	$-0.374093\pi$
$487$	17.0062	0.770626	0.385313	+	0.922786i	$0.374093\pi$
$488$	−30.2770	−1.37058
$489$	0	0
$490$	8.55318	0.386393
$491$	30.9009	1.39454	0.697270	−	0.716809i	$-0.254398\pi$
$491$	30.9009	1.39454	0.697270	+	0.716809i	$0.254398\pi$
$492$	0	0
$493$	1.97593	0.0889913
$494$	1.46869	0.0660793
$495$	0	0
$496$	−22.8683	−1.02682
$497$	−3.87533	−0.173832
$498$	0	0
$499$	−21.7368	−0.973073	−0.486537	−	0.873660i	$-0.661740\pi$
$499$	−21.7368	−0.973073	−0.486537	+	0.873660i	$0.661740\pi$
$500$	3.99554	0.178686
$501$	0	0
$502$	33.1820	1.48099
$503$	29.8741	1.33202	0.666011	−	0.745942i	$-0.268000\pi$
$503$	29.8741	1.33202	0.666011	+	0.745942i	$0.268000\pi$
$504$	0	0
$505$	6.65494	0.296141
$506$	−0.866467	−0.0385191
$507$	0	0
$508$	4.92052	0.218313
$509$	−18.3002	−0.811142	−0.405571	−	0.914064i	$-0.632927\pi$
$509$	−18.3002	−0.811142	−0.405571	+	0.914064i	$0.632927\pi$
$510$	0	0
$511$	−9.63524	−0.426238
$512$	−23.5049	−1.03878
$513$	0	0
$514$	18.8386	0.830937
$515$	−2.47812	−0.109199
$516$	0	0
$517$	−10.2318	−0.449996
$518$	55.0228	2.41756
$519$	0	0
$520$	2.56615	0.112533
$521$	−19.0384	−0.834086	−0.417043	−	0.908887i	$-0.636934\pi$
$521$	−19.0384	−0.834086	−0.417043	+	0.908887i	$0.636934\pi$
$522$	0	0
$523$	5.90351	0.258142	0.129071	−	0.991635i	$-0.458800\pi$
$523$	5.90351	0.258142	0.129071	+	0.991635i	$0.458800\pi$
$524$	−9.87249	−0.431282
$525$	0	0
$526$	−16.9296	−0.738167
$527$	−47.3859	−2.06416
$528$	0	0
$529$	−22.4756	−0.977200
$530$	5.55899	0.241467
$531$	0	0
$532$	2.53361	0.109846
$533$	7.62064	0.330087
$534$	0	0
$535$	2.56624	0.110948
$536$	3.07307	0.132736
$537$	0	0
$538$	−4.38886	−0.189217
$539$	−9.60475	−0.413706
$540$	0	0
$541$	−4.46800	−0.192094	−0.0960472	−	0.995377i	$-0.530620\pi$
$541$	−4.46800	−0.192094	−0.0960472	+	0.995377i	$0.530620\pi$
$542$	38.8498	1.66874
$543$	0	0
$544$	−16.3526	−0.701112
$545$	13.7820	0.590355
$546$	0	0
$547$	−0.520429	−0.0222519	−0.0111260	−	0.999938i	$-0.503542\pi$
$547$	−0.520429	−0.0222519	−0.0111260	+	0.999938i	$0.503542\pi$
$548$	12.1783	0.520230
$549$	0	0
$550$	5.31984	0.226839
$551$	0.410663	0.0174948
$552$	0	0
$553$	−13.2074	−0.561634
$554$	−9.90201	−0.420696
$555$	0	0
$556$	9.68057	0.410548
$557$	−28.6392	−1.21348	−0.606741	−	0.794900i	$-0.707523\pi$
$557$	−28.6392	−1.21348	−0.606741	+	0.794900i	$0.707523\pi$
$558$	0	0
$559$	−12.6376	−0.534513
$560$	−7.70419	−0.325561
$561$	0	0
$562$	20.4580	0.862967
$563$	11.4870	0.484120	0.242060	−	0.970261i	$-0.422177\pi$
$563$	11.4870	0.484120	0.242060	+	0.970261i	$0.422177\pi$
$564$	0	0
$565$	1.99411	0.0838930
$566$	−17.0717	−0.717576
$567$	0	0
$568$	2.92257	0.122628
$569$	−12.7634	−0.535069	−0.267535	−	0.963548i	$-0.586209\pi$
$569$	−12.7634	−0.535069	−0.267535	+	0.963548i	$0.586209\pi$
$570$	0	0
$571$	5.52770	0.231327	0.115664	−	0.993288i	$-0.463101\pi$
$571$	5.52770	0.231327	0.115664	+	0.993288i	$0.463101\pi$
$572$	−0.637664	−0.0266621
$573$	0	0
$574$	−33.1162	−1.38224
$575$	−3.21965	−0.134269
$576$	0	0
$577$	−10.8644	−0.452290	−0.226145	−	0.974094i	$-0.572612\pi$
$577$	−10.8644	−0.452290	−0.226145	+	0.974094i	$0.572612\pi$
$578$	12.8127	0.532940
$579$	0	0
$580$	0.158778	0.00659289
$581$	18.4917	0.767167
$582$	0	0
$583$	−6.24245	−0.258536
$584$	7.26639	0.300685
$585$	0	0
$586$	24.3662	1.00656
$587$	32.2354	1.33050	0.665249	−	0.746622i	$-0.268325\pi$
$587$	32.2354	1.33050	0.665249	+	0.746622i	$0.268325\pi$
$588$	0	0
$589$	−9.84836	−0.405794
$590$	−10.0191	−0.412480
$591$	0	0
$592$	−28.6679	−1.17824
$593$	−2.69079	−0.110497	−0.0552487	−	0.998473i	$-0.517595\pi$
$593$	−2.69079	−0.110497	−0.0552487	+	0.998473i	$0.517595\pi$
$594$	0	0
$595$	−15.9640	−0.654460
$596$	−11.7461	−0.481139
$597$	0	0
$598$	−0.972167	−0.0397548
$599$	23.7416	0.970057	0.485029	−	0.874498i	$-0.338809\pi$
$599$	23.7416	0.970057	0.485029	+	0.874498i	$0.338809\pi$
$600$	0	0
$601$	−32.7751	−1.33693	−0.668463	−	0.743746i	$-0.733047\pi$
$601$	−32.7751	−1.33693	−0.668463	+	0.743746i	$0.733047\pi$
$602$	54.9177	2.23828
$603$	0	0
$604$	−1.54227	−0.0627541
$605$	0.744253	0.0302582
$606$	0	0
$607$	−30.9890	−1.25780	−0.628902	−	0.777485i	$-0.716495\pi$
$607$	−30.9890	−1.25780	−0.628902	+	0.777485i	$0.716495\pi$
$608$	−3.39861	−0.137832
$609$	0	0
$610$	8.77368	0.355236
$611$	−11.4800	−0.464432
$612$	0	0
$613$	−41.2233	−1.66499	−0.832496	−	0.554031i	$-0.813089\pi$
$613$	−41.2233	−1.66499	−0.832496	+	0.554031i	$0.813089\pi$
$614$	10.4887	0.423289
$615$	0	0
$616$	12.5224	0.504543
$617$	−15.3401	−0.617567	−0.308784	−	0.951132i	$-0.599922\pi$
$617$	−15.3401	−0.617567	−0.308784	+	0.951132i	$0.599922\pi$
$618$	0	0
$619$	−34.2142	−1.37518	−0.687591	−	0.726098i	$-0.741332\pi$
$619$	−34.2142	−1.37518	−0.687591	+	0.726098i	$0.741332\pi$
$620$	−3.80775	−0.152923
$621$	0	0
$622$	−20.8659	−0.836646
$623$	21.1353	0.846767
$624$	0	0
$625$	16.9981	0.679925
$626$	23.4373	0.936741
$627$	0	0
$628$	11.2291	0.448090
$629$	−59.4033	−2.36856
$630$	0	0
$631$	−3.33447	−0.132743	−0.0663716	−	0.997795i	$-0.521142\pi$
$631$	−3.33447	−0.132743	−0.0663716	+	0.997795i	$0.521142\pi$
$632$	9.96030	0.396199
$633$	0	0
$634$	−18.0951	−0.718650
$635$	−6.44359	−0.255706
$636$	0	0
$637$	−10.7764	−0.426978
$638$	0.449145	0.0177818
$639$	0	0
$640$	3.21037	0.126901
$641$	−37.5285	−1.48229	−0.741143	−	0.671347i	$-0.765716\pi$
$641$	−37.5285	−1.48229	−0.741143	+	0.671347i	$0.765716\pi$
$642$	0	0
$643$	−2.91739	−0.115051	−0.0575253	−	0.998344i	$-0.518321\pi$
$643$	−2.91739	−0.115051	−0.0575253	+	0.998344i	$0.518321\pi$
$644$	−1.67707	−0.0660857
$645$	0	0
$646$	6.89042	0.271100
$647$	−25.3412	−0.996266	−0.498133	−	0.867101i	$-0.665981\pi$
$647$	−25.3412	−0.996266	−0.498133	+	0.867101i	$0.665981\pi$
$648$	0	0
$649$	11.2509	0.441637
$650$	5.96881	0.234116
$651$	0	0
$652$	−0.839665	−0.0328838
$653$	−2.51387	−0.0983754	−0.0491877	−	0.998790i	$-0.515663\pi$
$653$	−2.51387	−0.0983754	−0.0491877	+	0.998790i	$0.515663\pi$
$654$	0	0
$655$	12.9284	0.505153
$656$	17.2541	0.673660
$657$	0	0
$658$	49.8874	1.94481
$659$	−0.739150	−0.0287932	−0.0143966	−	0.999896i	$-0.504583\pi$
$659$	−0.739150	−0.0287932	−0.0143966	+	0.999896i	$0.504583\pi$
$660$	0	0
$661$	25.4426	0.989603	0.494802	−	0.869006i	$-0.335241\pi$
$661$	25.4426	0.989603	0.494802	+	0.869006i	$0.335241\pi$
$662$	38.8544	1.51012
$663$	0	0
$664$	−13.9455	−0.541190
$665$	−3.31785	−0.128661
$666$	0	0
$667$	−0.271830	−0.0105253
$668$	0.414095	0.0160218
$669$	0	0
$670$	−0.890515	−0.0344036
$671$	−9.85237	−0.380346
$672$	0	0
$673$	21.5175	0.829438	0.414719	−	0.909950i	$-0.363880\pi$
$673$	21.5175	0.829438	0.414719	+	0.909950i	$0.363880\pi$
$674$	8.93684	0.344234
$675$	0	0
$676$	6.67288	0.256649
$677$	41.6194	1.59956	0.799781	−	0.600292i	$-0.204949\pi$
$677$	41.6194	1.59956	0.799781	+	0.600292i	$0.204949\pi$
$678$	0	0
$679$	75.7531	2.90714
$680$	12.0392	0.461682
$681$	0	0
$682$	−10.7712	−0.412452
$683$	21.1009	0.807402	0.403701	−	0.914891i	$-0.367724\pi$
$683$	21.1009	0.807402	0.403701	+	0.914891i	$0.367724\pi$
$684$	0	0
$685$	−15.9479	−0.609337
$686$	12.7000	0.484888
$687$	0	0
$688$	−28.6132	−1.09087
$689$	−7.00396	−0.266830
$690$	0	0
$691$	−11.5428	−0.439108	−0.219554	−	0.975600i	$-0.570460\pi$
$691$	−11.5428	−0.439108	−0.219554	+	0.975600i	$0.570460\pi$
$692$	−6.29531	−0.239312
$693$	0	0
$694$	−9.90990	−0.376175
$695$	−12.6771	−0.480868
$696$	0	0
$697$	35.7526	1.35423
$698$	30.1376	1.14072
$699$	0	0
$700$	10.2967	0.389178
$701$	−28.6506	−1.08212	−0.541060	−	0.840984i	$-0.681977\pi$
$701$	−28.6506	−1.08212	−0.541060	+	0.840984i	$0.681977\pi$
$702$	0	0
$703$	−12.3460	−0.465637
$704$	−8.79775	−0.331578
$705$	0	0
$706$	0.546244	0.0205582
$707$	36.4368	1.37035
$708$	0	0
$709$	1.57846	0.0592804	0.0296402	−	0.999561i	$-0.490564\pi$
$709$	1.57846	0.0592804	0.0296402	+	0.999561i	$0.490564\pi$
$710$	−0.846904	−0.0317837
$711$	0	0
$712$	−15.9391	−0.597344
$713$	6.51891	0.244135
$714$	0	0
$715$	0.835044	0.0312289
$716$	3.47895	0.130015
$717$	0	0
$718$	6.34014	0.236612
$719$	25.9479	0.967694	0.483847	−	0.875152i	$-0.339239\pi$
$719$	25.9479	0.967694	0.483847	+	0.875152i	$0.339239\pi$
$720$	0	0
$721$	−13.5681	−0.505301
$722$	−21.3019	−0.792774
$723$	0	0
$724$	−9.15655	−0.340300
$725$	1.66895	0.0619834
$726$	0	0
$727$	8.36326	0.310176	0.155088	−	0.987901i	$-0.450434\pi$
$727$	8.36326	0.310176	0.155088	+	0.987901i	$0.450434\pi$
$728$	14.0500	0.520729
$729$	0	0
$730$	−2.10566	−0.0779339
$731$	−59.2899	−2.19292
$732$	0	0
$733$	−22.7609	−0.840694	−0.420347	−	0.907363i	$-0.638092\pi$
$733$	−22.7609	−0.840694	−0.420347	+	0.907363i	$0.638092\pi$
$734$	21.4071	0.790150
$735$	0	0
$736$	2.24964	0.0829228
$737$	1.00000	0.0368355
$738$	0	0
$739$	−20.1624	−0.741684	−0.370842	−	0.928696i	$-0.620931\pi$
$739$	−20.1624	−0.741684	−0.370842	+	0.928696i	$0.620931\pi$
$740$	−4.77341	−0.175474
$741$	0	0
$742$	30.4363	1.11735
$743$	−13.4527	−0.493530	−0.246765	−	0.969075i	$-0.579368\pi$
$743$	−13.4527	−0.493530	−0.246765	+	0.969075i	$0.579368\pi$
$744$	0	0
$745$	15.3819	0.563551
$746$	−22.6989	−0.831066
$747$	0	0
$748$	−2.99163	−0.109385
$749$	14.0505	0.513395
$750$	0	0
$751$	6.38246	0.232899	0.116450	−	0.993197i	$-0.462849\pi$
$751$	6.38246	0.232899	0.116450	+	0.993197i	$0.462849\pi$
$752$	−25.9923	−0.947840
$753$	0	0
$754$	0.503937	0.0183523
$755$	2.01966	0.0735028
$756$	0	0
$757$	−5.35799	−0.194740	−0.0973698	−	0.995248i	$-0.531043\pi$
$757$	−5.35799	−0.194740	−0.0973698	+	0.995248i	$0.531043\pi$
$758$	−15.1338	−0.549685
$759$	0	0
$760$	2.50214	0.0907623
$761$	12.4122	0.449943	0.224971	−	0.974365i	$-0.427771\pi$
$761$	12.4122	0.449943	0.224971	+	0.974365i	$0.427771\pi$
$762$	0	0
$763$	75.4583	2.73178
$764$	−2.11985	−0.0766934
$765$	0	0
$766$	−13.4394	−0.485586
$767$	12.6234	0.455805
$768$	0	0
$769$	−4.45773	−0.160750	−0.0803749	−	0.996765i	$-0.525612\pi$
$769$	−4.45773	−0.160750	−0.0803749	+	0.996765i	$0.525612\pi$
$770$	−3.62875	−0.130771
$771$	0	0
$772$	4.12158	0.148339
$773$	−33.4450	−1.20293	−0.601467	−	0.798898i	$-0.705417\pi$
$773$	−33.4450	−1.20293	−0.601467	+	0.798898i	$0.705417\pi$
$774$	0	0
$775$	−40.0242	−1.43771
$776$	−57.1290	−2.05081
$777$	0	0
$778$	37.3381	1.33863
$779$	7.43057	0.266228
$780$	0	0
$781$	0.951027	0.0340304
$782$	−4.56097	−0.163100
$783$	0	0
$784$	−24.3992	−0.871402
$785$	−14.7049	−0.524841
$786$	0	0
$787$	10.1233	0.360856	0.180428	−	0.983588i	$-0.442252\pi$
$787$	10.1233	0.360856	0.180428	+	0.983588i	$0.442252\pi$
$788$	−2.92589	−0.104231
$789$	0	0
$790$	−2.88630	−0.102690
$791$	10.9181	0.388202
$792$	0	0
$793$	−11.0543	−0.392548
$794$	−24.7129	−0.877027
$795$	0	0
$796$	1.94584	0.0689685
$797$	3.94823	0.139853	0.0699267	−	0.997552i	$-0.477723\pi$
$797$	3.94823	0.139853	0.0699267	+	0.997552i	$0.477723\pi$
$798$	0	0
$799$	−53.8591	−1.90540
$800$	−13.8121	−0.488332
$801$	0	0
$802$	37.8842	1.33774
$803$	2.36454	0.0834428
$804$	0	0
$805$	2.19618	0.0774051
$806$	−12.0852	−0.425683
$807$	0	0
$808$	−27.4787	−0.966697
$809$	33.0330	1.16138	0.580689	−	0.814125i	$-0.302783\pi$
$809$	33.0330	1.16138	0.580689	+	0.814125i	$0.302783\pi$
$810$	0	0
$811$	6.40582	0.224939	0.112469	−	0.993655i	$-0.464124\pi$
$811$	6.40582	0.224939	0.112469	+	0.993655i	$0.464124\pi$
$812$	0.869332	0.0305076
$813$	0	0
$814$	−13.5029	−0.473276
$815$	1.09957	0.0385163
$816$	0	0
$817$	−12.3224	−0.431106
$818$	2.83774	0.0992191
$819$	0	0
$820$	2.87294	0.100327
$821$	27.9057	0.973917	0.486958	−	0.873425i	$-0.338106\pi$
$821$	27.9057	0.973917	0.486958	+	0.873425i	$0.338106\pi$
$822$	0	0
$823$	40.0960	1.39766	0.698830	−	0.715288i	$-0.253704\pi$
$823$	40.0960	1.39766	0.698830	+	0.715288i	$0.253704\pi$
$824$	10.2323	0.356459
$825$	0	0
$826$	−54.8561	−1.90869
$827$	−8.17363	−0.284225	−0.142113	−	0.989851i	$-0.545389\pi$
$827$	−8.17363	−0.284225	−0.142113	+	0.989851i	$0.545389\pi$
$828$	0	0
$829$	−40.7296	−1.41460	−0.707299	−	0.706914i	$-0.750087\pi$
$829$	−40.7296	−1.41460	−0.707299	+	0.706914i	$0.750087\pi$
$830$	4.04113	0.140270
$831$	0	0
$832$	−9.87098	−0.342215
$833$	−50.5582	−1.75174
$834$	0	0
$835$	−0.542272	−0.0187661
$836$	−0.621760	−0.0215040
$837$	0	0
$838$	−18.4183	−0.636251
$839$	52.6913	1.81911	0.909553	−	0.415588i	$-0.136424\pi$
$839$	52.6913	1.81911	0.909553	+	0.415588i	$0.136424\pi$
$840$	0	0
$841$	−28.8591	−0.995141
$842$	−12.4798	−0.430084
$843$	0	0
$844$	10.4048	0.358148
$845$	−8.73838	−0.300609
$846$	0	0
$847$	4.07489	0.140015
$848$	−15.8579	−0.544562
$849$	0	0
$850$	28.0030	0.960494
$851$	8.17215	0.280138
$852$	0	0
$853$	−11.2478	−0.385117	−0.192559	−	0.981285i	$-0.561679\pi$
$853$	−11.2478	−0.385117	−0.192559	+	0.981285i	$0.561679\pi$
$854$	48.0372	1.64380
$855$	0	0
$856$	−10.5962	−0.362169
$857$	46.2429	1.57963	0.789814	−	0.613347i	$-0.210177\pi$
$857$	46.2429	1.57963	0.789814	+	0.613347i	$0.210177\pi$
$858$	0	0
$859$	−29.1473	−0.994494	−0.497247	−	0.867609i	$-0.665656\pi$
$859$	−29.1473	−0.994494	−0.497247	+	0.867609i	$0.665656\pi$
$860$	−4.76430	−0.162461
$861$	0	0
$862$	−10.5351	−0.358827
$863$	−24.6460	−0.838960	−0.419480	−	0.907765i	$-0.637787\pi$
$863$	−24.6460	−0.838960	−0.419480	+	0.907765i	$0.637787\pi$
$864$	0	0
$865$	8.24393	0.280302
$866$	−31.0971	−1.05672
$867$	0	0
$868$	−20.8480	−0.707626
$869$	3.24116	0.109949
$870$	0	0
$871$	1.12199	0.0380172
$872$	−56.9067	−1.92710
$873$	0	0
$874$	−0.947919	−0.0320638
$875$	−28.6476	−0.968467
$876$	0	0
$877$	−17.1256	−0.578292	−0.289146	−	0.957285i	$-0.593371\pi$
$877$	−17.1256	−0.578292	−0.289146	+	0.957285i	$0.593371\pi$
$878$	12.6969	0.428500
$879$	0	0
$880$	1.89065	0.0637337
$881$	−27.3376	−0.921027	−0.460514	−	0.887653i	$-0.652335\pi$
$881$	−27.3376	−0.921027	−0.460514	+	0.887653i	$0.652335\pi$
$882$	0	0
$883$	12.6231	0.424800	0.212400	−	0.977183i	$-0.431872\pi$
$883$	12.6231	0.424800	0.212400	+	0.977183i	$0.431872\pi$
$884$	−3.35658	−0.112894
$885$	0	0
$886$	−17.9150	−0.601866
$887$	−30.8390	−1.03547	−0.517735	−	0.855541i	$-0.673225\pi$
$887$	−30.8390	−1.03547	−0.517735	+	0.855541i	$0.673225\pi$
$888$	0	0
$889$	−35.2796	−1.18324
$890$	4.61884	0.154824
$891$	0	0
$892$	7.86073	0.263197
$893$	−11.1937	−0.374583
$894$	0	0
$895$	−4.55581	−0.152284
$896$	17.5773	0.587215
$897$	0	0
$898$	42.3541	1.41338
$899$	−3.37917	−0.112702
$900$	0	0
$901$	−32.8594	−1.09471
$902$	8.12688	0.270595
$903$	0	0
$904$	−8.23383	−0.273853
$905$	11.9908	0.398588
$906$	0	0
$907$	−36.9080	−1.22551	−0.612755	−	0.790273i	$-0.709939\pi$
$907$	−36.9080	−1.22551	−0.612755	+	0.790273i	$0.709939\pi$
$908$	−2.28792	−0.0759273
$909$	0	0
$910$	−4.07142	−0.134966
$911$	−41.5486	−1.37657	−0.688283	−	0.725442i	$-0.741635\pi$
$911$	−41.5486	−1.37657	−0.688283	+	0.725442i	$0.741635\pi$
$912$	0	0
$913$	−4.53797	−0.150185
$914$	48.9247	1.61828
$915$	0	0
$916$	16.3532	0.540325
$917$	70.7847	2.33752
$918$	0	0
$919$	12.0892	0.398787	0.199393	−	0.979920i	$-0.436103\pi$
$919$	12.0892	0.398787	0.199393	+	0.979920i	$0.436103\pi$
$920$	−1.65624	−0.0546047
$921$	0	0
$922$	−25.4625	−0.838563
$923$	1.06704	0.0351221
$924$	0	0
$925$	−50.1745	−1.64973
$926$	24.1083	0.792247
$927$	0	0
$928$	−1.16613	−0.0382802
$929$	−53.1500	−1.74380	−0.871898	−	0.489688i	$-0.837111\pi$
$929$	−53.1500	−1.74380	−0.871898	+	0.489688i	$0.837111\pi$
$930$	0	0
$931$	−10.5077	−0.344374
$932$	−6.29001	−0.206036
$933$	0	0
$934$	−1.61701	−0.0529101
$935$	3.91765	0.128121
$936$	0	0
$937$	−9.04574	−0.295511	−0.147756	−	0.989024i	$-0.547205\pi$
$937$	−9.04574	−0.295511	−0.147756	+	0.989024i	$0.547205\pi$
$938$	−4.87570	−0.159197
$939$	0	0
$940$	−4.32790	−0.141161
$941$	39.1751	1.27707	0.638536	−	0.769592i	$-0.279540\pi$
$941$	39.1751	1.27707	0.638536	+	0.769592i	$0.279540\pi$
$942$	0	0
$943$	−4.91851	−0.160169
$944$	28.5810	0.930234
$945$	0	0
$946$	−13.4771	−0.438178
$947$	8.49648	0.276099	0.138049	−	0.990425i	$-0.455917\pi$
$947$	8.49648	0.276099	0.138049	+	0.990425i	$0.455917\pi$
$948$	0	0
$949$	2.65299	0.0861196
$950$	5.81994	0.188824
$951$	0	0
$952$	65.9164	2.13636
$953$	10.0111	0.324290	0.162145	−	0.986767i	$-0.448159\pi$
$953$	10.0111	0.324290	0.162145	+	0.986767i	$0.448159\pi$
$954$	0	0
$955$	2.77601	0.0898297
$956$	13.5456	0.438097
$957$	0	0
$958$	−19.9522	−0.644627
$959$	−87.3170	−2.81961
$960$	0	0
$961$	50.0380	1.61413
$962$	−15.1501	−0.488458
$963$	0	0
$964$	−7.64920	−0.246364
$965$	−5.39735	−0.173747
$966$	0	0
$967$	−41.8887	−1.34705	−0.673525	−	0.739165i	$-0.735220\pi$
$967$	−41.8887	−1.34705	−0.673525	+	0.739165i	$0.735220\pi$
$968$	−3.07307	−0.0987722
$969$	0	0
$970$	16.5549	0.531544
$971$	29.1024	0.933941	0.466971	−	0.884273i	$-0.345345\pi$
$971$	29.1024	0.933941	0.466971	+	0.884273i	$0.345345\pi$
$972$	0	0
$973$	−69.4087	−2.22514
$974$	20.3484	0.652003
$975$	0	0
$976$	−25.0283	−0.801135
$977$	−6.91499	−0.221230	−0.110615	−	0.993863i	$-0.535282\pi$
$977$	−6.91499	−0.221230	−0.110615	+	0.993863i	$0.535282\pi$
$978$	0	0
$979$	−5.18671	−0.165768
$980$	−4.06266	−0.129777
$981$	0	0
$982$	36.9737	1.17988
$983$	3.48080	0.111020	0.0555102	−	0.998458i	$-0.482321\pi$
$983$	3.48080	0.111020	0.0555102	+	0.998458i	$0.482321\pi$
$984$	0	0
$985$	3.83156	0.122084
$986$	2.36424	0.0752928
$987$	0	0
$988$	−0.697608	−0.0221939
$989$	8.15655	0.259363
$990$	0	0
$991$	48.2549	1.53287	0.766434	−	0.642323i	$-0.222029\pi$
$991$	48.2549	1.53287	0.766434	+	0.642323i	$0.222029\pi$
$992$	27.9657	0.887913
$993$	0	0
$994$	−4.63692	−0.147074
$995$	−2.54815	−0.0807817
$996$	0	0
$997$	34.7487	1.10050	0.550251	−	0.834999i	$-0.314532\pi$
$997$	34.7487	1.10050	0.550251	+	0.834999i	$0.314532\pi$
$998$	−26.0086	−0.823287
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6633.2.a.w.1.12		17
3.2	odd	2		737.2.a.f.1.6	✓	17
33.32	even	2		8107.2.a.o.1.12		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
737.2.a.f.1.6	✓	17	3.2	odd	2
6633.2.a.w.1.12		17	1.1	even	1	trivial
8107.2.a.o.1.12		17	33.32	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:	\( N \)	\(=\)	\( 6633 = 3^{2} \cdot 11 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6633.a (trivial)

\(f(q)\)	\(=\)	\(q+1.19652 q^{2} -0.568334 q^{4} +0.744253 q^{5} +4.07489 q^{7} -3.07307 q^{8} +O(q^{10})\)	\(q+1.19652 q^{2} -0.568334 q^{4} +0.744253 q^{5} +4.07489 q^{7} -3.07307 q^{8} +0.890515 q^{10} -1.00000 q^{11} -1.12199 q^{13} +4.87570 q^{14} -2.54033 q^{16} -5.26387 q^{17} -1.09401 q^{19} -0.422984 q^{20} -1.19652 q^{22} +0.724154 q^{23} -4.44609 q^{25} -1.34249 q^{26} -2.31590 q^{28} -0.375376 q^{29} +9.00211 q^{31} +3.10658 q^{32} -6.29834 q^{34} +3.03275 q^{35} +11.2851 q^{37} -1.30900 q^{38} -2.28714 q^{40} -6.79208 q^{41} +11.2636 q^{43} +0.568334 q^{44} +0.866467 q^{46} +10.2318 q^{47} +9.60475 q^{49} -5.31984 q^{50} +0.637664 q^{52} +6.24245 q^{53} -0.744253 q^{55} -12.5224 q^{56} -0.449145 q^{58} -11.2509 q^{59} +9.85237 q^{61} +10.7712 q^{62} +8.79775 q^{64} -0.835044 q^{65} -1.00000 q^{67} +2.99163 q^{68} +3.62875 q^{70} -0.951027 q^{71} -2.36454 q^{73} +13.5029 q^{74} +0.621760 q^{76} -4.07489 q^{77} -3.24116 q^{79} -1.89065 q^{80} -8.12688 q^{82} +4.53797 q^{83} -3.91765 q^{85} +13.4771 q^{86} +3.07307 q^{88} +5.18671 q^{89} -4.57199 q^{91} -0.411561 q^{92} +12.2426 q^{94} -0.814216 q^{95} +18.5902 q^{97} +11.4923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * q^98

Introduction

L-functions

Modular forms

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Database

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