LMFDB - Embedded newform 2009.2.a.t.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2009,2,Mod(1,2009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2009.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:	$16.0419457661$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 27 x^{18} + 52 x^{17} + 302 x^{16} - 552 x^{15} - 1820 x^{14} + 3090 x^{13} + \cdots + 49 $ x^20 - 2x^19 - 27x^18 + 52x^17 + 302x^16 - 552x^15 - 1820x^14 + 3090x^13 + 6449x^12 - 9852x^11 - 13797x^10 + 18080x^9 + 17721x^8 - 18446x^7 - 13352x^6 + 9524x^5 + 5690x^4 - 1972x^3 - 1179x^2 + 42*x + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.41602$ of defining polynomial
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41602 q^{2} +2.14541 q^{3} +0.00509860 q^{4} +1.43123 q^{5} -3.03793 q^{6} +2.82481 q^{8} +1.60278 q^{9} +O(q^{10})$	\(q-1.41602 q^{2} +2.14541 q^{3} +0.00509860 q^{4} +1.43123 q^{5} -3.03793 q^{6} +2.82481 q^{8} +1.60278 q^{9} -2.02665 q^{10} -0.552813 q^{11} +0.0109386 q^{12} -5.69206 q^{13} +3.07058 q^{15} -4.01017 q^{16} -5.03615 q^{17} -2.26956 q^{18} +2.59844 q^{19} +0.00729727 q^{20} +0.782792 q^{22} +1.36689 q^{23} +6.06037 q^{24} -2.95158 q^{25} +8.06005 q^{26} -2.99761 q^{27} -9.99419 q^{29} -4.34798 q^{30} -2.88824 q^{31} +0.0288420 q^{32} -1.18601 q^{33} +7.13127 q^{34} +0.00817192 q^{36} +1.92052 q^{37} -3.67943 q^{38} -12.2118 q^{39} +4.04296 q^{40} -1.00000 q^{41} -3.41322 q^{43} -0.00281857 q^{44} +2.29395 q^{45} -1.93553 q^{46} +2.35904 q^{47} -8.60346 q^{48} +4.17948 q^{50} -10.8046 q^{51} -0.0290215 q^{52} -11.1018 q^{53} +4.24466 q^{54} -0.791204 q^{55} +5.57472 q^{57} +14.1519 q^{58} -7.91878 q^{59} +0.0156556 q^{60} +4.29043 q^{61} +4.08980 q^{62} +7.97950 q^{64} -8.14666 q^{65} +1.67941 q^{66} +5.65114 q^{67} -0.0256773 q^{68} +2.93253 q^{69} +6.45840 q^{71} +4.52754 q^{72} +3.87723 q^{73} -2.71948 q^{74} -6.33234 q^{75} +0.0132484 q^{76} +17.2921 q^{78} +6.32053 q^{79} -5.73948 q^{80} -11.2394 q^{81} +1.41602 q^{82} +11.3673 q^{83} -7.20790 q^{85} +4.83317 q^{86} -21.4416 q^{87} -1.56159 q^{88} -6.44925 q^{89} -3.24826 q^{90} +0.00696921 q^{92} -6.19646 q^{93} -3.34043 q^{94} +3.71897 q^{95} +0.0618778 q^{96} -0.377713 q^{97} -0.886037 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) $ 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9	$ 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9} - 16 q^{10} - 6 q^{12} - 12 q^{13} + 14 q^{16} - 8 q^{17} - 18 q^{18} - 36 q^{19} - 24 q^{20} - 8 q^{22} - 12 q^{23} - 36 q^{24} + 20 q^{25} + 22 q^{26} - 32 q^{27} + 4 q^{29} + 28 q^{30} - 80 q^{31} + 6 q^{32} + 12 q^{33} - 48 q^{34} + 26 q^{36} + 4 q^{37} - 12 q^{38} - 28 q^{39} + 4 q^{40} - 20 q^{41} - 20 q^{44} - 40 q^{45} + 8 q^{46} - 32 q^{47} - 16 q^{48} + 6 q^{50} - 20 q^{51} - 36 q^{52} + 4 q^{53} + 50 q^{54} - 64 q^{55} - 4 q^{57} - 32 q^{59} + 20 q^{60} - 44 q^{61} + 8 q^{62} - 30 q^{64} - 8 q^{65} - 32 q^{66} - 4 q^{67} + 48 q^{68} + 24 q^{69} + 8 q^{71} - 8 q^{72} - 48 q^{73} - 38 q^{74} - 24 q^{75} - 84 q^{76} + 30 q^{78} - 4 q^{79} - 56 q^{80} + 2 q^{82} - 8 q^{83} - 12 q^{85} - 24 q^{86} - 40 q^{87} - 48 q^{88} - 20 q^{89} - 48 q^{90} - 50 q^{92} + 48 q^{93} - 26 q^{94} + 20 q^{95} - 70 q^{96} - 8 q^{97} - 8 q^{99}+O(q^{100}) $ 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9 - 16 * q^10 - 6 * q^12 - 12 * q^13 + 14 * q^16 - 8 * q^17 - 18 * q^18 - 36 * q^19 - 24 * q^20 - 8 * q^22 - 12 * q^23 - 36 * q^24 + 20 * q^25 + 22 * q^26 - 32 * q^27 + 4 * q^29 + 28 * q^30 - 80 * q^31 + 6 * q^32 + 12 * q^33 - 48 * q^34 + 26 * q^36 + 4 * q^37 - 12 * q^38 - 28 * q^39 + 4 * q^40 - 20 * q^41 - 20 * q^44 - 40 * q^45 + 8 * q^46 - 32 * q^47 - 16 * q^48 + 6 * q^50 - 20 * q^51 - 36 * q^52 + 4 * q^53 + 50 * q^54 - 64 * q^55 - 4 * q^57 - 32 * q^59 + 20 * q^60 - 44 * q^61 + 8 * q^62 - 30 * q^64 - 8 * q^65 - 32 * q^66 - 4 * q^67 + 48 * q^68 + 24 * q^69 + 8 * q^71 - 8 * q^72 - 48 * q^73 - 38 * q^74 - 24 * q^75 - 84 * q^76 + 30 * q^78 - 4 * q^79 - 56 * q^80 + 2 * q^82 - 8 * q^83 - 12 * q^85 - 24 * q^86 - 40 * q^87 - 48 * q^88 - 20 * q^89 - 48 * q^90 - 50 * q^92 + 48 * q^93 - 26 * q^94 + 20 * q^95 - 70 * q^96 - 8 * q^97 - 8 * q^99

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 $

$2$	−1.41602	−1.00127	−0.500637	−	0.865657i	$-0.666901\pi$
$2$	−1.41602	−1.00127	−0.500637	+	0.865657i	$0.666901\pi$
$3$	2.14541	1.23865	0.619326	−	0.785134i	$-0.287406\pi$
$3$	2.14541	1.23865	0.619326	+	0.785134i	$0.287406\pi$
$4$	0.00509860	0.00254930
$5$	1.43123	0.640066	0.320033	−	0.947406i	$-0.396306\pi$
$5$	1.43123	0.640066	0.320033	+	0.947406i	$0.396306\pi$
$6$	−3.03793	−1.24023
$7$	0	0
$7$	0	0
$8$	2.82481	0.998721
$9$	1.60278	0.534259
$10$	−2.02665	−0.640882
$11$	−0.552813	−0.166679	−0.0833397	−	0.996521i	$-0.526559\pi$
$11$	−0.552813	−0.166679	−0.0833397	+	0.996521i	$0.526559\pi$
$12$	0.0109386	0.00315769
$13$	−5.69206	−1.57869	−0.789347	−	0.613947i	$-0.789581\pi$
$13$	−5.69206	−1.57869	−0.789347	+	0.613947i	$0.789581\pi$
$14$	0	0
$15$	3.07058	0.792820
$16$	−4.01017	−1.00254
$17$	−5.03615	−1.22145	−0.610723	−	0.791844i	$-0.709121\pi$
$17$	−5.03615	−1.22145	−0.610723	+	0.791844i	$0.709121\pi$
$18$	−2.26956	−0.534940
$19$	2.59844	0.596124	0.298062	−	0.954547i	$-0.403660\pi$
$19$	2.59844	0.596124	0.298062	+	0.954547i	$0.403660\pi$
$20$	0.00729727	0.00163172
$21$	0	0
$22$	0.782792	0.166892
$23$	1.36689	0.285016	0.142508	−	0.989794i	$-0.454483\pi$
$23$	1.36689	0.285016	0.142508	+	0.989794i	$0.454483\pi$
$24$	6.06037	1.23707
$25$	−2.95158	−0.590315
$26$	8.06005	1.58071
$27$	−2.99761	−0.576891
$28$	0	0
$29$	−9.99419	−1.85588	−0.927938	−	0.372736i	$-0.878420\pi$
$29$	−9.99419	−1.85588	−0.927938	+	0.372736i	$0.878420\pi$
$30$	−4.34798	−0.793829
$31$	−2.88824	−0.518744	−0.259372	−	0.965778i	$-0.583516\pi$
$31$	−2.88824	−0.518744	−0.259372	+	0.965778i	$0.583516\pi$
$32$	0.0288420	0.00509858
$33$	−1.18601	−0.206458
$34$	7.13127	1.22300
$35$	0	0
$36$	0.00817192	0.00136199
$37$	1.92052	0.315731	0.157865	−	0.987461i	$-0.449539\pi$
$37$	1.92052	0.315731	0.157865	+	0.987461i	$0.449539\pi$
$38$	−3.67943	−0.596883
$39$	−12.2118	−1.95545
$40$	4.04296	0.639248
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−3.41322	−0.520511	−0.260255	−	0.965540i	$-0.583807\pi$
$43$	−3.41322	−0.520511	−0.260255	+	0.965540i	$0.583807\pi$
$44$	−0.00281857	−0.000424916 0
$45$	2.29395	0.341961
$46$	−1.93553	−0.285379
$47$	2.35904	0.344101	0.172050	−	0.985088i	$-0.444961\pi$
$47$	2.35904	0.344101	0.172050	+	0.985088i	$0.444961\pi$
$48$	−8.60346	−1.24180
$49$	0	0
$50$	4.17948	0.591067
$51$	−10.8046	−1.51295
$52$	−0.0290215	−0.00402456
$53$	−11.1018	−1.52495	−0.762473	−	0.647020i	$-0.776015\pi$
$53$	−11.1018	−1.52495	−0.762473	+	0.647020i	$0.776015\pi$
$54$	4.24466	0.577626
$55$	−0.791204	−0.106686
$56$	0	0
$57$	5.57472	0.738390
$58$	14.1519	1.85824
$59$	−7.91878	−1.03094	−0.515468	−	0.856909i	$-0.672382\pi$
$59$	−7.91878	−1.03094	−0.515468	+	0.856909i	$0.672382\pi$
$60$	0.0156556	0.00202113
$61$	4.29043	0.549333	0.274667	−	0.961540i	$-0.411433\pi$
$61$	4.29043	0.549333	0.274667	+	0.961540i	$0.411433\pi$
$62$	4.08980	0.519404
$63$	0	0
$64$	7.97950	0.997438
$65$	−8.14666	−1.01047
$66$	1.67941	0.206721
$67$	5.65114	0.690396	0.345198	−	0.938530i	$-0.387812\pi$
$67$	5.65114	0.690396	0.345198	+	0.938530i	$0.387812\pi$
$68$	−0.0256773	−0.00311383
$69$	2.93253	0.353035
$70$	0	0
$71$	6.45840	0.766471	0.383236	−	0.923651i	$-0.374810\pi$
$71$	6.45840	0.766471	0.383236	+	0.923651i	$0.374810\pi$
$72$	4.52754	0.533576
$73$	3.87723	0.453795	0.226898	−	0.973919i	$-0.427142\pi$
$73$	3.87723	0.453795	0.226898	+	0.973919i	$0.427142\pi$
$74$	−2.71948	−0.316133
$75$	−6.33234	−0.731195
$76$	0.0132484	0.00151970
$77$	0	0
$78$	17.2921	1.95794
$79$	6.32053	0.711115	0.355557	−	0.934654i	$-0.384291\pi$
$79$	6.32053	0.711115	0.355557	+	0.934654i	$0.384291\pi$
$80$	−5.73948	−0.641694
$81$	−11.2394	−1.24883
$82$	1.41602	0.156373
$83$	11.3673	1.24772	0.623860	−	0.781536i	$-0.285564\pi$
$83$	11.3673	1.24772	0.623860	+	0.781536i	$0.285564\pi$
$84$	0	0
$85$	−7.20790	−0.781807
$86$	4.83317	0.521174
$87$	−21.4416	−2.29878
$88$	−1.56159	−0.166466
$89$	−6.44925	−0.683619	−0.341809	−	0.939769i	$-0.611040\pi$
$89$	−6.44925	−0.683619	−0.341809	+	0.939769i	$0.611040\pi$
$90$	−3.24826	−0.342397
$91$	0	0
$92$	0.00696921	0.000726590 0
$93$	−6.19646	−0.642543
$94$	−3.34043	−0.344539
$95$	3.71897	0.381559
$96$	0.0618778	0.00631537
$97$	−0.377713	−0.0383510	−0.0191755	−	0.999816i	$-0.506104\pi$
$97$	−0.377713	−0.0383510	−0.0191755	+	0.999816i	$0.506104\pi$
$98$	0	0
$99$	−0.886037	−0.0890500
$100$	−0.0150489	−0.00150489
$101$	12.8798	1.28159	0.640796	−	0.767712i	$-0.278605\pi$
$101$	12.8798	1.28159	0.640796	+	0.767712i	$0.278605\pi$
$102$	15.2995	1.51487
$103$	−10.9638	−1.08029	−0.540146	−	0.841571i	$-0.681631\pi$
$103$	−10.9638	−1.08029	−0.540146	+	0.841571i	$0.681631\pi$
$104$	−16.0790	−1.57668
$105$	0	0
$106$	15.7203	1.52689
$107$	15.8352	1.53085	0.765423	−	0.643527i	$-0.222530\pi$
$107$	15.8352	1.53085	0.765423	+	0.643527i	$0.222530\pi$
$108$	−0.0152836	−0.00147067
$109$	−10.7979	−1.03425	−0.517127	−	0.855909i	$-0.672999\pi$
$109$	−10.7979	−1.03425	−0.517127	+	0.855909i	$0.672999\pi$
$110$	1.12036	0.106822
$111$	4.12029	0.391081
$112$	0	0
$113$	−12.3115	−1.15817	−0.579086	−	0.815266i	$-0.696591\pi$
$113$	−12.3115	−1.15817	−0.579086	+	0.815266i	$0.696591\pi$
$114$	−7.89389	−0.739330
$115$	1.95633	0.182429
$116$	−0.0509564	−0.00473118
$117$	−9.12311	−0.843432
$118$	11.2131	1.03225
$119$	0	0
$120$	8.67380	0.791806
$121$	−10.6944	−0.972218
$122$	−6.07531	−0.550033
$123$	−2.14541	−0.193445
$124$	−0.0147260	−0.00132243
$125$	−11.3805	−1.01791
$126$	0	0
$127$	15.0224	1.33302	0.666509	−	0.745497i	$-0.267788\pi$
$127$	15.0224	1.33302	0.666509	+	0.745497i	$0.267788\pi$
$128$	−11.3568	−1.00381
$129$	−7.32274	−0.644732
$130$	11.5358	1.01176
$131$	4.74187	0.414299	0.207149	−	0.978309i	$-0.433581\pi$
$131$	4.74187	0.414299	0.207149	+	0.978309i	$0.433581\pi$
$132$	−0.00604699	−0.000526323 0
$133$	0	0
$134$	−8.00209	−0.691276
$135$	−4.29028	−0.369248
$136$	−14.2262	−1.21988
$137$	7.63212	0.652056	0.326028	−	0.945360i	$-0.394290\pi$
$137$	7.63212	0.652056	0.326028	+	0.945360i	$0.394290\pi$
$138$	−4.15251	−0.353485
$139$	−7.31877	−0.620770	−0.310385	−	0.950611i	$-0.600458\pi$
$139$	−7.31877	−0.620770	−0.310385	+	0.950611i	$0.600458\pi$
$140$	0	0
$141$	5.06110	0.426221
$142$	−9.14519	−0.767448
$143$	3.14665	0.263136
$144$	−6.42741	−0.535618
$145$	−14.3040	−1.18788
$146$	−5.49021	−0.454373
$147$	0	0
$148$	0.00979194	0.000804892 0
$149$	11.1149	0.910568	0.455284	−	0.890346i	$-0.349538\pi$
$149$	11.1149	0.910568	0.455284	+	0.890346i	$0.349538\pi$
$150$	8.96668	0.732127
$151$	4.12213	0.335455	0.167727	−	0.985833i	$-0.446357\pi$
$151$	4.12213	0.335455	0.167727	+	0.985833i	$0.446357\pi$
$152$	7.34011	0.595361
$153$	−8.07184	−0.652569
$154$	0	0
$155$	−4.13374	−0.332030
$156$	−0.0622630	−0.00498503
$157$	−17.4223	−1.39045	−0.695226	−	0.718791i	$-0.744696\pi$
$157$	−17.4223	−1.39045	−0.695226	+	0.718791i	$0.744696\pi$
$158$	−8.94996	−0.712020
$159$	−23.8178	−1.88888
$160$	0.0412795	0.00326343
$161$	0	0
$162$	15.9152	1.25042
$163$	1.68336	0.131851	0.0659254	−	0.997825i	$-0.479000\pi$
$163$	1.68336	0.131851	0.0659254	+	0.997825i	$0.479000\pi$
$164$	−0.00509860	−0.000398134 0
$165$	−1.69746	−0.132147
$166$	−16.0962	−1.24931
$167$	23.4238	1.81259	0.906293	−	0.422650i	$-0.138900\pi$
$167$	23.4238	1.81259	0.906293	+	0.422650i	$0.138900\pi$
$168$	0	0
$169$	19.3996	1.49227
$170$	10.2065	0.782803
$171$	4.16473	0.318485
$172$	−0.0174026	−0.00132694
$173$	−8.64584	−0.657331	−0.328666	−	0.944446i	$-0.606599\pi$
$173$	−8.64584	−0.657331	−0.328666	+	0.944446i	$0.606599\pi$
$174$	30.3617	2.30171
$175$	0	0
$176$	2.21688	0.167103
$177$	−16.9890	−1.27697
$178$	9.13223	0.684490
$179$	5.00616	0.374178	0.187089	−	0.982343i	$-0.440095\pi$
$179$	5.00616	0.374178	0.187089	+	0.982343i	$0.440095\pi$
$180$	0.0116959	0.000871762 0
$181$	−16.9453	−1.25953	−0.629766	−	0.776785i	$-0.716849\pi$
$181$	−16.9453	−1.25953	−0.629766	+	0.776785i	$0.716849\pi$
$182$	0	0
$183$	9.20472	0.680433
$184$	3.86120	0.284651
$185$	2.74870	0.202089
$186$	8.77428	0.643362
$187$	2.78405	0.203590
$188$	0.0120278	0.000877216 0
$189$	0	0
$190$	−5.26612	−0.382045
$191$	12.1453	0.878800	0.439400	−	0.898291i	$-0.355191\pi$
$191$	12.1453	0.878800	0.439400	+	0.898291i	$0.355191\pi$
$192$	17.1193	1.23548
$193$	18.0545	1.29959	0.649795	−	0.760109i	$-0.274855\pi$
$193$	18.0545	1.29959	0.649795	+	0.760109i	$0.274855\pi$
$194$	0.534848	0.0383998
$195$	−17.4779	−1.25162
$196$	0	0
$197$	−8.78247	−0.625725	−0.312862	−	0.949798i	$-0.601288\pi$
$197$	−8.78247	−0.625725	−0.312862	+	0.949798i	$0.601288\pi$
$198$	1.25464	0.0891635
$199$	−21.4452	−1.52021	−0.760105	−	0.649800i	$-0.774853\pi$
$199$	−21.4452	−1.52021	−0.760105	+	0.649800i	$0.774853\pi$
$200$	−8.33764	−0.589560
$201$	12.1240	0.855161
$202$	−18.2380	−1.28322
$203$	0	0
$204$	−0.0550883	−0.00385695
$205$	−1.43123	−0.0999616
$206$	15.5249	1.08167
$207$	2.19082	0.152272
$208$	22.8261	1.58271
$209$	−1.43645	−0.0993615
$210$	0	0
$211$	−11.2573	−0.774988	−0.387494	−	0.921872i	$-0.626659\pi$
$211$	−11.2573	−0.774988	−0.387494	+	0.921872i	$0.626659\pi$
$212$	−0.0566035	−0.00388754
$213$	13.8559	0.949391
$214$	−22.4229	−1.53280
$215$	−4.88510	−0.333161
$216$	−8.46768	−0.576153
$217$	0	0
$218$	15.2900	1.03557
$219$	8.31824	0.562094
$220$	−0.00403403	−0.000271974 0
$221$	28.6661	1.92829
$222$	−5.83439	−0.391579
$223$	−18.1563	−1.21584	−0.607918	−	0.794000i	$-0.707995\pi$
$223$	−18.1563	−1.21584	−0.607918	+	0.794000i	$0.707995\pi$
$224$	0	0
$225$	−4.73072	−0.315381
$226$	17.4333	1.15965
$227$	19.2410	1.27707	0.638534	−	0.769593i	$-0.279541\pi$
$227$	19.2410	1.27707	0.638534	+	0.769593i	$0.279541\pi$
$228$	0.0284233	0.00188238
$229$	9.15402	0.604915	0.302457	−	0.953163i	$-0.402193\pi$
$229$	9.15402	0.604915	0.302457	+	0.953163i	$0.402193\pi$
$230$	−2.77020	−0.182661
$231$	0	0
$232$	−28.2317	−1.85350
$233$	−17.1703	−1.12486	−0.562432	−	0.826843i	$-0.690134\pi$
$233$	−17.1703	−1.12486	−0.562432	+	0.826843i	$0.690134\pi$
$234$	12.9185	0.844507
$235$	3.37633	0.220247
$236$	−0.0403746	−0.00262817
$237$	13.5601	0.880824
$238$	0	0
$239$	−24.1714	−1.56351	−0.781757	−	0.623583i	$-0.785676\pi$
$239$	−24.1714	−1.56351	−0.781757	+	0.623583i	$0.785676\pi$
$240$	−12.3135	−0.794836
$241$	−0.248926	−0.0160347	−0.00801737	−	0.999968i	$-0.502552\pi$
$241$	−0.248926	−0.0160347	−0.00801737	+	0.999968i	$0.502552\pi$
$242$	15.1434	0.973456
$243$	−15.1203	−0.969971
$244$	0.0218752	0.00140041
$245$	0	0
$246$	3.03793	0.193691
$247$	−14.7905	−0.941097
$248$	−8.15874	−0.518080
$249$	24.3874	1.54549
$250$	16.1150	1.01920
$251$	7.05285	0.445172	0.222586	−	0.974913i	$-0.428550\pi$
$251$	7.05285	0.445172	0.222586	+	0.974913i	$0.428550\pi$
$252$	0	0
$253$	−0.755633	−0.0475063
$254$	−21.2719	−1.33472
$255$	−15.4639	−0.968387
$256$	0.122366	0.00764785
$257$	−27.8617	−1.73796	−0.868982	−	0.494844i	$-0.835225\pi$
$257$	−27.8617	−1.73796	−0.868982	+	0.494844i	$0.835225\pi$
$258$	10.3691	0.645553
$259$	0	0
$260$	−0.0415365	−0.00257599
$261$	−16.0185	−0.991519
$262$	−6.71456	−0.414827
$263$	−5.98411	−0.368996	−0.184498	−	0.982833i	$-0.559066\pi$
$263$	−5.98411	−0.368996	−0.184498	+	0.982833i	$0.559066\pi$
$264$	−3.35025	−0.206194
$265$	−15.8892	−0.976066
$266$	0	0
$267$	−13.8363	−0.846766
$268$	0.0288129	0.00176003
$269$	−18.8772	−1.15096	−0.575481	−	0.817815i	$-0.695185\pi$
$269$	−18.8772	−1.15096	−0.575481	+	0.817815i	$0.695185\pi$
$270$	6.07510	0.369719
$271$	−2.39515	−0.145495	−0.0727476	−	0.997350i	$-0.523177\pi$
$271$	−2.39515	−0.145495	−0.0727476	+	0.997350i	$0.523177\pi$
$272$	20.1958	1.22455
$273$	0	0
$274$	−10.8072	−0.652887
$275$	1.63167	0.0983934
$276$	0.0149518	0.000899992 0
$277$	3.59055	0.215735	0.107868	−	0.994165i	$-0.465598\pi$
$277$	3.59055	0.215735	0.107868	+	0.994165i	$0.465598\pi$
$278$	10.3635	0.621561
$279$	−4.62921	−0.277144
$280$	0	0
$281$	−5.10615	−0.304607	−0.152304	−	0.988334i	$-0.548669\pi$
$281$	−5.10615	−0.304607	−0.152304	+	0.988334i	$0.548669\pi$
$282$	−7.16659	−0.426764
$283$	−24.1802	−1.43737	−0.718683	−	0.695338i	$-0.755255\pi$
$283$	−24.1802	−1.43737	−0.718683	+	0.695338i	$0.755255\pi$
$284$	0.0329288	0.00195396
$285$	7.97872	0.472618
$286$	−4.45570	−0.263471
$287$	0	0
$288$	0.0462272	0.00272397
$289$	8.36283	0.491931
$290$	20.2547	1.18940
$291$	−0.810349	−0.0475035
$292$	0.0197684	0.00115686
$293$	6.91326	0.403877	0.201939	−	0.979398i	$-0.435276\pi$
$293$	6.91326	0.403877	0.201939	+	0.979398i	$0.435276\pi$
$294$	0	0
$295$	−11.3336	−0.659868
$296$	5.42509	0.315327
$297$	1.65712	0.0961558
$298$	−15.7389	−0.911728
$299$	−7.78041	−0.449953
$300$	−0.0322860	−0.00186403
$301$	0	0
$302$	−5.83700	−0.335882
$303$	27.6325	1.58745
$304$	−10.4202	−0.597639
$305$	6.14060	0.351610
$306$	11.4298	0.653401
$307$	12.9358	0.738285	0.369142	−	0.929373i	$-0.379651\pi$
$307$	12.9358	0.738285	0.369142	+	0.929373i	$0.379651\pi$
$308$	0	0
$309$	−23.5217	−1.33811
$310$	5.85344	0.332453
$311$	−25.2541	−1.43203	−0.716013	−	0.698087i	$-0.754035\pi$
$311$	−25.2541	−1.43203	−0.716013	+	0.698087i	$0.754035\pi$
$312$	−34.4960	−1.95295
$313$	11.0006	0.621789	0.310894	−	0.950444i	$-0.399371\pi$
$313$	11.0006	0.621789	0.310894	+	0.950444i	$0.399371\pi$
$314$	24.6703	1.39222
$315$	0	0
$316$	0.0322258	0.00181284
$317$	−0.242618	−0.0136268	−0.00681338	−	0.999977i	$-0.502169\pi$
$317$	−0.242618	−0.0136268	−0.00681338	+	0.999977i	$0.502169\pi$
$318$	33.7264	1.89128
$319$	5.52492	0.309336
$320$	11.4205	0.638426
$321$	33.9730	1.89619
$322$	0	0
$323$	−13.0862	−0.728133
$324$	−0.0573054	−0.00318363
$325$	16.8006	0.931927
$326$	−2.38366	−0.132019
$327$	−23.1660	−1.28108
$328$	−2.82481	−0.155974
$329$	0	0
$330$	2.40362	0.132315
$331$	35.8079	1.96818	0.984089	−	0.177675i	$-0.0568576\pi$
$331$	35.8079	1.96818	0.984089	+	0.177675i	$0.0568576\pi$
$332$	0.0579571	0.00318081
$333$	3.07816	0.168682
$334$	−33.1684	−1.81489
$335$	8.08808	0.441899
$336$	0	0
$337$	−16.3729	−0.891890	−0.445945	−	0.895060i	$-0.647132\pi$
$337$	−16.3729	−0.891890	−0.445945	+	0.895060i	$0.647132\pi$
$338$	−27.4701	−1.49418
$339$	−26.4133	−1.43457
$340$	−0.0367502	−0.00199306
$341$	1.59666	0.0864639
$342$	−5.89732	−0.318890
$343$	0	0
$344$	−9.64169	−0.519845
$345$	4.19713	0.225966
$346$	12.2426	0.658169
$347$	11.2954	0.606368	0.303184	−	0.952932i	$-0.401950\pi$
$347$	11.2954	0.606368	0.303184	+	0.952932i	$0.401950\pi$
$348$	−0.109322	−0.00586029
$349$	−7.43050	−0.397746	−0.198873	−	0.980025i	$-0.563728\pi$
$349$	−7.43050	−0.397746	−0.198873	+	0.980025i	$0.563728\pi$
$350$	0	0
$351$	17.0626	0.910734
$352$	−0.0159442	−0.000849829 0
$353$	11.1659	0.594302	0.297151	−	0.954830i	$-0.403963\pi$
$353$	11.1659	0.594302	0.297151	+	0.954830i	$0.403963\pi$
$354$	24.0567	1.27860
$355$	9.24347	0.490592
$356$	−0.0328821	−0.00174275
$357$	0	0
$358$	−7.08880	−0.374654
$359$	3.03157	0.160000	0.0800002	−	0.996795i	$-0.474508\pi$
$359$	3.03157	0.160000	0.0800002	+	0.996795i	$0.474508\pi$
$360$	6.47997	0.341524
$361$	−12.2481	−0.644637
$362$	23.9948	1.26114
$363$	−22.9439	−1.20424
$364$	0	0
$365$	5.54921	0.290459
$366$	−13.0340	−0.681299
$367$	−29.0499	−1.51639	−0.758196	−	0.652027i	$-0.773919\pi$
$367$	−29.0499	−1.51639	−0.758196	+	0.652027i	$0.773919\pi$
$368$	−5.48145	−0.285740
$369$	−1.60278	−0.0834373
$370$	−3.89220	−0.202346
$371$	0	0
$372$	−0.0315933	−0.00163803
$373$	−29.0333	−1.50329	−0.751644	−	0.659569i	$-0.770739\pi$
$373$	−29.0333	−1.50329	−0.751644	+	0.659569i	$0.770739\pi$
$374$	−3.94226	−0.203849
$375$	−24.4159	−1.26083
$376$	6.66383	0.343661
$377$	56.8876	2.92986
$378$	0	0
$379$	−5.33080	−0.273825	−0.136913	−	0.990583i	$-0.543718\pi$
$379$	−5.33080	−0.273825	−0.136913	+	0.990583i	$0.543718\pi$
$380$	0.0189615	0.000972707 0
$381$	32.2291	1.65115
$382$	−17.1979	−0.879920
$383$	13.6277	0.696343	0.348172	−	0.937431i	$-0.386803\pi$
$383$	13.6277	0.696343	0.348172	+	0.937431i	$0.386803\pi$
$384$	−24.3649	−1.24337
$385$	0	0
$386$	−25.5654	−1.30125
$387$	−5.47063	−0.278088
$388$	−0.00192581	−9.77681e−5 0
$389$	20.7943	1.05431	0.527155	−	0.849769i	$-0.323259\pi$
$389$	20.7943	1.05431	0.527155	+	0.849769i	$0.323259\pi$
$390$	24.7490	1.25321
$391$	−6.88385	−0.348131
$392$	0	0
$393$	10.1732	0.513172
$394$	12.4361	0.626522
$395$	9.04614	0.455160
$396$	−0.00451754	−0.000227015 0
$397$	9.03863	0.453636	0.226818	−	0.973937i	$-0.427168\pi$
$397$	9.03863	0.453636	0.226818	+	0.973937i	$0.427168\pi$
$398$	30.3667	1.52215
$399$	0	0
$400$	11.8363	0.591816
$401$	21.9783	1.09754	0.548772	−	0.835972i	$-0.315096\pi$
$401$	21.9783	1.09754	0.548772	+	0.835972i	$0.315096\pi$
$402$	−17.1678	−0.856250
$403$	16.4401	0.818938
$404$	0.0656691	0.00326716
$405$	−16.0862	−0.799332
$406$	0	0
$407$	−1.06169	−0.0526258
$408$	−30.5210	−1.51101
$409$	−10.3880	−0.513653	−0.256827	−	0.966457i	$-0.582677\pi$
$409$	−10.3880	−0.513653	−0.256827	+	0.966457i	$0.582677\pi$
$410$	2.02665	0.100089
$411$	16.3740	0.807671
$412$	−0.0558998	−0.00275399
$413$	0	0
$414$	−3.10223	−0.152466
$415$	16.2692	0.798623
$416$	−0.164170	−0.00804911
$417$	−15.7018	−0.768918
$418$	2.03404	0.0994881
$419$	−27.1441	−1.32608	−0.663038	−	0.748586i	$-0.730733\pi$
$419$	−27.1441	−1.32608	−0.663038	+	0.748586i	$0.730733\pi$
$420$	0	0
$421$	30.0270	1.46343	0.731713	−	0.681613i	$-0.238721\pi$
$421$	30.0270	1.46343	0.731713	+	0.681613i	$0.238721\pi$
$422$	15.9406	0.775975
$423$	3.78101	0.183839
$424$	−31.3604	−1.52300
$425$	14.8646	0.721038
$426$	−19.6202	−0.950601
$427$	0	0
$428$	0.0807373	0.00390258
$429$	6.75084	0.325934
$430$	6.91738	0.333586
$431$	27.3341	1.31664	0.658319	−	0.752739i	$-0.271268\pi$
$431$	27.3341	1.31664	0.658319	+	0.752739i	$0.271268\pi$
$432$	12.0209	0.578358
$433$	−28.6912	−1.37881	−0.689405	−	0.724376i	$-0.742128\pi$
$433$	−28.6912	−1.37881	−0.689405	+	0.724376i	$0.742128\pi$
$434$	0	0
$435$	−30.6879	−1.47137
$436$	−0.0550543	−0.00263662
$437$	3.55178	0.169905
$438$	−11.7787	−0.562810
$439$	0.318024	0.0151784	0.00758922	−	0.999971i	$-0.497584\pi$
$439$	0.318024	0.0151784	0.00758922	+	0.999971i	$0.497584\pi$
$440$	−2.23500	−0.106549
$441$	0	0
$442$	−40.5916	−1.93075
$443$	−21.5810	−1.02535	−0.512673	−	0.858584i	$-0.671345\pi$
$443$	−21.5810	−1.02535	−0.512673	+	0.858584i	$0.671345\pi$
$444$	0.0210077	0.000996981 0
$445$	−9.23037	−0.437561
$446$	25.7096	1.21739
$447$	23.8460	1.12788
$448$	0	0
$449$	28.6311	1.35119	0.675593	−	0.737274i	$-0.263888\pi$
$449$	28.6311	1.35119	0.675593	+	0.737274i	$0.263888\pi$
$450$	6.69877	0.315783
$451$	0.552813	0.0260310
$452$	−0.0627716	−0.00295253
$453$	8.84366	0.415512
$454$	−27.2455	−1.27870
$455$	0	0
$456$	15.7475	0.737446
$457$	37.8976	1.77277	0.886387	−	0.462945i	$-0.153207\pi$
$457$	37.8976	1.77277	0.886387	+	0.462945i	$0.153207\pi$
$458$	−12.9622	−0.605685
$459$	15.0964	0.704641
$460$	0.00997455	0.000465066 0
$461$	7.82313	0.364360	0.182180	−	0.983265i	$-0.441685\pi$
$461$	7.82313	0.364360	0.182180	+	0.983265i	$0.441685\pi$
$462$	0	0
$463$	−26.4698	−1.23015	−0.615077	−	0.788467i	$-0.710875\pi$
$463$	−26.4698	−1.23015	−0.615077	+	0.788467i	$0.710875\pi$
$464$	40.0784	1.86059
$465$	−8.86857	−0.411270
$466$	24.3134	1.12630
$467$	21.9676	1.01654	0.508270	−	0.861198i	$-0.330285\pi$
$467$	21.9676	1.01654	0.508270	+	0.861198i	$0.330285\pi$
$468$	−0.0465151	−0.00215016
$469$	0	0
$470$	−4.78093	−0.220528
$471$	−37.3780	−1.72229
$472$	−22.3690	−1.02962
$473$	1.88687	0.0867584
$474$	−19.2013	−0.881946
$475$	−7.66950	−0.351901
$476$	0	0
$477$	−17.7937	−0.814717
$478$	34.2270	1.56551
$479$	39.7740	1.81732	0.908661	−	0.417535i	$-0.137106\pi$
$479$	39.7740	1.81732	0.908661	+	0.417535i	$0.137106\pi$
$480$	0.0885614	0.00404226
$481$	−10.9317	−0.498442
$482$	0.352483	0.0160552
$483$	0	0
$484$	−0.0545264	−0.00247847
$485$	−0.540595	−0.0245472
$486$	21.4106	0.971206
$487$	−3.63186	−0.164575	−0.0822876	−	0.996609i	$-0.526223\pi$
$487$	−3.63186	−0.164575	−0.0822876	+	0.996609i	$0.526223\pi$
$488$	12.1196	0.548631
$489$	3.61149	0.163317
$490$	0	0
$491$	15.6542	0.706464	0.353232	−	0.935536i	$-0.385083\pi$
$491$	15.6542	0.706464	0.353232	+	0.935536i	$0.385083\pi$
$492$	−0.0109386	−0.000493149 0
$493$	50.3323	2.26685
$494$	20.9436	0.942296
$495$	−1.26812	−0.0569979
$496$	11.5823	0.520063
$497$	0	0
$498$	−34.5330	−1.54746
$499$	34.4953	1.54422	0.772110	−	0.635489i	$-0.219202\pi$
$499$	34.4953	1.54422	0.772110	+	0.635489i	$0.219202\pi$
$500$	−0.0580248	−0.00259495
$501$	50.2536	2.24516
$502$	−9.98694	−0.445739
$503$	18.3054	0.816199	0.408100	−	0.912937i	$-0.366192\pi$
$503$	18.3054	0.816199	0.408100	+	0.912937i	$0.366192\pi$
$504$	0	0
$505$	18.4340	0.820303
$506$	1.06999	0.0475668
$507$	41.6200	1.84841
$508$	0.0765929	0.00339826
$509$	14.0350	0.622092	0.311046	−	0.950395i	$-0.399321\pi$
$509$	14.0350	0.622092	0.311046	+	0.950395i	$0.399321\pi$
$510$	21.8971	0.969620
$511$	0	0
$512$	22.5403	0.996149
$513$	−7.78912	−0.343898
$514$	39.4526	1.74018
$515$	−15.6917	−0.691458
$516$	−0.0373357	−0.00164361
$517$	−1.30411	−0.0573545
$518$	0	0
$519$	−18.5489	−0.814205
$520$	−23.0128	−1.00918
$521$	17.0059	0.745042	0.372521	−	0.928024i	$-0.378493\pi$
$521$	17.0059	0.745042	0.372521	+	0.928024i	$0.378493\pi$
$522$	22.6824	0.992782
$523$	−10.8372	−0.473879	−0.236939	−	0.971524i	$-0.576144\pi$
$523$	−10.8372	−0.473879	−0.236939	+	0.971524i	$0.576144\pi$
$524$	0.0241769	0.00105617
$525$	0	0
$526$	8.47360	0.369466
$527$	14.5456	0.633618
$528$	4.75610	0.206983
$529$	−21.1316	−0.918766
$530$	22.4994	0.977310
$531$	−12.6920	−0.550788
$532$	0	0
$533$	5.69206	0.246551
$534$	19.5924	0.847845
$535$	22.6638	0.979843
$536$	15.9634	0.689513
$537$	10.7403	0.463476
$538$	26.7304	1.15243
$539$	0	0
$540$	−0.0218744	−0.000941324 0
$541$	−9.02376	−0.387962	−0.193981	−	0.981005i	$-0.562140\pi$
$541$	−9.02376	−0.387962	−0.193981	+	0.981005i	$0.562140\pi$
$542$	3.39157	0.145681
$543$	−36.3545	−1.56012
$544$	−0.145252	−0.00622765
$545$	−15.4543	−0.661991
$546$	0	0
$547$	28.0552	1.19955	0.599777	−	0.800167i	$-0.295256\pi$
$547$	28.0552	1.19955	0.599777	+	0.800167i	$0.295256\pi$
$548$	0.0389131	0.00166229
$549$	6.87660	0.293486
$550$	−2.31047	−0.0985187
$551$	−25.9693	−1.10633
$552$	8.28385	0.352584
$553$	0	0
$554$	−5.08427	−0.216010
$555$	5.89709	0.250318
$556$	−0.0373155	−0.00158253
$557$	−16.4895	−0.698681	−0.349340	−	0.936996i	$-0.613594\pi$
$557$	−16.4895	−0.698681	−0.349340	+	0.936996i	$0.613594\pi$
$558$	6.55503	0.277497
$559$	19.4282	0.821727
$560$	0	0
$561$	5.97293	0.252177
$562$	7.23038	0.304995
$563$	45.9206	1.93532	0.967662	−	0.252252i	$-0.0811712\pi$
$563$	45.9206	1.93532	0.967662	+	0.252252i	$0.0811712\pi$
$564$	0.0258045	0.00108657
$565$	−17.6207	−0.741307
$566$	34.2396	1.43920
$567$	0	0
$568$	18.2438	0.765491
$569$	−12.4407	−0.521541	−0.260771	−	0.965401i	$-0.583977\pi$
$569$	−12.4407	−0.521541	−0.260771	+	0.965401i	$0.583977\pi$
$570$	−11.2980	−0.473220
$571$	−2.71679	−0.113694	−0.0568472	−	0.998383i	$-0.518105\pi$
$571$	−2.71679	−0.113694	−0.0568472	+	0.998383i	$0.518105\pi$
$572$	0.0160435	0.000670812 0
$573$	26.0565	1.08853
$574$	0	0
$575$	−4.03447	−0.168249
$576$	12.7894	0.532890
$577$	33.4730	1.39350	0.696750	−	0.717314i	$-0.254629\pi$
$577$	33.4730	1.39350	0.696750	+	0.717314i	$0.254629\pi$
$578$	−11.8419	−0.492558
$579$	38.7343	1.60974
$580$	−0.0729304	−0.00302827
$581$	0	0
$582$	1.14747	0.0475640
$583$	6.13721	0.254177
$584$	10.9524	0.453215
$585$	−13.0573	−0.539852
$586$	−9.78928	−0.404392
$587$	−22.7767	−0.940095	−0.470047	−	0.882641i	$-0.655763\pi$
$587$	−22.7767	−0.940095	−0.470047	+	0.882641i	$0.655763\pi$
$588$	0	0
$589$	−7.50493	−0.309235
$590$	16.0486	0.660708
$591$	−18.8420	−0.775056
$592$	−7.70160	−0.316534
$593$	−44.3638	−1.82180	−0.910901	−	0.412624i	$-0.864612\pi$
$593$	−44.3638	−1.82180	−0.910901	+	0.412624i	$0.864612\pi$
$594$	−2.34651	−0.0962783
$595$	0	0
$596$	0.0566704	0.00232131
$597$	−46.0087	−1.88301
$598$	11.0172	0.450526
$599$	39.9945	1.63413	0.817066	−	0.576544i	$-0.195599\pi$
$599$	39.9945	1.63413	0.817066	+	0.576544i	$0.195599\pi$
$600$	−17.8876	−0.730260
$601$	−8.00232	−0.326421	−0.163211	−	0.986591i	$-0.552185\pi$
$601$	−8.00232	−0.326421	−0.163211	+	0.986591i	$0.552185\pi$
$602$	0	0
$603$	9.05752	0.368851
$604$	0.0210171	0.000855174 0
$605$	−15.3062	−0.622284
$606$	−39.1280	−1.58947
$607$	−46.3417	−1.88095	−0.940475	−	0.339862i	$-0.889620\pi$
$607$	−46.3417	−1.88095	−0.940475	+	0.339862i	$0.889620\pi$
$608$	0.0749441	0.00303939
$609$	0	0
$610$	−8.69518	−0.352057
$611$	−13.4278	−0.543230
$612$	−0.0411550	−0.00166359
$613$	5.62171	0.227059	0.113529	−	0.993535i	$-0.463784\pi$
$613$	5.62171	0.227059	0.113529	+	0.993535i	$0.463784\pi$
$614$	−18.3173	−0.739225
$615$	−3.07058	−0.123818
$616$	0	0
$617$	6.57467	0.264686	0.132343	−	0.991204i	$-0.457750\pi$
$617$	6.57467	0.264686	0.132343	+	0.991204i	$0.457750\pi$
$618$	33.3071	1.33981
$619$	−42.1396	−1.69373	−0.846867	−	0.531804i	$-0.821514\pi$
$619$	−42.1396	−1.69373	−0.846867	+	0.531804i	$0.821514\pi$
$620$	−0.0210763	−0.000846444 0
$621$	−4.09740	−0.164423
$622$	35.7601	1.43385
$623$	0	0
$624$	48.9714	1.96043
$625$	−1.53032	−0.0612128
$626$	−15.5770	−0.622581
$627$	−3.08178	−0.123074
$628$	−0.0888294	−0.00354468
$629$	−9.67201	−0.385648
$630$	0	0
$631$	−37.0747	−1.47592	−0.737959	−	0.674845i	$-0.764210\pi$
$631$	−37.0747	−1.47592	−0.737959	+	0.674845i	$0.764210\pi$
$632$	17.8543	0.710205
$633$	−24.1516	−0.959940
$634$	0.343550	0.0136441
$635$	21.5005	0.853220
$636$	−0.121438	−0.00481531
$637$	0	0
$638$	−7.82337	−0.309730
$639$	10.3514	0.409494
$640$	−16.2542	−0.642503
$641$	37.4356	1.47862	0.739309	−	0.673366i	$-0.235152\pi$
$641$	37.4356	1.47862	0.739309	+	0.673366i	$0.235152\pi$
$642$	−48.1062	−1.89860
$643$	22.2886	0.878977	0.439489	−	0.898248i	$-0.355160\pi$
$643$	22.2886	0.878977	0.439489	+	0.898248i	$0.355160\pi$
$644$	0	0
$645$	−10.4805	−0.412671
$646$	18.5302	0.729061
$647$	−32.1264	−1.26302	−0.631510	−	0.775368i	$-0.717564\pi$
$647$	−32.1264	−1.26302	−0.631510	+	0.775368i	$0.717564\pi$
$648$	−31.7493	−1.24723
$649$	4.37760	0.171836
$650$	−23.7898	−0.933114
$651$	0	0
$652$	0.00858276	0.000336127 0
$653$	−35.7213	−1.39788	−0.698941	−	0.715180i	$-0.746345\pi$
$653$	−35.7213	−1.39788	−0.698941	+	0.715180i	$0.746345\pi$
$654$	32.8034	1.28271
$655$	6.78671	0.265179
$656$	4.01017	0.156571
$657$	6.21433	0.242444
$658$	0	0
$659$	−10.2206	−0.398140	−0.199070	−	0.979985i	$-0.563792\pi$
$659$	−10.2206	−0.398140	−0.199070	+	0.979985i	$0.563792\pi$
$660$	−0.00865464	−0.000336881 0
$661$	10.7672	0.418796	0.209398	−	0.977831i	$-0.432850\pi$
$661$	10.7672	0.418796	0.209398	+	0.977831i	$0.432850\pi$
$662$	−50.7045	−1.97069
$663$	61.5005	2.38848
$664$	32.1104	1.24612
$665$	0	0
$666$	−4.35872	−0.168897
$667$	−13.6609	−0.528954
$668$	0.119428	0.00462082
$669$	−38.9527	−1.50600
$670$	−11.4528	−0.442462
$671$	−2.37180	−0.0915625
$672$	0	0
$673$	42.8951	1.65348	0.826742	−	0.562581i	$-0.190192\pi$
$673$	42.8951	1.65348	0.826742	+	0.562581i	$0.190192\pi$
$674$	23.1843	0.893026
$675$	8.84768	0.340547
$676$	0.0989106	0.00380425
$677$	−13.9704	−0.536926	−0.268463	−	0.963290i	$-0.586516\pi$
$677$	−13.9704	−0.536926	−0.268463	+	0.963290i	$0.586516\pi$
$678$	37.4016	1.43640
$679$	0	0
$680$	−20.3610	−0.780807
$681$	41.2798	1.58184
$682$	−2.26089	−0.0865740
$683$	0.891605	0.0341163	0.0170582	−	0.999854i	$-0.494570\pi$
$683$	0.891605	0.0341163	0.0170582	+	0.999854i	$0.494570\pi$
$684$	0.0212343	0.000811912 0
$685$	10.9233	0.417359
$686$	0	0
$687$	19.6391	0.749279
$688$	13.6876	0.521834
$689$	63.1920	2.40742
$690$	−5.94320	−0.226254
$691$	−34.9385	−1.32912	−0.664561	−	0.747234i	$-0.731381\pi$
$691$	−34.9385	−1.32912	−0.664561	+	0.747234i	$0.731381\pi$
$692$	−0.0440817	−0.00167573
$693$	0	0
$694$	−15.9944	−0.607141
$695$	−10.4749	−0.397334
$696$	−60.5685	−2.29584
$697$	5.03615	0.190758
$698$	10.5217	0.398252
$699$	−36.8373	−1.39332
$700$	0	0
$701$	29.4679	1.11299	0.556494	−	0.830851i	$-0.312146\pi$
$701$	29.4679	1.11299	0.556494	+	0.830851i	$0.312146\pi$
$702$	−24.1609	−0.911894
$703$	4.99035	0.188215
$704$	−4.41117	−0.166252
$705$	7.24360	0.272810
$706$	−15.8111	−0.595059
$707$	0	0
$708$	−0.0866201	−0.00325538
$709$	−36.0516	−1.35395	−0.676973	−	0.736008i	$-0.736709\pi$
$709$	−36.0516	−1.35395	−0.676973	+	0.736008i	$0.736709\pi$
$710$	−13.0889	−0.491217
$711$	10.1304	0.379920
$712$	−18.2179	−0.682745
$713$	−3.94790	−0.147850
$714$	0	0
$715$	4.50358	0.168424
$716$	0.0255244	0.000953891 0
$717$	−51.8574	−1.93665
$718$	−4.29276	−0.160204
$719$	31.7838	1.18533	0.592667	−	0.805448i	$-0.298075\pi$
$719$	31.7838	1.18533	0.592667	+	0.805448i	$0.298075\pi$
$720$	−9.19912	−0.342831
$721$	0	0
$722$	17.3435	0.645458
$723$	−0.534048	−0.0198615
$724$	−0.0863971	−0.00321092
$725$	29.4986	1.09555
$726$	32.4888	1.20577
$727$	−23.2937	−0.863915	−0.431958	−	0.901894i	$-0.642177\pi$
$727$	−23.2937	−0.863915	−0.431958	+	0.901894i	$0.642177\pi$
$728$	0	0
$729$	1.27898	0.0473697
$730$	−7.85776	−0.290829
$731$	17.1895	0.635776
$732$	0.0469312	0.00173463
$733$	−10.1095	−0.373404	−0.186702	−	0.982417i	$-0.559780\pi$
$733$	−10.1095	−0.373404	−0.186702	+	0.982417i	$0.559780\pi$
$734$	41.1351	1.51832
$735$	0	0
$736$	0.0394237	0.00145318
$737$	−3.12402	−0.115075
$738$	2.26956	0.0835436
$739$	−21.2112	−0.780267	−0.390134	−	0.920758i	$-0.627571\pi$
$739$	−21.2112	−0.780267	−0.390134	+	0.920758i	$0.627571\pi$
$740$	0.0140145	0.000515184 0
$741$	−31.7317	−1.16569
$742$	0	0
$743$	46.0372	1.68894	0.844471	−	0.535601i	$-0.179915\pi$
$743$	46.0372	1.68894	0.844471	+	0.535601i	$0.179915\pi$
$744$	−17.5038	−0.641721
$745$	15.9080	0.582824
$746$	41.1116	1.50520
$747$	18.2192	0.666606
$748$	0.0141948	0.000519012 0
$749$	0	0
$750$	34.5733	1.26244
$751$	43.3142	1.58056	0.790279	−	0.612747i	$-0.209936\pi$
$751$	43.3142	1.58056	0.790279	+	0.612747i	$0.209936\pi$
$752$	−9.46014	−0.344976
$753$	15.1312	0.551413
$754$	−80.5537	−2.93359
$755$	5.89973	0.214713
$756$	0	0
$757$	−8.94000	−0.324930	−0.162465	−	0.986714i	$-0.551944\pi$
$757$	−8.94000	−0.324930	−0.162465	+	0.986714i	$0.551944\pi$
$758$	7.54850	0.274174
$759$	−1.62114	−0.0588437
$760$	10.5054	0.381071
$761$	−52.0904	−1.88827	−0.944137	−	0.329554i	$-0.893102\pi$
$761$	−52.0904	−1.88827	−0.944137	+	0.329554i	$0.893102\pi$
$762$	−45.6369	−1.65325
$763$	0	0
$764$	0.0619238	0.00224032
$765$	−11.5527	−0.417688
$766$	−19.2970	−0.697230
$767$	45.0742	1.62753
$768$	0.262524	0.00947302
$769$	−41.6580	−1.50223	−0.751114	−	0.660173i	$-0.770483\pi$
$769$	−41.6580	−1.50223	−0.751114	+	0.660173i	$0.770483\pi$
$770$	0	0
$771$	−59.7747	−2.15273
$772$	0.0920526	0.00331304
$773$	−0.959021	−0.0344936	−0.0172468	−	0.999851i	$-0.505490\pi$
$773$	−0.959021	−0.0344936	−0.0172468	+	0.999851i	$0.505490\pi$
$774$	7.74649	0.278442
$775$	8.52487	0.306222
$776$	−1.06697	−0.0383019
$777$	0	0
$778$	−29.4450	−1.05565
$779$	−2.59844	−0.0930989
$780$	−0.0891128	−0.00319075
$781$	−3.57029	−0.127755
$782$	9.74764	0.348575
$783$	29.9587	1.07064
$784$	0	0
$785$	−24.9354	−0.889981
$786$	−14.4055	−0.513826
$787$	−31.1293	−1.10964	−0.554821	−	0.831970i	$-0.687213\pi$
$787$	−31.1293	−1.10964	−0.554821	+	0.831970i	$0.687213\pi$
$788$	−0.0447783	−0.00159516
$789$	−12.8384	−0.457058
$790$	−12.8095	−0.455740
$791$	0	0
$792$	−2.50289	−0.0889362
$793$	−24.4214	−0.867229
$794$	−12.7988	−0.454214
$795$	−34.0888	−1.20901
$796$	−0.109340	−0.00387547
$797$	27.5180	0.974737	0.487369	−	0.873196i	$-0.337957\pi$
$797$	27.5180	0.974737	0.487369	+	0.873196i	$0.337957\pi$
$798$	0	0
$799$	−11.8805	−0.420301
$800$	−0.0851292	−0.00300977
$801$	−10.3367	−0.365230
$802$	−31.1216	−1.09894
$803$	−2.14338	−0.0756383
$804$	0.0618154	0.00218006
$805$	0	0
$806$	−23.2794	−0.819981
$807$	−40.4992	−1.42564
$808$	36.3831	1.27995
$809$	−14.7737	−0.519414	−0.259707	−	0.965687i	$-0.583626\pi$
$809$	−14.7737	−0.519414	−0.259707	+	0.965687i	$0.583626\pi$
$810$	22.7784	0.800350
$811$	−1.05830	−0.0371620	−0.0185810	−	0.999827i	$-0.505915\pi$
$811$	−1.05830	−0.0371620	−0.0185810	+	0.999827i	$0.505915\pi$
$812$	0	0
$813$	−5.13858	−0.180218
$814$	1.50336	0.0526929
$815$	2.40928	0.0843932
$816$	43.3283	1.51679
$817$	−8.86905	−0.310289
$818$	14.7096	0.514308
$819$	0	0
$820$	−0.00729727	−0.000254832 0
$821$	−48.9249	−1.70749	−0.853745	−	0.520691i	$-0.825674\pi$
$821$	−48.9249	−1.70749	−0.853745	+	0.520691i	$0.825674\pi$
$822$	−23.1859	−0.808700
$823$	−16.1635	−0.563423	−0.281711	−	0.959499i	$-0.590902\pi$
$823$	−16.1635	−0.563423	−0.281711	+	0.959499i	$0.590902\pi$
$824$	−30.9705	−1.07891
$825$	3.50060	0.121875
$826$	0	0
$827$	2.79605	0.0972283	0.0486142	−	0.998818i	$-0.484520\pi$
$827$	2.79605	0.0972283	0.0486142	+	0.998818i	$0.484520\pi$
$828$	0.0111701	0.000388188 0
$829$	−48.2780	−1.67676	−0.838382	−	0.545084i	$-0.816498\pi$
$829$	−48.2780	−1.67676	−0.838382	+	0.545084i	$0.816498\pi$
$830$	−23.0374	−0.799641
$831$	7.70320	0.267221
$832$	−45.4198	−1.57465
$833$	0	0
$834$	22.2339	0.769898
$835$	33.5248	1.16017
$836$	−0.00732390	−0.000253302 0
$837$	8.65783	0.299258
$838$	38.4365	1.32777
$839$	−37.2433	−1.28578	−0.642890	−	0.765958i	$-0.722265\pi$
$839$	−37.2433	−1.28578	−0.642890	+	0.765958i	$0.722265\pi$
$840$	0	0
$841$	70.8839	2.44427
$842$	−42.5187	−1.46529
$843$	−10.9548	−0.377303
$844$	−0.0573967	−0.00197568
$845$	27.7653	0.955155
$846$	−5.35397	−0.184073
$847$	0	0
$848$	44.5200	1.52882
$849$	−51.8765	−1.78040
$850$	−21.0485	−0.721957
$851$	2.62513	0.0899882
$852$	0.0706457	0.00242028
$853$	20.3962	0.698352	0.349176	−	0.937057i	$-0.386462\pi$
$853$	20.3962	0.698352	0.349176	+	0.937057i	$0.386462\pi$
$854$	0	0
$855$	5.96069	0.203851
$856$	44.7314	1.52889
$857$	16.1256	0.550841	0.275420	−	0.961324i	$-0.411183\pi$
$857$	16.1256	0.550841	0.275420	+	0.961324i	$0.411183\pi$
$858$	−9.55930	−0.326349
$859$	3.57116	0.121846	0.0609231	−	0.998142i	$-0.480596\pi$
$859$	3.57116	0.121846	0.0609231	+	0.998142i	$0.480596\pi$
$860$	−0.0249072	−0.000849328 0
$861$	0	0
$862$	−38.7055	−1.31832
$863$	−10.1943	−0.347018	−0.173509	−	0.984832i	$-0.555511\pi$
$863$	−10.1943	−0.347018	−0.173509	+	0.984832i	$0.555511\pi$
$864$	−0.0864570	−0.00294133
$865$	−12.3742	−0.420736
$866$	40.6271	1.38057
$867$	17.9417	0.609332
$868$	0	0
$869$	−3.49407	−0.118528
$870$	43.4546	1.47325
$871$	−32.1666	−1.08992
$872$	−30.5021	−1.03293
$873$	−0.605391	−0.0204894
$874$	−5.02937	−0.170121
$875$	0	0
$876$	0.0424113	0.00143295
$877$	44.9014	1.51621	0.758106	−	0.652132i	$-0.226125\pi$
$877$	44.9014	1.51621	0.758106	+	0.652132i	$0.226125\pi$
$878$	−0.450326	−0.0151978
$879$	14.8318	0.500263
$880$	3.17286	0.106957
$881$	9.50009	0.320066	0.160033	−	0.987112i	$-0.448840\pi$
$881$	9.50009	0.320066	0.160033	+	0.987112i	$0.448840\pi$
$882$	0	0
$883$	55.1815	1.85700	0.928502	−	0.371328i	$-0.121097\pi$
$883$	55.1815	1.85700	0.928502	+	0.371328i	$0.121097\pi$
$884$	0.146157	0.00491579
$885$	−24.3152	−0.817347
$886$	30.5591	1.02665
$887$	−25.7396	−0.864252	−0.432126	−	0.901813i	$-0.642236\pi$
$887$	−25.7396	−0.864252	−0.432126	+	0.901813i	$0.642236\pi$
$888$	11.6390	0.390581
$889$	0	0
$890$	13.0703	0.438119
$891$	6.21331	0.208154
$892$	−0.0925717	−0.00309953
$893$	6.12982	0.205127
$894$	−33.7663	−1.12931
$895$	7.16497	0.239499
$896$	0	0
$897$	−16.6922	−0.557335
$898$	−40.5421	−1.35291
$899$	28.8657	0.962724
$900$	−0.0241200	−0.000804001 0
$901$	55.9102	1.86264
$902$	−0.782792	−0.0260641
$903$	0	0
$904$	−34.7778	−1.15669
$905$	−24.2526	−0.806184
$906$	−12.5228	−0.416041
$907$	33.4168	1.10959	0.554794	−	0.831988i	$-0.312797\pi$
$907$	33.4168	1.10959	0.554794	+	0.831988i	$0.312797\pi$
$908$	0.0981020	0.00325563
$909$	20.6435	0.684702
$910$	0	0
$911$	−43.4808	−1.44058	−0.720291	−	0.693672i	$-0.755992\pi$
$911$	−43.4808	−1.44058	−0.720291	+	0.693672i	$0.755992\pi$
$912$	−22.3556	−0.740267
$913$	−6.28397	−0.207969
$914$	−53.6635	−1.77503
$915$	13.1741	0.435522
$916$	0.0466727	0.00154211
$917$	0	0
$918$	−21.3768	−0.705539
$919$	57.6990	1.90331	0.951657	−	0.307164i	$-0.0993800\pi$
$919$	57.6990	1.90331	0.951657	+	0.307164i	$0.0993800\pi$
$920$	5.52627	0.182196
$921$	27.7526	0.914478
$922$	−11.0777	−0.364824
$923$	−36.7616	−1.21002
$924$	0	0
$925$	−5.66855	−0.186381
$926$	37.4816	1.23172
$927$	−17.5725	−0.577156
$928$	−0.288252	−0.00946234
$929$	−19.3277	−0.634123	−0.317061	−	0.948405i	$-0.602696\pi$
$929$	−19.3277	−0.634123	−0.317061	+	0.948405i	$0.602696\pi$
$930$	12.5580	0.411794
$931$	0	0
$932$	−0.0875445	−0.00286762
$933$	−54.1803	−1.77378
$934$	−31.1064	−1.01783
$935$	3.98462	0.130311
$936$	−25.7711	−0.842354
$937$	−27.7438	−0.906350	−0.453175	−	0.891422i	$-0.649709\pi$
$937$	−27.7438	−0.906350	−0.453175	+	0.891422i	$0.649709\pi$
$938$	0	0
$939$	23.6007	0.770180
$940$	0.0172145	0.000561476 0
$941$	−1.85933	−0.0606123	−0.0303062	−	0.999541i	$-0.509648\pi$
$941$	−1.85933	−0.0606123	−0.0303062	+	0.999541i	$0.509648\pi$
$942$	52.9278	1.72448
$943$	−1.36689	−0.0445120
$944$	31.7556	1.03356
$945$	0	0
$946$	−2.67184	−0.0868689
$947$	1.43869	0.0467511	0.0233756	−	0.999727i	$-0.492559\pi$
$947$	1.43869	0.0467511	0.0233756	+	0.999727i	$0.492559\pi$
$948$	0.0691375	0.00224548
$949$	−22.0694	−0.716404
$950$	10.8601	0.352349
$951$	−0.520514	−0.0168788
$952$	0	0
$953$	18.0423	0.584448	0.292224	−	0.956350i	$-0.405605\pi$
$953$	18.0423	0.584448	0.292224	+	0.956350i	$0.405605\pi$
$954$	25.1961	0.815754
$955$	17.3827	0.562490
$956$	−0.123240	−0.00398587
$957$	11.8532	0.383160
$958$	−56.3206	−1.81964
$959$	0	0
$960$	24.5017	0.790788
$961$	−22.6581	−0.730905
$962$	15.4794	0.499077
$963$	25.3803	0.817869
$964$	−0.00126917	−4.08773e−5 0
$965$	25.8402	0.831824
$966$	0	0
$967$	59.5923	1.91636	0.958179	−	0.286170i	$-0.0923822\pi$
$967$	59.5923	1.91636	0.958179	+	0.286170i	$0.0923822\pi$
$968$	−30.2096	−0.970975
$969$	−28.0751	−0.901904
$970$	0.765491	0.0245784
$971$	10.5438	0.338366	0.169183	−	0.985585i	$-0.445887\pi$
$971$	10.5438	0.338366	0.169183	+	0.985585i	$0.445887\pi$
$972$	−0.0770926	−0.00247275
$973$	0	0
$974$	5.14277	0.164785
$975$	36.0441	1.15433
$976$	−17.2054	−0.550730
$977$	52.6045	1.68297	0.841483	−	0.540283i	$-0.181683\pi$
$977$	52.6045	1.68297	0.841483	+	0.540283i	$0.181683\pi$
$978$	−5.11393	−0.163525
$979$	3.56523	0.113945
$980$	0	0
$981$	−17.3067	−0.552560
$982$	−22.1666	−0.707364
$983$	−29.0739	−0.927314	−0.463657	−	0.886015i	$-0.653463\pi$
$983$	−29.0739	−0.927314	−0.463657	+	0.886015i	$0.653463\pi$
$984$	−6.06037	−0.193198
$985$	−12.5697	−0.400505
$986$	−71.2713	−2.26974
$987$	0	0
$988$	−0.0754108	−0.00239914
$989$	−4.66548	−0.148354
$990$	1.79568	0.0570705
$991$	−8.64703	−0.274682	−0.137341	−	0.990524i	$-0.543856\pi$
$991$	−8.64703	−0.274682	−0.137341	+	0.990524i	$0.543856\pi$
$992$	−0.0833026	−0.00264486
$993$	76.8225	2.43789
$994$	0	0
$995$	−30.6931	−0.973035
$996$	0.124342	0.00393992
$997$	4.68164	0.148269	0.0741344	−	0.997248i	$-0.476381\pi$
$997$	4.68164	0.148269	0.0741344	+	0.997248i	$0.476381\pi$
$998$	−48.8458	−1.54619
$999$	−5.75696	−0.182142

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.t.1.6	✓	20
7.6	odd	2		2009.2.a.u.1.6	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.t.1.6	✓	20	1.1	even	1	trivial
2009.2.a.u.1.6	yes	20	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41602 q^{2} +2.14541 q^{3} +0.00509860 q^{4} +1.43123 q^{5} -3.03793 q^{6} +2.82481 q^{8} +1.60278 q^{9} +O(q^{10})\)	\(q-1.41602 q^{2} +2.14541 q^{3} +0.00509860 q^{4} +1.43123 q^{5} -3.03793 q^{6} +2.82481 q^{8} +1.60278 q^{9} -2.02665 q^{10} -0.552813 q^{11} +0.0109386 q^{12} -5.69206 q^{13} +3.07058 q^{15} -4.01017 q^{16} -5.03615 q^{17} -2.26956 q^{18} +2.59844 q^{19} +0.00729727 q^{20} +0.782792 q^{22} +1.36689 q^{23} +6.06037 q^{24} -2.95158 q^{25} +8.06005 q^{26} -2.99761 q^{27} -9.99419 q^{29} -4.34798 q^{30} -2.88824 q^{31} +0.0288420 q^{32} -1.18601 q^{33} +7.13127 q^{34} +0.00817192 q^{36} +1.92052 q^{37} -3.67943 q^{38} -12.2118 q^{39} +4.04296 q^{40} -1.00000 q^{41} -3.41322 q^{43} -0.00281857 q^{44} +2.29395 q^{45} -1.93553 q^{46} +2.35904 q^{47} -8.60346 q^{48} +4.17948 q^{50} -10.8046 q^{51} -0.0290215 q^{52} -11.1018 q^{53} +4.24466 q^{54} -0.791204 q^{55} +5.57472 q^{57} +14.1519 q^{58} -7.91878 q^{59} +0.0156556 q^{60} +4.29043 q^{61} +4.08980 q^{62} +7.97950 q^{64} -8.14666 q^{65} +1.67941 q^{66} +5.65114 q^{67} -0.0256773 q^{68} +2.93253 q^{69} +6.45840 q^{71} +4.52754 q^{72} +3.87723 q^{73} -2.71948 q^{74} -6.33234 q^{75} +0.0132484 q^{76} +17.2921 q^{78} +6.32053 q^{79} -5.73948 q^{80} -11.2394 q^{81} +1.41602 q^{82} +11.3673 q^{83} -7.20790 q^{85} +4.83317 q^{86} -21.4416 q^{87} -1.56159 q^{88} -6.44925 q^{89} -3.24826 q^{90} +0.00696921 q^{92} -6.19646 q^{93} -3.34043 q^{94} +3.71897 q^{95} +0.0618778 q^{96} -0.377713 q^{97} -0.886037 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9	\( 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9} - 16 q^{10} - 6 q^{12} - 12 q^{13} + 14 q^{16} - 8 q^{17} - 18 q^{18} - 36 q^{19} - 24 q^{20} - 8 q^{22} - 12 q^{23} - 36 q^{24} + 20 q^{25} + 22 q^{26} - 32 q^{27} + 4 q^{29} + 28 q^{30} - 80 q^{31} + 6 q^{32} + 12 q^{33} - 48 q^{34} + 26 q^{36} + 4 q^{37} - 12 q^{38} - 28 q^{39} + 4 q^{40} - 20 q^{41} - 20 q^{44} - 40 q^{45} + 8 q^{46} - 32 q^{47} - 16 q^{48} + 6 q^{50} - 20 q^{51} - 36 q^{52} + 4 q^{53} + 50 q^{54} - 64 q^{55} - 4 q^{57} - 32 q^{59} + 20 q^{60} - 44 q^{61} + 8 q^{62} - 30 q^{64} - 8 q^{65} - 32 q^{66} - 4 q^{67} + 48 q^{68} + 24 q^{69} + 8 q^{71} - 8 q^{72} - 48 q^{73} - 38 q^{74} - 24 q^{75} - 84 q^{76} + 30 q^{78} - 4 q^{79} - 56 q^{80} + 2 q^{82} - 8 q^{83} - 12 q^{85} - 24 q^{86} - 40 q^{87} - 48 q^{88} - 20 q^{89} - 48 q^{90} - 50 q^{92} + 48 q^{93} - 26 q^{94} + 20 q^{95} - 70 q^{96} - 8 q^{97} - 8 q^{99}+O(q^{100}) \) 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9 - 16 * q^10 - 6 * q^12 - 12 * q^13 + 14 * q^16 - 8 * q^17 - 18 * q^18 - 36 * q^19 - 24 * q^20 - 8 * q^22 - 12 * q^23 - 36 * q^24 + 20 * q^25 + 22 * q^26 - 32 * q^27 + 4 * q^29 + 28 * q^30 - 80 * q^31 + 6 * q^32 + 12 * q^33 - 48 * q^34 + 26 * q^36 + 4 * q^37 - 12 * q^38 - 28 * q^39 + 4 * q^40 - 20 * q^41 - 20 * q^44 - 40 * q^45 + 8 * q^46 - 32 * q^47 - 16 * q^48 + 6 * q^50 - 20 * q^51 - 36 * q^52 + 4 * q^53 + 50 * q^54 - 64 * q^55 - 4 * q^57 - 32 * q^59 + 20 * q^60 - 44 * q^61 + 8 * q^62 - 30 * q^64 - 8 * q^65 - 32 * q^66 - 4 * q^67 + 48 * q^68 + 24 * q^69 + 8 * q^71 - 8 * q^72 - 48 * q^73 - 38 * q^74 - 24 * q^75 - 84 * q^76 + 30 * q^78 - 4 * q^79 - 56 * q^80 + 2 * q^82 - 8 * q^83 - 12 * q^85 - 24 * q^86 - 40 * q^87 - 48 * q^88 - 20 * q^89 - 48 * q^90 - 50 * q^92 + 48 * q^93 - 26 * q^94 + 20 * q^95 - 70 * q^96 - 8 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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