LMFDB - Embedded newform 2009.2.a.u.1.5

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:	$0$
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			1.5
Root			$2.10538$ 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)

$f(q)$	$=$	$q-2.10538 q^{2} -1.20723 q^{3} +2.43261 q^{4} +1.24490 q^{5} +2.54167 q^{6} -0.910797 q^{8} -1.54260 q^{9} +O(q^{10})$	\(q-2.10538 q^{2} -1.20723 q^{3} +2.43261 q^{4} +1.24490 q^{5} +2.54167 q^{6} -0.910797 q^{8} -1.54260 q^{9} -2.62098 q^{10} +4.93810 q^{11} -2.93671 q^{12} -4.40174 q^{13} -1.50288 q^{15} -2.94764 q^{16} +3.97577 q^{17} +3.24775 q^{18} +4.63006 q^{19} +3.02835 q^{20} -10.3966 q^{22} -8.12991 q^{23} +1.09954 q^{24} -3.45022 q^{25} +9.26732 q^{26} +5.48396 q^{27} +7.06523 q^{29} +3.16412 q^{30} +6.58788 q^{31} +8.02749 q^{32} -5.96141 q^{33} -8.37048 q^{34} -3.75254 q^{36} +9.84697 q^{37} -9.74802 q^{38} +5.31391 q^{39} -1.13385 q^{40} +1.00000 q^{41} -12.0874 q^{43} +12.0124 q^{44} -1.92038 q^{45} +17.1165 q^{46} -11.1431 q^{47} +3.55848 q^{48} +7.26402 q^{50} -4.79966 q^{51} -10.7077 q^{52} +1.43780 q^{53} -11.5458 q^{54} +6.14744 q^{55} -5.58954 q^{57} -14.8750 q^{58} +5.44046 q^{59} -3.65591 q^{60} -0.694330 q^{61} -13.8700 q^{62} -11.0056 q^{64} -5.47973 q^{65} +12.5510 q^{66} +2.90092 q^{67} +9.67147 q^{68} +9.81466 q^{69} +1.06051 q^{71} +1.40500 q^{72} +7.26432 q^{73} -20.7316 q^{74} +4.16521 q^{75} +11.2631 q^{76} -11.1878 q^{78} -9.46563 q^{79} -3.66952 q^{80} -1.99259 q^{81} -2.10538 q^{82} +6.97005 q^{83} +4.94943 q^{85} +25.4486 q^{86} -8.52935 q^{87} -4.49761 q^{88} -13.9012 q^{89} +4.04313 q^{90} -19.7769 q^{92} -7.95307 q^{93} +23.4605 q^{94} +5.76397 q^{95} -9.69101 q^{96} +4.91665 q^{97} -7.61751 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.10538	−1.48873	−0.744363	−	0.667776i	$-0.767247\pi$
$2$	−2.10538	−1.48873	−0.744363	+	0.667776i	$0.767247\pi$
$3$	−1.20723	−0.696994	−0.348497	−	0.937310i	$-0.613308\pi$
$3$	−1.20723	−0.696994	−0.348497	+	0.937310i	$0.613308\pi$
$4$	2.43261	1.21630
$5$	1.24490	0.556736	0.278368	−	0.960474i	$-0.410206\pi$
$5$	1.24490	0.556736	0.278368	+	0.960474i	$0.410206\pi$
$6$	2.54167	1.03763
$7$	0	0
$7$	0	0
$8$	−0.910797	−0.322016
$9$	−1.54260	−0.514200
$10$	−2.62098	−0.828827
$11$	4.93810	1.48889	0.744446	−	0.667682i	$-0.232713\pi$
$11$	4.93810	1.48889	0.744446	+	0.667682i	$0.232713\pi$
$12$	−2.93671	−0.847755
$13$	−4.40174	−1.22082	−0.610412	−	0.792084i	$-0.708996\pi$
$13$	−4.40174	−1.22082	−0.610412	+	0.792084i	$0.708996\pi$
$14$	0	0
$15$	−1.50288	−0.388042
$16$	−2.94764	−0.736910
$17$	3.97577	0.964265	0.482132	−	0.876098i	$-0.339862\pi$
$17$	3.97577	0.964265	0.482132	+	0.876098i	$0.339862\pi$
$18$	3.24775	0.765502
$19$	4.63006	1.06221	0.531105	−	0.847306i	$-0.321777\pi$
$19$	4.63006	1.06221	0.531105	+	0.847306i	$0.321777\pi$
$20$	3.02835	0.677160
$21$	0	0
$22$	−10.3966	−2.21655
$23$	−8.12991	−1.69520	−0.847602	−	0.530633i	$-0.821954\pi$
$23$	−8.12991	−1.69520	−0.847602	+	0.530633i	$0.821954\pi$
$24$	1.09954	0.224443
$25$	−3.45022	−0.690045
$26$	9.26732	1.81747
$27$	5.48396	1.05539
$28$	0	0
$29$	7.06523	1.31198	0.655990	−	0.754769i	$-0.272251\pi$
$29$	7.06523	1.31198	0.655990	+	0.754769i	$0.272251\pi$
$30$	3.16412	0.577687
$31$	6.58788	1.18322	0.591609	−	0.806225i	$-0.298493\pi$
$31$	6.58788	1.18322	0.591609	+	0.806225i	$0.298493\pi$
$32$	8.02749	1.41907
$33$	−5.96141	−1.03775
$34$	−8.37048	−1.43553
$35$	0	0
$36$	−3.75254	−0.625423
$37$	9.84697	1.61883	0.809416	−	0.587236i	$-0.199784\pi$
$37$	9.84697	1.61883	0.809416	+	0.587236i	$0.199784\pi$
$38$	−9.74802	−1.58134
$39$	5.31391	0.850906
$40$	−1.13385	−0.179278
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−12.0874	−1.84332	−0.921659	−	0.388002i	$-0.873165\pi$
$43$	−12.0874	−1.84332	−0.921659	+	0.388002i	$0.873165\pi$
$44$	12.0124	1.81094
$45$	−1.92038	−0.286274
$46$	17.1165	2.52369
$47$	−11.1431	−1.62540	−0.812698	−	0.582686i	$-0.802002\pi$
$47$	−11.1431	−1.62540	−0.812698	+	0.582686i	$0.802002\pi$
$48$	3.55848	0.513622
$49$	0	0
$50$	7.26402	1.02729
$51$	−4.79966	−0.672086
$52$	−10.7077	−1.48489
$53$	1.43780	0.197497	0.0987487	−	0.995112i	$-0.468516\pi$
$53$	1.43780	0.197497	0.0987487	+	0.995112i	$0.468516\pi$
$54$	−11.5458	−1.57118
$55$	6.14744	0.828921
$56$	0	0
$57$	−5.58954	−0.740353
$58$	−14.8750	−1.95318
$59$	5.44046	0.708287	0.354143	−	0.935191i	$-0.384772\pi$
$59$	5.44046	0.708287	0.354143	+	0.935191i	$0.384772\pi$
$60$	−3.65591	−0.471976
$61$	−0.694330	−0.0888998	−0.0444499	−	0.999012i	$-0.514154\pi$
$61$	−0.694330	−0.0888998	−0.0444499	+	0.999012i	$0.514154\pi$
$62$	−13.8700	−1.76149
$63$	0	0
$64$	−11.0056	−1.37570
$65$	−5.47973	−0.679676
$66$	12.5510	1.54492
$67$	2.90092	0.354404	0.177202	−	0.984175i	$-0.443295\pi$
$67$	2.90092	0.354404	0.177202	+	0.984175i	$0.443295\pi$
$68$	9.67147	1.17284
$69$	9.81466	1.18155
$70$	0	0
$71$	1.06051	0.125859	0.0629296	−	0.998018i	$-0.479956\pi$
$71$	1.06051	0.125859	0.0629296	+	0.998018i	$0.479956\pi$
$72$	1.40500	0.165580
$73$	7.26432	0.850224	0.425112	−	0.905141i	$-0.360235\pi$
$73$	7.26432	0.850224	0.425112	+	0.905141i	$0.360235\pi$
$74$	−20.7316	−2.40999
$75$	4.16521	0.480957
$76$	11.2631	1.29197
$77$	0	0
$78$	−11.1878	−1.26677
$79$	−9.46563	−1.06497	−0.532483	−	0.846441i	$-0.678741\pi$
$79$	−9.46563	−1.06497	−0.532483	+	0.846441i	$0.678741\pi$
$80$	−3.66952	−0.410265
$81$	−1.99259	−0.221398
$82$	−2.10538	−0.232500
$83$	6.97005	0.765062	0.382531	−	0.923943i	$-0.375052\pi$
$83$	6.97005	0.765062	0.382531	+	0.923943i	$0.375052\pi$
$84$	0	0
$85$	4.94943	0.536841
$86$	25.4486	2.74419
$87$	−8.52935	−0.914442
$88$	−4.49761	−0.479447
$89$	−13.9012	−1.47352	−0.736760	−	0.676154i	$-0.763645\pi$
$89$	−13.9012	−1.47352	−0.736760	+	0.676154i	$0.763645\pi$
$90$	4.04313	0.426183
$91$	0	0
$92$	−19.7769	−2.06188
$93$	−7.95307	−0.824695
$94$	23.4605	2.41977
$95$	5.76397	0.591370
$96$	−9.69101	−0.989084
$97$	4.91665	0.499210	0.249605	−	0.968348i	$-0.419699\pi$
$97$	4.91665	0.499210	0.249605	+	0.968348i	$0.419699\pi$
$98$	0	0
$99$	−7.61751	−0.765589
$100$	−8.39304	−0.839304
$101$	−1.84241	−0.183327	−0.0916635	−	0.995790i	$-0.529218\pi$
$101$	−1.84241	−0.183327	−0.0916635	+	0.995790i	$0.529218\pi$
$102$	10.1051	1.00055
$103$	4.01077	0.395193	0.197596	−	0.980283i	$-0.436686\pi$
$103$	4.01077	0.395193	0.197596	+	0.980283i	$0.436686\pi$
$104$	4.00909	0.393124
$105$	0	0
$106$	−3.02711	−0.294019
$107$	−3.57127	−0.345247	−0.172624	−	0.984988i	$-0.555224\pi$
$107$	−3.57127	−0.345247	−0.172624	+	0.984988i	$0.555224\pi$
$108$	13.3403	1.28367
$109$	−11.0578	−1.05915	−0.529574	−	0.848264i	$-0.677648\pi$
$109$	−11.0578	−1.05915	−0.529574	+	0.848264i	$0.677648\pi$
$110$	−12.9427	−1.23403
$111$	−11.8875	−1.12832
$112$	0	0
$113$	−7.36228	−0.692585	−0.346293	−	0.938127i	$-0.612560\pi$
$113$	−7.36228	−0.692585	−0.346293	+	0.938127i	$0.612560\pi$
$114$	11.7681	1.10218
$115$	−10.1209	−0.943781
$116$	17.1869	1.59577
$117$	6.79012	0.627747
$118$	−11.4542	−1.05444
$119$	0	0
$120$	1.36882	0.124955
$121$	13.3848	1.21680
$122$	1.46182	0.132347
$123$	−1.20723	−0.108852
$124$	16.0257	1.43915
$125$	−10.5197	−0.940909
$126$	0	0
$127$	9.90066	0.878541	0.439271	−	0.898355i	$-0.355237\pi$
$127$	9.90066	0.878541	0.439271	+	0.898355i	$0.355237\pi$
$128$	7.11593	0.628965
$129$	14.5923	1.28478
$130$	11.5369	1.01185
$131$	20.0896	1.75523	0.877616	−	0.479364i	$-0.159133\pi$
$131$	20.0896	1.75523	0.877616	+	0.479364i	$0.159133\pi$
$132$	−14.5018	−1.26222
$133$	0	0
$134$	−6.10752	−0.527610
$135$	6.82697	0.587572
$136$	−3.62112	−0.310508
$137$	−10.7761	−0.920667	−0.460333	−	0.887746i	$-0.652270\pi$
$137$	−10.7761	−0.920667	−0.460333	+	0.887746i	$0.652270\pi$
$138$	−20.6635	−1.75900
$139$	16.5353	1.40251	0.701253	−	0.712912i	$-0.252624\pi$
$139$	16.5353	1.40251	0.701253	+	0.712912i	$0.252624\pi$
$140$	0	0
$141$	13.4523	1.13289
$142$	−2.23277	−0.187370
$143$	−21.7362	−1.81767
$144$	4.54703	0.378919
$145$	8.79551	0.730427
$146$	−15.2941	−1.26575
$147$	0	0
$148$	23.9538	1.96899
$149$	7.06477	0.578768	0.289384	−	0.957213i	$-0.406549\pi$
$149$	7.06477	0.578768	0.289384	+	0.957213i	$0.406549\pi$
$150$	−8.76933	−0.716013
$151$	−12.8381	−1.04475	−0.522374	−	0.852717i	$-0.674953\pi$
$151$	−12.8381	−1.04475	−0.522374	+	0.852717i	$0.674953\pi$
$152$	−4.21705	−0.342048
$153$	−6.13301	−0.495825
$154$	0	0
$155$	8.20125	0.658740
$156$	12.9266	1.03496
$157$	14.5838	1.16391	0.581956	−	0.813220i	$-0.302288\pi$
$157$	14.5838	1.16391	0.581956	+	0.813220i	$0.302288\pi$
$158$	19.9287	1.58544
$159$	−1.73576	−0.137654
$160$	9.99342	0.790049
$161$	0	0
$162$	4.19514	0.329602
$163$	19.0723	1.49386	0.746930	−	0.664903i	$-0.231527\pi$
$163$	19.0723	1.49386	0.746930	+	0.664903i	$0.231527\pi$
$164$	2.43261	0.189955
$165$	−7.42136	−0.577752
$166$	−14.6746	−1.13897
$167$	1.08381	0.0838676	0.0419338	−	0.999120i	$-0.486648\pi$
$167$	1.08381	0.0838676	0.0419338	+	0.999120i	$0.486648\pi$
$168$	0	0
$169$	6.37532	0.490409
$170$	−10.4204	−0.799209
$171$	−7.14233	−0.546188
$172$	−29.4040	−2.24203
$173$	24.9287	1.89529	0.947645	−	0.319325i	$-0.103456\pi$
$173$	24.9287	1.89529	0.947645	+	0.319325i	$0.103456\pi$
$174$	17.9575	1.36135
$175$	0	0
$176$	−14.5557	−1.09718
$177$	−6.56787	−0.493671
$178$	29.2672	2.19367
$179$	−10.5528	−0.788751	−0.394376	−	0.918949i	$-0.629039\pi$
$179$	−10.5528	−0.788751	−0.394376	+	0.918949i	$0.629039\pi$
$180$	−4.67153	−0.348196
$181$	19.7254	1.46618	0.733090	−	0.680131i	$-0.238077\pi$
$181$	19.7254	1.46618	0.733090	+	0.680131i	$0.238077\pi$
$182$	0	0
$183$	0.838215	0.0619626
$184$	7.40470	0.545882
$185$	12.2585	0.901262
$186$	16.7442	1.22774
$187$	19.6327	1.43569
$188$	−27.1069	−1.97697
$189$	0	0
$190$	−12.1353	−0.880388
$191$	13.6098	0.984774	0.492387	−	0.870376i	$-0.336125\pi$
$191$	13.6098	0.984774	0.492387	+	0.870376i	$0.336125\pi$
$192$	13.2863	0.958853
$193$	4.94271	0.355784	0.177892	−	0.984050i	$-0.443072\pi$
$193$	4.94271	0.355784	0.177892	+	0.984050i	$0.443072\pi$
$194$	−10.3514	−0.743186
$195$	6.61528	0.473730
$196$	0	0
$197$	5.67412	0.404264	0.202132	−	0.979358i	$-0.435213\pi$
$197$	5.67412	0.404264	0.202132	+	0.979358i	$0.435213\pi$
$198$	16.0377	1.13975
$199$	22.3034	1.58105	0.790524	−	0.612432i	$-0.209809\pi$
$199$	22.3034	1.58105	0.790524	+	0.612432i	$0.209809\pi$
$200$	3.14246	0.222205
$201$	−3.50207	−0.247017
$202$	3.87897	0.272924
$203$	0	0
$204$	−11.6757	−0.817460
$205$	1.24490	0.0869476
$206$	−8.44418	−0.588334
$207$	12.5412	0.871674
$208$	12.9747	0.899637
$209$	22.8637	1.58152
$210$	0	0
$211$	26.0908	1.79616	0.898082	−	0.439828i	$-0.144960\pi$
$211$	26.0908	1.79616	0.898082	+	0.439828i	$0.144960\pi$
$212$	3.49761	0.240217
$213$	−1.28028	−0.0877230
$214$	7.51886	0.513978
$215$	−15.0477	−1.02624
$216$	−4.99477	−0.339851
$217$	0	0
$218$	23.2809	1.57678
$219$	−8.76969	−0.592601
$220$	14.9543	1.00822
$221$	−17.5003	−1.17720
$222$	25.0277	1.67975
$223$	7.67045	0.513652	0.256826	−	0.966458i	$-0.417323\pi$
$223$	7.67045	0.513652	0.256826	+	0.966458i	$0.417323\pi$
$224$	0	0
$225$	5.32232	0.354821
$226$	15.5004	1.03107
$227$	−0.996678	−0.0661519	−0.0330759	−	0.999453i	$-0.510530\pi$
$227$	−0.996678	−0.0661519	−0.0330759	+	0.999453i	$0.510530\pi$
$228$	−13.5972	−0.900494
$229$	15.8892	1.04998	0.524992	−	0.851107i	$-0.324068\pi$
$229$	15.8892	1.04998	0.524992	+	0.851107i	$0.324068\pi$
$230$	21.3084	1.40503
$231$	0	0
$232$	−6.43499	−0.422478
$233$	−10.4386	−0.683858	−0.341929	−	0.939726i	$-0.611080\pi$
$233$	−10.4386	−0.683858	−0.341929	+	0.939726i	$0.611080\pi$
$234$	−14.2958	−0.934543
$235$	−13.8721	−0.904916
$236$	13.2345	0.861491
$237$	11.4272	0.742275
$238$	0	0
$239$	21.7897	1.40946	0.704730	−	0.709475i	$-0.251068\pi$
$239$	21.7897	1.40946	0.704730	+	0.709475i	$0.251068\pi$
$240$	4.42995	0.285952
$241$	−11.5883	−0.746466	−0.373233	−	0.927738i	$-0.621751\pi$
$241$	−11.5883	−0.746466	−0.373233	+	0.927738i	$0.621751\pi$
$242$	−28.1801	−1.81148
$243$	−14.0464	−0.901074
$244$	−1.68903	−0.108129
$245$	0	0
$246$	2.54167	0.162051
$247$	−20.3803	−1.29677
$248$	−6.00022	−0.381015
$249$	−8.41444	−0.533244
$250$	22.1479	1.40076
$251$	−19.2375	−1.21426	−0.607130	−	0.794603i	$-0.707679\pi$
$251$	−19.2375	−1.21426	−0.607130	+	0.794603i	$0.707679\pi$
$252$	0	0
$253$	−40.1463	−2.52398
$254$	−20.8446	−1.30791
$255$	−5.97509	−0.374175
$256$	7.02948	0.439343
$257$	−3.82504	−0.238600	−0.119300	−	0.992858i	$-0.538065\pi$
$257$	−3.82504	−0.238600	−0.119300	+	0.992858i	$0.538065\pi$
$258$	−30.7223	−1.91268
$259$	0	0
$260$	−13.3300	−0.826692
$261$	−10.8988	−0.674620
$262$	−42.2961	−2.61306
$263$	−1.72983	−0.106666	−0.0533330	−	0.998577i	$-0.516984\pi$
$263$	−1.72983	−0.106666	−0.0533330	+	0.998577i	$0.516984\pi$
$264$	5.42964	0.334171
$265$	1.78992	0.109954
$266$	0	0
$267$	16.7819	1.02703
$268$	7.05679	0.431062
$269$	−0.540992	−0.0329849	−0.0164924	−	0.999864i	$-0.505250\pi$
$269$	−0.540992	−0.0329849	−0.0164924	+	0.999864i	$0.505250\pi$
$270$	−14.3733	−0.874734
$271$	27.6713	1.68091	0.840455	−	0.541881i	$-0.182288\pi$
$271$	27.6713	1.68091	0.840455	+	0.541881i	$0.182288\pi$
$272$	−11.7191	−0.710576
$273$	0	0
$274$	22.6878	1.37062
$275$	−17.0376	−1.02740
$276$	23.8752	1.43712
$277$	12.8932	0.774676	0.387338	−	0.921938i	$-0.373395\pi$
$277$	12.8932	0.774676	0.387338	+	0.921938i	$0.373395\pi$
$278$	−34.8130	−2.08795
$279$	−10.1625	−0.608411
$280$	0	0
$281$	−3.96827	−0.236727	−0.118364	−	0.992970i	$-0.537765\pi$
$281$	−3.96827	−0.236727	−0.118364	+	0.992970i	$0.537765\pi$
$282$	−28.3222	−1.68656
$283$	−0.238130	−0.0141554	−0.00707769	−	0.999975i	$-0.502253\pi$
$283$	−0.238130	−0.0141554	−0.00707769	+	0.999975i	$0.502253\pi$
$284$	2.57980	0.153083
$285$	−6.95842	−0.412181
$286$	45.7629	2.70602
$287$	0	0
$288$	−12.3832	−0.729687
$289$	−1.19329	−0.0701937
$290$	−18.5178	−1.08741
$291$	−5.93552	−0.347946
$292$	17.6712	1.03413
$293$	1.72127	0.100558	0.0502789	−	0.998735i	$-0.483989\pi$
$293$	1.72127	0.100558	0.0502789	+	0.998735i	$0.483989\pi$
$294$	0	0
$295$	6.77282	0.394329
$296$	−8.96859	−0.521289
$297$	27.0803	1.57136
$298$	−14.8740	−0.861627
$299$	35.7858	2.06954
$300$	10.1323	0.584989
$301$	0	0
$302$	27.0290	1.55534
$303$	2.22421	0.127778
$304$	−13.6478	−0.782753
$305$	−0.864371	−0.0494937
$306$	12.9123	0.738147
$307$	28.3449	1.61773	0.808865	−	0.587995i	$-0.200083\pi$
$307$	28.3449	1.61773	0.808865	+	0.587995i	$0.200083\pi$
$308$	0	0
$309$	−4.84192	−0.275447
$310$	−17.2667	−0.980683
$311$	7.25401	0.411337	0.205669	−	0.978622i	$-0.434063\pi$
$311$	7.25401	0.411337	0.205669	+	0.978622i	$0.434063\pi$
$312$	−4.83989	−0.274005
$313$	−19.4856	−1.10139	−0.550697	−	0.834705i	$-0.685638\pi$
$313$	−19.4856	−1.10139	−0.550697	+	0.834705i	$0.685638\pi$
$314$	−30.7043	−1.73275
$315$	0	0
$316$	−23.0261	−1.29532
$317$	−15.4090	−0.865454	−0.432727	−	0.901525i	$-0.642449\pi$
$317$	−15.4090	−0.865454	−0.432727	+	0.901525i	$0.642449\pi$
$318$	3.65442	0.204930
$319$	34.8888	1.95340
$320$	−13.7009	−0.765901
$321$	4.31133	0.240635
$322$	0	0
$323$	18.4080	1.02425
$324$	−4.84718	−0.269288
$325$	15.1870	0.842423
$326$	−40.1544	−2.22395
$327$	13.3493	0.738219
$328$	−0.910797	−0.0502904
$329$	0	0
$330$	15.6248	0.860114
$331$	8.16298	0.448678	0.224339	−	0.974511i	$-0.427978\pi$
$331$	8.16298	0.448678	0.224339	+	0.974511i	$0.427978\pi$
$332$	16.9554	0.930548
$333$	−15.1899	−0.832403
$334$	−2.28182	−0.124856
$335$	3.61135	0.197309
$336$	0	0
$337$	9.95337	0.542194	0.271097	−	0.962552i	$-0.412614\pi$
$337$	9.95337	0.542194	0.271097	+	0.962552i	$0.412614\pi$
$338$	−13.4224	−0.730084
$339$	8.88796	0.482727
$340$	12.0400	0.652961
$341$	32.5316	1.76169
$342$	15.0373	0.813124
$343$	0	0
$344$	11.0092	0.593577
$345$	12.2183	0.657810
$346$	−52.4842	−2.82157
$347$	13.1812	0.707606	0.353803	−	0.935320i	$-0.384888\pi$
$347$	13.1812	0.707606	0.353803	+	0.935320i	$0.384888\pi$
$348$	−20.7485	−1.11224
$349$	30.3031	1.62209	0.811045	−	0.584984i	$-0.198899\pi$
$349$	30.3031	1.62209	0.811045	+	0.584984i	$0.198899\pi$
$350$	0	0
$351$	−24.1389	−1.28844
$352$	39.6405	2.11285
$353$	−28.9647	−1.54163	−0.770816	−	0.637057i	$-0.780151\pi$
$353$	−28.9647	−1.54163	−0.770816	+	0.637057i	$0.780151\pi$
$354$	13.8278	0.734941
$355$	1.32023	0.0700703
$356$	−33.8160	−1.79225
$357$	0	0
$358$	22.2175	1.17423
$359$	−1.33908	−0.0706739	−0.0353370	−	0.999375i	$-0.511250\pi$
$359$	−1.33908	−0.0706739	−0.0353370	+	0.999375i	$0.511250\pi$
$360$	1.74908	0.0921846
$361$	2.43749	0.128289
$362$	−41.5295	−2.18274
$363$	−16.1585	−0.848103
$364$	0	0
$365$	9.04335	0.473350
$366$	−1.76476	−0.0922453
$367$	12.3250	0.643360	0.321680	−	0.946848i	$-0.395752\pi$
$367$	12.3250	0.643360	0.321680	+	0.946848i	$0.395752\pi$
$368$	23.9641	1.24921
$369$	−1.54260	−0.0803045
$370$	−25.8087	−1.34173
$371$	0	0
$372$	−19.3467	−1.00308
$373$	−3.42648	−0.177416	−0.0887081	−	0.996058i	$-0.528274\pi$
$373$	−3.42648	−0.177416	−0.0887081	+	0.996058i	$0.528274\pi$
$374$	−41.3343	−2.13734
$375$	12.6997	0.655808
$376$	10.1492	0.523403
$377$	−31.0993	−1.60170
$378$	0	0
$379$	−29.1312	−1.49637	−0.748185	−	0.663490i	$-0.769075\pi$
$379$	−29.1312	−1.49637	−0.748185	+	0.663490i	$0.769075\pi$
$380$	14.0215	0.719285
$381$	−11.9524	−0.612337
$382$	−28.6538	−1.46606
$383$	−3.55509	−0.181657	−0.0908284	−	0.995867i	$-0.528951\pi$
$383$	−3.55509	−0.181657	−0.0908284	+	0.995867i	$0.528951\pi$
$384$	−8.59055	−0.438385
$385$	0	0
$386$	−10.4063	−0.529665
$387$	18.6461	0.947834
$388$	11.9603	0.607190
$389$	23.3905	1.18595	0.592974	−	0.805222i	$-0.297954\pi$
$389$	23.3905	1.18595	0.592974	+	0.805222i	$0.297954\pi$
$390$	−13.9276	−0.705254
$391$	−32.3226	−1.63463
$392$	0	0
$393$	−24.2527	−1.22339
$394$	−11.9461	−0.601838
$395$	−11.7838	−0.592905
$396$	−18.5304	−0.931188
$397$	−2.48701	−0.124820	−0.0624098	−	0.998051i	$-0.519879\pi$
$397$	−2.48701	−0.124820	−0.0624098	+	0.998051i	$0.519879\pi$
$398$	−46.9571	−2.35374
$399$	0	0
$400$	10.1700	0.508501
$401$	2.33080	0.116395	0.0581974	−	0.998305i	$-0.481465\pi$
$401$	2.33080	0.116395	0.0581974	+	0.998305i	$0.481465\pi$
$402$	7.37318	0.367741
$403$	−28.9981	−1.44450
$404$	−4.48187	−0.222981
$405$	−2.48057	−0.123261
$406$	0	0
$407$	48.6253	2.41027
$408$	4.37151	0.216422
$409$	−21.7705	−1.07648	−0.538241	−	0.842791i	$-0.680911\pi$
$409$	−21.7705	−1.07648	−0.538241	+	0.842791i	$0.680911\pi$
$410$	−2.62098	−0.129441
$411$	13.0092	0.641699
$412$	9.75662	0.480674
$413$	0	0
$414$	−26.4039	−1.29768
$415$	8.67701	0.425938
$416$	−35.3349	−1.73244
$417$	−19.9619	−0.977538
$418$	−48.1367	−2.35444
$419$	30.9212	1.51060	0.755299	−	0.655380i	$-0.227492\pi$
$419$	30.9212	1.51060	0.755299	+	0.655380i	$0.227492\pi$
$420$	0	0
$421$	−16.4091	−0.799728	−0.399864	−	0.916574i	$-0.630943\pi$
$421$	−16.4091	−0.799728	−0.399864	+	0.916574i	$0.630943\pi$
$422$	−54.9309	−2.67400
$423$	17.1894	0.835778
$424$	−1.30955	−0.0635972
$425$	−13.7173	−0.665386
$426$	2.69546	0.130595
$427$	0	0
$428$	−8.68748	−0.419925
$429$	26.2406	1.26691
$430$	31.6810	1.52779
$431$	0.818078	0.0394054	0.0197027	−	0.999806i	$-0.493728\pi$
$431$	0.818078	0.0394054	0.0197027	+	0.999806i	$0.493728\pi$
$432$	−16.1647	−0.777726
$433$	−10.5076	−0.504964	−0.252482	−	0.967602i	$-0.581247\pi$
$433$	−10.5076	−0.504964	−0.252482	+	0.967602i	$0.581247\pi$
$434$	0	0
$435$	−10.6182	−0.509103
$436$	−26.8993	−1.28824
$437$	−37.6420	−1.80066
$438$	18.4635	0.882220
$439$	−10.0621	−0.480236	−0.240118	−	0.970744i	$-0.577186\pi$
$439$	−10.0621	−0.480236	−0.240118	+	0.970744i	$0.577186\pi$
$440$	−5.59907	−0.266925
$441$	0	0
$442$	36.8447	1.75252
$443$	−10.8164	−0.513901	−0.256951	−	0.966425i	$-0.582718\pi$
$443$	−10.8164	−0.513901	−0.256951	+	0.966425i	$0.582718\pi$
$444$	−28.9177	−1.37237
$445$	−17.3055	−0.820362
$446$	−16.1492	−0.764686
$447$	−8.52879	−0.403398
$448$	0	0
$449$	−2.63583	−0.124392	−0.0621962	−	0.998064i	$-0.519810\pi$
$449$	−2.63583	−0.124392	−0.0621962	+	0.998064i	$0.519810\pi$
$450$	−11.2055	−0.528231
$451$	4.93810	0.232526
$452$	−17.9095	−0.842393
$453$	15.4985	0.728182
$454$	2.09838	0.0984820
$455$	0	0
$456$	5.09094	0.238405
$457$	21.2928	0.996034	0.498017	−	0.867167i	$-0.334062\pi$
$457$	21.2928	0.996034	0.498017	+	0.867167i	$0.334062\pi$
$458$	−33.4526	−1.56314
$459$	21.8029	1.01767
$460$	−24.6202	−1.14792
$461$	−38.1634	−1.77745	−0.888723	−	0.458445i	$-0.848406\pi$
$461$	−38.1634	−1.77745	−0.888723	+	0.458445i	$0.848406\pi$
$462$	0	0
$463$	−7.69282	−0.357516	−0.178758	−	0.983893i	$-0.557208\pi$
$463$	−7.69282	−0.357516	−0.178758	+	0.983893i	$0.557208\pi$
$464$	−20.8258	−0.966812
$465$	−9.90078	−0.459138
$466$	21.9772	1.01808
$467$	−19.0054	−0.879466	−0.439733	−	0.898129i	$-0.644927\pi$
$467$	−19.0054	−0.879466	−0.439733	+	0.898129i	$0.644927\pi$
$468$	16.5177	0.763531
$469$	0	0
$470$	29.2060	1.34717
$471$	−17.6060	−0.811239
$472$	−4.95515	−0.228079
$473$	−59.6890	−2.74450
$474$	−24.0585	−1.10504
$475$	−15.9748	−0.732972
$476$	0	0
$477$	−2.21795	−0.101553
$478$	−45.8756	−2.09830
$479$	28.6164	1.30751	0.653757	−	0.756704i	$-0.273192\pi$
$479$	28.6164	1.30751	0.653757	+	0.756704i	$0.273192\pi$
$480$	−12.0643	−0.550659
$481$	−43.3438	−1.97631
$482$	24.3977	1.11128
$483$	0	0
$484$	32.5600	1.48000
$485$	6.12073	0.277928
$486$	29.5729	1.34145
$487$	−5.39160	−0.244317	−0.122158	−	0.992511i	$-0.538982\pi$
$487$	−5.39160	−0.244317	−0.122158	+	0.992511i	$0.538982\pi$
$488$	0.632394	0.0286271
$489$	−23.0246	−1.04121
$490$	0	0
$491$	26.7914	1.20908	0.604540	−	0.796575i	$-0.293357\pi$
$491$	26.7914	1.20908	0.604540	+	0.796575i	$0.293357\pi$
$492$	−2.93671	−0.132397
$493$	28.0897	1.26510
$494$	42.9083	1.93053
$495$	−9.48304	−0.426231
$496$	−19.4187	−0.871925
$497$	0	0
$498$	17.7156	0.793853
$499$	−13.4220	−0.600852	−0.300426	−	0.953805i	$-0.597129\pi$
$499$	−13.4220	−0.600852	−0.300426	+	0.953805i	$0.597129\pi$
$500$	−25.5902	−1.14443
$501$	−1.30840	−0.0584552
$502$	40.5021	1.80770
$503$	1.62246	0.0723419	0.0361709	−	0.999346i	$-0.488484\pi$
$503$	1.62246	0.0723419	0.0361709	+	0.999346i	$0.488484\pi$
$504$	0	0
$505$	−2.29362	−0.102065
$506$	84.5231	3.75751
$507$	−7.69646	−0.341812
$508$	24.0844	1.06857
$509$	27.6041	1.22353	0.611765	−	0.791040i	$-0.290460\pi$
$509$	27.6041	1.22353	0.611765	+	0.791040i	$0.290460\pi$
$510$	12.5798	0.557043
$511$	0	0
$512$	−29.0316	−1.28303
$513$	25.3911	1.12104
$514$	8.05315	0.355209
$515$	4.99301	0.220018
$516$	35.4973	1.56268
$517$	−55.0260	−2.42004
$518$	0	0
$519$	−30.0946	−1.32101
$520$	4.99092	0.218866
$521$	19.2677	0.844131	0.422066	−	0.906565i	$-0.361305\pi$
$521$	19.2677	0.844131	0.422066	+	0.906565i	$0.361305\pi$
$522$	22.9461	1.00432
$523$	−7.85821	−0.343615	−0.171808	−	0.985131i	$-0.554961\pi$
$523$	−7.85821	−0.343615	−0.171808	+	0.985131i	$0.554961\pi$
$524$	48.8700	2.13489
$525$	0	0
$526$	3.64194	0.158796
$527$	26.1919	1.14094
$528$	17.5721	0.764728
$529$	43.0955	1.87372
$530$	−3.76845	−0.163691
$531$	−8.39245	−0.364201
$532$	0	0
$533$	−4.40174	−0.190661
$534$	−35.3321	−1.52897
$535$	−4.44587	−0.192212
$536$	−2.64215	−0.114123
$537$	12.7396	0.549755
$538$	1.13899	0.0491054
$539$	0	0
$540$	16.6073	0.714666
$541$	11.3629	0.488531	0.244265	−	0.969708i	$-0.421453\pi$
$541$	11.3629	0.488531	0.244265	+	0.969708i	$0.421453\pi$
$542$	−58.2584	−2.50241
$543$	−23.8131	−1.02192
$544$	31.9154	1.36836
$545$	−13.7659	−0.589666
$546$	0	0
$547$	−13.5137	−0.577805	−0.288902	−	0.957359i	$-0.593290\pi$
$547$	−13.5137	−0.577805	−0.288902	+	0.957359i	$0.593290\pi$
$548$	−26.2141	−1.11981
$549$	1.07107	0.0457123
$550$	35.8704	1.52952
$551$	32.7125	1.39360
$552$	−8.93917	−0.380476
$553$	0	0
$554$	−27.1450	−1.15328
$555$	−14.7988	−0.628174
$556$	40.2239	1.70587
$557$	28.7393	1.21772	0.608862	−	0.793276i	$-0.291626\pi$
$557$	28.7393	1.21772	0.608862	+	0.793276i	$0.291626\pi$
$558$	21.3958	0.905756
$559$	53.2058	2.25036
$560$	0	0
$561$	−23.7012	−1.00066
$562$	8.35470	0.352422
$563$	−6.90913	−0.291185	−0.145593	−	0.989345i	$-0.546509\pi$
$563$	−6.90913	−0.291185	−0.145593	+	0.989345i	$0.546509\pi$
$564$	32.7242	1.37794
$565$	−9.16530	−0.385587
$566$	0.501354	0.0210735
$567$	0	0
$568$	−0.965908	−0.0405286
$569$	−1.28781	−0.0539878	−0.0269939	−	0.999636i	$-0.508593\pi$
$569$	−1.28781	−0.0539878	−0.0269939	+	0.999636i	$0.508593\pi$
$570$	14.6501	0.613625
$571$	5.46139	0.228552	0.114276	−	0.993449i	$-0.463545\pi$
$571$	5.46139	0.228552	0.114276	+	0.993449i	$0.463545\pi$
$572$	−52.8757	−2.21084
$573$	−16.4302	−0.686381
$574$	0	0
$575$	28.0500	1.16977
$576$	16.9772	0.707384
$577$	−33.0280	−1.37497	−0.687486	−	0.726198i	$-0.741286\pi$
$577$	−33.0280	−1.37497	−0.687486	+	0.726198i	$0.741286\pi$
$578$	2.51233	0.104499
$579$	−5.96698	−0.247979
$580$	21.3960	0.888420
$581$	0	0
$582$	12.4965	0.517996
$583$	7.10001	0.294052
$584$	−6.61632	−0.273785
$585$	8.45302	0.349490
$586$	−3.62392	−0.149703
$587$	15.8653	0.654829	0.327415	−	0.944881i	$-0.393823\pi$
$587$	15.8653	0.654829	0.327415	+	0.944881i	$0.393823\pi$
$588$	0	0
$589$	30.5023	1.25683
$590$	−14.2593	−0.587047
$591$	−6.84995	−0.281769
$592$	−29.0253	−1.19293
$593$	−46.7201	−1.91856	−0.959282	−	0.282449i	$-0.908853\pi$
$593$	−46.7201	−1.91856	−0.959282	+	0.282449i	$0.908853\pi$
$594$	−57.0142	−2.33932
$595$	0	0
$596$	17.1858	0.703958
$597$	−26.9253	−1.10198
$598$	−75.3425	−3.08098
$599$	−2.06045	−0.0841876	−0.0420938	−	0.999114i	$-0.513403\pi$
$599$	−2.06045	−0.0841876	−0.0420938	+	0.999114i	$0.513403\pi$
$600$	−3.79366	−0.154876
$601$	−4.21587	−0.171969	−0.0859845	−	0.996296i	$-0.527404\pi$
$601$	−4.21587	−0.171969	−0.0859845	+	0.996296i	$0.527404\pi$
$602$	0	0
$603$	−4.47496	−0.182234
$604$	−31.2300	−1.27073
$605$	16.6628	0.677438
$606$	−4.68281	−0.190226
$607$	−0.116619	−0.00473340	−0.00236670	−	0.999997i	$-0.500753\pi$
$607$	−0.116619	−0.00473340	−0.00236670	+	0.999997i	$0.500753\pi$
$608$	37.1678	1.50735
$609$	0	0
$610$	1.81983	0.0736826
$611$	49.0492	1.98432
$612$	−14.9192	−0.603073
$613$	11.0244	0.445272	0.222636	−	0.974902i	$-0.428534\pi$
$613$	11.0244	0.445272	0.222636	+	0.974902i	$0.428534\pi$
$614$	−59.6767	−2.40835
$615$	−1.50288	−0.0606019
$616$	0	0
$617$	−10.3098	−0.415059	−0.207529	−	0.978229i	$-0.566542\pi$
$617$	−10.3098	−0.415059	−0.207529	+	0.978229i	$0.566542\pi$
$618$	10.1941	0.410065
$619$	9.95719	0.400213	0.200107	−	0.979774i	$-0.435871\pi$
$619$	9.95719	0.400213	0.200107	+	0.979774i	$0.435871\pi$
$620$	19.9504	0.801228
$621$	−44.5841	−1.78910
$622$	−15.2724	−0.612368
$623$	0	0
$624$	−15.6635	−0.627041
$625$	4.15517	0.166207
$626$	41.0246	1.63967
$627$	−27.6017	−1.10231
$628$	35.4766	1.41567
$629$	39.1492	1.56098
$630$	0	0
$631$	29.5350	1.17577	0.587884	−	0.808945i	$-0.299961\pi$
$631$	29.5350	1.17577	0.587884	+	0.808945i	$0.299961\pi$
$632$	8.62127	0.342936
$633$	−31.4975	−1.25192
$634$	32.4417	1.28842
$635$	12.3253	0.489116
$636$	−4.22241	−0.167429
$637$	0	0
$638$	−73.4540	−2.90807
$639$	−1.63594	−0.0647168
$640$	8.85862	0.350168
$641$	−26.6795	−1.05378	−0.526889	−	0.849934i	$-0.676642\pi$
$641$	−26.6795	−1.05378	−0.526889	+	0.849934i	$0.676642\pi$
$642$	−9.07698	−0.358240
$643$	23.2403	0.916507	0.458254	−	0.888822i	$-0.348475\pi$
$643$	23.2403	0.916507	0.458254	+	0.888822i	$0.348475\pi$
$644$	0	0
$645$	18.1659	0.715284
$646$	−38.7558	−1.52483
$647$	−5.15181	−0.202539	−0.101269	−	0.994859i	$-0.532290\pi$
$647$	−5.15181	−0.202539	−0.101269	+	0.994859i	$0.532290\pi$
$648$	1.81484	0.0712937
$649$	26.8655	1.05456
$650$	−31.9743	−1.25414
$651$	0	0
$652$	46.3954	1.81699
$653$	−15.5204	−0.607362	−0.303681	−	0.952774i	$-0.598216\pi$
$653$	−15.5204	−0.607362	−0.303681	+	0.952774i	$0.598216\pi$
$654$	−28.1053	−1.09901
$655$	25.0095	0.977201
$656$	−2.94764	−0.115086
$657$	−11.2059	−0.437185
$658$	0	0
$659$	21.4728	0.836460	0.418230	−	0.908341i	$-0.362651\pi$
$659$	21.4728	0.836460	0.418230	+	0.908341i	$0.362651\pi$
$660$	−18.0532	−0.702722
$661$	37.9385	1.47564	0.737819	−	0.674999i	$-0.235856\pi$
$661$	37.9385	1.47564	0.737819	+	0.674999i	$0.235856\pi$
$662$	−17.1861	−0.667958
$663$	21.1268	0.820498
$664$	−6.34830	−0.246362
$665$	0	0
$666$	31.9805	1.23922
$667$	−57.4397	−2.22407
$668$	2.63648	0.102008
$669$	−9.25999	−0.358012
$670$	−7.60325	−0.293739
$671$	−3.42867	−0.132362
$672$	0	0
$673$	36.2057	1.39563	0.697814	−	0.716279i	$-0.254156\pi$
$673$	36.2057	1.39563	0.697814	+	0.716279i	$0.254156\pi$
$674$	−20.9556	−0.807179
$675$	−18.9209	−0.728265
$676$	15.5086	0.596486
$677$	−32.5603	−1.25139	−0.625697	−	0.780066i	$-0.715185\pi$
$677$	−32.5603	−1.25139	−0.625697	+	0.780066i	$0.715185\pi$
$678$	−18.7125	−0.718649
$679$	0	0
$680$	−4.50793	−0.172871
$681$	1.20322	0.0461074
$682$	−68.4912	−2.62266
$683$	34.3943	1.31606	0.658032	−	0.752990i	$-0.271389\pi$
$683$	34.3943	1.31606	0.658032	+	0.752990i	$0.271389\pi$
$684$	−17.3745	−0.664330
$685$	−13.4152	−0.512568
$686$	0	0
$687$	−19.1818	−0.731833
$688$	35.6294	1.35836
$689$	−6.32883	−0.241109
$690$	−25.7240	−0.979298
$691$	−37.9358	−1.44315	−0.721573	−	0.692339i	$-0.756580\pi$
$691$	−37.9358	−1.44315	−0.721573	+	0.692339i	$0.756580\pi$
$692$	60.6416	2.30525
$693$	0	0
$694$	−27.7514	−1.05343
$695$	20.5848	0.780826
$696$	7.76851	0.294465
$697$	3.97577	0.150593
$698$	−63.7995	−2.41485
$699$	12.6018	0.476644
$700$	0	0
$701$	−21.1599	−0.799200	−0.399600	−	0.916690i	$-0.630851\pi$
$701$	−21.1599	−0.799200	−0.399600	+	0.916690i	$0.630851\pi$
$702$	50.8215	1.91814
$703$	45.5921	1.71954
$704$	−54.3467	−2.04827
$705$	16.7468	0.630721
$706$	60.9815	2.29507
$707$	0	0
$708$	−15.9770	−0.600454
$709$	−8.09913	−0.304169	−0.152085	−	0.988367i	$-0.548599\pi$
$709$	−8.09913	−0.304169	−0.152085	+	0.988367i	$0.548599\pi$
$710$	−2.77957	−0.104315
$711$	14.6017	0.547605
$712$	12.6611	0.474496
$713$	−53.5589	−2.00580
$714$	0	0
$715$	−27.0594	−1.01197
$716$	−25.6707	−0.959360
$717$	−26.3052	−0.982385
$718$	2.81927	0.105214
$719$	26.2662	0.979562	0.489781	−	0.871845i	$-0.337077\pi$
$719$	26.2662	0.979562	0.489781	+	0.871845i	$0.337077\pi$
$720$	5.66060	0.210958
$721$	0	0
$722$	−5.13182	−0.190987
$723$	13.9897	0.520282
$724$	47.9842	1.78332
$725$	−24.3766	−0.905325
$726$	34.0198	1.26259
$727$	−16.5222	−0.612774	−0.306387	−	0.951907i	$-0.599120\pi$
$727$	−16.5222	−0.612774	−0.306387	+	0.951907i	$0.599120\pi$
$728$	0	0
$729$	22.9349	0.849442
$730$	−19.0396	−0.704689
$731$	−48.0568	−1.77745
$732$	2.03905	0.0753653
$733$	3.76707	0.139140	0.0695700	−	0.997577i	$-0.477837\pi$
$733$	3.76707	0.139140	0.0695700	+	0.997577i	$0.477837\pi$
$734$	−25.9488	−0.957787
$735$	0	0
$736$	−65.2628	−2.40562
$737$	14.3250	0.527669
$738$	3.24775	0.119551
$739$	−27.6727	−1.01796	−0.508978	−	0.860779i	$-0.669977\pi$
$739$	−27.6727	−1.01796	−0.508978	+	0.860779i	$0.669977\pi$
$740$	29.8201	1.09621
$741$	24.6037	0.903840
$742$	0	0
$743$	26.9846	0.989968	0.494984	−	0.868902i	$-0.335174\pi$
$743$	26.9846	0.989968	0.494984	+	0.868902i	$0.335174\pi$
$744$	7.24364	0.265565
$745$	8.79493	0.322221
$746$	7.21402	0.264124
$747$	−10.7520	−0.393395
$748$	47.7587	1.74623
$749$	0	0
$750$	−26.7376	−0.976317
$751$	−3.33184	−0.121581	−0.0607903	−	0.998151i	$-0.519362\pi$
$751$	−3.33184	−0.121581	−0.0607903	+	0.998151i	$0.519362\pi$
$752$	32.8460	1.19777
$753$	23.2240	0.846331
$754$	65.4757	2.38449
$755$	−15.9821	−0.581649
$756$	0	0
$757$	33.4098	1.21430	0.607150	−	0.794587i	$-0.292313\pi$
$757$	33.4098	1.21430	0.607150	+	0.794587i	$0.292313\pi$
$758$	61.3322	2.22769
$759$	48.4658	1.75920
$760$	−5.24980	−0.190430
$761$	8.19935	0.297226	0.148613	−	0.988895i	$-0.452519\pi$
$761$	8.19935	0.297226	0.148613	+	0.988895i	$0.452519\pi$
$762$	25.1642	0.911602
$763$	0	0
$764$	33.1074	1.19778
$765$	−7.63499	−0.276044
$766$	7.48480	0.270437
$767$	−23.9475	−0.864693
$768$	−8.48619	−0.306219
$769$	−0.263684	−0.00950868	−0.00475434	−	0.999989i	$-0.501513\pi$
$769$	−0.263684	−0.00950868	−0.00475434	+	0.999989i	$0.501513\pi$
$770$	0	0
$771$	4.61770	0.166302
$772$	12.0237	0.432741
$773$	46.5505	1.67430	0.837152	−	0.546970i	$-0.184219\pi$
$773$	46.5505	1.67430	0.837152	+	0.546970i	$0.184219\pi$
$774$	−39.2570	−1.41106
$775$	−22.7297	−0.816474
$776$	−4.47807	−0.160753
$777$	0	0
$778$	−49.2459	−1.76555
$779$	4.63006	0.165889
$780$	16.0924	0.576199
$781$	5.23689	0.187391
$782$	68.0513	2.43351
$783$	38.7454	1.38465
$784$	0	0
$785$	18.1553	0.647992
$786$	51.0610	1.82129
$787$	−42.8680	−1.52808	−0.764040	−	0.645169i	$-0.776787\pi$
$787$	−42.8680	−1.52808	−0.764040	+	0.645169i	$0.776787\pi$
$788$	13.8029	0.491708
$789$	2.08830	0.0743455
$790$	24.8092	0.882673
$791$	0	0
$792$	6.93801	0.246531
$793$	3.05626	0.108531
$794$	5.23610	0.185822
$795$	−2.16084	−0.0766372
$796$	54.2554	1.92303
$797$	12.6903	0.449515	0.224758	−	0.974415i	$-0.427841\pi$
$797$	12.6903	0.449515	0.224758	+	0.974415i	$0.427841\pi$
$798$	0	0
$799$	−44.3025	−1.56731
$800$	−27.6966	−0.979224
$801$	21.4439	0.757684
$802$	−4.90722	−0.173280
$803$	35.8719	1.26589
$804$	−8.51916	−0.300448
$805$	0	0
$806$	61.0520	2.15046
$807$	0.653101	0.0229902
$808$	1.67807	0.0590341
$809$	−37.6877	−1.32503	−0.662515	−	0.749049i	$-0.730511\pi$
$809$	−37.6877	−1.32503	−0.662515	+	0.749049i	$0.730511\pi$
$810$	5.22253	0.183501
$811$	1.84218	0.0646875	0.0323438	−	0.999477i	$-0.489703\pi$
$811$	1.84218	0.0646875	0.0323438	+	0.999477i	$0.489703\pi$
$812$	0	0
$813$	−33.4056	−1.17158
$814$	−102.375	−3.58822
$815$	23.7431	0.831686
$816$	14.1477	0.495267
$817$	−55.9656	−1.95799
$818$	45.8351	1.60259
$819$	0	0
$820$	3.02835	0.105755
$821$	−11.8652	−0.414100	−0.207050	−	0.978330i	$-0.566386\pi$
$821$	−11.8652	−0.414100	−0.207050	+	0.978330i	$0.566386\pi$
$822$	−27.3893	−0.955313
$823$	−2.49518	−0.0869765	−0.0434883	−	0.999054i	$-0.513847\pi$
$823$	−2.49518	−0.0869765	−0.0434883	+	0.999054i	$0.513847\pi$
$824$	−3.65300	−0.127258
$825$	20.5682	0.716093
$826$	0	0
$827$	47.6293	1.65623	0.828117	−	0.560556i	$-0.189413\pi$
$827$	47.6293	1.65623	0.828117	+	0.560556i	$0.189413\pi$
$828$	30.5078	1.06022
$829$	−33.0278	−1.14710	−0.573551	−	0.819170i	$-0.694435\pi$
$829$	−33.0278	−1.14710	−0.573551	+	0.819170i	$0.694435\pi$
$830$	−18.2684	−0.634104
$831$	−15.5650	−0.539944
$832$	48.4437	1.67948
$833$	0	0
$834$	42.0273	1.45529
$835$	1.34923	0.0466921
$836$	55.6184	1.92360
$837$	36.1276	1.24875
$838$	−65.1007	−2.24887
$839$	21.8035	0.752739	0.376370	−	0.926470i	$-0.377172\pi$
$839$	21.8035	0.752739	0.376370	+	0.926470i	$0.377172\pi$
$840$	0	0
$841$	20.9175	0.721293
$842$	34.5472	1.19058
$843$	4.79061	0.164997
$844$	63.4686	2.18468
$845$	7.93663	0.273028
$846$	−36.1902	−1.24424
$847$	0	0
$848$	−4.23812	−0.145538
$849$	0.287478	0.00986620
$850$	28.8800	0.990577
$851$	−80.0550	−2.74425
$852$	−3.11441	−0.106698
$853$	−44.9671	−1.53964	−0.769821	−	0.638260i	$-0.779655\pi$
$853$	−44.9671	−1.53964	−0.769821	+	0.638260i	$0.779655\pi$
$854$	0	0
$855$	−8.89149	−0.304083
$856$	3.25270	0.111175
$857$	−10.5522	−0.360457	−0.180228	−	0.983625i	$-0.557684\pi$
$857$	−10.5522	−0.360457	−0.180228	+	0.983625i	$0.557684\pi$
$858$	−55.2463	−1.88608
$859$	24.7014	0.842802	0.421401	−	0.906874i	$-0.361539\pi$
$859$	24.7014	0.842802	0.421401	+	0.906874i	$0.361539\pi$
$860$	−36.6050	−1.24822
$861$	0	0
$862$	−1.72236	−0.0586639
$863$	13.8803	0.472491	0.236246	−	0.971693i	$-0.424083\pi$
$863$	13.8803	0.472491	0.236246	+	0.971693i	$0.424083\pi$
$864$	44.0224	1.49767
$865$	31.0337	1.05518
$866$	22.1225	0.751752
$867$	1.44058	0.0489245
$868$	0	0
$869$	−46.7422	−1.58562
$870$	22.3553	0.757914
$871$	−12.7691	−0.432664
$872$	10.0714	0.341062
$873$	−7.58442	−0.256694
$874$	79.2506	2.68069
$875$	0	0
$876$	−21.3332	−0.720782
$877$	−55.9633	−1.88975	−0.944874	−	0.327434i	$-0.893816\pi$
$877$	−55.9633	−1.88975	−0.944874	+	0.327434i	$0.893816\pi$
$878$	21.1844	0.714939
$879$	−2.07797	−0.0700881
$880$	−18.1204	−0.610840
$881$	−37.3545	−1.25850	−0.629252	−	0.777202i	$-0.716639\pi$
$881$	−37.3545	−1.25850	−0.629252	+	0.777202i	$0.716639\pi$
$882$	0	0
$883$	−16.1896	−0.544825	−0.272412	−	0.962181i	$-0.587821\pi$
$883$	−16.1896	−0.544825	−0.272412	+	0.962181i	$0.587821\pi$
$884$	−42.5713	−1.43183
$885$	−8.17634	−0.274845
$886$	22.7725	0.765058
$887$	5.57975	0.187350	0.0936748	−	0.995603i	$-0.470139\pi$
$887$	5.57975	0.187350	0.0936748	+	0.995603i	$0.470139\pi$
$888$	10.8271	0.363335
$889$	0	0
$890$	36.4347	1.22129
$891$	−9.83959	−0.329639
$892$	18.6592	0.624756
$893$	−51.5935	−1.72651
$894$	17.9563	0.600549
$895$	−13.1371	−0.439126
$896$	0	0
$897$	−43.2016	−1.44246
$898$	5.54941	0.185186
$899$	46.5449	1.55236
$900$	12.9471	0.431570
$901$	5.71636	0.190440
$902$	−10.3966	−0.346167
$903$	0	0
$904$	6.70555	0.223023
$905$	24.5562	0.816276
$906$	−32.6301	−1.08406
$907$	36.3346	1.20647	0.603236	−	0.797563i	$-0.293878\pi$
$907$	36.3346	1.20647	0.603236	+	0.797563i	$0.293878\pi$
$908$	−2.42453	−0.0804607
$909$	2.84211	0.0942667
$910$	0	0
$911$	−21.7625	−0.721022	−0.360511	−	0.932755i	$-0.617398\pi$
$911$	−21.7625	−0.721022	−0.360511	+	0.932755i	$0.617398\pi$
$912$	16.4760	0.545574
$913$	34.4188	1.13910
$914$	−44.8293	−1.48282
$915$	1.04349	0.0344968
$916$	38.6520	1.27710
$917$	0	0
$918$	−45.9033	−1.51504
$919$	−5.60305	−0.184827	−0.0924137	−	0.995721i	$-0.529458\pi$
$919$	−5.60305	−0.184827	−0.0924137	+	0.995721i	$0.529458\pi$
$920$	9.21811	0.303912
$921$	−34.2188	−1.12755
$922$	80.3482	2.64613
$923$	−4.66808	−0.153652
$924$	0	0
$925$	−33.9742	−1.11707
$926$	16.1963	0.532243
$927$	−6.18701	−0.203208
$928$	56.7160	1.86180
$929$	47.5945	1.56152	0.780762	−	0.624828i	$-0.214831\pi$
$929$	47.5945	1.56152	0.780762	+	0.624828i	$0.214831\pi$
$930$	20.8449	0.683530
$931$	0	0
$932$	−25.3931	−0.831778
$933$	−8.75725	−0.286699
$934$	40.0135	1.30928
$935$	24.4408	0.799299
$936$	−6.18443	−0.202144
$937$	−2.56846	−0.0839078	−0.0419539	−	0.999120i	$-0.513358\pi$
$937$	−2.56846	−0.0839078	−0.0419539	+	0.999120i	$0.513358\pi$
$938$	0	0
$939$	23.5236	0.767664
$940$	−33.7454	−1.10065
$941$	5.95773	0.194216	0.0971082	−	0.995274i	$-0.469041\pi$
$941$	5.95773	0.194216	0.0971082	+	0.995274i	$0.469041\pi$
$942$	37.0671	1.20771
$943$	−8.12991	−0.264746
$944$	−16.0365	−0.521944
$945$	0	0
$946$	125.668	4.08581
$947$	19.5564	0.635499	0.317750	−	0.948175i	$-0.397073\pi$
$947$	19.5564	0.635499	0.317750	+	0.948175i	$0.397073\pi$
$948$	27.7978	0.902831
$949$	−31.9756	−1.03797
$950$	33.6329	1.09119
$951$	18.6022	0.603216
$952$	0	0
$953$	−60.1357	−1.94799	−0.973993	−	0.226579i	$-0.927246\pi$
$953$	−60.1357	−1.94799	−0.973993	+	0.226579i	$0.927246\pi$
$954$	4.66962	0.151185
$955$	16.9429	0.548259
$956$	53.0058	1.71433
$957$	−42.1188	−1.36151
$958$	−60.2482	−1.94653
$959$	0	0
$960$	16.5401	0.533828
$961$	12.4002	0.400005
$962$	91.2549	2.94218
$963$	5.50903	0.177526
$964$	−28.1897	−0.907929
$965$	6.15318	0.198078
$966$	0	0
$967$	5.39745	0.173570	0.0867852	−	0.996227i	$-0.472341\pi$
$967$	5.39745	0.173570	0.0867852	+	0.996227i	$0.472341\pi$
$968$	−12.1909	−0.391829
$969$	−22.2227	−0.713896
$970$	−12.8864	−0.413759
$971$	−20.9887	−0.673559	−0.336780	−	0.941584i	$-0.609338\pi$
$971$	−20.9887	−0.673559	−0.336780	+	0.941584i	$0.609338\pi$
$972$	−34.1693	−1.09598
$973$	0	0
$974$	11.3513	0.363721
$975$	−18.3342	−0.587163
$976$	2.04663	0.0655112
$977$	−16.2311	−0.519278	−0.259639	−	0.965706i	$-0.583604\pi$
$977$	−16.2311	−0.519278	−0.259639	+	0.965706i	$0.583604\pi$
$978$	48.4755	1.55008
$979$	−68.6453	−2.19391
$980$	0	0
$981$	17.0578	0.544614
$982$	−56.4060	−1.79999
$983$	−22.2319	−0.709087	−0.354543	−	0.935040i	$-0.615364\pi$
$983$	−22.2319	−0.709087	−0.354543	+	0.935040i	$0.615364\pi$
$984$	1.09954	0.0350521
$985$	7.06371	0.225068
$986$	−59.1394	−1.88338
$987$	0	0
$988$	−49.5773	−1.57726
$989$	98.2698	3.12480
$990$	19.9654	0.634541
$991$	12.8582	0.408454	0.204227	−	0.978924i	$-0.434532\pi$
$991$	12.8582	0.408454	0.204227	+	0.978924i	$0.434532\pi$
$992$	52.8841	1.67907
$993$	−9.85458	−0.312726
$994$	0	0
$995$	27.7655	0.880226
$996$	−20.4690	−0.648586
$997$	−4.99391	−0.158159	−0.0790793	−	0.996868i	$-0.525198\pi$
$997$	−4.99391	−0.158159	−0.0790793	+	0.996868i	$0.525198\pi$
$998$	28.2584	0.894503
$999$	54.0003	1.70849

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.u.1.5	yes	20
7.6	odd	2		2009.2.a.t.1.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.t.1.5	✓	20	7.6	odd	2
2009.2.a.u.1.5	yes	20	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-2.10538 q^{2} -1.20723 q^{3} +2.43261 q^{4} +1.24490 q^{5} +2.54167 q^{6} -0.910797 q^{8} -1.54260 q^{9} +O(q^{10})\)	\(q-2.10538 q^{2} -1.20723 q^{3} +2.43261 q^{4} +1.24490 q^{5} +2.54167 q^{6} -0.910797 q^{8} -1.54260 q^{9} -2.62098 q^{10} +4.93810 q^{11} -2.93671 q^{12} -4.40174 q^{13} -1.50288 q^{15} -2.94764 q^{16} +3.97577 q^{17} +3.24775 q^{18} +4.63006 q^{19} +3.02835 q^{20} -10.3966 q^{22} -8.12991 q^{23} +1.09954 q^{24} -3.45022 q^{25} +9.26732 q^{26} +5.48396 q^{27} +7.06523 q^{29} +3.16412 q^{30} +6.58788 q^{31} +8.02749 q^{32} -5.96141 q^{33} -8.37048 q^{34} -3.75254 q^{36} +9.84697 q^{37} -9.74802 q^{38} +5.31391 q^{39} -1.13385 q^{40} +1.00000 q^{41} -12.0874 q^{43} +12.0124 q^{44} -1.92038 q^{45} +17.1165 q^{46} -11.1431 q^{47} +3.55848 q^{48} +7.26402 q^{50} -4.79966 q^{51} -10.7077 q^{52} +1.43780 q^{53} -11.5458 q^{54} +6.14744 q^{55} -5.58954 q^{57} -14.8750 q^{58} +5.44046 q^{59} -3.65591 q^{60} -0.694330 q^{61} -13.8700 q^{62} -11.0056 q^{64} -5.47973 q^{65} +12.5510 q^{66} +2.90092 q^{67} +9.67147 q^{68} +9.81466 q^{69} +1.06051 q^{71} +1.40500 q^{72} +7.26432 q^{73} -20.7316 q^{74} +4.16521 q^{75} +11.2631 q^{76} -11.1878 q^{78} -9.46563 q^{79} -3.66952 q^{80} -1.99259 q^{81} -2.10538 q^{82} +6.97005 q^{83} +4.94943 q^{85} +25.4486 q^{86} -8.52935 q^{87} -4.49761 q^{88} -13.9012 q^{89} +4.04313 q^{90} -19.7769 q^{92} -7.95307 q^{93} +23.4605 q^{94} +5.76397 q^{95} -9.69101 q^{96} +4.91665 q^{97} -7.61751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 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

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