LMFDB - Embedded newform 2009.2.a.p.1.4

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 10x^{5} - x^{4} + 30x^{3} + 7x^{2} - 25x - 11 $ x^7 - 10x^5 - x^4 + 30x^3 + 7x^2 - 25x - 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.84648$ 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)

$f(q)$	$=$	$q+0.409485 q^{2} -2.84648 q^{3} -1.83232 q^{4} +2.40160 q^{5} -1.16559 q^{6} -1.56928 q^{8} +5.10244 q^{9} +O(q^{10})$	\(q+0.409485 q^{2} -2.84648 q^{3} -1.83232 q^{4} +2.40160 q^{5} -1.16559 q^{6} -1.56928 q^{8} +5.10244 q^{9} +0.983421 q^{10} -0.464750 q^{11} +5.21567 q^{12} -2.07622 q^{13} -6.83611 q^{15} +3.02205 q^{16} +1.51591 q^{17} +2.08938 q^{18} -6.50122 q^{19} -4.40051 q^{20} -0.190308 q^{22} +3.71419 q^{23} +4.46692 q^{24} +0.767691 q^{25} -0.850180 q^{26} -5.98456 q^{27} +1.83639 q^{29} -2.79929 q^{30} +2.31777 q^{31} +4.37604 q^{32} +1.32290 q^{33} +0.620745 q^{34} -9.34932 q^{36} +9.96935 q^{37} -2.66216 q^{38} +5.90990 q^{39} -3.76879 q^{40} +1.00000 q^{41} +6.03402 q^{43} +0.851571 q^{44} +12.2540 q^{45} +1.52091 q^{46} -9.84846 q^{47} -8.60219 q^{48} +0.314358 q^{50} -4.31502 q^{51} +3.80429 q^{52} +1.60467 q^{53} -2.45059 q^{54} -1.11614 q^{55} +18.5056 q^{57} +0.751976 q^{58} -2.50393 q^{59} +12.5260 q^{60} -0.0952801 q^{61} +0.949095 q^{62} -4.25216 q^{64} -4.98624 q^{65} +0.541709 q^{66} -11.8237 q^{67} -2.77764 q^{68} -10.5724 q^{69} -15.9241 q^{71} -8.00716 q^{72} -16.3219 q^{73} +4.08230 q^{74} -2.18522 q^{75} +11.9123 q^{76} +2.42002 q^{78} +7.91940 q^{79} +7.25775 q^{80} +1.72760 q^{81} +0.409485 q^{82} -8.30396 q^{83} +3.64062 q^{85} +2.47084 q^{86} -5.22725 q^{87} +0.729323 q^{88} +10.5786 q^{89} +5.01785 q^{90} -6.80560 q^{92} -6.59750 q^{93} -4.03280 q^{94} -15.6134 q^{95} -12.4563 q^{96} -6.46102 q^{97} -2.37136 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9}+O(q^{10}) $ 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9	$ 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9} - 8 q^{10} - 12 q^{12} - q^{13} - 4 q^{15} + 5 q^{16} + 11 q^{17} + 16 q^{18} - 9 q^{19} - 12 q^{20} + 3 q^{23} + 23 q^{24} + 31 q^{25} - 10 q^{26} - 22 q^{27} - 14 q^{29} + 6 q^{30} - 34 q^{31} - 20 q^{32} - 12 q^{33} - 15 q^{34} + q^{36} + 7 q^{37} + 39 q^{38} - 22 q^{39} - 50 q^{40} + 7 q^{41} - 3 q^{43} + 26 q^{44} + 4 q^{45} - 8 q^{46} - 17 q^{47} - 17 q^{48} + 11 q^{50} + 8 q^{51} + 25 q^{52} + 24 q^{53} - 68 q^{54} - 48 q^{55} + 22 q^{57} - 38 q^{58} - 4 q^{59} - 6 q^{60} - 16 q^{61} - 24 q^{62} + 8 q^{64} - 6 q^{65} + 12 q^{66} - 24 q^{67} + 10 q^{68} - 35 q^{69} - 12 q^{71} - 11 q^{72} - 14 q^{73} - 6 q^{74} + 19 q^{75} - 42 q^{76} - 29 q^{78} - 8 q^{79} + 92 q^{80} + 15 q^{81} - q^{82} - 14 q^{83} + 16 q^{85} + 35 q^{86} + 20 q^{87} + 22 q^{88} + 19 q^{89} - 24 q^{90} - 10 q^{92} + 2 q^{93} + 20 q^{94} - 8 q^{95} + 25 q^{96} - 23 q^{97} + 24 q^{99}+O(q^{100}) $ 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9 - 8 * q^10 - 12 * q^12 - q^13 - 4 * q^15 + 5 * q^16 + 11 * q^17 + 16 * q^18 - 9 * q^19 - 12 * q^20 + 3 * q^23 + 23 * q^24 + 31 * q^25 - 10 * q^26 - 22 * q^27 - 14 * q^29 + 6 * q^30 - 34 * q^31 - 20 * q^32 - 12 * q^33 - 15 * q^34 + q^36 + 7 * q^37 + 39 * q^38 - 22 * q^39 - 50 * q^40 + 7 * q^41 - 3 * q^43 + 26 * q^44 + 4 * q^45 - 8 * q^46 - 17 * q^47 - 17 * q^48 + 11 * q^50 + 8 * q^51 + 25 * q^52 + 24 * q^53 - 68 * q^54 - 48 * q^55 + 22 * q^57 - 38 * q^58 - 4 * q^59 - 6 * q^60 - 16 * q^61 - 24 * q^62 + 8 * q^64 - 6 * q^65 + 12 * q^66 - 24 * q^67 + 10 * q^68 - 35 * q^69 - 12 * q^71 - 11 * q^72 - 14 * q^73 - 6 * q^74 + 19 * q^75 - 42 * q^76 - 29 * q^78 - 8 * q^79 + 92 * q^80 + 15 * q^81 - q^82 - 14 * q^83 + 16 * q^85 + 35 * q^86 + 20 * q^87 + 22 * q^88 + 19 * q^89 - 24 * q^90 - 10 * q^92 + 2 * q^93 + 20 * q^94 - 8 * q^95 + 25 * q^96 - 23 * q^97 + 24 * 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 $

$2$	0.409485	0.289550	0.144775	−	0.989465i	$-0.453754\pi$
$2$	0.409485	0.289550	0.144775	+	0.989465i	$0.453754\pi$
$3$	−2.84648	−1.64342	−0.821708	−	0.569909i	$-0.806978\pi$
$3$	−2.84648	−1.64342	−0.821708	+	0.569909i	$0.806978\pi$
$4$	−1.83232	−0.916161
$5$	2.40160	1.07403	0.537014	−	0.843573i	$-0.319552\pi$
$5$	2.40160	1.07403	0.537014	+	0.843573i	$0.319552\pi$
$6$	−1.16559	−0.475851
$7$	0	0
$7$	0	0
$8$	−1.56928	−0.554824
$9$	5.10244	1.70081
$10$	0.983421	0.310985
$11$	−0.464750	−0.140127	−0.0700637	−	0.997543i	$-0.522320\pi$
$11$	−0.464750	−0.140127	−0.0700637	+	0.997543i	$0.522320\pi$
$12$	5.21567	1.50563
$13$	−2.07622	−0.575838	−0.287919	−	0.957655i	$-0.592963\pi$
$13$	−2.07622	−0.575838	−0.287919	+	0.957655i	$0.592963\pi$
$14$	0	0
$15$	−6.83611	−1.76508
$16$	3.02205	0.755511
$17$	1.51591	0.367663	0.183832	−	0.982958i	$-0.441150\pi$
$17$	1.51591	0.367663	0.183832	+	0.982958i	$0.441150\pi$
$18$	2.08938	0.492471
$19$	−6.50122	−1.49148	−0.745742	−	0.666235i	$-0.767905\pi$
$19$	−6.50122	−1.49148	−0.745742	+	0.666235i	$0.767905\pi$
$20$	−4.40051	−0.983983
$21$	0	0
$22$	−0.190308	−0.0405739
$23$	3.71419	0.774463	0.387231	−	0.921983i	$-0.373431\pi$
$23$	3.71419	0.774463	0.387231	+	0.921983i	$0.373431\pi$
$24$	4.46692	0.911807
$25$	0.767691	0.153538
$26$	−0.850180	−0.166734
$27$	−5.98456	−1.15173
$28$	0	0
$29$	1.83639	0.341010	0.170505	−	0.985357i	$-0.445460\pi$
$29$	1.83639	0.341010	0.170505	+	0.985357i	$0.445460\pi$
$30$	−2.79929	−0.511078
$31$	2.31777	0.416285	0.208142	−	0.978099i	$-0.433258\pi$
$31$	2.31777	0.416285	0.208142	+	0.978099i	$0.433258\pi$
$32$	4.37604	0.773583
$33$	1.32290	0.230287
$34$	0.620745	0.106457
$35$	0	0
$36$	−9.34932	−1.55822
$37$	9.96935	1.63895	0.819476	−	0.573114i	$-0.194265\pi$
$37$	9.96935	1.63895	0.819476	+	0.573114i	$0.194265\pi$
$38$	−2.66216	−0.431859
$39$	5.90990	0.946342
$40$	−3.76879	−0.595897
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	6.03402	0.920179	0.460090	−	0.887873i	$-0.347817\pi$
$43$	6.03402	0.920179	0.460090	+	0.887873i	$0.347817\pi$
$44$	0.851571	0.128379
$45$	12.2540	1.82672
$46$	1.52091	0.224246
$47$	−9.84846	−1.43655	−0.718273	−	0.695761i	$-0.755067\pi$
$47$	−9.84846	−1.43655	−0.718273	+	0.695761i	$0.755067\pi$
$48$	−8.60219	−1.24162
$49$	0	0
$50$	0.314358	0.0444570
$51$	−4.31502	−0.604223
$52$	3.80429	0.527561
$53$	1.60467	0.220419	0.110209	−	0.993908i	$-0.464848\pi$
$53$	1.60467	0.220419	0.110209	+	0.993908i	$0.464848\pi$
$54$	−2.45059	−0.333483
$55$	−1.11614	−0.150501
$56$	0	0
$57$	18.5056	2.45113
$58$	0.751976	0.0987393
$59$	−2.50393	−0.325984	−0.162992	−	0.986627i	$-0.552114\pi$
$59$	−2.50393	−0.325984	−0.162992	+	0.986627i	$0.552114\pi$
$60$	12.5260	1.61709
$61$	−0.0952801	−0.0121994	−0.00609968	−	0.999981i	$-0.501942\pi$
$61$	−0.0952801	−0.0121994	−0.00609968	+	0.999981i	$0.501942\pi$
$62$	0.949095	0.120535
$63$	0	0
$64$	−4.25216	−0.531521
$65$	−4.98624	−0.618467
$66$	0.541709	0.0666797
$67$	−11.8237	−1.44450	−0.722248	−	0.691634i	$-0.756891\pi$
$67$	−11.8237	−1.44450	−0.722248	+	0.691634i	$0.756891\pi$
$68$	−2.77764	−0.336839
$69$	−10.5724	−1.27276
$70$	0	0
$71$	−15.9241	−1.88984	−0.944922	−	0.327295i	$-0.893863\pi$
$71$	−15.9241	−1.88984	−0.944922	+	0.327295i	$0.893863\pi$
$72$	−8.00716	−0.943653
$73$	−16.3219	−1.91033	−0.955167	−	0.296068i	$-0.904324\pi$
$73$	−16.3219	−1.91033	−0.955167	+	0.296068i	$0.904324\pi$
$74$	4.08230	0.474558
$75$	−2.18522	−0.252327
$76$	11.9123	1.36644
$77$	0	0
$78$	2.42002	0.274013
$79$	7.91940	0.891002	0.445501	−	0.895281i	$-0.353026\pi$
$79$	7.91940	0.891002	0.445501	+	0.895281i	$0.353026\pi$
$80$	7.25775	0.811441
$81$	1.72760	0.191956
$82$	0.409485	0.0452201
$83$	−8.30396	−0.911478	−0.455739	−	0.890113i	$-0.650625\pi$
$83$	−8.30396	−0.911478	−0.455739	+	0.890113i	$0.650625\pi$
$84$	0	0
$85$	3.64062	0.394881
$86$	2.47084	0.266438
$87$	−5.22725	−0.560420
$88$	0.729323	0.0777460
$89$	10.5786	1.12133	0.560664	−	0.828044i	$-0.310546\pi$
$89$	10.5786	1.12133	0.560664	+	0.828044i	$0.310546\pi$
$90$	5.01785	0.528928
$91$	0	0
$92$	−6.80560	−0.709532
$93$	−6.59750	−0.684129
$94$	−4.03280	−0.415952
$95$	−15.6134	−1.60190
$96$	−12.4563	−1.27132
$97$	−6.46102	−0.656017	−0.328009	−	0.944675i	$-0.606377\pi$
$97$	−6.46102	−0.656017	−0.328009	+	0.944675i	$0.606377\pi$
$98$	0	0
$99$	−2.37136	−0.238331
$100$	−1.40666	−0.140666
$101$	−3.67936	−0.366110	−0.183055	−	0.983103i	$-0.558599\pi$
$101$	−3.67936	−0.366110	−0.183055	+	0.983103i	$0.558599\pi$
$102$	−1.76694	−0.174953
$103$	−16.3989	−1.61583	−0.807915	−	0.589298i	$-0.799404\pi$
$103$	−16.3989	−1.61583	−0.807915	+	0.589298i	$0.799404\pi$
$104$	3.25816	0.319489
$105$	0	0
$106$	0.657090	0.0638222
$107$	−12.4349	−1.20213	−0.601064	−	0.799201i	$-0.705256\pi$
$107$	−12.4349	−1.20213	−0.601064	+	0.799201i	$0.705256\pi$
$108$	10.9656	1.05517
$109$	−6.44171	−0.617004	−0.308502	−	0.951224i	$-0.599828\pi$
$109$	−6.44171	−0.617004	−0.308502	+	0.951224i	$0.599828\pi$
$110$	−0.457045	−0.0435775
$111$	−28.3775	−2.69348
$112$	0	0
$113$	2.36569	0.222545	0.111273	−	0.993790i	$-0.464507\pi$
$113$	2.36569	0.222545	0.111273	+	0.993790i	$0.464507\pi$
$114$	7.57778	0.709724
$115$	8.92001	0.831795
$116$	−3.36486	−0.312420
$117$	−10.5938	−0.979395
$118$	−1.02532	−0.0943886
$119$	0	0
$120$	10.7278	0.979307
$121$	−10.7840	−0.980364
$122$	−0.0390158	−0.00353233
$123$	−2.84648	−0.256658
$124$	−4.24691	−0.381384
$125$	−10.1643	−0.909124
$126$	0	0
$127$	20.3511	1.80587	0.902934	−	0.429778i	$-0.141408\pi$
$127$	20.3511	1.80587	0.902934	+	0.429778i	$0.141408\pi$
$128$	−10.4933	−0.927484
$129$	−17.1757	−1.51224
$130$	−2.04179	−0.179077
$131$	3.01084	0.263059	0.131529	−	0.991312i	$-0.458011\pi$
$131$	3.01084	0.263059	0.131529	+	0.991312i	$0.458011\pi$
$132$	−2.42398	−0.210980
$133$	0	0
$134$	−4.84164	−0.418254
$135$	−14.3725	−1.23699
$136$	−2.37889	−0.203988
$137$	−11.9139	−1.01787	−0.508936	−	0.860805i	$-0.669961\pi$
$137$	−11.9139	−1.01787	−0.508936	+	0.860805i	$0.669961\pi$
$138$	−4.32923	−0.368529
$139$	10.0272	0.850498	0.425249	−	0.905076i	$-0.360186\pi$
$139$	10.0272	0.850498	0.425249	+	0.905076i	$0.360186\pi$
$140$	0	0
$141$	28.0334	2.36084
$142$	−6.52069	−0.547204
$143$	0.964921	0.0806907
$144$	15.4198	1.28498
$145$	4.41028	0.366254
$146$	−6.68358	−0.553137
$147$	0	0
$148$	−18.2671	−1.50154
$149$	8.00017	0.655400	0.327700	−	0.944782i	$-0.393727\pi$
$149$	8.00017	0.655400	0.327700	+	0.944782i	$0.393727\pi$
$150$	−0.894814	−0.0730613
$151$	8.10542	0.659610	0.329805	−	0.944049i	$-0.393017\pi$
$151$	8.10542	0.659610	0.329805	+	0.944049i	$0.393017\pi$
$152$	10.2022	0.827511
$153$	7.73486	0.625327
$154$	0	0
$155$	5.56637	0.447102
$156$	−10.8288	−0.867001
$157$	12.8499	1.02553	0.512767	−	0.858528i	$-0.328620\pi$
$157$	12.8499	1.02553	0.512767	+	0.858528i	$0.328620\pi$
$158$	3.24288	0.257990
$159$	−4.56767	−0.362239
$160$	10.5095	0.830850
$161$	0	0
$162$	0.707428	0.0555808
$163$	2.62755	0.205806	0.102903	−	0.994691i	$-0.467187\pi$
$163$	2.62755	0.205806	0.102903	+	0.994691i	$0.467187\pi$
$164$	−1.83232	−0.143080
$165$	3.17708	0.247335
$166$	−3.40035	−0.263919
$167$	−14.1200	−1.09264	−0.546321	−	0.837576i	$-0.683972\pi$
$167$	−14.1200	−1.09264	−0.546321	+	0.837576i	$0.683972\pi$
$168$	0	0
$169$	−8.68933	−0.668410
$170$	1.49078	0.114338
$171$	−33.1721	−2.53674
$172$	−11.0563	−0.843032
$173$	−0.473775	−0.0360205	−0.0180102	−	0.999838i	$-0.505733\pi$
$173$	−0.473775	−0.0360205	−0.0180102	+	0.999838i	$0.505733\pi$
$174$	−2.14048	−0.162270
$175$	0	0
$176$	−1.40450	−0.105868
$177$	7.12738	0.535727
$178$	4.33178	0.324680
$179$	−12.5812	−0.940364	−0.470182	−	0.882569i	$-0.655812\pi$
$179$	−12.5812	−0.940364	−0.470182	+	0.882569i	$0.655812\pi$
$180$	−22.4533	−1.67357
$181$	4.49429	0.334058	0.167029	−	0.985952i	$-0.446583\pi$
$181$	4.49429	0.334058	0.167029	+	0.985952i	$0.446583\pi$
$182$	0	0
$183$	0.271213	0.0200486
$184$	−5.82861	−0.429691
$185$	23.9424	1.76028
$186$	−2.70158	−0.198089
$187$	−0.704521	−0.0515196
$188$	18.0456	1.31611
$189$	0	0
$190$	−6.39344	−0.463829
$191$	−8.26581	−0.598093	−0.299046	−	0.954239i	$-0.596669\pi$
$191$	−8.26581	−0.598093	−0.299046	+	0.954239i	$0.596669\pi$
$192$	12.1037	0.873509
$193$	−20.7672	−1.49486	−0.747429	−	0.664342i	$-0.768712\pi$
$193$	−20.7672	−1.49486	−0.747429	+	0.664342i	$0.768712\pi$
$194$	−2.64569	−0.189950
$195$	14.1932	1.01640
$196$	0	0
$197$	15.2123	1.08383	0.541917	−	0.840432i	$-0.317699\pi$
$197$	15.2123	1.08383	0.541917	+	0.840432i	$0.317699\pi$
$198$	−0.971037	−0.0690086
$199$	2.77212	0.196510	0.0982551	−	0.995161i	$-0.468674\pi$
$199$	2.77212	0.196510	0.0982551	+	0.995161i	$0.468674\pi$
$200$	−1.20472	−0.0851867
$201$	33.6560	2.37391
$202$	−1.50664	−0.106007
$203$	0	0
$204$	7.90650	0.553566
$205$	2.40160	0.167735
$206$	−6.71511	−0.467864
$207$	18.9515	1.31722
$208$	−6.27442	−0.435053
$209$	3.02144	0.208998
$210$	0	0
$211$	−17.3666	−1.19557	−0.597783	−	0.801658i	$-0.703952\pi$
$211$	−17.3666	−1.19557	−0.597783	+	0.801658i	$0.703952\pi$
$212$	−2.94028	−0.201939
$213$	45.3277	3.10580
$214$	−5.09191	−0.348076
$215$	14.4913	0.988299
$216$	9.39146	0.639008
$217$	0	0
$218$	−2.63779	−0.178654
$219$	46.4599	3.13947
$220$	2.04513	0.137883
$221$	−3.14736	−0.211715
$222$	−11.6202	−0.779896
$223$	15.8882	1.06395	0.531975	−	0.846760i	$-0.321450\pi$
$223$	15.8882	1.06395	0.531975	+	0.846760i	$0.321450\pi$
$224$	0	0
$225$	3.91710	0.261140
$226$	0.968714	0.0644379
$227$	19.6744	1.30583	0.652917	−	0.757430i	$-0.273545\pi$
$227$	19.6744	1.30583	0.652917	+	0.757430i	$0.273545\pi$
$228$	−33.9082	−2.24563
$229$	−2.83991	−0.187667	−0.0938333	−	0.995588i	$-0.529912\pi$
$229$	−2.83991	−0.187667	−0.0938333	+	0.995588i	$0.529912\pi$
$230$	3.65262	0.240846
$231$	0	0
$232$	−2.88181	−0.189200
$233$	−13.5032	−0.884624	−0.442312	−	0.896861i	$-0.645842\pi$
$233$	−13.5032	−0.884624	−0.442312	+	0.896861i	$0.645842\pi$
$234$	−4.33800	−0.283584
$235$	−23.6521	−1.54289
$236$	4.58800	0.298654
$237$	−22.5424	−1.46429
$238$	0	0
$239$	−26.1208	−1.68962	−0.844808	−	0.535070i	$-0.820285\pi$
$239$	−26.1208	−1.68962	−0.844808	+	0.535070i	$0.820285\pi$
$240$	−20.6590	−1.33354
$241$	−8.28307	−0.533560	−0.266780	−	0.963758i	$-0.585960\pi$
$241$	−8.28307	−0.533560	−0.266780	+	0.963758i	$0.585960\pi$
$242$	−4.41589	−0.283864
$243$	13.0361	0.836267
$244$	0.174584	0.0111766
$245$	0	0
$246$	−1.16559	−0.0743154
$247$	13.4979	0.858853
$248$	−3.63724	−0.230965
$249$	23.6371	1.49794
$250$	−4.16214	−0.263237
$251$	−27.4692	−1.73384	−0.866920	−	0.498447i	$-0.833904\pi$
$251$	−27.4692	−1.73384	−0.866920	+	0.498447i	$0.833904\pi$
$252$	0	0
$253$	−1.72617	−0.108523
$254$	8.33348	0.522889
$255$	−10.3630	−0.648953
$256$	4.20748	0.262968
$257$	0.674497	0.0420740	0.0210370	−	0.999779i	$-0.493303\pi$
$257$	0.674497	0.0420740	0.0210370	+	0.999779i	$0.493303\pi$
$258$	−7.03320	−0.437868
$259$	0	0
$260$	9.13640	0.566615
$261$	9.37009	0.579994
$262$	1.23290	0.0761686
$263$	−14.7546	−0.909806	−0.454903	−	0.890541i	$-0.650326\pi$
$263$	−14.7546	−0.909806	−0.454903	+	0.890541i	$0.650326\pi$
$264$	−2.07600	−0.127769
$265$	3.85378	0.236736
$266$	0	0
$267$	−30.1117	−1.84281
$268$	21.6648	1.32339
$269$	−18.1949	−1.10936	−0.554682	−	0.832062i	$-0.687160\pi$
$269$	−18.1949	−1.10936	−0.554682	+	0.832062i	$0.687160\pi$
$270$	−5.88535	−0.358171
$271$	−24.9786	−1.51734	−0.758672	−	0.651473i	$-0.774151\pi$
$271$	−24.9786	−1.51734	−0.758672	+	0.651473i	$0.774151\pi$
$272$	4.58116	0.277774
$273$	0	0
$274$	−4.87856	−0.294725
$275$	−0.356784	−0.0215149
$276$	19.3720	1.16606
$277$	26.5015	1.59232	0.796159	−	0.605088i	$-0.206862\pi$
$277$	26.5015	1.59232	0.796159	+	0.605088i	$0.206862\pi$
$278$	4.10600	0.246262
$279$	11.8263	0.708023
$280$	0	0
$281$	−26.1604	−1.56060	−0.780298	−	0.625407i	$-0.784933\pi$
$281$	−26.1604	−1.56060	−0.780298	+	0.625407i	$0.784933\pi$
$282$	11.4793	0.683582
$283$	9.49962	0.564694	0.282347	−	0.959312i	$-0.408887\pi$
$283$	9.49962	0.564694	0.282347	+	0.959312i	$0.408887\pi$
$284$	29.1781	1.73140
$285$	44.4431	2.63258
$286$	0.395121	0.0233640
$287$	0	0
$288$	22.3285	1.31572
$289$	−14.7020	−0.864824
$290$	1.80595	0.106049
$291$	18.3912	1.07811
$292$	29.9070	1.75017
$293$	2.41855	0.141293	0.0706466	−	0.997501i	$-0.477494\pi$
$293$	2.41855	0.141293	0.0706466	+	0.997501i	$0.477494\pi$
$294$	0	0
$295$	−6.01344	−0.350116
$296$	−15.6447	−0.909330
$297$	2.78132	0.161389
$298$	3.27596	0.189771
$299$	−7.71146	−0.445965
$300$	4.00402	0.231172
$301$	0	0
$302$	3.31905	0.190990
$303$	10.4732	0.601671
$304$	−19.6470	−1.12683
$305$	−0.228825	−0.0131025
$306$	3.16731	0.181063
$307$	−7.66297	−0.437349	−0.218674	−	0.975798i	$-0.570173\pi$
$307$	−7.66297	−0.437349	−0.218674	+	0.975798i	$0.570173\pi$
$308$	0	0
$309$	46.6791	2.65548
$310$	2.27935	0.129458
$311$	−3.85635	−0.218674	−0.109337	−	0.994005i	$-0.534873\pi$
$311$	−3.85635	−0.218674	−0.109337	+	0.994005i	$0.534873\pi$
$312$	−9.27429	−0.525053
$313$	0.0837956	0.00473641	0.00236820	−	0.999997i	$-0.499246\pi$
$313$	0.0837956	0.00473641	0.00236820	+	0.999997i	$0.499246\pi$
$314$	5.26185	0.296944
$315$	0	0
$316$	−14.5109	−0.816301
$317$	31.9924	1.79687	0.898436	−	0.439104i	$-0.144704\pi$
$317$	31.9924	1.79687	0.898436	+	0.439104i	$0.144704\pi$
$318$	−1.87039	−0.104886
$319$	−0.853463	−0.0477848
$320$	−10.2120	−0.570869
$321$	35.3957	1.97559
$322$	0	0
$323$	−9.85530	−0.548363
$324$	−3.16552	−0.175862
$325$	−1.59389	−0.0884132
$326$	1.07594	0.0595910
$327$	18.3362	1.01399
$328$	−1.56928	−0.0866490
$329$	0	0
$330$	1.30097	0.0716160
$331$	−15.2661	−0.839099	−0.419550	−	0.907732i	$-0.637812\pi$
$331$	−15.2661	−0.839099	−0.419550	+	0.907732i	$0.637812\pi$
$332$	15.2155	0.835061
$333$	50.8681	2.78755
$334$	−5.78195	−0.316374
$335$	−28.3959	−1.55143
$336$	0	0
$337$	27.2117	1.48231	0.741157	−	0.671331i	$-0.234277\pi$
$337$	27.2117	1.48231	0.741157	+	0.671331i	$0.234277\pi$
$338$	−3.55815	−0.193538
$339$	−6.73388	−0.365734
$340$	−6.67079	−0.361774
$341$	−1.07719	−0.0583328
$342$	−13.5835	−0.734512
$343$	0	0
$344$	−9.46906	−0.510538
$345$	−25.3906	−1.36699
$346$	−0.194004	−0.0104297
$347$	20.7675	1.11486	0.557430	−	0.830224i	$-0.311788\pi$
$347$	20.7675	1.11486	0.557430	+	0.830224i	$0.311788\pi$
$348$	9.57801	0.513435
$349$	8.80291	0.471209	0.235604	−	0.971849i	$-0.424293\pi$
$349$	8.80291	0.471209	0.235604	+	0.971849i	$0.424293\pi$
$350$	0	0
$351$	12.4252	0.663210
$352$	−2.03377	−0.108400
$353$	−23.8923	−1.27166	−0.635829	−	0.771830i	$-0.719342\pi$
$353$	−23.8923	−1.27166	−0.635829	+	0.771830i	$0.719342\pi$
$354$	2.91856	0.155120
$355$	−38.2434	−2.02975
$356$	−19.3834	−1.02732
$357$	0	0
$358$	−5.15183	−0.272283
$359$	4.83937	0.255412	0.127706	−	0.991812i	$-0.459239\pi$
$359$	4.83937	0.255412	0.127706	+	0.991812i	$0.459239\pi$
$360$	−19.2300	−1.01351
$361$	23.2659	1.22452
$362$	1.84035	0.0967265
$363$	30.6965	1.61115
$364$	0	0
$365$	−39.1987	−2.05175
$366$	0.111058	0.00580508
$367$	−16.2409	−0.847771	−0.423885	−	0.905716i	$-0.639334\pi$
$367$	−16.2409	−0.847771	−0.423885	+	0.905716i	$0.639334\pi$
$368$	11.2245	0.585115
$369$	5.10244	0.265623
$370$	9.80407	0.509689
$371$	0	0
$372$	12.0887	0.626772
$373$	−9.51262	−0.492545	−0.246273	−	0.969201i	$-0.579206\pi$
$373$	−9.51262	−0.492545	−0.246273	+	0.969201i	$0.579206\pi$
$374$	−0.288491	−0.0149175
$375$	28.9325	1.49407
$376$	15.4550	0.797031
$377$	−3.81275	−0.196366
$378$	0	0
$379$	13.8654	0.712216	0.356108	−	0.934445i	$-0.384103\pi$
$379$	13.8654	0.712216	0.356108	+	0.934445i	$0.384103\pi$
$380$	28.6087	1.46759
$381$	−57.9290	−2.96779
$382$	−3.38473	−0.173178
$383$	38.0157	1.94251	0.971257	−	0.238035i	$-0.0765032\pi$
$383$	38.0157	1.94251	0.971257	+	0.238035i	$0.0765032\pi$
$384$	29.8689	1.52424
$385$	0	0
$386$	−8.50388	−0.432836
$387$	30.7882	1.56505
$388$	11.8387	0.601017
$389$	15.5785	0.789859	0.394930	−	0.918711i	$-0.370769\pi$
$389$	15.5785	0.789859	0.394930	+	0.918711i	$0.370769\pi$
$390$	5.81192	0.294298
$391$	5.63040	0.284741
$392$	0	0
$393$	−8.57030	−0.432314
$394$	6.22923	0.313824
$395$	19.0192	0.956962
$396$	4.34509	0.218349
$397$	28.5938	1.43508	0.717540	−	0.696517i	$-0.245268\pi$
$397$	28.5938	1.43508	0.717540	+	0.696517i	$0.245268\pi$
$398$	1.13514	0.0568996
$399$	0	0
$400$	2.32000	0.116000
$401$	7.02837	0.350980	0.175490	−	0.984481i	$-0.443849\pi$
$401$	7.02837	0.350980	0.175490	+	0.984481i	$0.443849\pi$
$402$	13.7816	0.687365
$403$	−4.81220	−0.239713
$404$	6.74177	0.335416
$405$	4.14901	0.206166
$406$	0	0
$407$	−4.63325	−0.229662
$408$	6.77147	0.335238
$409$	−19.7491	−0.976530	−0.488265	−	0.872695i	$-0.662370\pi$
$409$	−19.7491	−0.976530	−0.488265	+	0.872695i	$0.662370\pi$
$410$	0.983421	0.0485677
$411$	33.9126	1.67279
$412$	30.0480	1.48036
$413$	0	0
$414$	7.76035	0.381400
$415$	−19.9428	−0.978954
$416$	−9.08561	−0.445459
$417$	−28.5423	−1.39772
$418$	1.23724	0.0605152
$419$	27.7038	1.35342	0.676709	−	0.736251i	$-0.263406\pi$
$419$	27.7038	1.35342	0.676709	+	0.736251i	$0.263406\pi$
$420$	0	0
$421$	−4.50908	−0.219759	−0.109880	−	0.993945i	$-0.535047\pi$
$421$	−4.50908	−0.219759	−0.109880	+	0.993945i	$0.535047\pi$
$422$	−7.11137	−0.346176
$423$	−50.2512	−2.44330
$424$	−2.51818	−0.122294
$425$	1.16375	0.0564503
$426$	18.5610	0.899284
$427$	0	0
$428$	22.7847	1.10134
$429$	−2.74663	−0.132608
$430$	5.93398	0.286162
$431$	−35.1936	−1.69522	−0.847609	−	0.530621i	$-0.821959\pi$
$431$	−35.1936	−1.69522	−0.847609	+	0.530621i	$0.821959\pi$
$432$	−18.0856	−0.870145
$433$	23.7247	1.14014	0.570069	−	0.821597i	$-0.306917\pi$
$433$	23.7247	1.14014	0.570069	+	0.821597i	$0.306917\pi$
$434$	0	0
$435$	−12.5538	−0.601908
$436$	11.8033	0.565275
$437$	−24.1468	−1.15510
$438$	19.0247	0.909034
$439$	22.1486	1.05710	0.528548	−	0.848903i	$-0.322737\pi$
$439$	22.1486	1.05710	0.528548	+	0.848903i	$0.322737\pi$
$440$	1.75154	0.0835015
$441$	0	0
$442$	−1.28880	−0.0613019
$443$	1.18819	0.0564528	0.0282264	−	0.999602i	$-0.491014\pi$
$443$	1.18819	0.0564528	0.0282264	+	0.999602i	$0.491014\pi$
$444$	51.9968	2.46766
$445$	25.4055	1.20434
$446$	6.50597	0.308067
$447$	−22.7723	−1.07709
$448$	0	0
$449$	39.7423	1.87556	0.937778	−	0.347236i	$-0.112880\pi$
$449$	39.7423	1.87556	0.937778	+	0.347236i	$0.112880\pi$
$450$	1.60400	0.0756131
$451$	−0.464750	−0.0218842
$452$	−4.33470	−0.203887
$453$	−23.0719	−1.08401
$454$	8.05637	0.378104
$455$	0	0
$456$	−29.0405	−1.35994
$457$	23.9260	1.11921	0.559605	−	0.828760i	$-0.310953\pi$
$457$	23.9260	1.11921	0.559605	+	0.828760i	$0.310953\pi$
$458$	−1.16290	−0.0543389
$459$	−9.07208	−0.423448
$460$	−16.3443	−0.762058
$461$	−12.4303	−0.578935	−0.289467	−	0.957188i	$-0.593478\pi$
$461$	−12.4303	−0.578935	−0.289467	+	0.957188i	$0.593478\pi$
$462$	0	0
$463$	−15.4504	−0.718042	−0.359021	−	0.933329i	$-0.616889\pi$
$463$	−15.4504	−0.718042	−0.359021	+	0.933329i	$0.616889\pi$
$464$	5.54966	0.257637
$465$	−15.8446	−0.734774
$466$	−5.52936	−0.256143
$467$	−28.3941	−1.31392	−0.656961	−	0.753925i	$-0.728158\pi$
$467$	−28.3941	−1.31392	−0.656961	+	0.753925i	$0.728158\pi$
$468$	19.4112	0.897283
$469$	0	0
$470$	−9.68519	−0.446744
$471$	−36.5770	−1.68538
$472$	3.92937	0.180864
$473$	−2.80431	−0.128942
$474$	−9.23079	−0.423984
$475$	−4.99093	−0.229000
$476$	0	0
$477$	8.18775	0.374891
$478$	−10.6961	−0.489228
$479$	22.3577	1.02155	0.510774	−	0.859715i	$-0.329359\pi$
$479$	22.3577	1.02155	0.510774	+	0.859715i	$0.329359\pi$
$480$	−29.9151	−1.36543
$481$	−20.6985	−0.943771
$482$	−3.39180	−0.154492
$483$	0	0
$484$	19.7598	0.898171
$485$	−15.5168	−0.704581
$486$	5.33810	0.242141
$487$	−11.6088	−0.526045	−0.263022	−	0.964790i	$-0.584719\pi$
$487$	−11.6088	−0.526045	−0.263022	+	0.964790i	$0.584719\pi$
$488$	0.149521	0.00676850
$489$	−7.47927	−0.338224
$490$	0	0
$491$	−28.5561	−1.28872	−0.644359	−	0.764723i	$-0.722876\pi$
$491$	−28.5561	−1.28872	−0.644359	+	0.764723i	$0.722876\pi$
$492$	5.21567	0.235140
$493$	2.78381	0.125377
$494$	5.52721	0.248681
$495$	−5.69506	−0.255974
$496$	7.00442	0.314508
$497$	0	0
$498$	9.67903	0.433728
$499$	35.2150	1.57644	0.788219	−	0.615394i	$-0.211003\pi$
$499$	35.2150	1.57644	0.788219	+	0.615394i	$0.211003\pi$
$500$	18.6243	0.832904
$501$	40.1924	1.79566
$502$	−11.2482	−0.502034
$503$	39.6415	1.76753	0.883763	−	0.467935i	$-0.155002\pi$
$503$	39.6415	1.76753	0.883763	+	0.467935i	$0.155002\pi$
$504$	0	0
$505$	−8.83636	−0.393213
$506$	−0.706842	−0.0314229
$507$	24.7340	1.09848
$508$	−37.2898	−1.65447
$509$	−32.5126	−1.44109	−0.720547	−	0.693406i	$-0.756109\pi$
$509$	−32.5126	−1.44109	−0.720547	+	0.693406i	$0.756109\pi$
$510$	−4.24348	−0.187904
$511$	0	0
$512$	22.7095	1.00363
$513$	38.9070	1.71779
$514$	0.276197	0.0121825
$515$	−39.3836	−1.73545
$516$	31.4714	1.38545
$517$	4.57707	0.201299
$518$	0	0
$519$	1.34859	0.0591966
$520$	7.82481	0.343141
$521$	24.5649	1.07621	0.538104	−	0.842879i	$-0.319141\pi$
$521$	24.5649	1.07621	0.538104	+	0.842879i	$0.319141\pi$
$522$	3.83692	0.167937
$523$	−43.7840	−1.91454	−0.957270	−	0.289195i	$-0.906612\pi$
$523$	−43.7840	−1.91454	−0.957270	+	0.289195i	$0.906612\pi$
$524$	−5.51683	−0.241004
$525$	0	0
$526$	−6.04178	−0.263434
$527$	3.51355	0.153052
$528$	3.99787	0.173985
$529$	−9.20477	−0.400207
$530$	1.57807	0.0685469
$531$	−12.7762	−0.554438
$532$	0	0
$533$	−2.07622	−0.0899309
$534$	−12.3303	−0.533585
$535$	−29.8637	−1.29112
$536$	18.5547	0.801442
$537$	35.8122	1.54541
$538$	−7.45056	−0.321216
$539$	0	0
$540$	26.3351	1.13328
$541$	−0.926237	−0.0398220	−0.0199110	−	0.999802i	$-0.506338\pi$
$541$	−0.926237	−0.0398220	−0.0199110	+	0.999802i	$0.506338\pi$
$542$	−10.2284	−0.439347
$543$	−12.7929	−0.548996
$544$	6.63371	0.284418
$545$	−15.4704	−0.662680
$546$	0	0
$547$	−2.67232	−0.114260	−0.0571301	−	0.998367i	$-0.518195\pi$
$547$	−2.67232	−0.114260	−0.0571301	+	0.998367i	$0.518195\pi$
$548$	21.8301	0.932534
$549$	−0.486161	−0.0207489
$550$	−0.146098	−0.00622964
$551$	−11.9388	−0.508610
$552$	16.5910	0.706160
$553$	0	0
$554$	10.8520	0.461056
$555$	−68.1516	−2.89287
$556$	−18.3731	−0.779193
$557$	20.1948	0.855679	0.427840	−	0.903855i	$-0.359275\pi$
$557$	20.1948	0.855679	0.427840	+	0.903855i	$0.359275\pi$
$558$	4.84270	0.205008
$559$	−12.5279	−0.529875
$560$	0	0
$561$	2.00540	0.0846682
$562$	−10.7123	−0.451871
$563$	−18.6158	−0.784564	−0.392282	−	0.919845i	$-0.628314\pi$
$563$	−18.6158	−0.784564	−0.392282	+	0.919845i	$0.628314\pi$
$564$	−51.3663	−2.16291
$565$	5.68144	0.239020
$566$	3.88996	0.163507
$567$	0	0
$568$	24.9894	1.04853
$569$	−20.7427	−0.869578	−0.434789	−	0.900532i	$-0.643177\pi$
$569$	−20.7427	−0.869578	−0.434789	+	0.900532i	$0.643177\pi$
$570$	18.1988	0.762264
$571$	−2.47910	−0.103747	−0.0518736	−	0.998654i	$-0.516519\pi$
$571$	−2.47910	−0.103747	−0.0518736	+	0.998654i	$0.516519\pi$
$572$	−1.76804	−0.0739257
$573$	23.5285	0.982915
$574$	0	0
$575$	2.85135	0.118910
$576$	−21.6964	−0.904018
$577$	−30.0695	−1.25181	−0.625905	−	0.779899i	$-0.715270\pi$
$577$	−30.0695	−1.25181	−0.625905	+	0.779899i	$0.715270\pi$
$578$	−6.02026	−0.250410
$579$	59.1135	2.45667
$580$	−8.08106	−0.335548
$581$	0	0
$582$	7.53091	0.312166
$583$	−0.745771	−0.0308867
$584$	25.6136	1.05990
$585$	−25.4420	−1.05190
$586$	0.990361	0.0409114
$587$	20.8398	0.860150	0.430075	−	0.902793i	$-0.358487\pi$
$587$	20.8398	0.860150	0.430075	+	0.902793i	$0.358487\pi$
$588$	0	0
$589$	−15.0684	−0.620881
$590$	−2.46242	−0.101376
$591$	−43.3016	−1.78119
$592$	30.1278	1.23825
$593$	−9.57619	−0.393247	−0.196623	−	0.980479i	$-0.562998\pi$
$593$	−9.57619	−0.393247	−0.196623	+	0.980479i	$0.562998\pi$
$594$	1.13891	0.0467301
$595$	0	0
$596$	−14.6589	−0.600452
$597$	−7.89078	−0.322948
$598$	−3.15773	−0.129129
$599$	−29.0145	−1.18550	−0.592750	−	0.805386i	$-0.701958\pi$
$599$	−29.0145	−1.18550	−0.592750	+	0.805386i	$0.701958\pi$
$600$	3.42922	0.139997
$601$	−26.1987	−1.06867	−0.534335	−	0.845273i	$-0.679438\pi$
$601$	−26.1987	−1.06867	−0.534335	+	0.845273i	$0.679438\pi$
$602$	0	0
$603$	−60.3298	−2.45682
$604$	−14.8517	−0.604309
$605$	−25.8989	−1.05294
$606$	4.28863	0.174214
$607$	14.6916	0.596313	0.298156	−	0.954517i	$-0.403628\pi$
$607$	14.6916	0.596313	0.298156	+	0.954517i	$0.403628\pi$
$608$	−28.4496	−1.15379
$609$	0	0
$610$	−0.0937004	−0.00379382
$611$	20.4475	0.827219
$612$	−14.1728	−0.572900
$613$	−34.1216	−1.37816	−0.689080	−	0.724686i	$-0.741985\pi$
$613$	−34.1216	−1.37816	−0.689080	+	0.724686i	$0.741985\pi$
$614$	−3.13788	−0.126634
$615$	−6.83611	−0.275659
$616$	0	0
$617$	7.72876	0.311148	0.155574	−	0.987824i	$-0.450277\pi$
$617$	7.72876	0.311148	0.155574	+	0.987824i	$0.450277\pi$
$618$	19.1144	0.768895
$619$	−36.2400	−1.45661	−0.728305	−	0.685253i	$-0.759692\pi$
$619$	−36.2400	−1.45661	−0.728305	+	0.685253i	$0.759692\pi$
$620$	−10.1994	−0.409617
$621$	−22.2278	−0.891972
$622$	−1.57912	−0.0633170
$623$	0	0
$624$	17.8600	0.714972
$625$	−28.2491	−1.12996
$626$	0.0343131	0.00137143
$627$	−8.60047	−0.343470
$628$	−23.5452	−0.939555
$629$	15.1127	0.602582
$630$	0	0
$631$	11.9181	0.474453	0.237226	−	0.971454i	$-0.423762\pi$
$631$	11.9181	0.474453	0.237226	+	0.971454i	$0.423762\pi$
$632$	−12.4278	−0.494350
$633$	49.4337	1.96481
$634$	13.1004	0.520284
$635$	48.8753	1.93956
$636$	8.36943	0.331870
$637$	0	0
$638$	−0.349481	−0.0138361
$639$	−81.2519	−3.21428
$640$	−25.2007	−0.996145
$641$	−15.1173	−0.597096	−0.298548	−	0.954395i	$-0.596502\pi$
$641$	−15.1173	−0.597096	−0.298548	+	0.954395i	$0.596502\pi$
$642$	14.4940	0.572033
$643$	16.5899	0.654243	0.327122	−	0.944982i	$-0.393921\pi$
$643$	16.5899	0.654243	0.327122	+	0.944982i	$0.393921\pi$
$644$	0	0
$645$	−41.2492	−1.62419
$646$	−4.03560	−0.158779
$647$	−5.78551	−0.227452	−0.113726	−	0.993512i	$-0.536279\pi$
$647$	−5.78551	−0.227452	−0.113726	+	0.993512i	$0.536279\pi$
$648$	−2.71109	−0.106502
$649$	1.16370	0.0456792
$650$	−0.652676	−0.0256000
$651$	0	0
$652$	−4.81452	−0.188551
$653$	−17.3484	−0.678896	−0.339448	−	0.940625i	$-0.610240\pi$
$653$	−17.3484	−0.678896	−0.339448	+	0.940625i	$0.610240\pi$
$654$	7.50841	0.293602
$655$	7.23084	0.282532
$656$	3.02205	0.117991
$657$	−83.2816	−3.24912
$658$	0	0
$659$	25.0610	0.976239	0.488120	−	0.872777i	$-0.337683\pi$
$659$	25.0610	0.976239	0.488120	+	0.872777i	$0.337683\pi$
$660$	−5.82143	−0.226599
$661$	12.8355	0.499242	0.249621	−	0.968344i	$-0.419694\pi$
$661$	12.8355	0.499242	0.249621	+	0.968344i	$0.419694\pi$
$662$	−6.25123	−0.242961
$663$	8.95890	0.347935
$664$	13.0312	0.505710
$665$	0	0
$666$	20.8297	0.807136
$667$	6.82072	0.264099
$668$	25.8725	1.00104
$669$	−45.2253	−1.74851
$670$	−11.6277	−0.449217
$671$	0.0442814	0.00170946
$672$	0	0
$673$	−9.67527	−0.372954	−0.186477	−	0.982459i	$-0.559707\pi$
$673$	−9.67527	−0.372954	−0.186477	+	0.982459i	$0.559707\pi$
$674$	11.1428	0.429204
$675$	−4.59430	−0.176834
$676$	15.9216	0.612371
$677$	23.0833	0.887165	0.443582	−	0.896234i	$-0.353707\pi$
$677$	23.0833	0.887165	0.443582	+	0.896234i	$0.353707\pi$
$678$	−2.75742	−0.105898
$679$	0	0
$680$	−5.71315	−0.219089
$681$	−56.0027	−2.14603
$682$	−0.441092	−0.0168903
$683$	−1.21485	−0.0464851	−0.0232426	−	0.999730i	$-0.507399\pi$
$683$	−1.21485	−0.0464851	−0.0232426	+	0.999730i	$0.507399\pi$
$684$	60.7820	2.32406
$685$	−28.6124	−1.09322
$686$	0	0
$687$	8.08375	0.308414
$688$	18.2351	0.695206
$689$	−3.33164	−0.126926
$690$	−10.3971	−0.395811
$691$	−41.6161	−1.58315	−0.791576	−	0.611071i	$-0.790739\pi$
$691$	−41.6161	−1.58315	−0.791576	+	0.611071i	$0.790739\pi$
$692$	0.868109	0.0330005
$693$	0	0
$694$	8.50400	0.322808
$695$	24.0814	0.913460
$696$	8.20303	0.310935
$697$	1.51591	0.0574193
$698$	3.60466	0.136439
$699$	38.4366	1.45380
$700$	0	0
$701$	−0.256208	−0.00967683	−0.00483842	−	0.999988i	$-0.501540\pi$
$701$	−0.256208	−0.00967683	−0.00483842	+	0.999988i	$0.501540\pi$
$702$	5.08796	0.192033
$703$	−64.8130	−2.44447
$704$	1.97619	0.0744806
$705$	67.3252	2.53561
$706$	−9.78354	−0.368208
$707$	0	0
$708$	−13.0597	−0.490812
$709$	−1.86458	−0.0700259	−0.0350129	−	0.999387i	$-0.511147\pi$
$709$	−1.86458	−0.0700259	−0.0350129	+	0.999387i	$0.511147\pi$
$710$	−15.6601	−0.587713
$711$	40.4083	1.51543
$712$	−16.6008	−0.622140
$713$	8.60866	0.322397
$714$	0	0
$715$	2.31735	0.0866642
$716$	23.0528	0.861525
$717$	74.3524	2.77674
$718$	1.98165	0.0739546
$719$	22.5335	0.840359	0.420180	−	0.907441i	$-0.361967\pi$
$719$	22.5335	0.840359	0.420180	+	0.907441i	$0.361967\pi$
$720$	37.0323	1.38011
$721$	0	0
$722$	9.52706	0.354560
$723$	23.5776	0.876860
$724$	−8.23498	−0.306051
$725$	1.40978	0.0523580
$726$	12.5698	0.466507
$727$	23.7416	0.880527	0.440263	−	0.897869i	$-0.354885\pi$
$727$	23.7416	0.880527	0.440263	+	0.897869i	$0.354885\pi$
$728$	0	0
$729$	−42.2898	−1.56629
$730$	−16.0513	−0.594085
$731$	9.14705	0.338316
$732$	−0.496949	−0.0183678
$733$	−47.1895	−1.74299	−0.871493	−	0.490409i	$-0.836847\pi$
$733$	−47.1895	−1.74299	−0.871493	+	0.490409i	$0.836847\pi$
$734$	−6.65043	−0.245472
$735$	0	0
$736$	16.2535	0.599111
$737$	5.49507	0.202413
$738$	2.08938	0.0769110
$739$	−32.4338	−1.19310	−0.596548	−	0.802577i	$-0.703462\pi$
$739$	−32.4338	−1.19310	−0.596548	+	0.802577i	$0.703462\pi$
$740$	−43.8702	−1.61270
$741$	−38.4216	−1.41145
$742$	0	0
$743$	24.8191	0.910526	0.455263	−	0.890357i	$-0.349545\pi$
$743$	24.8191	0.910526	0.455263	+	0.890357i	$0.349545\pi$
$744$	10.3533	0.379571
$745$	19.2132	0.703918
$746$	−3.89528	−0.142616
$747$	−42.3705	−1.55026
$748$	1.29091	0.0472003
$749$	0	0
$750$	11.8475	0.432608
$751$	−25.6772	−0.936973	−0.468486	−	0.883471i	$-0.655201\pi$
$751$	−25.6772	−0.936973	−0.468486	+	0.883471i	$0.655201\pi$
$752$	−29.7625	−1.08533
$753$	78.1905	2.84942
$754$	−1.56126	−0.0568579
$755$	19.4660	0.708440
$756$	0	0
$757$	1.65695	0.0602227	0.0301114	−	0.999547i	$-0.490414\pi$
$757$	1.65695	0.0602227	0.0301114	+	0.999547i	$0.490414\pi$
$758$	5.67767	0.206222
$759$	4.91351	0.178349
$760$	24.5017	0.888771
$761$	40.1317	1.45477	0.727386	−	0.686228i	$-0.240735\pi$
$761$	40.1317	1.45477	0.727386	+	0.686228i	$0.240735\pi$
$762$	−23.7211	−0.859324
$763$	0	0
$764$	15.1456	0.547949
$765$	18.5761	0.671619
$766$	15.5669	0.562455
$767$	5.19870	0.187714
$768$	−11.9765	−0.432165
$769$	−17.2220	−0.621043	−0.310521	−	0.950566i	$-0.600504\pi$
$769$	−17.2220	−0.621043	−0.310521	+	0.950566i	$0.600504\pi$
$770$	0	0
$771$	−1.91994	−0.0691450
$772$	38.0522	1.36953
$773$	20.5506	0.739155	0.369578	−	0.929200i	$-0.379502\pi$
$773$	20.5506	0.739155	0.369578	+	0.929200i	$0.379502\pi$
$774$	12.6073	0.453161
$775$	1.77933	0.0639156
$776$	10.1392	0.363974
$777$	0	0
$778$	6.37915	0.228704
$779$	−6.50122	−0.232931
$780$	−26.0066	−0.931185
$781$	7.40073	0.264819
$782$	2.30557	0.0824469
$783$	−10.9900	−0.392751
$784$	0	0
$785$	30.8604	1.10145
$786$	−3.50941	−0.125177
$787$	22.4398	0.799891	0.399946	−	0.916539i	$-0.369029\pi$
$787$	22.4398	0.799891	0.399946	+	0.916539i	$0.369029\pi$
$788$	−27.8739	−0.992966
$789$	41.9986	1.49519
$790$	7.78811	0.277088
$791$	0	0
$792$	3.72133	0.132232
$793$	0.197822	0.00702486
$794$	11.7087	0.415527
$795$	−10.9697	−0.389056
$796$	−5.07941	−0.180035
$797$	−9.79677	−0.347020	−0.173510	−	0.984832i	$-0.555511\pi$
$797$	−9.79677	−0.347020	−0.173510	+	0.984832i	$0.555511\pi$
$798$	0	0
$799$	−14.9294	−0.528165
$800$	3.35945	0.118774
$801$	53.9766	1.90717
$802$	2.87802	0.101626
$803$	7.58560	0.267690
$804$	−61.6685	−2.17488
$805$	0	0
$806$	−1.97053	−0.0694088
$807$	51.7915	1.82315
$808$	5.77395	0.203127
$809$	−34.6288	−1.21748	−0.608741	−	0.793369i	$-0.708325\pi$
$809$	−34.6288	−1.21748	−0.608741	+	0.793369i	$0.708325\pi$
$810$	1.69896	0.0596954
$811$	37.0942	1.30255	0.651276	−	0.758841i	$-0.274234\pi$
$811$	37.0942	1.30255	0.651276	+	0.758841i	$0.274234\pi$
$812$	0	0
$813$	71.1011	2.49363
$814$	−1.89725	−0.0664986
$815$	6.31033	0.221041
$816$	−13.0402	−0.456498
$817$	−39.2285	−1.37243
$818$	−8.08697	−0.282754
$819$	0	0
$820$	−4.40051	−0.153672
$821$	−5.97987	−0.208699	−0.104349	−	0.994541i	$-0.533276\pi$
$821$	−5.97987	−0.208699	−0.104349	+	0.994541i	$0.533276\pi$
$822$	13.8867	0.484355
$823$	−5.09777	−0.177697	−0.0888486	−	0.996045i	$-0.528319\pi$
$823$	−5.09777	−0.177697	−0.0888486	+	0.996045i	$0.528319\pi$
$824$	25.7345	0.896502
$825$	1.01558	0.0353579
$826$	0	0
$827$	22.9685	0.798695	0.399347	−	0.916800i	$-0.369237\pi$
$827$	22.9685	0.798695	0.399347	+	0.916800i	$0.369237\pi$
$828$	−34.7252	−1.20678
$829$	15.1659	0.526733	0.263366	−	0.964696i	$-0.415167\pi$
$829$	15.1659	0.526733	0.263366	+	0.964696i	$0.415167\pi$
$830$	−8.16629	−0.283456
$831$	−75.4358	−2.61684
$832$	8.82841	0.306070
$833$	0	0
$834$	−11.6877	−0.404710
$835$	−33.9107	−1.17353
$836$	−5.53625	−0.191475
$837$	−13.8709	−0.479447
$838$	11.3443	0.391882
$839$	−19.5578	−0.675210	−0.337605	−	0.941288i	$-0.609617\pi$
$839$	−19.5578	−0.675210	−0.337605	+	0.941288i	$0.609617\pi$
$840$	0	0
$841$	−25.6277	−0.883712
$842$	−1.84640	−0.0636312
$843$	74.4649	2.56471
$844$	31.8212	1.09533
$845$	−20.8683	−0.717892
$846$	−20.5772	−0.707457
$847$	0	0
$848$	4.84939	0.166529
$849$	−27.0405	−0.928026
$850$	0.476540	0.0163452
$851$	37.0281	1.26931
$852$	−83.0548	−2.84541
$853$	−33.9176	−1.16132	−0.580658	−	0.814148i	$-0.697205\pi$
$853$	−33.9176	−1.16132	−0.580658	+	0.814148i	$0.697205\pi$
$854$	0	0
$855$	−79.6663	−2.72453
$856$	19.5138	0.666969
$857$	31.5306	1.07706	0.538532	−	0.842605i	$-0.318979\pi$
$857$	31.5306	1.07706	0.538532	+	0.842605i	$0.318979\pi$
$858$	−1.12470	−0.0383967
$859$	37.9122	1.29355	0.646774	−	0.762682i	$-0.276118\pi$
$859$	37.9122	1.29355	0.646774	+	0.762682i	$0.276118\pi$
$860$	−26.5527	−0.905441
$861$	0	0
$862$	−14.4113	−0.490850
$863$	−51.3427	−1.74772	−0.873862	−	0.486174i	$-0.838392\pi$
$863$	−51.3427	−1.74772	−0.873862	+	0.486174i	$0.838392\pi$
$864$	−26.1887	−0.890958
$865$	−1.13782	−0.0386870
$866$	9.71493	0.330127
$867$	41.8490	1.42126
$868$	0	0
$869$	−3.68054	−0.124854
$870$	−5.14059	−0.174282
$871$	24.5486	0.831797
$872$	10.1089	0.342329
$873$	−32.9670	−1.11576
$874$	−9.88777	−0.334459
$875$	0	0
$876$	−85.1296	−2.87626
$877$	16.7130	0.564356	0.282178	−	0.959362i	$-0.408943\pi$
$877$	16.7130	0.564356	0.282178	+	0.959362i	$0.408943\pi$
$878$	9.06954	0.306082
$879$	−6.88435	−0.232203
$880$	−3.37304	−0.113705
$881$	47.7445	1.60855	0.804276	−	0.594256i	$-0.202553\pi$
$881$	47.7445	1.60855	0.804276	+	0.594256i	$0.202553\pi$
$882$	0	0
$883$	−33.9356	−1.14202	−0.571012	−	0.820942i	$-0.693449\pi$
$883$	−33.9356	−1.14202	−0.571012	+	0.820942i	$0.693449\pi$
$884$	5.76698	0.193965
$885$	17.1171	0.575386
$886$	0.486549	0.0163459
$887$	−15.0209	−0.504351	−0.252175	−	0.967682i	$-0.581146\pi$
$887$	−15.0209	−0.504351	−0.252175	+	0.967682i	$0.581146\pi$
$888$	44.5323	1.49441
$889$	0	0
$890$	10.4032	0.348716
$891$	−0.802903	−0.0268983
$892$	−29.1122	−0.974749
$893$	64.0271	2.14258
$894$	−9.32494	−0.311873
$895$	−30.2151	−1.00998
$896$	0	0
$897$	21.9505	0.732907
$898$	16.2739	0.543067
$899$	4.25634	0.141957
$900$	−7.17739	−0.239246
$901$	2.43254	0.0810398
$902$	−0.190308	−0.00633657
$903$	0	0
$904$	−3.71242	−0.123473
$905$	10.7935	0.358788
$906$	−9.44762	−0.313876
$907$	4.81947	0.160028	0.0800139	−	0.996794i	$-0.474504\pi$
$907$	4.81947	0.160028	0.0800139	+	0.996794i	$0.474504\pi$
$908$	−36.0498	−1.19635
$909$	−18.7737	−0.622685
$910$	0	0
$911$	16.0532	0.531867	0.265934	−	0.963991i	$-0.414320\pi$
$911$	16.0532	0.531867	0.265934	+	0.963991i	$0.414320\pi$
$912$	55.9248	1.85185
$913$	3.85927	0.127723
$914$	9.79733	0.324067
$915$	0.651345	0.0215328
$916$	5.20363	0.171933
$917$	0	0
$918$	−3.71489	−0.122609
$919$	29.0320	0.957677	0.478838	−	0.877903i	$-0.341058\pi$
$919$	29.0320	0.957677	0.478838	+	0.877903i	$0.341058\pi$
$920$	−13.9980	−0.461500
$921$	21.8125	0.718746
$922$	−5.09001	−0.167631
$923$	33.0619	1.08825
$924$	0	0
$925$	7.65338	0.251642
$926$	−6.32672	−0.207909
$927$	−83.6744	−2.74823
$928$	8.03614	0.263799
$929$	−32.7020	−1.07292	−0.536460	−	0.843926i	$-0.680239\pi$
$929$	−32.7020	−1.07292	−0.536460	+	0.843926i	$0.680239\pi$
$930$	−6.48812	−0.212754
$931$	0	0
$932$	24.7422	0.810457
$933$	10.9770	0.359372
$934$	−11.6270	−0.380446
$935$	−1.69198	−0.0553336
$936$	16.6246	0.543392
$937$	−0.290452	−0.00948864	−0.00474432	−	0.999989i	$-0.501510\pi$
$937$	−0.290452	−0.00948864	−0.00474432	+	0.999989i	$0.501510\pi$
$938$	0	0
$939$	−0.238523	−0.00778389
$940$	43.3382	1.41354
$941$	44.4426	1.44879	0.724394	−	0.689386i	$-0.242120\pi$
$941$	44.4426	1.44879	0.724394	+	0.689386i	$0.242120\pi$
$942$	−14.9778	−0.488002
$943$	3.71419	0.120951
$944$	−7.56699	−0.246285
$945$	0	0
$946$	−1.14832	−0.0373352
$947$	6.78398	0.220450	0.110225	−	0.993907i	$-0.464843\pi$
$947$	6.78398	0.220450	0.110225	+	0.993907i	$0.464843\pi$
$948$	41.3049	1.34152
$949$	33.8878	1.10004
$950$	−2.04371	−0.0663068
$951$	−91.0657	−2.95301
$952$	0	0
$953$	49.2075	1.59399	0.796993	−	0.603989i	$-0.206423\pi$
$953$	49.2075	1.59399	0.796993	+	0.603989i	$0.206423\pi$
$954$	3.35276	0.108550
$955$	−19.8512	−0.642369
$956$	47.8617	1.54796
$957$	2.42937	0.0785302
$958$	9.15515	0.295789
$959$	0	0
$960$	29.0683	0.938174
$961$	−25.6279	−0.826707
$962$	−8.47574	−0.273269
$963$	−63.4484	−2.04460
$964$	15.1773	0.488826
$965$	−49.8746	−1.60552
$966$	0	0
$967$	19.5993	0.630270	0.315135	−	0.949047i	$-0.397950\pi$
$967$	19.5993	0.630270	0.315135	+	0.949047i	$0.397950\pi$
$968$	16.9231	0.543930
$969$	28.0529	0.901189
$970$	−6.35390	−0.204012
$971$	−5.41565	−0.173796	−0.0868982	−	0.996217i	$-0.527695\pi$
$971$	−5.41565	−0.173796	−0.0868982	+	0.996217i	$0.527695\pi$
$972$	−23.8863	−0.766155
$973$	0	0
$974$	−4.75363	−0.152316
$975$	4.53698	0.145300
$976$	−0.287941	−0.00921676
$977$	−32.8672	−1.05152	−0.525758	−	0.850634i	$-0.676218\pi$
$977$	−32.8672	−1.05152	−0.525758	+	0.850634i	$0.676218\pi$
$978$	−3.06265	−0.0979328
$979$	−4.91639	−0.157129
$980$	0	0
$981$	−32.8685	−1.04941
$982$	−11.6933	−0.373149
$983$	54.0793	1.72486	0.862431	−	0.506175i	$-0.168941\pi$
$983$	54.0793	1.72486	0.862431	+	0.506175i	$0.168941\pi$
$984$	4.46692	0.142400
$985$	36.5340	1.16407
$986$	1.13993	0.0363028
$987$	0	0
$988$	−24.7326	−0.786848
$989$	22.4115	0.712644
$990$	−2.33205	−0.0741173
$991$	28.3248	0.899766	0.449883	−	0.893087i	$-0.351466\pi$
$991$	28.3248	0.899766	0.449883	+	0.893087i	$0.351466\pi$
$992$	10.1427	0.322031
$993$	43.4545	1.37899
$994$	0	0
$995$	6.65753	0.211058
$996$	−43.3107	−1.37235
$997$	−17.0080	−0.538649	−0.269325	−	0.963049i	$-0.586800\pi$
$997$	−17.0080	−0.538649	−0.269325	+	0.963049i	$0.586800\pi$
$998$	14.4200	0.456458
$999$	−59.6622	−1.88763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.p.1.4	✓	7
7.6	odd	2		2009.2.a.q.1.4	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.p.1.4	✓	7	1.1	even	1	trivial
2009.2.a.q.1.4	yes	7	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)

\(f(q)\)	\(=\)	\(q+0.409485 q^{2} -2.84648 q^{3} -1.83232 q^{4} +2.40160 q^{5} -1.16559 q^{6} -1.56928 q^{8} +5.10244 q^{9} +O(q^{10})\)	\(q+0.409485 q^{2} -2.84648 q^{3} -1.83232 q^{4} +2.40160 q^{5} -1.16559 q^{6} -1.56928 q^{8} +5.10244 q^{9} +0.983421 q^{10} -0.464750 q^{11} +5.21567 q^{12} -2.07622 q^{13} -6.83611 q^{15} +3.02205 q^{16} +1.51591 q^{17} +2.08938 q^{18} -6.50122 q^{19} -4.40051 q^{20} -0.190308 q^{22} +3.71419 q^{23} +4.46692 q^{24} +0.767691 q^{25} -0.850180 q^{26} -5.98456 q^{27} +1.83639 q^{29} -2.79929 q^{30} +2.31777 q^{31} +4.37604 q^{32} +1.32290 q^{33} +0.620745 q^{34} -9.34932 q^{36} +9.96935 q^{37} -2.66216 q^{38} +5.90990 q^{39} -3.76879 q^{40} +1.00000 q^{41} +6.03402 q^{43} +0.851571 q^{44} +12.2540 q^{45} +1.52091 q^{46} -9.84846 q^{47} -8.60219 q^{48} +0.314358 q^{50} -4.31502 q^{51} +3.80429 q^{52} +1.60467 q^{53} -2.45059 q^{54} -1.11614 q^{55} +18.5056 q^{57} +0.751976 q^{58} -2.50393 q^{59} +12.5260 q^{60} -0.0952801 q^{61} +0.949095 q^{62} -4.25216 q^{64} -4.98624 q^{65} +0.541709 q^{66} -11.8237 q^{67} -2.77764 q^{68} -10.5724 q^{69} -15.9241 q^{71} -8.00716 q^{72} -16.3219 q^{73} +4.08230 q^{74} -2.18522 q^{75} +11.9123 q^{76} +2.42002 q^{78} +7.91940 q^{79} +7.25775 q^{80} +1.72760 q^{81} +0.409485 q^{82} -8.30396 q^{83} +3.64062 q^{85} +2.47084 q^{86} -5.22725 q^{87} +0.729323 q^{88} +10.5786 q^{89} +5.01785 q^{90} -6.80560 q^{92} -6.59750 q^{93} -4.03280 q^{94} -15.6134 q^{95} -12.4563 q^{96} -6.46102 q^{97} -2.37136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9}+O(q^{10}) \) 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9	\( 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9} - 8 q^{10} - 12 q^{12} - q^{13} - 4 q^{15} + 5 q^{16} + 11 q^{17} + 16 q^{18} - 9 q^{19} - 12 q^{20} + 3 q^{23} + 23 q^{24} + 31 q^{25} - 10 q^{26} - 22 q^{27} - 14 q^{29} + 6 q^{30} - 34 q^{31} - 20 q^{32} - 12 q^{33} - 15 q^{34} + q^{36} + 7 q^{37} + 39 q^{38} - 22 q^{39} - 50 q^{40} + 7 q^{41} - 3 q^{43} + 26 q^{44} + 4 q^{45} - 8 q^{46} - 17 q^{47} - 17 q^{48} + 11 q^{50} + 8 q^{51} + 25 q^{52} + 24 q^{53} - 68 q^{54} - 48 q^{55} + 22 q^{57} - 38 q^{58} - 4 q^{59} - 6 q^{60} - 16 q^{61} - 24 q^{62} + 8 q^{64} - 6 q^{65} + 12 q^{66} - 24 q^{67} + 10 q^{68} - 35 q^{69} - 12 q^{71} - 11 q^{72} - 14 q^{73} - 6 q^{74} + 19 q^{75} - 42 q^{76} - 29 q^{78} - 8 q^{79} + 92 q^{80} + 15 q^{81} - q^{82} - 14 q^{83} + 16 q^{85} + 35 q^{86} + 20 q^{87} + 22 q^{88} + 19 q^{89} - 24 q^{90} - 10 q^{92} + 2 q^{93} + 20 q^{94} - 8 q^{95} + 25 q^{96} - 23 q^{97} + 24 q^{99}+O(q^{100}) \) 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9 - 8 * q^10 - 12 * q^12 - q^13 - 4 * q^15 + 5 * q^16 + 11 * q^17 + 16 * q^18 - 9 * q^19 - 12 * q^20 + 3 * q^23 + 23 * q^24 + 31 * q^25 - 10 * q^26 - 22 * q^27 - 14 * q^29 + 6 * q^30 - 34 * q^31 - 20 * q^32 - 12 * q^33 - 15 * q^34 + q^36 + 7 * q^37 + 39 * q^38 - 22 * q^39 - 50 * q^40 + 7 * q^41 - 3 * q^43 + 26 * q^44 + 4 * q^45 - 8 * q^46 - 17 * q^47 - 17 * q^48 + 11 * q^50 + 8 * q^51 + 25 * q^52 + 24 * q^53 - 68 * q^54 - 48 * q^55 + 22 * q^57 - 38 * q^58 - 4 * q^59 - 6 * q^60 - 16 * q^61 - 24 * q^62 + 8 * q^64 - 6 * q^65 + 12 * q^66 - 24 * q^67 + 10 * q^68 - 35 * q^69 - 12 * q^71 - 11 * q^72 - 14 * q^73 - 6 * q^74 + 19 * q^75 - 42 * q^76 - 29 * q^78 - 8 * q^79 + 92 * q^80 + 15 * q^81 - q^82 - 14 * q^83 + 16 * q^85 + 35 * q^86 + 20 * q^87 + 22 * q^88 + 19 * q^89 - 24 * q^90 - 10 * q^92 + 2 * q^93 + 20 * q^94 - 8 * q^95 + 25 * q^96 - 23 * q^97 + 24 * q^99

Introduction

L-functions

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