LMFDB - Embedded newform 4805.2.a.y.1.17

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4805,2,Mod(1,4805)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4805.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4805, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4805 = 5 \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4805.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20,-1,4,15,-20,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$38.3681181712$
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} - x^{19} - 27 x^{18} + 29 x^{17} + 298 x^{16} - 343 x^{15} - 1729 x^{14} + 2127 x^{13} + 5634 x^{12} + \cdots + 1 $ x^20 - x^19 - 27x^18 + 29x^17 + 298x^16 - 343x^15 - 1729x^14 + 2127x^13 + 5634x^12 - 7404x^11 - 10174x^10 + 14381x^9 + 9333x^8 - 14641x^7 - 3440x^6 + 6885x^5 - 5x^4 - 1073x^3 + 89x^2 + 30x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$-2.09217$ of defining polynomial
Character	$\chi$	$=$	4805.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.09217 q^{2} -1.52922 q^{3} +2.37719 q^{4} -1.00000 q^{5} -3.19940 q^{6} +3.03261 q^{7} +0.789144 q^{8} -0.661478 q^{9} -2.09217 q^{10} +6.10532 q^{11} -3.63525 q^{12} +5.76122 q^{13} +6.34475 q^{14} +1.52922 q^{15} -3.10335 q^{16} +5.74835 q^{17} -1.38393 q^{18} -2.04231 q^{19} -2.37719 q^{20} -4.63754 q^{21} +12.7734 q^{22} -6.12364 q^{23} -1.20678 q^{24} +1.00000 q^{25} +12.0535 q^{26} +5.59922 q^{27} +7.20909 q^{28} -5.75211 q^{29} +3.19940 q^{30} -8.07104 q^{32} -9.33639 q^{33} +12.0266 q^{34} -3.03261 q^{35} -1.57246 q^{36} -2.36580 q^{37} -4.27287 q^{38} -8.81019 q^{39} -0.789144 q^{40} +0.924914 q^{41} -9.70253 q^{42} +4.26449 q^{43} +14.5135 q^{44} +0.661478 q^{45} -12.8117 q^{46} +1.56258 q^{47} +4.74571 q^{48} +2.19673 q^{49} +2.09217 q^{50} -8.79051 q^{51} +13.6955 q^{52} +2.87837 q^{53} +11.7145 q^{54} -6.10532 q^{55} +2.39317 q^{56} +3.12315 q^{57} -12.0344 q^{58} +0.194720 q^{59} +3.63525 q^{60} +9.68296 q^{61} -2.00601 q^{63} -10.6793 q^{64} -5.76122 q^{65} -19.5333 q^{66} +6.44649 q^{67} +13.6649 q^{68} +9.36440 q^{69} -6.34475 q^{70} +0.958977 q^{71} -0.522002 q^{72} +1.07913 q^{73} -4.94966 q^{74} -1.52922 q^{75} -4.85496 q^{76} +18.5151 q^{77} -18.4325 q^{78} +9.64623 q^{79} +3.10335 q^{80} -6.57801 q^{81} +1.93508 q^{82} -7.99699 q^{83} -11.0243 q^{84} -5.74835 q^{85} +8.92205 q^{86} +8.79626 q^{87} +4.81798 q^{88} -10.6879 q^{89} +1.38393 q^{90} +17.4716 q^{91} -14.5570 q^{92} +3.26919 q^{94} +2.04231 q^{95} +12.3424 q^{96} +2.42074 q^{97} +4.59595 q^{98} -4.03853 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} + 4 q^{3} + 15 q^{4} - 20 q^{5} + 4 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} + q^{10} + 17 q^{11} - 13 q^{12} + 9 q^{13} + 6 q^{14} - 4 q^{15} + 9 q^{16} - 20 q^{17} - 28 q^{18} - 7 q^{19} - 15 q^{20}+ \cdots + 53 q^{99}+O(q^{100}) $ 20 * q - q^2 + 4 * q^3 + 15 * q^4 - 20 * q^5 + 4 * q^6 + q^7 + 9 * q^8 + 8 * q^9 + q^10 + 17 * q^11 - 13 * q^12 + 9 * q^13 + 6 * q^14 - 4 * q^15 + 9 * q^16 - 20 * q^17 - 28 * q^18 - 7 * q^19 - 15 * q^20 + 22 * q^21 + 22 * q^22 - 9 * q^23 + 28 * q^24 + 20 * q^25 + 32 * q^26 + 13 * q^27 + 8 * q^28 + 4 * q^29 - 4 * q^30 + 2 * q^32 - 26 * q^33 + 27 * q^34 - q^35 + 13 * q^36 + 12 * q^37 + 15 * q^38 + 11 * q^39 - 9 * q^40 + 9 * q^41 - 14 * q^42 + 42 * q^43 + 49 * q^44 - 8 * q^45 - 24 * q^46 - 32 * q^47 - 31 * q^48 + q^49 - q^50 + 2 * q^51 + 29 * q^52 + 9 * q^53 + 44 * q^54 - 17 * q^55 - 9 * q^56 + 38 * q^57 + 47 * q^58 + 6 * q^59 + 13 * q^60 + 50 * q^61 + 55 * q^63 - 19 * q^64 - 9 * q^65 + 15 * q^66 - 30 * q^67 - 34 * q^68 + 2 * q^69 - 6 * q^70 + 7 * q^71 - 16 * q^72 + 32 * q^73 - 22 * q^74 + 4 * q^75 + 23 * q^76 - 8 * q^77 + 29 * q^78 + 63 * q^79 - 9 * q^80 + 8 * q^81 - 12 * q^82 - 35 * q^83 + 71 * q^84 + 20 * q^85 + 65 * q^86 + 46 * q^87 + 70 * q^88 + 40 * q^89 + 28 * q^90 + 28 * q^91 + 5 * q^92 + 65 * q^94 + 7 * q^95 + 46 * q^96 + 41 * q^97 + 65 * q^98 + 53 * q^99

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 $

$2$	2.09217	1.47939	0.739695	−	0.672942i	$-0.234970\pi$
$2$	2.09217	1.47939	0.739695	+	0.672942i	$0.234970\pi$
$3$	−1.52922	−0.882897	−0.441449	−	0.897287i	$-0.645535\pi$
$3$	−1.52922	−0.882897	−0.441449	+	0.897287i	$0.645535\pi$
$4$	2.37719	1.18859
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.19940	−1.30615
$7$	3.03261	1.14622	0.573110	−	0.819479i	$-0.305737\pi$
$7$	3.03261	1.14622	0.573110	+	0.819479i	$0.305737\pi$
$8$	0.789144	0.279005
$9$	−0.661478	−0.220493
$10$	−2.09217	−0.661603
$11$	6.10532	1.84082	0.920411	−	0.390952i	$-0.127854\pi$
$11$	6.10532	1.84082	0.920411	+	0.390952i	$0.127854\pi$
$12$	−3.63525	−1.04941
$13$	5.76122	1.59788	0.798938	−	0.601413i	$-0.205396\pi$
$13$	5.76122	1.59788	0.798938	+	0.601413i	$0.205396\pi$
$14$	6.34475	1.69571
$15$	1.52922	0.394844
$16$	−3.10335	−0.775838
$17$	5.74835	1.39418	0.697090	−	0.716983i	$-0.254478\pi$
$17$	5.74835	1.39418	0.697090	+	0.716983i	$0.254478\pi$
$18$	−1.38393	−0.326195
$19$	−2.04231	−0.468538	−0.234269	−	0.972172i	$-0.575270\pi$
$19$	−2.04231	−0.468538	−0.234269	+	0.972172i	$0.575270\pi$
$20$	−2.37719	−0.531556
$21$	−4.63754	−1.01199
$22$	12.7734	2.72329
$23$	−6.12364	−1.27687	−0.638433	−	0.769677i	$-0.720417\pi$
$23$	−6.12364	−1.27687	−0.638433	+	0.769677i	$0.720417\pi$
$24$	−1.20678	−0.246332
$25$	1.00000	0.200000
$26$	12.0535	2.36388
$27$	5.59922	1.07757
$28$	7.20909	1.36239
$29$	−5.75211	−1.06814	−0.534070	−	0.845440i	$-0.679338\pi$
$29$	−5.75211	−1.06814	−0.534070	+	0.845440i	$0.679338\pi$
$30$	3.19940	0.584128
$31$	0	0
$31$	0	0
$32$	−8.07104	−1.42677
$33$	−9.33639	−1.62526
$34$	12.0266	2.06254
$35$	−3.03261	−0.512605
$36$	−1.57246	−0.262076
$37$	−2.36580	−0.388935	−0.194468	−	0.980909i	$-0.562298\pi$
$37$	−2.36580	−0.388935	−0.194468	+	0.980909i	$0.562298\pi$
$38$	−4.27287	−0.693150
$39$	−8.81019	−1.41076
$40$	−0.789144	−0.124775
$41$	0.924914	0.144447	0.0722236	−	0.997388i	$-0.476990\pi$
$41$	0.924914	0.144447	0.0722236	+	0.997388i	$0.476990\pi$
$42$	−9.70253	−1.49713
$43$	4.26449	0.650328	0.325164	−	0.945658i	$-0.394580\pi$
$43$	4.26449	0.650328	0.325164	+	0.945658i	$0.394580\pi$
$44$	14.5135	2.18799
$45$	0.661478	0.0986073
$46$	−12.8117	−1.88898
$47$	1.56258	0.227926	0.113963	−	0.993485i	$-0.463645\pi$
$47$	1.56258	0.227926	0.113963	+	0.993485i	$0.463645\pi$
$48$	4.74571	0.684985
$49$	2.19673	0.313819
$50$	2.09217	0.295878
$51$	−8.79051	−1.23092
$52$	13.6955	1.89923
$53$	2.87837	0.395375	0.197687	−	0.980265i	$-0.436657\pi$
$53$	2.87837	0.395375	0.197687	+	0.980265i	$0.436657\pi$
$54$	11.7145	1.59415
$55$	−6.10532	−0.823241
$56$	2.39317	0.319801
$57$	3.12315	0.413671
$58$	−12.0344	−1.58020
$59$	0.194720	0.0253504	0.0126752	−	0.999920i	$-0.495965\pi$
$59$	0.194720	0.0253504	0.0126752	+	0.999920i	$0.495965\pi$
$60$	3.63525	0.469309
$61$	9.68296	1.23978	0.619888	−	0.784690i	$-0.287178\pi$
$61$	9.68296	1.23978	0.619888	+	0.784690i	$0.287178\pi$
$62$	0	0
$63$	−2.00601	−0.252733
$64$	−10.6793	−1.33491
$65$	−5.76122	−0.714592
$66$	−19.5333	−2.40439
$67$	6.44649	0.787564	0.393782	−	0.919204i	$-0.371167\pi$
$67$	6.44649	0.787564	0.393782	+	0.919204i	$0.371167\pi$
$68$	13.6649	1.65712
$69$	9.36440	1.12734
$70$	−6.34475	−0.758343
$71$	0.958977	0.113810	0.0569048	−	0.998380i	$-0.481877\pi$
$71$	0.958977	0.113810	0.0569048	+	0.998380i	$0.481877\pi$
$72$	−0.522002	−0.0615185
$73$	1.07913	0.126303	0.0631515	−	0.998004i	$-0.479885\pi$
$73$	1.07913	0.126303	0.0631515	+	0.998004i	$0.479885\pi$
$74$	−4.94966	−0.575387
$75$	−1.52922	−0.176579
$76$	−4.85496	−0.556902
$77$	18.5151	2.10999
$78$	−18.4325	−2.08706
$79$	9.64623	1.08529	0.542643	−	0.839963i	$-0.317424\pi$
$79$	9.64623	1.08529	0.542643	+	0.839963i	$0.317424\pi$
$80$	3.10335	0.346965
$81$	−6.57801	−0.730890
$82$	1.93508	0.213694
$83$	−7.99699	−0.877783	−0.438892	−	0.898540i	$-0.644629\pi$
$83$	−7.99699	−0.877783	−0.438892	+	0.898540i	$0.644629\pi$
$84$	−11.0243	−1.20285
$85$	−5.74835	−0.623497
$86$	8.92205	0.962089
$87$	8.79626	0.943058
$88$	4.81798	0.513598
$89$	−10.6879	−1.13292	−0.566458	−	0.824091i	$-0.691686\pi$
$89$	−10.6879	−1.13292	−0.566458	+	0.824091i	$0.691686\pi$
$90$	1.38393	0.145879
$91$	17.4716	1.83152
$92$	−14.5570	−1.51768
$93$	0	0
$94$	3.26919	0.337191
$95$	2.04231	0.209537
$96$	12.3424	1.25969
$97$	2.42074	0.245789	0.122895	−	0.992420i	$-0.460782\pi$
$97$	2.42074	0.245789	0.122895	+	0.992420i	$0.460782\pi$
$98$	4.59595	0.464261
$99$	−4.03853	−0.405888
$100$	2.37719	0.237719
$101$	−1.17406	−0.116824	−0.0584119	−	0.998293i	$-0.518604\pi$
$101$	−1.17406	−0.116824	−0.0584119	+	0.998293i	$0.518604\pi$
$102$	−18.3913	−1.82101
$103$	12.0361	1.18595	0.592977	−	0.805219i	$-0.297952\pi$
$103$	12.0361	1.18595	0.592977	+	0.805219i	$0.297952\pi$
$104$	4.54644	0.445815
$105$	4.63754	0.452577
$106$	6.02205	0.584913
$107$	−3.65289	−0.353138	−0.176569	−	0.984288i	$-0.556500\pi$
$107$	−3.65289	−0.353138	−0.176569	+	0.984288i	$0.556500\pi$
$108$	13.3104	1.28079
$109$	−2.91891	−0.279581	−0.139790	−	0.990181i	$-0.544643\pi$
$109$	−2.91891	−0.279581	−0.139790	+	0.990181i	$0.544643\pi$
$110$	−12.7734	−1.21789
$111$	3.61783	0.343390
$112$	−9.41126	−0.889280
$113$	17.2796	1.62553	0.812765	−	0.582592i	$-0.197961\pi$
$113$	17.2796	1.62553	0.812765	+	0.582592i	$0.197961\pi$
$114$	6.53416	0.611980
$115$	6.12364	0.571032
$116$	−13.6739	−1.26959
$117$	−3.81092	−0.352320
$118$	0.407388	0.0375031
$119$	17.4325	1.59804
$120$	1.20678	0.110163
$121$	26.2749	2.38863
$122$	20.2584	1.83411
$123$	−1.41440	−0.127532
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−4.19691	−0.373891
$127$	9.93372	0.881475	0.440738	−	0.897636i	$-0.354717\pi$
$127$	9.93372	0.881475	0.440738	+	0.897636i	$0.354717\pi$
$128$	−6.20088	−0.548086
$129$	−6.52135	−0.574173
$130$	−12.0535	−1.05716
$131$	5.95405	0.520208	0.260104	−	0.965581i	$-0.416243\pi$
$131$	5.95405	0.520208	0.260104	+	0.965581i	$0.416243\pi$
$132$	−22.1944	−1.93177
$133$	−6.19353	−0.537047
$134$	13.4872	1.16511
$135$	−5.59922	−0.481904
$136$	4.53628	0.388983
$137$	22.3422	1.90882	0.954410	−	0.298498i	$-0.0964857\pi$
$137$	22.3422	1.90882	0.954410	+	0.298498i	$0.0964857\pi$
$138$	19.5920	1.66778
$139$	6.76957	0.574188	0.287094	−	0.957902i	$-0.407311\pi$
$139$	6.76957	0.574188	0.287094	+	0.957902i	$0.407311\pi$
$140$	−7.20909	−0.609279
$141$	−2.38953	−0.201235
$142$	2.00635	0.168369
$143$	35.1741	2.94141
$144$	2.05280	0.171067
$145$	5.75211	0.477687
$146$	2.25773	0.186851
$147$	−3.35929	−0.277070
$148$	−5.62395	−0.462286
$149$	−21.4057	−1.75362	−0.876812	−	0.480833i	$-0.840334\pi$
$149$	−21.4057	−1.75362	−0.876812	+	0.480833i	$0.840334\pi$
$150$	−3.19940	−0.261230
$151$	3.40039	0.276720	0.138360	−	0.990382i	$-0.455817\pi$
$151$	3.40039	0.276720	0.138360	+	0.990382i	$0.455817\pi$
$152$	−1.61168	−0.130724
$153$	−3.80241	−0.307407
$154$	38.7367	3.12149
$155$	0	0
$156$	−20.9435	−1.67682
$157$	9.41684	0.751545	0.375773	−	0.926712i	$-0.377377\pi$
$157$	9.41684	0.751545	0.375773	+	0.926712i	$0.377377\pi$
$158$	20.1816	1.60556
$159$	−4.40167	−0.349075
$160$	8.07104	0.638071
$161$	−18.5706	−1.46357
$162$	−13.7623	−1.08127
$163$	−11.4367	−0.895788	−0.447894	−	0.894087i	$-0.647826\pi$
$163$	−11.4367	−0.895788	−0.447894	+	0.894087i	$0.647826\pi$
$164$	2.19870	0.171689
$165$	9.33639	0.726837
$166$	−16.7311	−1.29858
$167$	−7.71810	−0.597245	−0.298622	−	0.954371i	$-0.596527\pi$
$167$	−7.71810	−0.597245	−0.298622	+	0.954371i	$0.596527\pi$
$168$	−3.65969	−0.282351
$169$	20.1917	1.55321
$170$	−12.0266	−0.922394
$171$	1.35094	0.103309
$172$	10.1375	0.772977
$173$	−4.65205	−0.353689	−0.176844	−	0.984239i	$-0.556589\pi$
$173$	−4.65205	−0.353689	−0.176844	+	0.984239i	$0.556589\pi$
$174$	18.4033	1.39515
$175$	3.03261	0.229244
$176$	−18.9469	−1.42818
$177$	−0.297770	−0.0223818
$178$	−22.3609	−1.67602
$179$	5.68439	0.424871	0.212436	−	0.977175i	$-0.431860\pi$
$179$	5.68439	0.424871	0.212436	+	0.977175i	$0.431860\pi$
$180$	1.57246	0.117204
$181$	1.13672	0.0844915	0.0422457	−	0.999107i	$-0.486549\pi$
$181$	1.13672	0.0844915	0.0422457	+	0.999107i	$0.486549\pi$
$182$	36.5535	2.70953
$183$	−14.8074	−1.09459
$184$	−4.83243	−0.356252
$185$	2.36580	0.173937
$186$	0	0
$187$	35.0955	2.56644
$188$	3.71455	0.270911
$189$	16.9802	1.23513
$190$	4.27287	0.309986
$191$	−19.4059	−1.40416	−0.702079	−	0.712099i	$-0.747745\pi$
$191$	−19.4059	−1.40416	−0.702079	+	0.712099i	$0.747745\pi$
$192$	16.3310	1.17859
$193$	−16.1656	−1.16363	−0.581813	−	0.813322i	$-0.697657\pi$
$193$	−16.1656	−1.16363	−0.581813	+	0.813322i	$0.697657\pi$
$194$	5.06461	0.363618
$195$	8.81019	0.630911
$196$	5.22205	0.373004
$197$	7.73728	0.551258	0.275629	−	0.961264i	$-0.411114\pi$
$197$	7.73728	0.551258	0.275629	+	0.961264i	$0.411114\pi$
$198$	−8.44931	−0.600466
$199$	−26.5342	−1.88096	−0.940481	−	0.339847i	$-0.889625\pi$
$199$	−26.5342	−1.88096	−0.940481	+	0.339847i	$0.889625\pi$
$200$	0.789144	0.0558009
$201$	−9.85811	−0.695338
$202$	−2.45634	−0.172828
$203$	−17.4439	−1.22432
$204$	−20.8967	−1.46306
$205$	−0.924914	−0.0645988
$206$	25.1817	1.75449
$207$	4.05065	0.281540
$208$	−17.8791	−1.23969
$209$	−12.4689	−0.862495
$210$	9.70253	0.669538
$211$	16.2521	1.11884	0.559419	−	0.828885i	$-0.311024\pi$
$211$	16.2521	1.11884	0.559419	+	0.828885i	$0.311024\pi$
$212$	6.84243	0.469940
$213$	−1.46649	−0.100482
$214$	−7.64247	−0.522428
$215$	−4.26449	−0.290836
$216$	4.41859	0.300647
$217$	0	0
$218$	−6.10686	−0.413609
$219$	−1.65024	−0.111513
$220$	−14.5135	−0.978499
$221$	33.1176	2.22773
$222$	7.56914	0.508007
$223$	4.85815	0.325326	0.162663	−	0.986682i	$-0.447992\pi$
$223$	4.85815	0.325326	0.162663	+	0.986682i	$0.447992\pi$
$224$	−24.4763	−1.63539
$225$	−0.661478	−0.0440985
$226$	36.1519	2.40479
$227$	4.88513	0.324237	0.162119	−	0.986771i	$-0.448167\pi$
$227$	4.88513	0.324237	0.162119	+	0.986771i	$0.448167\pi$
$228$	7.42431	0.491687
$229$	−3.42385	−0.226254	−0.113127	−	0.993581i	$-0.536087\pi$
$229$	−3.42385	−0.226254	−0.113127	+	0.993581i	$0.536087\pi$
$230$	12.8117	0.844779
$231$	−28.3136	−1.86290
$232$	−4.53925	−0.298016
$233$	14.9111	0.976861	0.488431	−	0.872603i	$-0.337569\pi$
$233$	14.9111	0.976861	0.488431	+	0.872603i	$0.337569\pi$
$234$	−7.97311	−0.521219
$235$	−1.56258	−0.101932
$236$	0.462886	0.0301313
$237$	−14.7512	−0.958196
$238$	36.4719	2.36412
$239$	7.50199	0.485264	0.242632	−	0.970118i	$-0.421989\pi$
$239$	7.50199	0.485264	0.242632	+	0.970118i	$0.421989\pi$
$240$	−4.74571	−0.306335
$241$	6.09736	0.392765	0.196383	−	0.980527i	$-0.437081\pi$
$241$	6.09736	0.392765	0.196383	+	0.980527i	$0.437081\pi$
$242$	54.9716	3.53371
$243$	−6.73840	−0.432269
$244$	23.0182	1.47359
$245$	−2.19673	−0.140344
$246$	−2.95917	−0.188670
$247$	−11.7662	−0.748666
$248$	0	0
$249$	12.2292	0.774992
$250$	−2.09217	−0.132321
$251$	−5.29608	−0.334285	−0.167143	−	0.985933i	$-0.553454\pi$
$251$	−5.29608	−0.334285	−0.167143	+	0.985933i	$0.553454\pi$
$252$	−4.76866	−0.300397
$253$	−37.3867	−2.35048
$254$	20.7831	1.30405
$255$	8.79051	0.550483
$256$	8.38529	0.524081
$257$	−7.33587	−0.457599	−0.228800	−	0.973474i	$-0.573480\pi$
$257$	−7.33587	−0.457599	−0.228800	+	0.973474i	$0.573480\pi$
$258$	−13.6438	−0.849426
$259$	−7.17455	−0.445805
$260$	−13.6955	−0.849360
$261$	3.80490	0.235517
$262$	12.4569	0.769590
$263$	17.6857	1.09055	0.545274	−	0.838258i	$-0.316426\pi$
$263$	17.6857	1.09055	0.545274	+	0.838258i	$0.316426\pi$
$264$	−7.36776	−0.453454
$265$	−2.87837	−0.176817
$266$	−12.9579	−0.794502
$267$	16.3442	1.00025
$268$	15.3245	0.936094
$269$	14.7535	0.899535	0.449767	−	0.893146i	$-0.351507\pi$
$269$	14.7535	0.899535	0.449767	+	0.893146i	$0.351507\pi$
$270$	−11.7145	−0.712923
$271$	−11.7078	−0.711200	−0.355600	−	0.934638i	$-0.615723\pi$
$271$	−11.7078	−0.711200	−0.355600	+	0.934638i	$0.615723\pi$
$272$	−17.8392	−1.08166
$273$	−26.7179	−1.61704
$274$	46.7437	2.82389
$275$	6.10532	0.368164
$276$	22.2610	1.33995
$277$	10.2231	0.614244	0.307122	−	0.951670i	$-0.400634\pi$
$277$	10.2231	0.614244	0.307122	+	0.951670i	$0.400634\pi$
$278$	14.1631	0.849448
$279$	0	0
$280$	−2.39317	−0.143019
$281$	−4.27915	−0.255272	−0.127636	−	0.991821i	$-0.540739\pi$
$281$	−4.27915	−0.255272	−0.127636	+	0.991821i	$0.540739\pi$
$282$	−4.99932	−0.297705
$283$	−20.5689	−1.22269	−0.611347	−	0.791363i	$-0.709372\pi$
$283$	−20.5689	−1.22269	−0.611347	+	0.791363i	$0.709372\pi$
$284$	2.27967	0.135274
$285$	−3.12315	−0.184999
$286$	73.5903	4.35149
$287$	2.80490	0.165568
$288$	5.33881	0.314593
$289$	16.0436	0.943740
$290$	12.0344	0.706685
$291$	−3.70186	−0.217007
$292$	2.56530	0.150123
$293$	−6.54979	−0.382643	−0.191321	−	0.981527i	$-0.561277\pi$
$293$	−6.54979	−0.382643	−0.191321	+	0.981527i	$0.561277\pi$
$294$	−7.02823	−0.409894
$295$	−0.194720	−0.0113370
$296$	−1.86696	−0.108515
$297$	34.1850	1.98361
$298$	−44.7845	−2.59429
$299$	−35.2796	−2.04027
$300$	−3.63525	−0.209881
$301$	12.9325	0.745419
$302$	7.11420	0.409376
$303$	1.79541	0.103143
$304$	6.33800	0.363509
$305$	−9.68296	−0.554445
$306$	−7.95530	−0.454774
$307$	−31.4457	−1.79470	−0.897351	−	0.441317i	$-0.854512\pi$
$307$	−31.4457	−1.79470	−0.897351	+	0.441317i	$0.854512\pi$
$308$	44.0138	2.50792
$309$	−18.4059	−1.04708
$310$	0	0
$311$	1.01917	0.0577920	0.0288960	−	0.999582i	$-0.490801\pi$
$311$	1.01917	0.0577920	0.0288960	+	0.999582i	$0.490801\pi$
$312$	−6.95251	−0.393609
$313$	1.77163	0.100138	0.0500692	−	0.998746i	$-0.484056\pi$
$313$	1.77163	0.100138	0.0500692	+	0.998746i	$0.484056\pi$
$314$	19.7017	1.11183
$315$	2.00601	0.113026
$316$	22.9309	1.28996
$317$	−20.3671	−1.14393	−0.571966	−	0.820277i	$-0.693819\pi$
$317$	−20.3671	−1.14393	−0.571966	+	0.820277i	$0.693819\pi$
$318$	−9.20906	−0.516418
$319$	−35.1185	−1.96626
$320$	10.6793	0.596991
$321$	5.58608	0.311784
$322$	−38.8529	−2.16519
$323$	−11.7399	−0.653227
$324$	−15.6372	−0.868732
$325$	5.76122	0.319575
$326$	−23.9275	−1.32522
$327$	4.46366	0.246841
$328$	0.729891	0.0403015
$329$	4.73870	0.261253
$330$	19.5333	1.07528
$331$	−17.9721	−0.987835	−0.493917	−	0.869509i	$-0.664436\pi$
$331$	−17.9721	−0.987835	−0.493917	+	0.869509i	$0.664436\pi$
$332$	−19.0103	−1.04333
$333$	1.56492	0.0857573
$334$	−16.1476	−0.883558
$335$	−6.44649	−0.352209
$336$	14.3919	0.785143
$337$	−9.06753	−0.493940	−0.246970	−	0.969023i	$-0.579435\pi$
$337$	−9.06753	−0.493940	−0.246970	+	0.969023i	$0.579435\pi$
$338$	42.2445	2.29780
$339$	−26.4244	−1.43518
$340$	−13.6649	−0.741084
$341$	0	0
$342$	2.82641	0.152835
$343$	−14.5664	−0.786514
$344$	3.36530	0.181445
$345$	−9.36440	−0.504163
$346$	−9.73289	−0.523244
$347$	9.60134	0.515427	0.257714	−	0.966221i	$-0.417031\pi$
$347$	9.60134	0.515427	0.257714	+	0.966221i	$0.417031\pi$
$348$	20.9104	1.12091
$349$	9.09027	0.486591	0.243295	−	0.969952i	$-0.421772\pi$
$349$	9.09027	0.486591	0.243295	+	0.969952i	$0.421772\pi$
$350$	6.34475	0.339141
$351$	32.2583	1.72182
$352$	−49.2762	−2.62643
$353$	1.21444	0.0646382	0.0323191	−	0.999478i	$-0.489711\pi$
$353$	1.21444	0.0646382	0.0323191	+	0.999478i	$0.489711\pi$
$354$	−0.622987	−0.0331114
$355$	−0.958977	−0.0508972
$356$	−25.4072	−1.34658
$357$	−26.6582	−1.41090
$358$	11.8927	0.628550
$359$	1.57185	0.0829593	0.0414797	−	0.999139i	$-0.486793\pi$
$359$	1.57185	0.0829593	0.0414797	+	0.999139i	$0.486793\pi$
$360$	0.522002	0.0275119
$361$	−14.8290	−0.780472
$362$	2.37821	0.124996
$363$	−40.1802	−2.10891
$364$	41.5332	2.17693
$365$	−1.07913	−0.0564844
$366$	−30.9796	−1.61933
$367$	−2.31454	−0.120818	−0.0604091	−	0.998174i	$-0.519241\pi$
$367$	−2.31454	−0.120818	−0.0604091	+	0.998174i	$0.519241\pi$
$368$	19.0038	0.990641
$369$	−0.611810	−0.0318496
$370$	4.94966	0.257321
$371$	8.72898	0.453186
$372$	0	0
$373$	−30.7941	−1.59446	−0.797230	−	0.603676i	$-0.793702\pi$
$373$	−30.7941	−1.59446	−0.797230	+	0.603676i	$0.793702\pi$
$374$	73.4259	3.79676
$375$	1.52922	0.0789687
$376$	1.23310	0.0635924
$377$	−33.1392	−1.70676
$378$	35.5256	1.82724
$379$	−1.99947	−0.102706	−0.0513528	−	0.998681i	$-0.516353\pi$
$379$	−1.99947	−0.102706	−0.0513528	+	0.998681i	$0.516353\pi$
$380$	4.85496	0.249054
$381$	−15.1909	−0.778252
$382$	−40.6004	−2.07730
$383$	−15.2148	−0.777441	−0.388721	−	0.921356i	$-0.627083\pi$
$383$	−15.2148	−0.777441	−0.388721	+	0.921356i	$0.627083\pi$
$384$	9.48253	0.483903
$385$	−18.5151	−0.943615
$386$	−33.8213	−1.72146
$387$	−2.82087	−0.143393
$388$	5.75456	0.292144
$389$	22.0976	1.12039	0.560195	−	0.828361i	$-0.310726\pi$
$389$	22.0976	1.12039	0.560195	+	0.828361i	$0.310726\pi$
$390$	18.4325	0.933364
$391$	−35.2008	−1.78018
$392$	1.73354	0.0875570
$393$	−9.10507	−0.459290
$394$	16.1877	0.815526
$395$	−9.64623	−0.485355
$396$	−9.60036	−0.482436
$397$	−5.91908	−0.297070	−0.148535	−	0.988907i	$-0.547456\pi$
$397$	−5.91908	−0.297070	−0.148535	+	0.988907i	$0.547456\pi$
$398$	−55.5142	−2.78267
$399$	9.47129	0.474158
$400$	−3.10335	−0.155168
$401$	−22.0363	−1.10044	−0.550220	−	0.835020i	$-0.685456\pi$
$401$	−22.0363	−1.10044	−0.550220	+	0.835020i	$0.685456\pi$
$402$	−20.6249	−1.02868
$403$	0	0
$404$	−2.79097	−0.138856
$405$	6.57801	0.326864
$406$	−36.4957	−1.81125
$407$	−14.4440	−0.715960
$408$	−6.93698	−0.343432
$409$	31.5501	1.56005	0.780026	−	0.625748i	$-0.215206\pi$
$409$	31.5501	1.56005	0.780026	+	0.625748i	$0.215206\pi$
$410$	−1.93508	−0.0955668
$411$	−34.1662	−1.68529
$412$	28.6121	1.40962
$413$	0.590510	0.0290571
$414$	8.47466	0.416507
$415$	7.99699	0.392557
$416$	−46.4990	−2.27980
$417$	−10.3522	−0.506949
$418$	−26.0872	−1.27597
$419$	4.23282	0.206787	0.103393	−	0.994641i	$-0.467030\pi$
$419$	4.23282	0.206787	0.103393	+	0.994641i	$0.467030\pi$
$420$	11.0243	0.537931
$421$	0.821194	0.0400225	0.0200113	−	0.999800i	$-0.493630\pi$
$421$	0.821194	0.0400225	0.0200113	+	0.999800i	$0.493630\pi$
$422$	34.0022	1.65520
$423$	−1.03361	−0.0502560
$424$	2.27145	0.110311
$425$	5.74835	0.278836
$426$	−3.06815	−0.148652
$427$	29.3647	1.42106
$428$	−8.68360	−0.419738
$429$	−53.7890	−2.59696
$430$	−8.92205	−0.430259
$431$	36.7895	1.77209	0.886044	−	0.463602i	$-0.153443\pi$
$431$	36.7895	1.77209	0.886044	+	0.463602i	$0.153443\pi$
$432$	−17.3763	−0.836019
$433$	8.37708	0.402577	0.201288	−	0.979532i	$-0.435487\pi$
$433$	8.37708	0.402577	0.201288	+	0.979532i	$0.435487\pi$
$434$	0	0
$435$	−8.79626	−0.421749
$436$	−6.93880	−0.332308
$437$	12.5064	0.598261
$438$	−3.45258	−0.164971
$439$	−38.3505	−1.83037	−0.915184	−	0.403036i	$-0.867955\pi$
$439$	−38.3505	−1.83037	−0.915184	+	0.403036i	$0.867955\pi$
$440$	−4.81798	−0.229688
$441$	−1.45309	−0.0691948
$442$	69.2877	3.29568
$443$	13.8338	0.657263	0.328631	−	0.944458i	$-0.393413\pi$
$443$	13.8338	0.657263	0.328631	+	0.944458i	$0.393413\pi$
$444$	8.60028	0.408151
$445$	10.6879	0.506655
$446$	10.1641	0.481284
$447$	32.7341	1.54827
$448$	−32.3862	−1.53010
$449$	−23.7051	−1.11871	−0.559357	−	0.828927i	$-0.688952\pi$
$449$	−23.7051	−1.11871	−0.559357	+	0.828927i	$0.688952\pi$
$450$	−1.38393	−0.0652389
$451$	5.64689	0.265902
$452$	41.0769	1.93210
$453$	−5.19995	−0.244315
$454$	10.2205	0.479673
$455$	−17.4716	−0.819079
$456$	2.46461	0.115416
$457$	29.2220	1.36695	0.683474	−	0.729975i	$-0.260468\pi$
$457$	29.2220	1.36695	0.683474	+	0.729975i	$0.260468\pi$
$458$	−7.16329	−0.334718
$459$	32.1863	1.50233
$460$	14.5570	0.678726
$461$	23.4272	1.09111	0.545556	−	0.838074i	$-0.316318\pi$
$461$	23.4272	1.09111	0.545556	+	0.838074i	$0.316318\pi$
$462$	−59.2370	−2.75596
$463$	−24.0254	−1.11655	−0.558277	−	0.829655i	$-0.688537\pi$
$463$	−24.0254	−1.11655	−0.558277	+	0.829655i	$0.688537\pi$
$464$	17.8508	0.828704
$465$	0	0
$466$	31.1967	1.44516
$467$	−22.4026	−1.03667	−0.518334	−	0.855178i	$-0.673448\pi$
$467$	−22.4026	−1.03667	−0.518334	+	0.855178i	$0.673448\pi$
$468$	−9.05928	−0.418766
$469$	19.5497	0.902721
$470$	−3.26919	−0.150797
$471$	−14.4004	−0.663537
$472$	0.153662	0.00707288
$473$	26.0361	1.19714
$474$	−30.8621	−1.41755
$475$	−2.04231	−0.0937076
$476$	41.4404	1.89942
$477$	−1.90398	−0.0871772
$478$	15.6955	0.717894
$479$	22.1983	1.01427	0.507134	−	0.861867i	$-0.330705\pi$
$479$	22.1983	1.01427	0.507134	+	0.861867i	$0.330705\pi$
$480$	−12.3424	−0.563351
$481$	−13.6299	−0.621470
$482$	12.7567	0.581053
$483$	28.3986	1.29218
$484$	62.4604	2.83911
$485$	−2.42074	−0.109920
$486$	−14.0979	−0.639494
$487$	20.9757	0.950498	0.475249	−	0.879851i	$-0.342358\pi$
$487$	20.9757	0.950498	0.475249	+	0.879851i	$0.342358\pi$
$488$	7.64125	0.345903
$489$	17.4892	0.790889
$490$	−4.59595	−0.207624
$491$	−22.7700	−1.02760	−0.513798	−	0.857911i	$-0.671762\pi$
$491$	−22.7700	−1.02760	−0.513798	+	0.857911i	$0.671762\pi$
$492$	−3.36229	−0.151584
$493$	−33.0652	−1.48918
$494$	−24.6169	−1.10757
$495$	4.03853	0.181519
$496$	0	0
$497$	2.90821	0.130451
$498$	25.5855	1.14652
$499$	−6.78106	−0.303562	−0.151781	−	0.988414i	$-0.548501\pi$
$499$	−6.78106	−0.303562	−0.151781	+	0.988414i	$0.548501\pi$
$500$	−2.37719	−0.106311
$501$	11.8027	0.527306
$502$	−11.0803	−0.494538
$503$	−18.6512	−0.831616	−0.415808	−	0.909452i	$-0.636501\pi$
$503$	−18.6512	−0.831616	−0.415808	+	0.909452i	$0.636501\pi$
$504$	−1.58303	−0.0705137
$505$	1.17406	0.0522452
$506$	−78.2195	−3.47728
$507$	−30.8776	−1.37132
$508$	23.6143	1.04772
$509$	19.3528	0.857796	0.428898	−	0.903353i	$-0.358902\pi$
$509$	19.3528	0.857796	0.428898	+	0.903353i	$0.358902\pi$
$510$	18.3913	0.814379
$511$	3.27259	0.144771
$512$	29.9452	1.32341
$513$	−11.4353	−0.504882
$514$	−15.3479	−0.676968
$515$	−12.0361	−0.530375
$516$	−15.5025	−0.682459
$517$	9.54005	0.419571
$518$	−15.0104	−0.659519
$519$	7.11402	0.312271
$520$	−4.54644	−0.199374
$521$	−35.7981	−1.56834	−0.784172	−	0.620543i	$-0.786912\pi$
$521$	−35.7981	−1.56834	−0.784172	+	0.620543i	$0.786912\pi$
$522$	7.96050	0.348422
$523$	16.5770	0.724861	0.362431	−	0.932011i	$-0.381947\pi$
$523$	16.5770	0.724861	0.362431	+	0.932011i	$0.381947\pi$
$524$	14.1539	0.618316
$525$	−4.63754	−0.202399
$526$	37.0016	1.61334
$527$	0	0
$528$	28.9741	1.26094
$529$	14.4989	0.630388
$530$	−6.02205	−0.261581
$531$	−0.128803	−0.00558957
$532$	−14.7232	−0.638332
$533$	5.32864	0.230809
$534$	34.1949	1.47976
$535$	3.65289	0.157928
$536$	5.08721	0.219734
$537$	−8.69270	−0.375118
$538$	30.8668	1.33076
$539$	13.4118	0.577685
$540$	−13.3104	−0.572788
$541$	−20.0984	−0.864096	−0.432048	−	0.901851i	$-0.642209\pi$
$541$	−20.0984	−0.864096	−0.432048	+	0.901851i	$0.642209\pi$
$542$	−24.4948	−1.05214
$543$	−1.73829	−0.0745973
$544$	−46.3952	−1.98918
$545$	2.91891	0.125032
$546$	−55.8985	−2.39223
$547$	10.4942	0.448700	0.224350	−	0.974509i	$-0.427974\pi$
$547$	10.4942	0.448700	0.224350	+	0.974509i	$0.427974\pi$
$548$	53.1116	2.26881
$549$	−6.40507	−0.273362
$550$	12.7734	0.544659
$551$	11.7476	0.500465
$552$	7.38987	0.314534
$553$	29.2533	1.24398
$554$	21.3884	0.908707
$555$	−3.61783	−0.153569
$556$	16.0926	0.682476
$557$	−30.1350	−1.27686	−0.638431	−	0.769679i	$-0.720416\pi$
$557$	−30.1350	−1.27686	−0.638431	+	0.769679i	$0.720416\pi$
$558$	0	0
$559$	24.5687	1.03914
$560$	9.41126	0.397698
$561$	−53.6689	−2.26590
$562$	−8.95271	−0.377647
$563$	41.1744	1.73530	0.867648	−	0.497180i	$-0.165631\pi$
$563$	41.1744	1.73530	0.867648	+	0.497180i	$0.165631\pi$
$564$	−5.68038	−0.239187
$565$	−17.2796	−0.726959
$566$	−43.0337	−1.80884
$567$	−19.9486	−0.837761
$568$	0.756772	0.0317534
$569$	38.1405	1.59893	0.799466	−	0.600711i	$-0.205116\pi$
$569$	38.1405	1.59893	0.799466	+	0.600711i	$0.205116\pi$
$570$	−6.53416	−0.273686
$571$	−12.7050	−0.531688	−0.265844	−	0.964016i	$-0.585651\pi$
$571$	−12.7050	−0.531688	−0.265844	+	0.964016i	$0.585651\pi$
$572$	83.6155	3.49614
$573$	29.6759	1.23973
$574$	5.86835	0.244940
$575$	−6.12364	−0.255373
$576$	7.06413	0.294339
$577$	−9.63495	−0.401108	−0.200554	−	0.979683i	$-0.564274\pi$
$577$	−9.63495	−0.401108	−0.200554	+	0.979683i	$0.564274\pi$
$578$	33.5659	1.39616
$579$	24.7208	1.02736
$580$	13.6739	0.567776
$581$	−24.2518	−1.00613
$582$	−7.74492	−0.321037
$583$	17.5734	0.727815
$584$	0.851592	0.0352391
$585$	3.81092	0.157562
$586$	−13.7033	−0.566078
$587$	−36.4858	−1.50593	−0.752966	−	0.658060i	$-0.771378\pi$
$587$	−36.4858	−1.50593	−0.752966	+	0.658060i	$0.771378\pi$
$588$	−7.98568	−0.329324
$589$	0	0
$590$	−0.407388	−0.0167719
$591$	−11.8320	−0.486704
$592$	7.34191	0.301751
$593$	−28.0679	−1.15261	−0.576305	−	0.817234i	$-0.695506\pi$
$593$	−28.0679	−1.15261	−0.576305	+	0.817234i	$0.695506\pi$
$594$	71.5209	2.93454
$595$	−17.4325	−0.714664
$596$	−50.8854	−2.08435
$597$	40.5767	1.66070
$598$	−73.8111	−3.01836
$599$	−13.9582	−0.570319	−0.285159	−	0.958480i	$-0.592047\pi$
$599$	−13.9582	−0.570319	−0.285159	+	0.958480i	$0.592047\pi$
$600$	−1.20678	−0.0492665
$601$	−28.8591	−1.17719	−0.588593	−	0.808429i	$-0.700318\pi$
$601$	−28.8591	−1.17719	−0.588593	+	0.808429i	$0.700318\pi$
$602$	27.0571	1.10277
$603$	−4.26421	−0.173652
$604$	8.08337	0.328907
$605$	−26.2749	−1.06823
$606$	3.75630	0.152589
$607$	16.1794	0.656701	0.328350	−	0.944556i	$-0.393507\pi$
$607$	16.1794	0.656701	0.328350	+	0.944556i	$0.393507\pi$
$608$	16.4836	0.668497
$609$	26.6757	1.08095
$610$	−20.2584	−0.820240
$611$	9.00238	0.364197
$612$	−9.03905	−0.365382
$613$	−10.0615	−0.406379	−0.203189	−	0.979139i	$-0.565131\pi$
$613$	−10.0615	−0.406379	−0.203189	+	0.979139i	$0.565131\pi$
$614$	−65.7899	−2.65507
$615$	1.41440	0.0570341
$616$	14.6111	0.588696
$617$	31.6993	1.27616	0.638082	−	0.769968i	$-0.279728\pi$
$617$	31.6993	1.27616	0.638082	+	0.769968i	$0.279728\pi$
$618$	−38.5084	−1.54903
$619$	8.46480	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$619$	8.46480	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$620$	0	0
$621$	−34.2876	−1.37591
$622$	2.13229	0.0854970
$623$	−32.4122	−1.29857
$624$	27.3411	1.09452
$625$	1.00000	0.0400000
$626$	3.70656	0.148144
$627$	19.0678	0.761495
$628$	22.3856	0.893283
$629$	−13.5995	−0.542246
$630$	4.19691	0.167209
$631$	46.3316	1.84443	0.922216	−	0.386675i	$-0.126376\pi$
$631$	46.3316	1.84443	0.922216	+	0.386675i	$0.126376\pi$
$632$	7.61227	0.302800
$633$	−24.8530	−0.987820
$634$	−42.6116	−1.69232
$635$	−9.93372	−0.394208
$636$	−10.4636	−0.414909
$637$	12.6559	0.501444
$638$	−73.4739	−2.90886
$639$	−0.634343	−0.0250942
$640$	6.20088	0.245111
$641$	−27.1226	−1.07128	−0.535639	−	0.844447i	$-0.679929\pi$
$641$	−27.1226	−1.07128	−0.535639	+	0.844447i	$0.679929\pi$
$642$	11.6870	0.461251
$643$	−3.28555	−0.129569	−0.0647847	−	0.997899i	$-0.520636\pi$
$643$	−3.28555	−0.129569	−0.0647847	+	0.997899i	$0.520636\pi$
$644$	−44.1459	−1.73959
$645$	6.52135	0.256778
$646$	−24.5619	−0.966377
$647$	−49.6701	−1.95273	−0.976366	−	0.216125i	$-0.930658\pi$
$647$	−49.6701	−1.95273	−0.976366	+	0.216125i	$0.930658\pi$
$648$	−5.19100	−0.203922
$649$	1.18883	0.0466656
$650$	12.0535	0.472776
$651$	0	0
$652$	−27.1871	−1.06473
$653$	−10.9619	−0.428974	−0.214487	−	0.976727i	$-0.568808\pi$
$653$	−10.9619	−0.428974	−0.214487	+	0.976727i	$0.568808\pi$
$654$	9.33875	0.365174
$655$	−5.95405	−0.232644
$656$	−2.87033	−0.112068
$657$	−0.713823	−0.0278489
$658$	9.91419	0.386495
$659$	44.7513	1.74326	0.871631	−	0.490162i	$-0.163062\pi$
$659$	44.7513	1.74326	0.871631	+	0.490162i	$0.163062\pi$
$660$	22.1944	0.863914
$661$	21.3978	0.832279	0.416140	−	0.909301i	$-0.363383\pi$
$661$	21.3978	0.832279	0.416140	+	0.909301i	$0.363383\pi$
$662$	−37.6007	−1.46139
$663$	−50.6441	−1.96685
$664$	−6.31078	−0.244906
$665$	6.19353	0.240175
$666$	3.27409	0.126869
$667$	35.2239	1.36387
$668$	−18.3474	−0.709882
$669$	−7.42920	−0.287229
$670$	−13.4872	−0.521055
$671$	59.1175	2.28221
$672$	37.4297	1.44388
$673$	38.2044	1.47267	0.736335	−	0.676617i	$-0.236555\pi$
$673$	38.2044	1.47267	0.736335	+	0.676617i	$0.236555\pi$
$674$	−18.9708	−0.730729
$675$	5.59922	0.215514
$676$	47.9995	1.84613
$677$	−37.6813	−1.44821	−0.724105	−	0.689690i	$-0.757747\pi$
$677$	−37.6813	−1.44821	−0.724105	+	0.689690i	$0.757747\pi$
$678$	−55.2844	−2.12318
$679$	7.34117	0.281728
$680$	−4.53628	−0.173958
$681$	−7.47045	−0.286268
$682$	0	0
$683$	1.48318	0.0567524	0.0283762	−	0.999597i	$-0.490966\pi$
$683$	1.48318	0.0567524	0.0283762	+	0.999597i	$0.490966\pi$
$684$	3.21145	0.122793
$685$	−22.3422	−0.853650
$686$	−30.4755	−1.16356
$687$	5.23583	0.199759
$688$	−13.2342	−0.504549
$689$	16.5829	0.631760
$690$	−19.5920	−0.745853
$691$	32.8917	1.25126	0.625630	−	0.780120i	$-0.284842\pi$
$691$	32.8917	1.25126	0.625630	+	0.780120i	$0.284842\pi$
$692$	−11.0588	−0.420392
$693$	−12.2473	−0.465237
$694$	20.0877	0.762518
$695$	−6.76957	−0.256785
$696$	6.94152	0.263118
$697$	5.31673	0.201386
$698$	19.0184	0.719858
$699$	−22.8025	−0.862468
$700$	7.20909	0.272478
$701$	3.37012	0.127288	0.0636438	−	0.997973i	$-0.479728\pi$
$701$	3.37012	0.127288	0.0636438	+	0.997973i	$0.479728\pi$
$702$	67.4900	2.54725
$703$	4.83170	0.182231
$704$	−65.2005	−2.45734
$705$	2.38953	0.0899951
$706$	2.54082	0.0956251
$707$	−3.56048	−0.133906
$708$	−0.707856	−0.0266029
$709$	14.6848	0.551498	0.275749	−	0.961230i	$-0.411074\pi$
$709$	14.6848	0.551498	0.275749	+	0.961230i	$0.411074\pi$
$710$	−2.00635	−0.0752969
$711$	−6.38077	−0.239298
$712$	−8.43430	−0.316089
$713$	0	0
$714$	−55.7736	−2.08727
$715$	−35.1741	−1.31544
$716$	13.5129	0.505000
$717$	−11.4722	−0.428438
$718$	3.28859	0.122729
$719$	−53.3561	−1.98985	−0.994924	−	0.100628i	$-0.967915\pi$
$719$	−53.3561	−1.98985	−0.994924	+	0.100628i	$0.967915\pi$
$720$	−2.05280	−0.0765033
$721$	36.5009	1.35936
$722$	−31.0248	−1.15462
$723$	−9.32422	−0.346771
$724$	2.70219	0.100426
$725$	−5.75211	−0.213628
$726$	−84.0638	−3.11990
$727$	25.0988	0.930864	0.465432	−	0.885084i	$-0.345899\pi$
$727$	25.0988	0.930864	0.465432	+	0.885084i	$0.345899\pi$
$728$	13.7876	0.511002
$729$	30.0386	1.11254
$730$	−2.25773	−0.0835625
$731$	24.5138	0.906675
$732$	−35.2000	−1.30103
$733$	−51.4926	−1.90192	−0.950961	−	0.309310i	$-0.899902\pi$
$733$	−51.4926	−1.90192	−0.950961	+	0.309310i	$0.899902\pi$
$734$	−4.84242	−0.178737
$735$	3.35929	0.123909
$736$	49.4241	1.82180
$737$	39.3578	1.44976
$738$	−1.28001	−0.0471179
$739$	38.4925	1.41597	0.707985	−	0.706228i	$-0.249604\pi$
$739$	38.4925	1.41597	0.707985	+	0.706228i	$0.249604\pi$
$740$	5.62395	0.206741
$741$	17.9931	0.660995
$742$	18.2625	0.670439
$743$	14.9183	0.547299	0.273650	−	0.961829i	$-0.411769\pi$
$743$	14.9183	0.547299	0.273650	+	0.961829i	$0.411769\pi$
$744$	0	0
$745$	21.4057	0.784245
$746$	−64.4266	−2.35883
$747$	5.28983	0.193545
$748$	83.4287	3.05045
$749$	−11.0778	−0.404773
$750$	3.19940	0.116826
$751$	4.41599	0.161142	0.0805709	−	0.996749i	$-0.474326\pi$
$751$	4.41599	0.161142	0.0805709	+	0.996749i	$0.474326\pi$
$752$	−4.84924	−0.176834
$753$	8.09888	0.295140
$754$	−69.3330	−2.52496
$755$	−3.40039	−0.123753
$756$	40.3652	1.46807
$757$	−26.0148	−0.945525	−0.472763	−	0.881190i	$-0.656743\pi$
$757$	−26.0148	−0.945525	−0.472763	+	0.881190i	$0.656743\pi$
$758$	−4.18323	−0.151942
$759$	57.1727	2.07524
$760$	1.61168	0.0584617
$761$	−40.9330	−1.48382	−0.741911	−	0.670499i	$-0.766080\pi$
$761$	−40.9330	−1.48382	−0.741911	+	0.670499i	$0.766080\pi$
$762$	−31.7819	−1.15134
$763$	−8.85192	−0.320461
$764$	−46.1314	−1.66898
$765$	3.80241	0.137476
$766$	−31.8321	−1.15014
$767$	1.12183	0.0405068
$768$	−12.8230	−0.462709
$769$	27.6555	0.997282	0.498641	−	0.866809i	$-0.333833\pi$
$769$	27.6555	0.997282	0.498641	+	0.866809i	$0.333833\pi$
$770$	−38.7367	−1.39597
$771$	11.2182	0.404013
$772$	−38.4287	−1.38308
$773$	−31.9791	−1.15021	−0.575103	−	0.818081i	$-0.695038\pi$
$773$	−31.9791	−1.15021	−0.575103	+	0.818081i	$0.695038\pi$
$774$	−5.90174	−0.212134
$775$	0	0
$776$	1.91032	0.0685763
$777$	10.9715	0.393600
$778$	46.2319	1.65749
$779$	−1.88896	−0.0676791
$780$	20.9435	0.749897
$781$	5.85486	0.209503
$782$	−73.6462	−2.63358
$783$	−32.2073	−1.15100
$784$	−6.81723	−0.243473
$785$	−9.41684	−0.336101
$786$	−19.0494	−0.679469
$787$	41.9937	1.49692	0.748458	−	0.663183i	$-0.230795\pi$
$787$	41.9937	1.49692	0.748458	+	0.663183i	$0.230795\pi$
$788$	18.3930	0.655222
$789$	−27.0454	−0.962841
$790$	−20.1816	−0.718029
$791$	52.4024	1.86321
$792$	−3.18699	−0.113245
$793$	55.7857	1.98101
$794$	−12.3837	−0.439482
$795$	4.40167	0.156111
$796$	−63.0769	−2.23570
$797$	−23.3624	−0.827540	−0.413770	−	0.910381i	$-0.635788\pi$
$797$	−23.3624	−0.827540	−0.413770	+	0.910381i	$0.635788\pi$
$798$	19.8156	0.701464
$799$	8.98227	0.317770
$800$	−8.07104	−0.285354
$801$	7.06981	0.249799
$802$	−46.1037	−1.62798
$803$	6.58845	0.232501
$804$	−23.4346	−0.826475
$805$	18.5706	0.654528
$806$	0	0
$807$	−22.5613	−0.794197
$808$	−0.926506	−0.0325944
$809$	−49.4055	−1.73701	−0.868503	−	0.495685i	$-0.834917\pi$
$809$	−49.4055	−1.73701	−0.868503	+	0.495685i	$0.834917\pi$
$810$	13.7623	0.483559
$811$	30.2735	1.06305	0.531524	−	0.847043i	$-0.321620\pi$
$811$	30.2735	1.06305	0.531524	+	0.847043i	$0.321620\pi$
$812$	−41.4675	−1.45522
$813$	17.9039	0.627916
$814$	−30.2193	−1.05918
$815$	11.4367	0.400609
$816$	27.2800	0.954993
$817$	−8.70941	−0.304704
$818$	66.0082	2.30792
$819$	−11.5570	−0.403836
$820$	−2.19870	−0.0767818
$821$	−11.4745	−0.400463	−0.200231	−	0.979749i	$-0.564169\pi$
$821$	−11.4745	−0.400463	−0.200231	+	0.979749i	$0.564169\pi$
$822$	−71.4815	−2.49320
$823$	−19.0833	−0.665202	−0.332601	−	0.943068i	$-0.607926\pi$
$823$	−19.0833	−0.665202	−0.332601	+	0.943068i	$0.607926\pi$
$824$	9.49824	0.330887
$825$	−9.33639	−0.325051
$826$	1.23545	0.0429868
$827$	15.7481	0.547614	0.273807	−	0.961785i	$-0.411717\pi$
$827$	15.7481	0.547614	0.273807	+	0.961785i	$0.411717\pi$
$828$	9.62916	0.334637
$829$	3.00255	0.104283	0.0521414	−	0.998640i	$-0.483395\pi$
$829$	3.00255	0.104283	0.0521414	+	0.998640i	$0.483395\pi$
$830$	16.7311	0.580744
$831$	−15.6333	−0.542315
$832$	−61.5259	−2.13303
$833$	12.6276	0.437520
$834$	−21.6586	−0.749975
$835$	7.71810	0.267096
$836$	−29.6410	−1.02516
$837$	0	0
$838$	8.85580	0.305918
$839$	−38.3073	−1.32252	−0.661258	−	0.750159i	$-0.729977\pi$
$839$	−38.3073	−1.32252	−0.661258	+	0.750159i	$0.729977\pi$
$840$	3.65969	0.126271
$841$	4.08682	0.140925
$842$	1.71808	0.0592089
$843$	6.54377	0.225379
$844$	38.6343	1.32985
$845$	−20.1917	−0.694616
$846$	−2.16250	−0.0743482
$847$	79.6815	2.73789
$848$	−8.93260	−0.306747
$849$	31.4544	1.07951
$850$	12.0266	0.412507
$851$	14.4873	0.496618
$852$	−3.48612	−0.119433
$853$	11.1609	0.382142	0.191071	−	0.981576i	$-0.438804\pi$
$853$	11.1609	0.382142	0.191071	+	0.981576i	$0.438804\pi$
$854$	61.4359	2.10229
$855$	−1.35094	−0.0462013
$856$	−2.88265	−0.0985271
$857$	−39.2227	−1.33982	−0.669911	−	0.742441i	$-0.733668\pi$
$857$	−39.2227	−1.33982	−0.669911	+	0.742441i	$0.733668\pi$
$858$	−112.536	−3.84191
$859$	28.2330	0.963297	0.481648	−	0.876365i	$-0.340038\pi$
$859$	28.2330	0.963297	0.481648	+	0.876365i	$0.340038\pi$
$860$	−10.1375	−0.345686
$861$	−4.28932	−0.146180
$862$	76.9700	2.62161
$863$	27.9495	0.951412	0.475706	−	0.879604i	$-0.342193\pi$
$863$	27.9495	0.951412	0.475706	+	0.879604i	$0.342193\pi$
$864$	−45.1915	−1.53745
$865$	4.65205	0.158174
$866$	17.5263	0.595568
$867$	−24.5342	−0.833225
$868$	0	0
$869$	58.8933	1.99782
$870$	−18.4033	−0.623931
$871$	37.1397	1.25843
$872$	−2.30344	−0.0780043
$873$	−1.60127	−0.0541947
$874$	26.1655	0.885061
$875$	−3.03261	−0.102521
$876$	−3.92292	−0.132543
$877$	−41.9821	−1.41763	−0.708817	−	0.705392i	$-0.750771\pi$
$877$	−41.9821	−1.41763	−0.708817	+	0.705392i	$0.750771\pi$
$878$	−80.2358	−2.70783
$879$	10.0161	0.337834
$880$	18.9469	0.638701
$881$	52.1568	1.75721	0.878603	−	0.477553i	$-0.158476\pi$
$881$	52.1568	1.75721	0.878603	+	0.477553i	$0.158476\pi$
$882$	−3.04012	−0.102366
$883$	7.94214	0.267274	0.133637	−	0.991030i	$-0.457334\pi$
$883$	7.94214	0.267274	0.133637	+	0.991030i	$0.457334\pi$
$884$	78.7267	2.64786
$885$	0.297770	0.0100094
$886$	28.9427	0.972348
$887$	8.05328	0.270403	0.135201	−	0.990818i	$-0.456832\pi$
$887$	8.05328	0.270403	0.135201	+	0.990818i	$0.456832\pi$
$888$	2.85499	0.0958073
$889$	30.1251	1.01036
$890$	22.3609	0.749540
$891$	−40.1608	−1.34544
$892$	11.5487	0.386681
$893$	−3.19128	−0.106792
$894$	68.4854	2.29049
$895$	−5.68439	−0.190008
$896$	−18.8049	−0.628227
$897$	53.9504	1.80135
$898$	−49.5952	−1.65501
$899$	0	0
$900$	−1.57246	−0.0524153
$901$	16.5459	0.551224
$902$	11.8143	0.393372
$903$	−19.7767	−0.658128
$904$	13.6361	0.453530
$905$	−1.13672	−0.0377857
$906$	−10.8792	−0.361437
$907$	−21.5284	−0.714839	−0.357419	−	0.933944i	$-0.616343\pi$
$907$	−21.5284	−0.714839	−0.357419	+	0.933944i	$0.616343\pi$
$908$	11.6129	0.385387
$909$	0.776617	0.0257588
$910$	−36.5535	−1.21174
$911$	5.48759	0.181812	0.0909059	−	0.995859i	$-0.471024\pi$
$911$	5.48759	0.181812	0.0909059	+	0.995859i	$0.471024\pi$
$912$	−9.69222	−0.320941
$913$	−48.8241	−1.61584
$914$	61.1375	2.02225
$915$	14.8074	0.489518
$916$	−8.13914	−0.268925
$917$	18.0563	0.596272
$918$	67.3393	2.22253
$919$	−43.0407	−1.41978	−0.709891	−	0.704312i	$-0.751256\pi$
$919$	−43.0407	−1.41978	−0.709891	+	0.704312i	$0.751256\pi$
$920$	4.83243	0.159321
$921$	48.0875	1.58454
$922$	49.0137	1.61418
$923$	5.52488	0.181854
$924$	−67.3069	−2.21423
$925$	−2.36580	−0.0777870
$926$	−50.2653	−1.65182
$927$	−7.96163	−0.261494
$928$	46.4255	1.52399
$929$	−1.87481	−0.0615104	−0.0307552	−	0.999527i	$-0.509791\pi$
$929$	−1.87481	−0.0615104	−0.0307552	+	0.999527i	$0.509791\pi$
$930$	0	0
$931$	−4.48641	−0.147036
$932$	35.4466	1.16109
$933$	−1.55854	−0.0510244
$934$	−46.8701	−1.53364
$935$	−35.0955	−1.14775
$936$	−3.00737	−0.0982989
$937$	−54.4428	−1.77857	−0.889285	−	0.457353i	$-0.848798\pi$
$937$	−54.4428	−1.77857	−0.889285	+	0.457353i	$0.848798\pi$
$938$	40.9013	1.33548
$939$	−2.70922	−0.0884119
$940$	−3.71455	−0.121155
$941$	24.7622	0.807226	0.403613	−	0.914930i	$-0.367754\pi$
$941$	24.7622	0.807226	0.403613	+	0.914930i	$0.367754\pi$
$942$	−30.1282	−0.981630
$943$	−5.66384	−0.184440
$944$	−0.604285	−0.0196678
$945$	−16.9802	−0.552367
$946$	54.4719	1.77104
$947$	−14.4139	−0.468389	−0.234194	−	0.972190i	$-0.575245\pi$
$947$	−14.4139	−0.468389	−0.234194	+	0.972190i	$0.575245\pi$
$948$	−35.0665	−1.13891
$949$	6.21713	0.201817
$950$	−4.27287	−0.138630
$951$	31.1459	1.00997
$952$	13.7568	0.445860
$953$	−28.7339	−0.930782	−0.465391	−	0.885105i	$-0.654086\pi$
$953$	−28.7339	−0.930782	−0.465391	+	0.885105i	$0.654086\pi$
$954$	−3.98346	−0.128969
$955$	19.4059	0.627959
$956$	17.8337	0.576782
$957$	53.7040	1.73600
$958$	46.4428	1.50050
$959$	67.7551	2.18793
$960$	−16.3310	−0.527082
$961$	0	0
$962$	−28.5161	−0.919397
$963$	2.41630	0.0778643
$964$	14.4946	0.466839
$965$	16.1656	0.520390
$966$	59.4148	1.91164
$967$	−29.9451	−0.962970	−0.481485	−	0.876454i	$-0.659902\pi$
$967$	−29.9451	−0.962970	−0.481485	+	0.876454i	$0.659902\pi$
$968$	20.7347	0.666438
$969$	17.9530	0.576732
$970$	−5.06461	−0.162615
$971$	0.308178	0.00988990	0.00494495	−	0.999988i	$-0.498426\pi$
$971$	0.308178	0.00988990	0.00494495	+	0.999988i	$0.498426\pi$
$972$	−16.0184	−0.513792
$973$	20.5295	0.658145
$974$	43.8847	1.40616
$975$	−8.81019	−0.282152
$976$	−30.0496	−0.961865
$977$	−27.3170	−0.873947	−0.436974	−	0.899474i	$-0.643950\pi$
$977$	−27.3170	−0.873947	−0.436974	+	0.899474i	$0.643950\pi$
$978$	36.5904	1.17003
$979$	−65.2530	−2.08550
$980$	−5.22205	−0.166812
$981$	1.93079	0.0616455
$982$	−47.6388	−1.52022
$983$	−14.6961	−0.468734	−0.234367	−	0.972148i	$-0.575302\pi$
$983$	−14.6961	−0.468734	−0.234367	+	0.972148i	$0.575302\pi$
$984$	−1.11617	−0.0355821
$985$	−7.73728	−0.246530
$986$	−69.1781	−2.20308
$987$	−7.24653	−0.230660
$988$	−27.9705	−0.889860
$989$	−26.1142	−0.830383
$990$	8.44931	0.268537
$991$	−23.9105	−0.759542	−0.379771	−	0.925081i	$-0.623997\pi$
$991$	−23.9105	−0.759542	−0.379771	+	0.925081i	$0.623997\pi$
$992$	0	0
$993$	27.4833	0.872157
$994$	6.08447	0.192988
$995$	26.5342	0.841191
$996$	29.0710	0.921151
$997$	23.2785	0.737236	0.368618	−	0.929581i	$-0.379831\pi$
$997$	23.2785	0.737236	0.368618	+	0.929581i	$0.379831\pi$
$998$	−14.1871	−0.449086
$999$	−13.2466	−0.419105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4805.2.a.y.1.17		20
31.12	odd	30		155.2.q.a.51.5	✓	40
31.13	odd	30		155.2.q.a.76.5	yes	40
31.30	odd	2		4805.2.a.x.1.17		20
155.12	even	60		775.2.ck.c.299.1		80
155.13	even	60		775.2.ck.c.324.1		80
155.43	even	60		775.2.ck.c.299.10		80
155.44	odd	30		775.2.bl.c.76.1		40
155.74	odd	30		775.2.bl.c.51.1		40
155.137	even	60		775.2.ck.c.324.10		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.q.a.51.5	✓	40	31.12	odd	30
155.2.q.a.76.5	yes	40	31.13	odd	30
775.2.bl.c.51.1		40	155.74	odd	30
775.2.bl.c.76.1		40	155.44	odd	30
775.2.ck.c.299.1		80	155.12	even	60
775.2.ck.c.299.10		80	155.43	even	60
775.2.ck.c.324.1		80	155.13	even	60
775.2.ck.c.324.10		80	155.137	even	60
4805.2.a.x.1.17		20	31.30	odd	2
4805.2.a.y.1.17		20	1.1	even	1	trivial

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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4805 = 5 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4805.a (trivial)

\(f(q)\)	\(=\)	\(q+2.09217 q^{2} -1.52922 q^{3} +2.37719 q^{4} -1.00000 q^{5} -3.19940 q^{6} +3.03261 q^{7} +0.789144 q^{8} -0.661478 q^{9} -2.09217 q^{10} +6.10532 q^{11} -3.63525 q^{12} +5.76122 q^{13} +6.34475 q^{14} +1.52922 q^{15} -3.10335 q^{16} +5.74835 q^{17} -1.38393 q^{18} -2.04231 q^{19} -2.37719 q^{20} -4.63754 q^{21} +12.7734 q^{22} -6.12364 q^{23} -1.20678 q^{24} +1.00000 q^{25} +12.0535 q^{26} +5.59922 q^{27} +7.20909 q^{28} -5.75211 q^{29} +3.19940 q^{30} -8.07104 q^{32} -9.33639 q^{33} +12.0266 q^{34} -3.03261 q^{35} -1.57246 q^{36} -2.36580 q^{37} -4.27287 q^{38} -8.81019 q^{39} -0.789144 q^{40} +0.924914 q^{41} -9.70253 q^{42} +4.26449 q^{43} +14.5135 q^{44} +0.661478 q^{45} -12.8117 q^{46} +1.56258 q^{47} +4.74571 q^{48} +2.19673 q^{49} +2.09217 q^{50} -8.79051 q^{51} +13.6955 q^{52} +2.87837 q^{53} +11.7145 q^{54} -6.10532 q^{55} +2.39317 q^{56} +3.12315 q^{57} -12.0344 q^{58} +0.194720 q^{59} +3.63525 q^{60} +9.68296 q^{61} -2.00601 q^{63} -10.6793 q^{64} -5.76122 q^{65} -19.5333 q^{66} +6.44649 q^{67} +13.6649 q^{68} +9.36440 q^{69} -6.34475 q^{70} +0.958977 q^{71} -0.522002 q^{72} +1.07913 q^{73} -4.94966 q^{74} -1.52922 q^{75} -4.85496 q^{76} +18.5151 q^{77} -18.4325 q^{78} +9.64623 q^{79} +3.10335 q^{80} -6.57801 q^{81} +1.93508 q^{82} -7.99699 q^{83} -11.0243 q^{84} -5.74835 q^{85} +8.92205 q^{86} +8.79626 q^{87} +4.81798 q^{88} -10.6879 q^{89} +1.38393 q^{90} +17.4716 q^{91} -14.5570 q^{92} +3.26919 q^{94} +2.04231 q^{95} +12.3424 q^{96} +2.42074 q^{97} +4.59595 q^{98} -4.03853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} + 4 q^{3} + 15 q^{4} - 20 q^{5} + 4 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} + q^{10} + 17 q^{11} - 13 q^{12} + 9 q^{13} + 6 q^{14} - 4 q^{15} + 9 q^{16} - 20 q^{17} - 28 q^{18} - 7 q^{19} - 15 q^{20}+ \cdots + 53 q^{99}+O(q^{100}) \) 20 * q - q^2 + 4 * q^3 + 15 * q^4 - 20 * q^5 + 4 * q^6 + q^7 + 9 * q^8 + 8 * q^9 + q^10 + 17 * q^11 - 13 * q^12 + 9 * q^13 + 6 * q^14 - 4 * q^15 + 9 * q^16 - 20 * q^17 - 28 * q^18 - 7 * q^19 - 15 * q^20 + 22 * q^21 + 22 * q^22 - 9 * q^23 + 28 * q^24 + 20 * q^25 + 32 * q^26 + 13 * q^27 + 8 * q^28 + 4 * q^29 - 4 * q^30 + 2 * q^32 - 26 * q^33 + 27 * q^34 - q^35 + 13 * q^36 + 12 * q^37 + 15 * q^38 + 11 * q^39 - 9 * q^40 + 9 * q^41 - 14 * q^42 + 42 * q^43 + 49 * q^44 - 8 * q^45 - 24 * q^46 - 32 * q^47 - 31 * q^48 + q^49 - q^50 + 2 * q^51 + 29 * q^52 + 9 * q^53 + 44 * q^54 - 17 * q^55 - 9 * q^56 + 38 * q^57 + 47 * q^58 + 6 * q^59 + 13 * q^60 + 50 * q^61 + 55 * q^63 - 19 * q^64 - 9 * q^65 + 15 * q^66 - 30 * q^67 - 34 * q^68 + 2 * q^69 - 6 * q^70 + 7 * q^71 - 16 * q^72 + 32 * q^73 - 22 * q^74 + 4 * q^75 + 23 * q^76 - 8 * q^77 + 29 * q^78 + 63 * q^79 - 9 * q^80 + 8 * q^81 - 12 * q^82 - 35 * q^83 + 71 * q^84 + 20 * q^85 + 65 * q^86 + 46 * q^87 + 70 * q^88 + 40 * q^89 + 28 * q^90 + 28 * q^91 + 5 * q^92 + 65 * q^94 + 7 * q^95 + 46 * q^96 + 41 * q^97 + 65 * q^98 + 53 * q^99

Introduction

L-functions

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Database

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