LMFDB - Embedded newform 4025.2.a.s.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4025,2,Mod(1,4025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4025, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4025 = 5^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4025.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:	$32.1397868136$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 19x^{5} + 12x^{4} - 34x^{3} - 12x^{2} + 17x + 7 $ x^8 - 3x^7 - 6x^6 + 19x^5 + 12x^4 - 34x^3 - 12x^2 + 17*x + 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.86556$ of defining polynomial
Character	$\chi$	$=$	4025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.86556 q^{2} -1.47264 q^{3} +1.48032 q^{4} -2.74730 q^{6} +1.00000 q^{7} -0.969501 q^{8} -0.831331 q^{9} +O(q^{10})$	\(q+1.86556 q^{2} -1.47264 q^{3} +1.48032 q^{4} -2.74730 q^{6} +1.00000 q^{7} -0.969501 q^{8} -0.831331 q^{9} +2.38861 q^{11} -2.17997 q^{12} +5.15152 q^{13} +1.86556 q^{14} -4.76930 q^{16} -3.94710 q^{17} -1.55090 q^{18} -8.03699 q^{19} -1.47264 q^{21} +4.45610 q^{22} +1.00000 q^{23} +1.42773 q^{24} +9.61048 q^{26} +5.64217 q^{27} +1.48032 q^{28} -3.71096 q^{29} +4.20350 q^{31} -6.95841 q^{32} -3.51756 q^{33} -7.36356 q^{34} -1.23063 q^{36} +2.12680 q^{37} -14.9935 q^{38} -7.58634 q^{39} +2.12162 q^{41} -2.74730 q^{42} -5.36459 q^{43} +3.53590 q^{44} +1.86556 q^{46} -6.57020 q^{47} +7.02346 q^{48} +1.00000 q^{49} +5.81267 q^{51} +7.62589 q^{52} -12.6510 q^{53} +10.5258 q^{54} -0.969501 q^{56} +11.8356 q^{57} -6.92302 q^{58} +2.07665 q^{59} -2.74616 q^{61} +7.84188 q^{62} -0.831331 q^{63} -3.44274 q^{64} -6.56223 q^{66} -5.56259 q^{67} -5.84296 q^{68} -1.47264 q^{69} +6.20033 q^{71} +0.805976 q^{72} -5.19586 q^{73} +3.96767 q^{74} -11.8973 q^{76} +2.38861 q^{77} -14.1528 q^{78} -6.41433 q^{79} -5.81490 q^{81} +3.95801 q^{82} +6.39627 q^{83} -2.17997 q^{84} -10.0080 q^{86} +5.46491 q^{87} -2.31576 q^{88} -1.93591 q^{89} +5.15152 q^{91} +1.48032 q^{92} -6.19024 q^{93} -12.2571 q^{94} +10.2472 q^{96} -5.62379 q^{97} +1.86556 q^{98} -1.98572 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{2} - 2 q^{3} + 5 q^{4} - q^{6} + 8 q^{7} - 12 q^{8}+O(q^{10}) $ 8 * q - 3 * q^2 - 2 * q^3 + 5 * q^4 - q^6 + 8 * q^7 - 12 * q^8	$ 8 q - 3 q^{2} - 2 q^{3} + 5 q^{4} - q^{6} + 8 q^{7} - 12 q^{8} - 5 q^{11} - 3 q^{12} - 9 q^{13} - 3 q^{14} - q^{16} + q^{17} - 6 q^{18} - 4 q^{19} - 2 q^{21} + 5 q^{22} + 8 q^{23} + 16 q^{24} + 22 q^{26} + q^{27} + 5 q^{28} - 5 q^{29} - q^{31} - 2 q^{32} - 12 q^{33} - 2 q^{34} + 16 q^{36} - 18 q^{37} - 14 q^{38} + 14 q^{39} - q^{41} - q^{42} - 20 q^{43} - 3 q^{46} - 10 q^{47} - 31 q^{48} + 8 q^{49} - 4 q^{51} - 11 q^{52} - 11 q^{53} + 29 q^{54} - 12 q^{56} - 8 q^{57} - 24 q^{58} + 20 q^{59} - 6 q^{61} - 2 q^{62} + 8 q^{64} - 37 q^{66} - 23 q^{67} - 9 q^{68} - 2 q^{69} + 3 q^{71} - 29 q^{72} + 8 q^{73} + 35 q^{74} - 29 q^{76} - 5 q^{77} - 31 q^{78} + 4 q^{79} - 44 q^{81} - 27 q^{82} - 4 q^{83} - 3 q^{84} - 18 q^{86} - 27 q^{87} + 4 q^{88} - 17 q^{89} - 9 q^{91} + 5 q^{92} + 7 q^{93} + 13 q^{94} + 22 q^{96} - 41 q^{97} - 3 q^{98} + 7 q^{99}+O(q^{100}) $ 8 * q - 3 * q^2 - 2 * q^3 + 5 * q^4 - q^6 + 8 * q^7 - 12 * q^8 - 5 * q^11 - 3 * q^12 - 9 * q^13 - 3 * q^14 - q^16 + q^17 - 6 * q^18 - 4 * q^19 - 2 * q^21 + 5 * q^22 + 8 * q^23 + 16 * q^24 + 22 * q^26 + q^27 + 5 * q^28 - 5 * q^29 - q^31 - 2 * q^32 - 12 * q^33 - 2 * q^34 + 16 * q^36 - 18 * q^37 - 14 * q^38 + 14 * q^39 - q^41 - q^42 - 20 * q^43 - 3 * q^46 - 10 * q^47 - 31 * q^48 + 8 * q^49 - 4 * q^51 - 11 * q^52 - 11 * q^53 + 29 * q^54 - 12 * q^56 - 8 * q^57 - 24 * q^58 + 20 * q^59 - 6 * q^61 - 2 * q^62 + 8 * q^64 - 37 * q^66 - 23 * q^67 - 9 * q^68 - 2 * q^69 + 3 * q^71 - 29 * q^72 + 8 * q^73 + 35 * q^74 - 29 * q^76 - 5 * q^77 - 31 * q^78 + 4 * q^79 - 44 * q^81 - 27 * q^82 - 4 * q^83 - 3 * q^84 - 18 * q^86 - 27 * q^87 + 4 * q^88 - 17 * q^89 - 9 * q^91 + 5 * q^92 + 7 * q^93 + 13 * q^94 + 22 * q^96 - 41 * q^97 - 3 * q^98 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.86556	1.31915	0.659575	−	0.751639i	$-0.270736\pi$
$2$	1.86556	1.31915	0.659575	+	0.751639i	$0.270736\pi$
$3$	−1.47264	−0.850229	−0.425115	−	0.905140i	$-0.639766\pi$
$3$	−1.47264	−0.850229	−0.425115	+	0.905140i	$0.639766\pi$
$4$	1.48032	0.740158
$5$	0	0
$5$	0	0
$6$	−2.74730	−1.12158
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.969501	−0.342770
$9$	−0.831331	−0.277110
$10$	0	0
$11$	2.38861	0.720193	0.360096	−	0.932915i	$-0.382744\pi$
$11$	2.38861	0.720193	0.360096	+	0.932915i	$0.382744\pi$
$12$	−2.17997	−0.629304
$13$	5.15152	1.42878	0.714388	−	0.699750i	$-0.246705\pi$
$13$	5.15152	1.42878	0.714388	+	0.699750i	$0.246705\pi$
$14$	1.86556	0.498592
$15$	0	0
$16$	−4.76930	−1.19232
$17$	−3.94710	−0.957313	−0.478657	−	0.878002i	$-0.658876\pi$
$17$	−3.94710	−0.957313	−0.478657	+	0.878002i	$0.658876\pi$
$18$	−1.55090	−0.365550
$19$	−8.03699	−1.84381	−0.921907	−	0.387412i	$-0.873369\pi$
$19$	−8.03699	−1.84381	−0.921907	+	0.387412i	$0.873369\pi$
$20$	0	0
$21$	−1.47264	−0.321356
$22$	4.45610	0.950043
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.42773	0.291433
$25$	0	0
$26$	9.61048	1.88477
$27$	5.64217	1.08584
$28$	1.48032	0.279754
$29$	−3.71096	−0.689107	−0.344554	−	0.938767i	$-0.611970\pi$
$29$	−3.71096	−0.689107	−0.344554	+	0.938767i	$0.611970\pi$
$30$	0	0
$31$	4.20350	0.754971	0.377485	−	0.926016i	$-0.376789\pi$
$31$	4.20350	0.754971	0.377485	+	0.926016i	$0.376789\pi$
$32$	−6.95841	−1.23008
$33$	−3.51756	−0.612329
$34$	−7.36356	−1.26284
$35$	0	0
$36$	−1.23063	−0.205105
$37$	2.12680	0.349644	0.174822	−	0.984600i	$-0.444065\pi$
$37$	2.12680	0.349644	0.174822	+	0.984600i	$0.444065\pi$
$38$	−14.9935	−2.43227
$39$	−7.58634	−1.21479
$40$	0	0
$41$	2.12162	0.331341	0.165671	−	0.986181i	$-0.447021\pi$
$41$	2.12162	0.331341	0.165671	+	0.986181i	$0.447021\pi$
$42$	−2.74730	−0.423918
$43$	−5.36459	−0.818092	−0.409046	−	0.912514i	$-0.634138\pi$
$43$	−5.36459	−0.818092	−0.409046	+	0.912514i	$0.634138\pi$
$44$	3.53590	0.533057
$45$	0	0
$46$	1.86556	0.275062
$47$	−6.57020	−0.958362	−0.479181	−	0.877716i	$-0.659066\pi$
$47$	−6.57020	−0.958362	−0.479181	+	0.877716i	$0.659066\pi$
$48$	7.02346	1.01375
$49$	1.00000	0.142857
$50$	0	0
$51$	5.81267	0.813936
$52$	7.62589	1.05752
$53$	−12.6510	−1.73775	−0.868875	−	0.495032i	$-0.835156\pi$
$53$	−12.6510	−1.73775	−0.868875	+	0.495032i	$0.835156\pi$
$54$	10.5258	1.43238
$55$	0	0
$56$	−0.969501	−0.129555
$57$	11.8356	1.56766
$58$	−6.92302	−0.909036
$59$	2.07665	0.270357	0.135178	−	0.990821i	$-0.456839\pi$
$59$	2.07665	0.270357	0.135178	+	0.990821i	$0.456839\pi$
$60$	0	0
$61$	−2.74616	−0.351610	−0.175805	−	0.984425i	$-0.556253\pi$
$61$	−2.74616	−0.351610	−0.175805	+	0.984425i	$0.556253\pi$
$62$	7.84188	0.995920
$63$	−0.831331	−0.104738
$64$	−3.44274	−0.430343
$65$	0	0
$66$	−6.56223	−0.807754
$67$	−5.56259	−0.679579	−0.339789	−	0.940502i	$-0.610356\pi$
$67$	−5.56259	−0.679579	−0.339789	+	0.940502i	$0.610356\pi$
$68$	−5.84296	−0.708563
$69$	−1.47264	−0.177285
$70$	0	0
$71$	6.20033	0.735843	0.367922	−	0.929857i	$-0.380069\pi$
$71$	6.20033	0.735843	0.367922	+	0.929857i	$0.380069\pi$
$72$	0.805976	0.0949852
$73$	−5.19586	−0.608130	−0.304065	−	0.952651i	$-0.598344\pi$
$73$	−5.19586	−0.608130	−0.304065	+	0.952651i	$0.598344\pi$
$74$	3.96767	0.461233
$75$	0	0
$76$	−11.8973	−1.36471
$77$	2.38861	0.272207
$78$	−14.1528	−1.60249
$79$	−6.41433	−0.721668	−0.360834	−	0.932630i	$-0.617508\pi$
$79$	−6.41433	−0.721668	−0.360834	+	0.932630i	$0.617508\pi$
$80$	0	0
$81$	−5.81490	−0.646100
$82$	3.95801	0.437089
$83$	6.39627	0.702082	0.351041	−	0.936360i	$-0.385828\pi$
$83$	6.39627	0.702082	0.351041	+	0.936360i	$0.385828\pi$
$84$	−2.17997	−0.237855
$85$	0	0
$86$	−10.0080	−1.07919
$87$	5.46491	0.585899
$88$	−2.31576	−0.246861
$89$	−1.93591	−0.205206	−0.102603	−	0.994722i	$-0.532717\pi$
$89$	−1.93591	−0.205206	−0.102603	+	0.994722i	$0.532717\pi$
$90$	0	0
$91$	5.15152	0.540026
$92$	1.48032	0.154334
$93$	−6.19024	−0.641898
$94$	−12.2571	−1.26422
$95$	0	0
$96$	10.2472	1.04585
$97$	−5.62379	−0.571009	−0.285505	−	0.958377i	$-0.592161\pi$
$97$	−5.62379	−0.571009	−0.285505	+	0.958377i	$0.592161\pi$
$98$	1.86556	0.188450
$99$	−1.98572	−0.199573
$100$	0	0
$101$	14.0961	1.40262	0.701308	−	0.712858i	$-0.252600\pi$
$101$	14.0961	1.40262	0.701308	+	0.712858i	$0.252600\pi$
$102$	10.8439	1.07370
$103$	−13.6737	−1.34731	−0.673653	−	0.739048i	$-0.735276\pi$
$103$	−13.6737	−1.34731	−0.673653	+	0.739048i	$0.735276\pi$
$104$	−4.99441	−0.489742
$105$	0	0
$106$	−23.6012	−2.29235
$107$	−14.0571	−1.35895	−0.679477	−	0.733697i	$-0.737793\pi$
$107$	−14.0571	−1.35895	−0.679477	+	0.733697i	$0.737793\pi$
$108$	8.35220	0.803691
$109$	−9.50172	−0.910100	−0.455050	−	0.890466i	$-0.650379\pi$
$109$	−9.50172	−0.910100	−0.455050	+	0.890466i	$0.650379\pi$
$110$	0	0
$111$	−3.13201	−0.297277
$112$	−4.76930	−0.450656
$113$	−17.4801	−1.64439	−0.822196	−	0.569204i	$-0.807251\pi$
$113$	−17.4801	−1.64439	−0.822196	+	0.569204i	$0.807251\pi$
$114$	22.0800	2.06798
$115$	0	0
$116$	−5.49339	−0.510049
$117$	−4.28262	−0.395928
$118$	3.87412	0.356641
$119$	−3.94710	−0.361830
$120$	0	0
$121$	−5.29454	−0.481322
$122$	−5.12313	−0.463826
$123$	−3.12438	−0.281716
$124$	6.22251	0.558798
$125$	0	0
$126$	−1.55090	−0.138165
$127$	−1.58046	−0.140243	−0.0701216	−	0.997538i	$-0.522339\pi$
$127$	−1.58046	−0.140243	−0.0701216	+	0.997538i	$0.522339\pi$
$128$	7.49418	0.662398
$129$	7.90011	0.695566
$130$	0	0
$131$	−3.80847	−0.332748	−0.166374	−	0.986063i	$-0.553206\pi$
$131$	−3.80847	−0.332748	−0.166374	+	0.986063i	$0.553206\pi$
$132$	−5.20711	−0.453220
$133$	−8.03699	−0.696896
$134$	−10.3774	−0.896467
$135$	0	0
$136$	3.82672	0.328139
$137$	4.78795	0.409062	0.204531	−	0.978860i	$-0.434433\pi$
$137$	4.78795	0.409062	0.204531	+	0.978860i	$0.434433\pi$
$138$	−2.74730	−0.233866
$139$	−2.18731	−0.185525	−0.0927625	−	0.995688i	$-0.529570\pi$
$139$	−2.18731	−0.185525	−0.0927625	+	0.995688i	$0.529570\pi$
$140$	0	0
$141$	9.67554	0.814827
$142$	11.5671	0.970688
$143$	12.3050	1.02899
$144$	3.96486	0.330405
$145$	0	0
$146$	−9.69320	−0.802215
$147$	−1.47264	−0.121461
$148$	3.14834	0.258792
$149$	9.84123	0.806225	0.403112	−	0.915150i	$-0.367928\pi$
$149$	9.84123	0.806225	0.403112	+	0.915150i	$0.367928\pi$
$150$	0	0
$151$	−8.29915	−0.675375	−0.337687	−	0.941258i	$-0.609645\pi$
$151$	−8.29915	−0.675375	−0.337687	+	0.941258i	$0.609645\pi$
$152$	7.79188	0.632005
$153$	3.28135	0.265281
$154$	4.45610	0.359082
$155$	0	0
$156$	−11.2302	−0.899135
$157$	4.29360	0.342667	0.171333	−	0.985213i	$-0.445192\pi$
$157$	4.29360	0.342667	0.171333	+	0.985213i	$0.445192\pi$
$158$	−11.9663	−0.951989
$159$	18.6304	1.47749
$160$	0	0
$161$	1.00000	0.0788110
$162$	−10.8480	−0.852303
$163$	7.31735	0.573139	0.286570	−	0.958059i	$-0.407485\pi$
$163$	7.31735	0.573139	0.286570	+	0.958059i	$0.407485\pi$
$164$	3.14067	0.245245
$165$	0	0
$166$	11.9326	0.926152
$167$	−17.4368	−1.34930	−0.674648	−	0.738139i	$-0.735705\pi$
$167$	−17.4368	−1.34930	−0.674648	+	0.738139i	$0.735705\pi$
$168$	1.42773	0.110151
$169$	13.5382	1.04140
$170$	0	0
$171$	6.68140	0.510939
$172$	−7.94129	−0.605517
$173$	−1.62465	−0.123520	−0.0617598	−	0.998091i	$-0.519671\pi$
$173$	−1.62465	−0.123520	−0.0617598	+	0.998091i	$0.519671\pi$
$174$	10.1951	0.772889
$175$	0	0
$176$	−11.3920	−0.858703
$177$	−3.05816	−0.229865
$178$	−3.61155	−0.270697
$179$	−9.29389	−0.694658	−0.347329	−	0.937743i	$-0.612911\pi$
$179$	−9.29389	−0.694658	−0.347329	+	0.937743i	$0.612911\pi$
$180$	0	0
$181$	2.59757	0.193076	0.0965379	−	0.995329i	$-0.469223\pi$
$181$	2.59757	0.193076	0.0965379	+	0.995329i	$0.469223\pi$
$182$	9.61048	0.712376
$183$	4.04411	0.298949
$184$	−0.969501	−0.0714726
$185$	0	0
$186$	−11.5483	−0.846760
$187$	−9.42809	−0.689450
$188$	−9.72597	−0.709339
$189$	5.64217	0.410408
$190$	0	0
$191$	−12.8844	−0.932281	−0.466141	−	0.884711i	$-0.654356\pi$
$191$	−12.8844	−0.932281	−0.466141	+	0.884711i	$0.654356\pi$
$192$	5.06992	0.365890
$193$	−8.49047	−0.611157	−0.305579	−	0.952167i	$-0.598850\pi$
$193$	−8.49047	−0.611157	−0.305579	+	0.952167i	$0.598850\pi$
$194$	−10.4915	−0.753247
$195$	0	0
$196$	1.48032	0.105737
$197$	−10.0052	−0.712842	−0.356421	−	0.934325i	$-0.616003\pi$
$197$	−10.0052	−0.712842	−0.356421	+	0.934325i	$0.616003\pi$
$198$	−3.70449	−0.263267
$199$	25.1848	1.78530	0.892652	−	0.450746i	$-0.148842\pi$
$199$	25.1848	1.78530	0.892652	+	0.450746i	$0.148842\pi$
$200$	0	0
$201$	8.19170	0.577798
$202$	26.2972	1.85026
$203$	−3.71096	−0.260458
$204$	8.60458	0.602441
$205$	0	0
$206$	−25.5090	−1.77730
$207$	−0.831331	−0.0577815
$208$	−24.5691	−1.70356
$209$	−19.1972	−1.32790
$210$	0	0
$211$	−22.2507	−1.53180	−0.765900	−	0.642960i	$-0.777706\pi$
$211$	−22.2507	−1.53180	−0.765900	+	0.642960i	$0.777706\pi$
$212$	−18.7275	−1.28621
$213$	−9.13085	−0.625635
$214$	−26.2244	−1.79266
$215$	0	0
$216$	−5.47009	−0.372193
$217$	4.20350	0.285352
$218$	−17.7260	−1.20056
$219$	7.65164	0.517050
$220$	0	0
$221$	−20.3336	−1.36779
$222$	−5.84295	−0.392153
$223$	−24.7280	−1.65591	−0.827955	−	0.560795i	$-0.810496\pi$
$223$	−24.7280	−1.65591	−0.827955	+	0.560795i	$0.810496\pi$
$224$	−6.95841	−0.464928
$225$	0	0
$226$	−32.6102	−2.16920
$227$	−17.6036	−1.16840	−0.584198	−	0.811611i	$-0.698591\pi$
$227$	−17.6036	−1.16840	−0.584198	+	0.811611i	$0.698591\pi$
$228$	17.5204	1.16032
$229$	8.24072	0.544562	0.272281	−	0.962218i	$-0.412222\pi$
$229$	8.24072	0.544562	0.272281	+	0.962218i	$0.412222\pi$
$230$	0	0
$231$	−3.51756	−0.231439
$232$	3.59778	0.236206
$233$	7.67428	0.502759	0.251380	−	0.967889i	$-0.419116\pi$
$233$	7.67428	0.502759	0.251380	+	0.967889i	$0.419116\pi$
$234$	−7.98949	−0.522289
$235$	0	0
$236$	3.07410	0.200107
$237$	9.44600	0.613584
$238$	−7.36356	−0.477309
$239$	25.7901	1.66822	0.834112	−	0.551595i	$-0.185981\pi$
$239$	25.7901	1.66822	0.834112	+	0.551595i	$0.185981\pi$
$240$	0	0
$241$	18.2500	1.17559	0.587793	−	0.809012i	$-0.299997\pi$
$241$	18.2500	1.17559	0.587793	+	0.809012i	$0.299997\pi$
$242$	−9.87729	−0.634937
$243$	−8.36326	−0.536504
$244$	−4.06519	−0.260247
$245$	0	0
$246$	−5.82872	−0.371626
$247$	−41.4028	−2.63440
$248$	−4.07530	−0.258782
$249$	−9.41941	−0.596931
$250$	0	0
$251$	10.4199	0.657696	0.328848	−	0.944383i	$-0.393340\pi$
$251$	10.4199	0.657696	0.328848	+	0.944383i	$0.393340\pi$
$252$	−1.23063	−0.0775225
$253$	2.38861	0.150171
$254$	−2.94844	−0.185002
$255$	0	0
$256$	20.8663	1.30415
$257$	29.6739	1.85101	0.925505	−	0.378737i	$-0.123641\pi$
$257$	29.6739	1.85101	0.925505	+	0.378737i	$0.123641\pi$
$258$	14.7381	0.917556
$259$	2.12680	0.132153
$260$	0	0
$261$	3.08503	0.190959
$262$	−7.10494	−0.438945
$263$	25.3969	1.56604	0.783020	−	0.621996i	$-0.213678\pi$
$263$	25.3969	1.56604	0.783020	+	0.621996i	$0.213678\pi$
$264$	3.41028	0.209888
$265$	0	0
$266$	−14.9935	−0.919311
$267$	2.85090	0.174472
$268$	−8.23440	−0.502996
$269$	−9.01494	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$269$	−9.01494	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$270$	0	0
$271$	8.89439	0.540296	0.270148	−	0.962819i	$-0.412927\pi$
$271$	8.89439	0.540296	0.270148	+	0.962819i	$0.412927\pi$
$272$	18.8249	1.14143
$273$	−7.58634	−0.459146
$274$	8.93221	0.539614
$275$	0	0
$276$	−2.17997	−0.131219
$277$	−1.59622	−0.0959075	−0.0479538	−	0.998850i	$-0.515270\pi$
$277$	−1.59622	−0.0959075	−0.0479538	+	0.998850i	$0.515270\pi$
$278$	−4.08056	−0.244735
$279$	−3.49450	−0.209210
$280$	0	0
$281$	−28.0273	−1.67197	−0.835986	−	0.548751i	$-0.815103\pi$
$281$	−28.0273	−1.67197	−0.835986	+	0.548751i	$0.815103\pi$
$282$	18.0503	1.07488
$283$	9.37346	0.557194	0.278597	−	0.960408i	$-0.410131\pi$
$283$	9.37346	0.557194	0.278597	+	0.960408i	$0.410131\pi$
$284$	9.17844	0.544640
$285$	0	0
$286$	22.9557	1.35740
$287$	2.12162	0.125235
$288$	5.78474	0.340869
$289$	−1.42037	−0.0835510
$290$	0	0
$291$	8.28182	0.485489
$292$	−7.69152	−0.450112
$293$	8.11404	0.474027	0.237014	−	0.971506i	$-0.423831\pi$
$293$	8.11404	0.474027	0.237014	+	0.971506i	$0.423831\pi$
$294$	−2.74730	−0.160226
$295$	0	0
$296$	−2.06193	−0.119847
$297$	13.4769	0.782012
$298$	18.3594	1.06353
$299$	5.15152	0.297920
$300$	0	0
$301$	−5.36459	−0.309210
$302$	−15.4826	−0.890921
$303$	−20.7585	−1.19255
$304$	38.3308	2.19842
$305$	0	0
$306$	6.12155	0.349946
$307$	31.8495	1.81775	0.908873	−	0.417073i	$-0.136944\pi$
$307$	31.8495	1.81775	0.908873	+	0.417073i	$0.136944\pi$
$308$	3.53590	0.201476
$309$	20.1364	1.14552
$310$	0	0
$311$	3.13562	0.177805	0.0889024	−	0.996040i	$-0.471664\pi$
$311$	3.13562	0.177805	0.0889024	+	0.996040i	$0.471664\pi$
$312$	7.35497	0.416393
$313$	−23.2352	−1.31333	−0.656666	−	0.754181i	$-0.728034\pi$
$313$	−23.2352	−1.31333	−0.656666	+	0.754181i	$0.728034\pi$
$314$	8.00998	0.452029
$315$	0	0
$316$	−9.49524	−0.534149
$317$	15.6598	0.879540	0.439770	−	0.898110i	$-0.355060\pi$
$317$	15.6598	0.879540	0.439770	+	0.898110i	$0.355060\pi$
$318$	34.7561	1.94903
$319$	−8.86403	−0.496290
$320$	0	0
$321$	20.7011	1.15542
$322$	1.86556	0.103964
$323$	31.7229	1.76511
$324$	−8.60789	−0.478216
$325$	0	0
$326$	13.6510	0.756057
$327$	13.9926	0.773794
$328$	−2.05691	−0.113574
$329$	−6.57020	−0.362227
$330$	0	0
$331$	−21.8143	−1.19902	−0.599512	−	0.800365i	$-0.704639\pi$
$331$	−21.8143	−1.19902	−0.599512	+	0.800365i	$0.704639\pi$
$332$	9.46851	0.519652
$333$	−1.76807	−0.0968898
$334$	−32.5293	−1.77993
$335$	0	0
$336$	7.02346	0.383161
$337$	−5.42470	−0.295502	−0.147751	−	0.989025i	$-0.547203\pi$
$337$	−5.42470	−0.295502	−0.147751	+	0.989025i	$0.547203\pi$
$338$	25.2563	1.37376
$339$	25.7419	1.39811
$340$	0	0
$341$	10.0405	0.543724
$342$	12.4646	0.674006
$343$	1.00000	0.0539949
$344$	5.20097	0.280418
$345$	0	0
$346$	−3.03088	−0.162941
$347$	−11.4060	−0.612309	−0.306154	−	0.951982i	$-0.599042\pi$
$347$	−11.4060	−0.612309	−0.306154	+	0.951982i	$0.599042\pi$
$348$	8.08979	0.433658
$349$	13.3270	0.713379	0.356690	−	0.934223i	$-0.383905\pi$
$349$	13.3270	0.713379	0.356690	+	0.934223i	$0.383905\pi$
$350$	0	0
$351$	29.0658	1.55142
$352$	−16.6209	−0.885898
$353$	27.3740	1.45697	0.728485	−	0.685062i	$-0.240225\pi$
$353$	27.3740	1.45697	0.728485	+	0.685062i	$0.240225\pi$
$354$	−5.70518	−0.303227
$355$	0	0
$356$	−2.86576	−0.151885
$357$	5.81267	0.307639
$358$	−17.3383	−0.916359
$359$	−14.0136	−0.739612	−0.369806	−	0.929109i	$-0.620576\pi$
$359$	−14.0136	−0.739612	−0.369806	+	0.929109i	$0.620576\pi$
$360$	0	0
$361$	45.5933	2.39965
$362$	4.84592	0.254696
$363$	7.79696	0.409234
$364$	7.62589	0.399705
$365$	0	0
$366$	7.54453	0.394359
$367$	−6.01486	−0.313973	−0.156986	−	0.987601i	$-0.550178\pi$
$367$	−6.01486	−0.313973	−0.156986	+	0.987601i	$0.550178\pi$
$368$	−4.76930	−0.248617
$369$	−1.76377	−0.0918180
$370$	0	0
$371$	−12.6510	−0.656807
$372$	−9.16352	−0.475106
$373$	−8.76458	−0.453813	−0.226906	−	0.973917i	$-0.572861\pi$
$373$	−8.76458	−0.453813	−0.226906	+	0.973917i	$0.572861\pi$
$374$	−17.5887	−0.909489
$375$	0	0
$376$	6.36982	0.328498
$377$	−19.1171	−0.984580
$378$	10.5258	0.541389
$379$	−15.7948	−0.811325	−0.405662	−	0.914023i	$-0.632959\pi$
$379$	−15.7948	−0.811325	−0.405662	+	0.914023i	$0.632959\pi$
$380$	0	0
$381$	2.32745	0.119239
$382$	−24.0366	−1.22982
$383$	31.6206	1.61574	0.807869	−	0.589362i	$-0.200621\pi$
$383$	31.6206	1.61574	0.807869	+	0.589362i	$0.200621\pi$
$384$	−11.0362	−0.563190
$385$	0	0
$386$	−15.8395	−0.806208
$387$	4.45974	0.226702
$388$	−8.32498	−0.422637
$389$	17.1433	0.869198	0.434599	−	0.900624i	$-0.356890\pi$
$389$	17.1433	0.869198	0.434599	+	0.900624i	$0.356890\pi$
$390$	0	0
$391$	−3.94710	−0.199614
$392$	−0.969501	−0.0489672
$393$	5.60851	0.282912
$394$	−18.6653	−0.940346
$395$	0	0
$396$	−2.93950	−0.147715
$397$	−32.4863	−1.63044	−0.815221	−	0.579150i	$-0.803385\pi$
$397$	−32.4863	−1.63044	−0.815221	+	0.579150i	$0.803385\pi$
$398$	46.9838	2.35509
$399$	11.8356	0.592521
$400$	0	0
$401$	14.5317	0.725679	0.362840	−	0.931852i	$-0.381807\pi$
$401$	14.5317	0.725679	0.362840	+	0.931852i	$0.381807\pi$
$402$	15.2821	0.762202
$403$	21.6544	1.07868
$404$	20.8667	1.03816
$405$	0	0
$406$	−6.92302	−0.343584
$407$	5.08009	0.251811
$408$	−5.63539	−0.278993
$409$	19.6266	0.970474	0.485237	−	0.874383i	$-0.338733\pi$
$409$	19.6266	0.970474	0.485237	+	0.874383i	$0.338733\pi$
$410$	0	0
$411$	−7.05092	−0.347796
$412$	−20.2413	−0.997219
$413$	2.07665	0.102185
$414$	−1.55090	−0.0762225
$415$	0	0
$416$	−35.8464	−1.75751
$417$	3.22112	0.157739
$418$	−35.8136	−1.75170
$419$	27.1779	1.32773	0.663863	−	0.747854i	$-0.268916\pi$
$419$	27.1779	1.32773	0.663863	+	0.747854i	$0.268916\pi$
$420$	0	0
$421$	32.6505	1.59129	0.795645	−	0.605764i	$-0.207132\pi$
$421$	32.6505	1.59129	0.795645	+	0.605764i	$0.207132\pi$
$422$	−41.5100	−2.02067
$423$	5.46201	0.265572
$424$	12.2652	0.595649
$425$	0	0
$426$	−17.0342	−0.825307
$427$	−2.74616	−0.132896
$428$	−20.8090	−1.00584
$429$	−18.1208	−0.874881
$430$	0	0
$431$	22.5655	1.08694	0.543471	−	0.839428i	$-0.317110\pi$
$431$	22.5655	1.08694	0.543471	+	0.839428i	$0.317110\pi$
$432$	−26.9092	−1.29467
$433$	5.41260	0.260113	0.130056	−	0.991507i	$-0.458484\pi$
$433$	5.41260	0.260113	0.130056	+	0.991507i	$0.458484\pi$
$434$	7.84188	0.376422
$435$	0	0
$436$	−14.0656	−0.673618
$437$	−8.03699	−0.384462
$438$	14.2746	0.682067
$439$	19.5191	0.931598	0.465799	−	0.884891i	$-0.345767\pi$
$439$	19.5191	0.931598	0.465799	+	0.884891i	$0.345767\pi$
$440$	0	0
$441$	−0.831331	−0.0395872
$442$	−37.9336	−1.80432
$443$	−22.2864	−1.05886	−0.529430	−	0.848354i	$-0.677594\pi$
$443$	−22.2864	−1.05886	−0.529430	+	0.848354i	$0.677594\pi$
$444$	−4.63637	−0.220032
$445$	0	0
$446$	−46.1316	−2.18439
$447$	−14.4926	−0.685476
$448$	−3.44274	−0.162654
$449$	18.3447	0.865740	0.432870	−	0.901456i	$-0.357501\pi$
$449$	18.3447	0.865740	0.432870	+	0.901456i	$0.357501\pi$
$450$	0	0
$451$	5.06772	0.238629
$452$	−25.8761	−1.21711
$453$	12.2217	0.574224
$454$	−32.8407	−1.54129
$455$	0	0
$456$	−11.4746	−0.537349
$457$	−30.4321	−1.42355	−0.711776	−	0.702406i	$-0.752109\pi$
$457$	−30.4321	−1.42355	−0.711776	+	0.702406i	$0.752109\pi$
$458$	15.3736	0.718360
$459$	−22.2702	−1.03949
$460$	0	0
$461$	33.5427	1.56224	0.781119	−	0.624383i	$-0.214649\pi$
$461$	33.5427	1.56224	0.781119	+	0.624383i	$0.214649\pi$
$462$	−6.56223	−0.305302
$463$	−20.3855	−0.947394	−0.473697	−	0.880688i	$-0.657081\pi$
$463$	−20.3855	−0.947394	−0.473697	+	0.880688i	$0.657081\pi$
$464$	17.6987	0.821639
$465$	0	0
$466$	14.3168	0.663215
$467$	−15.3079	−0.708363	−0.354182	−	0.935177i	$-0.615241\pi$
$467$	−15.3079	−0.708363	−0.354182	+	0.935177i	$0.615241\pi$
$468$	−6.33963	−0.293050
$469$	−5.56259	−0.256857
$470$	0	0
$471$	−6.32293	−0.291345
$472$	−2.01331	−0.0926703
$473$	−12.8139	−0.589184
$474$	17.6221	0.809409
$475$	0	0
$476$	−5.84296	−0.267812
$477$	10.5172	0.481548
$478$	48.1130	2.20064
$479$	−18.3531	−0.838575	−0.419288	−	0.907853i	$-0.637720\pi$
$479$	−18.3531	−0.838575	−0.419288	+	0.907853i	$0.637720\pi$
$480$	0	0
$481$	10.9563	0.499562
$482$	34.0465	1.55077
$483$	−1.47264	−0.0670075
$484$	−7.83760	−0.356255
$485$	0	0
$486$	−15.6022	−0.707729
$487$	8.20282	0.371705	0.185853	−	0.982578i	$-0.440495\pi$
$487$	8.20282	0.371705	0.185853	+	0.982578i	$0.440495\pi$
$488$	2.66241	0.120521
$489$	−10.7758	−0.487300
$490$	0	0
$491$	11.0280	0.497686	0.248843	−	0.968544i	$-0.419950\pi$
$491$	11.0280	0.497686	0.248843	+	0.968544i	$0.419950\pi$
$492$	−4.62507	−0.208514
$493$	14.6475	0.659692
$494$	−77.2394	−3.47516
$495$	0	0
$496$	−20.0477	−0.900170
$497$	6.20033	0.278123
$498$	−17.5725	−0.787442
$499$	10.6322	0.475965	0.237982	−	0.971269i	$-0.423514\pi$
$499$	10.6322	0.475965	0.237982	+	0.971269i	$0.423514\pi$
$500$	0	0
$501$	25.6781	1.14721
$502$	19.4389	0.867600
$503$	17.0269	0.759191	0.379595	−	0.925153i	$-0.376063\pi$
$503$	17.0269	0.759191	0.379595	+	0.925153i	$0.376063\pi$
$504$	0.805976	0.0359010
$505$	0	0
$506$	4.45610	0.198098
$507$	−19.9369	−0.885429
$508$	−2.33958	−0.103802
$509$	−7.96156	−0.352890	−0.176445	−	0.984311i	$-0.556460\pi$
$509$	−7.96156	−0.352890	−0.176445	+	0.984311i	$0.556460\pi$
$510$	0	0
$511$	−5.19586	−0.229852
$512$	23.9390	1.05797
$513$	−45.3461	−2.00208
$514$	55.3585	2.44176
$515$	0	0
$516$	11.6947	0.514829
$517$	−15.6936	−0.690205
$518$	3.96767	0.174330
$519$	2.39252	0.105020
$520$	0	0
$521$	7.19516	0.315226	0.157613	−	0.987501i	$-0.449620\pi$
$521$	7.19516	0.315226	0.157613	+	0.987501i	$0.449620\pi$
$522$	5.75531	0.251903
$523$	−45.4879	−1.98905	−0.994523	−	0.104516i	$-0.966671\pi$
$523$	−45.4879	−1.98905	−0.994523	+	0.104516i	$0.966671\pi$
$524$	−5.63775	−0.246286
$525$	0	0
$526$	47.3795	2.06584
$527$	−16.5916	−0.722744
$528$	16.7763	0.730095
$529$	1.00000	0.0434783
$530$	0	0
$531$	−1.72638	−0.0749186
$532$	−11.8973	−0.515813
$533$	10.9296	0.473412
$534$	5.31852	0.230155
$535$	0	0
$536$	5.39294	0.232940
$537$	13.6866	0.590619
$538$	−16.8179	−0.725072
$539$	2.38861	0.102885
$540$	0	0
$541$	−38.9932	−1.67645	−0.838225	−	0.545325i	$-0.816406\pi$
$541$	−38.9932	−1.67645	−0.838225	+	0.545325i	$0.816406\pi$
$542$	16.5930	0.712731
$543$	−3.82529	−0.164159
$544$	27.4656	1.17758
$545$	0	0
$546$	−14.1528	−0.605683
$547$	42.0336	1.79723	0.898613	−	0.438742i	$-0.144576\pi$
$547$	42.0336	1.79723	0.898613	+	0.438742i	$0.144576\pi$
$548$	7.08768	0.302771
$549$	2.28297	0.0974347
$550$	0	0
$551$	29.8249	1.27059
$552$	1.42773	0.0607681
$553$	−6.41433	−0.272765
$554$	−2.97784	−0.126516
$555$	0	0
$556$	−3.23791	−0.137318
$557$	−1.72572	−0.0731213	−0.0365606	−	0.999331i	$-0.511640\pi$
$557$	−1.72572	−0.0731213	−0.0365606	+	0.999331i	$0.511640\pi$
$558$	−6.51920	−0.275980
$559$	−27.6358	−1.16887
$560$	0	0
$561$	13.8842	0.586191
$562$	−52.2867	−2.20558
$563$	−6.91648	−0.291495	−0.145747	−	0.989322i	$-0.546559\pi$
$563$	−6.91648	−0.291495	−0.145747	+	0.989322i	$0.546559\pi$
$564$	14.3229	0.603101
$565$	0	0
$566$	17.4868	0.735023
$567$	−5.81490	−0.244203
$568$	−6.01122	−0.252225
$569$	33.4704	1.40315	0.701577	−	0.712594i	$-0.252480\pi$
$569$	33.4704	1.40315	0.701577	+	0.712594i	$0.252480\pi$
$570$	0	0
$571$	−17.7343	−0.742158	−0.371079	−	0.928601i	$-0.621012\pi$
$571$	−17.7343	−0.742158	−0.371079	+	0.928601i	$0.621012\pi$
$572$	18.2153	0.761618
$573$	18.9741	0.792653
$574$	3.95801	0.165204
$575$	0	0
$576$	2.86206	0.119252
$577$	14.3568	0.597682	0.298841	−	0.954303i	$-0.403400\pi$
$577$	14.3568	0.597682	0.298841	+	0.954303i	$0.403400\pi$
$578$	−2.64978	−0.110216
$579$	12.5034	0.519624
$580$	0	0
$581$	6.39627	0.265362
$582$	15.4502	0.640433
$583$	−30.2183	−1.25151
$584$	5.03740	0.208449
$585$	0	0
$586$	15.1372	0.625313
$587$	37.6419	1.55365	0.776823	−	0.629718i	$-0.216830\pi$
$587$	37.6419	1.55365	0.776823	+	0.629718i	$0.216830\pi$
$588$	−2.17997	−0.0899006
$589$	−33.7835	−1.39202
$590$	0	0
$591$	14.7341	0.606079
$592$	−10.1433	−0.416888
$593$	−44.3928	−1.82299	−0.911497	−	0.411307i	$-0.865073\pi$
$593$	−44.3928	−1.82299	−0.911497	+	0.411307i	$0.865073\pi$
$594$	25.1421	1.03159
$595$	0	0
$596$	14.5681	0.596734
$597$	−37.0882	−1.51792
$598$	9.61048	0.393002
$599$	7.74272	0.316359	0.158180	−	0.987410i	$-0.449438\pi$
$599$	7.74272	0.316359	0.158180	+	0.987410i	$0.449438\pi$
$600$	0	0
$601$	31.9476	1.30317	0.651584	−	0.758576i	$-0.274105\pi$
$601$	31.9476	1.30317	0.651584	+	0.758576i	$0.274105\pi$
$602$	−10.0080	−0.407894
$603$	4.62435	0.188318
$604$	−12.2854	−0.499884
$605$	0	0
$606$	−38.7263	−1.57315
$607$	−21.1266	−0.857501	−0.428750	−	0.903423i	$-0.641046\pi$
$607$	−21.1266	−0.857501	−0.428750	+	0.903423i	$0.641046\pi$
$608$	55.9247	2.26805
$609$	5.46491	0.221449
$610$	0	0
$611$	−33.8465	−1.36928
$612$	4.85743	0.196350
$613$	39.7435	1.60523	0.802613	−	0.596500i	$-0.203442\pi$
$613$	39.7435	1.60523	0.802613	+	0.596500i	$0.203442\pi$
$614$	59.4171	2.39788
$615$	0	0
$616$	−2.31576	−0.0933046
$617$	7.62182	0.306843	0.153421	−	0.988161i	$-0.450971\pi$
$617$	7.62182	0.306843	0.153421	+	0.988161i	$0.450971\pi$
$618$	37.5656	1.51111
$619$	−7.69341	−0.309224	−0.154612	−	0.987975i	$-0.549413\pi$
$619$	−7.69341	−0.309224	−0.154612	+	0.987975i	$0.549413\pi$
$620$	0	0
$621$	5.64217	0.226413
$622$	5.84969	0.234551
$623$	−1.93591	−0.0775605
$624$	36.1815	1.44842
$625$	0	0
$626$	−43.3467	−1.73248
$627$	28.2706	1.12902
$628$	6.35589	0.253628
$629$	−8.39470	−0.334719
$630$	0	0
$631$	32.2726	1.28475	0.642377	−	0.766389i	$-0.277949\pi$
$631$	32.2726	1.28475	0.642377	+	0.766389i	$0.277949\pi$
$632$	6.21870	0.247367
$633$	32.7672	1.30238
$634$	29.2142	1.16025
$635$	0	0
$636$	27.5789	1.09357
$637$	5.15152	0.204111
$638$	−16.5364	−0.654682
$639$	−5.15452	−0.203910
$640$	0	0
$641$	−43.4456	−1.71600	−0.857999	−	0.513651i	$-0.828293\pi$
$641$	−43.4456	−1.71600	−0.857999	+	0.513651i	$0.828293\pi$
$642$	38.6191	1.52418
$643$	−38.9653	−1.53664	−0.768321	−	0.640065i	$-0.778907\pi$
$643$	−38.9653	−1.53664	−0.768321	+	0.640065i	$0.778907\pi$
$644$	1.48032	0.0583326
$645$	0	0
$646$	59.1809	2.32844
$647$	−37.1483	−1.46045	−0.730226	−	0.683206i	$-0.760585\pi$
$647$	−37.1483	−1.46045	−0.730226	+	0.683206i	$0.760585\pi$
$648$	5.63755	0.221464
$649$	4.96031	0.194709
$650$	0	0
$651$	−6.19024	−0.242615
$652$	10.8320	0.424214
$653$	12.1524	0.475560	0.237780	−	0.971319i	$-0.423580\pi$
$653$	12.1524	0.475560	0.237780	+	0.971319i	$0.423580\pi$
$654$	26.1041	1.02075
$655$	0	0
$656$	−10.1186	−0.395066
$657$	4.31948	0.168519
$658$	−12.2571	−0.477832
$659$	−31.5160	−1.22769	−0.613845	−	0.789427i	$-0.710378\pi$
$659$	−31.5160	−1.22769	−0.613845	+	0.789427i	$0.710378\pi$
$660$	0	0
$661$	1.13154	0.0440117	0.0220058	−	0.999758i	$-0.492995\pi$
$661$	1.13154	0.0440117	0.0220058	+	0.999758i	$0.492995\pi$
$662$	−40.6960	−1.58169
$663$	29.9441	1.16293
$664$	−6.20119	−0.240653
$665$	0	0
$666$	−3.29845	−0.127812
$667$	−3.71096	−0.143689
$668$	−25.8119	−0.998693
$669$	36.4155	1.40790
$670$	0	0
$671$	−6.55951	−0.253227
$672$	10.2472	0.395296
$673$	−38.2300	−1.47366	−0.736829	−	0.676080i	$-0.763677\pi$
$673$	−38.2300	−1.47366	−0.736829	+	0.676080i	$0.763677\pi$
$674$	−10.1201	−0.389812
$675$	0	0
$676$	20.0408	0.770801
$677$	29.5706	1.13649	0.568245	−	0.822859i	$-0.307622\pi$
$677$	29.5706	1.13649	0.568245	+	0.822859i	$0.307622\pi$
$678$	48.0232	1.84432
$679$	−5.62379	−0.215821
$680$	0	0
$681$	25.9238	0.993404
$682$	18.7312	0.717254
$683$	25.5066	0.975982	0.487991	−	0.872849i	$-0.337730\pi$
$683$	25.5066	0.975982	0.487991	+	0.872849i	$0.337730\pi$
$684$	9.89058	0.378176
$685$	0	0
$686$	1.86556	0.0712274
$687$	−12.1356	−0.463003
$688$	25.5853	0.975431
$689$	−65.1720	−2.48285
$690$	0	0
$691$	0.00938603	0.000357061 0	0.000178531	−	1.00000i	$-0.499943\pi$
$691$	0.00938603	0.000357061 0	0.000178531		1.00000i	$0.499943\pi$
$692$	−2.40499	−0.0914241
$693$	−1.98572	−0.0754314
$694$	−21.2787	−0.807727
$695$	0	0
$696$	−5.29823	−0.200829
$697$	−8.37425	−0.317197
$698$	24.8624	0.941055
$699$	−11.3015	−0.427460
$700$	0	0
$701$	−2.19980	−0.0830852	−0.0415426	−	0.999137i	$-0.513227\pi$
$701$	−2.19980	−0.0830852	−0.0415426	+	0.999137i	$0.513227\pi$
$702$	54.2240	2.04655
$703$	−17.0931	−0.644677
$704$	−8.22336	−0.309930
$705$	0	0
$706$	51.0678	1.92196
$707$	14.0961	0.530139
$708$	−4.52704	−0.170137
$709$	38.2506	1.43653	0.718266	−	0.695768i	$-0.244936\pi$
$709$	38.2506	1.43653	0.718266	+	0.695768i	$0.244936\pi$
$710$	0	0
$711$	5.33243	0.199982
$712$	1.87686	0.0703385
$713$	4.20350	0.157422
$714$	10.8439	0.405822
$715$	0	0
$716$	−13.7579	−0.514157
$717$	−37.9796	−1.41837
$718$	−26.1433	−0.975659
$719$	−16.5752	−0.618149	−0.309074	−	0.951038i	$-0.600019\pi$
$719$	−16.5752	−0.618149	−0.309074	+	0.951038i	$0.600019\pi$
$720$	0	0
$721$	−13.6737	−0.509234
$722$	85.0570	3.16549
$723$	−26.8757	−0.999517
$724$	3.84523	0.142907
$725$	0	0
$726$	14.5457	0.539842
$727$	51.1494	1.89703	0.948513	−	0.316739i	$-0.102588\pi$
$727$	51.1494	1.89703	0.948513	+	0.316739i	$0.102588\pi$
$728$	−4.99441	−0.185105
$729$	29.7608	1.10225
$730$	0	0
$731$	21.1746	0.783170
$732$	5.98656	0.221270
$733$	22.7843	0.841557	0.420778	−	0.907163i	$-0.361757\pi$
$733$	22.7843	0.841557	0.420778	+	0.907163i	$0.361757\pi$
$734$	−11.2211	−0.414178
$735$	0	0
$736$	−6.95841	−0.256490
$737$	−13.2869	−0.489428
$738$	−3.29041	−0.121122
$739$	−28.0978	−1.03360	−0.516798	−	0.856107i	$-0.672876\pi$
$739$	−28.0978	−1.03360	−0.516798	+	0.856107i	$0.672876\pi$
$740$	0	0
$741$	60.9714	2.23984
$742$	−23.6012	−0.866428
$743$	−40.5765	−1.48861	−0.744303	−	0.667842i	$-0.767218\pi$
$743$	−40.5765	−1.48861	−0.744303	+	0.667842i	$0.767218\pi$
$744$	6.00145	0.220024
$745$	0	0
$746$	−16.3508	−0.598647
$747$	−5.31742	−0.194554
$748$	−13.9566	−0.510302
$749$	−14.0571	−0.513636
$750$	0	0
$751$	−13.1626	−0.480309	−0.240154	−	0.970735i	$-0.577198\pi$
$751$	−13.1626	−0.480309	−0.240154	+	0.970735i	$0.577198\pi$
$752$	31.3352	1.14268
$753$	−15.3447	−0.559192
$754$	−35.6641	−1.29881
$755$	0	0
$756$	8.35220	0.303767
$757$	−7.26122	−0.263914	−0.131957	−	0.991255i	$-0.542126\pi$
$757$	−7.26122	−0.263914	−0.131957	+	0.991255i	$0.542126\pi$
$758$	−29.4662	−1.07026
$759$	−3.51756	−0.127679
$760$	0	0
$761$	−10.4997	−0.380614	−0.190307	−	0.981725i	$-0.560948\pi$
$761$	−10.4997	−0.380614	−0.190307	+	0.981725i	$0.560948\pi$
$762$	4.34200	0.157294
$763$	−9.50172	−0.343986
$764$	−19.0730	−0.690036
$765$	0	0
$766$	58.9902	2.13140
$767$	10.6979	0.386279
$768$	−30.7286	−1.10882
$769$	7.08830	0.255610	0.127805	−	0.991799i	$-0.459207\pi$
$769$	7.08830	0.255610	0.127805	+	0.991799i	$0.459207\pi$
$770$	0	0
$771$	−43.6990	−1.57378
$772$	−12.5686	−0.452353
$773$	1.64999	0.0593460	0.0296730	−	0.999560i	$-0.490553\pi$
$773$	1.64999	0.0593460	0.0296730	+	0.999560i	$0.490553\pi$
$774$	8.31992	0.299054
$775$	0	0
$776$	5.45227	0.195725
$777$	−3.13201	−0.112360
$778$	31.9818	1.14660
$779$	−17.0514	−0.610931
$780$	0	0
$781$	14.8102	0.529949
$782$	−7.36356	−0.263320
$783$	−20.9379	−0.748258
$784$	−4.76930	−0.170332
$785$	0	0
$786$	10.4630	0.373204
$787$	−2.18169	−0.0777690	−0.0388845	−	0.999244i	$-0.512380\pi$
$787$	−2.18169	−0.0777690	−0.0388845	+	0.999244i	$0.512380\pi$
$788$	−14.8109	−0.527616
$789$	−37.4005	−1.33149
$790$	0	0
$791$	−17.4801	−0.621522
$792$	1.92516	0.0684076
$793$	−14.1469	−0.502372
$794$	−60.6052	−2.15080
$795$	0	0
$796$	37.2815	1.32141
$797$	−36.9437	−1.30861	−0.654306	−	0.756230i	$-0.727039\pi$
$797$	−36.9437	−1.30861	−0.654306	+	0.756230i	$0.727039\pi$
$798$	22.0800	0.781625
$799$	25.9333	0.917453
$800$	0	0
$801$	1.60938	0.0568646
$802$	27.1098	0.957280
$803$	−12.4109	−0.437971
$804$	12.1263	0.427662
$805$	0	0
$806$	40.3976	1.42295
$807$	13.2758	0.467329
$808$	−13.6662	−0.480775
$809$	−21.5021	−0.755974	−0.377987	−	0.925811i	$-0.623384\pi$
$809$	−21.5021	−0.755974	−0.377987	+	0.925811i	$0.623384\pi$
$810$	0	0
$811$	−36.5149	−1.28221	−0.641106	−	0.767452i	$-0.721524\pi$
$811$	−36.5149	−1.28221	−0.641106	+	0.767452i	$0.721524\pi$
$812$	−5.49339	−0.192780
$813$	−13.0982	−0.459375
$814$	9.47722	0.332176
$815$	0	0
$816$	−27.7223	−0.970475
$817$	43.1152	1.50841
$818$	36.6147	1.28020
$819$	−4.28262	−0.149647
$820$	0	0
$821$	44.8174	1.56414	0.782068	−	0.623193i	$-0.214165\pi$
$821$	44.8174	1.56414	0.782068	+	0.623193i	$0.214165\pi$
$822$	−13.1539	−0.458796
$823$	23.5737	0.821727	0.410863	−	0.911697i	$-0.365227\pi$
$823$	23.5737	0.821727	0.410863	+	0.911697i	$0.365227\pi$
$824$	13.2566	0.461817
$825$	0	0
$826$	3.87412	0.134798
$827$	−15.2355	−0.529791	−0.264895	−	0.964277i	$-0.585337\pi$
$827$	−15.2355	−0.529791	−0.264895	+	0.964277i	$0.585337\pi$
$828$	−1.23063	−0.0427674
$829$	−45.0464	−1.56453	−0.782263	−	0.622949i	$-0.785934\pi$
$829$	−45.0464	−1.56453	−0.782263	+	0.622949i	$0.785934\pi$
$830$	0	0
$831$	2.35066	0.0815434
$832$	−17.7354	−0.614863
$833$	−3.94710	−0.136759
$834$	6.00919	0.208081
$835$	0	0
$836$	−28.4180	−0.982857
$837$	23.7169	0.819775
$838$	50.7020	1.75147
$839$	41.1443	1.42046	0.710230	−	0.703970i	$-0.248591\pi$
$839$	41.1443	1.42046	0.710230	+	0.703970i	$0.248591\pi$
$840$	0	0
$841$	−15.2288	−0.525131
$842$	60.9115	2.09915
$843$	41.2742	1.42156
$844$	−32.9380	−1.13377
$845$	0	0
$846$	10.1897	0.350329
$847$	−5.29454	−0.181923
$848$	60.3364	2.07196
$849$	−13.8037	−0.473743
$850$	0	0
$851$	2.12680	0.0729057
$852$	−13.5165	−0.463069
$853$	44.1920	1.51311	0.756553	−	0.653933i	$-0.226882\pi$
$853$	44.1920	1.51311	0.756553	+	0.653933i	$0.226882\pi$
$854$	−5.12313	−0.175310
$855$	0	0
$856$	13.6284	0.465809
$857$	−5.66995	−0.193682	−0.0968410	−	0.995300i	$-0.530874\pi$
$857$	−5.66995	−0.193682	−0.0968410	+	0.995300i	$0.530874\pi$
$858$	−33.8055	−1.15410
$859$	−24.7661	−0.845008	−0.422504	−	0.906361i	$-0.638849\pi$
$859$	−24.7661	−0.845008	−0.422504	+	0.906361i	$0.638849\pi$
$860$	0	0
$861$	−3.12438	−0.106479
$862$	42.0973	1.43384
$863$	−51.6782	−1.75915	−0.879574	−	0.475763i	$-0.842172\pi$
$863$	−51.6782	−1.75915	−0.879574	+	0.475763i	$0.842172\pi$
$864$	−39.2605	−1.33567
$865$	0	0
$866$	10.0975	0.343128
$867$	2.09169	0.0710375
$868$	6.22251	0.211206
$869$	−15.3213	−0.519740
$870$	0	0
$871$	−28.6558	−0.970966
$872$	9.21193	0.311955
$873$	4.67523	0.158232
$874$	−14.9935	−0.507163
$875$	0	0
$876$	11.3268	0.382699
$877$	22.5801	0.762477	0.381238	−	0.924477i	$-0.375498\pi$
$877$	22.5801	0.762477	0.381238	+	0.924477i	$0.375498\pi$
$878$	36.4141	1.22892
$879$	−11.9491	−0.403032
$880$	0	0
$881$	−23.2804	−0.784337	−0.392169	−	0.919893i	$-0.628275\pi$
$881$	−23.2804	−0.784337	−0.392169	+	0.919893i	$0.628275\pi$
$882$	−1.55090	−0.0522214
$883$	−18.3267	−0.616741	−0.308371	−	0.951266i	$-0.599784\pi$
$883$	−18.3267	−0.616741	−0.308371	+	0.951266i	$0.599784\pi$
$884$	−30.1002	−1.01238
$885$	0	0
$886$	−41.5767	−1.39680
$887$	−24.7614	−0.831406	−0.415703	−	0.909500i	$-0.636464\pi$
$887$	−24.7614	−0.831406	−0.415703	+	0.909500i	$0.636464\pi$
$888$	3.03649	0.101898
$889$	−1.58046	−0.0530069
$890$	0	0
$891$	−13.8895	−0.465316
$892$	−36.6053	−1.22564
$893$	52.8046	1.76704
$894$	−27.0368	−0.904246
$895$	0	0
$896$	7.49418	0.250363
$897$	−7.58634	−0.253301
$898$	34.2232	1.14204
$899$	−15.5990	−0.520256
$900$	0	0
$901$	49.9348	1.66357
$902$	9.45413	0.314788
$903$	7.90011	0.262899
$904$	16.9470	0.563649
$905$	0	0
$906$	22.8002	0.757487
$907$	−45.5765	−1.51334	−0.756672	−	0.653795i	$-0.773176\pi$
$907$	−45.5765	−1.51334	−0.756672	+	0.653795i	$0.773176\pi$
$908$	−26.0590	−0.864797
$909$	−11.7185	−0.388679
$910$	0	0
$911$	−19.7278	−0.653611	−0.326805	−	0.945092i	$-0.605972\pi$
$911$	−19.7278	−0.653611	−0.326805	+	0.945092i	$0.605972\pi$
$912$	−56.4475	−1.86916
$913$	15.2782	0.505635
$914$	−56.7729	−1.87788
$915$	0	0
$916$	12.1989	0.403062
$917$	−3.80847	−0.125767
$918$	−41.5465	−1.37124
$919$	31.0813	1.02528	0.512639	−	0.858604i	$-0.328668\pi$
$919$	31.0813	1.02528	0.512639	+	0.858604i	$0.328668\pi$
$920$	0	0
$921$	−46.9028	−1.54550
$922$	62.5759	2.06083
$923$	31.9411	1.05135
$924$	−5.20711	−0.171301
$925$	0	0
$926$	−38.0304	−1.24976
$927$	11.3673	0.373352
$928$	25.8224	0.847660
$929$	25.4983	0.836571	0.418286	−	0.908316i	$-0.362631\pi$
$929$	25.4983	0.836571	0.418286	+	0.908316i	$0.362631\pi$
$930$	0	0
$931$	−8.03699	−0.263402
$932$	11.3604	0.372121
$933$	−4.61764	−0.151175
$934$	−28.5577	−0.934438
$935$	0	0
$936$	4.15200	0.135713
$937$	16.3831	0.535213	0.267607	−	0.963528i	$-0.413767\pi$
$937$	16.3831	0.535213	0.267607	+	0.963528i	$0.413767\pi$
$938$	−10.3774	−0.338833
$939$	34.2171	1.11663
$940$	0	0
$941$	−24.4129	−0.795839	−0.397920	−	0.917420i	$-0.630268\pi$
$941$	−24.4129	−0.795839	−0.397920	+	0.917420i	$0.630268\pi$
$942$	−11.7958	−0.384328
$943$	2.12162	0.0690894
$944$	−9.90416	−0.322353
$945$	0	0
$946$	−23.9051	−0.777222
$947$	38.2048	1.24149	0.620745	−	0.784012i	$-0.286830\pi$
$947$	38.2048	1.24149	0.620745	+	0.784012i	$0.286830\pi$
$948$	13.9831	0.454149
$949$	−26.7666	−0.868881
$950$	0	0
$951$	−23.0612	−0.747811
$952$	3.82672	0.124025
$953$	12.1510	0.393610	0.196805	−	0.980443i	$-0.436943\pi$
$953$	12.1510	0.393610	0.196805	+	0.980443i	$0.436943\pi$
$954$	19.6204	0.635234
$955$	0	0
$956$	38.1775	1.23475
$957$	13.0535	0.421961
$958$	−34.2389	−1.10621
$959$	4.78795	0.154611
$960$	0	0
$961$	−13.3306	−0.430019
$962$	20.4396	0.658998
$963$	11.6861	0.376580
$964$	27.0158	0.870119
$965$	0	0
$966$	−2.74730	−0.0883929
$967$	−54.2381	−1.74418	−0.872089	−	0.489347i	$-0.837235\pi$
$967$	−54.2381	−1.74418	−0.872089	+	0.489347i	$0.837235\pi$
$968$	5.13307	0.164983
$969$	−46.7164	−1.50075
$970$	0	0
$971$	1.88823	0.0605962	0.0302981	−	0.999541i	$-0.490354\pi$
$971$	1.88823	0.0605962	0.0302981	+	0.999541i	$0.490354\pi$
$972$	−12.3803	−0.397097
$973$	−2.18731	−0.0701219
$974$	15.3029	0.490335
$975$	0	0
$976$	13.0973	0.419233
$977$	−22.4378	−0.717850	−0.358925	−	0.933366i	$-0.616857\pi$
$977$	−22.4378	−0.717850	−0.358925	+	0.933366i	$0.616857\pi$
$978$	−20.1030	−0.642822
$979$	−4.62413	−0.147788
$980$	0	0
$981$	7.89907	0.252198
$982$	20.5734	0.656523
$983$	60.1504	1.91850	0.959250	−	0.282558i	$-0.0911830\pi$
$983$	60.1504	1.91850	0.959250	+	0.282558i	$0.0911830\pi$
$984$	3.02909	0.0965639
$985$	0	0
$986$	27.3259	0.870233
$987$	9.67554	0.307976
$988$	−61.2892	−1.94987
$989$	−5.36459	−0.170584
$990$	0	0
$991$	48.0838	1.52743	0.763717	−	0.645551i	$-0.223372\pi$
$991$	48.0838	1.52743	0.763717	+	0.645551i	$0.223372\pi$
$992$	−29.2497	−0.928678
$993$	32.1247	1.01945
$994$	11.5671	0.366886
$995$	0	0
$996$	−13.9437	−0.441823
$997$	37.0280	1.17269	0.586344	−	0.810062i	$-0.300567\pi$
$997$	37.0280	1.17269	0.586344	+	0.810062i	$0.300567\pi$
$998$	19.8351	0.627869
$999$	11.9998	0.379656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.s.1.8	✓	8
5.4	even	2		4025.2.a.w.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4025.2.a.s.1.8	✓	8	1.1	even	1	trivial
4025.2.a.w.1.1	yes	8	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.86556 q^{2} -1.47264 q^{3} +1.48032 q^{4} -2.74730 q^{6} +1.00000 q^{7} -0.969501 q^{8} -0.831331 q^{9} +O(q^{10})\)	\(q+1.86556 q^{2} -1.47264 q^{3} +1.48032 q^{4} -2.74730 q^{6} +1.00000 q^{7} -0.969501 q^{8} -0.831331 q^{9} +2.38861 q^{11} -2.17997 q^{12} +5.15152 q^{13} +1.86556 q^{14} -4.76930 q^{16} -3.94710 q^{17} -1.55090 q^{18} -8.03699 q^{19} -1.47264 q^{21} +4.45610 q^{22} +1.00000 q^{23} +1.42773 q^{24} +9.61048 q^{26} +5.64217 q^{27} +1.48032 q^{28} -3.71096 q^{29} +4.20350 q^{31} -6.95841 q^{32} -3.51756 q^{33} -7.36356 q^{34} -1.23063 q^{36} +2.12680 q^{37} -14.9935 q^{38} -7.58634 q^{39} +2.12162 q^{41} -2.74730 q^{42} -5.36459 q^{43} +3.53590 q^{44} +1.86556 q^{46} -6.57020 q^{47} +7.02346 q^{48} +1.00000 q^{49} +5.81267 q^{51} +7.62589 q^{52} -12.6510 q^{53} +10.5258 q^{54} -0.969501 q^{56} +11.8356 q^{57} -6.92302 q^{58} +2.07665 q^{59} -2.74616 q^{61} +7.84188 q^{62} -0.831331 q^{63} -3.44274 q^{64} -6.56223 q^{66} -5.56259 q^{67} -5.84296 q^{68} -1.47264 q^{69} +6.20033 q^{71} +0.805976 q^{72} -5.19586 q^{73} +3.96767 q^{74} -11.8973 q^{76} +2.38861 q^{77} -14.1528 q^{78} -6.41433 q^{79} -5.81490 q^{81} +3.95801 q^{82} +6.39627 q^{83} -2.17997 q^{84} -10.0080 q^{86} +5.46491 q^{87} -2.31576 q^{88} -1.93591 q^{89} +5.15152 q^{91} +1.48032 q^{92} -6.19024 q^{93} -12.2571 q^{94} +10.2472 q^{96} -5.62379 q^{97} +1.86556 q^{98} -1.98572 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} - 2 q^{3} + 5 q^{4} - q^{6} + 8 q^{7} - 12 q^{8}+O(q^{10}) \) 8 * q - 3 * q^2 - 2 * q^3 + 5 * q^4 - q^6 + 8 * q^7 - 12 * q^8	\( 8 q - 3 q^{2} - 2 q^{3} + 5 q^{4} - q^{6} + 8 q^{7} - 12 q^{8} - 5 q^{11} - 3 q^{12} - 9 q^{13} - 3 q^{14} - q^{16} + q^{17} - 6 q^{18} - 4 q^{19} - 2 q^{21} + 5 q^{22} + 8 q^{23} + 16 q^{24} + 22 q^{26} + q^{27} + 5 q^{28} - 5 q^{29} - q^{31} - 2 q^{32} - 12 q^{33} - 2 q^{34} + 16 q^{36} - 18 q^{37} - 14 q^{38} + 14 q^{39} - q^{41} - q^{42} - 20 q^{43} - 3 q^{46} - 10 q^{47} - 31 q^{48} + 8 q^{49} - 4 q^{51} - 11 q^{52} - 11 q^{53} + 29 q^{54} - 12 q^{56} - 8 q^{57} - 24 q^{58} + 20 q^{59} - 6 q^{61} - 2 q^{62} + 8 q^{64} - 37 q^{66} - 23 q^{67} - 9 q^{68} - 2 q^{69} + 3 q^{71} - 29 q^{72} + 8 q^{73} + 35 q^{74} - 29 q^{76} - 5 q^{77} - 31 q^{78} + 4 q^{79} - 44 q^{81} - 27 q^{82} - 4 q^{83} - 3 q^{84} - 18 q^{86} - 27 q^{87} + 4 q^{88} - 17 q^{89} - 9 q^{91} + 5 q^{92} + 7 q^{93} + 13 q^{94} + 22 q^{96} - 41 q^{97} - 3 q^{98} + 7 q^{99}+O(q^{100}) \) 8 * q - 3 * q^2 - 2 * q^3 + 5 * q^4 - q^6 + 8 * q^7 - 12 * q^8 - 5 * q^11 - 3 * q^12 - 9 * q^13 - 3 * q^14 - q^16 + q^17 - 6 * q^18 - 4 * q^19 - 2 * q^21 + 5 * q^22 + 8 * q^23 + 16 * q^24 + 22 * q^26 + q^27 + 5 * q^28 - 5 * q^29 - q^31 - 2 * q^32 - 12 * q^33 - 2 * q^34 + 16 * q^36 - 18 * q^37 - 14 * q^38 + 14 * q^39 - q^41 - q^42 - 20 * q^43 - 3 * q^46 - 10 * q^47 - 31 * q^48 + 8 * q^49 - 4 * q^51 - 11 * q^52 - 11 * q^53 + 29 * q^54 - 12 * q^56 - 8 * q^57 - 24 * q^58 + 20 * q^59 - 6 * q^61 - 2 * q^62 + 8 * q^64 - 37 * q^66 - 23 * q^67 - 9 * q^68 - 2 * q^69 + 3 * q^71 - 29 * q^72 + 8 * q^73 + 35 * q^74 - 29 * q^76 - 5 * q^77 - 31 * q^78 + 4 * q^79 - 44 * q^81 - 27 * q^82 - 4 * q^83 - 3 * q^84 - 18 * q^86 - 27 * q^87 + 4 * q^88 - 17 * q^89 - 9 * q^91 + 5 * q^92 + 7 * q^93 + 13 * q^94 + 22 * q^96 - 41 * q^97 - 3 * q^98 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more