LMFDB - Embedded newform 722.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [722,2,Mod(1,722)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(722, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("722.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 722 = 2 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	722.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:	$5.76519902594$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.90211$ of defining polynomial
Character	$\chi$	$=$	722.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.79360 q^{3} +1.00000 q^{4} -2.34458 q^{5} +2.79360 q^{6} -1.28408 q^{7} -1.00000 q^{8} +4.80423 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.79360 q^{3} +1.00000 q^{4} -2.34458 q^{5} +2.79360 q^{6} -1.28408 q^{7} -1.00000 q^{8} +4.80423 q^{9} +2.34458 q^{10} +5.75621 q^{11} -2.79360 q^{12} -0.304282 q^{13} +1.28408 q^{14} +6.54982 q^{15} +1.00000 q^{16} +4.18619 q^{17} -4.80423 q^{18} -2.34458 q^{20} +3.58721 q^{21} -5.75621 q^{22} -6.47684 q^{23} +2.79360 q^{24} +0.497039 q^{25} +0.304282 q^{26} -5.04029 q^{27} -1.28408 q^{28} +3.12756 q^{29} -6.54982 q^{30} -6.44246 q^{31} -1.00000 q^{32} -16.0806 q^{33} -4.18619 q^{34} +3.01062 q^{35} +4.80423 q^{36} +3.97980 q^{37} +0.850045 q^{39} +2.34458 q^{40} +5.01719 q^{41} -3.58721 q^{42} -0.989378 q^{43} +5.75621 q^{44} -11.2639 q^{45} +6.47684 q^{46} -4.39445 q^{47} -2.79360 q^{48} -5.35114 q^{49} -0.497039 q^{50} -11.6946 q^{51} -0.304282 q^{52} -3.29064 q^{53} +5.04029 q^{54} -13.4959 q^{55} +1.28408 q^{56} -3.12756 q^{58} -3.31375 q^{59} +6.54982 q^{60} -10.9615 q^{61} +6.44246 q^{62} -6.16901 q^{63} +1.00000 q^{64} +0.713414 q^{65} +16.0806 q^{66} +4.38081 q^{67} +4.18619 q^{68} +18.0937 q^{69} -3.01062 q^{70} +4.41570 q^{71} -4.80423 q^{72} -2.26689 q^{73} -3.97980 q^{74} -1.38853 q^{75} -7.39144 q^{77} -0.850045 q^{78} -8.57659 q^{79} -2.34458 q^{80} -0.332090 q^{81} -5.01719 q^{82} -9.76464 q^{83} +3.58721 q^{84} -9.81485 q^{85} +0.989378 q^{86} -8.73716 q^{87} -5.75621 q^{88} +15.0765 q^{89} +11.2639 q^{90} +0.390723 q^{91} -6.47684 q^{92} +17.9977 q^{93} +4.39445 q^{94} +2.79360 q^{96} -17.3522 q^{97} +5.35114 q^{98} +27.6542 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 18 q^{13} + 2 q^{14} - 4 q^{15} + 4 q^{16} + 6 q^{17} - 4 q^{18} - 2 q^{20} - 4 q^{21} - 2 q^{22} - 10 q^{23} + 2 q^{24} + 6 q^{25} + 18 q^{26} + 4 q^{27} - 2 q^{28} + 2 q^{29} + 4 q^{30} - 26 q^{31} - 4 q^{32} - 16 q^{33} - 6 q^{34} + 6 q^{35} + 4 q^{36} - 4 q^{37} - 6 q^{39} + 2 q^{40} + 12 q^{41} + 4 q^{42} - 10 q^{43} + 2 q^{44} - 22 q^{45} + 10 q^{46} - 12 q^{47} - 2 q^{48} - 12 q^{49} - 6 q^{50} + 2 q^{51} - 18 q^{52} - 8 q^{53} - 4 q^{54} - 26 q^{55} + 2 q^{56} - 2 q^{58} + 8 q^{59} - 4 q^{60} + 26 q^{62} - 22 q^{63} + 4 q^{64} + 4 q^{65} + 16 q^{66} - 10 q^{67} + 6 q^{68} + 20 q^{69} - 6 q^{70} - 4 q^{72} - 14 q^{73} + 4 q^{74} - 8 q^{75} + 4 q^{77} + 6 q^{78} - 22 q^{79} - 2 q^{80} - 4 q^{81} - 12 q^{82} - 12 q^{83} - 4 q^{84} - 18 q^{85} + 10 q^{86} - 26 q^{87} - 2 q^{88} + 16 q^{89} + 22 q^{90} + 4 q^{91} - 10 q^{92} + 8 q^{93} + 12 q^{94} + 2 q^{96} - 28 q^{97} + 12 q^{98} + 22 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 + 2 * q^10 + 2 * q^11 - 2 * q^12 - 18 * q^13 + 2 * q^14 - 4 * q^15 + 4 * q^16 + 6 * q^17 - 4 * q^18 - 2 * q^20 - 4 * q^21 - 2 * q^22 - 10 * q^23 + 2 * q^24 + 6 * q^25 + 18 * q^26 + 4 * q^27 - 2 * q^28 + 2 * q^29 + 4 * q^30 - 26 * q^31 - 4 * q^32 - 16 * q^33 - 6 * q^34 + 6 * q^35 + 4 * q^36 - 4 * q^37 - 6 * q^39 + 2 * q^40 + 12 * q^41 + 4 * q^42 - 10 * q^43 + 2 * q^44 - 22 * q^45 + 10 * q^46 - 12 * q^47 - 2 * q^48 - 12 * q^49 - 6 * q^50 + 2 * q^51 - 18 * q^52 - 8 * q^53 - 4 * q^54 - 26 * q^55 + 2 * q^56 - 2 * q^58 + 8 * q^59 - 4 * q^60 + 26 * q^62 - 22 * q^63 + 4 * q^64 + 4 * q^65 + 16 * q^66 - 10 * q^67 + 6 * q^68 + 20 * q^69 - 6 * q^70 - 4 * q^72 - 14 * q^73 + 4 * q^74 - 8 * q^75 + 4 * q^77 + 6 * q^78 - 22 * q^79 - 2 * q^80 - 4 * q^81 - 12 * q^82 - 12 * q^83 - 4 * q^84 - 18 * q^85 + 10 * q^86 - 26 * q^87 - 2 * q^88 + 16 * q^89 + 22 * q^90 + 4 * q^91 - 10 * q^92 + 8 * q^93 + 12 * q^94 + 2 * q^96 - 28 * q^97 + 12 * q^98 + 22 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.79360	−1.61289	−0.806444	−	0.591310i	$-0.798611\pi$
$3$	−2.79360	−1.61289	−0.806444	+	0.591310i	$0.798611\pi$
$4$	1.00000	0.500000
$5$	−2.34458	−1.04853	−0.524263	−	0.851556i	$-0.675659\pi$
$5$	−2.34458	−1.04853	−0.524263	+	0.851556i	$0.675659\pi$
$6$	2.79360	1.14048
$7$	−1.28408	−0.485336	−0.242668	−	0.970109i	$-0.578023\pi$
$7$	−1.28408	−0.485336	−0.242668	+	0.970109i	$0.578023\pi$
$8$	−1.00000	−0.353553
$9$	4.80423	1.60141
$10$	2.34458	0.741420
$11$	5.75621	1.73556	0.867782	−	0.496945i	$-0.165545\pi$
$11$	5.75621	1.73556	0.867782	+	0.496945i	$0.165545\pi$
$12$	−2.79360	−0.806444
$13$	−0.304282	−0.0843928	−0.0421964	−	0.999109i	$-0.513436\pi$
$13$	−0.304282	−0.0843928	−0.0421964	+	0.999109i	$0.513436\pi$
$14$	1.28408	0.343185
$15$	6.54982	1.69116
$16$	1.00000	0.250000
$17$	4.18619	1.01530	0.507650	−	0.861563i	$-0.330514\pi$
$17$	4.18619	1.01530	0.507650	+	0.861563i	$0.330514\pi$
$18$	−4.80423	−1.13237
$19$	0	0
$19$	0	0
$20$	−2.34458	−0.524263
$21$	3.58721	0.782793
$22$	−5.75621	−1.22723
$23$	−6.47684	−1.35051	−0.675257	−	0.737583i	$-0.735967\pi$
$23$	−6.47684	−1.35051	−0.675257	+	0.737583i	$0.735967\pi$
$24$	2.79360	0.570242
$25$	0.497039	0.0994078
$26$	0.304282	0.0596747
$27$	−5.04029	−0.970005
$28$	−1.28408	−0.242668
$29$	3.12756	0.580773	0.290387	−	0.956909i	$-0.406216\pi$
$29$	3.12756	0.580773	0.290387	+	0.956909i	$0.406216\pi$
$30$	−6.54982	−1.19583
$31$	−6.44246	−1.15710	−0.578550	−	0.815647i	$-0.696381\pi$
$31$	−6.44246	−1.15710	−0.578550	+	0.815647i	$0.696381\pi$
$32$	−1.00000	−0.176777
$33$	−16.0806	−2.79927
$34$	−4.18619	−0.717926
$35$	3.01062	0.508888
$36$	4.80423	0.800704
$37$	3.97980	0.654275	0.327137	−	0.944977i	$-0.393916\pi$
$37$	3.97980	0.654275	0.327137	+	0.944977i	$0.393916\pi$
$38$	0	0
$39$	0.850045	0.136116
$40$	2.34458	0.370710
$41$	5.01719	0.783553	0.391776	−	0.920060i	$-0.371861\pi$
$41$	5.01719	0.783553	0.391776	+	0.920060i	$0.371861\pi$
$42$	−3.58721	−0.553518
$43$	−0.989378	−0.150879	−0.0754394	−	0.997150i	$-0.524036\pi$
$43$	−0.989378	−0.150879	−0.0754394	+	0.997150i	$0.524036\pi$
$44$	5.75621	0.867782
$45$	−11.2639	−1.67912
$46$	6.47684	0.954957
$47$	−4.39445	−0.640997	−0.320498	−	0.947249i	$-0.603850\pi$
$47$	−4.39445	−0.640997	−0.320498	+	0.947249i	$0.603850\pi$
$48$	−2.79360	−0.403222
$49$	−5.35114	−0.764449
$50$	−0.497039	−0.0702919
$51$	−11.6946	−1.63757
$52$	−0.304282	−0.0421964
$53$	−3.29064	−0.452005	−0.226002	−	0.974127i	$-0.572566\pi$
$53$	−3.29064	−0.452005	−0.226002	+	0.974127i	$0.572566\pi$
$54$	5.04029	0.685897
$55$	−13.4959	−1.81978
$56$	1.28408	0.171592
$57$	0	0
$58$	−3.12756	−0.410669
$59$	−3.31375	−0.431414	−0.215707	−	0.976458i	$-0.569206\pi$
$59$	−3.31375	−0.431414	−0.215707	+	0.976458i	$0.569206\pi$
$60$	6.54982	0.845578
$61$	−10.9615	−1.40347	−0.701735	−	0.712438i	$-0.747591\pi$
$61$	−10.9615	−1.40347	−0.701735	+	0.712438i	$0.747591\pi$
$62$	6.44246	0.818194
$63$	−6.16901	−0.777222
$64$	1.00000	0.125000
$65$	0.713414	0.0884881
$66$	16.0806	1.97938
$67$	4.38081	0.535202	0.267601	−	0.963530i	$-0.413769\pi$
$67$	4.38081	0.535202	0.267601	+	0.963530i	$0.413769\pi$
$68$	4.18619	0.507650
$69$	18.0937	2.17823
$70$	−3.01062	−0.359838
$71$	4.41570	0.524047	0.262023	−	0.965062i	$-0.415610\pi$
$71$	4.41570	0.524047	0.262023	+	0.965062i	$0.415610\pi$
$72$	−4.80423	−0.566183
$73$	−2.26689	−0.265320	−0.132660	−	0.991162i	$-0.542352\pi$
$73$	−2.26689	−0.265320	−0.132660	+	0.991162i	$0.542352\pi$
$74$	−3.97980	−0.462642
$75$	−1.38853	−0.160334
$76$	0	0
$77$	−7.39144	−0.842332
$78$	−0.850045	−0.0962486
$79$	−8.57659	−0.964941	−0.482471	−	0.875912i	$-0.660261\pi$
$79$	−8.57659	−0.964941	−0.482471	+	0.875912i	$0.660261\pi$
$80$	−2.34458	−0.262132
$81$	−0.332090	−0.0368989
$82$	−5.01719	−0.554056
$83$	−9.76464	−1.07181	−0.535904	−	0.844279i	$-0.680029\pi$
$83$	−9.76464	−1.07181	−0.535904	+	0.844279i	$0.680029\pi$
$84$	3.58721	0.391397
$85$	−9.81485	−1.06457
$86$	0.989378	0.106687
$87$	−8.73716	−0.936722
$88$	−5.75621	−0.613615
$89$	15.0765	1.59811	0.799055	−	0.601259i	$-0.205334\pi$
$89$	15.0765	1.59811	0.799055	+	0.601259i	$0.205334\pi$
$90$	11.2639	1.18732
$91$	0.390723	0.0409589
$92$	−6.47684	−0.675257
$93$	17.9977	1.86627
$94$	4.39445	0.453253
$95$	0	0
$96$	2.79360	0.285121
$97$	−17.3522	−1.76185	−0.880924	−	0.473259i	$-0.843078\pi$
$97$	−17.3522	−1.76185	−0.880924	+	0.473259i	$0.843078\pi$
$98$	5.35114	0.540547
$99$	27.6542	2.77935
$100$	0.497039	0.0497039
$101$	3.67667	0.365842	0.182921	−	0.983128i	$-0.441445\pi$
$101$	3.67667	0.365842	0.182921	+	0.983128i	$0.441445\pi$
$102$	11.6946	1.15793
$103$	−19.0135	−1.87346	−0.936729	−	0.350055i	$-0.886163\pi$
$103$	−19.0135	−1.87346	−0.936729	+	0.350055i	$0.886163\pi$
$104$	0.304282	0.0298374
$105$	−8.41049	−0.820779
$106$	3.29064	0.319616
$107$	−10.0373	−0.970340	−0.485170	−	0.874420i	$-0.661242\pi$
$107$	−10.0373	−0.970340	−0.485170	+	0.874420i	$0.661242\pi$
$108$	−5.04029	−0.485003
$109$	8.84348	0.847052	0.423526	−	0.905884i	$-0.360792\pi$
$109$	8.84348	0.847052	0.423526	+	0.905884i	$0.360792\pi$
$110$	13.4959	1.28678
$111$	−11.1180	−1.05527
$112$	−1.28408	−0.121334
$113$	14.2502	1.34055	0.670275	−	0.742113i	$-0.266176\pi$
$113$	14.2502	1.34055	0.670275	+	0.742113i	$0.266176\pi$
$114$	0	0
$115$	15.1854	1.41605
$116$	3.12756	0.290387
$117$	−1.46184	−0.135147
$118$	3.31375	0.305056
$119$	−5.37540	−0.492762
$120$	−6.54982	−0.597914
$121$	22.1340	2.01218
$122$	10.9615	0.992404
$123$	−14.0160	−1.26378
$124$	−6.44246	−0.578550
$125$	10.5575	0.944295
$126$	6.16901	0.549579
$127$	−2.23015	−0.197893	−0.0989467	−	0.995093i	$-0.531547\pi$
$127$	−2.23015	−0.197893	−0.0989467	+	0.995093i	$0.531547\pi$
$128$	−1.00000	−0.0883883
$129$	2.76393	0.243351
$130$	−0.713414	−0.0625705
$131$	1.52265	0.133035	0.0665175	−	0.997785i	$-0.478811\pi$
$131$	1.52265	0.133035	0.0665175	+	0.997785i	$0.478811\pi$
$132$	−16.0806	−1.39964
$133$	0	0
$134$	−4.38081	−0.378445
$135$	11.8174	1.01708
$136$	−4.18619	−0.358963
$137$	10.7118	0.915167	0.457583	−	0.889167i	$-0.348715\pi$
$137$	10.7118	0.915167	0.457583	+	0.889167i	$0.348715\pi$
$138$	−18.0937	−1.54024
$139$	−10.8143	−0.917260	−0.458630	−	0.888627i	$-0.651660\pi$
$139$	−10.8143	−0.917260	−0.458630	+	0.888627i	$0.651660\pi$
$140$	3.01062	0.254444
$141$	12.2764	1.03386
$142$	−4.41570	−0.370557
$143$	−1.75152	−0.146469
$144$	4.80423	0.400352
$145$	−7.33280	−0.608956
$146$	2.26689	0.187609
$147$	14.9490	1.23297
$148$	3.97980	0.327137
$149$	−9.33384	−0.764658	−0.382329	−	0.924026i	$-0.624878\pi$
$149$	−9.33384	−0.764658	−0.382329	+	0.924026i	$0.624878\pi$
$150$	1.38853	0.113373
$151$	−19.8232	−1.61319	−0.806593	−	0.591107i	$-0.798691\pi$
$151$	−19.8232	−1.61319	−0.806593	+	0.591107i	$0.798691\pi$
$152$	0	0
$153$	20.1114	1.62591
$154$	7.39144	0.595619
$155$	15.1048	1.21325
$156$	0.850045	0.0680581
$157$	13.8676	1.10676	0.553379	−	0.832930i	$-0.313338\pi$
$157$	13.8676	1.10676	0.553379	+	0.832930i	$0.313338\pi$
$158$	8.57659	0.682317
$159$	9.19276	0.729033
$160$	2.34458	0.185355
$161$	8.31677	0.655453
$162$	0.332090	0.0260915
$163$	14.2496	1.11611	0.558057	−	0.829803i	$-0.311547\pi$
$163$	14.2496	1.11611	0.558057	+	0.829803i	$0.311547\pi$
$164$	5.01719	0.391776
$165$	37.7022	2.93511
$166$	9.76464	0.757883
$167$	−11.5901	−0.896870	−0.448435	−	0.893815i	$-0.648018\pi$
$167$	−11.5901	−0.896870	−0.448435	+	0.893815i	$0.648018\pi$
$168$	−3.58721	−0.276759
$169$	−12.9074	−0.992878
$170$	9.81485	0.752764
$171$	0	0
$172$	−0.989378	−0.0754394
$173$	−3.98353	−0.302862	−0.151431	−	0.988468i	$-0.548388\pi$
$173$	−3.98353	−0.302862	−0.151431	+	0.988468i	$0.548388\pi$
$174$	8.73716	0.662363
$175$	−0.638237	−0.0482462
$176$	5.75621	0.433891
$177$	9.25731	0.695822
$178$	−15.0765	−1.13003
$179$	8.45089	0.631649	0.315825	−	0.948818i	$-0.397719\pi$
$179$	8.45089	0.631649	0.315825	+	0.948818i	$0.397719\pi$
$180$	−11.2639	−0.839560
$181$	−8.48526	−0.630705	−0.315352	−	0.948975i	$-0.602123\pi$
$181$	−8.48526	−0.630705	−0.315352	+	0.948975i	$0.602123\pi$
$182$	−0.390723	−0.0289623
$183$	30.6220	2.26364
$184$	6.47684	0.477479
$185$	−9.33094	−0.686024
$186$	−17.9977	−1.31966
$187$	24.0966	1.76212
$188$	−4.39445	−0.320498
$189$	6.47214	0.470779
$190$	0	0
$191$	−22.3790	−1.61929	−0.809644	−	0.586921i	$-0.800340\pi$
$191$	−22.3790	−1.61929	−0.809644	+	0.586921i	$0.800340\pi$
$192$	−2.79360	−0.201611
$193$	−7.42811	−0.534687	−0.267344	−	0.963601i	$-0.586146\pi$
$193$	−7.42811	−0.534687	−0.267344	+	0.963601i	$0.586146\pi$
$194$	17.3522	1.24581
$195$	−1.99300	−0.142721
$196$	−5.35114	−0.382224
$197$	8.53554	0.608132	0.304066	−	0.952651i	$-0.401656\pi$
$197$	8.53554	0.608132	0.304066	+	0.952651i	$0.401656\pi$
$198$	−27.6542	−1.96530
$199$	−6.35926	−0.450796	−0.225398	−	0.974267i	$-0.572368\pi$
$199$	−6.35926	−0.450796	−0.225398	+	0.974267i	$0.572368\pi$
$200$	−0.497039	−0.0351460
$201$	−12.2383	−0.863220
$202$	−3.67667	−0.258689
$203$	−4.01603	−0.281870
$204$	−11.6946	−0.818783
$205$	−11.7632	−0.821576
$206$	19.0135	1.32474
$207$	−31.1162	−2.16272
$208$	−0.304282	−0.0210982
$209$	0	0
$210$	8.41049	0.580379
$211$	13.9578	0.960894	0.480447	−	0.877024i	$-0.340474\pi$
$211$	13.9578	0.960894	0.480447	+	0.877024i	$0.340474\pi$
$212$	−3.29064	−0.226002
$213$	−12.3357	−0.845229
$214$	10.0373	0.686134
$215$	2.31967	0.158200
$216$	5.04029	0.342949
$217$	8.27263	0.561583
$218$	−8.84348	−0.598956
$219$	6.33280	0.427931
$220$	−13.4959	−0.909892
$221$	−1.27378	−0.0856840
$222$	11.1180	0.746190
$223$	−8.92400	−0.597595	−0.298798	−	0.954317i	$-0.596586\pi$
$223$	−8.92400	−0.597595	−0.298798	+	0.954317i	$0.596586\pi$
$224$	1.28408	0.0857961
$225$	2.38789	0.159193
$226$	−14.2502	−0.947912
$227$	0.659796	0.0437922	0.0218961	−	0.999760i	$-0.493030\pi$
$227$	0.659796	0.0437922	0.0218961	+	0.999760i	$0.493030\pi$
$228$	0	0
$229$	−6.97208	−0.460728	−0.230364	−	0.973105i	$-0.573992\pi$
$229$	−6.97208	−0.460728	−0.230364	+	0.973105i	$0.573992\pi$
$230$	−15.1854	−1.00130
$231$	20.6487	1.35859
$232$	−3.12756	−0.205334
$233$	−29.6097	−1.93980	−0.969898	−	0.243512i	$-0.921700\pi$
$233$	−29.6097	−1.93980	−0.969898	+	0.243512i	$0.921700\pi$
$234$	1.46184	0.0955636
$235$	10.3031	0.672102
$236$	−3.31375	−0.215707
$237$	23.9596	1.55634
$238$	5.37540	0.348436
$239$	−24.7034	−1.59793	−0.798965	−	0.601378i	$-0.794619\pi$
$239$	−24.7034	−1.59793	−0.798965	+	0.601378i	$0.794619\pi$
$240$	6.54982	0.422789
$241$	−27.5586	−1.77521	−0.887604	−	0.460607i	$-0.847632\pi$
$241$	−27.5586	−1.77521	−0.887604	+	0.460607i	$0.847632\pi$
$242$	−22.1340	−1.42283
$243$	16.0486	1.02952
$244$	−10.9615	−0.701735
$245$	12.5462	0.801545
$246$	14.0160	0.893630
$247$	0	0
$248$	6.44246	0.409097
$249$	27.2786	1.72871
$250$	−10.5575	−0.667717
$251$	−2.98855	−0.188636	−0.0943179	−	0.995542i	$-0.530067\pi$
$251$	−2.98855	−0.188636	−0.0943179	+	0.995542i	$0.530067\pi$
$252$	−6.16901	−0.388611
$253$	−37.2821	−2.34390
$254$	2.23015	0.139932
$255$	27.4188	1.71703
$256$	1.00000	0.0625000
$257$	−14.1323	−0.881546	−0.440773	−	0.897619i	$-0.645296\pi$
$257$	−14.1323	−0.881546	−0.440773	+	0.897619i	$0.645296\pi$
$258$	−2.76393	−0.172075
$259$	−5.11037	−0.317543
$260$	0.713414	0.0442440
$261$	15.0255	0.930055
$262$	−1.52265	−0.0940699
$263$	7.56887	0.466717	0.233358	−	0.972391i	$-0.425029\pi$
$263$	7.56887	0.466717	0.233358	+	0.972391i	$0.425029\pi$
$264$	16.0806	0.989692
$265$	7.71517	0.473939
$266$	0	0
$267$	−42.1179	−2.57757
$268$	4.38081	0.267601
$269$	7.67074	0.467694	0.233847	−	0.972273i	$-0.424869\pi$
$269$	7.67074	0.467694	0.233847	+	0.972273i	$0.424869\pi$
$270$	−11.8174	−0.719181
$271$	11.6649	0.708592	0.354296	−	0.935133i	$-0.384721\pi$
$271$	11.6649	0.708592	0.354296	+	0.935133i	$0.384721\pi$
$272$	4.18619	0.253825
$273$	−1.09152	−0.0660621
$274$	−10.7118	−0.647121
$275$	2.86106	0.172529
$276$	18.0937	1.08911
$277$	27.4405	1.64874	0.824370	−	0.566052i	$-0.191530\pi$
$277$	27.4405	1.64874	0.824370	+	0.566052i	$0.191530\pi$
$278$	10.8143	0.648601
$279$	−30.9511	−1.85299
$280$	−3.01062	−0.179919
$281$	−16.2715	−0.970675	−0.485338	−	0.874327i	$-0.661303\pi$
$281$	−16.2715	−0.970675	−0.485338	+	0.874327i	$0.661303\pi$
$282$	−12.2764	−0.731047
$283$	3.39374	0.201737	0.100868	−	0.994900i	$-0.467838\pi$
$283$	3.39374	0.201737	0.100868	+	0.994900i	$0.467838\pi$
$284$	4.41570	0.262023
$285$	0	0
$286$	1.75152	0.103569
$287$	−6.44246	−0.380287
$288$	−4.80423	−0.283092
$289$	0.524204	0.0308355
$290$	7.33280	0.430597
$291$	48.4751	2.84166
$292$	−2.26689	−0.132660
$293$	−23.3783	−1.36578	−0.682888	−	0.730523i	$-0.739276\pi$
$293$	−23.3783	−1.36578	−0.682888	+	0.730523i	$0.739276\pi$
$294$	−14.9490	−0.871842
$295$	7.76934	0.452349
$296$	−3.97980	−0.231321
$297$	−29.0130	−1.68351
$298$	9.33384	0.540695
$299$	1.97079	0.113974
$300$	−1.38853	−0.0801668
$301$	1.27044	0.0732270
$302$	19.8232	1.14069
$303$	−10.2712	−0.590062
$304$	0	0
$305$	25.7000	1.47158
$306$	−20.1114	−1.14969
$307$	1.38802	0.0792185	0.0396093	−	0.999215i	$-0.487389\pi$
$307$	1.38802	0.0792185	0.0396093	+	0.999215i	$0.487389\pi$
$308$	−7.39144	−0.421166
$309$	53.1163	3.02168
$310$	−15.1048	−0.857898
$311$	−12.1902	−0.691246	−0.345623	−	0.938374i	$-0.612332\pi$
$311$	−12.1902	−0.691246	−0.345623	+	0.938374i	$0.612332\pi$
$312$	−0.850045	−0.0481243
$313$	16.5314	0.934411	0.467205	−	0.884149i	$-0.345261\pi$
$313$	16.5314	0.934411	0.467205	+	0.884149i	$0.345261\pi$
$314$	−13.8676	−0.782596
$315$	14.4637	0.814938
$316$	−8.57659	−0.482471
$317$	−2.68811	−0.150979	−0.0754897	−	0.997147i	$-0.524052\pi$
$317$	−2.68811	−0.150979	−0.0754897	+	0.997147i	$0.524052\pi$
$318$	−9.19276	−0.515504
$319$	18.0029	1.00797
$320$	−2.34458	−0.131066
$321$	28.0402	1.56505
$322$	−8.31677	−0.463475
$323$	0	0
$324$	−0.332090	−0.0184495
$325$	−0.151240	−0.00838930
$326$	−14.2496	−0.789212
$327$	−24.7052	−1.36620
$328$	−5.01719	−0.277028
$329$	5.64282	0.311099
$330$	−37.7022	−2.07544
$331$	−5.21303	−0.286534	−0.143267	−	0.989684i	$-0.545761\pi$
$331$	−5.21303	−0.286534	−0.143267	+	0.989684i	$0.545761\pi$
$332$	−9.76464	−0.535904
$333$	19.1198	1.04776
$334$	11.5901	0.634183
$335$	−10.2712	−0.561173
$336$	3.58721	0.195698
$337$	26.7275	1.45594	0.727969	−	0.685610i	$-0.240464\pi$
$337$	26.7275	1.45594	0.727969	+	0.685610i	$0.240464\pi$
$338$	12.9074	0.702071
$339$	−39.8095	−2.16216
$340$	−9.81485	−0.532285
$341$	−37.0842	−2.00822
$342$	0	0
$343$	15.8598	0.856351
$344$	0.989378	0.0533437
$345$	−42.4221	−2.28393
$346$	3.98353	0.214156
$347$	−32.6682	−1.75372	−0.876860	−	0.480745i	$-0.840366\pi$
$347$	−32.6682	−1.75372	−0.876860	+	0.480745i	$0.840366\pi$
$348$	−8.73716	−0.468361
$349$	−29.1058	−1.55800	−0.778998	−	0.627026i	$-0.784272\pi$
$349$	−29.1058	−1.55800	−0.778998	+	0.627026i	$0.784272\pi$
$350$	0.638237	0.0341152
$351$	1.53367	0.0818614
$352$	−5.75621	−0.306807
$353$	−14.3049	−0.761372	−0.380686	−	0.924704i	$-0.624312\pi$
$353$	−14.3049	−0.761372	−0.380686	+	0.924704i	$0.624312\pi$
$354$	−9.25731	−0.492021
$355$	−10.3529	−0.549477
$356$	15.0765	0.799055
$357$	15.0167	0.794770
$358$	−8.45089	−0.446644
$359$	19.6707	1.03818	0.519090	−	0.854720i	$-0.326271\pi$
$359$	19.6707	1.03818	0.519090	+	0.854720i	$0.326271\pi$
$360$	11.2639	0.593658
$361$	0	0
$362$	8.48526	0.445976
$363$	−61.8337	−3.24543
$364$	0.390723	0.0204794
$365$	5.31490	0.278195
$366$	−30.6220	−1.60064
$367$	9.55053	0.498534	0.249267	−	0.968435i	$-0.419810\pi$
$367$	9.55053	0.498534	0.249267	+	0.968435i	$0.419810\pi$
$368$	−6.47684	−0.337628
$369$	24.1037	1.25479
$370$	9.33094	0.485092
$371$	4.22545	0.219374
$372$	17.9977	0.933137
$373$	−3.64010	−0.188477	−0.0942387	−	0.995550i	$-0.530042\pi$
$373$	−3.64010	−0.188477	−0.0942387	+	0.995550i	$0.530042\pi$
$374$	−24.0966	−1.24601
$375$	−29.4936	−1.52304
$376$	4.39445	0.226627
$377$	−0.951662	−0.0490131
$378$	−6.47214	−0.332891
$379$	−8.34741	−0.428778	−0.214389	−	0.976748i	$-0.568776\pi$
$379$	−8.34741	−0.428778	−0.214389	+	0.976748i	$0.568776\pi$
$380$	0	0
$381$	6.23015	0.319180
$382$	22.3790	1.14501
$383$	9.48566	0.484695	0.242347	−	0.970190i	$-0.422083\pi$
$383$	9.48566	0.484695	0.242347	+	0.970190i	$0.422083\pi$
$384$	2.79360	0.142561
$385$	17.3298	0.883208
$386$	7.42811	0.378081
$387$	−4.75320	−0.241619
$388$	−17.3522	−0.880924
$389$	36.0480	1.82770	0.913852	−	0.406047i	$-0.133093\pi$
$389$	36.0480	1.82770	0.913852	+	0.406047i	$0.133093\pi$
$390$	1.99300	0.100919
$391$	−27.1133	−1.37118
$392$	5.35114	0.270273
$393$	−4.25369	−0.214570
$394$	−8.53554	−0.430014
$395$	20.1085	1.01177
$396$	27.6542	1.38967
$397$	−6.40906	−0.321662	−0.160831	−	0.986982i	$-0.551417\pi$
$397$	−6.40906	−0.321662	−0.160831	+	0.986982i	$0.551417\pi$
$398$	6.35926	0.318761
$399$	0	0
$400$	0.497039	0.0248520
$401$	−0.187660	−0.00937128	−0.00468564	−	0.999989i	$-0.501491\pi$
$401$	−0.187660	−0.00937128	−0.00468564	+	0.999989i	$0.501491\pi$
$402$	12.2383	0.610389
$403$	1.96033	0.0976509
$404$	3.67667	0.182921
$405$	0.778611	0.0386895
$406$	4.01603	0.199312
$407$	22.9086	1.13554
$408$	11.6946	0.578967
$409$	−17.2663	−0.853762	−0.426881	−	0.904308i	$-0.640388\pi$
$409$	−17.2663	−0.853762	−0.426881	+	0.904308i	$0.640388\pi$
$410$	11.7632	0.580942
$411$	−29.9244	−1.47606
$412$	−19.0135	−0.936729
$413$	4.25512	0.209381
$414$	31.1162	1.52928
$415$	22.8940	1.12382
$416$	0.304282	0.0149187
$417$	30.2110	1.47944
$418$	0	0
$419$	−1.72572	−0.0843068	−0.0421534	−	0.999111i	$-0.513422\pi$
$419$	−1.72572	−0.0843068	−0.0421534	+	0.999111i	$0.513422\pi$
$420$	−8.41049	−0.410390
$421$	−32.7546	−1.59636	−0.798180	−	0.602420i	$-0.794203\pi$
$421$	−32.7546	−1.59636	−0.798180	+	0.602420i	$0.794203\pi$
$422$	−13.9578	−0.679455
$423$	−21.1119	−1.02650
$424$	3.29064	0.159808
$425$	2.08070	0.100929
$426$	12.3357	0.597667
$427$	14.0754	0.681155
$428$	−10.0373	−0.485170
$429$	4.89304	0.236238
$430$	−2.31967	−0.111865
$431$	25.5119	1.22887	0.614433	−	0.788969i	$-0.289385\pi$
$431$	25.5119	1.22887	0.614433	+	0.788969i	$0.289385\pi$
$432$	−5.04029	−0.242501
$433$	−24.4834	−1.17660	−0.588299	−	0.808644i	$-0.700202\pi$
$433$	−24.4834	−1.17660	−0.588299	+	0.808644i	$0.700202\pi$
$434$	−8.27263	−0.397099
$435$	20.4849	0.982178
$436$	8.84348	0.423526
$437$	0	0
$438$	−6.33280	−0.302593
$439$	30.8388	1.47186	0.735928	−	0.677060i	$-0.236746\pi$
$439$	30.8388	1.47186	0.735928	+	0.677060i	$0.236746\pi$
$440$	13.4959	0.643391
$441$	−25.7081	−1.22419
$442$	1.27378	0.0605878
$443$	10.7047	0.508595	0.254297	−	0.967126i	$-0.418156\pi$
$443$	10.7047	0.508595	0.254297	+	0.967126i	$0.418156\pi$
$444$	−11.1180	−0.527636
$445$	−35.3481	−1.67566
$446$	8.92400	0.422564
$447$	26.0751	1.23331
$448$	−1.28408	−0.0606670
$449$	−2.31562	−0.109281	−0.0546403	−	0.998506i	$-0.517401\pi$
$449$	−2.31562	−0.109281	−0.0546403	+	0.998506i	$0.517401\pi$
$450$	−2.38789	−0.112566
$451$	28.8800	1.35991
$452$	14.2502	0.670275
$453$	55.3781	2.60189
$454$	−0.659796	−0.0309657
$455$	−0.916079	−0.0429465
$456$	0	0
$457$	−0.0521810	−0.00244092	−0.00122046	−	0.999999i	$-0.500388\pi$
$457$	−0.0521810	−0.00244092	−0.00122046	+	0.999999i	$0.500388\pi$
$458$	6.97208	0.325784
$459$	−21.0996	−0.984847
$460$	15.1854	0.708025
$461$	39.4720	1.83840	0.919198	−	0.393796i	$-0.128838\pi$
$461$	39.4720	1.83840	0.919198	+	0.393796i	$0.128838\pi$
$462$	−20.6487	−0.960667
$463$	−6.19056	−0.287700	−0.143850	−	0.989600i	$-0.545948\pi$
$463$	−6.19056	−0.287700	−0.143850	+	0.989600i	$0.545948\pi$
$464$	3.12756	0.145193
$465$	−42.1970	−1.95684
$466$	29.6097	1.37164
$467$	−20.9962	−0.971586	−0.485793	−	0.874074i	$-0.661469\pi$
$467$	−20.9962	−0.971586	−0.485793	+	0.874074i	$0.661469\pi$
$468$	−1.46184	−0.0675737
$469$	−5.62531	−0.259753
$470$	−10.3031	−0.475248
$471$	−38.7407	−1.78508
$472$	3.31375	0.152528
$473$	−5.69507	−0.261860
$474$	−23.9596	−1.10050
$475$	0	0
$476$	−5.37540	−0.246381
$477$	−15.8090	−0.723844
$478$	24.7034	1.12991
$479$	15.5229	0.709261	0.354631	−	0.935006i	$-0.384607\pi$
$479$	15.5229	0.709261	0.354631	+	0.935006i	$0.384607\pi$
$480$	−6.54982	−0.298957
$481$	−1.21098	−0.0552160
$482$	27.5586	1.25526
$483$	−23.2338	−1.05717
$484$	22.1340	1.00609
$485$	40.6835	1.84734
$486$	−16.0486	−0.727980
$487$	28.6065	1.29629	0.648143	−	0.761519i	$-0.275546\pi$
$487$	28.6065	1.29629	0.648143	+	0.761519i	$0.275546\pi$
$488$	10.9615	0.496202
$489$	−39.8077	−1.80017
$490$	−12.5462	−0.566778
$491$	−30.3488	−1.36962	−0.684812	−	0.728720i	$-0.740116\pi$
$491$	−30.3488	−1.36962	−0.684812	+	0.728720i	$0.740116\pi$
$492$	−14.0160	−0.631892
$493$	13.0926	0.589659
$494$	0	0
$495$	−64.8373	−2.91422
$496$	−6.44246	−0.289275
$497$	−5.67010	−0.254339
$498$	−27.2786	−1.22238
$499$	8.77624	0.392878	0.196439	−	0.980516i	$-0.437062\pi$
$499$	8.77624	0.392878	0.196439	+	0.980516i	$0.437062\pi$
$500$	10.5575	0.472147
$501$	32.3782	1.44655
$502$	2.98855	0.133386
$503$	−21.1043	−0.940996	−0.470498	−	0.882401i	$-0.655926\pi$
$503$	−21.1043	−0.940996	−0.470498	+	0.882401i	$0.655926\pi$
$504$	6.16901	0.274789
$505$	−8.62023	−0.383595
$506$	37.2821	1.65739
$507$	36.0582	1.60140
$508$	−2.23015	−0.0989467
$509$	−22.2212	−0.984936	−0.492468	−	0.870331i	$-0.663905\pi$
$509$	−22.2212	−0.984936	−0.492468	+	0.870331i	$0.663905\pi$
$510$	−27.4188	−1.21412
$511$	2.91087	0.128769
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	14.1323	0.623347
$515$	44.5787	1.96437
$516$	2.76393	0.121675
$517$	−25.2954	−1.11249
$518$	5.11037	0.224537
$519$	11.1284	0.488482
$520$	−0.713414	−0.0312853
$521$	−10.0332	−0.439563	−0.219782	−	0.975549i	$-0.570534\pi$
$521$	−10.0332	−0.439563	−0.219782	+	0.975549i	$0.570534\pi$
$522$	−15.0255	−0.657648
$523$	−29.2823	−1.28042	−0.640212	−	0.768198i	$-0.721154\pi$
$523$	−29.2823	−1.28042	−0.640212	+	0.768198i	$0.721154\pi$
$524$	1.52265	0.0665175
$525$	1.78298	0.0778158
$526$	−7.56887	−0.330018
$527$	−26.9694	−1.17481
$528$	−16.0806	−0.699818
$529$	18.9494	0.823887
$530$	−7.71517	−0.335125
$531$	−15.9200	−0.690870
$532$	0	0
$533$	−1.52664	−0.0661262
$534$	42.1179	1.82262
$535$	23.5332	1.01743
$536$	−4.38081	−0.189222
$537$	−23.6085	−1.01878
$538$	−7.67074	−0.330709
$539$	−30.8023	−1.32675
$540$	11.8174	0.508538
$541$	−37.8404	−1.62688	−0.813442	−	0.581646i	$-0.802409\pi$
$541$	−37.8404	−1.62688	−0.813442	+	0.581646i	$0.802409\pi$
$542$	−11.6649	−0.501050
$543$	23.7045	1.01726
$544$	−4.18619	−0.179482
$545$	−20.7342	−0.888156
$546$	1.09152	0.0467130
$547$	14.4976	0.619871	0.309936	−	0.950758i	$-0.399692\pi$
$547$	14.4976	0.619871	0.309936	+	0.950758i	$0.399692\pi$
$548$	10.7118	0.457583
$549$	−52.6613	−2.24753
$550$	−2.86106	−0.121996
$551$	0	0
$552$	−18.0937	−0.770120
$553$	11.0130	0.468321
$554$	−27.4405	−1.16583
$555$	26.0669	1.10648
$556$	−10.8143	−0.458630
$557$	−4.33801	−0.183807	−0.0919037	−	0.995768i	$-0.529295\pi$
$557$	−4.33801	−0.183807	−0.0919037	+	0.995768i	$0.529295\pi$
$558$	30.9511	1.31026
$559$	0.301051	0.0127331
$560$	3.01062	0.127222
$561$	−67.3164	−2.84210
$562$	16.2715	0.686371
$563$	27.2936	1.15029	0.575144	−	0.818052i	$-0.304946\pi$
$563$	27.2936	1.15029	0.575144	+	0.818052i	$0.304946\pi$
$564$	12.2764	0.516928
$565$	−33.4108	−1.40560
$566$	−3.39374	−0.142650
$567$	0.426430	0.0179084
$568$	−4.41570	−0.185278
$569$	7.19043	0.301439	0.150719	−	0.988577i	$-0.451841\pi$
$569$	7.19043	0.301439	0.150719	+	0.988577i	$0.451841\pi$
$570$	0	0
$571$	42.0134	1.75821	0.879103	−	0.476631i	$-0.158142\pi$
$571$	42.0134	1.75821	0.879103	+	0.476631i	$0.158142\pi$
$572$	−1.75152	−0.0732345
$573$	62.5181	2.61173
$574$	6.44246	0.268903
$575$	−3.21924	−0.134252
$576$	4.80423	0.200176
$577$	21.6713	0.902190	0.451095	−	0.892476i	$-0.351034\pi$
$577$	21.6713	0.902190	0.451095	+	0.892476i	$0.351034\pi$
$578$	−0.524204	−0.0218040
$579$	20.7512	0.862391
$580$	−7.33280	−0.304478
$581$	12.5386	0.520188
$582$	−48.4751	−2.00936
$583$	−18.9417	−0.784483
$584$	2.26689	0.0938047
$585$	3.42740	0.141706
$586$	23.3783	0.965749
$587$	−1.26864	−0.0523626	−0.0261813	−	0.999657i	$-0.508335\pi$
$587$	−1.26864	−0.0523626	−0.0261813	+	0.999657i	$0.508335\pi$
$588$	14.9490	0.616485
$589$	0	0
$590$	−7.76934	−0.319859
$591$	−23.8449	−0.980849
$592$	3.97980	0.163569
$593$	21.5600	0.885363	0.442682	−	0.896679i	$-0.354027\pi$
$593$	21.5600	0.885363	0.442682	+	0.896679i	$0.354027\pi$
$594$	29.0130	1.19042
$595$	12.6030	0.516674
$596$	−9.33384	−0.382329
$597$	17.7652	0.727083
$598$	−1.97079	−0.0805915
$599$	28.6417	1.17027	0.585135	−	0.810936i	$-0.301042\pi$
$599$	28.6417	1.17027	0.585135	+	0.810936i	$0.301042\pi$
$600$	1.38853	0.0566865
$601$	−7.55865	−0.308324	−0.154162	−	0.988046i	$-0.549268\pi$
$601$	−7.55865	−0.308324	−0.154162	+	0.988046i	$0.549268\pi$
$602$	−1.27044	−0.0517793
$603$	21.0464	0.857076
$604$	−19.8232	−0.806593
$605$	−51.8949	−2.10983
$606$	10.2712	0.417237
$607$	−1.72737	−0.0701117	−0.0350558	−	0.999385i	$-0.511161\pi$
$607$	−1.72737	−0.0701117	−0.0350558	+	0.999385i	$0.511161\pi$
$608$	0	0
$609$	11.2192	0.454625
$610$	−25.7000	−1.04056
$611$	1.33715	0.0540955
$612$	20.1114	0.812956
$613$	−27.4169	−1.10736	−0.553680	−	0.832730i	$-0.686777\pi$
$613$	−27.4169	−1.10736	−0.553680	+	0.832730i	$0.686777\pi$
$614$	−1.38802	−0.0560160
$615$	32.8617	1.32511
$616$	7.39144	0.297809
$617$	10.6275	0.427847	0.213924	−	0.976850i	$-0.431376\pi$
$617$	10.6275	0.427847	0.213924	+	0.976850i	$0.431376\pi$
$618$	−53.1163	−2.13665
$619$	8.36008	0.336020	0.168010	−	0.985785i	$-0.446266\pi$
$619$	8.36008	0.336020	0.168010	+	0.985785i	$0.446266\pi$
$620$	15.1048	0.606625
$621$	32.6452	1.31000
$622$	12.1902	0.488784
$623$	−19.3595	−0.775620
$624$	0.850045	0.0340290
$625$	−27.2381	−1.08953
$626$	−16.5314	−0.660728
$627$	0	0
$628$	13.8676	0.553379
$629$	16.6602	0.664285
$630$	−14.4637	−0.576248
$631$	24.5301	0.976530	0.488265	−	0.872696i	$-0.337630\pi$
$631$	24.5301	0.976530	0.488265	+	0.872696i	$0.337630\pi$
$632$	8.57659	0.341158
$633$	−38.9926	−1.54982
$634$	2.68811	0.106759
$635$	5.22875	0.207497
$636$	9.19276	0.364517
$637$	1.62826	0.0645139
$638$	−18.0029	−0.712742
$639$	21.2140	0.839213
$640$	2.34458	0.0926775
$641$	6.97270	0.275405	0.137703	−	0.990474i	$-0.456028\pi$
$641$	6.97270	0.275405	0.137703	+	0.990474i	$0.456028\pi$
$642$	−28.0402	−1.10666
$643$	−9.09024	−0.358484	−0.179242	−	0.983805i	$-0.557365\pi$
$643$	−9.09024	−0.358484	−0.179242	+	0.983805i	$0.557365\pi$
$644$	8.31677	0.327727
$645$	−6.48025	−0.255160
$646$	0	0
$647$	30.0074	1.17971	0.589856	−	0.807509i	$-0.299185\pi$
$647$	30.0074	1.17971	0.589856	+	0.807509i	$0.299185\pi$
$648$	0.332090	0.0130457
$649$	−19.0747	−0.748746
$650$	0.151240	0.00593213
$651$	−23.1105	−0.905770
$652$	14.2496	0.558057
$653$	10.4191	0.407732	0.203866	−	0.978999i	$-0.434649\pi$
$653$	10.4191	0.407732	0.203866	+	0.978999i	$0.434649\pi$
$654$	24.7052	0.966049
$655$	−3.56998	−0.139491
$656$	5.01719	0.195888
$657$	−10.8907	−0.424885
$658$	−5.64282	−0.219980
$659$	26.3756	1.02745	0.513724	−	0.857956i	$-0.328266\pi$
$659$	26.3756	1.02745	0.513724	+	0.857956i	$0.328266\pi$
$660$	37.7022	1.46755
$661$	24.6939	0.960483	0.480242	−	0.877136i	$-0.340549\pi$
$661$	24.6939	0.960483	0.480242	+	0.877136i	$0.340549\pi$
$662$	5.21303	0.202610
$663$	3.55845	0.138199
$664$	9.76464	0.378942
$665$	0	0
$666$	−19.1198	−0.740879
$667$	−20.2567	−0.784342
$668$	−11.5901	−0.448435
$669$	24.9301	0.963854
$670$	10.2712	0.396809
$671$	−63.0965	−2.43581
$672$	−3.58721	−0.138380
$673$	9.77380	0.376752	0.188376	−	0.982097i	$-0.439678\pi$
$673$	9.77380	0.376752	0.188376	+	0.982097i	$0.439678\pi$
$674$	−26.7275	−1.02950
$675$	−2.50522	−0.0964261
$676$	−12.9074	−0.496439
$677$	21.5747	0.829184	0.414592	−	0.910007i	$-0.363924\pi$
$677$	21.5747	0.829184	0.414592	+	0.910007i	$0.363924\pi$
$678$	39.8095	1.52888
$679$	22.2816	0.855088
$680$	9.81485	0.376382
$681$	−1.84321	−0.0706319
$682$	37.0842	1.42003
$683$	−33.2509	−1.27231	−0.636155	−	0.771561i	$-0.719476\pi$
$683$	−33.2509	−1.27231	−0.636155	+	0.771561i	$0.719476\pi$
$684$	0	0
$685$	−25.1145	−0.959576
$686$	−15.8598	−0.605532
$687$	19.4772	0.743103
$688$	−0.989378	−0.0377197
$689$	1.00129	0.0381459
$690$	42.4221	1.61498
$691$	4.80932	0.182955	0.0914776	−	0.995807i	$-0.470841\pi$
$691$	4.80932	0.182955	0.0914776	+	0.995807i	$0.470841\pi$
$692$	−3.98353	−0.151431
$693$	−35.5101	−1.34892
$694$	32.6682	1.24007
$695$	25.3550	0.961772
$696$	8.73716	0.331181
$697$	21.0029	0.795542
$698$	29.1058	1.10167
$699$	82.7178	3.12867
$700$	−0.638237	−0.0241231
$701$	3.46466	0.130859	0.0654293	−	0.997857i	$-0.479158\pi$
$701$	3.46466	0.130859	0.0654293	+	0.997857i	$0.479158\pi$
$702$	−1.53367	−0.0578848
$703$	0	0
$704$	5.75621	0.216946
$705$	−28.7829	−1.08403
$706$	14.3049	0.538371
$707$	−4.72113	−0.177556
$708$	9.25731	0.347911
$709$	20.5743	0.772684	0.386342	−	0.922356i	$-0.373738\pi$
$709$	20.5743	0.772684	0.386342	+	0.922356i	$0.373738\pi$
$710$	10.3529	0.388539
$711$	−41.2039	−1.54527
$712$	−15.0765	−0.565017
$713$	41.7268	1.56268
$714$	−15.0167	−0.561988
$715$	4.10656	0.153577
$716$	8.45089	0.315825
$717$	69.0115	2.57728
$718$	−19.6707	−0.734104
$719$	24.1885	0.902080	0.451040	−	0.892504i	$-0.351053\pi$
$719$	24.1885	0.902080	0.451040	+	0.892504i	$0.351053\pi$
$720$	−11.2639	−0.419780
$721$	24.4149	0.909257
$722$	0	0
$723$	76.9880	2.86321
$724$	−8.48526	−0.315352
$725$	1.55452	0.0577334
$726$	61.8337	2.29486
$727$	18.1276	0.672317	0.336158	−	0.941806i	$-0.390872\pi$
$727$	18.1276	0.672317	0.336158	+	0.941806i	$0.390872\pi$
$728$	−0.390723	−0.0144811
$729$	−43.8372	−1.62360
$730$	−5.31490	−0.196713
$731$	−4.14173	−0.153187
$732$	30.6220	1.13182
$733$	22.7363	0.839785	0.419893	−	0.907574i	$-0.362068\pi$
$733$	22.7363	0.839785	0.419893	+	0.907574i	$0.362068\pi$
$734$	−9.55053	−0.352517
$735$	−35.0490	−1.29280
$736$	6.47684	0.238739
$737$	25.2169	0.928877
$738$	−24.1037	−0.887269
$739$	15.1787	0.558356	0.279178	−	0.960239i	$-0.409938\pi$
$739$	15.1787	0.558356	0.279178	+	0.960239i	$0.409938\pi$
$740$	−9.33094	−0.343012
$741$	0	0
$742$	−4.22545	−0.155121
$743$	3.27625	0.120194	0.0600970	−	0.998193i	$-0.480859\pi$
$743$	3.27625	0.120194	0.0600970	+	0.998193i	$0.480859\pi$
$744$	−17.9977	−0.659828
$745$	21.8839	0.801764
$746$	3.64010	0.133274
$747$	−46.9116	−1.71640
$748$	24.0966	0.881060
$749$	12.8887	0.470941
$750$	29.4936	1.07695
$751$	−52.2323	−1.90598	−0.952992	−	0.302995i	$-0.902013\pi$
$751$	−52.2323	−1.90598	−0.952992	+	0.302995i	$0.902013\pi$
$752$	−4.39445	−0.160249
$753$	8.34884	0.304248
$754$	0.951662	0.0346575
$755$	46.4769	1.69147
$756$	6.47214	0.235389
$757$	−24.2342	−0.880807	−0.440404	−	0.897800i	$-0.645165\pi$
$757$	−24.2342	−0.880807	−0.440404	+	0.897800i	$0.645165\pi$
$758$	8.34741	0.303192
$759$	104.151	3.78045
$760$	0	0
$761$	−24.8054	−0.899195	−0.449597	−	0.893231i	$-0.648433\pi$
$761$	−24.8054	−0.899195	−0.449597	+	0.893231i	$0.648433\pi$
$762$	−6.23015	−0.225694
$763$	−11.3557	−0.411105
$764$	−22.3790	−0.809644
$765$	−47.1527	−1.70481
$766$	−9.48566	−0.342731
$767$	1.00832	0.0364082
$768$	−2.79360	−0.100806
$769$	9.64289	0.347732	0.173866	−	0.984769i	$-0.444374\pi$
$769$	9.64289	0.347732	0.173866	+	0.984769i	$0.444374\pi$
$770$	−17.3298	−0.624522
$771$	39.4799	1.42184
$772$	−7.42811	−0.267344
$773$	−1.28731	−0.0463014	−0.0231507	−	0.999732i	$-0.507370\pi$
$773$	−1.28731	−0.0463014	−0.0231507	+	0.999732i	$0.507370\pi$
$774$	4.75320	0.170850
$775$	−3.20216	−0.115025
$776$	17.3522	0.622907
$777$	14.2764	0.512162
$778$	−36.0480	−1.29238
$779$	0	0
$780$	−1.99300	−0.0713607
$781$	25.4177	0.909517
$782$	27.1133	0.969569
$783$	−15.7638	−0.563353
$784$	−5.35114	−0.191112
$785$	−32.5137	−1.16046
$786$	4.25369	0.151724
$787$	16.3446	0.582624	0.291312	−	0.956628i	$-0.405908\pi$
$787$	16.3446	0.582624	0.291312	+	0.956628i	$0.405908\pi$
$788$	8.53554	0.304066
$789$	−21.1444	−0.752762
$790$	−20.1085	−0.715427
$791$	−18.2984	−0.650617
$792$	−27.6542	−0.982648
$793$	3.33538	0.118443
$794$	6.40906	0.227449
$795$	−21.5531	−0.764410
$796$	−6.35926	−0.225398
$797$	15.6884	0.555712	0.277856	−	0.960623i	$-0.410376\pi$
$797$	15.6884	0.555712	0.277856	+	0.960623i	$0.410376\pi$
$798$	0	0
$799$	−18.3960	−0.650804
$800$	−0.497039	−0.0175730
$801$	72.4311	2.55923
$802$	0.187660	0.00662650
$803$	−13.0487	−0.460479
$804$	−12.2383	−0.431610
$805$	−19.4993	−0.687260
$806$	−1.96033	−0.0690496
$807$	−21.4290	−0.754337
$808$	−3.67667	−0.129345
$809$	−11.5248	−0.405192	−0.202596	−	0.979262i	$-0.564938\pi$
$809$	−11.5248	−0.405192	−0.202596	+	0.979262i	$0.564938\pi$
$810$	−0.778611	−0.0273576
$811$	8.30102	0.291488	0.145744	−	0.989322i	$-0.453442\pi$
$811$	8.30102	0.291488	0.145744	+	0.989322i	$0.453442\pi$
$812$	−4.01603	−0.140935
$813$	−32.5871	−1.14288
$814$	−22.9086	−0.802945
$815$	−33.4093	−1.17028
$816$	−11.6946	−0.409392
$817$	0	0
$818$	17.2663	0.603701
$819$	1.87712	0.0655919
$820$	−11.7632	−0.410788
$821$	−33.7112	−1.17653	−0.588265	−	0.808668i	$-0.700189\pi$
$821$	−33.7112	−1.17653	−0.588265	+	0.808668i	$0.700189\pi$
$822$	29.9244	1.04373
$823$	21.8678	0.762262	0.381131	−	0.924521i	$-0.375535\pi$
$823$	21.8678	0.762262	0.381131	+	0.924521i	$0.375535\pi$
$824$	19.0135	0.662368
$825$	−7.99268	−0.278269
$826$	−4.25512	−0.148055
$827$	−39.1695	−1.36206	−0.681028	−	0.732257i	$-0.738467\pi$
$827$	−39.1695	−1.36206	−0.681028	+	0.732257i	$0.738467\pi$
$828$	−31.1162	−1.08136
$829$	14.4176	0.500743	0.250371	−	0.968150i	$-0.419447\pi$
$829$	14.4176	0.500743	0.250371	+	0.968150i	$0.419447\pi$
$830$	−22.8940	−0.794661
$831$	−76.6579	−2.65923
$832$	−0.304282	−0.0105491
$833$	−22.4009	−0.776145
$834$	−30.2110	−1.04612
$835$	27.1739	0.940392
$836$	0	0
$837$	32.4719	1.12239
$838$	1.72572	0.0596139
$839$	−20.9532	−0.723385	−0.361693	−	0.932297i	$-0.617801\pi$
$839$	−20.9532	−0.723385	−0.361693	+	0.932297i	$0.617801\pi$
$840$	8.41049	0.290189
$841$	−19.2184	−0.662702
$842$	32.7546	1.12880
$843$	45.4561	1.56559
$844$	13.9578	0.480447
$845$	30.2624	1.04106
$846$	21.1119	0.725844
$847$	−28.4218	−0.976585
$848$	−3.29064	−0.113001
$849$	−9.48077	−0.325379
$850$	−2.08070	−0.0713675
$851$	−25.7765	−0.883607
$852$	−12.3357	−0.422614
$853$	51.9553	1.77892	0.889458	−	0.457017i	$-0.151082\pi$
$853$	51.9553	1.77892	0.889458	+	0.457017i	$0.151082\pi$
$854$	−14.0754	−0.481650
$855$	0	0
$856$	10.0373	0.343067
$857$	−29.1835	−0.996888	−0.498444	−	0.866922i	$-0.666095\pi$
$857$	−29.1835	−0.996888	−0.498444	+	0.866922i	$0.666095\pi$
$858$	−4.89304	−0.167046
$859$	21.9913	0.750335	0.375168	−	0.926957i	$-0.377585\pi$
$859$	21.9913	0.750335	0.375168	+	0.926957i	$0.377585\pi$
$860$	2.31967	0.0791002
$861$	17.9977	0.613360
$862$	−25.5119	−0.868939
$863$	35.8157	1.21918	0.609590	−	0.792717i	$-0.291334\pi$
$863$	35.8157	1.21918	0.609590	+	0.792717i	$0.291334\pi$
$864$	5.04029	0.171474
$865$	9.33968	0.317559
$866$	24.4834	0.831980
$867$	−1.46442	−0.0497343
$868$	8.27263	0.280791
$869$	−49.3687	−1.67472
$870$	−20.4849	−0.694505
$871$	−1.33300	−0.0451671
$872$	−8.84348	−0.299478
$873$	−83.3638	−2.82144
$874$	0	0
$875$	−13.5567	−0.458300
$876$	6.33280	0.213966
$877$	−7.82483	−0.264226	−0.132113	−	0.991235i	$-0.542176\pi$
$877$	−7.82483	−0.264226	−0.132113	+	0.991235i	$0.542176\pi$
$878$	−30.8388	−1.04076
$879$	65.3097	2.20284
$880$	−13.4959	−0.454946
$881$	−14.4829	−0.487941	−0.243970	−	0.969783i	$-0.578450\pi$
$881$	−14.4829	−0.487941	−0.243970	+	0.969783i	$0.578450\pi$
$882$	25.7081	0.865636
$883$	−23.5529	−0.792619	−0.396310	−	0.918117i	$-0.629709\pi$
$883$	−23.5529	−0.792619	−0.396310	+	0.918117i	$0.629709\pi$
$884$	−1.27378	−0.0428420
$885$	−21.7045	−0.729588
$886$	−10.7047	−0.359631
$887$	15.8736	0.532982	0.266491	−	0.963837i	$-0.414136\pi$
$887$	15.8736	0.532982	0.266491	+	0.963837i	$0.414136\pi$
$888$	11.1180	0.373095
$889$	2.86368	0.0960449
$890$	35.3481	1.18487
$891$	−1.91158	−0.0640404
$892$	−8.92400	−0.298798
$893$	0	0
$894$	−26.0751	−0.872081
$895$	−19.8138	−0.662301
$896$	1.28408	0.0428981
$897$	−5.50560	−0.183827
$898$	2.31562	0.0772731
$899$	−20.1492	−0.672013
$900$	2.38789	0.0795963
$901$	−13.7753	−0.458921
$902$	−28.8800	−0.961599
$903$	−3.54911	−0.118107
$904$	−14.2502	−0.473956
$905$	19.8944	0.661311
$906$	−55.3781	−1.83981
$907$	−45.0127	−1.49462	−0.747311	−	0.664474i	$-0.768656\pi$
$907$	−45.0127	−1.49462	−0.747311	+	0.664474i	$0.768656\pi$
$908$	0.659796	0.0218961
$909$	17.6635	0.585863
$910$	0.916079	0.0303677
$911$	2.75901	0.0914100	0.0457050	−	0.998955i	$-0.485447\pi$
$911$	2.75901	0.0914100	0.0457050	+	0.998955i	$0.485447\pi$
$912$	0	0
$913$	−56.2074	−1.86019
$914$	0.0521810	0.00172599
$915$	−71.7956	−2.37349
$916$	−6.97208	−0.230364
$917$	−1.95521	−0.0645667
$918$	21.0996	0.696392
$919$	−39.3083	−1.29666	−0.648331	−	0.761359i	$-0.724533\pi$
$919$	−39.3083	−1.29666	−0.648331	+	0.761359i	$0.724533\pi$
$920$	−15.1854	−0.500649
$921$	−3.87758	−0.127771
$922$	−39.4720	−1.29994
$923$	−1.34362	−0.0442258
$924$	20.6487	0.679294
$925$	1.97811	0.0650400
$926$	6.19056	0.203435
$927$	−91.3453	−3.00017
$928$	−3.12756	−0.102667
$929$	−29.7417	−0.975795	−0.487897	−	0.872901i	$-0.662236\pi$
$929$	−29.7417	−0.975795	−0.487897	+	0.872901i	$0.662236\pi$
$930$	42.1970	1.38369
$931$	0	0
$932$	−29.6097	−0.969898
$933$	34.0547	1.11490
$934$	20.9962	0.687015
$935$	−56.4964	−1.84763
$936$	1.46184	0.0477818
$937$	37.7211	1.23229	0.616147	−	0.787631i	$-0.288693\pi$
$937$	37.7211	1.23229	0.616147	+	0.787631i	$0.288693\pi$
$938$	5.62531	0.183673
$939$	−46.1822	−1.50710
$940$	10.3031	0.336051
$941$	6.47652	0.211129	0.105564	−	0.994412i	$-0.466335\pi$
$941$	6.47652	0.211129	0.105564	+	0.994412i	$0.466335\pi$
$942$	38.7407	1.26224
$943$	−32.4955	−1.05820
$944$	−3.31375	−0.107853
$945$	−15.1744	−0.493624
$946$	5.69507	0.185163
$947$	16.2913	0.529396	0.264698	−	0.964331i	$-0.414728\pi$
$947$	16.2913	0.529396	0.264698	+	0.964331i	$0.414728\pi$
$948$	23.9596	0.778171
$949$	0.689776	0.0223911
$950$	0	0
$951$	7.50953	0.243513
$952$	5.37540	0.174218
$953$	34.6213	1.12150	0.560748	−	0.827987i	$-0.310514\pi$
$953$	34.6213	1.12150	0.560748	+	0.827987i	$0.310514\pi$
$954$	15.8090	0.511835
$955$	52.4693	1.69787
$956$	−24.7034	−0.798965
$957$	−50.2930	−1.62574
$958$	−15.5229	−0.501523
$959$	−13.7547	−0.444164
$960$	6.54982	0.211395
$961$	10.5053	0.338882
$962$	1.21098	0.0390436
$963$	−48.2213	−1.55391
$964$	−27.5586	−0.887604
$965$	17.4158	0.560634
$966$	23.2338	0.747534
$967$	−48.9105	−1.57286	−0.786428	−	0.617682i	$-0.788072\pi$
$967$	−48.9105	−1.57286	−0.786428	+	0.617682i	$0.788072\pi$
$968$	−22.1340	−0.711414
$969$	0	0
$970$	−40.6835	−1.30627
$971$	−6.07592	−0.194986	−0.0974928	−	0.995236i	$-0.531082\pi$
$971$	−6.07592	−0.194986	−0.0974928	+	0.995236i	$0.531082\pi$
$972$	16.0486	0.514759
$973$	13.8865	0.445180
$974$	−28.6065	−0.916613
$975$	0.422505	0.0135310
$976$	−10.9615	−0.350868
$977$	−57.0477	−1.82512	−0.912559	−	0.408944i	$-0.865897\pi$
$977$	−57.0477	−1.82512	−0.912559	+	0.408944i	$0.865897\pi$
$978$	39.8077	1.27291
$979$	86.7838	2.77362
$980$	12.5462	0.400772
$981$	42.4861	1.35648
$982$	30.3488	0.968470
$983$	−0.186519	−0.00594905	−0.00297452	−	0.999996i	$-0.500947\pi$
$983$	−0.186519	−0.00594905	−0.00297452	+	0.999996i	$0.500947\pi$
$984$	14.0160	0.446815
$985$	−20.0122	−0.637642
$986$	−13.0926	−0.416952
$987$	−15.7638	−0.501768
$988$	0	0
$989$	6.40804	0.203764
$990$	64.8373	2.06066
$991$	46.2398	1.46886	0.734428	−	0.678686i	$-0.237450\pi$
$991$	46.2398	1.46886	0.734428	+	0.678686i	$0.237450\pi$
$992$	6.44246	0.204548
$993$	14.5631	0.462147
$994$	5.67010	0.179845
$995$	14.9098	0.472671
$996$	27.2786	0.864354
$997$	−20.8490	−0.660295	−0.330147	−	0.943929i	$-0.607098\pi$
$997$	−20.8490	−0.660295	−0.330147	+	0.943929i	$0.607098\pi$
$998$	−8.77624	−0.277807
$999$	−20.0593	−0.634650

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.2.a.m.1.1	✓	4
3.2	odd	2		6498.2.a.ca.1.3		4
4.3	odd	2		5776.2.a.bv.1.4		4
19.2	odd	18		722.2.e.r.99.4		24
19.3	odd	18		722.2.e.r.389.4		24
19.4	even	9		722.2.e.s.415.4		24
19.5	even	9		722.2.e.s.595.4		24
19.6	even	9		722.2.e.s.245.1		24
19.7	even	3		722.2.c.n.429.4		8
19.8	odd	6		722.2.c.m.653.1		8
19.9	even	9		722.2.e.s.423.1		24
19.10	odd	18		722.2.e.r.423.4		24
19.11	even	3		722.2.c.n.653.4		8
19.12	odd	6		722.2.c.m.429.1		8
19.13	odd	18		722.2.e.r.245.4		24
19.14	odd	18		722.2.e.r.595.1		24
19.15	odd	18		722.2.e.r.415.1		24
19.16	even	9		722.2.e.s.389.1		24
19.17	even	9		722.2.e.s.99.1		24
19.18	odd	2		722.2.a.n.1.4	yes	4
57.56	even	2		6498.2.a.bx.1.3		4
76.75	even	2		5776.2.a.bt.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.a.m.1.1	✓	4	1.1	even	1	trivial
722.2.a.n.1.4	yes	4	19.18	odd	2
722.2.c.m.429.1		8	19.12	odd	6
722.2.c.m.653.1		8	19.8	odd	6
722.2.c.n.429.4		8	19.7	even	3
722.2.c.n.653.4		8	19.11	even	3
722.2.e.r.99.4		24	19.2	odd	18
722.2.e.r.245.4		24	19.13	odd	18
722.2.e.r.389.4		24	19.3	odd	18
722.2.e.r.415.1		24	19.15	odd	18
722.2.e.r.423.4		24	19.10	odd	18
722.2.e.r.595.1		24	19.14	odd	18
722.2.e.s.99.1		24	19.17	even	9
722.2.e.s.245.1		24	19.6	even	9
722.2.e.s.389.1		24	19.16	even	9
722.2.e.s.415.4		24	19.4	even	9
722.2.e.s.423.1		24	19.9	even	9
722.2.e.s.595.4		24	19.5	even	9
5776.2.a.bt.1.1		4	76.75	even	2
5776.2.a.bv.1.4		4	4.3	odd	2
6498.2.a.bx.1.3		4	57.56	even	2
6498.2.a.ca.1.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	722.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.79360 q^{3} +1.00000 q^{4} -2.34458 q^{5} +2.79360 q^{6} -1.28408 q^{7} -1.00000 q^{8} +4.80423 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.79360 q^{3} +1.00000 q^{4} -2.34458 q^{5} +2.79360 q^{6} -1.28408 q^{7} -1.00000 q^{8} +4.80423 q^{9} +2.34458 q^{10} +5.75621 q^{11} -2.79360 q^{12} -0.304282 q^{13} +1.28408 q^{14} +6.54982 q^{15} +1.00000 q^{16} +4.18619 q^{17} -4.80423 q^{18} -2.34458 q^{20} +3.58721 q^{21} -5.75621 q^{22} -6.47684 q^{23} +2.79360 q^{24} +0.497039 q^{25} +0.304282 q^{26} -5.04029 q^{27} -1.28408 q^{28} +3.12756 q^{29} -6.54982 q^{30} -6.44246 q^{31} -1.00000 q^{32} -16.0806 q^{33} -4.18619 q^{34} +3.01062 q^{35} +4.80423 q^{36} +3.97980 q^{37} +0.850045 q^{39} +2.34458 q^{40} +5.01719 q^{41} -3.58721 q^{42} -0.989378 q^{43} +5.75621 q^{44} -11.2639 q^{45} +6.47684 q^{46} -4.39445 q^{47} -2.79360 q^{48} -5.35114 q^{49} -0.497039 q^{50} -11.6946 q^{51} -0.304282 q^{52} -3.29064 q^{53} +5.04029 q^{54} -13.4959 q^{55} +1.28408 q^{56} -3.12756 q^{58} -3.31375 q^{59} +6.54982 q^{60} -10.9615 q^{61} +6.44246 q^{62} -6.16901 q^{63} +1.00000 q^{64} +0.713414 q^{65} +16.0806 q^{66} +4.38081 q^{67} +4.18619 q^{68} +18.0937 q^{69} -3.01062 q^{70} +4.41570 q^{71} -4.80423 q^{72} -2.26689 q^{73} -3.97980 q^{74} -1.38853 q^{75} -7.39144 q^{77} -0.850045 q^{78} -8.57659 q^{79} -2.34458 q^{80} -0.332090 q^{81} -5.01719 q^{82} -9.76464 q^{83} +3.58721 q^{84} -9.81485 q^{85} +0.989378 q^{86} -8.73716 q^{87} -5.75621 q^{88} +15.0765 q^{89} +11.2639 q^{90} +0.390723 q^{91} -6.47684 q^{92} +17.9977 q^{93} +4.39445 q^{94} +2.79360 q^{96} -17.3522 q^{97} +5.35114 q^{98} +27.6542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 18 q^{13} + 2 q^{14} - 4 q^{15} + 4 q^{16} + 6 q^{17} - 4 q^{18} - 2 q^{20} - 4 q^{21} - 2 q^{22} - 10 q^{23} + 2 q^{24} + 6 q^{25} + 18 q^{26} + 4 q^{27} - 2 q^{28} + 2 q^{29} + 4 q^{30} - 26 q^{31} - 4 q^{32} - 16 q^{33} - 6 q^{34} + 6 q^{35} + 4 q^{36} - 4 q^{37} - 6 q^{39} + 2 q^{40} + 12 q^{41} + 4 q^{42} - 10 q^{43} + 2 q^{44} - 22 q^{45} + 10 q^{46} - 12 q^{47} - 2 q^{48} - 12 q^{49} - 6 q^{50} + 2 q^{51} - 18 q^{52} - 8 q^{53} - 4 q^{54} - 26 q^{55} + 2 q^{56} - 2 q^{58} + 8 q^{59} - 4 q^{60} + 26 q^{62} - 22 q^{63} + 4 q^{64} + 4 q^{65} + 16 q^{66} - 10 q^{67} + 6 q^{68} + 20 q^{69} - 6 q^{70} - 4 q^{72} - 14 q^{73} + 4 q^{74} - 8 q^{75} + 4 q^{77} + 6 q^{78} - 22 q^{79} - 2 q^{80} - 4 q^{81} - 12 q^{82} - 12 q^{83} - 4 q^{84} - 18 q^{85} + 10 q^{86} - 26 q^{87} - 2 q^{88} + 16 q^{89} + 22 q^{90} + 4 q^{91} - 10 q^{92} + 8 q^{93} + 12 q^{94} + 2 q^{96} - 28 q^{97} + 12 q^{98} + 22 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 + 2 * q^10 + 2 * q^11 - 2 * q^12 - 18 * q^13 + 2 * q^14 - 4 * q^15 + 4 * q^16 + 6 * q^17 - 4 * q^18 - 2 * q^20 - 4 * q^21 - 2 * q^22 - 10 * q^23 + 2 * q^24 + 6 * q^25 + 18 * q^26 + 4 * q^27 - 2 * q^28 + 2 * q^29 + 4 * q^30 - 26 * q^31 - 4 * q^32 - 16 * q^33 - 6 * q^34 + 6 * q^35 + 4 * q^36 - 4 * q^37 - 6 * q^39 + 2 * q^40 + 12 * q^41 + 4 * q^42 - 10 * q^43 + 2 * q^44 - 22 * q^45 + 10 * q^46 - 12 * q^47 - 2 * q^48 - 12 * q^49 - 6 * q^50 + 2 * q^51 - 18 * q^52 - 8 * q^53 - 4 * q^54 - 26 * q^55 + 2 * q^56 - 2 * q^58 + 8 * q^59 - 4 * q^60 + 26 * q^62 - 22 * q^63 + 4 * q^64 + 4 * q^65 + 16 * q^66 - 10 * q^67 + 6 * q^68 + 20 * q^69 - 6 * q^70 - 4 * q^72 - 14 * q^73 + 4 * q^74 - 8 * q^75 + 4 * q^77 + 6 * q^78 - 22 * q^79 - 2 * q^80 - 4 * q^81 - 12 * q^82 - 12 * q^83 - 4 * q^84 - 18 * q^85 + 10 * q^86 - 26 * q^87 - 2 * q^88 + 16 * q^89 + 22 * q^90 + 4 * q^91 - 10 * q^92 + 8 * q^93 + 12 * q^94 + 2 * q^96 - 28 * q^97 + 12 * q^98 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more