LMFDB - Embedded newform 354.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,2,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("354.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	354.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:	$2.82670423155$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
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			$-3.31662$ of defining polynomial
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.31662 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.31662 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.31662 q^{10} -2.00000 q^{11} +1.00000 q^{12} +2.31662 q^{13} -4.00000 q^{14} -2.31662 q^{15} +1.00000 q^{16} +6.63325 q^{17} -1.00000 q^{18} -2.00000 q^{19} -2.31662 q^{20} +4.00000 q^{21} +2.00000 q^{22} +8.63325 q^{23} -1.00000 q^{24} +0.366750 q^{25} -2.31662 q^{26} +1.00000 q^{27} +4.00000 q^{28} -1.68338 q^{29} +2.31662 q^{30} -4.31662 q^{31} -1.00000 q^{32} -2.00000 q^{33} -6.63325 q^{34} -9.26650 q^{35} +1.00000 q^{36} +2.31662 q^{37} +2.00000 q^{38} +2.31662 q^{39} +2.31662 q^{40} +10.6332 q^{41} -4.00000 q^{42} +4.00000 q^{43} -2.00000 q^{44} -2.31662 q^{45} -8.63325 q^{46} -4.63325 q^{47} +1.00000 q^{48} +9.00000 q^{49} -0.366750 q^{50} +6.63325 q^{51} +2.31662 q^{52} -10.3166 q^{53} -1.00000 q^{54} +4.63325 q^{55} -4.00000 q^{56} -2.00000 q^{57} +1.68338 q^{58} -1.00000 q^{59} -2.31662 q^{60} +2.31662 q^{61} +4.31662 q^{62} +4.00000 q^{63} +1.00000 q^{64} -5.36675 q^{65} +2.00000 q^{66} -8.63325 q^{67} +6.63325 q^{68} +8.63325 q^{69} +9.26650 q^{70} -0.316625 q^{71} -1.00000 q^{72} -15.2665 q^{73} -2.31662 q^{74} +0.366750 q^{75} -2.00000 q^{76} -8.00000 q^{77} -2.31662 q^{78} -2.31662 q^{80} +1.00000 q^{81} -10.6332 q^{82} -4.00000 q^{83} +4.00000 q^{84} -15.3668 q^{85} -4.00000 q^{86} -1.68338 q^{87} +2.00000 q^{88} -4.63325 q^{89} +2.31662 q^{90} +9.26650 q^{91} +8.63325 q^{92} -4.31662 q^{93} +4.63325 q^{94} +4.63325 q^{95} -1.00000 q^{96} -18.6332 q^{97} -9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 8 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 8 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 8 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} + 2 q^{12} - 2 q^{13} - 8 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{18} - 4 q^{19} + 2 q^{20} + 8 q^{21} + 4 q^{22} + 4 q^{23} - 2 q^{24} + 14 q^{25} + 2 q^{26} + 2 q^{27} + 8 q^{28} - 10 q^{29} - 2 q^{30} - 2 q^{31} - 2 q^{32} - 4 q^{33} + 8 q^{35} + 2 q^{36} - 2 q^{37} + 4 q^{38} - 2 q^{39} - 2 q^{40} + 8 q^{41} - 8 q^{42} + 8 q^{43} - 4 q^{44} + 2 q^{45} - 4 q^{46} + 4 q^{47} + 2 q^{48} + 18 q^{49} - 14 q^{50} - 2 q^{52} - 14 q^{53} - 2 q^{54} - 4 q^{55} - 8 q^{56} - 4 q^{57} + 10 q^{58} - 2 q^{59} + 2 q^{60} - 2 q^{61} + 2 q^{62} + 8 q^{63} + 2 q^{64} - 24 q^{65} + 4 q^{66} - 4 q^{67} + 4 q^{69} - 8 q^{70} + 6 q^{71} - 2 q^{72} - 4 q^{73} + 2 q^{74} + 14 q^{75} - 4 q^{76} - 16 q^{77} + 2 q^{78} + 2 q^{80} + 2 q^{81} - 8 q^{82} - 8 q^{83} + 8 q^{84} - 44 q^{85} - 8 q^{86} - 10 q^{87} + 4 q^{88} + 4 q^{89} - 2 q^{90} - 8 q^{91} + 4 q^{92} - 2 q^{93} - 4 q^{94} - 4 q^{95} - 2 q^{96} - 24 q^{97} - 18 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 8 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 + 2 * q^12 - 2 * q^13 - 8 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^18 - 4 * q^19 + 2 * q^20 + 8 * q^21 + 4 * q^22 + 4 * q^23 - 2 * q^24 + 14 * q^25 + 2 * q^26 + 2 * q^27 + 8 * q^28 - 10 * q^29 - 2 * q^30 - 2 * q^31 - 2 * q^32 - 4 * q^33 + 8 * q^35 + 2 * q^36 - 2 * q^37 + 4 * q^38 - 2 * q^39 - 2 * q^40 + 8 * q^41 - 8 * q^42 + 8 * q^43 - 4 * q^44 + 2 * q^45 - 4 * q^46 + 4 * q^47 + 2 * q^48 + 18 * q^49 - 14 * q^50 - 2 * q^52 - 14 * q^53 - 2 * q^54 - 4 * q^55 - 8 * q^56 - 4 * q^57 + 10 * q^58 - 2 * q^59 + 2 * q^60 - 2 * q^61 + 2 * q^62 + 8 * q^63 + 2 * q^64 - 24 * q^65 + 4 * q^66 - 4 * q^67 + 4 * q^69 - 8 * q^70 + 6 * q^71 - 2 * q^72 - 4 * q^73 + 2 * q^74 + 14 * q^75 - 4 * q^76 - 16 * q^77 + 2 * q^78 + 2 * q^80 + 2 * q^81 - 8 * q^82 - 8 * q^83 + 8 * q^84 - 44 * q^85 - 8 * q^86 - 10 * q^87 + 4 * q^88 + 4 * q^89 - 2 * q^90 - 8 * q^91 + 4 * q^92 - 2 * q^93 - 4 * q^94 - 4 * q^95 - 2 * q^96 - 24 * q^97 - 18 * q^98 - 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−2.31662	−1.03603	−0.518013	−	0.855373i	$-0.673328\pi$
$5$	−2.31662	−1.03603	−0.518013	+	0.855373i	$0.673328\pi$
$6$	−1.00000	−0.408248
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.31662	0.732581
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.00000	0.288675
$13$	2.31662	0.642516	0.321258	−	0.946992i	$-0.395894\pi$
$13$	2.31662	0.642516	0.321258	+	0.946992i	$0.395894\pi$
$14$	−4.00000	−1.06904
$15$	−2.31662	−0.598150
$16$	1.00000	0.250000
$17$	6.63325	1.60880	0.804400	−	0.594089i	$-0.202487\pi$
$17$	6.63325	1.60880	0.804400	+	0.594089i	$0.202487\pi$
$18$	−1.00000	−0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−2.31662	−0.518013
$21$	4.00000	0.872872
$22$	2.00000	0.426401
$23$	8.63325	1.80016	0.900078	−	0.435728i	$-0.143509\pi$
$23$	8.63325	1.80016	0.900078	+	0.435728i	$0.143509\pi$
$24$	−1.00000	−0.204124
$25$	0.366750	0.0733501
$26$	−2.31662	−0.454328
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	−1.68338	−0.312595	−0.156297	−	0.987710i	$-0.549956\pi$
$29$	−1.68338	−0.312595	−0.156297	+	0.987710i	$0.549956\pi$
$30$	2.31662	0.422956
$31$	−4.31662	−0.775289	−0.387644	−	0.921809i	$-0.626711\pi$
$31$	−4.31662	−0.775289	−0.387644	+	0.921809i	$0.626711\pi$
$32$	−1.00000	−0.176777
$33$	−2.00000	−0.348155
$34$	−6.63325	−1.13759
$35$	−9.26650	−1.56632
$36$	1.00000	0.166667
$37$	2.31662	0.380851	0.190425	−	0.981702i	$-0.439013\pi$
$37$	2.31662	0.380851	0.190425	+	0.981702i	$0.439013\pi$
$38$	2.00000	0.324443
$39$	2.31662	0.370957
$40$	2.31662	0.366291
$41$	10.6332	1.66063	0.830317	−	0.557291i	$-0.188159\pi$
$41$	10.6332	1.66063	0.830317	+	0.557291i	$0.188159\pi$
$42$	−4.00000	−0.617213
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−2.00000	−0.301511
$45$	−2.31662	−0.345342
$46$	−8.63325	−1.27290
$47$	−4.63325	−0.675829	−0.337914	−	0.941177i	$-0.609721\pi$
$47$	−4.63325	−0.675829	−0.337914	+	0.941177i	$0.609721\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	−0.366750	−0.0518663
$51$	6.63325	0.928841
$52$	2.31662	0.321258
$53$	−10.3166	−1.41710	−0.708549	−	0.705662i	$-0.750650\pi$
$53$	−10.3166	−1.41710	−0.708549	+	0.705662i	$0.750650\pi$
$54$	−1.00000	−0.136083
$55$	4.63325	0.624747
$56$	−4.00000	−0.534522
$57$	−2.00000	−0.264906
$58$	1.68338	0.221038
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	−2.31662	−0.299075
$61$	2.31662	0.296613	0.148307	−	0.988941i	$-0.452618\pi$
$61$	2.31662	0.296613	0.148307	+	0.988941i	$0.452618\pi$
$62$	4.31662	0.548212
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	−5.36675	−0.665663
$66$	2.00000	0.246183
$67$	−8.63325	−1.05472	−0.527360	−	0.849642i	$-0.676818\pi$
$67$	−8.63325	−1.05472	−0.527360	+	0.849642i	$0.676818\pi$
$68$	6.63325	0.804400
$69$	8.63325	1.03932
$70$	9.26650	1.10756
$71$	−0.316625	−0.0375764	−0.0187882	−	0.999823i	$-0.505981\pi$
$71$	−0.316625	−0.0375764	−0.0187882	+	0.999823i	$0.505981\pi$
$72$	−1.00000	−0.117851
$73$	−15.2665	−1.78681	−0.893404	−	0.449254i	$-0.851690\pi$
$73$	−15.2665	−1.78681	−0.893404	+	0.449254i	$0.851690\pi$
$74$	−2.31662	−0.269302
$75$	0.366750	0.0423487
$76$	−2.00000	−0.229416
$77$	−8.00000	−0.911685
$78$	−2.31662	−0.262306
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−2.31662	−0.259007
$81$	1.00000	0.111111
$82$	−10.6332	−1.17425
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	4.00000	0.436436
$85$	−15.3668	−1.66676
$86$	−4.00000	−0.431331
$87$	−1.68338	−0.180477
$88$	2.00000	0.213201
$89$	−4.63325	−0.491123	−0.245562	−	0.969381i	$-0.578972\pi$
$89$	−4.63325	−0.491123	−0.245562	+	0.969381i	$0.578972\pi$
$90$	2.31662	0.244194
$91$	9.26650	0.971393
$92$	8.63325	0.900078
$93$	−4.31662	−0.447613
$94$	4.63325	0.477883
$95$	4.63325	0.475361
$96$	−1.00000	−0.102062
$97$	−18.6332	−1.89192	−0.945960	−	0.324284i	$-0.894877\pi$
$97$	−18.6332	−1.89192	−0.945960	+	0.324284i	$0.894877\pi$
$98$	−9.00000	−0.909137
$99$	−2.00000	−0.201008
$100$	0.366750	0.0366750
$101$	−14.6332	−1.45606	−0.728031	−	0.685544i	$-0.759565\pi$
$101$	−14.6332	−1.45606	−0.728031	+	0.685544i	$0.759565\pi$
$102$	−6.63325	−0.656790
$103$	8.31662	0.819461	0.409731	−	0.912207i	$-0.365623\pi$
$103$	8.31662	0.819461	0.409731	+	0.912207i	$0.365623\pi$
$104$	−2.31662	−0.227164
$105$	−9.26650	−0.904318
$106$	10.3166	1.00204
$107$	−0.633250	−0.0612185	−0.0306093	−	0.999531i	$-0.509745\pi$
$107$	−0.633250	−0.0612185	−0.0306093	+	0.999531i	$0.509745\pi$
$108$	1.00000	0.0962250
$109$	−6.94987	−0.665677	−0.332839	−	0.942984i	$-0.608006\pi$
$109$	−6.94987	−0.665677	−0.332839	+	0.942984i	$0.608006\pi$
$110$	−4.63325	−0.441763
$111$	2.31662	0.219884
$112$	4.00000	0.377964
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	2.00000	0.187317
$115$	−20.0000	−1.86501
$116$	−1.68338	−0.156297
$117$	2.31662	0.214172
$118$	1.00000	0.0920575
$119$	26.5330	2.43228
$120$	2.31662	0.211478
$121$	−7.00000	−0.636364
$122$	−2.31662	−0.209737
$123$	10.6332	0.958768
$124$	−4.31662	−0.387644
$125$	10.7335	0.960034
$126$	−4.00000	−0.356348
$127$	21.8997	1.94329	0.971644	−	0.236448i	$-0.0759832\pi$
$127$	21.8997	1.94329	0.971644	+	0.236448i	$0.0759832\pi$
$128$	−1.00000	−0.0883883
$129$	4.00000	0.352180
$130$	5.36675	0.470695
$131$	−7.26650	−0.634877	−0.317438	−	0.948279i	$-0.602823\pi$
$131$	−7.26650	−0.634877	−0.317438	+	0.948279i	$0.602823\pi$
$132$	−2.00000	−0.174078
$133$	−8.00000	−0.693688
$134$	8.63325	0.745799
$135$	−2.31662	−0.199383
$136$	−6.63325	−0.568796
$137$	1.36675	0.116769	0.0583847	−	0.998294i	$-0.481405\pi$
$137$	1.36675	0.116769	0.0583847	+	0.998294i	$0.481405\pi$
$138$	−8.63325	−0.734911
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	−9.26650	−0.783162
$141$	−4.63325	−0.390190
$142$	0.316625	0.0265706
$143$	−4.63325	−0.387452
$144$	1.00000	0.0833333
$145$	3.89975	0.323857
$146$	15.2665	1.26346
$147$	9.00000	0.742307
$148$	2.31662	0.190425
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	−0.366750	−0.0299450
$151$	7.68338	0.625264	0.312632	−	0.949874i	$-0.398789\pi$
$151$	7.68338	0.625264	0.312632	+	0.949874i	$0.398789\pi$
$152$	2.00000	0.162221
$153$	6.63325	0.536266
$154$	8.00000	0.644658
$155$	10.0000	0.803219
$156$	2.31662	0.185478
$157$	19.5831	1.56290	0.781452	−	0.623966i	$-0.214480\pi$
$157$	19.5831	1.56290	0.781452	+	0.623966i	$0.214480\pi$
$158$	0	0
$159$	−10.3166	−0.818162
$160$	2.31662	0.183145
$161$	34.5330	2.72158
$162$	−1.00000	−0.0785674
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	10.6332	0.830317
$165$	4.63325	0.360698
$166$	4.00000	0.310460
$167$	11.6834	0.904087	0.452043	−	0.891996i	$-0.350695\pi$
$167$	11.6834	0.904087	0.452043	+	0.891996i	$0.350695\pi$
$168$	−4.00000	−0.308607
$169$	−7.63325	−0.587173
$170$	15.3668	1.17858
$171$	−2.00000	−0.152944
$172$	4.00000	0.304997
$173$	−9.36675	−0.712141	−0.356070	−	0.934459i	$-0.615884\pi$
$173$	−9.36675	−0.712141	−0.356070	+	0.934459i	$0.615884\pi$
$174$	1.68338	0.127616
$175$	1.46700	0.110895
$176$	−2.00000	−0.150756
$177$	−1.00000	−0.0751646
$178$	4.63325	0.347277
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−2.31662	−0.172671
$181$	−23.2665	−1.72939	−0.864693	−	0.502301i	$-0.832487\pi$
$181$	−23.2665	−1.72939	−0.864693	+	0.502301i	$0.832487\pi$
$182$	−9.26650	−0.686879
$183$	2.31662	0.171250
$184$	−8.63325	−0.636452
$185$	−5.36675	−0.394571
$186$	4.31662	0.316510
$187$	−13.2665	−0.970143
$188$	−4.63325	−0.337914
$189$	4.00000	0.290957
$190$	−4.63325	−0.336131
$191$	−12.6332	−0.914110	−0.457055	−	0.889438i	$-0.651096\pi$
$191$	−12.6332	−0.914110	−0.457055	+	0.889438i	$0.651096\pi$
$192$	1.00000	0.0721688
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	18.6332	1.33779
$195$	−5.36675	−0.384321
$196$	9.00000	0.642857
$197$	22.3166	1.58999	0.794997	−	0.606614i	$-0.207472\pi$
$197$	22.3166	1.58999	0.794997	+	0.606614i	$0.207472\pi$
$198$	2.00000	0.142134
$199$	11.3668	0.805768	0.402884	−	0.915251i	$-0.368008\pi$
$199$	11.3668	0.805768	0.402884	+	0.915251i	$0.368008\pi$
$200$	−0.366750	−0.0259332
$201$	−8.63325	−0.608942
$202$	14.6332	1.02959
$203$	−6.73350	−0.472599
$204$	6.63325	0.464420
$205$	−24.6332	−1.72046
$206$	−8.31662	−0.579447
$207$	8.63325	0.600052
$208$	2.31662	0.160629
$209$	4.00000	0.276686
$210$	9.26650	0.639449
$211$	3.36675	0.231777	0.115888	−	0.993262i	$-0.463029\pi$
$211$	3.36675	0.231777	0.115888	+	0.993262i	$0.463029\pi$
$212$	−10.3166	−0.708549
$213$	−0.316625	−0.0216948
$214$	0.633250	0.0432881
$215$	−9.26650	−0.631970
$216$	−1.00000	−0.0680414
$217$	−17.2665	−1.17213
$218$	6.94987	0.470705
$219$	−15.2665	−1.03161
$220$	4.63325	0.312374
$221$	15.3668	1.03368
$222$	−2.31662	−0.155482
$223$	13.2665	0.888390	0.444195	−	0.895930i	$-0.353490\pi$
$223$	13.2665	0.888390	0.444195	+	0.895930i	$0.353490\pi$
$224$	−4.00000	−0.267261
$225$	0.366750	0.0244500
$226$	14.0000	0.931266
$227$	5.26650	0.349550	0.174775	−	0.984608i	$-0.444080\pi$
$227$	5.26650	0.349550	0.174775	+	0.984608i	$0.444080\pi$
$228$	−2.00000	−0.132453
$229$	27.5831	1.82274	0.911372	−	0.411583i	$-0.135024\pi$
$229$	27.5831	1.82274	0.911372	+	0.411583i	$0.135024\pi$
$230$	20.0000	1.31876
$231$	−8.00000	−0.526361
$232$	1.68338	0.110519
$233$	−7.36675	−0.482612	−0.241306	−	0.970449i	$-0.577576\pi$
$233$	−7.36675	−0.482612	−0.241306	+	0.970449i	$0.577576\pi$
$234$	−2.31662	−0.151443
$235$	10.7335	0.700176
$236$	−1.00000	−0.0650945
$237$	0	0
$238$	−26.5330	−1.71988
$239$	−16.9499	−1.09640	−0.548198	−	0.836349i	$-0.684686\pi$
$239$	−16.9499	−1.09640	−0.548198	+	0.836349i	$0.684686\pi$
$240$	−2.31662	−0.149537
$241$	−24.6332	−1.58677	−0.793384	−	0.608722i	$-0.791682\pi$
$241$	−24.6332	−1.58677	−0.793384	+	0.608722i	$0.791682\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	2.31662	0.148307
$245$	−20.8496	−1.33203
$246$	−10.6332	−0.677951
$247$	−4.63325	−0.294807
$248$	4.31662	0.274106
$249$	−4.00000	−0.253490
$250$	−10.7335	−0.678846
$251$	30.5330	1.92723	0.963613	−	0.267302i	$-0.0861320\pi$
$251$	30.5330	1.92723	0.963613	+	0.267302i	$0.0861320\pi$
$252$	4.00000	0.251976
$253$	−17.2665	−1.08554
$254$	−21.8997	−1.37411
$255$	−15.3668	−0.962303
$256$	1.00000	0.0625000
$257$	19.2665	1.20181	0.600906	−	0.799320i	$-0.294807\pi$
$257$	19.2665	1.20181	0.600906	+	0.799320i	$0.294807\pi$
$258$	−4.00000	−0.249029
$259$	9.26650	0.575792
$260$	−5.36675	−0.332832
$261$	−1.68338	−0.104198
$262$	7.26650	0.448926
$263$	−25.5831	−1.57752	−0.788762	−	0.614699i	$-0.789277\pi$
$263$	−25.5831	−1.57752	−0.788762	+	0.614699i	$0.789277\pi$
$264$	2.00000	0.123091
$265$	23.8997	1.46815
$266$	8.00000	0.490511
$267$	−4.63325	−0.283550
$268$	−8.63325	−0.527360
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	2.31662	0.140985
$271$	26.5330	1.61176	0.805882	−	0.592076i	$-0.201691\pi$
$271$	26.5330	1.61176	0.805882	+	0.592076i	$0.201691\pi$
$272$	6.63325	0.402200
$273$	9.26650	0.560834
$274$	−1.36675	−0.0825684
$275$	−0.733501	−0.0442318
$276$	8.63325	0.519661
$277$	−5.36675	−0.322457	−0.161228	−	0.986917i	$-0.551546\pi$
$277$	−5.36675	−0.322457	−0.161228	+	0.986917i	$0.551546\pi$
$278$	−10.0000	−0.599760
$279$	−4.31662	−0.258430
$280$	9.26650	0.553779
$281$	−11.2665	−0.672103	−0.336051	−	0.941844i	$-0.609092\pi$
$281$	−11.2665	−0.672103	−0.336051	+	0.941844i	$0.609092\pi$
$282$	4.63325	0.275906
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	−0.316625	−0.0187882
$285$	4.63325	0.274450
$286$	4.63325	0.273970
$287$	42.5330	2.51064
$288$	−1.00000	−0.0589256
$289$	27.0000	1.58824
$290$	−3.89975	−0.229001
$291$	−18.6332	−1.09230
$292$	−15.2665	−0.893404
$293$	−1.05013	−0.0613490	−0.0306745	−	0.999529i	$-0.509766\pi$
$293$	−1.05013	−0.0613490	−0.0306745	+	0.999529i	$0.509766\pi$
$294$	−9.00000	−0.524891
$295$	2.31662	0.134879
$296$	−2.31662	−0.134651
$297$	−2.00000	−0.116052
$298$	−22.0000	−1.27443
$299$	20.0000	1.15663
$300$	0.366750	0.0211743
$301$	16.0000	0.922225
$302$	−7.68338	−0.442129
$303$	−14.6332	−0.840658
$304$	−2.00000	−0.114708
$305$	−5.36675	−0.307299
$306$	−6.63325	−0.379198
$307$	−7.26650	−0.414721	−0.207361	−	0.978265i	$-0.566487\pi$
$307$	−7.26650	−0.414721	−0.207361	+	0.978265i	$0.566487\pi$
$308$	−8.00000	−0.455842
$309$	8.31662	0.473116
$310$	−10.0000	−0.567962
$311$	−4.31662	−0.244773	−0.122387	−	0.992482i	$-0.539055\pi$
$311$	−4.31662	−0.244773	−0.122387	+	0.992482i	$0.539055\pi$
$312$	−2.31662	−0.131153
$313$	−5.36675	−0.303347	−0.151673	−	0.988431i	$-0.548466\pi$
$313$	−5.36675	−0.303347	−0.151673	+	0.988431i	$0.548466\pi$
$314$	−19.5831	−1.10514
$315$	−9.26650	−0.522108
$316$	0	0
$317$	−18.3166	−1.02876	−0.514382	−	0.857561i	$-0.671979\pi$
$317$	−18.3166	−1.02876	−0.514382	+	0.857561i	$0.671979\pi$
$318$	10.3166	0.578528
$319$	3.36675	0.188502
$320$	−2.31662	−0.129503
$321$	−0.633250	−0.0353445
$322$	−34.5330	−1.92445
$323$	−13.2665	−0.738168
$324$	1.00000	0.0555556
$325$	0.849623	0.0471286
$326$	10.0000	0.553849
$327$	−6.94987	−0.384329
$328$	−10.6332	−0.587123
$329$	−18.5330	−1.02176
$330$	−4.63325	−0.255052
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	−4.00000	−0.219529
$333$	2.31662	0.126950
$334$	−11.6834	−0.639286
$335$	20.0000	1.09272
$336$	4.00000	0.218218
$337$	−7.26650	−0.395831	−0.197916	−	0.980219i	$-0.563417\pi$
$337$	−7.26650	−0.395831	−0.197916	+	0.980219i	$0.563417\pi$
$338$	7.63325	0.415194
$339$	−14.0000	−0.760376
$340$	−15.3668	−0.833379
$341$	8.63325	0.467517
$342$	2.00000	0.108148
$343$	8.00000	0.431959
$344$	−4.00000	−0.215666
$345$	−20.0000	−1.07676
$346$	9.36675	0.503560
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	−1.68338	−0.0902384
$349$	9.05013	0.484442	0.242221	−	0.970221i	$-0.422124\pi$
$349$	9.05013	0.484442	0.242221	+	0.970221i	$0.422124\pi$
$350$	−1.46700	−0.0784145
$351$	2.31662	0.123652
$352$	2.00000	0.106600
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	1.00000	0.0531494
$355$	0.733501	0.0389302
$356$	−4.63325	−0.245562
$357$	26.5330	1.40428
$358$	4.00000	0.211407
$359$	−11.6834	−0.616625	−0.308312	−	0.951285i	$-0.599764\pi$
$359$	−11.6834	−0.616625	−0.308312	+	0.951285i	$0.599764\pi$
$360$	2.31662	0.122097
$361$	−15.0000	−0.789474
$362$	23.2665	1.22286
$363$	−7.00000	−0.367405
$364$	9.26650	0.485697
$365$	35.3668	1.85118
$366$	−2.31662	−0.121092
$367$	15.6834	0.818666	0.409333	−	0.912385i	$-0.365762\pi$
$367$	15.6834	0.818666	0.409333	+	0.912385i	$0.365762\pi$
$368$	8.63325	0.450039
$369$	10.6332	0.553545
$370$	5.36675	0.279004
$371$	−41.2665	−2.14245
$372$	−4.31662	−0.223807
$373$	23.2665	1.20469	0.602347	−	0.798234i	$-0.294232\pi$
$373$	23.2665	1.20469	0.602347	+	0.798234i	$0.294232\pi$
$374$	13.2665	0.685994
$375$	10.7335	0.554276
$376$	4.63325	0.238942
$377$	−3.89975	−0.200847
$378$	−4.00000	−0.205738
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	4.63325	0.237681
$381$	21.8997	1.12196
$382$	12.6332	0.646373
$383$	−14.2164	−0.726423	−0.363211	−	0.931707i	$-0.618320\pi$
$383$	−14.2164	−0.726423	−0.363211	+	0.931707i	$0.618320\pi$
$384$	−1.00000	−0.0510310
$385$	18.5330	0.944529
$386$	10.0000	0.508987
$387$	4.00000	0.203331
$388$	−18.6332	−0.945960
$389$	5.68338	0.288159	0.144079	−	0.989566i	$-0.453978\pi$
$389$	5.68338	0.288159	0.144079	+	0.989566i	$0.453978\pi$
$390$	5.36675	0.271756
$391$	57.2665	2.89609
$392$	−9.00000	−0.454569
$393$	−7.26650	−0.366546
$394$	−22.3166	−1.12430
$395$	0	0
$396$	−2.00000	−0.100504
$397$	−6.31662	−0.317022	−0.158511	−	0.987357i	$-0.550669\pi$
$397$	−6.31662	−0.317022	−0.158511	+	0.987357i	$0.550669\pi$
$398$	−11.3668	−0.569764
$399$	−8.00000	−0.400501
$400$	0.366750	0.0183375
$401$	−17.8997	−0.893871	−0.446935	−	0.894566i	$-0.647485\pi$
$401$	−17.8997	−0.893871	−0.446935	+	0.894566i	$0.647485\pi$
$402$	8.63325	0.430587
$403$	−10.0000	−0.498135
$404$	−14.6332	−0.728031
$405$	−2.31662	−0.115114
$406$	6.73350	0.334178
$407$	−4.63325	−0.229662
$408$	−6.63325	−0.328395
$409$	6.63325	0.327993	0.163997	−	0.986461i	$-0.447561\pi$
$409$	6.63325	0.327993	0.163997	+	0.986461i	$0.447561\pi$
$410$	24.6332	1.21655
$411$	1.36675	0.0674168
$412$	8.31662	0.409731
$413$	−4.00000	−0.196827
$414$	−8.63325	−0.424301
$415$	9.26650	0.454875
$416$	−2.31662	−0.113582
$417$	10.0000	0.489702
$418$	−4.00000	−0.195646
$419$	5.26650	0.257285	0.128643	−	0.991691i	$-0.458938\pi$
$419$	5.26650	0.257285	0.128643	+	0.991691i	$0.458938\pi$
$420$	−9.26650	−0.452159
$421$	−22.3166	−1.08765	−0.543823	−	0.839200i	$-0.683024\pi$
$421$	−22.3166	−1.08765	−0.543823	+	0.839200i	$0.683024\pi$
$422$	−3.36675	−0.163891
$423$	−4.63325	−0.225276
$424$	10.3166	0.501020
$425$	2.43275	0.118006
$426$	0.316625	0.0153405
$427$	9.26650	0.448437
$428$	−0.633250	−0.0306093
$429$	−4.63325	−0.223695
$430$	9.26650	0.446870
$431$	4.63325	0.223176	0.111588	−	0.993755i	$-0.464406\pi$
$431$	4.63325	0.223176	0.111588	+	0.993755i	$0.464406\pi$
$432$	1.00000	0.0481125
$433$	7.36675	0.354023	0.177012	−	0.984209i	$-0.443357\pi$
$433$	7.36675	0.354023	0.177012	+	0.984209i	$0.443357\pi$
$434$	17.2665	0.828818
$435$	3.89975	0.186979
$436$	−6.94987	−0.332839
$437$	−17.2665	−0.825969
$438$	15.2665	0.729462
$439$	−37.8997	−1.80886	−0.904428	−	0.426626i	$-0.859702\pi$
$439$	−37.8997	−1.80886	−0.904428	+	0.426626i	$0.859702\pi$
$440$	−4.63325	−0.220882
$441$	9.00000	0.428571
$442$	−15.3668	−0.730922
$443$	5.26650	0.250219	0.125109	−	0.992143i	$-0.460072\pi$
$443$	5.26650	0.250219	0.125109	+	0.992143i	$0.460072\pi$
$444$	2.31662	0.109942
$445$	10.7335	0.508817
$446$	−13.2665	−0.628187
$447$	22.0000	1.04056
$448$	4.00000	0.188982
$449$	1.36675	0.0645009	0.0322505	−	0.999480i	$-0.489733\pi$
$449$	1.36675	0.0645009	0.0322505	+	0.999480i	$0.489733\pi$
$450$	−0.366750	−0.0172888
$451$	−21.2665	−1.00140
$452$	−14.0000	−0.658505
$453$	7.68338	0.360996
$454$	−5.26650	−0.247169
$455$	−21.4670	−1.00639
$456$	2.00000	0.0936586
$457$	−33.1662	−1.55145	−0.775726	−	0.631070i	$-0.782616\pi$
$457$	−33.1662	−1.55145	−0.775726	+	0.631070i	$0.782616\pi$
$458$	−27.5831	−1.28887
$459$	6.63325	0.309614
$460$	−20.0000	−0.932505
$461$	−1.68338	−0.0784026	−0.0392013	−	0.999231i	$-0.512481\pi$
$461$	−1.68338	−0.0784026	−0.0392013	+	0.999231i	$0.512481\pi$
$462$	8.00000	0.372194
$463$	−0.949874	−0.0441444	−0.0220722	−	0.999756i	$-0.507026\pi$
$463$	−0.949874	−0.0441444	−0.0220722	+	0.999756i	$0.507026\pi$
$464$	−1.68338	−0.0781487
$465$	10.0000	0.463739
$466$	7.36675	0.341258
$467$	11.2665	0.521351	0.260676	−	0.965426i	$-0.416055\pi$
$467$	11.2665	0.521351	0.260676	+	0.965426i	$0.416055\pi$
$468$	2.31662	0.107086
$469$	−34.5330	−1.59459
$470$	−10.7335	−0.495099
$471$	19.5831	0.902343
$472$	1.00000	0.0460287
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−0.733501	−0.0336553
$476$	26.5330	1.21614
$477$	−10.3166	−0.472366
$478$	16.9499	0.775269
$479$	38.2164	1.74615	0.873075	−	0.487585i	$-0.162122\pi$
$479$	38.2164	1.74615	0.873075	+	0.487585i	$0.162122\pi$
$480$	2.31662	0.105739
$481$	5.36675	0.244703
$482$	24.6332	1.12201
$483$	34.5330	1.57131
$484$	−7.00000	−0.318182
$485$	43.1662	1.96008
$486$	−1.00000	−0.0453609
$487$	−12.6332	−0.572467	−0.286234	−	0.958160i	$-0.592403\pi$
$487$	−12.6332	−0.572467	−0.286234	+	0.958160i	$0.592403\pi$
$488$	−2.31662	−0.104869
$489$	−10.0000	−0.452216
$490$	20.8496	0.941890
$491$	−37.8997	−1.71039	−0.855196	−	0.518305i	$-0.826563\pi$
$491$	−37.8997	−1.71039	−0.855196	+	0.518305i	$0.826563\pi$
$492$	10.6332	0.479384
$493$	−11.1662	−0.502903
$494$	4.63325	0.208460
$495$	4.63325	0.208249
$496$	−4.31662	−0.193822
$497$	−1.26650	−0.0568103
$498$	4.00000	0.179244
$499$	38.5330	1.72497	0.862487	−	0.506079i	$-0.168906\pi$
$499$	38.5330	1.72497	0.862487	+	0.506079i	$0.168906\pi$
$500$	10.7335	0.480017
$501$	11.6834	0.521975
$502$	−30.5330	−1.36275
$503$	−27.1662	−1.21128	−0.605642	−	0.795738i	$-0.707083\pi$
$503$	−27.1662	−1.21128	−0.605642	+	0.795738i	$0.707083\pi$
$504$	−4.00000	−0.178174
$505$	33.8997	1.50852
$506$	17.2665	0.767590
$507$	−7.63325	−0.339005
$508$	21.8997	0.971644
$509$	−27.8997	−1.23663	−0.618317	−	0.785929i	$-0.712185\pi$
$509$	−27.8997	−1.23663	−0.618317	+	0.785929i	$0.712185\pi$
$510$	15.3668	0.680451
$511$	−61.0660	−2.70140
$512$	−1.00000	−0.0441942
$513$	−2.00000	−0.0883022
$514$	−19.2665	−0.849809
$515$	−19.2665	−0.848983
$516$	4.00000	0.176090
$517$	9.26650	0.407540
$518$	−9.26650	−0.407147
$519$	−9.36675	−0.411155
$520$	5.36675	0.235348
$521$	34.6332	1.51731	0.758655	−	0.651492i	$-0.225857\pi$
$521$	34.6332	1.51731	0.758655	+	0.651492i	$0.225857\pi$
$522$	1.68338	0.0736793
$523$	38.5330	1.68493	0.842465	−	0.538751i	$-0.181104\pi$
$523$	38.5330	1.68493	0.842465	+	0.538751i	$0.181104\pi$
$524$	−7.26650	−0.317438
$525$	1.46700	0.0640252
$526$	25.5831	1.11548
$527$	−28.6332	−1.24728
$528$	−2.00000	−0.0870388
$529$	51.5330	2.24057
$530$	−23.8997	−1.03814
$531$	−1.00000	−0.0433963
$532$	−8.00000	−0.346844
$533$	24.6332	1.06698
$534$	4.63325	0.200500
$535$	1.46700	0.0634240
$536$	8.63325	0.372900
$537$	−4.00000	−0.172613
$538$	−2.00000	−0.0862261
$539$	−18.0000	−0.775315
$540$	−2.31662	−0.0996917
$541$	−22.9499	−0.986692	−0.493346	−	0.869833i	$-0.664226\pi$
$541$	−22.9499	−0.986692	−0.493346	+	0.869833i	$0.664226\pi$
$542$	−26.5330	−1.13969
$543$	−23.2665	−0.998461
$544$	−6.63325	−0.284398
$545$	16.1003	0.689659
$546$	−9.26650	−0.396570
$547$	22.5330	0.963441	0.481721	−	0.876325i	$-0.340012\pi$
$547$	22.5330	0.963441	0.481721	+	0.876325i	$0.340012\pi$
$548$	1.36675	0.0583847
$549$	2.31662	0.0988711
$550$	0.733501	0.0312766
$551$	3.36675	0.143428
$552$	−8.63325	−0.367456
$553$	0	0
$554$	5.36675	0.228011
$555$	−5.36675	−0.227806
$556$	10.0000	0.424094
$557$	5.68338	0.240812	0.120406	−	0.992725i	$-0.461580\pi$
$557$	5.68338	0.240812	0.120406	+	0.992725i	$0.461580\pi$
$558$	4.31662	0.182737
$559$	9.26650	0.391931
$560$	−9.26650	−0.391581
$561$	−13.2665	−0.560112
$562$	11.2665	0.475249
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	−4.63325	−0.195095
$565$	32.4327	1.36446
$566$	24.0000	1.00880
$567$	4.00000	0.167984
$568$	0.316625	0.0132853
$569$	7.26650	0.304627	0.152314	−	0.988332i	$-0.451328\pi$
$569$	7.26650	0.304627	0.152314	+	0.988332i	$0.451328\pi$
$570$	−4.63325	−0.194065
$571$	9.26650	0.387791	0.193895	−	0.981022i	$-0.437888\pi$
$571$	9.26650	0.387791	0.193895	+	0.981022i	$0.437888\pi$
$572$	−4.63325	−0.193726
$573$	−12.6332	−0.527762
$574$	−42.5330	−1.77529
$575$	3.16625	0.132042
$576$	1.00000	0.0416667
$577$	−16.6332	−0.692451	−0.346226	−	0.938151i	$-0.612537\pi$
$577$	−16.6332	−0.692451	−0.346226	+	0.938151i	$0.612537\pi$
$578$	−27.0000	−1.12305
$579$	−10.0000	−0.415586
$580$	3.89975	0.161928
$581$	−16.0000	−0.663792
$582$	18.6332	0.772373
$583$	20.6332	0.854542
$584$	15.2665	0.631732
$585$	−5.36675	−0.221888
$586$	1.05013	0.0433803
$587$	47.2665	1.95090	0.975449	−	0.220227i	$-0.0706799\pi$
$587$	47.2665	1.95090	0.975449	+	0.220227i	$0.0706799\pi$
$588$	9.00000	0.371154
$589$	8.63325	0.355727
$590$	−2.31662	−0.0953739
$591$	22.3166	0.917983
$592$	2.31662	0.0952127
$593$	−7.26650	−0.298399	−0.149200	−	0.988807i	$-0.547670\pi$
$593$	−7.26650	−0.298399	−0.149200	+	0.988807i	$0.547670\pi$
$594$	2.00000	0.0820610
$595$	−61.4670	−2.51990
$596$	22.0000	0.901155
$597$	11.3668	0.465210
$598$	−20.0000	−0.817861
$599$	20.9499	0.855989	0.427994	−	0.903781i	$-0.359220\pi$
$599$	20.9499	0.855989	0.427994	+	0.903781i	$0.359220\pi$
$600$	−0.366750	−0.0149725
$601$	47.2665	1.92804	0.964020	−	0.265828i	$-0.0856455\pi$
$601$	47.2665	1.92804	0.964020	+	0.265828i	$0.0856455\pi$
$602$	−16.0000	−0.652111
$603$	−8.63325	−0.351573
$604$	7.68338	0.312632
$605$	16.2164	0.659289
$606$	14.6332	0.594435
$607$	−33.8997	−1.37595	−0.687974	−	0.725735i	$-0.741500\pi$
$607$	−33.8997	−1.37595	−0.687974	+	0.725735i	$0.741500\pi$
$608$	2.00000	0.0811107
$609$	−6.73350	−0.272855
$610$	5.36675	0.217293
$611$	−10.7335	−0.434231
$612$	6.63325	0.268133
$613$	1.05013	0.0424142	0.0212071	−	0.999775i	$-0.493249\pi$
$613$	1.05013	0.0424142	0.0212071	+	0.999775i	$0.493249\pi$
$614$	7.26650	0.293252
$615$	−24.6332	−0.993308
$616$	8.00000	0.322329
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	−8.31662	−0.334544
$619$	−45.2665	−1.81941	−0.909707	−	0.415251i	$-0.863694\pi$
$619$	−45.2665	−1.81941	−0.909707	+	0.415251i	$0.863694\pi$
$620$	10.0000	0.401610
$621$	8.63325	0.346440
$622$	4.31662	0.173081
$623$	−18.5330	−0.742509
$624$	2.31662	0.0927392
$625$	−26.6992	−1.06797
$626$	5.36675	0.214498
$627$	4.00000	0.159745
$628$	19.5831	0.781452
$629$	15.3668	0.612712
$630$	9.26650	0.369186
$631$	18.5330	0.737787	0.368894	−	0.929472i	$-0.379737\pi$
$631$	18.5330	0.737787	0.368894	+	0.929472i	$0.379737\pi$
$632$	0	0
$633$	3.36675	0.133816
$634$	18.3166	0.727446
$635$	−50.7335	−2.01330
$636$	−10.3166	−0.409081
$637$	20.8496	0.826092
$638$	−3.36675	−0.133291
$639$	−0.316625	−0.0125255
$640$	2.31662	0.0915726
$641$	13.3668	0.527955	0.263977	−	0.964529i	$-0.414966\pi$
$641$	13.3668	0.527955	0.263977	+	0.964529i	$0.414966\pi$
$642$	0.633250	0.0249924
$643$	21.2665	0.838669	0.419334	−	0.907832i	$-0.362263\pi$
$643$	21.2665	0.838669	0.419334	+	0.907832i	$0.362263\pi$
$644$	34.5330	1.36079
$645$	−9.26650	−0.364868
$646$	13.2665	0.521963
$647$	33.5831	1.32029	0.660144	−	0.751139i	$-0.270495\pi$
$647$	33.5831	1.32029	0.660144	+	0.751139i	$0.270495\pi$
$648$	−1.00000	−0.0392837
$649$	2.00000	0.0785069
$650$	−0.849623	−0.0333250
$651$	−17.2665	−0.676727
$652$	−10.0000	−0.391630
$653$	−18.3166	−0.716785	−0.358392	−	0.933571i	$-0.616675\pi$
$653$	−18.3166	−0.716785	−0.358392	+	0.933571i	$0.616675\pi$
$654$	6.94987	0.271762
$655$	16.8338	0.657749
$656$	10.6332	0.415159
$657$	−15.2665	−0.595603
$658$	18.5330	0.722491
$659$	−21.2665	−0.828425	−0.414213	−	0.910180i	$-0.635943\pi$
$659$	−21.2665	−0.828425	−0.414213	+	0.910180i	$0.635943\pi$
$660$	4.63325	0.180349
$661$	−11.2665	−0.438216	−0.219108	−	0.975701i	$-0.570315\pi$
$661$	−11.2665	−0.438216	−0.219108	+	0.975701i	$0.570315\pi$
$662$	14.0000	0.544125
$663$	15.3668	0.596795
$664$	4.00000	0.155230
$665$	18.5330	0.718679
$666$	−2.31662	−0.0897674
$667$	−14.5330	−0.562720
$668$	11.6834	0.452043
$669$	13.2665	0.512912
$670$	−20.0000	−0.772667
$671$	−4.63325	−0.178865
$672$	−4.00000	−0.154303
$673$	33.1662	1.27846	0.639232	−	0.769014i	$-0.279252\pi$
$673$	33.1662	1.27846	0.639232	+	0.769014i	$0.279252\pi$
$674$	7.26650	0.279895
$675$	0.366750	0.0141162
$676$	−7.63325	−0.293587
$677$	−26.3166	−1.01143	−0.505715	−	0.862701i	$-0.668771\pi$
$677$	−26.3166	−1.01143	−0.505715	+	0.862701i	$0.668771\pi$
$678$	14.0000	0.537667
$679$	−74.5330	−2.86031
$680$	15.3668	0.589288
$681$	5.26650	0.201813
$682$	−8.63325	−0.330584
$683$	12.5330	0.479562	0.239781	−	0.970827i	$-0.422924\pi$
$683$	12.5330	0.479562	0.239781	+	0.970827i	$0.422924\pi$
$684$	−2.00000	−0.0764719
$685$	−3.16625	−0.120976
$686$	−8.00000	−0.305441
$687$	27.5831	1.05236
$688$	4.00000	0.152499
$689$	−23.8997	−0.910508
$690$	20.0000	0.761387
$691$	44.4327	1.69030	0.845151	−	0.534528i	$-0.179511\pi$
$691$	44.4327	1.69030	0.845151	+	0.534528i	$0.179511\pi$
$692$	−9.36675	−0.356070
$693$	−8.00000	−0.303895
$694$	−2.00000	−0.0759190
$695$	−23.1662	−0.878746
$696$	1.68338	0.0638082
$697$	70.5330	2.67163
$698$	−9.05013	−0.342552
$699$	−7.36675	−0.278636
$700$	1.46700	0.0554475
$701$	−29.1662	−1.10159	−0.550797	−	0.834639i	$-0.685676\pi$
$701$	−29.1662	−1.10159	−0.550797	+	0.834639i	$0.685676\pi$
$702$	−2.31662	−0.0874354
$703$	−4.63325	−0.174746
$704$	−2.00000	−0.0753778
$705$	10.7335	0.404247
$706$	14.0000	0.526897
$707$	−58.5330	−2.20136
$708$	−1.00000	−0.0375823
$709$	−4.73350	−0.177770	−0.0888852	−	0.996042i	$-0.528330\pi$
$709$	−4.73350	−0.177770	−0.0888852	+	0.996042i	$0.528330\pi$
$710$	−0.733501	−0.0275278
$711$	0	0
$712$	4.63325	0.173638
$713$	−37.2665	−1.39564
$714$	−26.5330	−0.992973
$715$	10.7335	0.401410
$716$	−4.00000	−0.149487
$717$	−16.9499	−0.633005
$718$	11.6834	0.436020
$719$	2.10025	0.0783262	0.0391631	−	0.999233i	$-0.487531\pi$
$719$	2.10025	0.0783262	0.0391631	+	0.999233i	$0.487531\pi$
$720$	−2.31662	−0.0863355
$721$	33.2665	1.23891
$722$	15.0000	0.558242
$723$	−24.6332	−0.916120
$724$	−23.2665	−0.864693
$725$	−0.617379	−0.0229289
$726$	7.00000	0.259794
$727$	31.1662	1.15589	0.577946	−	0.816075i	$-0.303854\pi$
$727$	31.1662	1.15589	0.577946	+	0.816075i	$0.303854\pi$
$728$	−9.26650	−0.343439
$729$	1.00000	0.0370370
$730$	−35.3668	−1.30898
$731$	26.5330	0.981358
$732$	2.31662	0.0856249
$733$	−8.10025	−0.299190	−0.149595	−	0.988747i	$-0.547797\pi$
$733$	−8.10025	−0.299190	−0.149595	+	0.988747i	$0.547797\pi$
$734$	−15.6834	−0.578884
$735$	−20.8496	−0.769050
$736$	−8.63325	−0.318226
$737$	17.2665	0.636020
$738$	−10.6332	−0.391415
$739$	−34.5330	−1.27032	−0.635158	−	0.772382i	$-0.719065\pi$
$739$	−34.5330	−1.27032	−0.635158	+	0.772382i	$0.719065\pi$
$740$	−5.36675	−0.197286
$741$	−4.63325	−0.170207
$742$	41.2665	1.51494
$743$	26.2164	0.961785	0.480893	−	0.876779i	$-0.340313\pi$
$743$	26.2164	0.961785	0.480893	+	0.876779i	$0.340313\pi$
$744$	4.31662	0.158255
$745$	−50.9657	−1.86724
$746$	−23.2665	−0.851847
$747$	−4.00000	−0.146352
$748$	−13.2665	−0.485071
$749$	−2.53300	−0.0925537
$750$	−10.7335	−0.391932
$751$	38.8496	1.41764	0.708821	−	0.705388i	$-0.249227\pi$
$751$	38.8496	1.41764	0.708821	+	0.705388i	$0.249227\pi$
$752$	−4.63325	−0.168957
$753$	30.5330	1.11268
$754$	3.89975	0.142020
$755$	−17.7995	−0.647790
$756$	4.00000	0.145479
$757$	−15.2665	−0.554870	−0.277435	−	0.960744i	$-0.589484\pi$
$757$	−15.2665	−0.554870	−0.277435	+	0.960744i	$0.589484\pi$
$758$	−20.0000	−0.726433
$759$	−17.2665	−0.626734
$760$	−4.63325	−0.168066
$761$	33.1662	1.20228	0.601138	−	0.799145i	$-0.294714\pi$
$761$	33.1662	1.20228	0.601138	+	0.799145i	$0.294714\pi$
$762$	−21.8997	−0.793344
$763$	−27.7995	−1.00641
$764$	−12.6332	−0.457055
$765$	−15.3668	−0.555586
$766$	14.2164	0.513658
$767$	−2.31662	−0.0836485
$768$	1.00000	0.0360844
$769$	−5.36675	−0.193530	−0.0967650	−	0.995307i	$-0.530850\pi$
$769$	−5.36675	−0.193530	−0.0967650	+	0.995307i	$0.530850\pi$
$770$	−18.5330	−0.667883
$771$	19.2665	0.693866
$772$	−10.0000	−0.359908
$773$	8.10025	0.291346	0.145673	−	0.989333i	$-0.453465\pi$
$773$	8.10025	0.291346	0.145673	+	0.989333i	$0.453465\pi$
$774$	−4.00000	−0.143777
$775$	−1.58312	−0.0568675
$776$	18.6332	0.668895
$777$	9.26650	0.332434
$778$	−5.68338	−0.203759
$779$	−21.2665	−0.761951
$780$	−5.36675	−0.192160
$781$	0.633250	0.0226595
$782$	−57.2665	−2.04785
$783$	−1.68338	−0.0601589
$784$	9.00000	0.321429
$785$	−45.3668	−1.61921
$786$	7.26650	0.259187
$787$	36.5330	1.30226	0.651130	−	0.758966i	$-0.274295\pi$
$787$	36.5330	1.30226	0.651130	+	0.758966i	$0.274295\pi$
$788$	22.3166	0.794997
$789$	−25.5831	−0.910783
$790$	0	0
$791$	−56.0000	−1.99113
$792$	2.00000	0.0710669
$793$	5.36675	0.190579
$794$	6.31662	0.224169
$795$	23.8997	0.847637
$796$	11.3668	0.402884
$797$	−39.2665	−1.39089	−0.695445	−	0.718579i	$-0.744793\pi$
$797$	−39.2665	−1.39089	−0.695445	+	0.718579i	$0.744793\pi$
$798$	8.00000	0.283197
$799$	−30.7335	−1.08727
$800$	−0.366750	−0.0129666
$801$	−4.63325	−0.163708
$802$	17.8997	0.632062
$803$	30.5330	1.07749
$804$	−8.63325	−0.304471
$805$	−80.0000	−2.81963
$806$	10.0000	0.352235
$807$	2.00000	0.0704033
$808$	14.6332	0.514796
$809$	−29.8997	−1.05122	−0.525610	−	0.850726i	$-0.676163\pi$
$809$	−29.8997	−1.05122	−0.525610	+	0.850726i	$0.676163\pi$
$810$	2.31662	0.0813979
$811$	−13.8997	−0.488086	−0.244043	−	0.969764i	$-0.578474\pi$
$811$	−13.8997	−0.488086	−0.244043	+	0.969764i	$0.578474\pi$
$812$	−6.73350	−0.236300
$813$	26.5330	0.930553
$814$	4.63325	0.162395
$815$	23.1662	0.811478
$816$	6.63325	0.232210
$817$	−8.00000	−0.279885
$818$	−6.63325	−0.231926
$819$	9.26650	0.323798
$820$	−24.6332	−0.860230
$821$	55.0660	1.92182	0.960908	−	0.276867i	$-0.0892961\pi$
$821$	55.0660	1.92182	0.960908	+	0.276867i	$0.0892961\pi$
$822$	−1.36675	−0.0476709
$823$	46.8496	1.63308	0.816538	−	0.577292i	$-0.195891\pi$
$823$	46.8496	1.63308	0.816538	+	0.577292i	$0.195891\pi$
$824$	−8.31662	−0.289723
$825$	−0.733501	−0.0255372
$826$	4.00000	0.139178
$827$	−37.2665	−1.29588	−0.647942	−	0.761690i	$-0.724370\pi$
$827$	−37.2665	−1.29588	−0.647942	+	0.761690i	$0.724370\pi$
$828$	8.63325	0.300026
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	−9.26650	−0.321645
$831$	−5.36675	−0.186171
$832$	2.31662	0.0803145
$833$	59.6992	2.06846
$834$	−10.0000	−0.346272
$835$	−27.0660	−0.936657
$836$	4.00000	0.138343
$837$	−4.31662	−0.149204
$838$	−5.26650	−0.181928
$839$	37.8997	1.30844	0.654222	−	0.756302i	$-0.272996\pi$
$839$	37.8997	1.30844	0.654222	+	0.756302i	$0.272996\pi$
$840$	9.26650	0.319725
$841$	−26.1662	−0.902284
$842$	22.3166	0.769082
$843$	−11.2665	−0.388039
$844$	3.36675	0.115888
$845$	17.6834	0.608327
$846$	4.63325	0.159294
$847$	−28.0000	−0.962091
$848$	−10.3166	−0.354274
$849$	−24.0000	−0.823678
$850$	−2.43275	−0.0834425
$851$	20.0000	0.685591
$852$	−0.316625	−0.0108474
$853$	−49.1662	−1.68342	−0.841710	−	0.539930i	$-0.818451\pi$
$853$	−49.1662	−1.68342	−0.841710	+	0.539930i	$0.818451\pi$
$854$	−9.26650	−0.317093
$855$	4.63325	0.158454
$856$	0.633250	0.0216440
$857$	−25.8997	−0.884719	−0.442359	−	0.896838i	$-0.645858\pi$
$857$	−25.8997	−0.884719	−0.442359	+	0.896838i	$0.645858\pi$
$858$	4.63325	0.158177
$859$	−13.8997	−0.474253	−0.237127	−	0.971479i	$-0.576206\pi$
$859$	−13.8997	−0.474253	−0.237127	+	0.971479i	$0.576206\pi$
$860$	−9.26650	−0.315985
$861$	42.5330	1.44952
$862$	−4.63325	−0.157809
$863$	−53.8997	−1.83477	−0.917384	−	0.398002i	$-0.869704\pi$
$863$	−53.8997	−1.83477	−0.917384	+	0.398002i	$0.869704\pi$
$864$	−1.00000	−0.0340207
$865$	21.6992	0.737797
$866$	−7.36675	−0.250332
$867$	27.0000	0.916968
$868$	−17.2665	−0.586063
$869$	0	0
$870$	−3.89975	−0.132214
$871$	−20.0000	−0.677674
$872$	6.94987	0.235352
$873$	−18.6332	−0.630640
$874$	17.2665	0.584048
$875$	42.9340	1.45143
$876$	−15.2665	−0.515807
$877$	−5.36675	−0.181222	−0.0906111	−	0.995886i	$-0.528882\pi$
$877$	−5.36675	−0.181222	−0.0906111	+	0.995886i	$0.528882\pi$
$878$	37.8997	1.27905
$879$	−1.05013	−0.0354199
$880$	4.63325	0.156187
$881$	−1.89975	−0.0640042	−0.0320021	−	0.999488i	$-0.510188\pi$
$881$	−1.89975	−0.0640042	−0.0320021	+	0.999488i	$0.510188\pi$
$882$	−9.00000	−0.303046
$883$	−13.2665	−0.446453	−0.223227	−	0.974767i	$-0.571659\pi$
$883$	−13.2665	−0.446453	−0.223227	+	0.974767i	$0.571659\pi$
$884$	15.3668	0.516840
$885$	2.31662	0.0778725
$886$	−5.26650	−0.176931
$887$	41.2665	1.38559	0.692797	−	0.721133i	$-0.256378\pi$
$887$	41.2665	1.38559	0.692797	+	0.721133i	$0.256378\pi$
$888$	−2.31662	−0.0777408
$889$	87.5990	2.93798
$890$	−10.7335	−0.359788
$891$	−2.00000	−0.0670025
$892$	13.2665	0.444195
$893$	9.26650	0.310092
$894$	−22.0000	−0.735790
$895$	9.26650	0.309745
$896$	−4.00000	−0.133631
$897$	20.0000	0.667781
$898$	−1.36675	−0.0456091
$899$	7.26650	0.242351
$900$	0.366750	0.0122250
$901$	−68.4327	−2.27983
$902$	21.2665	0.708097
$903$	16.0000	0.532447
$904$	14.0000	0.465633
$905$	53.8997	1.79169
$906$	−7.68338	−0.255263
$907$	−4.73350	−0.157173	−0.0785867	−	0.996907i	$-0.525041\pi$
$907$	−4.73350	−0.157173	−0.0785867	+	0.996907i	$0.525041\pi$
$908$	5.26650	0.174775
$909$	−14.6332	−0.485354
$910$	21.4670	0.711624
$911$	14.4169	0.477652	0.238826	−	0.971062i	$-0.423237\pi$
$911$	14.4169	0.477652	0.238826	+	0.971062i	$0.423237\pi$
$912$	−2.00000	−0.0662266
$913$	8.00000	0.264761
$914$	33.1662	1.09704
$915$	−5.36675	−0.177419
$916$	27.5831	0.911372
$917$	−29.0660	−0.959844
$918$	−6.63325	−0.218930
$919$	−33.5831	−1.10781	−0.553903	−	0.832582i	$-0.686862\pi$
$919$	−33.5831	−1.10781	−0.553903	+	0.832582i	$0.686862\pi$
$920$	20.0000	0.659380
$921$	−7.26650	−0.239439
$922$	1.68338	0.0554390
$923$	−0.733501	−0.0241435
$924$	−8.00000	−0.263181
$925$	0.849623	0.0279354
$926$	0.949874	0.0312148
$927$	8.31662	0.273154
$928$	1.68338	0.0552595
$929$	44.6332	1.46437	0.732185	−	0.681106i	$-0.238501\pi$
$929$	44.6332	1.46437	0.732185	+	0.681106i	$0.238501\pi$
$930$	−10.0000	−0.327913
$931$	−18.0000	−0.589926
$932$	−7.36675	−0.241306
$933$	−4.31662	−0.141320
$934$	−11.2665	−0.368651
$935$	30.7335	1.00509
$936$	−2.31662	−0.0757213
$937$	39.2665	1.28278	0.641390	−	0.767215i	$-0.278358\pi$
$937$	39.2665	1.28278	0.641390	+	0.767215i	$0.278358\pi$
$938$	34.5330	1.12754
$939$	−5.36675	−0.175137
$940$	10.7335	0.350088
$941$	6.63325	0.216238	0.108119	−	0.994138i	$-0.465517\pi$
$941$	6.63325	0.216238	0.108119	+	0.994138i	$0.465517\pi$
$942$	−19.5831	−0.638053
$943$	91.7995	2.98940
$944$	−1.00000	−0.0325472
$945$	−9.26650	−0.301439
$946$	8.00000	0.260102
$947$	−35.1662	−1.14275	−0.571375	−	0.820689i	$-0.693590\pi$
$947$	−35.1662	−1.14275	−0.571375	+	0.820689i	$0.693590\pi$
$948$	0	0
$949$	−35.3668	−1.14805
$950$	0.733501	0.0237979
$951$	−18.3166	−0.593957
$952$	−26.5330	−0.859939
$953$	−1.36675	−0.0442734	−0.0221367	−	0.999755i	$-0.507047\pi$
$953$	−1.36675	−0.0442734	−0.0221367	+	0.999755i	$0.507047\pi$
$954$	10.3166	0.334013
$955$	29.2665	0.947042
$956$	−16.9499	−0.548198
$957$	3.36675	0.108832
$958$	−38.2164	−1.23471
$959$	5.46700	0.176539
$960$	−2.31662	−0.0747687
$961$	−12.3668	−0.398927
$962$	−5.36675	−0.173031
$963$	−0.633250	−0.0204062
$964$	−24.6332	−0.793384
$965$	23.1662	0.745748
$966$	−34.5330	−1.11108
$967$	−11.6834	−0.375712	−0.187856	−	0.982197i	$-0.560154\pi$
$967$	−11.6834	−0.375712	−0.187856	+	0.982197i	$0.560154\pi$
$968$	7.00000	0.224989
$969$	−13.2665	−0.426181
$970$	−43.1662	−1.38598
$971$	−15.1662	−0.486708	−0.243354	−	0.969938i	$-0.578248\pi$
$971$	−15.1662	−0.486708	−0.243354	+	0.969938i	$0.578248\pi$
$972$	1.00000	0.0320750
$973$	40.0000	1.28234
$974$	12.6332	0.404795
$975$	0.849623	0.0272097
$976$	2.31662	0.0741534
$977$	32.5330	1.04082	0.520411	−	0.853916i	$-0.325779\pi$
$977$	32.5330	1.04082	0.520411	+	0.853916i	$0.325779\pi$
$978$	10.0000	0.319765
$979$	9.26650	0.296159
$980$	−20.8496	−0.666017
$981$	−6.94987	−0.221892
$982$	37.8997	1.20943
$983$	5.89975	0.188173	0.0940864	−	0.995564i	$-0.470007\pi$
$983$	5.89975	0.188173	0.0940864	+	0.995564i	$0.470007\pi$
$984$	−10.6332	−0.338976
$985$	−51.6992	−1.64727
$986$	11.1662	0.355606
$987$	−18.5330	−0.589912
$988$	−4.63325	−0.147403
$989$	34.5330	1.09809
$990$	−4.63325	−0.147254
$991$	16.9499	0.538431	0.269215	−	0.963080i	$-0.413236\pi$
$991$	16.9499	0.538431	0.269215	+	0.963080i	$0.413236\pi$
$992$	4.31662	0.137053
$993$	−14.0000	−0.444277
$994$	1.26650	0.0401709
$995$	−26.3325	−0.834796
$996$	−4.00000	−0.126745
$997$	17.3668	0.550011	0.275005	−	0.961443i	$-0.411320\pi$
$997$	17.3668	0.550011	0.275005	+	0.961443i	$0.411320\pi$
$998$	−38.5330	−1.21974
$999$	2.31662	0.0732948

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.a.g.1.1	✓	2
3.2	odd	2		1062.2.a.m.1.2		2
4.3	odd	2		2832.2.a.l.1.1		2
5.4	even	2		8850.2.a.bm.1.1		2
12.11	even	2		8496.2.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.a.g.1.1	✓	2	1.1	even	1	trivial
1062.2.a.m.1.2		2	3.2	odd	2
2832.2.a.l.1.1		2	4.3	odd	2
8496.2.a.z.1.2		2	12.11	even	2
8850.2.a.bm.1.1		2	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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.31662 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.31662 q^{5} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.31662 q^{10} -2.00000 q^{11} +1.00000 q^{12} +2.31662 q^{13} -4.00000 q^{14} -2.31662 q^{15} +1.00000 q^{16} +6.63325 q^{17} -1.00000 q^{18} -2.00000 q^{19} -2.31662 q^{20} +4.00000 q^{21} +2.00000 q^{22} +8.63325 q^{23} -1.00000 q^{24} +0.366750 q^{25} -2.31662 q^{26} +1.00000 q^{27} +4.00000 q^{28} -1.68338 q^{29} +2.31662 q^{30} -4.31662 q^{31} -1.00000 q^{32} -2.00000 q^{33} -6.63325 q^{34} -9.26650 q^{35} +1.00000 q^{36} +2.31662 q^{37} +2.00000 q^{38} +2.31662 q^{39} +2.31662 q^{40} +10.6332 q^{41} -4.00000 q^{42} +4.00000 q^{43} -2.00000 q^{44} -2.31662 q^{45} -8.63325 q^{46} -4.63325 q^{47} +1.00000 q^{48} +9.00000 q^{49} -0.366750 q^{50} +6.63325 q^{51} +2.31662 q^{52} -10.3166 q^{53} -1.00000 q^{54} +4.63325 q^{55} -4.00000 q^{56} -2.00000 q^{57} +1.68338 q^{58} -1.00000 q^{59} -2.31662 q^{60} +2.31662 q^{61} +4.31662 q^{62} +4.00000 q^{63} +1.00000 q^{64} -5.36675 q^{65} +2.00000 q^{66} -8.63325 q^{67} +6.63325 q^{68} +8.63325 q^{69} +9.26650 q^{70} -0.316625 q^{71} -1.00000 q^{72} -15.2665 q^{73} -2.31662 q^{74} +0.366750 q^{75} -2.00000 q^{76} -8.00000 q^{77} -2.31662 q^{78} -2.31662 q^{80} +1.00000 q^{81} -10.6332 q^{82} -4.00000 q^{83} +4.00000 q^{84} -15.3668 q^{85} -4.00000 q^{86} -1.68338 q^{87} +2.00000 q^{88} -4.63325 q^{89} +2.31662 q^{90} +9.26650 q^{91} +8.63325 q^{92} -4.31662 q^{93} +4.63325 q^{94} +4.63325 q^{95} -1.00000 q^{96} -18.6332 q^{97} -9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 8 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 8 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 8 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} + 2 q^{12} - 2 q^{13} - 8 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{18} - 4 q^{19} + 2 q^{20} + 8 q^{21} + 4 q^{22} + 4 q^{23} - 2 q^{24} + 14 q^{25} + 2 q^{26} + 2 q^{27} + 8 q^{28} - 10 q^{29} - 2 q^{30} - 2 q^{31} - 2 q^{32} - 4 q^{33} + 8 q^{35} + 2 q^{36} - 2 q^{37} + 4 q^{38} - 2 q^{39} - 2 q^{40} + 8 q^{41} - 8 q^{42} + 8 q^{43} - 4 q^{44} + 2 q^{45} - 4 q^{46} + 4 q^{47} + 2 q^{48} + 18 q^{49} - 14 q^{50} - 2 q^{52} - 14 q^{53} - 2 q^{54} - 4 q^{55} - 8 q^{56} - 4 q^{57} + 10 q^{58} - 2 q^{59} + 2 q^{60} - 2 q^{61} + 2 q^{62} + 8 q^{63} + 2 q^{64} - 24 q^{65} + 4 q^{66} - 4 q^{67} + 4 q^{69} - 8 q^{70} + 6 q^{71} - 2 q^{72} - 4 q^{73} + 2 q^{74} + 14 q^{75} - 4 q^{76} - 16 q^{77} + 2 q^{78} + 2 q^{80} + 2 q^{81} - 8 q^{82} - 8 q^{83} + 8 q^{84} - 44 q^{85} - 8 q^{86} - 10 q^{87} + 4 q^{88} + 4 q^{89} - 2 q^{90} - 8 q^{91} + 4 q^{92} - 2 q^{93} - 4 q^{94} - 4 q^{95} - 2 q^{96} - 24 q^{97} - 18 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 8 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 + 2 * q^12 - 2 * q^13 - 8 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^18 - 4 * q^19 + 2 * q^20 + 8 * q^21 + 4 * q^22 + 4 * q^23 - 2 * q^24 + 14 * q^25 + 2 * q^26 + 2 * q^27 + 8 * q^28 - 10 * q^29 - 2 * q^30 - 2 * q^31 - 2 * q^32 - 4 * q^33 + 8 * q^35 + 2 * q^36 - 2 * q^37 + 4 * q^38 - 2 * q^39 - 2 * q^40 + 8 * q^41 - 8 * q^42 + 8 * q^43 - 4 * q^44 + 2 * q^45 - 4 * q^46 + 4 * q^47 + 2 * q^48 + 18 * q^49 - 14 * q^50 - 2 * q^52 - 14 * q^53 - 2 * q^54 - 4 * q^55 - 8 * q^56 - 4 * q^57 + 10 * q^58 - 2 * q^59 + 2 * q^60 - 2 * q^61 + 2 * q^62 + 8 * q^63 + 2 * q^64 - 24 * q^65 + 4 * q^66 - 4 * q^67 + 4 * q^69 - 8 * q^70 + 6 * q^71 - 2 * q^72 - 4 * q^73 + 2 * q^74 + 14 * q^75 - 4 * q^76 - 16 * q^77 + 2 * q^78 + 2 * q^80 + 2 * q^81 - 8 * q^82 - 8 * q^83 + 8 * q^84 - 44 * q^85 - 8 * q^86 - 10 * q^87 + 4 * q^88 + 4 * q^89 - 2 * q^90 - 8 * q^91 + 4 * q^92 - 2 * q^93 - 4 * q^94 - 4 * q^95 - 2 * q^96 - 24 * q^97 - 18 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more