LMFDB - Embedded newform 2898.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2898,2,Mod(1,2898)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2898, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2898.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2898.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:	$23.1406465058$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2898.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.41421 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.41421 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.41421 q^{10} +4.24264 q^{11} +0.828427 q^{13} -1.00000 q^{14} +1.00000 q^{16} -0.828427 q^{17} -7.07107 q^{19} -3.41421 q^{20} +4.24264 q^{22} +1.00000 q^{23} +6.65685 q^{25} +0.828427 q^{26} -1.00000 q^{28} +2.00000 q^{29} +0.828427 q^{31} +1.00000 q^{32} -0.828427 q^{34} +3.41421 q^{35} -10.2426 q^{37} -7.07107 q^{38} -3.41421 q^{40} -2.00000 q^{41} -6.58579 q^{43} +4.24264 q^{44} +1.00000 q^{46} -3.17157 q^{47} +1.00000 q^{49} +6.65685 q^{50} +0.828427 q^{52} -1.75736 q^{53} -14.4853 q^{55} -1.00000 q^{56} +2.00000 q^{58} +2.82843 q^{59} -5.75736 q^{61} +0.828427 q^{62} +1.00000 q^{64} -2.82843 q^{65} -3.75736 q^{67} -0.828427 q^{68} +3.41421 q^{70} +12.8284 q^{71} -12.0000 q^{73} -10.2426 q^{74} -7.07107 q^{76} -4.24264 q^{77} -11.3137 q^{79} -3.41421 q^{80} -2.00000 q^{82} -6.58579 q^{83} +2.82843 q^{85} -6.58579 q^{86} +4.24264 q^{88} -3.65685 q^{89} -0.828427 q^{91} +1.00000 q^{92} -3.17157 q^{94} +24.1421 q^{95} -3.17157 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{7} + 2 q^{8} - 4 q^{10} - 4 q^{13} - 2 q^{14} + 2 q^{16} + 4 q^{17} - 4 q^{20} + 2 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{28} + 4 q^{29} - 4 q^{31} + 2 q^{32} + 4 q^{34} + 4 q^{35} - 12 q^{37} - 4 q^{40} - 4 q^{41} - 16 q^{43} + 2 q^{46} - 12 q^{47} + 2 q^{49} + 2 q^{50} - 4 q^{52} - 12 q^{53} - 12 q^{55} - 2 q^{56} + 4 q^{58} - 20 q^{61} - 4 q^{62} + 2 q^{64} - 16 q^{67} + 4 q^{68} + 4 q^{70} + 20 q^{71} - 24 q^{73} - 12 q^{74} - 4 q^{80} - 4 q^{82} - 16 q^{83} - 16 q^{86} + 4 q^{89} + 4 q^{91} + 2 q^{92} - 12 q^{94} + 20 q^{95} - 12 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^7 + 2 * q^8 - 4 * q^10 - 4 * q^13 - 2 * q^14 + 2 * q^16 + 4 * q^17 - 4 * q^20 + 2 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^28 + 4 * q^29 - 4 * q^31 + 2 * q^32 + 4 * q^34 + 4 * q^35 - 12 * q^37 - 4 * q^40 - 4 * q^41 - 16 * q^43 + 2 * q^46 - 12 * q^47 + 2 * q^49 + 2 * q^50 - 4 * q^52 - 12 * q^53 - 12 * q^55 - 2 * q^56 + 4 * q^58 - 20 * q^61 - 4 * q^62 + 2 * q^64 - 16 * q^67 + 4 * q^68 + 4 * q^70 + 20 * q^71 - 24 * q^73 - 12 * q^74 - 4 * q^80 - 4 * q^82 - 16 * q^83 - 16 * q^86 + 4 * q^89 + 4 * q^91 + 2 * q^92 - 12 * q^94 + 20 * q^95 - 12 * q^97 + 2 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.41421	−1.07967
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	0.828427	0.229764	0.114882	−	0.993379i	$-0.463351\pi$
$13$	0.828427	0.229764	0.114882	+	0.993379i	$0.463351\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	0	0
$19$	−7.07107	−1.62221	−0.811107	−	0.584898i	$-0.801135\pi$
$19$	−7.07107	−1.62221	−0.811107	+	0.584898i	$0.801135\pi$
$20$	−3.41421	−0.763441
$21$	0	0
$22$	4.24264	0.904534
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	6.65685	1.33137
$26$	0.828427	0.162468
$27$	0	0
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0.828427	0.148790	0.0743950	−	0.997229i	$-0.476297\pi$
$31$	0.828427	0.148790	0.0743950	+	0.997229i	$0.476297\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.828427	−0.142074
$35$	3.41421	0.577107
$36$	0	0
$37$	−10.2426	−1.68388	−0.841940	−	0.539571i	$-0.818586\pi$
$37$	−10.2426	−1.68388	−0.841940	+	0.539571i	$0.818586\pi$
$38$	−7.07107	−1.14708
$39$	0	0
$40$	−3.41421	−0.539835
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−6.58579	−1.00432	−0.502162	−	0.864774i	$-0.667462\pi$
$43$	−6.58579	−1.00432	−0.502162	+	0.864774i	$0.667462\pi$
$44$	4.24264	0.639602
$45$	0	0
$46$	1.00000	0.147442
$47$	−3.17157	−0.462621	−0.231311	−	0.972880i	$-0.574301\pi$
$47$	−3.17157	−0.462621	−0.231311	+	0.972880i	$0.574301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	6.65685	0.941421
$51$	0	0
$52$	0.828427	0.114882
$53$	−1.75736	−0.241392	−0.120696	−	0.992690i	$-0.538513\pi$
$53$	−1.75736	−0.241392	−0.120696	+	0.992690i	$0.538513\pi$
$54$	0	0
$55$	−14.4853	−1.95319
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.00000	0.262613
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	−5.75736	−0.737154	−0.368577	−	0.929597i	$-0.620155\pi$
$61$	−5.75736	−0.737154	−0.368577	+	0.929597i	$0.620155\pi$
$62$	0.828427	0.105210
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.82843	−0.350823
$66$	0	0
$67$	−3.75736	−0.459034	−0.229517	−	0.973305i	$-0.573715\pi$
$67$	−3.75736	−0.459034	−0.229517	+	0.973305i	$0.573715\pi$
$68$	−0.828427	−0.100462
$69$	0	0
$70$	3.41421	0.408077
$71$	12.8284	1.52245	0.761227	−	0.648485i	$-0.224597\pi$
$71$	12.8284	1.52245	0.761227	+	0.648485i	$0.224597\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	−10.2426	−1.19068
$75$	0	0
$76$	−7.07107	−0.811107
$77$	−4.24264	−0.483494
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	−3.41421	−0.381721
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−6.58579	−0.722884	−0.361442	−	0.932395i	$-0.617715\pi$
$83$	−6.58579	−0.722884	−0.361442	+	0.932395i	$0.617715\pi$
$84$	0	0
$85$	2.82843	0.306786
$86$	−6.58579	−0.710164
$87$	0	0
$88$	4.24264	0.452267
$89$	−3.65685	−0.387626	−0.193813	−	0.981039i	$-0.562085\pi$
$89$	−3.65685	−0.387626	−0.193813	+	0.981039i	$0.562085\pi$
$90$	0	0
$91$	−0.828427	−0.0868428
$92$	1.00000	0.104257
$93$	0	0
$94$	−3.17157	−0.327123
$95$	24.1421	2.47693
$96$	0	0
$97$	−3.17157	−0.322024	−0.161012	−	0.986952i	$-0.551476\pi$
$97$	−3.17157	−0.322024	−0.161012	+	0.986952i	$0.551476\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	6.65685	0.665685
$101$	−7.65685	−0.761885	−0.380943	−	0.924599i	$-0.624401\pi$
$101$	−7.65685	−0.761885	−0.380943	+	0.924599i	$0.624401\pi$
$102$	0	0
$103$	4.48528	0.441948	0.220974	−	0.975280i	$-0.429076\pi$
$103$	4.48528	0.441948	0.220974	+	0.975280i	$0.429076\pi$
$104$	0.828427	0.0812340
$105$	0	0
$106$	−1.75736	−0.170690
$107$	−18.3848	−1.77732	−0.888662	−	0.458563i	$-0.848364\pi$
$107$	−18.3848	−1.77732	−0.888662	+	0.458563i	$0.848364\pi$
$108$	0	0
$109$	15.4142	1.47641	0.738207	−	0.674574i	$-0.235673\pi$
$109$	15.4142	1.47641	0.738207	+	0.674574i	$0.235673\pi$
$110$	−14.4853	−1.38112
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	9.31371	0.876160	0.438080	−	0.898936i	$-0.355659\pi$
$113$	9.31371	0.876160	0.438080	+	0.898936i	$0.355659\pi$
$114$	0	0
$115$	−3.41421	−0.318377
$116$	2.00000	0.185695
$117$	0	0
$118$	2.82843	0.260378
$119$	0.828427	0.0759418
$120$	0	0
$121$	7.00000	0.636364
$122$	−5.75736	−0.521247
$123$	0	0
$124$	0.828427	0.0743950
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−16.9706	−1.50589	−0.752947	−	0.658081i	$-0.771368\pi$
$127$	−16.9706	−1.50589	−0.752947	+	0.658081i	$0.771368\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.82843	−0.248069
$131$	−9.17157	−0.801324	−0.400662	−	0.916226i	$-0.631220\pi$
$131$	−9.17157	−0.801324	−0.400662	+	0.916226i	$0.631220\pi$
$132$	0	0
$133$	7.07107	0.613139
$134$	−3.75736	−0.324586
$135$	0	0
$136$	−0.828427	−0.0710370
$137$	−18.4853	−1.57930	−0.789652	−	0.613555i	$-0.789739\pi$
$137$	−18.4853	−1.57930	−0.789652	+	0.613555i	$0.789739\pi$
$138$	0	0
$139$	−20.4853	−1.73754	−0.868769	−	0.495217i	$-0.835089\pi$
$139$	−20.4853	−1.73754	−0.868769	+	0.495217i	$0.835089\pi$
$140$	3.41421	0.288554
$141$	0	0
$142$	12.8284	1.07654
$143$	3.51472	0.293916
$144$	0	0
$145$	−6.82843	−0.567070
$146$	−12.0000	−0.993127
$147$	0	0
$148$	−10.2426	−0.841940
$149$	−2.72792	−0.223480	−0.111740	−	0.993737i	$-0.535642\pi$
$149$	−2.72792	−0.223480	−0.111740	+	0.993737i	$0.535642\pi$
$150$	0	0
$151$	7.17157	0.583614	0.291807	−	0.956477i	$-0.405743\pi$
$151$	7.17157	0.583614	0.291807	+	0.956477i	$0.405743\pi$
$152$	−7.07107	−0.573539
$153$	0	0
$154$	−4.24264	−0.341882
$155$	−2.82843	−0.227185
$156$	0	0
$157$	7.41421	0.591719	0.295859	−	0.955232i	$-0.404394\pi$
$157$	7.41421	0.591719	0.295859	+	0.955232i	$0.404394\pi$
$158$	−11.3137	−0.900070
$159$	0	0
$160$	−3.41421	−0.269917
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−5.17157	−0.405069	−0.202534	−	0.979275i	$-0.564918\pi$
$163$	−5.17157	−0.405069	−0.202534	+	0.979275i	$0.564918\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−6.58579	−0.511156
$167$	−8.14214	−0.630057	−0.315029	−	0.949082i	$-0.602014\pi$
$167$	−8.14214	−0.630057	−0.315029	+	0.949082i	$0.602014\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	2.82843	0.216930
$171$	0	0
$172$	−6.58579	−0.502162
$173$	−8.14214	−0.619035	−0.309518	−	0.950894i	$-0.600168\pi$
$173$	−8.14214	−0.619035	−0.309518	+	0.950894i	$0.600168\pi$
$174$	0	0
$175$	−6.65685	−0.503211
$176$	4.24264	0.319801
$177$	0	0
$178$	−3.65685	−0.274093
$179$	1.17157	0.0875675	0.0437837	−	0.999041i	$-0.486059\pi$
$179$	1.17157	0.0875675	0.0437837	+	0.999041i	$0.486059\pi$
$180$	0	0
$181$	20.3848	1.51519	0.757594	−	0.652726i	$-0.226375\pi$
$181$	20.3848	1.51519	0.757594	+	0.652726i	$0.226375\pi$
$182$	−0.828427	−0.0614071
$183$	0	0
$184$	1.00000	0.0737210
$185$	34.9706	2.57109
$186$	0	0
$187$	−3.51472	−0.257022
$188$	−3.17157	−0.231311
$189$	0	0
$190$	24.1421	1.75145
$191$	2.82843	0.204658	0.102329	−	0.994751i	$-0.467371\pi$
$191$	2.82843	0.204658	0.102329	+	0.994751i	$0.467371\pi$
$192$	0	0
$193$	−7.65685	−0.551152	−0.275576	−	0.961279i	$-0.588869\pi$
$193$	−7.65685	−0.551152	−0.275576	+	0.961279i	$0.588869\pi$
$194$	−3.17157	−0.227706
$195$	0	0
$196$	1.00000	0.0714286
$197$	19.6569	1.40049	0.700246	−	0.713901i	$-0.253073\pi$
$197$	19.6569	1.40049	0.700246	+	0.713901i	$0.253073\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	6.65685	0.470711
$201$	0	0
$202$	−7.65685	−0.538734
$203$	−2.00000	−0.140372
$204$	0	0
$205$	6.82843	0.476918
$206$	4.48528	0.312504
$207$	0	0
$208$	0.828427	0.0574411
$209$	−30.0000	−2.07514
$210$	0	0
$211$	−9.65685	−0.664805	−0.332403	−	0.943138i	$-0.607859\pi$
$211$	−9.65685	−0.664805	−0.332403	+	0.943138i	$0.607859\pi$
$212$	−1.75736	−0.120696
$213$	0	0
$214$	−18.3848	−1.25676
$215$	22.4853	1.53348
$216$	0	0
$217$	−0.828427	−0.0562373
$218$	15.4142	1.04398
$219$	0	0
$220$	−14.4853	−0.976597
$221$	−0.686292	−0.0461650
$222$	0	0
$223$	28.1421	1.88454	0.942268	−	0.334859i	$-0.108689\pi$
$223$	28.1421	1.88454	0.942268	+	0.334859i	$0.108689\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	9.31371	0.619539
$227$	3.75736	0.249385	0.124692	−	0.992195i	$-0.460206\pi$
$227$	3.75736	0.249385	0.124692	+	0.992195i	$0.460206\pi$
$228$	0	0
$229$	20.3848	1.34706	0.673532	−	0.739158i	$-0.264776\pi$
$229$	20.3848	1.34706	0.673532	+	0.739158i	$0.264776\pi$
$230$	−3.41421	−0.225127
$231$	0	0
$232$	2.00000	0.131306
$233$	22.6274	1.48237	0.741186	−	0.671300i	$-0.234264\pi$
$233$	22.6274	1.48237	0.741186	+	0.671300i	$0.234264\pi$
$234$	0	0
$235$	10.8284	0.706369
$236$	2.82843	0.184115
$237$	0	0
$238$	0.828427	0.0536990
$239$	10.3431	0.669042	0.334521	−	0.942388i	$-0.391425\pi$
$239$	10.3431	0.669042	0.334521	+	0.942388i	$0.391425\pi$
$240$	0	0
$241$	6.48528	0.417754	0.208877	−	0.977942i	$-0.433019\pi$
$241$	6.48528	0.417754	0.208877	+	0.977942i	$0.433019\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−5.75736	−0.368577
$245$	−3.41421	−0.218126
$246$	0	0
$247$	−5.85786	−0.372727
$248$	0.828427	0.0526052
$249$	0	0
$250$	−5.65685	−0.357771
$251$	18.3848	1.16044	0.580218	−	0.814461i	$-0.302967\pi$
$251$	18.3848	1.16044	0.580218	+	0.814461i	$0.302967\pi$
$252$	0	0
$253$	4.24264	0.266733
$254$	−16.9706	−1.06483
$255$	0	0
$256$	1.00000	0.0625000
$257$	29.3137	1.82854	0.914269	−	0.405107i	$-0.132766\pi$
$257$	29.3137	1.82854	0.914269	+	0.405107i	$0.132766\pi$
$258$	0	0
$259$	10.2426	0.636447
$260$	−2.82843	−0.175412
$261$	0	0
$262$	−9.17157	−0.566622
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	7.07107	0.433555
$267$	0	0
$268$	−3.75736	−0.229517
$269$	13.7990	0.841339	0.420670	−	0.907214i	$-0.361795\pi$
$269$	13.7990	0.841339	0.420670	+	0.907214i	$0.361795\pi$
$270$	0	0
$271$	−10.3431	−0.628301	−0.314151	−	0.949373i	$-0.601720\pi$
$271$	−10.3431	−0.628301	−0.314151	+	0.949373i	$0.601720\pi$
$272$	−0.828427	−0.0502308
$273$	0	0
$274$	−18.4853	−1.11674
$275$	28.2426	1.70310
$276$	0	0
$277$	−0.828427	−0.0497754	−0.0248877	−	0.999690i	$-0.507923\pi$
$277$	−0.828427	−0.0497754	−0.0248877	+	0.999690i	$0.507923\pi$
$278$	−20.4853	−1.22863
$279$	0	0
$280$	3.41421	0.204038
$281$	18.9706	1.13169	0.565844	−	0.824512i	$-0.308550\pi$
$281$	18.9706	1.13169	0.565844	+	0.824512i	$0.308550\pi$
$282$	0	0
$283$	1.41421	0.0840663	0.0420331	−	0.999116i	$-0.486616\pi$
$283$	1.41421	0.0840663	0.0420331	+	0.999116i	$0.486616\pi$
$284$	12.8284	0.761227
$285$	0	0
$286$	3.51472	0.207830
$287$	2.00000	0.118056
$288$	0	0
$289$	−16.3137	−0.959630
$290$	−6.82843	−0.400979
$291$	0	0
$292$	−12.0000	−0.702247
$293$	−27.2132	−1.58981	−0.794906	−	0.606732i	$-0.792480\pi$
$293$	−27.2132	−1.58981	−0.794906	+	0.606732i	$0.792480\pi$
$294$	0	0
$295$	−9.65685	−0.562244
$296$	−10.2426	−0.595341
$297$	0	0
$298$	−2.72792	−0.158024
$299$	0.828427	0.0479092
$300$	0	0
$301$	6.58579	0.379598
$302$	7.17157	0.412678
$303$	0	0
$304$	−7.07107	−0.405554
$305$	19.6569	1.12555
$306$	0	0
$307$	11.3137	0.645707	0.322854	−	0.946449i	$-0.395358\pi$
$307$	11.3137	0.645707	0.322854	+	0.946449i	$0.395358\pi$
$308$	−4.24264	−0.241747
$309$	0	0
$310$	−2.82843	−0.160644
$311$	−0.970563	−0.0550356	−0.0275178	−	0.999621i	$-0.508760\pi$
$311$	−0.970563	−0.0550356	−0.0275178	+	0.999621i	$0.508760\pi$
$312$	0	0
$313$	−13.7990	−0.779965	−0.389983	−	0.920822i	$-0.627519\pi$
$313$	−13.7990	−0.779965	−0.389983	+	0.920822i	$0.627519\pi$
$314$	7.41421	0.418408
$315$	0	0
$316$	−11.3137	−0.636446
$317$	6.97056	0.391506	0.195753	−	0.980653i	$-0.437285\pi$
$317$	6.97056	0.391506	0.195753	+	0.980653i	$0.437285\pi$
$318$	0	0
$319$	8.48528	0.475085
$320$	−3.41421	−0.190860
$321$	0	0
$322$	−1.00000	−0.0557278
$323$	5.85786	0.325940
$324$	0	0
$325$	5.51472	0.305902
$326$	−5.17157	−0.286427
$327$	0	0
$328$	−2.00000	−0.110432
$329$	3.17157	0.174854
$330$	0	0
$331$	−18.6274	−1.02386	−0.511928	−	0.859029i	$-0.671068\pi$
$331$	−18.6274	−1.02386	−0.511928	+	0.859029i	$0.671068\pi$
$332$	−6.58579	−0.361442
$333$	0	0
$334$	−8.14214	−0.445518
$335$	12.8284	0.700892
$336$	0	0
$337$	−23.6569	−1.28867	−0.644335	−	0.764743i	$-0.722866\pi$
$337$	−23.6569	−1.28867	−0.644335	+	0.764743i	$0.722866\pi$
$338$	−12.3137	−0.669777
$339$	0	0
$340$	2.82843	0.153393
$341$	3.51472	0.190333
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−6.58579	−0.355082
$345$	0	0
$346$	−8.14214	−0.437724
$347$	−15.7990	−0.848134	−0.424067	−	0.905631i	$-0.639398\pi$
$347$	−15.7990	−0.848134	−0.424067	+	0.905631i	$0.639398\pi$
$348$	0	0
$349$	−23.6569	−1.26632	−0.633161	−	0.774020i	$-0.718243\pi$
$349$	−23.6569	−1.26632	−0.633161	+	0.774020i	$0.718243\pi$
$350$	−6.65685	−0.355824
$351$	0	0
$352$	4.24264	0.226134
$353$	3.31371	0.176371	0.0881855	−	0.996104i	$-0.471893\pi$
$353$	3.31371	0.176371	0.0881855	+	0.996104i	$0.471893\pi$
$354$	0	0
$355$	−43.7990	−2.32461
$356$	−3.65685	−0.193813
$357$	0	0
$358$	1.17157	0.0619196
$359$	15.5147	0.818836	0.409418	−	0.912347i	$-0.365732\pi$
$359$	15.5147	0.818836	0.409418	+	0.912347i	$0.365732\pi$
$360$	0	0
$361$	31.0000	1.63158
$362$	20.3848	1.07140
$363$	0	0
$364$	−0.828427	−0.0434214
$365$	40.9706	2.14450
$366$	0	0
$367$	31.3137	1.63456	0.817281	−	0.576239i	$-0.195480\pi$
$367$	31.3137	1.63456	0.817281	+	0.576239i	$0.195480\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	34.9706	1.81803
$371$	1.75736	0.0912375
$372$	0	0
$373$	−13.7574	−0.712329	−0.356165	−	0.934423i	$-0.615916\pi$
$373$	−13.7574	−0.712329	−0.356165	+	0.934423i	$0.615916\pi$
$374$	−3.51472	−0.181742
$375$	0	0
$376$	−3.17157	−0.163561
$377$	1.65685	0.0853323
$378$	0	0
$379$	32.5269	1.67080	0.835398	−	0.549646i	$-0.185237\pi$
$379$	32.5269	1.67080	0.835398	+	0.549646i	$0.185237\pi$
$380$	24.1421	1.23847
$381$	0	0
$382$	2.82843	0.144715
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	0	0
$385$	14.4853	0.738238
$386$	−7.65685	−0.389724
$387$	0	0
$388$	−3.17157	−0.161012
$389$	−31.6985	−1.60718	−0.803588	−	0.595185i	$-0.797079\pi$
$389$	−31.6985	−1.60718	−0.803588	+	0.595185i	$0.797079\pi$
$390$	0	0
$391$	−0.828427	−0.0418954
$392$	1.00000	0.0505076
$393$	0	0
$394$	19.6569	0.990298
$395$	38.6274	1.94356
$396$	0	0
$397$	7.17157	0.359931	0.179965	−	0.983673i	$-0.442401\pi$
$397$	7.17157	0.359931	0.179965	+	0.983673i	$0.442401\pi$
$398$	0	0
$399$	0	0
$400$	6.65685	0.332843
$401$	4.82843	0.241120	0.120560	−	0.992706i	$-0.461531\pi$
$401$	4.82843	0.241120	0.120560	+	0.992706i	$0.461531\pi$
$402$	0	0
$403$	0.686292	0.0341866
$404$	−7.65685	−0.380943
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	−43.4558	−2.15403
$408$	0	0
$409$	−5.31371	−0.262746	−0.131373	−	0.991333i	$-0.541939\pi$
$409$	−5.31371	−0.262746	−0.131373	+	0.991333i	$0.541939\pi$
$410$	6.82843	0.337232
$411$	0	0
$412$	4.48528	0.220974
$413$	−2.82843	−0.139178
$414$	0	0
$415$	22.4853	1.10376
$416$	0.828427	0.0406170
$417$	0	0
$418$	−30.0000	−1.46735
$419$	−39.5563	−1.93245	−0.966227	−	0.257692i	$-0.917038\pi$
$419$	−39.5563	−1.93245	−0.966227	+	0.257692i	$0.917038\pi$
$420$	0	0
$421$	−8.10051	−0.394794	−0.197397	−	0.980324i	$-0.563249\pi$
$421$	−8.10051	−0.394794	−0.197397	+	0.980324i	$0.563249\pi$
$422$	−9.65685	−0.470088
$423$	0	0
$424$	−1.75736	−0.0853449
$425$	−5.51472	−0.267503
$426$	0	0
$427$	5.75736	0.278618
$428$	−18.3848	−0.888662
$429$	0	0
$430$	22.4853	1.08434
$431$	−38.8284	−1.87030	−0.935150	−	0.354253i	$-0.884735\pi$
$431$	−38.8284	−1.87030	−0.935150	+	0.354253i	$0.884735\pi$
$432$	0	0
$433$	8.14214	0.391286	0.195643	−	0.980675i	$-0.437321\pi$
$433$	8.14214	0.391286	0.195643	+	0.980675i	$0.437321\pi$
$434$	−0.828427	−0.0397658
$435$	0	0
$436$	15.4142	0.738207
$437$	−7.07107	−0.338255
$438$	0	0
$439$	−13.6569	−0.651806	−0.325903	−	0.945403i	$-0.605668\pi$
$439$	−13.6569	−0.651806	−0.325903	+	0.945403i	$0.605668\pi$
$440$	−14.4853	−0.690559
$441$	0	0
$442$	−0.686292	−0.0326436
$443$	28.4853	1.35338	0.676688	−	0.736270i	$-0.263414\pi$
$443$	28.4853	1.35338	0.676688	+	0.736270i	$0.263414\pi$
$444$	0	0
$445$	12.4853	0.591859
$446$	28.1421	1.33257
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−30.9706	−1.46159	−0.730796	−	0.682596i	$-0.760851\pi$
$449$	−30.9706	−1.46159	−0.730796	+	0.682596i	$0.760851\pi$
$450$	0	0
$451$	−8.48528	−0.399556
$452$	9.31371	0.438080
$453$	0	0
$454$	3.75736	0.176342
$455$	2.82843	0.132599
$456$	0	0
$457$	20.8284	0.974313	0.487156	−	0.873315i	$-0.338034\pi$
$457$	20.8284	0.974313	0.487156	+	0.873315i	$0.338034\pi$
$458$	20.3848	0.952518
$459$	0	0
$460$	−3.41421	−0.159189
$461$	−3.17157	−0.147715	−0.0738574	−	0.997269i	$-0.523531\pi$
$461$	−3.17157	−0.147715	−0.0738574	+	0.997269i	$0.523531\pi$
$462$	0	0
$463$	27.3137	1.26938	0.634688	−	0.772769i	$-0.281129\pi$
$463$	27.3137	1.26938	0.634688	+	0.772769i	$0.281129\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	22.6274	1.04819
$467$	−9.41421	−0.435638	−0.217819	−	0.975989i	$-0.569894\pi$
$467$	−9.41421	−0.435638	−0.217819	+	0.975989i	$0.569894\pi$
$468$	0	0
$469$	3.75736	0.173499
$470$	10.8284	0.499478
$471$	0	0
$472$	2.82843	0.130189
$473$	−27.9411	−1.28473
$474$	0	0
$475$	−47.0711	−2.15977
$476$	0.828427	0.0379709
$477$	0	0
$478$	10.3431	0.473084
$479$	−8.97056	−0.409875	−0.204938	−	0.978775i	$-0.565699\pi$
$479$	−8.97056	−0.409875	−0.204938	+	0.978775i	$0.565699\pi$
$480$	0	0
$481$	−8.48528	−0.386896
$482$	6.48528	0.295396
$483$	0	0
$484$	7.00000	0.318182
$485$	10.8284	0.491694
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−5.75736	−0.260623
$489$	0	0
$490$	−3.41421	−0.154238
$491$	18.1421	0.818743	0.409372	−	0.912368i	$-0.365748\pi$
$491$	18.1421	0.818743	0.409372	+	0.912368i	$0.365748\pi$
$492$	0	0
$493$	−1.65685	−0.0746210
$494$	−5.85786	−0.263558
$495$	0	0
$496$	0.828427	0.0371975
$497$	−12.8284	−0.575434
$498$	0	0
$499$	26.1421	1.17028	0.585141	−	0.810931i	$-0.301039\pi$
$499$	26.1421	1.17028	0.585141	+	0.810931i	$0.301039\pi$
$500$	−5.65685	−0.252982
$501$	0	0
$502$	18.3848	0.820553
$503$	44.7696	1.99618	0.998088	−	0.0618114i	$-0.0196877\pi$
$503$	44.7696	1.99618	0.998088	+	0.0618114i	$0.0196877\pi$
$504$	0	0
$505$	26.1421	1.16331
$506$	4.24264	0.188608
$507$	0	0
$508$	−16.9706	−0.752947
$509$	−29.1127	−1.29040	−0.645199	−	0.764015i	$-0.723225\pi$
$509$	−29.1127	−1.29040	−0.645199	+	0.764015i	$0.723225\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	1.00000	0.0441942
$513$	0	0
$514$	29.3137	1.29297
$515$	−15.3137	−0.674803
$516$	0	0
$517$	−13.4558	−0.591787
$518$	10.2426	0.450036
$519$	0	0
$520$	−2.82843	−0.124035
$521$	−5.51472	−0.241604	−0.120802	−	0.992677i	$-0.538547\pi$
$521$	−5.51472	−0.241604	−0.120802	+	0.992677i	$0.538547\pi$
$522$	0	0
$523$	−19.7574	−0.863929	−0.431965	−	0.901891i	$-0.642179\pi$
$523$	−19.7574	−0.863929	−0.431965	+	0.901891i	$0.642179\pi$
$524$	−9.17157	−0.400662
$525$	0	0
$526$	12.0000	0.523225
$527$	−0.686292	−0.0298953
$528$	0	0
$529$	1.00000	0.0434783
$530$	6.00000	0.260623
$531$	0	0
$532$	7.07107	0.306570
$533$	−1.65685	−0.0717663
$534$	0	0
$535$	62.7696	2.71376
$536$	−3.75736	−0.162293
$537$	0	0
$538$	13.7990	0.594917
$539$	4.24264	0.182743
$540$	0	0
$541$	7.17157	0.308330	0.154165	−	0.988045i	$-0.450731\pi$
$541$	7.17157	0.308330	0.154165	+	0.988045i	$0.450731\pi$
$542$	−10.3431	−0.444276
$543$	0	0
$544$	−0.828427	−0.0355185
$545$	−52.6274	−2.25431
$546$	0	0
$547$	−22.6274	−0.967478	−0.483739	−	0.875212i	$-0.660722\pi$
$547$	−22.6274	−0.967478	−0.483739	+	0.875212i	$0.660722\pi$
$548$	−18.4853	−0.789652
$549$	0	0
$550$	28.2426	1.20427
$551$	−14.1421	−0.602475
$552$	0	0
$553$	11.3137	0.481108
$554$	−0.828427	−0.0351965
$555$	0	0
$556$	−20.4853	−0.868769
$557$	−1.27208	−0.0538997	−0.0269498	−	0.999637i	$-0.508579\pi$
$557$	−1.27208	−0.0538997	−0.0269498	+	0.999637i	$0.508579\pi$
$558$	0	0
$559$	−5.45584	−0.230758
$560$	3.41421	0.144277
$561$	0	0
$562$	18.9706	0.800225
$563$	13.2132	0.556870	0.278435	−	0.960455i	$-0.410184\pi$
$563$	13.2132	0.556870	0.278435	+	0.960455i	$0.410184\pi$
$564$	0	0
$565$	−31.7990	−1.33779
$566$	1.41421	0.0594438
$567$	0	0
$568$	12.8284	0.538269
$569$	−13.5147	−0.566566	−0.283283	−	0.959036i	$-0.591424\pi$
$569$	−13.5147	−0.566566	−0.283283	+	0.959036i	$0.591424\pi$
$570$	0	0
$571$	23.0711	0.965494	0.482747	−	0.875760i	$-0.339639\pi$
$571$	23.0711	0.965494	0.482747	+	0.875760i	$0.339639\pi$
$572$	3.51472	0.146958
$573$	0	0
$574$	2.00000	0.0834784
$575$	6.65685	0.277610
$576$	0	0
$577$	−21.9411	−0.913421	−0.456711	−	0.889615i	$-0.650972\pi$
$577$	−21.9411	−0.913421	−0.456711	+	0.889615i	$0.650972\pi$
$578$	−16.3137	−0.678561
$579$	0	0
$580$	−6.82843	−0.283535
$581$	6.58579	0.273224
$582$	0	0
$583$	−7.45584	−0.308790
$584$	−12.0000	−0.496564
$585$	0	0
$586$	−27.2132	−1.12417
$587$	25.4558	1.05068	0.525338	−	0.850894i	$-0.323939\pi$
$587$	25.4558	1.05068	0.525338	+	0.850894i	$0.323939\pi$
$588$	0	0
$589$	−5.85786	−0.241369
$590$	−9.65685	−0.397566
$591$	0	0
$592$	−10.2426	−0.420970
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	−2.82843	−0.115954
$596$	−2.72792	−0.111740
$597$	0	0
$598$	0.828427	0.0338769
$599$	33.9411	1.38680	0.693398	−	0.720554i	$-0.256113\pi$
$599$	33.9411	1.38680	0.693398	+	0.720554i	$0.256113\pi$
$600$	0	0
$601$	−10.6274	−0.433501	−0.216751	−	0.976227i	$-0.569546\pi$
$601$	−10.6274	−0.433501	−0.216751	+	0.976227i	$0.569546\pi$
$602$	6.58579	0.268417
$603$	0	0
$604$	7.17157	0.291807
$605$	−23.8995	−0.971653
$606$	0	0
$607$	−32.8284	−1.33246	−0.666232	−	0.745744i	$-0.732094\pi$
$607$	−32.8284	−1.33246	−0.666232	+	0.745744i	$0.732094\pi$
$608$	−7.07107	−0.286770
$609$	0	0
$610$	19.6569	0.795883
$611$	−2.62742	−0.106294
$612$	0	0
$613$	−43.6985	−1.76497	−0.882483	−	0.470345i	$-0.844129\pi$
$613$	−43.6985	−1.76497	−0.882483	+	0.470345i	$0.844129\pi$
$614$	11.3137	0.456584
$615$	0	0
$616$	−4.24264	−0.170941
$617$	−35.6569	−1.43549	−0.717745	−	0.696306i	$-0.754826\pi$
$617$	−35.6569	−1.43549	−0.717745	+	0.696306i	$0.754826\pi$
$618$	0	0
$619$	18.8701	0.758452	0.379226	−	0.925304i	$-0.376190\pi$
$619$	18.8701	0.758452	0.379226	+	0.925304i	$0.376190\pi$
$620$	−2.82843	−0.113592
$621$	0	0
$622$	−0.970563	−0.0389160
$623$	3.65685	0.146509
$624$	0	0
$625$	−13.9706	−0.558823
$626$	−13.7990	−0.551519
$627$	0	0
$628$	7.41421	0.295859
$629$	8.48528	0.338330
$630$	0	0
$631$	25.1716	1.00206	0.501032	−	0.865429i	$-0.332954\pi$
$631$	25.1716	1.00206	0.501032	+	0.865429i	$0.332954\pi$
$632$	−11.3137	−0.450035
$633$	0	0
$634$	6.97056	0.276836
$635$	57.9411	2.29932
$636$	0	0
$637$	0.828427	0.0328235
$638$	8.48528	0.335936
$639$	0	0
$640$	−3.41421	−0.134959
$641$	−11.8579	−0.468357	−0.234179	−	0.972194i	$-0.575240\pi$
$641$	−11.8579	−0.468357	−0.234179	+	0.972194i	$0.575240\pi$
$642$	0	0
$643$	−13.6985	−0.540216	−0.270108	−	0.962830i	$-0.587059\pi$
$643$	−13.6985	−0.540216	−0.270108	+	0.962830i	$0.587059\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	5.85786	0.230475
$647$	−47.5980	−1.87127	−0.935635	−	0.352969i	$-0.885172\pi$
$647$	−47.5980	−1.87127	−0.935635	+	0.352969i	$0.885172\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	5.51472	0.216305
$651$	0	0
$652$	−5.17157	−0.202534
$653$	25.1127	0.982736	0.491368	−	0.870952i	$-0.336497\pi$
$653$	25.1127	0.982736	0.491368	+	0.870952i	$0.336497\pi$
$654$	0	0
$655$	31.3137	1.22353
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	3.17157	0.123641
$659$	12.7279	0.495809	0.247905	−	0.968784i	$-0.420258\pi$
$659$	12.7279	0.495809	0.247905	+	0.968784i	$0.420258\pi$
$660$	0	0
$661$	−10.2426	−0.398393	−0.199196	−	0.979960i	$-0.563833\pi$
$661$	−10.2426	−0.398393	−0.199196	+	0.979960i	$0.563833\pi$
$662$	−18.6274	−0.723975
$663$	0	0
$664$	−6.58579	−0.255578
$665$	−24.1421	−0.936192
$666$	0	0
$667$	2.00000	0.0774403
$668$	−8.14214	−0.315029
$669$	0	0
$670$	12.8284	0.495605
$671$	−24.4264	−0.942971
$672$	0	0
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	−23.6569	−0.911228
$675$	0	0
$676$	−12.3137	−0.473604
$677$	−28.5858	−1.09864	−0.549321	−	0.835612i	$-0.685113\pi$
$677$	−28.5858	−1.09864	−0.549321	+	0.835612i	$0.685113\pi$
$678$	0	0
$679$	3.17157	0.121714
$680$	2.82843	0.108465
$681$	0	0
$682$	3.51472	0.134586
$683$	−15.0294	−0.575085	−0.287543	−	0.957768i	$-0.592838\pi$
$683$	−15.0294	−0.575085	−0.287543	+	0.957768i	$0.592838\pi$
$684$	0	0
$685$	63.1127	2.41141
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−6.58579	−0.251081
$689$	−1.45584	−0.0554632
$690$	0	0
$691$	34.1421	1.29883	0.649414	−	0.760435i	$-0.275014\pi$
$691$	34.1421	1.29883	0.649414	+	0.760435i	$0.275014\pi$
$692$	−8.14214	−0.309518
$693$	0	0
$694$	−15.7990	−0.599721
$695$	69.9411	2.65302
$696$	0	0
$697$	1.65685	0.0627578
$698$	−23.6569	−0.895425
$699$	0	0
$700$	−6.65685	−0.251605
$701$	−25.0711	−0.946921	−0.473461	−	0.880815i	$-0.656995\pi$
$701$	−25.0711	−0.946921	−0.473461	+	0.880815i	$0.656995\pi$
$702$	0	0
$703$	72.4264	2.73161
$704$	4.24264	0.159901
$705$	0	0
$706$	3.31371	0.124713
$707$	7.65685	0.287966
$708$	0	0
$709$	25.2721	0.949113	0.474556	−	0.880225i	$-0.342608\pi$
$709$	25.2721	0.949113	0.474556	+	0.880225i	$0.342608\pi$
$710$	−43.7990	−1.64375
$711$	0	0
$712$	−3.65685	−0.137046
$713$	0.828427	0.0310248
$714$	0	0
$715$	−12.0000	−0.448775
$716$	1.17157	0.0437837
$717$	0	0
$718$	15.5147	0.579004
$719$	24.9706	0.931245	0.465622	−	0.884983i	$-0.345830\pi$
$719$	24.9706	0.931245	0.465622	+	0.884983i	$0.345830\pi$
$720$	0	0
$721$	−4.48528	−0.167041
$722$	31.0000	1.15370
$723$	0	0
$724$	20.3848	0.757594
$725$	13.3137	0.494459
$726$	0	0
$727$	−36.2843	−1.34571	−0.672855	−	0.739775i	$-0.734932\pi$
$727$	−36.2843	−1.34571	−0.672855	+	0.739775i	$0.734932\pi$
$728$	−0.828427	−0.0307036
$729$	0	0
$730$	40.9706	1.51639
$731$	5.45584	0.201792
$732$	0	0
$733$	−20.1005	−0.742429	−0.371215	−	0.928547i	$-0.621059\pi$
$733$	−20.1005	−0.742429	−0.371215	+	0.928547i	$0.621059\pi$
$734$	31.3137	1.15581
$735$	0	0
$736$	1.00000	0.0368605
$737$	−15.9411	−0.587199
$738$	0	0
$739$	5.85786	0.215485	0.107743	−	0.994179i	$-0.465638\pi$
$739$	5.85786	0.215485	0.107743	+	0.994179i	$0.465638\pi$
$740$	34.9706	1.28554
$741$	0	0
$742$	1.75736	0.0645147
$743$	−42.6274	−1.56385	−0.781924	−	0.623374i	$-0.785762\pi$
$743$	−42.6274	−1.56385	−0.781924	+	0.623374i	$0.785762\pi$
$744$	0	0
$745$	9.31371	0.341228
$746$	−13.7574	−0.503693
$747$	0	0
$748$	−3.51472	−0.128511
$749$	18.3848	0.671765
$750$	0	0
$751$	23.1127	0.843394	0.421697	−	0.906737i	$-0.361435\pi$
$751$	23.1127	0.843394	0.421697	+	0.906737i	$0.361435\pi$
$752$	−3.17157	−0.115655
$753$	0	0
$754$	1.65685	0.0603391
$755$	−24.4853	−0.891111
$756$	0	0
$757$	5.07107	0.184311	0.0921555	−	0.995745i	$-0.470624\pi$
$757$	5.07107	0.184311	0.0921555	+	0.995745i	$0.470624\pi$
$758$	32.5269	1.18143
$759$	0	0
$760$	24.1421	0.875727
$761$	25.6569	0.930060	0.465030	−	0.885295i	$-0.346043\pi$
$761$	25.6569	0.930060	0.465030	+	0.885295i	$0.346043\pi$
$762$	0	0
$763$	−15.4142	−0.558032
$764$	2.82843	0.102329
$765$	0	0
$766$	20.0000	0.722629
$767$	2.34315	0.0846061
$768$	0	0
$769$	5.79899	0.209117	0.104558	−	0.994519i	$-0.466657\pi$
$769$	5.79899	0.209117	0.104558	+	0.994519i	$0.466657\pi$
$770$	14.4853	0.522013
$771$	0	0
$772$	−7.65685	−0.275576
$773$	5.07107	0.182394	0.0911968	−	0.995833i	$-0.470931\pi$
$773$	5.07107	0.182394	0.0911968	+	0.995833i	$0.470931\pi$
$774$	0	0
$775$	5.51472	0.198095
$776$	−3.17157	−0.113853
$777$	0	0
$778$	−31.6985	−1.13645
$779$	14.1421	0.506695
$780$	0	0
$781$	54.4264	1.94753
$782$	−0.828427	−0.0296245
$783$	0	0
$784$	1.00000	0.0357143
$785$	−25.3137	−0.903485
$786$	0	0
$787$	14.1005	0.502629	0.251314	−	0.967906i	$-0.419137\pi$
$787$	14.1005	0.502629	0.251314	+	0.967906i	$0.419137\pi$
$788$	19.6569	0.700246
$789$	0	0
$790$	38.6274	1.37430
$791$	−9.31371	−0.331157
$792$	0	0
$793$	−4.76955	−0.169372
$794$	7.17157	0.254510
$795$	0	0
$796$	0	0
$797$	4.87006	0.172506	0.0862531	−	0.996273i	$-0.472511\pi$
$797$	4.87006	0.172506	0.0862531	+	0.996273i	$0.472511\pi$
$798$	0	0
$799$	2.62742	0.0929513
$800$	6.65685	0.235355
$801$	0	0
$802$	4.82843	0.170498
$803$	−50.9117	−1.79663
$804$	0	0
$805$	3.41421	0.120335
$806$	0.686292	0.0241736
$807$	0	0
$808$	−7.65685	−0.269367
$809$	−33.9411	−1.19331	−0.596653	−	0.802499i	$-0.703503\pi$
$809$	−33.9411	−1.19331	−0.596653	+	0.802499i	$0.703503\pi$
$810$	0	0
$811$	−2.82843	−0.0993195	−0.0496598	−	0.998766i	$-0.515814\pi$
$811$	−2.82843	−0.0993195	−0.0496598	+	0.998766i	$0.515814\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	−43.4558	−1.52313
$815$	17.6569	0.618493
$816$	0	0
$817$	46.5685	1.62923
$818$	−5.31371	−0.185789
$819$	0	0
$820$	6.82843	0.238459
$821$	−25.1127	−0.876439	−0.438220	−	0.898868i	$-0.644391\pi$
$821$	−25.1127	−0.876439	−0.438220	+	0.898868i	$0.644391\pi$
$822$	0	0
$823$	5.51472	0.192231	0.0961155	−	0.995370i	$-0.469358\pi$
$823$	5.51472	0.192231	0.0961155	+	0.995370i	$0.469358\pi$
$824$	4.48528	0.156252
$825$	0	0
$826$	−2.82843	−0.0984136
$827$	5.21320	0.181281	0.0906404	−	0.995884i	$-0.471109\pi$
$827$	5.21320	0.181281	0.0906404	+	0.995884i	$0.471109\pi$
$828$	0	0
$829$	36.6274	1.27212	0.636061	−	0.771638i	$-0.280562\pi$
$829$	36.6274	1.27212	0.636061	+	0.771638i	$0.280562\pi$
$830$	22.4853	0.780476
$831$	0	0
$832$	0.828427	0.0287205
$833$	−0.828427	−0.0287033
$834$	0	0
$835$	27.7990	0.962024
$836$	−30.0000	−1.03757
$837$	0	0
$838$	−39.5563	−1.36645
$839$	0.686292	0.0236934	0.0118467	−	0.999930i	$-0.496229\pi$
$839$	0.686292	0.0236934	0.0118467	+	0.999930i	$0.496229\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−8.10051	−0.279162
$843$	0	0
$844$	−9.65685	−0.332403
$845$	42.0416	1.44628
$846$	0	0
$847$	−7.00000	−0.240523
$848$	−1.75736	−0.0603480
$849$	0	0
$850$	−5.51472	−0.189153
$851$	−10.2426	−0.351113
$852$	0	0
$853$	−17.5147	−0.599693	−0.299846	−	0.953988i	$-0.596935\pi$
$853$	−17.5147	−0.599693	−0.299846	+	0.953988i	$0.596935\pi$
$854$	5.75736	0.197013
$855$	0	0
$856$	−18.3848	−0.628379
$857$	12.6863	0.433355	0.216678	−	0.976243i	$-0.430478\pi$
$857$	12.6863	0.433355	0.216678	+	0.976243i	$0.430478\pi$
$858$	0	0
$859$	35.5980	1.21459	0.607294	−	0.794477i	$-0.292255\pi$
$859$	35.5980	1.21459	0.607294	+	0.794477i	$0.292255\pi$
$860$	22.4853	0.766742
$861$	0	0
$862$	−38.8284	−1.32250
$863$	−8.97056	−0.305362	−0.152681	−	0.988276i	$-0.548791\pi$
$863$	−8.97056	−0.305362	−0.152681	+	0.988276i	$0.548791\pi$
$864$	0	0
$865$	27.7990	0.945194
$866$	8.14214	0.276681
$867$	0	0
$868$	−0.828427	−0.0281186
$869$	−48.0000	−1.62829
$870$	0	0
$871$	−3.11270	−0.105470
$872$	15.4142	0.521991
$873$	0	0
$874$	−7.07107	−0.239182
$875$	5.65685	0.191237
$876$	0	0
$877$	8.82843	0.298115	0.149057	−	0.988829i	$-0.452376\pi$
$877$	8.82843	0.298115	0.149057	+	0.988829i	$0.452376\pi$
$878$	−13.6569	−0.460897
$879$	0	0
$880$	−14.4853	−0.488299
$881$	9.31371	0.313787	0.156893	−	0.987616i	$-0.449852\pi$
$881$	9.31371	0.313787	0.156893	+	0.987616i	$0.449852\pi$
$882$	0	0
$883$	−33.9411	−1.14221	−0.571105	−	0.820877i	$-0.693485\pi$
$883$	−33.9411	−1.14221	−0.571105	+	0.820877i	$0.693485\pi$
$884$	−0.686292	−0.0230825
$885$	0	0
$886$	28.4853	0.956982
$887$	40.1421	1.34784	0.673921	−	0.738804i	$-0.264609\pi$
$887$	40.1421	1.34784	0.673921	+	0.738804i	$0.264609\pi$
$888$	0	0
$889$	16.9706	0.569174
$890$	12.4853	0.418508
$891$	0	0
$892$	28.1421	0.942268
$893$	22.4264	0.750471
$894$	0	0
$895$	−4.00000	−0.133705
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−30.9706	−1.03350
$899$	1.65685	0.0552592
$900$	0	0
$901$	1.45584	0.0485012
$902$	−8.48528	−0.282529
$903$	0	0
$904$	9.31371	0.309769
$905$	−69.5980	−2.31352
$906$	0	0
$907$	−41.0122	−1.36179	−0.680894	−	0.732382i	$-0.738408\pi$
$907$	−41.0122	−1.36179	−0.680894	+	0.732382i	$0.738408\pi$
$908$	3.75736	0.124692
$909$	0	0
$910$	2.82843	0.0937614
$911$	−42.1421	−1.39623	−0.698116	−	0.715985i	$-0.745978\pi$
$911$	−42.1421	−1.39623	−0.698116	+	0.715985i	$0.745978\pi$
$912$	0	0
$913$	−27.9411	−0.924716
$914$	20.8284	0.688943
$915$	0	0
$916$	20.3848	0.673532
$917$	9.17157	0.302872
$918$	0	0
$919$	54.6274	1.80199	0.900996	−	0.433827i	$-0.142837\pi$
$919$	54.6274	1.80199	0.900996	+	0.433827i	$0.142837\pi$
$920$	−3.41421	−0.112563
$921$	0	0
$922$	−3.17157	−0.104450
$923$	10.6274	0.349806
$924$	0	0
$925$	−68.1838	−2.24187
$926$	27.3137	0.897584
$927$	0	0
$928$	2.00000	0.0656532
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	0	0
$931$	−7.07107	−0.231745
$932$	22.6274	0.741186
$933$	0	0
$934$	−9.41421	−0.308042
$935$	12.0000	0.392442
$936$	0	0
$937$	−50.2843	−1.64272	−0.821358	−	0.570413i	$-0.806783\pi$
$937$	−50.2843	−1.64272	−0.821358	+	0.570413i	$0.806783\pi$
$938$	3.75736	0.122682
$939$	0	0
$940$	10.8284	0.353184
$941$	−7.89949	−0.257516	−0.128758	−	0.991676i	$-0.541099\pi$
$941$	−7.89949	−0.257516	−0.128758	+	0.991676i	$0.541099\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	2.82843	0.0920575
$945$	0	0
$946$	−27.9411	−0.908444
$947$	19.5147	0.634143	0.317072	−	0.948402i	$-0.397300\pi$
$947$	19.5147	0.634143	0.317072	+	0.948402i	$0.397300\pi$
$948$	0	0
$949$	−9.94113	−0.322703
$950$	−47.0711	−1.52719
$951$	0	0
$952$	0.828427	0.0268495
$953$	11.1716	0.361883	0.180941	−	0.983494i	$-0.442086\pi$
$953$	11.1716	0.361883	0.180941	+	0.983494i	$0.442086\pi$
$954$	0	0
$955$	−9.65685	−0.312488
$956$	10.3431	0.334521
$957$	0	0
$958$	−8.97056	−0.289826
$959$	18.4853	0.596921
$960$	0	0
$961$	−30.3137	−0.977862
$962$	−8.48528	−0.273576
$963$	0	0
$964$	6.48528	0.208877
$965$	26.1421	0.841545
$966$	0	0
$967$	16.8284	0.541166	0.270583	−	0.962697i	$-0.412784\pi$
$967$	16.8284	0.541166	0.270583	+	0.962697i	$0.412784\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	10.8284	0.347680
$971$	−32.5269	−1.04384	−0.521919	−	0.852995i	$-0.674784\pi$
$971$	−32.5269	−1.04384	−0.521919	+	0.852995i	$0.674784\pi$
$972$	0	0
$973$	20.4853	0.656728
$974$	0	0
$975$	0	0
$976$	−5.75736	−0.184289
$977$	9.11270	0.291541	0.145771	−	0.989318i	$-0.453434\pi$
$977$	9.11270	0.291541	0.145771	+	0.989318i	$0.453434\pi$
$978$	0	0
$979$	−15.5147	−0.495853
$980$	−3.41421	−0.109063
$981$	0	0
$982$	18.1421	0.578939
$983$	−38.1421	−1.21655	−0.608273	−	0.793728i	$-0.708137\pi$
$983$	−38.1421	−1.21655	−0.608273	+	0.793728i	$0.708137\pi$
$984$	0	0
$985$	−67.1127	−2.13839
$986$	−1.65685	−0.0527650
$987$	0	0
$988$	−5.85786	−0.186363
$989$	−6.58579	−0.209416
$990$	0	0
$991$	40.1421	1.27516	0.637578	−	0.770385i	$-0.279936\pi$
$991$	40.1421	1.27516	0.637578	+	0.770385i	$0.279936\pi$
$992$	0.828427	0.0263026
$993$	0	0
$994$	−12.8284	−0.406893
$995$	0	0
$996$	0	0
$997$	−14.9706	−0.474122	−0.237061	−	0.971495i	$-0.576184\pi$
$997$	−14.9706	−0.474122	−0.237061	+	0.971495i	$0.576184\pi$
$998$	26.1421	0.827515
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.z.1.1	yes	2
3.2	odd	2		2898.2.a.y.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2898.2.a.y.1.2	✓	2	3.2	odd	2
2898.2.a.z.1.1	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.41421 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.41421 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.41421 q^{10} +4.24264 q^{11} +0.828427 q^{13} -1.00000 q^{14} +1.00000 q^{16} -0.828427 q^{17} -7.07107 q^{19} -3.41421 q^{20} +4.24264 q^{22} +1.00000 q^{23} +6.65685 q^{25} +0.828427 q^{26} -1.00000 q^{28} +2.00000 q^{29} +0.828427 q^{31} +1.00000 q^{32} -0.828427 q^{34} +3.41421 q^{35} -10.2426 q^{37} -7.07107 q^{38} -3.41421 q^{40} -2.00000 q^{41} -6.58579 q^{43} +4.24264 q^{44} +1.00000 q^{46} -3.17157 q^{47} +1.00000 q^{49} +6.65685 q^{50} +0.828427 q^{52} -1.75736 q^{53} -14.4853 q^{55} -1.00000 q^{56} +2.00000 q^{58} +2.82843 q^{59} -5.75736 q^{61} +0.828427 q^{62} +1.00000 q^{64} -2.82843 q^{65} -3.75736 q^{67} -0.828427 q^{68} +3.41421 q^{70} +12.8284 q^{71} -12.0000 q^{73} -10.2426 q^{74} -7.07107 q^{76} -4.24264 q^{77} -11.3137 q^{79} -3.41421 q^{80} -2.00000 q^{82} -6.58579 q^{83} +2.82843 q^{85} -6.58579 q^{86} +4.24264 q^{88} -3.65685 q^{89} -0.828427 q^{91} +1.00000 q^{92} -3.17157 q^{94} +24.1421 q^{95} -3.17157 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{7} + 2 q^{8} - 4 q^{10} - 4 q^{13} - 2 q^{14} + 2 q^{16} + 4 q^{17} - 4 q^{20} + 2 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{28} + 4 q^{29} - 4 q^{31} + 2 q^{32} + 4 q^{34} + 4 q^{35} - 12 q^{37} - 4 q^{40} - 4 q^{41} - 16 q^{43} + 2 q^{46} - 12 q^{47} + 2 q^{49} + 2 q^{50} - 4 q^{52} - 12 q^{53} - 12 q^{55} - 2 q^{56} + 4 q^{58} - 20 q^{61} - 4 q^{62} + 2 q^{64} - 16 q^{67} + 4 q^{68} + 4 q^{70} + 20 q^{71} - 24 q^{73} - 12 q^{74} - 4 q^{80} - 4 q^{82} - 16 q^{83} - 16 q^{86} + 4 q^{89} + 4 q^{91} + 2 q^{92} - 12 q^{94} + 20 q^{95} - 12 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^7 + 2 * q^8 - 4 * q^10 - 4 * q^13 - 2 * q^14 + 2 * q^16 + 4 * q^17 - 4 * q^20 + 2 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^28 + 4 * q^29 - 4 * q^31 + 2 * q^32 + 4 * q^34 + 4 * q^35 - 12 * q^37 - 4 * q^40 - 4 * q^41 - 16 * q^43 + 2 * q^46 - 12 * q^47 + 2 * q^49 + 2 * q^50 - 4 * q^52 - 12 * q^53 - 12 * q^55 - 2 * q^56 + 4 * q^58 - 20 * q^61 - 4 * q^62 + 2 * q^64 - 16 * q^67 + 4 * q^68 + 4 * q^70 + 20 * q^71 - 24 * q^73 - 12 * q^74 - 4 * q^80 - 4 * q^82 - 16 * q^83 - 16 * q^86 + 4 * q^89 + 4 * q^91 + 2 * q^92 - 12 * q^94 + 20 * q^95 - 12 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more