LMFDB - Embedded newform 2898.2.a.be.1.3

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:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81361$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.10278 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.10278 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.10278 q^{10} +5.62721 q^{11} -3.62721 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.52444 q^{17} -0.578337 q^{19} +1.10278 q^{20} -5.62721 q^{22} +1.00000 q^{23} -3.78389 q^{25} +3.62721 q^{26} -1.00000 q^{28} -5.83276 q^{29} -2.52444 q^{31} -1.00000 q^{32} +4.52444 q^{34} -1.10278 q^{35} -7.04888 q^{37} +0.578337 q^{38} -1.10278 q^{40} -3.15667 q^{41} +7.25443 q^{43} +5.62721 q^{44} -1.00000 q^{46} -2.52444 q^{47} +1.00000 q^{49} +3.78389 q^{50} -3.62721 q^{52} -3.04888 q^{53} +6.20555 q^{55} +1.00000 q^{56} +5.83276 q^{58} -9.30833 q^{59} +8.35720 q^{61} +2.52444 q^{62} +1.00000 q^{64} -4.00000 q^{65} +5.62721 q^{67} -4.52444 q^{68} +1.10278 q^{70} +13.0489 q^{71} -14.3033 q^{73} +7.04888 q^{74} -0.578337 q^{76} -5.62721 q^{77} -16.9894 q^{79} +1.10278 q^{80} +3.15667 q^{82} -13.6272 q^{83} -4.98944 q^{85} -7.25443 q^{86} -5.62721 q^{88} +6.72999 q^{89} +3.62721 q^{91} +1.00000 q^{92} +2.52444 q^{94} -0.637776 q^{95} +16.9355 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} + 4 q^{10} + 4 q^{11} + 2 q^{13} + 3 q^{14} + 3 q^{16} - 8 q^{17} - 4 q^{20} - 4 q^{22} + 3 q^{23} + 5 q^{25} - 2 q^{26} - 3 q^{28} + 10 q^{29} - 2 q^{31} - 3 q^{32} + 8 q^{34} + 4 q^{35} - 10 q^{37} + 4 q^{40} - 6 q^{41} - 4 q^{43} + 4 q^{44} - 3 q^{46} - 2 q^{47} + 3 q^{49} - 5 q^{50} + 2 q^{52} + 2 q^{53} + 4 q^{55} + 3 q^{56} - 10 q^{58} - 6 q^{59} - 8 q^{61} + 2 q^{62} + 3 q^{64} - 12 q^{65} + 4 q^{67} - 8 q^{68} - 4 q^{70} + 28 q^{71} - 6 q^{73} + 10 q^{74} - 4 q^{77} - 20 q^{79} - 4 q^{80} + 6 q^{82} - 28 q^{83} + 16 q^{85} + 4 q^{86} - 4 q^{88} - 2 q^{91} + 3 q^{92} + 2 q^{94} - 20 q^{95} + 16 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 + 4 * q^10 + 4 * q^11 + 2 * q^13 + 3 * q^14 + 3 * q^16 - 8 * q^17 - 4 * q^20 - 4 * q^22 + 3 * q^23 + 5 * q^25 - 2 * q^26 - 3 * q^28 + 10 * q^29 - 2 * q^31 - 3 * q^32 + 8 * q^34 + 4 * q^35 - 10 * q^37 + 4 * q^40 - 6 * q^41 - 4 * q^43 + 4 * q^44 - 3 * q^46 - 2 * q^47 + 3 * q^49 - 5 * q^50 + 2 * q^52 + 2 * q^53 + 4 * q^55 + 3 * q^56 - 10 * q^58 - 6 * q^59 - 8 * q^61 + 2 * q^62 + 3 * q^64 - 12 * q^65 + 4 * q^67 - 8 * q^68 - 4 * q^70 + 28 * q^71 - 6 * q^73 + 10 * q^74 - 4 * q^77 - 20 * q^79 - 4 * q^80 + 6 * q^82 - 28 * q^83 + 16 * q^85 + 4 * q^86 - 4 * q^88 - 2 * q^91 + 3 * q^92 + 2 * q^94 - 20 * q^95 + 16 * q^97 - 3 * 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
$5$	1.10278	0.493176	0.246588	−	0.969120i	$-0.420691\pi$
$5$	1.10278	0.493176	0.246588	+	0.969120i	$0.420691\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.10278	−0.348728
$11$	5.62721	1.69667	0.848334	−	0.529461i	$-0.177606\pi$
$11$	5.62721	1.69667	0.848334	+	0.529461i	$0.177606\pi$
$12$	0	0
$13$	−3.62721	−1.00601	−0.503004	−	0.864284i	$-0.667772\pi$
$13$	−3.62721	−1.00601	−0.503004	+	0.864284i	$0.667772\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.52444	−1.09734	−0.548669	−	0.836040i	$-0.684865\pi$
$17$	−4.52444	−1.09734	−0.548669	+	0.836040i	$0.684865\pi$
$18$	0	0
$19$	−0.578337	−0.132680	−0.0663398	−	0.997797i	$-0.521132\pi$
$19$	−0.578337	−0.132680	−0.0663398	+	0.997797i	$0.521132\pi$
$20$	1.10278	0.246588
$21$	0	0
$22$	−5.62721	−1.19973
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.78389	−0.756777
$26$	3.62721	0.711355
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−5.83276	−1.08312	−0.541558	−	0.840663i	$-0.682166\pi$
$29$	−5.83276	−1.08312	−0.541558	+	0.840663i	$0.682166\pi$
$30$	0	0
$31$	−2.52444	−0.453402	−0.226701	−	0.973964i	$-0.572794\pi$
$31$	−2.52444	−0.453402	−0.226701	+	0.973964i	$0.572794\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.52444	0.775935
$35$	−1.10278	−0.186403
$36$	0	0
$37$	−7.04888	−1.15883	−0.579414	−	0.815033i	$-0.696719\pi$
$37$	−7.04888	−1.15883	−0.579414	+	0.815033i	$0.696719\pi$
$38$	0.578337	0.0938187
$39$	0	0
$40$	−1.10278	−0.174364
$41$	−3.15667	−0.492990	−0.246495	−	0.969144i	$-0.579279\pi$
$41$	−3.15667	−0.492990	−0.246495	+	0.969144i	$0.579279\pi$
$42$	0	0
$43$	7.25443	1.10629	0.553145	−	0.833085i	$-0.313428\pi$
$43$	7.25443	1.10629	0.553145	+	0.833085i	$0.313428\pi$
$44$	5.62721	0.848334
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−2.52444	−0.368227	−0.184114	−	0.982905i	$-0.558941\pi$
$47$	−2.52444	−0.368227	−0.184114	+	0.982905i	$0.558941\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.78389	0.535122
$51$	0	0
$52$	−3.62721	−0.503004
$53$	−3.04888	−0.418795	−0.209398	−	0.977831i	$-0.567150\pi$
$53$	−3.04888	−0.418795	−0.209398	+	0.977831i	$0.567150\pi$
$54$	0	0
$55$	6.20555	0.836756
$56$	1.00000	0.133631
$57$	0	0
$58$	5.83276	0.765879
$59$	−9.30833	−1.21184	−0.605920	−	0.795525i	$-0.707195\pi$
$59$	−9.30833	−1.21184	−0.605920	+	0.795525i	$0.707195\pi$
$60$	0	0
$61$	8.35720	1.07003	0.535015	−	0.844843i	$-0.320306\pi$
$61$	8.35720	1.07003	0.535015	+	0.844843i	$0.320306\pi$
$62$	2.52444	0.320604
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	5.62721	0.687473	0.343737	−	0.939066i	$-0.388307\pi$
$67$	5.62721	0.687473	0.343737	+	0.939066i	$0.388307\pi$
$68$	−4.52444	−0.548669
$69$	0	0
$70$	1.10278	0.131807
$71$	13.0489	1.54862	0.774308	−	0.632809i	$-0.218098\pi$
$71$	13.0489	1.54862	0.774308	+	0.632809i	$0.218098\pi$
$72$	0	0
$73$	−14.3033	−1.67407	−0.837037	−	0.547146i	$-0.815714\pi$
$73$	−14.3033	−1.67407	−0.837037	+	0.547146i	$0.815714\pi$
$74$	7.04888	0.819415
$75$	0	0
$76$	−0.578337	−0.0663398
$77$	−5.62721	−0.641280
$78$	0	0
$79$	−16.9894	−1.91146	−0.955731	−	0.294243i	$-0.904932\pi$
$79$	−16.9894	−1.91146	−0.955731	+	0.294243i	$0.904932\pi$
$80$	1.10278	0.123294
$81$	0	0
$82$	3.15667	0.348596
$83$	−13.6272	−1.49578	−0.747890	−	0.663822i	$-0.768933\pi$
$83$	−13.6272	−1.49578	−0.747890	+	0.663822i	$0.768933\pi$
$84$	0	0
$85$	−4.98944	−0.541180
$86$	−7.25443	−0.782265
$87$	0	0
$88$	−5.62721	−0.599863
$89$	6.72999	0.713377	0.356689	−	0.934223i	$-0.383906\pi$
$89$	6.72999	0.713377	0.356689	+	0.934223i	$0.383906\pi$
$90$	0	0
$91$	3.62721	0.380235
$92$	1.00000	0.104257
$93$	0	0
$94$	2.52444	0.260376
$95$	−0.637776	−0.0654344
$96$	0	0
$97$	16.9355	1.71954	0.859772	−	0.510679i	$-0.170606\pi$
$97$	16.9355	1.71954	0.859772	+	0.510679i	$0.170606\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−3.78389	−0.378389
$101$	−4.37279	−0.435109	−0.217554	−	0.976048i	$-0.569808\pi$
$101$	−4.37279	−0.435109	−0.217554	+	0.976048i	$0.569808\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	3.62721	0.355677
$105$	0	0
$106$	3.04888	0.296133
$107$	18.5089	1.78932	0.894659	−	0.446749i	$-0.147418\pi$
$107$	18.5089	1.78932	0.894659	+	0.446749i	$0.147418\pi$
$108$	0	0
$109$	−0.843326	−0.0807760	−0.0403880	−	0.999184i	$-0.512859\pi$
$109$	−0.843326	−0.0807760	−0.0403880	+	0.999184i	$0.512859\pi$
$110$	−6.20555	−0.591676
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−3.79445	−0.356952	−0.178476	−	0.983944i	$-0.557117\pi$
$113$	−3.79445	−0.356952	−0.178476	+	0.983944i	$0.557117\pi$
$114$	0	0
$115$	1.10278	0.102834
$116$	−5.83276	−0.541558
$117$	0	0
$118$	9.30833	0.856901
$119$	4.52444	0.414755
$120$	0	0
$121$	20.6655	1.87868
$122$	−8.35720	−0.756625
$123$	0	0
$124$	−2.52444	−0.226701
$125$	−9.68665	−0.866400
$126$	0	0
$127$	−14.2056	−1.26054	−0.630269	−	0.776377i	$-0.717056\pi$
$127$	−14.2056	−1.26054	−0.630269	+	0.776377i	$0.717056\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	4.00000	0.350823
$131$	−22.3572	−1.95336	−0.976679	−	0.214705i	$-0.931121\pi$
$131$	−22.3572	−1.95336	−0.976679	+	0.214705i	$0.931121\pi$
$132$	0	0
$133$	0.578337	0.0501482
$134$	−5.62721	−0.486117
$135$	0	0
$136$	4.52444	0.387967
$137$	1.89220	0.161662	0.0808309	−	0.996728i	$-0.474243\pi$
$137$	1.89220	0.161662	0.0808309	+	0.996728i	$0.474243\pi$
$138$	0	0
$139$	−6.35720	−0.539211	−0.269605	−	0.962971i	$-0.586893\pi$
$139$	−6.35720	−0.539211	−0.269605	+	0.962971i	$0.586893\pi$
$140$	−1.10278	−0.0932015
$141$	0	0
$142$	−13.0489	−1.09504
$143$	−20.4111	−1.70686
$144$	0	0
$145$	−6.43223	−0.534167
$146$	14.3033	1.18375
$147$	0	0
$148$	−7.04888	−0.579414
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	18.6167	1.51500	0.757501	−	0.652834i	$-0.226420\pi$
$151$	18.6167	1.51500	0.757501	+	0.652834i	$0.226420\pi$
$152$	0.578337	0.0469093
$153$	0	0
$154$	5.62721	0.453454
$155$	−2.78389	−0.223607
$156$	0	0
$157$	6.89722	0.550458	0.275229	−	0.961379i	$-0.411246\pi$
$157$	6.89722	0.550458	0.275229	+	0.961379i	$0.411246\pi$
$158$	16.9894	1.35161
$159$	0	0
$160$	−1.10278	−0.0871820
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−19.3622	−1.51657	−0.758283	−	0.651925i	$-0.773962\pi$
$163$	−19.3622	−1.51657	−0.758283	+	0.651925i	$0.773962\pi$
$164$	−3.15667	−0.246495
$165$	0	0
$166$	13.6272	1.05768
$167$	10.5244	0.814405	0.407203	−	0.913338i	$-0.366504\pi$
$167$	10.5244	0.814405	0.407203	+	0.913338i	$0.366504\pi$
$168$	0	0
$169$	0.156674	0.0120519
$170$	4.98944	0.382672
$171$	0	0
$172$	7.25443	0.553145
$173$	−17.4217	−1.32454	−0.662272	−	0.749263i	$-0.730408\pi$
$173$	−17.4217	−1.32454	−0.662272	+	0.749263i	$0.730408\pi$
$174$	0	0
$175$	3.78389	0.286035
$176$	5.62721	0.424167
$177$	0	0
$178$	−6.72999	−0.504434
$179$	11.6655	0.871922	0.435961	−	0.899965i	$-0.356408\pi$
$179$	11.6655	0.871922	0.435961	+	0.899965i	$0.356408\pi$
$180$	0	0
$181$	6.25945	0.465261	0.232631	−	0.972565i	$-0.425267\pi$
$181$	6.25945	0.465261	0.232631	+	0.972565i	$0.425267\pi$
$182$	−3.62721	−0.268867
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−7.77332	−0.571506
$186$	0	0
$187$	−25.4600	−1.86182
$188$	−2.52444	−0.184114
$189$	0	0
$190$	0.637776	0.0462691
$191$	2.31335	0.167388	0.0836940	−	0.996492i	$-0.473328\pi$
$191$	2.31335	0.167388	0.0836940	+	0.996492i	$0.473328\pi$
$192$	0	0
$193$	5.58890	0.402298	0.201149	−	0.979561i	$-0.435532\pi$
$193$	5.58890	0.402298	0.201149	+	0.979561i	$0.435532\pi$
$194$	−16.9355	−1.21590
$195$	0	0
$196$	1.00000	0.0714286
$197$	−18.4111	−1.31174	−0.655868	−	0.754875i	$-0.727697\pi$
$197$	−18.4111	−1.31174	−0.655868	+	0.754875i	$0.727697\pi$
$198$	0	0
$199$	4.41110	0.312695	0.156347	−	0.987702i	$-0.450028\pi$
$199$	4.41110	0.312695	0.156347	+	0.987702i	$0.450028\pi$
$200$	3.78389	0.267561
$201$	0	0
$202$	4.37279	0.307668
$203$	5.83276	0.409380
$204$	0	0
$205$	−3.48110	−0.243131
$206$	8.00000	0.557386
$207$	0	0
$208$	−3.62721	−0.251502
$209$	−3.25443	−0.225113
$210$	0	0
$211$	10.8433	0.746485	0.373243	−	0.927734i	$-0.378246\pi$
$211$	10.8433	0.746485	0.373243	+	0.927734i	$0.378246\pi$
$212$	−3.04888	−0.209398
$213$	0	0
$214$	−18.5089	−1.26524
$215$	8.00000	0.545595
$216$	0	0
$217$	2.52444	0.171370
$218$	0.843326	0.0571172
$219$	0	0
$220$	6.20555	0.418378
$221$	16.4111	1.10393
$222$	0	0
$223$	−19.9844	−1.33826	−0.669128	−	0.743147i	$-0.733332\pi$
$223$	−19.9844	−1.33826	−0.669128	+	0.743147i	$0.733332\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	3.79445	0.252403
$227$	−4.98944	−0.331161	−0.165580	−	0.986196i	$-0.552950\pi$
$227$	−4.98944	−0.331161	−0.165580	+	0.986196i	$0.552950\pi$
$228$	0	0
$229$	3.94610	0.260766	0.130383	−	0.991464i	$-0.458379\pi$
$229$	3.94610	0.260766	0.130383	+	0.991464i	$0.458379\pi$
$230$	−1.10278	−0.0727148
$231$	0	0
$232$	5.83276	0.382940
$233$	2.88164	0.188782	0.0943912	−	0.995535i	$-0.469910\pi$
$233$	2.88164	0.188782	0.0943912	+	0.995535i	$0.469910\pi$
$234$	0	0
$235$	−2.78389	−0.181601
$236$	−9.30833	−0.605920
$237$	0	0
$238$	−4.52444	−0.293276
$239$	−20.7144	−1.33990	−0.669952	−	0.742405i	$-0.733685\pi$
$239$	−20.7144	−1.33990	−0.669952	+	0.742405i	$0.733685\pi$
$240$	0	0
$241$	11.0333	0.710717	0.355358	−	0.934730i	$-0.384359\pi$
$241$	11.0333	0.710717	0.355358	+	0.934730i	$0.384359\pi$
$242$	−20.6655	−1.32843
$243$	0	0
$244$	8.35720	0.535015
$245$	1.10278	0.0704537
$246$	0	0
$247$	2.09775	0.133477
$248$	2.52444	0.160302
$249$	0	0
$250$	9.68665	0.612638
$251$	−16.2439	−1.02530	−0.512652	−	0.858597i	$-0.671337\pi$
$251$	−16.2439	−1.02530	−0.512652	+	0.858597i	$0.671337\pi$
$252$	0	0
$253$	5.62721	0.353780
$254$	14.2056	0.891335
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.6655	0.852432	0.426216	−	0.904621i	$-0.359846\pi$
$257$	13.6655	0.852432	0.426216	+	0.904621i	$0.359846\pi$
$258$	0	0
$259$	7.04888	0.437996
$260$	−4.00000	−0.248069
$261$	0	0
$262$	22.3572	1.38123
$263$	−24.9894	−1.54091	−0.770457	−	0.637492i	$-0.779972\pi$
$263$	−24.9894	−1.54091	−0.770457	+	0.637492i	$0.779972\pi$
$264$	0	0
$265$	−3.36222	−0.206540
$266$	−0.578337	−0.0354601
$267$	0	0
$268$	5.62721	0.343737
$269$	17.4983	1.06689	0.533445	−	0.845835i	$-0.320897\pi$
$269$	17.4983	1.06689	0.533445	+	0.845835i	$0.320897\pi$
$270$	0	0
$271$	−18.5244	−1.12528	−0.562640	−	0.826702i	$-0.690214\pi$
$271$	−18.5244	−1.12528	−0.562640	+	0.826702i	$0.690214\pi$
$272$	−4.52444	−0.274334
$273$	0	0
$274$	−1.89220	−0.114312
$275$	−21.2927	−1.28400
$276$	0	0
$277$	−10.9894	−0.660291	−0.330146	−	0.943930i	$-0.607098\pi$
$277$	−10.9894	−0.660291	−0.330146	+	0.943930i	$0.607098\pi$
$278$	6.35720	0.381280
$279$	0	0
$280$	1.10278	0.0659034
$281$	7.45998	0.445025	0.222512	−	0.974930i	$-0.428574\pi$
$281$	7.45998	0.445025	0.222512	+	0.974930i	$0.428574\pi$
$282$	0	0
$283$	1.51941	0.0903198	0.0451599	−	0.998980i	$-0.485620\pi$
$283$	1.51941	0.0903198	0.0451599	+	0.998980i	$0.485620\pi$
$284$	13.0489	0.774308
$285$	0	0
$286$	20.4111	1.20693
$287$	3.15667	0.186333
$288$	0	0
$289$	3.47054	0.204149
$290$	6.43223	0.377713
$291$	0	0
$292$	−14.3033	−0.837037
$293$	1.10278	0.0644248	0.0322124	−	0.999481i	$-0.489745\pi$
$293$	1.10278	0.0644248	0.0322124	+	0.999481i	$0.489745\pi$
$294$	0	0
$295$	−10.2650	−0.597651
$296$	7.04888	0.409708
$297$	0	0
$298$	−2.00000	−0.115857
$299$	−3.62721	−0.209767
$300$	0	0
$301$	−7.25443	−0.418138
$302$	−18.6167	−1.07127
$303$	0	0
$304$	−0.578337	−0.0331699
$305$	9.21611	0.527713
$306$	0	0
$307$	−11.1028	−0.633669	−0.316834	−	0.948481i	$-0.602620\pi$
$307$	−11.1028	−0.633669	−0.316834	+	0.948481i	$0.602620\pi$
$308$	−5.62721	−0.320640
$309$	0	0
$310$	2.78389	0.158114
$311$	14.2978	0.810752	0.405376	−	0.914150i	$-0.367141\pi$
$311$	14.2978	0.810752	0.405376	+	0.914150i	$0.367141\pi$
$312$	0	0
$313$	−19.4756	−1.10082	−0.550412	−	0.834893i	$-0.685529\pi$
$313$	−19.4756	−1.10082	−0.550412	+	0.834893i	$0.685529\pi$
$314$	−6.89722	−0.389233
$315$	0	0
$316$	−16.9894	−0.955731
$317$	−28.3416	−1.59182	−0.795912	−	0.605413i	$-0.793008\pi$
$317$	−28.3416	−1.59182	−0.795912	+	0.605413i	$0.793008\pi$
$318$	0	0
$319$	−32.8222	−1.83769
$320$	1.10278	0.0616470
$321$	0	0
$322$	1.00000	0.0557278
$323$	2.61665	0.145594
$324$	0	0
$325$	13.7250	0.761324
$326$	19.3622	1.07237
$327$	0	0
$328$	3.15667	0.174298
$329$	2.52444	0.139177
$330$	0	0
$331$	−34.8122	−1.91345	−0.956725	−	0.290995i	$-0.906014\pi$
$331$	−34.8122	−1.91345	−0.956725	+	0.290995i	$0.906014\pi$
$332$	−13.6272	−0.747890
$333$	0	0
$334$	−10.5244	−0.575872
$335$	6.20555	0.339045
$336$	0	0
$337$	−4.84333	−0.263833	−0.131916	−	0.991261i	$-0.542113\pi$
$337$	−4.84333	−0.263833	−0.131916	+	0.991261i	$0.542113\pi$
$338$	−0.156674	−0.00852195
$339$	0	0
$340$	−4.98944	−0.270590
$341$	−14.2056	−0.769274
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−7.25443	−0.391132
$345$	0	0
$346$	17.4217	0.936594
$347$	23.7733	1.27622	0.638109	−	0.769946i	$-0.279717\pi$
$347$	23.7733	1.27622	0.638109	+	0.769946i	$0.279717\pi$
$348$	0	0
$349$	32.7527	1.75321	0.876606	−	0.481208i	$-0.159802\pi$
$349$	32.7527	1.75321	0.876606	+	0.481208i	$0.159802\pi$
$350$	−3.78389	−0.202257
$351$	0	0
$352$	−5.62721	−0.299931
$353$	−13.5577	−0.721605	−0.360803	−	0.932642i	$-0.617497\pi$
$353$	−13.5577	−0.721605	−0.360803	+	0.932642i	$0.617497\pi$
$354$	0	0
$355$	14.3900	0.763741
$356$	6.72999	0.356689
$357$	0	0
$358$	−11.6655	−0.616542
$359$	−6.67609	−0.352350	−0.176175	−	0.984359i	$-0.556373\pi$
$359$	−6.67609	−0.352350	−0.176175	+	0.984359i	$0.556373\pi$
$360$	0	0
$361$	−18.6655	−0.982396
$362$	−6.25945	−0.328989
$363$	0	0
$364$	3.62721	0.190118
$365$	−15.7733	−0.825614
$366$	0	0
$367$	−24.3033	−1.26862	−0.634311	−	0.773078i	$-0.718716\pi$
$367$	−24.3033	−1.26862	−0.634311	+	0.773078i	$0.718716\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	7.77332	0.404116
$371$	3.04888	0.158290
$372$	0	0
$373$	−8.50885	−0.440572	−0.220286	−	0.975435i	$-0.570699\pi$
$373$	−8.50885	−0.440572	−0.220286	+	0.975435i	$0.570699\pi$
$374$	25.4600	1.31650
$375$	0	0
$376$	2.52444	0.130188
$377$	21.1567	1.08962
$378$	0	0
$379$	−9.21611	−0.473400	−0.236700	−	0.971583i	$-0.576066\pi$
$379$	−9.21611	−0.473400	−0.236700	+	0.971583i	$0.576066\pi$
$380$	−0.637776	−0.0327172
$381$	0	0
$382$	−2.31335	−0.118361
$383$	23.9688	1.22475	0.612375	−	0.790567i	$-0.290214\pi$
$383$	23.9688	1.22475	0.612375	+	0.790567i	$0.290214\pi$
$384$	0	0
$385$	−6.20555	−0.316264
$386$	−5.58890	−0.284468
$387$	0	0
$388$	16.9355	0.859772
$389$	2.33447	0.118363	0.0591813	−	0.998247i	$-0.481151\pi$
$389$	2.33447	0.118363	0.0591813	+	0.998247i	$0.481151\pi$
$390$	0	0
$391$	−4.52444	−0.228811
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	18.4111	0.927538
$395$	−18.7355	−0.942687
$396$	0	0
$397$	−21.9406	−1.10117	−0.550583	−	0.834781i	$-0.685594\pi$
$397$	−21.9406	−1.10117	−0.550583	+	0.834781i	$0.685594\pi$
$398$	−4.41110	−0.221108
$399$	0	0
$400$	−3.78389	−0.189194
$401$	27.3522	1.36590	0.682951	−	0.730464i	$-0.260696\pi$
$401$	27.3522	1.36590	0.682951	+	0.730464i	$0.260696\pi$
$402$	0	0
$403$	9.15667	0.456126
$404$	−4.37279	−0.217554
$405$	0	0
$406$	−5.83276	−0.289475
$407$	−39.6655	−1.96615
$408$	0	0
$409$	4.95112	0.244817	0.122409	−	0.992480i	$-0.460938\pi$
$409$	4.95112	0.244817	0.122409	+	0.992480i	$0.460938\pi$
$410$	3.48110	0.171919
$411$	0	0
$412$	−8.00000	−0.394132
$413$	9.30833	0.458033
$414$	0	0
$415$	−15.0278	−0.737683
$416$	3.62721	0.177839
$417$	0	0
$418$	3.25443	0.159179
$419$	26.3416	1.28687	0.643436	−	0.765500i	$-0.277508\pi$
$419$	26.3416	1.28687	0.643436	+	0.765500i	$0.277508\pi$
$420$	0	0
$421$	−9.36222	−0.456287	−0.228143	−	0.973628i	$-0.573266\pi$
$421$	−9.36222	−0.456287	−0.228143	+	0.973628i	$0.573266\pi$
$422$	−10.8433	−0.527845
$423$	0	0
$424$	3.04888	0.148067
$425$	17.1200	0.830440
$426$	0	0
$427$	−8.35720	−0.404433
$428$	18.5089	0.894659
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−8.82220	−0.424950	−0.212475	−	0.977166i	$-0.568152\pi$
$431$	−8.82220	−0.424950	−0.212475	+	0.977166i	$0.568152\pi$
$432$	0	0
$433$	14.6222	0.702698	0.351349	−	0.936245i	$-0.385723\pi$
$433$	14.6222	0.702698	0.351349	+	0.936245i	$0.385723\pi$
$434$	−2.52444	−0.121177
$435$	0	0
$436$	−0.843326	−0.0403880
$437$	−0.578337	−0.0276656
$438$	0	0
$439$	−30.2978	−1.44603	−0.723017	−	0.690831i	$-0.757245\pi$
$439$	−30.2978	−1.44603	−0.723017	+	0.690831i	$0.757245\pi$
$440$	−6.20555	−0.295838
$441$	0	0
$442$	−16.4111	−0.780596
$443$	−17.5678	−0.834670	−0.417335	−	0.908753i	$-0.637036\pi$
$443$	−17.5678	−0.834670	−0.417335	+	0.908753i	$0.637036\pi$
$444$	0	0
$445$	7.42166	0.351821
$446$	19.9844	0.946289
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−11.2927	−0.532937	−0.266469	−	0.963844i	$-0.585857\pi$
$449$	−11.2927	−0.532937	−0.266469	+	0.963844i	$0.585857\pi$
$450$	0	0
$451$	−17.7633	−0.836440
$452$	−3.79445	−0.178476
$453$	0	0
$454$	4.98944	0.234166
$455$	4.00000	0.187523
$456$	0	0
$457$	27.4600	1.28452	0.642262	−	0.766485i	$-0.277996\pi$
$457$	27.4600	1.28452	0.642262	+	0.766485i	$0.277996\pi$
$458$	−3.94610	−0.184389
$459$	0	0
$460$	1.10278	0.0514172
$461$	−29.4983	−1.37387	−0.686936	−	0.726718i	$-0.741045\pi$
$461$	−29.4983	−1.37387	−0.686936	+	0.726718i	$0.741045\pi$
$462$	0	0
$463$	14.7244	0.684303	0.342152	−	0.939645i	$-0.388844\pi$
$463$	14.7244	0.684303	0.342152	+	0.939645i	$0.388844\pi$
$464$	−5.83276	−0.270779
$465$	0	0
$466$	−2.88164	−0.133489
$467$	−8.57834	−0.396958	−0.198479	−	0.980105i	$-0.563600\pi$
$467$	−8.57834	−0.396958	−0.198479	+	0.980105i	$0.563600\pi$
$468$	0	0
$469$	−5.62721	−0.259841
$470$	2.78389	0.128411
$471$	0	0
$472$	9.30833	0.428450
$473$	40.8222	1.87701
$474$	0	0
$475$	2.18836	0.100409
$476$	4.52444	0.207377
$477$	0	0
$478$	20.7144	0.947455
$479$	28.0766	1.28285	0.641427	−	0.767184i	$-0.278343\pi$
$479$	28.0766	1.28285	0.641427	+	0.767184i	$0.278343\pi$
$480$	0	0
$481$	25.5678	1.16579
$482$	−11.0333	−0.502553
$483$	0	0
$484$	20.6655	0.939342
$485$	18.6761	0.848038
$486$	0	0
$487$	22.7244	1.02974	0.514872	−	0.857267i	$-0.327840\pi$
$487$	22.7244	1.02974	0.514872	+	0.857267i	$0.327840\pi$
$488$	−8.35720	−0.378313
$489$	0	0
$490$	−1.10278	−0.0498183
$491$	18.7244	0.845023	0.422511	−	0.906358i	$-0.361149\pi$
$491$	18.7244	0.845023	0.422511	+	0.906358i	$0.361149\pi$
$492$	0	0
$493$	26.3900	1.18854
$494$	−2.09775	−0.0943823
$495$	0	0
$496$	−2.52444	−0.113351
$497$	−13.0489	−0.585322
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−9.68665	−0.433200
$501$	0	0
$502$	16.2439	0.724999
$503$	−13.5678	−0.604957	−0.302479	−	0.953156i	$-0.597814\pi$
$503$	−13.5678	−0.604957	−0.302479	+	0.953156i	$0.597814\pi$
$504$	0	0
$505$	−4.82220	−0.214585
$506$	−5.62721	−0.250160
$507$	0	0
$508$	−14.2056	−0.630269
$509$	4.14611	0.183773	0.0918866	−	0.995769i	$-0.470710\pi$
$509$	4.14611	0.183773	0.0918866	+	0.995769i	$0.470710\pi$
$510$	0	0
$511$	14.3033	0.632741
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−13.6655	−0.602761
$515$	−8.82220	−0.388753
$516$	0	0
$517$	−14.2056	−0.624759
$518$	−7.04888	−0.309710
$519$	0	0
$520$	4.00000	0.175412
$521$	28.2978	1.23975	0.619874	−	0.784702i	$-0.287184\pi$
$521$	28.2978	1.23975	0.619874	+	0.784702i	$0.287184\pi$
$522$	0	0
$523$	−4.98944	−0.218173	−0.109086	−	0.994032i	$-0.534793\pi$
$523$	−4.98944	−0.218173	−0.109086	+	0.994032i	$0.534793\pi$
$524$	−22.3572	−0.976679
$525$	0	0
$526$	24.9894	1.08959
$527$	11.4217	0.497535
$528$	0	0
$529$	1.00000	0.0434783
$530$	3.36222	0.146046
$531$	0	0
$532$	0.578337	0.0250741
$533$	11.4499	0.495952
$534$	0	0
$535$	20.4111	0.882449
$536$	−5.62721	−0.243059
$537$	0	0
$538$	−17.4983	−0.754405
$539$	5.62721	0.242381
$540$	0	0
$541$	28.3416	1.21850	0.609251	−	0.792978i	$-0.291470\pi$
$541$	28.3416	1.21850	0.609251	+	0.792978i	$0.291470\pi$
$542$	18.5244	0.795693
$543$	0	0
$544$	4.52444	0.193984
$545$	−0.929999	−0.0398368
$546$	0	0
$547$	1.26447	0.0540649	0.0270325	−	0.999635i	$-0.491394\pi$
$547$	1.26447	0.0540649	0.0270325	+	0.999635i	$0.491394\pi$
$548$	1.89220	0.0808309
$549$	0	0
$550$	21.2927	0.907925
$551$	3.37330	0.143708
$552$	0	0
$553$	16.9894	0.722464
$554$	10.9894	0.466896
$555$	0	0
$556$	−6.35720	−0.269605
$557$	2.21560	0.0938778	0.0469389	−	0.998898i	$-0.485053\pi$
$557$	2.21560	0.0938778	0.0469389	+	0.998898i	$0.485053\pi$
$558$	0	0
$559$	−26.3133	−1.11294
$560$	−1.10278	−0.0466008
$561$	0	0
$562$	−7.45998	−0.314680
$563$	20.7738	0.875513	0.437757	−	0.899094i	$-0.355773\pi$
$563$	20.7738	0.875513	0.437757	+	0.899094i	$0.355773\pi$
$564$	0	0
$565$	−4.18442	−0.176040
$566$	−1.51941	−0.0638658
$567$	0	0
$568$	−13.0489	−0.547519
$569$	35.0177	1.46802	0.734009	−	0.679139i	$-0.237647\pi$
$569$	35.0177	1.46802	0.734009	+	0.679139i	$0.237647\pi$
$570$	0	0
$571$	35.6655	1.49256	0.746278	−	0.665634i	$-0.231839\pi$
$571$	35.6655	1.49256	0.746278	+	0.665634i	$0.231839\pi$
$572$	−20.4111	−0.853431
$573$	0	0
$574$	−3.15667	−0.131757
$575$	−3.78389	−0.157799
$576$	0	0
$577$	12.6167	0.525238	0.262619	−	0.964900i	$-0.415414\pi$
$577$	12.6167	0.525238	0.262619	+	0.964900i	$0.415414\pi$
$578$	−3.47054	−0.144355
$579$	0	0
$580$	−6.43223	−0.267084
$581$	13.6272	0.565352
$582$	0	0
$583$	−17.1567	−0.710557
$584$	14.3033	0.591875
$585$	0	0
$586$	−1.10278	−0.0455552
$587$	21.0816	0.870133	0.435066	−	0.900398i	$-0.356725\pi$
$587$	21.0816	0.870133	0.435066	+	0.900398i	$0.356725\pi$
$588$	0	0
$589$	1.45998	0.0601573
$590$	10.2650	0.422603
$591$	0	0
$592$	−7.04888	−0.289707
$593$	1.25443	0.0515131	0.0257566	−	0.999668i	$-0.491801\pi$
$593$	1.25443	0.0515131	0.0257566	+	0.999668i	$0.491801\pi$
$594$	0	0
$595$	4.98944	0.204547
$596$	2.00000	0.0819232
$597$	0	0
$598$	3.62721	0.148328
$599$	38.3900	1.56857	0.784286	−	0.620400i	$-0.213030\pi$
$599$	38.3900	1.56857	0.784286	+	0.620400i	$0.213030\pi$
$600$	0	0
$601$	14.7144	0.600213	0.300106	−	0.953906i	$-0.402978\pi$
$601$	14.7144	0.600213	0.300106	+	0.953906i	$0.402978\pi$
$602$	7.25443	0.295668
$603$	0	0
$604$	18.6167	0.757501
$605$	22.7894	0.926522
$606$	0	0
$607$	29.5633	1.19994	0.599968	−	0.800024i	$-0.295180\pi$
$607$	29.5633	1.19994	0.599968	+	0.800024i	$0.295180\pi$
$608$	0.578337	0.0234547
$609$	0	0
$610$	−9.21611	−0.373150
$611$	9.15667	0.370439
$612$	0	0
$613$	−15.3522	−0.620069	−0.310034	−	0.950725i	$-0.600341\pi$
$613$	−15.3522	−0.620069	−0.310034	+	0.950725i	$0.600341\pi$
$614$	11.1028	0.448072
$615$	0	0
$616$	5.62721	0.226727
$617$	−41.3311	−1.66393	−0.831963	−	0.554831i	$-0.812783\pi$
$617$	−41.3311	−1.66393	−0.831963	+	0.554831i	$0.812783\pi$
$618$	0	0
$619$	−5.84281	−0.234842	−0.117421	−	0.993082i	$-0.537463\pi$
$619$	−5.84281	−0.234842	−0.117421	+	0.993082i	$0.537463\pi$
$620$	−2.78389	−0.111804
$621$	0	0
$622$	−14.2978	−0.573288
$623$	−6.72999	−0.269631
$624$	0	0
$625$	8.23724	0.329490
$626$	19.4756	0.778400
$627$	0	0
$628$	6.89722	0.275229
$629$	31.8922	1.27163
$630$	0	0
$631$	6.34162	0.252456	0.126228	−	0.992001i	$-0.459713\pi$
$631$	6.34162	0.252456	0.126228	+	0.992001i	$0.459713\pi$
$632$	16.9894	0.675804
$633$	0	0
$634$	28.3416	1.12559
$635$	−15.6655	−0.621667
$636$	0	0
$637$	−3.62721	−0.143715
$638$	32.8222	1.29944
$639$	0	0
$640$	−1.10278	−0.0435910
$641$	32.3133	1.27630	0.638150	−	0.769912i	$-0.279700\pi$
$641$	32.3133	1.27630	0.638150	+	0.769912i	$0.279700\pi$
$642$	0	0
$643$	28.1672	1.11081	0.555404	−	0.831581i	$-0.312564\pi$
$643$	28.1672	1.11081	0.555404	+	0.831581i	$0.312564\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	−2.61665	−0.102951
$647$	−30.2978	−1.19113	−0.595564	−	0.803308i	$-0.703071\pi$
$647$	−30.2978	−1.19113	−0.595564	+	0.803308i	$0.703071\pi$
$648$	0	0
$649$	−52.3799	−2.05609
$650$	−13.7250	−0.538337
$651$	0	0
$652$	−19.3622	−0.758283
$653$	4.64782	0.181883	0.0909417	−	0.995856i	$-0.471012\pi$
$653$	4.64782	0.181883	0.0909417	+	0.995856i	$0.471012\pi$
$654$	0	0
$655$	−24.6550	−0.963349
$656$	−3.15667	−0.123247
$657$	0	0
$658$	−2.52444	−0.0984128
$659$	11.8811	0.462823	0.231411	−	0.972856i	$-0.425666\pi$
$659$	11.8811	0.462823	0.231411	+	0.972856i	$0.425666\pi$
$660$	0	0
$661$	3.94610	0.153486	0.0767428	−	0.997051i	$-0.475548\pi$
$661$	3.94610	0.153486	0.0767428	+	0.997051i	$0.475548\pi$
$662$	34.8122	1.35301
$663$	0	0
$664$	13.6272	0.528838
$665$	0.637776	0.0247319
$666$	0	0
$667$	−5.83276	−0.225845
$668$	10.5244	0.407203
$669$	0	0
$670$	−6.20555	−0.239741
$671$	47.0278	1.81549
$672$	0	0
$673$	24.0383	0.926609	0.463304	−	0.886199i	$-0.346664\pi$
$673$	24.0383	0.926609	0.463304	+	0.886199i	$0.346664\pi$
$674$	4.84333	0.186558
$675$	0	0
$676$	0.156674	0.00602593
$677$	−39.2005	−1.50660	−0.753299	−	0.657678i	$-0.771539\pi$
$677$	−39.2005	−1.50660	−0.753299	+	0.657678i	$0.771539\pi$
$678$	0	0
$679$	−16.9355	−0.649926
$680$	4.98944	0.191336
$681$	0	0
$682$	14.2056	0.543959
$683$	0.745574	0.0285286	0.0142643	−	0.999898i	$-0.495459\pi$
$683$	0.745574	0.0285286	0.0142643	+	0.999898i	$0.495459\pi$
$684$	0	0
$685$	2.08667	0.0797277
$686$	1.00000	0.0381802
$687$	0	0
$688$	7.25443	0.276572
$689$	11.0589	0.421311
$690$	0	0
$691$	−11.7094	−0.445446	−0.222723	−	0.974882i	$-0.571495\pi$
$691$	−11.7094	−0.445446	−0.222723	+	0.974882i	$0.571495\pi$
$692$	−17.4217	−0.662272
$693$	0	0
$694$	−23.7733	−0.902423
$695$	−7.01056	−0.265926
$696$	0	0
$697$	14.2822	0.540976
$698$	−32.7527	−1.23971
$699$	0	0
$700$	3.78389	0.143017
$701$	−7.45998	−0.281759	−0.140880	−	0.990027i	$-0.544993\pi$
$701$	−7.45998	−0.281759	−0.140880	+	0.990027i	$0.544993\pi$
$702$	0	0
$703$	4.07663	0.153753
$704$	5.62721	0.212084
$705$	0	0
$706$	13.5577	0.510252
$707$	4.37279	0.164456
$708$	0	0
$709$	5.69670	0.213944	0.106972	−	0.994262i	$-0.465884\pi$
$709$	5.69670	0.213944	0.106972	+	0.994262i	$0.465884\pi$
$710$	−14.3900	−0.540046
$711$	0	0
$712$	−6.72999	−0.252217
$713$	−2.52444	−0.0945409
$714$	0	0
$715$	−22.5089	−0.841783
$716$	11.6655	0.435961
$717$	0	0
$718$	6.67609	0.249149
$719$	−1.67107	−0.0623202	−0.0311601	−	0.999514i	$-0.509920\pi$
$719$	−1.67107	−0.0623202	−0.0311601	+	0.999514i	$0.509920\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	18.6655	0.694659
$723$	0	0
$724$	6.25945	0.232631
$725$	22.0705	0.819678
$726$	0	0
$727$	36.0766	1.33801	0.669004	−	0.743259i	$-0.266721\pi$
$727$	36.0766	1.33801	0.669004	+	0.743259i	$0.266721\pi$
$728$	−3.62721	−0.134433
$729$	0	0
$730$	15.7733	0.583797
$731$	−32.8222	−1.21397
$732$	0	0
$733$	−12.0227	−0.444070	−0.222035	−	0.975039i	$-0.571270\pi$
$733$	−12.0227	−0.444070	−0.222035	+	0.975039i	$0.571270\pi$
$734$	24.3033	0.897051
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	31.6655	1.16641
$738$	0	0
$739$	−41.5366	−1.52795	−0.763974	−	0.645247i	$-0.776755\pi$
$739$	−41.5366	−1.52795	−0.763974	+	0.645247i	$0.776755\pi$
$740$	−7.77332	−0.285753
$741$	0	0
$742$	−3.04888	−0.111928
$743$	13.1849	0.483709	0.241854	−	0.970313i	$-0.422244\pi$
$743$	13.1849	0.483709	0.241854	+	0.970313i	$0.422244\pi$
$744$	0	0
$745$	2.20555	0.0808051
$746$	8.50885	0.311531
$747$	0	0
$748$	−25.4600	−0.930909
$749$	−18.5089	−0.676299
$750$	0	0
$751$	−7.83276	−0.285822	−0.142911	−	0.989736i	$-0.545646\pi$
$751$	−7.83276	−0.285822	−0.142911	+	0.989736i	$0.545646\pi$
$752$	−2.52444	−0.0920568
$753$	0	0
$754$	−21.1567	−0.770481
$755$	20.5300	0.747162
$756$	0	0
$757$	36.5089	1.32694	0.663468	−	0.748204i	$-0.269084\pi$
$757$	36.5089	1.32694	0.663468	+	0.748204i	$0.269084\pi$
$758$	9.21611	0.334744
$759$	0	0
$760$	0.637776	0.0231346
$761$	−35.2444	−1.27761	−0.638804	−	0.769370i	$-0.720570\pi$
$761$	−35.2444	−1.27761	−0.638804	+	0.769370i	$0.720570\pi$
$762$	0	0
$763$	0.843326	0.0305304
$764$	2.31335	0.0836940
$765$	0	0
$766$	−23.9688	−0.866029
$767$	33.7633	1.21912
$768$	0	0
$769$	−24.4933	−0.883250	−0.441625	−	0.897200i	$-0.645598\pi$
$769$	−24.4933	−0.883250	−0.441625	+	0.897200i	$0.645598\pi$
$770$	6.20555	0.223633
$771$	0	0
$772$	5.58890	0.201149
$773$	7.94610	0.285801	0.142901	−	0.989737i	$-0.454357\pi$
$773$	7.94610	0.285801	0.142901	+	0.989737i	$0.454357\pi$
$774$	0	0
$775$	9.55219	0.343125
$776$	−16.9355	−0.607950
$777$	0	0
$778$	−2.33447	−0.0836949
$779$	1.82562	0.0654097
$780$	0	0
$781$	73.4288	2.62749
$782$	4.52444	0.161794
$783$	0	0
$784$	1.00000	0.0357143
$785$	7.60609	0.271473
$786$	0	0
$787$	19.5295	0.696150	0.348075	−	0.937467i	$-0.386835\pi$
$787$	19.5295	0.696150	0.348075	+	0.937467i	$0.386835\pi$
$788$	−18.4111	−0.655868
$789$	0	0
$790$	18.7355	0.666580
$791$	3.79445	0.134915
$792$	0	0
$793$	−30.3133	−1.07646
$794$	21.9406	0.778641
$795$	0	0
$796$	4.41110	0.156347
$797$	17.7406	0.628403	0.314201	−	0.949356i	$-0.398263\pi$
$797$	17.7406	0.628403	0.314201	+	0.949356i	$0.398263\pi$
$798$	0	0
$799$	11.4217	0.404069
$800$	3.78389	0.133781
$801$	0	0
$802$	−27.3522	−0.965839
$803$	−80.4877	−2.84035
$804$	0	0
$805$	−1.10278	−0.0388677
$806$	−9.15667	−0.322530
$807$	0	0
$808$	4.37279	0.153834
$809$	18.4111	0.647300	0.323650	−	0.946177i	$-0.395090\pi$
$809$	18.4111	0.647300	0.323650	+	0.946177i	$0.395090\pi$
$810$	0	0
$811$	20.2594	0.711405	0.355703	−	0.934599i	$-0.384242\pi$
$811$	20.2594	0.711405	0.355703	+	0.934599i	$0.384242\pi$
$812$	5.83276	0.204690
$813$	0	0
$814$	39.6655	1.39028
$815$	−21.3522	−0.747934
$816$	0	0
$817$	−4.19550	−0.146782
$818$	−4.95112	−0.173112
$819$	0	0
$820$	−3.48110	−0.121565
$821$	−31.8116	−1.11023	−0.555117	−	0.831772i	$-0.687326\pi$
$821$	−31.8116	−1.11023	−0.555117	+	0.831772i	$0.687326\pi$
$822$	0	0
$823$	32.9099	1.14717	0.573584	−	0.819147i	$-0.305553\pi$
$823$	32.9099	1.14717	0.573584	+	0.819147i	$0.305553\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−9.30833	−0.323878
$827$	37.4288	1.30153	0.650764	−	0.759280i	$-0.274449\pi$
$827$	37.4288	1.30153	0.650764	+	0.759280i	$0.274449\pi$
$828$	0	0
$829$	28.5572	0.991833	0.495916	−	0.868370i	$-0.334832\pi$
$829$	28.5572	0.991833	0.495916	+	0.868370i	$0.334832\pi$
$830$	15.0278	0.521621
$831$	0	0
$832$	−3.62721	−0.125751
$833$	−4.52444	−0.156762
$834$	0	0
$835$	11.6061	0.401645
$836$	−3.25443	−0.112557
$837$	0	0
$838$	−26.3416	−0.909956
$839$	−13.3833	−0.462045	−0.231022	−	0.972948i	$-0.574207\pi$
$839$	−13.3833	−0.462045	−0.231022	+	0.972948i	$0.574207\pi$
$840$	0	0
$841$	5.02113	0.173142
$842$	9.36222	0.322644
$843$	0	0
$844$	10.8433	0.373243
$845$	0.172776	0.00594369
$846$	0	0
$847$	−20.6655	−0.710076
$848$	−3.04888	−0.104699
$849$	0	0
$850$	−17.1200	−0.587210
$851$	−7.04888	−0.241632
$852$	0	0
$853$	−31.4882	−1.07814	−0.539068	−	0.842262i	$-0.681224\pi$
$853$	−31.4882	−1.07814	−0.539068	+	0.842262i	$0.681224\pi$
$854$	8.35720	0.285978
$855$	0	0
$856$	−18.5089	−0.632620
$857$	−31.2333	−1.06691	−0.533455	−	0.845829i	$-0.679106\pi$
$857$	−31.2333	−1.06691	−0.533455	+	0.845829i	$0.679106\pi$
$858$	0	0
$859$	−53.0816	−1.81112	−0.905561	−	0.424216i	$-0.860550\pi$
$859$	−53.0816	−1.81112	−0.905561	+	0.424216i	$0.860550\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	8.82220	0.300485
$863$	−5.04888	−0.171866	−0.0859329	−	0.996301i	$-0.527387\pi$
$863$	−5.04888	−0.171866	−0.0859329	+	0.996301i	$0.527387\pi$
$864$	0	0
$865$	−19.2122	−0.653234
$866$	−14.6222	−0.496882
$867$	0	0
$868$	2.52444	0.0856850
$869$	−95.6032	−3.24312
$870$	0	0
$871$	−20.4111	−0.691604
$872$	0.843326	0.0285586
$873$	0	0
$874$	0.578337	0.0195625
$875$	9.68665	0.327469
$876$	0	0
$877$	11.7350	0.396263	0.198132	−	0.980175i	$-0.436513\pi$
$877$	11.7350	0.396263	0.198132	+	0.980175i	$0.436513\pi$
$878$	30.2978	1.02250
$879$	0	0
$880$	6.20555	0.209189
$881$	−38.9255	−1.31143	−0.655717	−	0.755007i	$-0.727633\pi$
$881$	−38.9255	−1.31143	−0.655717	+	0.755007i	$0.727633\pi$
$882$	0	0
$883$	17.3522	0.583947	0.291974	−	0.956426i	$-0.405688\pi$
$883$	17.3522	0.583947	0.291974	+	0.956426i	$0.405688\pi$
$884$	16.4111	0.551965
$885$	0	0
$886$	17.5678	0.590201
$887$	1.55219	0.0521174	0.0260587	−	0.999660i	$-0.491704\pi$
$887$	1.55219	0.0521174	0.0260587	+	0.999660i	$0.491704\pi$
$888$	0	0
$889$	14.2056	0.476439
$890$	−7.42166	−0.248775
$891$	0	0
$892$	−19.9844	−0.669128
$893$	1.45998	0.0488562
$894$	0	0
$895$	12.8645	0.430011
$896$	1.00000	0.0334077
$897$	0	0
$898$	11.2927	0.376844
$899$	14.7244	0.491088
$900$	0	0
$901$	13.7944	0.459560
$902$	17.7633	0.591452
$903$	0	0
$904$	3.79445	0.126202
$905$	6.90276	0.229456
$906$	0	0
$907$	43.5466	1.44594	0.722971	−	0.690878i	$-0.242776\pi$
$907$	43.5466	1.44594	0.722971	+	0.690878i	$0.242776\pi$
$908$	−4.98944	−0.165580
$909$	0	0
$910$	−4.00000	−0.132599
$911$	25.8116	0.855178	0.427589	−	0.903973i	$-0.359363\pi$
$911$	25.8116	0.855178	0.427589	+	0.903973i	$0.359363\pi$
$912$	0	0
$913$	−76.6832	−2.53784
$914$	−27.4600	−0.908295
$915$	0	0
$916$	3.94610	0.130383
$917$	22.3572	0.738300
$918$	0	0
$919$	−17.1083	−0.564351	−0.282176	−	0.959363i	$-0.591056\pi$
$919$	−17.1083	−0.564351	−0.282176	+	0.959363i	$0.591056\pi$
$920$	−1.10278	−0.0363574
$921$	0	0
$922$	29.4983	0.971474
$923$	−47.3311	−1.55792
$924$	0	0
$925$	26.6722	0.876975
$926$	−14.7244	−0.483875
$927$	0	0
$928$	5.83276	0.191470
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	−0.578337	−0.0189542
$932$	2.88164	0.0943912
$933$	0	0
$934$	8.57834	0.280692
$935$	−28.0766	−0.918204
$936$	0	0
$937$	39.1411	1.27868	0.639342	−	0.768923i	$-0.279207\pi$
$937$	39.1411	1.27868	0.639342	+	0.768923i	$0.279207\pi$
$938$	5.62721	0.183735
$939$	0	0
$940$	−2.78389	−0.0908004
$941$	17.4061	0.567422	0.283711	−	0.958910i	$-0.408434\pi$
$941$	17.4061	0.567422	0.283711	+	0.958910i	$0.408434\pi$
$942$	0	0
$943$	−3.15667	−0.102795
$944$	−9.30833	−0.302960
$945$	0	0
$946$	−40.8222	−1.32724
$947$	7.89220	0.256462	0.128231	−	0.991744i	$-0.459070\pi$
$947$	7.89220	0.256462	0.128231	+	0.991744i	$0.459070\pi$
$948$	0	0
$949$	51.8811	1.68413
$950$	−2.18836	−0.0709998
$951$	0	0
$952$	−4.52444	−0.146638
$953$	−22.7456	−0.736801	−0.368401	−	0.929667i	$-0.620095\pi$
$953$	−22.7456	−0.736801	−0.368401	+	0.929667i	$0.620095\pi$
$954$	0	0
$955$	2.55110	0.0825518
$956$	−20.7144	−0.669952
$957$	0	0
$958$	−28.0766	−0.907115
$959$	−1.89220	−0.0611024
$960$	0	0
$961$	−24.6272	−0.794426
$962$	−25.5678	−0.824338
$963$	0	0
$964$	11.0333	0.355358
$965$	6.16330	0.198404
$966$	0	0
$967$	39.6655	1.27556	0.637779	−	0.770220i	$-0.279853\pi$
$967$	39.6655	1.27556	0.637779	+	0.770220i	$0.279853\pi$
$968$	−20.6655	−0.664215
$969$	0	0
$970$	−18.6761	−0.599653
$971$	−32.7628	−1.05141	−0.525704	−	0.850668i	$-0.676198\pi$
$971$	−32.7628	−1.05141	−0.525704	+	0.850668i	$0.676198\pi$
$972$	0	0
$973$	6.35720	0.203803
$974$	−22.7244	−0.728138
$975$	0	0
$976$	8.35720	0.267507
$977$	−13.5577	−0.433750	−0.216875	−	0.976199i	$-0.569586\pi$
$977$	−13.5577	−0.433750	−0.216875	+	0.976199i	$0.569586\pi$
$978$	0	0
$979$	37.8711	1.21036
$980$	1.10278	0.0352269
$981$	0	0
$982$	−18.7244	−0.597521
$983$	47.0278	1.49995	0.749976	−	0.661465i	$-0.230065\pi$
$983$	47.0278	1.49995	0.749976	+	0.661465i	$0.230065\pi$
$984$	0	0
$985$	−20.3033	−0.646917
$986$	−26.3900	−0.840428
$987$	0	0
$988$	2.09775	0.0667384
$989$	7.25443	0.230677
$990$	0	0
$991$	42.8888	1.36241	0.681204	−	0.732094i	$-0.261457\pi$
$991$	42.8888	1.36241	0.681204	+	0.732094i	$0.261457\pi$
$992$	2.52444	0.0801510
$993$	0	0
$994$	13.0489	0.413885
$995$	4.86445	0.154213
$996$	0	0
$997$	7.10831	0.225123	0.112561	−	0.993645i	$-0.464095\pi$
$997$	7.10831	0.225123	0.112561	+	0.993645i	$0.464095\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.be.1.3		3
3.2	odd	2		322.2.a.g.1.3	✓	3
12.11	even	2		2576.2.a.w.1.1		3
15.14	odd	2		8050.2.a.bh.1.1		3
21.20	even	2		2254.2.a.p.1.1		3
69.68	even	2		7406.2.a.x.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.a.g.1.3	✓	3	3.2	odd	2
2254.2.a.p.1.1		3	21.20	even	2
2576.2.a.w.1.1		3	12.11	even	2
2898.2.a.be.1.3		3	1.1	even	1	trivial
7406.2.a.x.1.3		3	69.68	even	2
8050.2.a.bh.1.1		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.10278 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.10278 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.10278 q^{10} +5.62721 q^{11} -3.62721 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.52444 q^{17} -0.578337 q^{19} +1.10278 q^{20} -5.62721 q^{22} +1.00000 q^{23} -3.78389 q^{25} +3.62721 q^{26} -1.00000 q^{28} -5.83276 q^{29} -2.52444 q^{31} -1.00000 q^{32} +4.52444 q^{34} -1.10278 q^{35} -7.04888 q^{37} +0.578337 q^{38} -1.10278 q^{40} -3.15667 q^{41} +7.25443 q^{43} +5.62721 q^{44} -1.00000 q^{46} -2.52444 q^{47} +1.00000 q^{49} +3.78389 q^{50} -3.62721 q^{52} -3.04888 q^{53} +6.20555 q^{55} +1.00000 q^{56} +5.83276 q^{58} -9.30833 q^{59} +8.35720 q^{61} +2.52444 q^{62} +1.00000 q^{64} -4.00000 q^{65} +5.62721 q^{67} -4.52444 q^{68} +1.10278 q^{70} +13.0489 q^{71} -14.3033 q^{73} +7.04888 q^{74} -0.578337 q^{76} -5.62721 q^{77} -16.9894 q^{79} +1.10278 q^{80} +3.15667 q^{82} -13.6272 q^{83} -4.98944 q^{85} -7.25443 q^{86} -5.62721 q^{88} +6.72999 q^{89} +3.62721 q^{91} +1.00000 q^{92} +2.52444 q^{94} -0.637776 q^{95} +16.9355 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} + 4 q^{10} + 4 q^{11} + 2 q^{13} + 3 q^{14} + 3 q^{16} - 8 q^{17} - 4 q^{20} - 4 q^{22} + 3 q^{23} + 5 q^{25} - 2 q^{26} - 3 q^{28} + 10 q^{29} - 2 q^{31} - 3 q^{32} + 8 q^{34} + 4 q^{35} - 10 q^{37} + 4 q^{40} - 6 q^{41} - 4 q^{43} + 4 q^{44} - 3 q^{46} - 2 q^{47} + 3 q^{49} - 5 q^{50} + 2 q^{52} + 2 q^{53} + 4 q^{55} + 3 q^{56} - 10 q^{58} - 6 q^{59} - 8 q^{61} + 2 q^{62} + 3 q^{64} - 12 q^{65} + 4 q^{67} - 8 q^{68} - 4 q^{70} + 28 q^{71} - 6 q^{73} + 10 q^{74} - 4 q^{77} - 20 q^{79} - 4 q^{80} + 6 q^{82} - 28 q^{83} + 16 q^{85} + 4 q^{86} - 4 q^{88} - 2 q^{91} + 3 q^{92} + 2 q^{94} - 20 q^{95} + 16 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 + 4 * q^10 + 4 * q^11 + 2 * q^13 + 3 * q^14 + 3 * q^16 - 8 * q^17 - 4 * q^20 - 4 * q^22 + 3 * q^23 + 5 * q^25 - 2 * q^26 - 3 * q^28 + 10 * q^29 - 2 * q^31 - 3 * q^32 + 8 * q^34 + 4 * q^35 - 10 * q^37 + 4 * q^40 - 6 * q^41 - 4 * q^43 + 4 * q^44 - 3 * q^46 - 2 * q^47 + 3 * q^49 - 5 * q^50 + 2 * q^52 + 2 * q^53 + 4 * q^55 + 3 * q^56 - 10 * q^58 - 6 * q^59 - 8 * q^61 + 2 * q^62 + 3 * q^64 - 12 * q^65 + 4 * q^67 - 8 * q^68 - 4 * q^70 + 28 * q^71 - 6 * q^73 + 10 * q^74 - 4 * q^77 - 20 * q^79 - 4 * q^80 + 6 * q^82 - 28 * q^83 + 16 * q^85 + 4 * q^86 - 4 * q^88 - 2 * q^91 + 3 * q^92 + 2 * q^94 - 20 * q^95 + 16 * q^97 - 3 * q^98

Introduction

L-functions

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