LMFDB - Embedded newform 2898.2.a.x.1.2

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(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +4.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +4.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} -4.37228 q^{10} -4.00000 q^{11} -4.37228 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.74456 q^{17} +4.00000 q^{19} +4.37228 q^{20} +4.00000 q^{22} +1.00000 q^{23} +14.1168 q^{25} +4.37228 q^{26} -1.00000 q^{28} -3.62772 q^{29} -4.74456 q^{31} -1.00000 q^{32} +6.74456 q^{34} -4.37228 q^{35} +8.37228 q^{37} -4.00000 q^{38} -4.37228 q^{40} -9.11684 q^{41} -11.1168 q^{43} -4.00000 q^{44} -1.00000 q^{46} -10.3723 q^{47} +1.00000 q^{49} -14.1168 q^{50} -4.37228 q^{52} +6.74456 q^{53} -17.4891 q^{55} +1.00000 q^{56} +3.62772 q^{58} -8.74456 q^{59} -2.00000 q^{61} +4.74456 q^{62} +1.00000 q^{64} -19.1168 q^{65} -4.00000 q^{67} -6.74456 q^{68} +4.37228 q^{70} -3.25544 q^{71} -1.25544 q^{73} -8.37228 q^{74} +4.00000 q^{76} +4.00000 q^{77} +9.48913 q^{79} +4.37228 q^{80} +9.11684 q^{82} -4.00000 q^{83} -29.4891 q^{85} +11.1168 q^{86} +4.00000 q^{88} +15.4891 q^{89} +4.37228 q^{91} +1.00000 q^{92} +10.3723 q^{94} +17.4891 q^{95} -3.62772 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 3 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 3 q^{5} - 2 q^{7} - 2 q^{8} - 3 q^{10} - 8 q^{11} - 3 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 8 q^{19} + 3 q^{20} + 8 q^{22} + 2 q^{23} + 11 q^{25} + 3 q^{26} - 2 q^{28} - 13 q^{29} + 2 q^{31} - 2 q^{32} + 2 q^{34} - 3 q^{35} + 11 q^{37} - 8 q^{38} - 3 q^{40} - q^{41} - 5 q^{43} - 8 q^{44} - 2 q^{46} - 15 q^{47} + 2 q^{49} - 11 q^{50} - 3 q^{52} + 2 q^{53} - 12 q^{55} + 2 q^{56} + 13 q^{58} - 6 q^{59} - 4 q^{61} - 2 q^{62} + 2 q^{64} - 21 q^{65} - 8 q^{67} - 2 q^{68} + 3 q^{70} - 18 q^{71} - 14 q^{73} - 11 q^{74} + 8 q^{76} + 8 q^{77} - 4 q^{79} + 3 q^{80} + q^{82} - 8 q^{83} - 36 q^{85} + 5 q^{86} + 8 q^{88} + 8 q^{89} + 3 q^{91} + 2 q^{92} + 15 q^{94} + 12 q^{95} - 13 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 - 2 * q^7 - 2 * q^8 - 3 * q^10 - 8 * q^11 - 3 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 + 8 * q^19 + 3 * q^20 + 8 * q^22 + 2 * q^23 + 11 * q^25 + 3 * q^26 - 2 * q^28 - 13 * q^29 + 2 * q^31 - 2 * q^32 + 2 * q^34 - 3 * q^35 + 11 * q^37 - 8 * q^38 - 3 * q^40 - q^41 - 5 * q^43 - 8 * q^44 - 2 * q^46 - 15 * q^47 + 2 * q^49 - 11 * q^50 - 3 * q^52 + 2 * q^53 - 12 * q^55 + 2 * q^56 + 13 * q^58 - 6 * q^59 - 4 * q^61 - 2 * q^62 + 2 * q^64 - 21 * q^65 - 8 * q^67 - 2 * q^68 + 3 * q^70 - 18 * q^71 - 14 * q^73 - 11 * q^74 + 8 * q^76 + 8 * q^77 - 4 * q^79 + 3 * q^80 + q^82 - 8 * q^83 - 36 * q^85 + 5 * q^86 + 8 * q^88 + 8 * q^89 + 3 * q^91 + 2 * q^92 + 15 * q^94 + 12 * q^95 - 13 * 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
$5$	4.37228	1.95534	0.977672	−	0.210138i	$-0.0673912\pi$
$5$	4.37228	1.95534	0.977672	+	0.210138i	$0.0673912\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−4.37228	−1.38264
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−4.37228	−1.21265	−0.606326	−	0.795216i	$-0.707357\pi$
$13$	−4.37228	−1.21265	−0.606326	+	0.795216i	$0.707357\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.74456	−1.63580	−0.817898	−	0.575363i	$-0.804861\pi$
$17$	−6.74456	−1.63580	−0.817898	+	0.575363i	$0.804861\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	4.37228	0.977672
$21$	0	0
$22$	4.00000	0.852803
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	14.1168	2.82337
$26$	4.37228	0.857475
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−3.62772	−0.673650	−0.336825	−	0.941567i	$-0.609353\pi$
$29$	−3.62772	−0.673650	−0.336825	+	0.941567i	$0.609353\pi$
$30$	0	0
$31$	−4.74456	−0.852149	−0.426074	−	0.904688i	$-0.640104\pi$
$31$	−4.74456	−0.852149	−0.426074	+	0.904688i	$0.640104\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.74456	1.15668
$35$	−4.37228	−0.739050
$36$	0	0
$37$	8.37228	1.37639	0.688197	−	0.725524i	$-0.258402\pi$
$37$	8.37228	1.37639	0.688197	+	0.725524i	$0.258402\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−4.37228	−0.691318
$41$	−9.11684	−1.42381	−0.711906	−	0.702275i	$-0.752168\pi$
$41$	−9.11684	−1.42381	−0.711906	+	0.702275i	$0.752168\pi$
$42$	0	0
$43$	−11.1168	−1.69530	−0.847651	−	0.530554i	$-0.821984\pi$
$43$	−11.1168	−1.69530	−0.847651	+	0.530554i	$0.821984\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−10.3723	−1.51295	−0.756476	−	0.654021i	$-0.773081\pi$
$47$	−10.3723	−1.51295	−0.756476	+	0.654021i	$0.773081\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−14.1168	−1.99642
$51$	0	0
$52$	−4.37228	−0.606326
$53$	6.74456	0.926437	0.463218	−	0.886244i	$-0.346695\pi$
$53$	6.74456	0.926437	0.463218	+	0.886244i	$0.346695\pi$
$54$	0	0
$55$	−17.4891	−2.35823
$56$	1.00000	0.133631
$57$	0	0
$58$	3.62772	0.476343
$59$	−8.74456	−1.13845	−0.569223	−	0.822183i	$-0.692756\pi$
$59$	−8.74456	−1.13845	−0.569223	+	0.822183i	$0.692756\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.74456	0.602560
$63$	0	0
$64$	1.00000	0.125000
$65$	−19.1168	−2.37115
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−6.74456	−0.817898
$69$	0	0
$70$	4.37228	0.522588
$71$	−3.25544	−0.386349	−0.193175	−	0.981164i	$-0.561878\pi$
$71$	−3.25544	−0.386349	−0.193175	+	0.981164i	$0.561878\pi$
$72$	0	0
$73$	−1.25544	−0.146938	−0.0734689	−	0.997298i	$-0.523407\pi$
$73$	−1.25544	−0.146938	−0.0734689	+	0.997298i	$0.523407\pi$
$74$	−8.37228	−0.973258
$75$	0	0
$76$	4.00000	0.458831
$77$	4.00000	0.455842
$78$	0	0
$79$	9.48913	1.06761	0.533805	−	0.845608i	$-0.320762\pi$
$79$	9.48913	1.06761	0.533805	+	0.845608i	$0.320762\pi$
$80$	4.37228	0.488836
$81$	0	0
$82$	9.11684	1.00679
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−29.4891	−3.19854
$86$	11.1168	1.19876
$87$	0	0
$88$	4.00000	0.426401
$89$	15.4891	1.64184	0.820922	−	0.571040i	$-0.193460\pi$
$89$	15.4891	1.64184	0.820922	+	0.571040i	$0.193460\pi$
$90$	0	0
$91$	4.37228	0.458340
$92$	1.00000	0.104257
$93$	0	0
$94$	10.3723	1.06982
$95$	17.4891	1.79435
$96$	0	0
$97$	−3.62772	−0.368339	−0.184170	−	0.982894i	$-0.558960\pi$
$97$	−3.62772	−0.368339	−0.184170	+	0.982894i	$0.558960\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	14.1168	1.41168
$101$	−7.48913	−0.745196	−0.372598	−	0.927993i	$-0.621533\pi$
$101$	−7.48913	−0.745196	−0.372598	+	0.927993i	$0.621533\pi$
$102$	0	0
$103$	7.11684	0.701243	0.350622	−	0.936517i	$-0.385970\pi$
$103$	7.11684	0.701243	0.350622	+	0.936517i	$0.385970\pi$
$104$	4.37228	0.428737
$105$	0	0
$106$	−6.74456	−0.655090
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−9.11684	−0.873235	−0.436618	−	0.899647i	$-0.643824\pi$
$109$	−9.11684	−0.873235	−0.436618	+	0.899647i	$0.643824\pi$
$110$	17.4891	1.66752
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−1.11684	−0.105064	−0.0525319	−	0.998619i	$-0.516729\pi$
$113$	−1.11684	−0.105064	−0.0525319	+	0.998619i	$0.516729\pi$
$114$	0	0
$115$	4.37228	0.407717
$116$	−3.62772	−0.336825
$117$	0	0
$118$	8.74456	0.805002
$119$	6.74456	0.618273
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	0	0
$124$	−4.74456	−0.426074
$125$	39.8614	3.56531
$126$	0	0
$127$	−5.62772	−0.499379	−0.249690	−	0.968326i	$-0.580329\pi$
$127$	−5.62772	−0.499379	−0.249690	+	0.968326i	$0.580329\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	19.1168	1.67666
$131$	−7.25544	−0.633911	−0.316955	−	0.948440i	$-0.602661\pi$
$131$	−7.25544	−0.633911	−0.316955	+	0.948440i	$0.602661\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	4.00000	0.345547
$135$	0	0
$136$	6.74456	0.578341
$137$	−9.11684	−0.778905	−0.389452	−	0.921047i	$-0.627336\pi$
$137$	−9.11684	−0.778905	−0.389452	+	0.921047i	$0.627336\pi$
$138$	0	0
$139$	14.3723	1.21904	0.609520	−	0.792770i	$-0.291362\pi$
$139$	14.3723	1.21904	0.609520	+	0.792770i	$0.291362\pi$
$140$	−4.37228	−0.369525
$141$	0	0
$142$	3.25544	0.273190
$143$	17.4891	1.46251
$144$	0	0
$145$	−15.8614	−1.31722
$146$	1.25544	0.103901
$147$	0	0
$148$	8.37228	0.688197
$149$	16.2337	1.32992	0.664958	−	0.746881i	$-0.268450\pi$
$149$	16.2337	1.32992	0.664958	+	0.746881i	$0.268450\pi$
$150$	0	0
$151$	2.37228	0.193054	0.0965268	−	0.995330i	$-0.469227\pi$
$151$	2.37228	0.193054	0.0965268	+	0.995330i	$0.469227\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	−4.00000	−0.322329
$155$	−20.7446	−1.66624
$156$	0	0
$157$	−22.7446	−1.81521	−0.907607	−	0.419821i	$-0.862093\pi$
$157$	−22.7446	−1.81521	−0.907607	+	0.419821i	$0.862093\pi$
$158$	−9.48913	−0.754914
$159$	0	0
$160$	−4.37228	−0.345659
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−9.11684	−0.711906
$165$	0	0
$166$	4.00000	0.310460
$167$	−17.4891	−1.35335	−0.676675	−	0.736282i	$-0.736580\pi$
$167$	−17.4891	−1.35335	−0.676675	+	0.736282i	$0.736580\pi$
$168$	0	0
$169$	6.11684	0.470526
$170$	29.4891	2.26171
$171$	0	0
$172$	−11.1168	−0.847651
$173$	−15.4891	−1.17762	−0.588808	−	0.808273i	$-0.700403\pi$
$173$	−15.4891	−1.17762	−0.588808	+	0.808273i	$0.700403\pi$
$174$	0	0
$175$	−14.1168	−1.06713
$176$	−4.00000	−0.301511
$177$	0	0
$178$	−15.4891	−1.16096
$179$	−7.86141	−0.587589	−0.293795	−	0.955869i	$-0.594918\pi$
$179$	−7.86141	−0.587589	−0.293795	+	0.955869i	$0.594918\pi$
$180$	0	0
$181$	24.9783	1.85662	0.928309	−	0.371809i	$-0.121262\pi$
$181$	24.9783	1.85662	0.928309	+	0.371809i	$0.121262\pi$
$182$	−4.37228	−0.324095
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	36.6060	2.69132
$186$	0	0
$187$	26.9783	1.97285
$188$	−10.3723	−0.756476
$189$	0	0
$190$	−17.4891	−1.26879
$191$	−25.4891	−1.84433	−0.922164	−	0.386799i	$-0.873581\pi$
$191$	−25.4891	−1.84433	−0.922164	+	0.386799i	$0.873581\pi$
$192$	0	0
$193$	17.1168	1.23210	0.616049	−	0.787708i	$-0.288732\pi$
$193$	17.1168	1.23210	0.616049	+	0.787708i	$0.288732\pi$
$194$	3.62772	0.260455
$195$	0	0
$196$	1.00000	0.0714286
$197$	15.6277	1.11343	0.556714	−	0.830704i	$-0.312062\pi$
$197$	15.6277	1.11343	0.556714	+	0.830704i	$0.312062\pi$
$198$	0	0
$199$	11.8614	0.840833	0.420416	−	0.907331i	$-0.361884\pi$
$199$	11.8614	0.840833	0.420416	+	0.907331i	$0.361884\pi$
$200$	−14.1168	−0.998212
$201$	0	0
$202$	7.48913	0.526933
$203$	3.62772	0.254616
$204$	0	0
$205$	−39.8614	−2.78404
$206$	−7.11684	−0.495854
$207$	0	0
$208$	−4.37228	−0.303163
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.74456	0.463218
$213$	0	0
$214$	4.00000	0.273434
$215$	−48.6060	−3.31490
$216$	0	0
$217$	4.74456	0.322082
$218$	9.11684	0.617471
$219$	0	0
$220$	−17.4891	−1.17912
$221$	29.4891	1.98365
$222$	0	0
$223$	−20.7446	−1.38916	−0.694579	−	0.719416i	$-0.744409\pi$
$223$	−20.7446	−1.38916	−0.694579	+	0.719416i	$0.744409\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	1.11684	0.0742914
$227$	11.1168	0.737851	0.368925	−	0.929459i	$-0.379726\pi$
$227$	11.1168	0.737851	0.368925	+	0.929459i	$0.379726\pi$
$228$	0	0
$229$	10.7446	0.710021	0.355010	−	0.934862i	$-0.384477\pi$
$229$	10.7446	0.710021	0.355010	+	0.934862i	$0.384477\pi$
$230$	−4.37228	−0.288300
$231$	0	0
$232$	3.62772	0.238171
$233$	−0.510875	−0.0334685	−0.0167343	−	0.999860i	$-0.505327\pi$
$233$	−0.510875	−0.0334685	−0.0167343	+	0.999860i	$0.505327\pi$
$234$	0	0
$235$	−45.3505	−2.95834
$236$	−8.74456	−0.569223
$237$	0	0
$238$	−6.74456	−0.437185
$239$	9.48913	0.613800	0.306900	−	0.951742i	$-0.400708\pi$
$239$	9.48913	0.613800	0.306900	+	0.951742i	$0.400708\pi$
$240$	0	0
$241$	−8.37228	−0.539306	−0.269653	−	0.962958i	$-0.586909\pi$
$241$	−8.37228	−0.539306	−0.269653	+	0.962958i	$0.586909\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−2.00000	−0.128037
$245$	4.37228	0.279335
$246$	0	0
$247$	−17.4891	−1.11281
$248$	4.74456	0.301280
$249$	0	0
$250$	−39.8614	−2.52106
$251$	19.1168	1.20664	0.603322	−	0.797498i	$-0.293843\pi$
$251$	19.1168	1.20664	0.603322	+	0.797498i	$0.293843\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	5.62772	0.353114
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.9783	1.05907	0.529537	−	0.848287i	$-0.322366\pi$
$257$	16.9783	1.05907	0.529537	+	0.848287i	$0.322366\pi$
$258$	0	0
$259$	−8.37228	−0.520228
$260$	−19.1168	−1.18558
$261$	0	0
$262$	7.25544	0.448242
$263$	7.11684	0.438843	0.219422	−	0.975630i	$-0.429583\pi$
$263$	7.11684	0.438843	0.219422	+	0.975630i	$0.429583\pi$
$264$	0	0
$265$	29.4891	1.81150
$266$	4.00000	0.245256
$267$	0	0
$268$	−4.00000	−0.244339
$269$	−24.9783	−1.52295	−0.761475	−	0.648194i	$-0.775525\pi$
$269$	−24.9783	−1.52295	−0.761475	+	0.648194i	$0.775525\pi$
$270$	0	0
$271$	18.9783	1.15285	0.576423	−	0.817151i	$-0.304448\pi$
$271$	18.9783	1.15285	0.576423	+	0.817151i	$0.304448\pi$
$272$	−6.74456	−0.408949
$273$	0	0
$274$	9.11684	0.550769
$275$	−56.4674	−3.40511
$276$	0	0
$277$	17.2554	1.03678	0.518389	−	0.855145i	$-0.326532\pi$
$277$	17.2554	1.03678	0.518389	+	0.855145i	$0.326532\pi$
$278$	−14.3723	−0.861992
$279$	0	0
$280$	4.37228	0.261294
$281$	11.6277	0.693652	0.346826	−	0.937930i	$-0.387260\pi$
$281$	11.6277	0.693652	0.346826	+	0.937930i	$0.387260\pi$
$282$	0	0
$283$	−24.7446	−1.47091	−0.735456	−	0.677573i	$-0.763032\pi$
$283$	−24.7446	−1.47091	−0.735456	+	0.677573i	$0.763032\pi$
$284$	−3.25544	−0.193175
$285$	0	0
$286$	−17.4891	−1.03415
$287$	9.11684	0.538150
$288$	0	0
$289$	28.4891	1.67583
$290$	15.8614	0.931414
$291$	0	0
$292$	−1.25544	−0.0734689
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	−38.2337	−2.22605
$296$	−8.37228	−0.486629
$297$	0	0
$298$	−16.2337	−0.940392
$299$	−4.37228	−0.252856
$300$	0	0
$301$	11.1168	0.640764
$302$	−2.37228	−0.136509
$303$	0	0
$304$	4.00000	0.229416
$305$	−8.74456	−0.500712
$306$	0	0
$307$	1.62772	0.0928988	0.0464494	−	0.998921i	$-0.485209\pi$
$307$	1.62772	0.0928988	0.0464494	+	0.998921i	$0.485209\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	20.7446	1.17821
$311$	17.4891	0.991717	0.495859	−	0.868403i	$-0.334853\pi$
$311$	17.4891	0.991717	0.495859	+	0.868403i	$0.334853\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	22.7446	1.28355
$315$	0	0
$316$	9.48913	0.533805
$317$	1.11684	0.0627282	0.0313641	−	0.999508i	$-0.490015\pi$
$317$	1.11684	0.0627282	0.0313641	+	0.999508i	$0.490015\pi$
$318$	0	0
$319$	14.5109	0.812453
$320$	4.37228	0.244418
$321$	0	0
$322$	1.00000	0.0557278
$323$	−26.9783	−1.50111
$324$	0	0
$325$	−61.7228	−3.42377
$326$	−4.00000	−0.221540
$327$	0	0
$328$	9.11684	0.503393
$329$	10.3723	0.571842
$330$	0	0
$331$	−13.4891	−0.741429	−0.370715	−	0.928747i	$-0.620887\pi$
$331$	−13.4891	−0.741429	−0.370715	+	0.928747i	$0.620887\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	17.4891	0.956962
$335$	−17.4891	−0.955533
$336$	0	0
$337$	−12.2337	−0.666411	−0.333206	−	0.942854i	$-0.608130\pi$
$337$	−12.2337	−0.666411	−0.333206	+	0.942854i	$0.608130\pi$
$338$	−6.11684	−0.332712
$339$	0	0
$340$	−29.4891	−1.59927
$341$	18.9783	1.02773
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	11.1168	0.599380
$345$	0	0
$346$	15.4891	0.832701
$347$	−22.3723	−1.20101	−0.600503	−	0.799622i	$-0.705033\pi$
$347$	−22.3723	−1.20101	−0.600503	+	0.799622i	$0.705033\pi$
$348$	0	0
$349$	12.5109	0.669692	0.334846	−	0.942273i	$-0.391316\pi$
$349$	12.5109	0.669692	0.334846	+	0.942273i	$0.391316\pi$
$350$	14.1168	0.754577
$351$	0	0
$352$	4.00000	0.213201
$353$	−18.8832	−1.00505	−0.502524	−	0.864563i	$-0.667595\pi$
$353$	−18.8832	−1.00505	−0.502524	+	0.864563i	$0.667595\pi$
$354$	0	0
$355$	−14.2337	−0.755446
$356$	15.4891	0.820922
$357$	0	0
$358$	7.86141	0.415488
$359$	−13.6277	−0.719243	−0.359622	−	0.933098i	$-0.617094\pi$
$359$	−13.6277	−0.719243	−0.359622	+	0.933098i	$0.617094\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−24.9783	−1.31283
$363$	0	0
$364$	4.37228	0.229170
$365$	−5.48913	−0.287314
$366$	0	0
$367$	−5.62772	−0.293765	−0.146882	−	0.989154i	$-0.546924\pi$
$367$	−5.62772	−0.293765	−0.146882	+	0.989154i	$0.546924\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−36.6060	−1.90305
$371$	−6.74456	−0.350160
$372$	0	0
$373$	−8.51087	−0.440676	−0.220338	−	0.975424i	$-0.570716\pi$
$373$	−8.51087	−0.440676	−0.220338	+	0.975424i	$0.570716\pi$
$374$	−26.9783	−1.39501
$375$	0	0
$376$	10.3723	0.534910
$377$	15.8614	0.816904
$378$	0	0
$379$	9.62772	0.494543	0.247271	−	0.968946i	$-0.420466\pi$
$379$	9.62772	0.494543	0.247271	+	0.968946i	$0.420466\pi$
$380$	17.4891	0.897173
$381$	0	0
$382$	25.4891	1.30414
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	17.4891	0.891328
$386$	−17.1168	−0.871224
$387$	0	0
$388$	−3.62772	−0.184170
$389$	11.4891	0.582522	0.291261	−	0.956644i	$-0.405925\pi$
$389$	11.4891	0.582522	0.291261	+	0.956644i	$0.405925\pi$
$390$	0	0
$391$	−6.74456	−0.341087
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−15.6277	−0.787313
$395$	41.4891	2.08754
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	−11.8614	−0.594559
$399$	0	0
$400$	14.1168	0.705842
$401$	−20.9783	−1.04760	−0.523802	−	0.851840i	$-0.675487\pi$
$401$	−20.9783	−1.04760	−0.523802	+	0.851840i	$0.675487\pi$
$402$	0	0
$403$	20.7446	1.03336
$404$	−7.48913	−0.372598
$405$	0	0
$406$	−3.62772	−0.180041
$407$	−33.4891	−1.65999
$408$	0	0
$409$	−10.7446	−0.531284	−0.265642	−	0.964072i	$-0.585584\pi$
$409$	−10.7446	−0.531284	−0.265642	+	0.964072i	$0.585584\pi$
$410$	39.8614	1.96861
$411$	0	0
$412$	7.11684	0.350622
$413$	8.74456	0.430292
$414$	0	0
$415$	−17.4891	−0.858507
$416$	4.37228	0.214369
$417$	0	0
$418$	16.0000	0.782586
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	1.86141	0.0907194	0.0453597	−	0.998971i	$-0.485557\pi$
$421$	1.86141	0.0907194	0.0453597	+	0.998971i	$0.485557\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	−6.74456	−0.327545
$425$	−95.2119	−4.61846
$426$	0	0
$427$	2.00000	0.0967868
$428$	−4.00000	−0.193347
$429$	0	0
$430$	48.6060	2.34399
$431$	−19.8614	−0.956690	−0.478345	−	0.878172i	$-0.658763\pi$
$431$	−19.8614	−0.956690	−0.478345	+	0.878172i	$0.658763\pi$
$432$	0	0
$433$	−17.8614	−0.858364	−0.429182	−	0.903218i	$-0.641198\pi$
$433$	−17.8614	−0.858364	−0.429182	+	0.903218i	$0.641198\pi$
$434$	−4.74456	−0.227746
$435$	0	0
$436$	−9.11684	−0.436618
$437$	4.00000	0.191346
$438$	0	0
$439$	19.2554	0.919012	0.459506	−	0.888175i	$-0.348026\pi$
$439$	19.2554	0.919012	0.459506	+	0.888175i	$0.348026\pi$
$440$	17.4891	0.833761
$441$	0	0
$442$	−29.4891	−1.40265
$443$	14.3723	0.682848	0.341424	−	0.939909i	$-0.389091\pi$
$443$	14.3723	0.682848	0.341424	+	0.939909i	$0.389091\pi$
$444$	0	0
$445$	67.7228	3.21037
$446$	20.7446	0.982284
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	16.9783	0.801253	0.400627	−	0.916241i	$-0.368792\pi$
$449$	16.9783	0.801253	0.400627	+	0.916241i	$0.368792\pi$
$450$	0	0
$451$	36.4674	1.71718
$452$	−1.11684	−0.0525319
$453$	0	0
$454$	−11.1168	−0.521739
$455$	19.1168	0.896211
$456$	0	0
$457$	24.2337	1.13360	0.566802	−	0.823854i	$-0.308180\pi$
$457$	24.2337	1.13360	0.566802	+	0.823854i	$0.308180\pi$
$458$	−10.7446	−0.502060
$459$	0	0
$460$	4.37228	0.203859
$461$	35.4891	1.65289	0.826447	−	0.563015i	$-0.190359\pi$
$461$	35.4891	1.65289	0.826447	+	0.563015i	$0.190359\pi$
$462$	0	0
$463$	15.1168	0.702539	0.351270	−	0.936274i	$-0.385750\pi$
$463$	15.1168	0.702539	0.351270	+	0.936274i	$0.385750\pi$
$464$	−3.62772	−0.168413
$465$	0	0
$466$	0.510875	0.0236658
$467$	−0.138593	−0.00641334	−0.00320667	−	0.999995i	$-0.501021\pi$
$467$	−0.138593	−0.00641334	−0.00320667	+	0.999995i	$0.501021\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	45.3505	2.09186
$471$	0	0
$472$	8.74456	0.402501
$473$	44.4674	2.04461
$474$	0	0
$475$	56.4674	2.59090
$476$	6.74456	0.309137
$477$	0	0
$478$	−9.48913	−0.434022
$479$	−4.74456	−0.216785	−0.108392	−	0.994108i	$-0.534570\pi$
$479$	−4.74456	−0.216785	−0.108392	+	0.994108i	$0.534570\pi$
$480$	0	0
$481$	−36.6060	−1.66909
$482$	8.37228	0.381347
$483$	0	0
$484$	5.00000	0.227273
$485$	−15.8614	−0.720229
$486$	0	0
$487$	23.1168	1.04752	0.523762	−	0.851865i	$-0.324528\pi$
$487$	23.1168	1.04752	0.523762	+	0.851865i	$0.324528\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	−4.37228	−0.197520
$491$	−14.9783	−0.675959	−0.337979	−	0.941153i	$-0.609743\pi$
$491$	−14.9783	−0.675959	−0.337979	+	0.941153i	$0.609743\pi$
$492$	0	0
$493$	24.4674	1.10196
$494$	17.4891	0.786873
$495$	0	0
$496$	−4.74456	−0.213037
$497$	3.25544	0.146026
$498$	0	0
$499$	−35.7228	−1.59917	−0.799586	−	0.600551i	$-0.794948\pi$
$499$	−35.7228	−1.59917	−0.799586	+	0.600551i	$0.794948\pi$
$500$	39.8614	1.78266
$501$	0	0
$502$	−19.1168	−0.853227
$503$	12.7446	0.568252	0.284126	−	0.958787i	$-0.408297\pi$
$503$	12.7446	0.568252	0.284126	+	0.958787i	$0.408297\pi$
$504$	0	0
$505$	−32.7446	−1.45711
$506$	4.00000	0.177822
$507$	0	0
$508$	−5.62772	−0.249690
$509$	8.23369	0.364952	0.182476	−	0.983210i	$-0.441589\pi$
$509$	8.23369	0.364952	0.182476	+	0.983210i	$0.441589\pi$
$510$	0	0
$511$	1.25544	0.0555373
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−16.9783	−0.748879
$515$	31.1168	1.37117
$516$	0	0
$517$	41.4891	1.82469
$518$	8.37228	0.367857
$519$	0	0
$520$	19.1168	0.838329
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	−5.76631	−0.252143	−0.126072	−	0.992021i	$-0.540237\pi$
$523$	−5.76631	−0.252143	−0.126072	+	0.992021i	$0.540237\pi$
$524$	−7.25544	−0.316955
$525$	0	0
$526$	−7.11684	−0.310309
$527$	32.0000	1.39394
$528$	0	0
$529$	1.00000	0.0434783
$530$	−29.4891	−1.28093
$531$	0	0
$532$	−4.00000	−0.173422
$533$	39.8614	1.72659
$534$	0	0
$535$	−17.4891	−0.756121
$536$	4.00000	0.172774
$537$	0	0
$538$	24.9783	1.07689
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−16.2337	−0.697941	−0.348970	−	0.937134i	$-0.613469\pi$
$541$	−16.2337	−0.697941	−0.348970	+	0.937134i	$0.613469\pi$
$542$	−18.9783	−0.815186
$543$	0	0
$544$	6.74456	0.289171
$545$	−39.8614	−1.70748
$546$	0	0
$547$	−32.7446	−1.40006	−0.700028	−	0.714115i	$-0.746829\pi$
$547$	−32.7446	−1.40006	−0.700028	+	0.714115i	$0.746829\pi$
$548$	−9.11684	−0.389452
$549$	0	0
$550$	56.4674	2.40778
$551$	−14.5109	−0.618184
$552$	0	0
$553$	−9.48913	−0.403519
$554$	−17.2554	−0.733113
$555$	0	0
$556$	14.3723	0.609520
$557$	−20.2337	−0.857329	−0.428664	−	0.903464i	$-0.641016\pi$
$557$	−20.2337	−0.857329	−0.428664	+	0.903464i	$0.641016\pi$
$558$	0	0
$559$	48.6060	2.05581
$560$	−4.37228	−0.184763
$561$	0	0
$562$	−11.6277	−0.490486
$563$	−30.3723	−1.28004	−0.640020	−	0.768359i	$-0.721074\pi$
$563$	−30.3723	−1.28004	−0.640020	+	0.768359i	$0.721074\pi$
$564$	0	0
$565$	−4.88316	−0.205436
$566$	24.7446	1.04009
$567$	0	0
$568$	3.25544	0.136595
$569$	37.1168	1.55602	0.778010	−	0.628252i	$-0.216230\pi$
$569$	37.1168	1.55602	0.778010	+	0.628252i	$0.216230\pi$
$570$	0	0
$571$	22.9783	0.961610	0.480805	−	0.876828i	$-0.340345\pi$
$571$	22.9783	0.961610	0.480805	+	0.876828i	$0.340345\pi$
$572$	17.4891	0.731257
$573$	0	0
$574$	−9.11684	−0.380530
$575$	14.1168	0.588713
$576$	0	0
$577$	−26.4674	−1.10185	−0.550926	−	0.834554i	$-0.685725\pi$
$577$	−26.4674	−1.10185	−0.550926	+	0.834554i	$0.685725\pi$
$578$	−28.4891	−1.18499
$579$	0	0
$580$	−15.8614	−0.658609
$581$	4.00000	0.165948
$582$	0	0
$583$	−26.9783	−1.11732
$584$	1.25544	0.0519504
$585$	0	0
$586$	−10.0000	−0.413096
$587$	−29.4891	−1.21715	−0.608573	−	0.793498i	$-0.708258\pi$
$587$	−29.4891	−1.21715	−0.608573	+	0.793498i	$0.708258\pi$
$588$	0	0
$589$	−18.9783	−0.781985
$590$	38.2337	1.57406
$591$	0	0
$592$	8.37228	0.344099
$593$	8.37228	0.343808	0.171904	−	0.985114i	$-0.445008\pi$
$593$	8.37228	0.343808	0.171904	+	0.985114i	$0.445008\pi$
$594$	0	0
$595$	29.4891	1.20894
$596$	16.2337	0.664958
$597$	0	0
$598$	4.37228	0.178796
$599$	−20.4674	−0.836274	−0.418137	−	0.908384i	$-0.637317\pi$
$599$	−20.4674	−0.836274	−0.418137	+	0.908384i	$0.637317\pi$
$600$	0	0
$601$	−4.23369	−0.172696	−0.0863479	−	0.996265i	$-0.527520\pi$
$601$	−4.23369	−0.172696	−0.0863479	+	0.996265i	$0.527520\pi$
$602$	−11.1168	−0.453089
$603$	0	0
$604$	2.37228	0.0965268
$605$	21.8614	0.888793
$606$	0	0
$607$	11.2554	0.456844	0.228422	−	0.973562i	$-0.426643\pi$
$607$	11.2554	0.456844	0.228422	+	0.973562i	$0.426643\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	8.74456	0.354057
$611$	45.3505	1.83469
$612$	0	0
$613$	45.1168	1.82225	0.911126	−	0.412128i	$-0.135214\pi$
$613$	45.1168	1.82225	0.911126	+	0.412128i	$0.135214\pi$
$614$	−1.62772	−0.0656894
$615$	0	0
$616$	−4.00000	−0.161165
$617$	12.5109	0.503669	0.251834	−	0.967770i	$-0.418966\pi$
$617$	12.5109	0.503669	0.251834	+	0.967770i	$0.418966\pi$
$618$	0	0
$619$	23.2554	0.934715	0.467357	−	0.884068i	$-0.345206\pi$
$619$	23.2554	0.934715	0.467357	+	0.884068i	$0.345206\pi$
$620$	−20.7446	−0.833122
$621$	0	0
$622$	−17.4891	−0.701250
$623$	−15.4891	−0.620559
$624$	0	0
$625$	103.701	4.14804
$626$	6.00000	0.239808
$627$	0	0
$628$	−22.7446	−0.907607
$629$	−56.4674	−2.25150
$630$	0	0
$631$	10.9783	0.437037	0.218519	−	0.975833i	$-0.429878\pi$
$631$	10.9783	0.437037	0.218519	+	0.975833i	$0.429878\pi$
$632$	−9.48913	−0.377457
$633$	0	0
$634$	−1.11684	−0.0443555
$635$	−24.6060	−0.976458
$636$	0	0
$637$	−4.37228	−0.173236
$638$	−14.5109	−0.574491
$639$	0	0
$640$	−4.37228	−0.172830
$641$	−31.3505	−1.23827	−0.619136	−	0.785284i	$-0.712517\pi$
$641$	−31.3505	−1.23827	−0.619136	+	0.785284i	$0.712517\pi$
$642$	0	0
$643$	−42.2337	−1.66553	−0.832767	−	0.553624i	$-0.813245\pi$
$643$	−42.2337	−1.66553	−0.832767	+	0.553624i	$0.813245\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	26.9783	1.06145
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	34.9783	1.37302
$650$	61.7228	2.42097
$651$	0	0
$652$	4.00000	0.156652
$653$	−32.0951	−1.25598	−0.627989	−	0.778222i	$-0.716122\pi$
$653$	−32.0951	−1.25598	−0.627989	+	0.778222i	$0.716122\pi$
$654$	0	0
$655$	−31.7228	−1.23951
$656$	−9.11684	−0.355953
$657$	0	0
$658$	−10.3723	−0.404354
$659$	22.9783	0.895106	0.447553	−	0.894258i	$-0.352296\pi$
$659$	22.9783	0.895106	0.447553	+	0.894258i	$0.352296\pi$
$660$	0	0
$661$	−19.4891	−0.758039	−0.379020	−	0.925389i	$-0.623739\pi$
$661$	−19.4891	−0.758039	−0.379020	+	0.925389i	$0.623739\pi$
$662$	13.4891	0.524270
$663$	0	0
$664$	4.00000	0.155230
$665$	−17.4891	−0.678199
$666$	0	0
$667$	−3.62772	−0.140466
$668$	−17.4891	−0.676675
$669$	0	0
$670$	17.4891	0.675664
$671$	8.00000	0.308837
$672$	0	0
$673$	26.6060	1.02558	0.512792	−	0.858513i	$-0.328611\pi$
$673$	26.6060	1.02558	0.512792	+	0.858513i	$0.328611\pi$
$674$	12.2337	0.471224
$675$	0	0
$676$	6.11684	0.235263
$677$	12.9783	0.498795	0.249397	−	0.968401i	$-0.419767\pi$
$677$	12.9783	0.498795	0.249397	+	0.968401i	$0.419767\pi$
$678$	0	0
$679$	3.62772	0.139219
$680$	29.4891	1.13086
$681$	0	0
$682$	−18.9783	−0.726715
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−39.8614	−1.52303
$686$	1.00000	0.0381802
$687$	0	0
$688$	−11.1168	−0.423826
$689$	−29.4891	−1.12345
$690$	0	0
$691$	−42.8397	−1.62970	−0.814849	−	0.579674i	$-0.803180\pi$
$691$	−42.8397	−1.62970	−0.814849	+	0.579674i	$0.803180\pi$
$692$	−15.4891	−0.588808
$693$	0	0
$694$	22.3723	0.849240
$695$	62.8397	2.38364
$696$	0	0
$697$	61.4891	2.32907
$698$	−12.5109	−0.473544
$699$	0	0
$700$	−14.1168	−0.533567
$701$	40.2337	1.51961	0.759803	−	0.650154i	$-0.225296\pi$
$701$	40.2337	1.51961	0.759803	+	0.650154i	$0.225296\pi$
$702$	0	0
$703$	33.4891	1.26307
$704$	−4.00000	−0.150756
$705$	0	0
$706$	18.8832	0.710677
$707$	7.48913	0.281658
$708$	0	0
$709$	−27.4891	−1.03238	−0.516188	−	0.856475i	$-0.672649\pi$
$709$	−27.4891	−1.03238	−0.516188	+	0.856475i	$0.672649\pi$
$710$	14.2337	0.534181
$711$	0	0
$712$	−15.4891	−0.580480
$713$	−4.74456	−0.177685
$714$	0	0
$715$	76.4674	2.85972
$716$	−7.86141	−0.293795
$717$	0	0
$718$	13.6277	0.508582
$719$	5.62772	0.209878	0.104939	−	0.994479i	$-0.466535\pi$
$719$	5.62772	0.209878	0.104939	+	0.994479i	$0.466535\pi$
$720$	0	0
$721$	−7.11684	−0.265045
$722$	3.00000	0.111648
$723$	0	0
$724$	24.9783	0.928309
$725$	−51.2119	−1.90196
$726$	0	0
$727$	33.4891	1.24204	0.621021	−	0.783794i	$-0.286718\pi$
$727$	33.4891	1.24204	0.621021	+	0.783794i	$0.286718\pi$
$728$	−4.37228	−0.162048
$729$	0	0
$730$	5.48913	0.203162
$731$	74.9783	2.77317
$732$	0	0
$733$	−16.2337	−0.599605	−0.299802	−	0.954001i	$-0.596921\pi$
$733$	−16.2337	−0.599605	−0.299802	+	0.954001i	$0.596921\pi$
$734$	5.62772	0.207723
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	16.0000	0.589368
$738$	0	0
$739$	−16.7446	−0.615959	−0.307979	−	0.951393i	$-0.599653\pi$
$739$	−16.7446	−0.615959	−0.307979	+	0.951393i	$0.599653\pi$
$740$	36.6060	1.34566
$741$	0	0
$742$	6.74456	0.247601
$743$	45.9565	1.68598	0.842990	−	0.537929i	$-0.180793\pi$
$743$	45.9565	1.68598	0.842990	+	0.537929i	$0.180793\pi$
$744$	0	0
$745$	70.9783	2.60044
$746$	8.51087	0.311605
$747$	0	0
$748$	26.9783	0.986423
$749$	4.00000	0.146157
$750$	0	0
$751$	−9.48913	−0.346263	−0.173132	−	0.984899i	$-0.555389\pi$
$751$	−9.48913	−0.346263	−0.173132	+	0.984899i	$0.555389\pi$
$752$	−10.3723	−0.378238
$753$	0	0
$754$	−15.8614	−0.577638
$755$	10.3723	0.377486
$756$	0	0
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	−9.62772	−0.349694
$759$	0	0
$760$	−17.4891	−0.634397
$761$	40.9783	1.48546	0.742730	−	0.669591i	$-0.233531\pi$
$761$	40.9783	1.48546	0.742730	+	0.669591i	$0.233531\pi$
$762$	0	0
$763$	9.11684	0.330052
$764$	−25.4891	−0.922164
$765$	0	0
$766$	16.0000	0.578103
$767$	38.2337	1.38054
$768$	0	0
$769$	−45.1168	−1.62696	−0.813478	−	0.581596i	$-0.802428\pi$
$769$	−45.1168	−1.62696	−0.813478	+	0.581596i	$0.802428\pi$
$770$	−17.4891	−0.630264
$771$	0	0
$772$	17.1168	0.616049
$773$	1.39403	0.0501398	0.0250699	−	0.999686i	$-0.492019\pi$
$773$	1.39403	0.0501398	0.0250699	+	0.999686i	$0.492019\pi$
$774$	0	0
$775$	−66.9783	−2.40593
$776$	3.62772	0.130228
$777$	0	0
$778$	−11.4891	−0.411905
$779$	−36.4674	−1.30658
$780$	0	0
$781$	13.0217	0.465955
$782$	6.74456	0.241185
$783$	0	0
$784$	1.00000	0.0357143
$785$	−99.4456	−3.54937
$786$	0	0
$787$	29.4891	1.05117	0.525587	−	0.850740i	$-0.323846\pi$
$787$	29.4891	1.05117	0.525587	+	0.850740i	$0.323846\pi$
$788$	15.6277	0.556714
$789$	0	0
$790$	−41.4891	−1.47612
$791$	1.11684	0.0397104
$792$	0	0
$793$	8.74456	0.310529
$794$	10.0000	0.354887
$795$	0	0
$796$	11.8614	0.420416
$797$	21.8614	0.774371	0.387185	−	0.922002i	$-0.373447\pi$
$797$	21.8614	0.774371	0.387185	+	0.922002i	$0.373447\pi$
$798$	0	0
$799$	69.9565	2.47488
$800$	−14.1168	−0.499106
$801$	0	0
$802$	20.9783	0.740768
$803$	5.02175	0.177214
$804$	0	0
$805$	−4.37228	−0.154103
$806$	−20.7446	−0.730696
$807$	0	0
$808$	7.48913	0.263467
$809$	1.25544	0.0441388	0.0220694	−	0.999756i	$-0.492975\pi$
$809$	1.25544	0.0441388	0.0220694	+	0.999756i	$0.492975\pi$
$810$	0	0
$811$	−20.6060	−0.723573	−0.361787	−	0.932261i	$-0.617833\pi$
$811$	−20.6060	−0.723573	−0.361787	+	0.932261i	$0.617833\pi$
$812$	3.62772	0.127308
$813$	0	0
$814$	33.4891	1.17379
$815$	17.4891	0.612617
$816$	0	0
$817$	−44.4674	−1.55572
$818$	10.7446	0.375675
$819$	0	0
$820$	−39.8614	−1.39202
$821$	0.510875	0.0178297	0.00891483	−	0.999960i	$-0.497162\pi$
$821$	0.510875	0.0178297	0.00891483	+	0.999960i	$0.497162\pi$
$822$	0	0
$823$	5.35053	0.186508	0.0932539	−	0.995642i	$-0.470273\pi$
$823$	5.35053	0.186508	0.0932539	+	0.995642i	$0.470273\pi$
$824$	−7.11684	−0.247927
$825$	0	0
$826$	−8.74456	−0.304262
$827$	−18.2337	−0.634047	−0.317024	−	0.948418i	$-0.602683\pi$
$827$	−18.2337	−0.634047	−0.317024	+	0.948418i	$0.602683\pi$
$828$	0	0
$829$	47.4891	1.64937	0.824683	−	0.565596i	$-0.191354\pi$
$829$	47.4891	1.64937	0.824683	+	0.565596i	$0.191354\pi$
$830$	17.4891	0.607056
$831$	0	0
$832$	−4.37228	−0.151582
$833$	−6.74456	−0.233685
$834$	0	0
$835$	−76.4674	−2.64626
$836$	−16.0000	−0.553372
$837$	0	0
$838$	4.00000	0.138178
$839$	−44.7446	−1.54475	−0.772377	−	0.635164i	$-0.780932\pi$
$839$	−44.7446	−1.54475	−0.772377	+	0.635164i	$0.780932\pi$
$840$	0	0
$841$	−15.8397	−0.546195
$842$	−1.86141	−0.0641483
$843$	0	0
$844$	−12.0000	−0.413057
$845$	26.7446	0.920041
$846$	0	0
$847$	−5.00000	−0.171802
$848$	6.74456	0.231609
$849$	0	0
$850$	95.2119	3.26574
$851$	8.37228	0.286998
$852$	0	0
$853$	−45.5842	−1.56077	−0.780387	−	0.625297i	$-0.784978\pi$
$853$	−45.5842	−1.56077	−0.780387	+	0.625297i	$0.784978\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	4.00000	0.136717
$857$	6.88316	0.235124	0.117562	−	0.993066i	$-0.462492\pi$
$857$	6.88316	0.235124	0.117562	+	0.993066i	$0.462492\pi$
$858$	0	0
$859$	4.88316	0.166611	0.0833056	−	0.996524i	$-0.473452\pi$
$859$	4.88316	0.166611	0.0833056	+	0.996524i	$0.473452\pi$
$860$	−48.6060	−1.65745
$861$	0	0
$862$	19.8614	0.676482
$863$	9.48913	0.323014	0.161507	−	0.986872i	$-0.448365\pi$
$863$	9.48913	0.323014	0.161507	+	0.986872i	$0.448365\pi$
$864$	0	0
$865$	−67.7228	−2.30264
$866$	17.8614	0.606955
$867$	0	0
$868$	4.74456	0.161041
$869$	−37.9565	−1.28759
$870$	0	0
$871$	17.4891	0.592596
$872$	9.11684	0.308735
$873$	0	0
$874$	−4.00000	−0.135302
$875$	−39.8614	−1.34756
$876$	0	0
$877$	12.2337	0.413102	0.206551	−	0.978436i	$-0.433776\pi$
$877$	12.2337	0.413102	0.206551	+	0.978436i	$0.433776\pi$
$878$	−19.2554	−0.649840
$879$	0	0
$880$	−17.4891	−0.589558
$881$	−4.97825	−0.167722	−0.0838608	−	0.996477i	$-0.526725\pi$
$881$	−4.97825	−0.167722	−0.0838608	+	0.996477i	$0.526725\pi$
$882$	0	0
$883$	−32.7446	−1.10194	−0.550971	−	0.834524i	$-0.685743\pi$
$883$	−32.7446	−1.10194	−0.550971	+	0.834524i	$0.685743\pi$
$884$	29.4891	0.991827
$885$	0	0
$886$	−14.3723	−0.482846
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	5.62772	0.188748
$890$	−67.7228	−2.27007
$891$	0	0
$892$	−20.7446	−0.694579
$893$	−41.4891	−1.38838
$894$	0	0
$895$	−34.3723	−1.14894
$896$	1.00000	0.0334077
$897$	0	0
$898$	−16.9783	−0.566572
$899$	17.2119	0.574050
$900$	0	0
$901$	−45.4891	−1.51546
$902$	−36.4674	−1.21423
$903$	0	0
$904$	1.11684	0.0371457
$905$	109.212	3.63033
$906$	0	0
$907$	22.0951	0.733656	0.366828	−	0.930289i	$-0.380444\pi$
$907$	22.0951	0.733656	0.366828	+	0.930289i	$0.380444\pi$
$908$	11.1168	0.368925
$909$	0	0
$910$	−19.1168	−0.633717
$911$	48.3288	1.60120	0.800602	−	0.599196i	$-0.204513\pi$
$911$	48.3288	1.60120	0.800602	+	0.599196i	$0.204513\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	−24.2337	−0.801579
$915$	0	0
$916$	10.7446	0.355010
$917$	7.25544	0.239596
$918$	0	0
$919$	45.9565	1.51597	0.757983	−	0.652275i	$-0.226185\pi$
$919$	45.9565	1.51597	0.757983	+	0.652275i	$0.226185\pi$
$920$	−4.37228	−0.144150
$921$	0	0
$922$	−35.4891	−1.16877
$923$	14.2337	0.468508
$924$	0	0
$925$	118.190	3.88607
$926$	−15.1168	−0.496770
$927$	0	0
$928$	3.62772	0.119086
$929$	−33.1168	−1.08653	−0.543264	−	0.839562i	$-0.682812\pi$
$929$	−33.1168	−1.08653	−0.543264	+	0.839562i	$0.682812\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	−0.510875	−0.0167343
$933$	0	0
$934$	0.138593	0.00453491
$935$	117.957	3.85759
$936$	0	0
$937$	−27.6277	−0.902558	−0.451279	−	0.892383i	$-0.649032\pi$
$937$	−27.6277	−0.902558	−0.451279	+	0.892383i	$0.649032\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	−45.3505	−1.47917
$941$	60.3723	1.96808	0.984040	−	0.177947i	$-0.0569456\pi$
$941$	60.3723	1.96808	0.984040	+	0.177947i	$0.0569456\pi$
$942$	0	0
$943$	−9.11684	−0.296885
$944$	−8.74456	−0.284611
$945$	0	0
$946$	−44.4674	−1.44576
$947$	57.3505	1.86364	0.931821	−	0.362918i	$-0.118220\pi$
$947$	57.3505	1.86364	0.931821	+	0.362918i	$0.118220\pi$
$948$	0	0
$949$	5.48913	0.178185
$950$	−56.4674	−1.83204
$951$	0	0
$952$	−6.74456	−0.218593
$953$	−0.510875	−0.0165489	−0.00827443	−	0.999966i	$-0.502634\pi$
$953$	−0.510875	−0.0165489	−0.00827443	+	0.999966i	$0.502634\pi$
$954$	0	0
$955$	−111.446	−3.60630
$956$	9.48913	0.306900
$957$	0	0
$958$	4.74456	0.153290
$959$	9.11684	0.294398
$960$	0	0
$961$	−8.48913	−0.273843
$962$	36.6060	1.18022
$963$	0	0
$964$	−8.37228	−0.269653
$965$	74.8397	2.40917
$966$	0	0
$967$	−30.5109	−0.981164	−0.490582	−	0.871395i	$-0.663216\pi$
$967$	−30.5109	−0.981164	−0.490582	+	0.871395i	$0.663216\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	15.8614	0.509279
$971$	−40.4674	−1.29866	−0.649330	−	0.760507i	$-0.724951\pi$
$971$	−40.4674	−1.29866	−0.649330	+	0.760507i	$0.724951\pi$
$972$	0	0
$973$	−14.3723	−0.460754
$974$	−23.1168	−0.740711
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	43.3505	1.38691	0.693453	−	0.720502i	$-0.256088\pi$
$977$	43.3505	1.38691	0.693453	+	0.720502i	$0.256088\pi$
$978$	0	0
$979$	−61.9565	−1.98014
$980$	4.37228	0.139667
$981$	0	0
$982$	14.9783	0.477975
$983$	38.2337	1.21947	0.609733	−	0.792607i	$-0.291277\pi$
$983$	38.2337	1.21947	0.609733	+	0.792607i	$0.291277\pi$
$984$	0	0
$985$	68.3288	2.17714
$986$	−24.4674	−0.779200
$987$	0	0
$988$	−17.4891	−0.556403
$989$	−11.1168	−0.353495
$990$	0	0
$991$	9.48913	0.301432	0.150716	−	0.988577i	$-0.451842\pi$
$991$	9.48913	0.301432	0.150716	+	0.988577i	$0.451842\pi$
$992$	4.74456	0.150640
$993$	0	0
$994$	−3.25544	−0.103256
$995$	51.8614	1.64412
$996$	0	0
$997$	−50.0000	−1.58352	−0.791758	−	0.610835i	$-0.790834\pi$
$997$	−50.0000	−1.58352	−0.791758	+	0.610835i	$0.790834\pi$
$998$	35.7228	1.13079
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.x.1.2		2
3.2	odd	2		966.2.a.o.1.1	✓	2
12.11	even	2		7728.2.a.bh.1.1		2
21.20	even	2		6762.2.a.cd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.o.1.1	✓	2	3.2	odd	2
2898.2.a.x.1.2		2	1.1	even	1	trivial
6762.2.a.cd.1.2		2	21.20	even	2
7728.2.a.bh.1.1		2	12.11	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:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} -4.37228 q^{10} -4.00000 q^{11} -4.37228 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.74456 q^{17} +4.00000 q^{19} +4.37228 q^{20} +4.00000 q^{22} +1.00000 q^{23} +14.1168 q^{25} +4.37228 q^{26} -1.00000 q^{28} -3.62772 q^{29} -4.74456 q^{31} -1.00000 q^{32} +6.74456 q^{34} -4.37228 q^{35} +8.37228 q^{37} -4.00000 q^{38} -4.37228 q^{40} -9.11684 q^{41} -11.1168 q^{43} -4.00000 q^{44} -1.00000 q^{46} -10.3723 q^{47} +1.00000 q^{49} -14.1168 q^{50} -4.37228 q^{52} +6.74456 q^{53} -17.4891 q^{55} +1.00000 q^{56} +3.62772 q^{58} -8.74456 q^{59} -2.00000 q^{61} +4.74456 q^{62} +1.00000 q^{64} -19.1168 q^{65} -4.00000 q^{67} -6.74456 q^{68} +4.37228 q^{70} -3.25544 q^{71} -1.25544 q^{73} -8.37228 q^{74} +4.00000 q^{76} +4.00000 q^{77} +9.48913 q^{79} +4.37228 q^{80} +9.11684 q^{82} -4.00000 q^{83} -29.4891 q^{85} +11.1168 q^{86} +4.00000 q^{88} +15.4891 q^{89} +4.37228 q^{91} +1.00000 q^{92} +10.3723 q^{94} +17.4891 q^{95} -3.62772 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 3 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 3 q^{5} - 2 q^{7} - 2 q^{8} - 3 q^{10} - 8 q^{11} - 3 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 8 q^{19} + 3 q^{20} + 8 q^{22} + 2 q^{23} + 11 q^{25} + 3 q^{26} - 2 q^{28} - 13 q^{29} + 2 q^{31} - 2 q^{32} + 2 q^{34} - 3 q^{35} + 11 q^{37} - 8 q^{38} - 3 q^{40} - q^{41} - 5 q^{43} - 8 q^{44} - 2 q^{46} - 15 q^{47} + 2 q^{49} - 11 q^{50} - 3 q^{52} + 2 q^{53} - 12 q^{55} + 2 q^{56} + 13 q^{58} - 6 q^{59} - 4 q^{61} - 2 q^{62} + 2 q^{64} - 21 q^{65} - 8 q^{67} - 2 q^{68} + 3 q^{70} - 18 q^{71} - 14 q^{73} - 11 q^{74} + 8 q^{76} + 8 q^{77} - 4 q^{79} + 3 q^{80} + q^{82} - 8 q^{83} - 36 q^{85} + 5 q^{86} + 8 q^{88} + 8 q^{89} + 3 q^{91} + 2 q^{92} + 15 q^{94} + 12 q^{95} - 13 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 - 2 * q^7 - 2 * q^8 - 3 * q^10 - 8 * q^11 - 3 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 + 8 * q^19 + 3 * q^20 + 8 * q^22 + 2 * q^23 + 11 * q^25 + 3 * q^26 - 2 * q^28 - 13 * q^29 + 2 * q^31 - 2 * q^32 + 2 * q^34 - 3 * q^35 + 11 * q^37 - 8 * q^38 - 3 * q^40 - q^41 - 5 * q^43 - 8 * q^44 - 2 * q^46 - 15 * q^47 + 2 * q^49 - 11 * q^50 - 3 * q^52 + 2 * q^53 - 12 * q^55 + 2 * q^56 + 13 * q^58 - 6 * q^59 - 4 * q^61 - 2 * q^62 + 2 * q^64 - 21 * q^65 - 8 * q^67 - 2 * q^68 + 3 * q^70 - 18 * q^71 - 14 * q^73 - 11 * q^74 + 8 * q^76 + 8 * q^77 - 4 * q^79 + 3 * q^80 + q^82 - 8 * q^83 - 36 * q^85 + 5 * q^86 + 8 * q^88 + 8 * q^89 + 3 * q^91 + 2 * q^92 + 15 * q^94 + 12 * q^95 - 13 * q^97 - 2 * q^98

Introduction

L-functions

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