LMFDB - Embedded newform 2898.2.a.x.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(\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			1.1
Root			$-2.37228$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.37228 q^{10} -4.00000 q^{11} +1.37228 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.74456 q^{17} +4.00000 q^{19} -1.37228 q^{20} +4.00000 q^{22} +1.00000 q^{23} -3.11684 q^{25} -1.37228 q^{26} -1.00000 q^{28} -9.37228 q^{29} +6.74456 q^{31} -1.00000 q^{32} -4.74456 q^{34} +1.37228 q^{35} +2.62772 q^{37} -4.00000 q^{38} +1.37228 q^{40} +8.11684 q^{41} +6.11684 q^{43} -4.00000 q^{44} -1.00000 q^{46} -4.62772 q^{47} +1.00000 q^{49} +3.11684 q^{50} +1.37228 q^{52} -4.74456 q^{53} +5.48913 q^{55} +1.00000 q^{56} +9.37228 q^{58} +2.74456 q^{59} -2.00000 q^{61} -6.74456 q^{62} +1.00000 q^{64} -1.88316 q^{65} -4.00000 q^{67} +4.74456 q^{68} -1.37228 q^{70} -14.7446 q^{71} -12.7446 q^{73} -2.62772 q^{74} +4.00000 q^{76} +4.00000 q^{77} -13.4891 q^{79} -1.37228 q^{80} -8.11684 q^{82} -4.00000 q^{83} -6.51087 q^{85} -6.11684 q^{86} +4.00000 q^{88} -7.48913 q^{89} -1.37228 q^{91} +1.00000 q^{92} +4.62772 q^{94} -5.48913 q^{95} -9.37228 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.37228	−0.613703	−0.306851	−	0.951757i	$-0.599275\pi$
$5$	−1.37228	−0.613703	−0.306851	+	0.951757i	$0.599275\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.37228	0.433953
$11$	−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
$13$	1.37228	0.380602	0.190301	−	0.981726i	$-0.439054\pi$
$13$	1.37228	0.380602	0.190301	+	0.981726i	$0.439054\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.74456	1.15073	0.575363	−	0.817898i	$-0.304861\pi$
$17$	4.74456	1.15073	0.575363	+	0.817898i	$0.304861\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.37228	−0.306851
$21$	0	0
$22$	4.00000	0.852803
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.11684	−0.623369
$26$	−1.37228	−0.269127
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−9.37228	−1.74039	−0.870194	−	0.492708i	$-0.836007\pi$
$29$	−9.37228	−1.74039	−0.870194	+	0.492708i	$0.836007\pi$
$30$	0	0
$31$	6.74456	1.21136	0.605680	−	0.795709i	$-0.292901\pi$
$31$	6.74456	1.21136	0.605680	+	0.795709i	$0.292901\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.74456	−0.813686
$35$	1.37228	0.231958
$36$	0	0
$37$	2.62772	0.431994	0.215997	−	0.976394i	$-0.430700\pi$
$37$	2.62772	0.431994	0.215997	+	0.976394i	$0.430700\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	1.37228	0.216977
$41$	8.11684	1.26764	0.633819	−	0.773481i	$-0.281486\pi$
$41$	8.11684	1.26764	0.633819	+	0.773481i	$0.281486\pi$
$42$	0	0
$43$	6.11684	0.932810	0.466405	−	0.884571i	$-0.345549\pi$
$43$	6.11684	0.932810	0.466405	+	0.884571i	$0.345549\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−4.62772	−0.675022	−0.337511	−	0.941322i	$-0.609585\pi$
$47$	−4.62772	−0.675022	−0.337511	+	0.941322i	$0.609585\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.11684	0.440788
$51$	0	0
$52$	1.37228	0.190301
$53$	−4.74456	−0.651716	−0.325858	−	0.945419i	$-0.605653\pi$
$53$	−4.74456	−0.651716	−0.325858	+	0.945419i	$0.605653\pi$
$54$	0	0
$55$	5.48913	0.740154
$56$	1.00000	0.133631
$57$	0	0
$58$	9.37228	1.23064
$59$	2.74456	0.357312	0.178656	−	0.983912i	$-0.442825\pi$
$59$	2.74456	0.357312	0.178656	+	0.983912i	$0.442825\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−6.74456	−0.856560
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.88316	−0.233577
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	4.74456	0.575363
$69$	0	0
$70$	−1.37228	−0.164019
$71$	−14.7446	−1.74986	−0.874929	−	0.484252i	$-0.839092\pi$
$71$	−14.7446	−1.74986	−0.874929	+	0.484252i	$0.839092\pi$
$72$	0	0
$73$	−12.7446	−1.49164	−0.745819	−	0.666149i	$-0.767942\pi$
$73$	−12.7446	−1.49164	−0.745819	+	0.666149i	$0.767942\pi$
$74$	−2.62772	−0.305466
$75$	0	0
$76$	4.00000	0.458831
$77$	4.00000	0.455842
$78$	0	0
$79$	−13.4891	−1.51765	−0.758823	−	0.651297i	$-0.774225\pi$
$79$	−13.4891	−1.51765	−0.758823	+	0.651297i	$0.774225\pi$
$80$	−1.37228	−0.153426
$81$	0	0
$82$	−8.11684	−0.896355
$83$	−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
$85$	−6.51087	−0.706204
$86$	−6.11684	−0.659596
$87$	0	0
$88$	4.00000	0.426401
$89$	−7.48913	−0.793846	−0.396923	−	0.917852i	$-0.629922\pi$
$89$	−7.48913	−0.793846	−0.396923	+	0.917852i	$0.629922\pi$
$90$	0	0
$91$	−1.37228	−0.143854
$92$	1.00000	0.104257
$93$	0	0
$94$	4.62772	0.477313
$95$	−5.48913	−0.563172
$96$	0	0
$97$	−9.37228	−0.951611	−0.475805	−	0.879551i	$-0.657843\pi$
$97$	−9.37228	−0.951611	−0.475805	+	0.879551i	$0.657843\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−3.11684	−0.311684
$101$	15.4891	1.54123	0.770613	−	0.637304i	$-0.219950\pi$
$101$	15.4891	1.54123	0.770613	+	0.637304i	$0.219950\pi$
$102$	0	0
$103$	−10.1168	−0.996842	−0.498421	−	0.866935i	$-0.666087\pi$
$103$	−10.1168	−0.996842	−0.498421	+	0.866935i	$0.666087\pi$
$104$	−1.37228	−0.134563
$105$	0	0
$106$	4.74456	0.460833
$107$	−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
$109$	8.11684	0.777453	0.388726	−	0.921353i	$-0.372915\pi$
$109$	8.11684	0.777453	0.388726	+	0.921353i	$0.372915\pi$
$110$	−5.48913	−0.523368
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	16.1168	1.51615	0.758073	−	0.652170i	$-0.226141\pi$
$113$	16.1168	1.51615	0.758073	+	0.652170i	$0.226141\pi$
$114$	0	0
$115$	−1.37228	−0.127966
$116$	−9.37228	−0.870194
$117$	0	0
$118$	−2.74456	−0.252657
$119$	−4.74456	−0.434933
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	0	0
$124$	6.74456	0.605680
$125$	11.1386	0.996266
$126$	0	0
$127$	−11.3723	−1.00913	−0.504563	−	0.863375i	$-0.668347\pi$
$127$	−11.3723	−1.00913	−0.504563	+	0.863375i	$0.668347\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.88316	0.165164
$131$	−18.7446	−1.63772	−0.818860	−	0.573993i	$-0.805394\pi$
$131$	−18.7446	−1.63772	−0.818860	+	0.573993i	$0.805394\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	4.00000	0.345547
$135$	0	0
$136$	−4.74456	−0.406843
$137$	8.11684	0.693469	0.346734	−	0.937963i	$-0.387291\pi$
$137$	8.11684	0.693469	0.346734	+	0.937963i	$0.387291\pi$
$138$	0	0
$139$	8.62772	0.731794	0.365897	−	0.930655i	$-0.380762\pi$
$139$	8.62772	0.731794	0.365897	+	0.930655i	$0.380762\pi$
$140$	1.37228	0.115979
$141$	0	0
$142$	14.7446	1.23734
$143$	−5.48913	−0.459024
$144$	0	0
$145$	12.8614	1.06808
$146$	12.7446	1.05475
$147$	0	0
$148$	2.62772	0.215997
$149$	−18.2337	−1.49376	−0.746881	−	0.664958i	$-0.768450\pi$
$149$	−18.2337	−1.49376	−0.746881	+	0.664958i	$0.768450\pi$
$150$	0	0
$151$	−3.37228	−0.274432	−0.137216	−	0.990541i	$-0.543816\pi$
$151$	−3.37228	−0.274432	−0.137216	+	0.990541i	$0.543816\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	−4.00000	−0.322329
$155$	−9.25544	−0.743415
$156$	0	0
$157$	−11.2554	−0.898282	−0.449141	−	0.893461i	$-0.648270\pi$
$157$	−11.2554	−0.898282	−0.449141	+	0.893461i	$0.648270\pi$
$158$	13.4891	1.07314
$159$	0	0
$160$	1.37228	0.108488
$161$	−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	$0.449930\pi$
$164$	8.11684	0.633819
$165$	0	0
$166$	4.00000	0.310460
$167$	5.48913	0.424761	0.212381	−	0.977187i	$-0.431878\pi$
$167$	5.48913	0.424761	0.212381	+	0.977187i	$0.431878\pi$
$168$	0	0
$169$	−11.1168	−0.855142
$170$	6.51087	0.499361
$171$	0	0
$172$	6.11684	0.466405
$173$	7.48913	0.569388	0.284694	−	0.958618i	$-0.408108\pi$
$173$	7.48913	0.569388	0.284694	+	0.958618i	$0.408108\pi$
$174$	0	0
$175$	3.11684	0.235611
$176$	−4.00000	−0.301511
$177$	0	0
$178$	7.48913	0.561334
$179$	20.8614	1.55925	0.779627	−	0.626244i	$-0.215408\pi$
$179$	20.8614	1.55925	0.779627	+	0.626244i	$0.215408\pi$
$180$	0	0
$181$	−20.9783	−1.55930	−0.779651	−	0.626215i	$-0.784603\pi$
$181$	−20.9783	−1.55930	−0.779651	+	0.626215i	$0.784603\pi$
$182$	1.37228	0.101720
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−3.60597	−0.265116
$186$	0	0
$187$	−18.9783	−1.38783
$188$	−4.62772	−0.337511
$189$	0	0
$190$	5.48913	0.398223
$191$	−2.51087	−0.181681	−0.0908403	−	0.995865i	$-0.528955\pi$
$191$	−2.51087	−0.181681	−0.0908403	+	0.995865i	$0.528955\pi$
$192$	0	0
$193$	−0.116844	−0.00841061	−0.00420531	−	0.999991i	$-0.501339\pi$
$193$	−0.116844	−0.00841061	−0.00420531	+	0.999991i	$0.501339\pi$
$194$	9.37228	0.672891
$195$	0	0
$196$	1.00000	0.0714286
$197$	21.3723	1.52271	0.761356	−	0.648334i	$-0.224534\pi$
$197$	21.3723	1.52271	0.761356	+	0.648334i	$0.224534\pi$
$198$	0	0
$199$	−16.8614	−1.19527	−0.597637	−	0.801767i	$-0.703893\pi$
$199$	−16.8614	−1.19527	−0.597637	+	0.801767i	$0.703893\pi$
$200$	3.11684	0.220394
$201$	0	0
$202$	−15.4891	−1.08981
$203$	9.37228	0.657805
$204$	0	0
$205$	−11.1386	−0.777953
$206$	10.1168	0.704874
$207$	0	0
$208$	1.37228	0.0951506
$209$	−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	$0.635539\pi$
$212$	−4.74456	−0.325858
$213$	0	0
$214$	4.00000	0.273434
$215$	−8.39403	−0.572468
$216$	0	0
$217$	−6.74456	−0.457851
$218$	−8.11684	−0.549742
$219$	0	0
$220$	5.48913	0.370077
$221$	6.51087	0.437969
$222$	0	0
$223$	−9.25544	−0.619790	−0.309895	−	0.950771i	$-0.600294\pi$
$223$	−9.25544	−0.619790	−0.309895	+	0.950771i	$0.600294\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−16.1168	−1.07208
$227$	−6.11684	−0.405989	−0.202995	−	0.979180i	$-0.565067\pi$
$227$	−6.11684	−0.405989	−0.202995	+	0.979180i	$0.565067\pi$
$228$	0	0
$229$	−0.744563	−0.0492021	−0.0246010	−	0.999697i	$-0.507832\pi$
$229$	−0.744563	−0.0492021	−0.0246010	+	0.999697i	$0.507832\pi$
$230$	1.37228	0.0904856
$231$	0	0
$232$	9.37228	0.615320
$233$	−23.4891	−1.53882	−0.769412	−	0.638753i	$-0.779451\pi$
$233$	−23.4891	−1.53882	−0.769412	+	0.638753i	$0.779451\pi$
$234$	0	0
$235$	6.35053	0.414263
$236$	2.74456	0.178656
$237$	0	0
$238$	4.74456	0.307544
$239$	−13.4891	−0.872539	−0.436269	−	0.899816i	$-0.643701\pi$
$239$	−13.4891	−0.872539	−0.436269	+	0.899816i	$0.643701\pi$
$240$	0	0
$241$	−2.62772	−0.169266	−0.0846331	−	0.996412i	$-0.526972\pi$
$241$	−2.62772	−0.169266	−0.0846331	+	0.996412i	$0.526972\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−2.00000	−0.128037
$245$	−1.37228	−0.0876718
$246$	0	0
$247$	5.48913	0.349265
$248$	−6.74456	−0.428280
$249$	0	0
$250$	−11.1386	−0.704467
$251$	1.88316	0.118864	0.0594319	−	0.998232i	$-0.481071\pi$
$251$	1.88316	0.118864	0.0594319	+	0.998232i	$0.481071\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	11.3723	0.713560
$255$	0	0
$256$	1.00000	0.0625000
$257$	−28.9783	−1.80761	−0.903807	−	0.427941i	$-0.859239\pi$
$257$	−28.9783	−1.80761	−0.903807	+	0.427941i	$0.859239\pi$
$258$	0	0
$259$	−2.62772	−0.163278
$260$	−1.88316	−0.116788
$261$	0	0
$262$	18.7446	1.15804
$263$	−10.1168	−0.623831	−0.311916	−	0.950110i	$-0.600971\pi$
$263$	−10.1168	−0.623831	−0.311916	+	0.950110i	$0.600971\pi$
$264$	0	0
$265$	6.51087	0.399960
$266$	4.00000	0.245256
$267$	0	0
$268$	−4.00000	−0.244339
$269$	20.9783	1.27907	0.639533	−	0.768763i	$-0.279128\pi$
$269$	20.9783	1.27907	0.639533	+	0.768763i	$0.279128\pi$
$270$	0	0
$271$	−26.9783	−1.63881	−0.819406	−	0.573214i	$-0.805697\pi$
$271$	−26.9783	−1.63881	−0.819406	+	0.573214i	$0.805697\pi$
$272$	4.74456	0.287681
$273$	0	0
$274$	−8.11684	−0.490356
$275$	12.4674	0.751811
$276$	0	0
$277$	28.7446	1.72709	0.863547	−	0.504269i	$-0.168238\pi$
$277$	28.7446	1.72709	0.863547	+	0.504269i	$0.168238\pi$
$278$	−8.62772	−0.517456
$279$	0	0
$280$	−1.37228	−0.0820095
$281$	17.3723	1.03634	0.518172	−	0.855277i	$-0.326613\pi$
$281$	17.3723	1.03634	0.518172	+	0.855277i	$0.326613\pi$
$282$	0	0
$283$	−13.2554	−0.787954	−0.393977	−	0.919120i	$-0.628901\pi$
$283$	−13.2554	−0.787954	−0.393977	+	0.919120i	$0.628901\pi$
$284$	−14.7446	−0.874929
$285$	0	0
$286$	5.48913	0.324579
$287$	−8.11684	−0.479122
$288$	0	0
$289$	5.51087	0.324169
$290$	−12.8614	−0.755248
$291$	0	0
$292$	−12.7446	−0.745819
$293$	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
$295$	−3.76631	−0.219283
$296$	−2.62772	−0.152733
$297$	0	0
$298$	18.2337	1.05625
$299$	1.37228	0.0793611
$300$	0	0
$301$	−6.11684	−0.352569
$302$	3.37228	0.194053
$303$	0	0
$304$	4.00000	0.229416
$305$	2.74456	0.157153
$306$	0	0
$307$	7.37228	0.420758	0.210379	−	0.977620i	$-0.432530\pi$
$307$	7.37228	0.420758	0.210379	+	0.977620i	$0.432530\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	9.25544	0.525674
$311$	−5.48913	−0.311260	−0.155630	−	0.987815i	$-0.549741\pi$
$311$	−5.48913	−0.311260	−0.155630	+	0.987815i	$0.549741\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	11.2554	0.635181
$315$	0	0
$316$	−13.4891	−0.758823
$317$	−16.1168	−0.905212	−0.452606	−	0.891711i	$-0.649506\pi$
$317$	−16.1168	−0.905212	−0.452606	+	0.891711i	$0.649506\pi$
$318$	0	0
$319$	37.4891	2.09899
$320$	−1.37228	−0.0767129
$321$	0	0
$322$	1.00000	0.0557278
$323$	18.9783	1.05598
$324$	0	0
$325$	−4.27719	−0.237256
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−8.11684	−0.448178
$329$	4.62772	0.255134
$330$	0	0
$331$	9.48913	0.521569	0.260785	−	0.965397i	$-0.416019\pi$
$331$	9.48913	0.521569	0.260785	+	0.965397i	$0.416019\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	−5.48913	−0.300352
$335$	5.48913	0.299903
$336$	0	0
$337$	22.2337	1.21115	0.605573	−	0.795790i	$-0.292944\pi$
$337$	22.2337	1.21115	0.605573	+	0.795790i	$0.292944\pi$
$338$	11.1168	0.604677
$339$	0	0
$340$	−6.51087	−0.353102
$341$	−26.9783	−1.46095
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−6.11684	−0.329798
$345$	0	0
$346$	−7.48913	−0.402618
$347$	−16.6277	−0.892623	−0.446311	−	0.894878i	$-0.647263\pi$
$347$	−16.6277	−0.892623	−0.446311	+	0.894878i	$0.647263\pi$
$348$	0	0
$349$	35.4891	1.89969	0.949845	−	0.312722i	$-0.101241\pi$
$349$	35.4891	1.89969	0.949845	+	0.312722i	$0.101241\pi$
$350$	−3.11684	−0.166602
$351$	0	0
$352$	4.00000	0.213201
$353$	−36.1168	−1.92231	−0.961153	−	0.276017i	$-0.910985\pi$
$353$	−36.1168	−1.92231	−0.961153	+	0.276017i	$0.910985\pi$
$354$	0	0
$355$	20.2337	1.07389
$356$	−7.48913	−0.396923
$357$	0	0
$358$	−20.8614	−1.10256
$359$	−19.3723	−1.02243	−0.511215	−	0.859453i	$-0.670804\pi$
$359$	−19.3723	−1.02243	−0.511215	+	0.859453i	$0.670804\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	20.9783	1.10259
$363$	0	0
$364$	−1.37228	−0.0719271
$365$	17.4891	0.915423
$366$	0	0
$367$	−11.3723	−0.593628	−0.296814	−	0.954935i	$-0.595924\pi$
$367$	−11.3723	−0.593628	−0.296814	+	0.954935i	$0.595924\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	3.60597	0.187465
$371$	4.74456	0.246325
$372$	0	0
$373$	−31.4891	−1.63045	−0.815223	−	0.579148i	$-0.803385\pi$
$373$	−31.4891	−1.63045	−0.815223	+	0.579148i	$0.803385\pi$
$374$	18.9783	0.981342
$375$	0	0
$376$	4.62772	0.238656
$377$	−12.8614	−0.662396
$378$	0	0
$379$	15.3723	0.789621	0.394811	−	0.918763i	$-0.370810\pi$
$379$	15.3723	0.789621	0.394811	+	0.918763i	$0.370810\pi$
$380$	−5.48913	−0.281586
$381$	0	0
$382$	2.51087	0.128468
$383$	−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
$385$	−5.48913	−0.279752
$386$	0.116844	0.00594720
$387$	0	0
$388$	−9.37228	−0.475805
$389$	−11.4891	−0.582522	−0.291261	−	0.956644i	$-0.594075\pi$
$389$	−11.4891	−0.582522	−0.291261	+	0.956644i	$0.594075\pi$
$390$	0	0
$391$	4.74456	0.239943
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−21.3723	−1.07672
$395$	18.5109	0.931383
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	16.8614	0.845186
$399$	0	0
$400$	−3.11684	−0.155842
$401$	24.9783	1.24735	0.623677	−	0.781682i	$-0.285638\pi$
$401$	24.9783	1.24735	0.623677	+	0.781682i	$0.285638\pi$
$402$	0	0
$403$	9.25544	0.461046
$404$	15.4891	0.770613
$405$	0	0
$406$	−9.37228	−0.465139
$407$	−10.5109	−0.521005
$408$	0	0
$409$	0.744563	0.0368163	0.0184081	−	0.999831i	$-0.494140\pi$
$409$	0.744563	0.0368163	0.0184081	+	0.999831i	$0.494140\pi$
$410$	11.1386	0.550096
$411$	0	0
$412$	−10.1168	−0.498421
$413$	−2.74456	−0.135051
$414$	0	0
$415$	5.48913	0.269451
$416$	−1.37228	−0.0672816
$417$	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
$421$	−26.8614	−1.30914	−0.654572	−	0.755999i	$-0.727151\pi$
$421$	−26.8614	−1.30914	−0.654572	+	0.755999i	$0.727151\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	4.74456	0.230416
$425$	−14.7881	−0.717326
$426$	0	0
$427$	2.00000	0.0967868
$428$	−4.00000	−0.193347
$429$	0	0
$430$	8.39403	0.404796
$431$	8.86141	0.426839	0.213419	−	0.976961i	$-0.431540\pi$
$431$	8.86141	0.426839	0.213419	+	0.976961i	$0.431540\pi$
$432$	0	0
$433$	10.8614	0.521966	0.260983	−	0.965343i	$-0.415953\pi$
$433$	10.8614	0.521966	0.260983	+	0.965343i	$0.415953\pi$
$434$	6.74456	0.323749
$435$	0	0
$436$	8.11684	0.388726
$437$	4.00000	0.191346
$438$	0	0
$439$	30.7446	1.46736	0.733679	−	0.679496i	$-0.237802\pi$
$439$	30.7446	1.46736	0.733679	+	0.679496i	$0.237802\pi$
$440$	−5.48913	−0.261684
$441$	0	0
$442$	−6.51087	−0.309691
$443$	8.62772	0.409915	0.204958	−	0.978771i	$-0.434294\pi$
$443$	8.62772	0.409915	0.204958	+	0.978771i	$0.434294\pi$
$444$	0	0
$445$	10.2772	0.487185
$446$	9.25544	0.438258
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−28.9783	−1.36757	−0.683784	−	0.729684i	$-0.739667\pi$
$449$	−28.9783	−1.36757	−0.683784	+	0.729684i	$0.739667\pi$
$450$	0	0
$451$	−32.4674	−1.52883
$452$	16.1168	0.758073
$453$	0	0
$454$	6.11684	0.287078
$455$	1.88316	0.0882837
$456$	0	0
$457$	−10.2337	−0.478712	−0.239356	−	0.970932i	$-0.576936\pi$
$457$	−10.2337	−0.478712	−0.239356	+	0.970932i	$0.576936\pi$
$458$	0.744563	0.0347911
$459$	0	0
$460$	−1.37228	−0.0639829
$461$	12.5109	0.582690	0.291345	−	0.956618i	$-0.405897\pi$
$461$	12.5109	0.582690	0.291345	+	0.956618i	$0.405897\pi$
$462$	0	0
$463$	−2.11684	−0.0983781	−0.0491890	−	0.998789i	$-0.515664\pi$
$463$	−2.11684	−0.0983781	−0.0491890	+	0.998789i	$0.515664\pi$
$464$	−9.37228	−0.435097
$465$	0	0
$466$	23.4891	1.08811
$467$	−28.8614	−1.33555	−0.667773	−	0.744365i	$-0.732752\pi$
$467$	−28.8614	−1.33555	−0.667773	+	0.744365i	$0.732752\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	−6.35053	−0.292928
$471$	0	0
$472$	−2.74456	−0.126329
$473$	−24.4674	−1.12501
$474$	0	0
$475$	−12.4674	−0.572042
$476$	−4.74456	−0.217467
$477$	0	0
$478$	13.4891	0.616978
$479$	6.74456	0.308167	0.154083	−	0.988058i	$-0.450758\pi$
$479$	6.74456	0.308167	0.154083	+	0.988058i	$0.450758\pi$
$480$	0	0
$481$	3.60597	0.164418
$482$	2.62772	0.119689
$483$	0	0
$484$	5.00000	0.227273
$485$	12.8614	0.584006
$486$	0	0
$487$	5.88316	0.266591	0.133296	−	0.991076i	$-0.457444\pi$
$487$	5.88316	0.266591	0.133296	+	0.991076i	$0.457444\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	1.37228	0.0619934
$491$	30.9783	1.39803	0.699014	−	0.715108i	$-0.253622\pi$
$491$	30.9783	1.39803	0.699014	+	0.715108i	$0.253622\pi$
$492$	0	0
$493$	−44.4674	−2.00271
$494$	−5.48913	−0.246967
$495$	0	0
$496$	6.74456	0.302840
$497$	14.7446	0.661384
$498$	0	0
$499$	21.7228	0.972447	0.486223	−	0.873835i	$-0.338374\pi$
$499$	21.7228	0.972447	0.486223	+	0.873835i	$0.338374\pi$
$500$	11.1386	0.498133
$501$	0	0
$502$	−1.88316	−0.0840494
$503$	1.25544	0.0559772	0.0279886	−	0.999608i	$-0.491090\pi$
$503$	1.25544	0.0559772	0.0279886	+	0.999608i	$0.491090\pi$
$504$	0	0
$505$	−21.2554	−0.945855
$506$	4.00000	0.177822
$507$	0	0
$508$	−11.3723	−0.504563
$509$	−26.2337	−1.16279	−0.581394	−	0.813622i	$-0.697492\pi$
$509$	−26.2337	−1.16279	−0.581394	+	0.813622i	$0.697492\pi$
$510$	0	0
$511$	12.7446	0.563786
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	28.9783	1.27818
$515$	13.8832	0.611765
$516$	0	0
$517$	18.5109	0.814107
$518$	2.62772	0.115455
$519$	0	0
$520$	1.88316	0.0825819
$521$	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
$523$	−40.2337	−1.75930	−0.879648	−	0.475625i	$-0.842222\pi$
$523$	−40.2337	−1.75930	−0.879648	+	0.475625i	$0.842222\pi$
$524$	−18.7446	−0.818860
$525$	0	0
$526$	10.1168	0.441115
$527$	32.0000	1.39394
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.51087	−0.282814
$531$	0	0
$532$	−4.00000	−0.173422
$533$	11.1386	0.482466
$534$	0	0
$535$	5.48913	0.237316
$536$	4.00000	0.172774
$537$	0	0
$538$	−20.9783	−0.904437
$539$	−4.00000	−0.172292
$540$	0	0
$541$	18.2337	0.783927	0.391964	−	0.919981i	$-0.371796\pi$
$541$	18.2337	0.783927	0.391964	+	0.919981i	$0.371796\pi$
$542$	26.9783	1.15882
$543$	0	0
$544$	−4.74456	−0.203421
$545$	−11.1386	−0.477125
$546$	0	0
$547$	−21.2554	−0.908817	−0.454408	−	0.890793i	$-0.650149\pi$
$547$	−21.2554	−0.908817	−0.454408	+	0.890793i	$0.650149\pi$
$548$	8.11684	0.346734
$549$	0	0
$550$	−12.4674	−0.531611
$551$	−37.4891	−1.59709
$552$	0	0
$553$	13.4891	0.573616
$554$	−28.7446	−1.22124
$555$	0	0
$556$	8.62772	0.365897
$557$	14.2337	0.603101	0.301550	−	0.953450i	$-0.402496\pi$
$557$	14.2337	0.603101	0.301550	+	0.953450i	$0.402496\pi$
$558$	0	0
$559$	8.39403	0.355030
$560$	1.37228	0.0579895
$561$	0	0
$562$	−17.3723	−0.732805
$563$	−24.6277	−1.03793	−0.518967	−	0.854794i	$-0.673683\pi$
$563$	−24.6277	−1.03793	−0.518967	+	0.854794i	$0.673683\pi$
$564$	0	0
$565$	−22.1168	−0.930463
$566$	13.2554	0.557168
$567$	0	0
$568$	14.7446	0.618668
$569$	19.8832	0.833545	0.416773	−	0.909011i	$-0.363161\pi$
$569$	19.8832	0.833545	0.416773	+	0.909011i	$0.363161\pi$
$570$	0	0
$571$	−22.9783	−0.961610	−0.480805	−	0.876828i	$-0.659655\pi$
$571$	−22.9783	−0.961610	−0.480805	+	0.876828i	$0.659655\pi$
$572$	−5.48913	−0.229512
$573$	0	0
$574$	8.11684	0.338791
$575$	−3.11684	−0.129981
$576$	0	0
$577$	42.4674	1.76794	0.883970	−	0.467544i	$-0.154861\pi$
$577$	42.4674	1.76794	0.883970	+	0.467544i	$0.154861\pi$
$578$	−5.51087	−0.229222
$579$	0	0
$580$	12.8614	0.534041
$581$	4.00000	0.165948
$582$	0	0
$583$	18.9783	0.785999
$584$	12.7446	0.527374
$585$	0	0
$586$	−10.0000	−0.413096
$587$	−6.51087	−0.268733	−0.134366	−	0.990932i	$-0.542900\pi$
$587$	−6.51087	−0.268733	−0.134366	+	0.990932i	$0.542900\pi$
$588$	0	0
$589$	26.9783	1.11162
$590$	3.76631	0.155057
$591$	0	0
$592$	2.62772	0.107999
$593$	2.62772	0.107907	0.0539537	−	0.998543i	$-0.482818\pi$
$593$	2.62772	0.107907	0.0539537	+	0.998543i	$0.482818\pi$
$594$	0	0
$595$	6.51087	0.266920
$596$	−18.2337	−0.746881
$597$	0	0
$598$	−1.37228	−0.0561168
$599$	48.4674	1.98032	0.990162	−	0.139928i	$-0.0446872\pi$
$599$	48.4674	1.98032	0.990162	+	0.139928i	$0.0446872\pi$
$600$	0	0
$601$	30.2337	1.23326	0.616629	−	0.787254i	$-0.288498\pi$
$601$	30.2337	1.23326	0.616629	+	0.787254i	$0.288498\pi$
$602$	6.11684	0.249304
$603$	0	0
$604$	−3.37228	−0.137216
$605$	−6.86141	−0.278956
$606$	0	0
$607$	22.7446	0.923173	0.461587	−	0.887095i	$-0.347280\pi$
$607$	22.7446	0.923173	0.461587	+	0.887095i	$0.347280\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−2.74456	−0.111124
$611$	−6.35053	−0.256915
$612$	0	0
$613$	27.8832	1.12619	0.563095	−	0.826392i	$-0.309610\pi$
$613$	27.8832	1.12619	0.563095	+	0.826392i	$0.309610\pi$
$614$	−7.37228	−0.297521
$615$	0	0
$616$	−4.00000	−0.161165
$617$	35.4891	1.42874	0.714369	−	0.699769i	$-0.246714\pi$
$617$	35.4891	1.42874	0.714369	+	0.699769i	$0.246714\pi$
$618$	0	0
$619$	34.7446	1.39650	0.698251	−	0.715853i	$-0.253962\pi$
$619$	34.7446	1.39650	0.698251	+	0.715853i	$0.253962\pi$
$620$	−9.25544	−0.371707
$621$	0	0
$622$	5.48913	0.220094
$623$	7.48913	0.300045
$624$	0	0
$625$	0.298936	0.0119574
$626$	6.00000	0.239808
$627$	0	0
$628$	−11.2554	−0.449141
$629$	12.4674	0.497107
$630$	0	0
$631$	−34.9783	−1.39246	−0.696231	−	0.717818i	$-0.745141\pi$
$631$	−34.9783	−1.39246	−0.696231	+	0.717818i	$0.745141\pi$
$632$	13.4891	0.536569
$633$	0	0
$634$	16.1168	0.640082
$635$	15.6060	0.619304
$636$	0	0
$637$	1.37228	0.0543718
$638$	−37.4891	−1.48421
$639$	0	0
$640$	1.37228	0.0542442
$641$	20.3505	0.803798	0.401899	−	0.915684i	$-0.368350\pi$
$641$	20.3505	0.803798	0.401899	+	0.915684i	$0.368350\pi$
$642$	0	0
$643$	−7.76631	−0.306273	−0.153137	−	0.988205i	$-0.548937\pi$
$643$	−7.76631	−0.306273	−0.153137	+	0.988205i	$0.548937\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	−18.9783	−0.746689
$647$	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
$649$	−10.9783	−0.430934
$650$	4.27719	0.167765
$651$	0	0
$652$	4.00000	0.156652
$653$	31.0951	1.21685	0.608423	−	0.793613i	$-0.291803\pi$
$653$	31.0951	1.21685	0.608423	+	0.793613i	$0.291803\pi$
$654$	0	0
$655$	25.7228	1.00507
$656$	8.11684	0.316910
$657$	0	0
$658$	−4.62772	−0.180407
$659$	−22.9783	−0.895106	−0.447553	−	0.894258i	$-0.647704\pi$
$659$	−22.9783	−0.895106	−0.447553	+	0.894258i	$0.647704\pi$
$660$	0	0
$661$	3.48913	0.135711	0.0678556	−	0.997695i	$-0.478384\pi$
$661$	3.48913	0.135711	0.0678556	+	0.997695i	$0.478384\pi$
$662$	−9.48913	−0.368805
$663$	0	0
$664$	4.00000	0.155230
$665$	5.48913	0.212859
$666$	0	0
$667$	−9.37228	−0.362896
$668$	5.48913	0.212381
$669$	0	0
$670$	−5.48913	−0.212063
$671$	8.00000	0.308837
$672$	0	0
$673$	−13.6060	−0.524472	−0.262236	−	0.965004i	$-0.584460\pi$
$673$	−13.6060	−0.524472	−0.262236	+	0.965004i	$0.584460\pi$
$674$	−22.2337	−0.856410
$675$	0	0
$676$	−11.1168	−0.427571
$677$	−32.9783	−1.26746	−0.633729	−	0.773555i	$-0.718476\pi$
$677$	−32.9783	−1.26746	−0.633729	+	0.773555i	$0.718476\pi$
$678$	0	0
$679$	9.37228	0.359675
$680$	6.51087	0.249681
$681$	0	0
$682$	26.9783	1.03305
$683$	−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
$685$	−11.1386	−0.425584
$686$	1.00000	0.0381802
$687$	0	0
$688$	6.11684	0.233202
$689$	−6.51087	−0.248045
$690$	0	0
$691$	31.8397	1.21124	0.605619	−	0.795755i	$-0.292926\pi$
$691$	31.8397	1.21124	0.605619	+	0.795755i	$0.292926\pi$
$692$	7.48913	0.284694
$693$	0	0
$694$	16.6277	0.631180
$695$	−11.8397	−0.449104
$696$	0	0
$697$	38.5109	1.45870
$698$	−35.4891	−1.34328
$699$	0	0
$700$	3.11684	0.117806
$701$	5.76631	0.217791	0.108895	−	0.994053i	$-0.465269\pi$
$701$	5.76631	0.217791	0.108895	+	0.994053i	$0.465269\pi$
$702$	0	0
$703$	10.5109	0.396425
$704$	−4.00000	−0.150756
$705$	0	0
$706$	36.1168	1.35928
$707$	−15.4891	−0.582529
$708$	0	0
$709$	−4.51087	−0.169409	−0.0847047	−	0.996406i	$-0.526995\pi$
$709$	−4.51087	−0.169409	−0.0847047	+	0.996406i	$0.526995\pi$
$710$	−20.2337	−0.759357
$711$	0	0
$712$	7.48913	0.280667
$713$	6.74456	0.252586
$714$	0	0
$715$	7.53262	0.281704
$716$	20.8614	0.779627
$717$	0	0
$718$	19.3723	0.722967
$719$	11.3723	0.424115	0.212057	−	0.977257i	$-0.431984\pi$
$719$	11.3723	0.424115	0.212057	+	0.977257i	$0.431984\pi$
$720$	0	0
$721$	10.1168	0.376771
$722$	3.00000	0.111648
$723$	0	0
$724$	−20.9783	−0.779651
$725$	29.2119	1.08490
$726$	0	0
$727$	10.5109	0.389827	0.194913	−	0.980820i	$-0.437557\pi$
$727$	10.5109	0.389827	0.194913	+	0.980820i	$0.437557\pi$
$728$	1.37228	0.0508601
$729$	0	0
$730$	−17.4891	−0.647302
$731$	29.0217	1.07341
$732$	0	0
$733$	18.2337	0.673477	0.336738	−	0.941598i	$-0.390676\pi$
$733$	18.2337	0.673477	0.336738	+	0.941598i	$0.390676\pi$
$734$	11.3723	0.419759
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	16.0000	0.589368
$738$	0	0
$739$	−5.25544	−0.193324	−0.0966622	−	0.995317i	$-0.530817\pi$
$739$	−5.25544	−0.193324	−0.0966622	+	0.995317i	$0.530817\pi$
$740$	−3.60597	−0.132558
$741$	0	0
$742$	−4.74456	−0.174178
$743$	−45.9565	−1.68598	−0.842990	−	0.537929i	$-0.819207\pi$
$743$	−45.9565	−1.68598	−0.842990	+	0.537929i	$0.819207\pi$
$744$	0	0
$745$	25.0217	0.916726
$746$	31.4891	1.15290
$747$	0	0
$748$	−18.9783	−0.693914
$749$	4.00000	0.146157
$750$	0	0
$751$	13.4891	0.492225	0.246113	−	0.969241i	$-0.420847\pi$
$751$	13.4891	0.492225	0.246113	+	0.969241i	$0.420847\pi$
$752$	−4.62772	−0.168756
$753$	0	0
$754$	12.8614	0.468385
$755$	4.62772	0.168420
$756$	0	0
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	−15.3723	−0.558346
$759$	0	0
$760$	5.48913	0.199112
$761$	−4.97825	−0.180461	−0.0902307	−	0.995921i	$-0.528760\pi$
$761$	−4.97825	−0.180461	−0.0902307	+	0.995921i	$0.528760\pi$
$762$	0	0
$763$	−8.11684	−0.293849
$764$	−2.51087	−0.0908403
$765$	0	0
$766$	16.0000	0.578103
$767$	3.76631	0.135994
$768$	0	0
$769$	−27.8832	−1.00549	−0.502746	−	0.864434i	$-0.667677\pi$
$769$	−27.8832	−1.00549	−0.502746	+	0.864434i	$0.667677\pi$
$770$	5.48913	0.197814
$771$	0	0
$772$	−0.116844	−0.00420531
$773$	41.6060	1.49646	0.748231	−	0.663438i	$-0.230903\pi$
$773$	41.6060	1.49646	0.748231	+	0.663438i	$0.230903\pi$
$774$	0	0
$775$	−21.0217	−0.755124
$776$	9.37228	0.336445
$777$	0	0
$778$	11.4891	0.411905
$779$	32.4674	1.16326
$780$	0	0
$781$	58.9783	2.11041
$782$	−4.74456	−0.169665
$783$	0	0
$784$	1.00000	0.0357143
$785$	15.4456	0.551278
$786$	0	0
$787$	6.51087	0.232088	0.116044	−	0.993244i	$-0.462979\pi$
$787$	6.51087	0.232088	0.116044	+	0.993244i	$0.462979\pi$
$788$	21.3723	0.761356
$789$	0	0
$790$	−18.5109	−0.658587
$791$	−16.1168	−0.573049
$792$	0	0
$793$	−2.74456	−0.0974623
$794$	10.0000	0.354887
$795$	0	0
$796$	−16.8614	−0.597637
$797$	−6.86141	−0.243043	−0.121522	−	0.992589i	$-0.538777\pi$
$797$	−6.86141	−0.243043	−0.121522	+	0.992589i	$0.538777\pi$
$798$	0	0
$799$	−21.9565	−0.776765
$800$	3.11684	0.110197
$801$	0	0
$802$	−24.9783	−0.882013
$803$	50.9783	1.79898
$804$	0	0
$805$	1.37228	0.0483666
$806$	−9.25544	−0.326009
$807$	0	0
$808$	−15.4891	−0.544906
$809$	12.7446	0.448075	0.224037	−	0.974581i	$-0.428076\pi$
$809$	12.7446	0.448075	0.224037	+	0.974581i	$0.428076\pi$
$810$	0	0
$811$	19.6060	0.688459	0.344229	−	0.938886i	$-0.388140\pi$
$811$	19.6060	0.688459	0.344229	+	0.938886i	$0.388140\pi$
$812$	9.37228	0.328903
$813$	0	0
$814$	10.5109	0.368406
$815$	−5.48913	−0.192276
$816$	0	0
$817$	24.4674	0.856005
$818$	−0.744563	−0.0260330
$819$	0	0
$820$	−11.1386	−0.388977
$821$	23.4891	0.819776	0.409888	−	0.912136i	$-0.365568\pi$
$821$	23.4891	0.819776	0.409888	+	0.912136i	$0.365568\pi$
$822$	0	0
$823$	−46.3505	−1.61568	−0.807839	−	0.589403i	$-0.799363\pi$
$823$	−46.3505	−1.61568	−0.807839	+	0.589403i	$0.799363\pi$
$824$	10.1168	0.352437
$825$	0	0
$826$	2.74456	0.0954955
$827$	16.2337	0.564501	0.282250	−	0.959341i	$-0.408919\pi$
$827$	16.2337	0.564501	0.282250	+	0.959341i	$0.408919\pi$
$828$	0	0
$829$	24.5109	0.851298	0.425649	−	0.904888i	$-0.360046\pi$
$829$	24.5109	0.851298	0.425649	+	0.904888i	$0.360046\pi$
$830$	−5.48913	−0.190530
$831$	0	0
$832$	1.37228	0.0475753
$833$	4.74456	0.164389
$834$	0	0
$835$	−7.53262	−0.260677
$836$	−16.0000	−0.553372
$837$	0	0
$838$	4.00000	0.138178
$839$	−33.2554	−1.14811	−0.574053	−	0.818818i	$-0.694630\pi$
$839$	−33.2554	−1.14811	−0.574053	+	0.818818i	$0.694630\pi$
$840$	0	0
$841$	58.8397	2.02895
$842$	26.8614	0.925705
$843$	0	0
$844$	−12.0000	−0.413057
$845$	15.2554	0.524803
$846$	0	0
$847$	−5.00000	−0.171802
$848$	−4.74456	−0.162929
$849$	0	0
$850$	14.7881	0.507226
$851$	2.62772	0.0900770
$852$	0	0
$853$	40.5842	1.38958	0.694789	−	0.719214i	$-0.255498\pi$
$853$	40.5842	1.38958	0.694789	+	0.719214i	$0.255498\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	4.00000	0.136717
$857$	24.1168	0.823816	0.411908	−	0.911226i	$-0.364863\pi$
$857$	24.1168	0.823816	0.411908	+	0.911226i	$0.364863\pi$
$858$	0	0
$859$	22.1168	0.754617	0.377308	−	0.926088i	$-0.376850\pi$
$859$	22.1168	0.754617	0.377308	+	0.926088i	$0.376850\pi$
$860$	−8.39403	−0.286234
$861$	0	0
$862$	−8.86141	−0.301821
$863$	−13.4891	−0.459175	−0.229588	−	0.973288i	$-0.573738\pi$
$863$	−13.4891	−0.459175	−0.229588	+	0.973288i	$0.573738\pi$
$864$	0	0
$865$	−10.2772	−0.349435
$866$	−10.8614	−0.369086
$867$	0	0
$868$	−6.74456	−0.228925
$869$	53.9565	1.83035
$870$	0	0
$871$	−5.48913	−0.185992
$872$	−8.11684	−0.274871
$873$	0	0
$874$	−4.00000	−0.135302
$875$	−11.1386	−0.376553
$876$	0	0
$877$	−22.2337	−0.750778	−0.375389	−	0.926867i	$-0.622491\pi$
$877$	−22.2337	−0.750778	−0.375389	+	0.926867i	$0.622491\pi$
$878$	−30.7446	−1.03758
$879$	0	0
$880$	5.48913	0.185038
$881$	40.9783	1.38059	0.690296	−	0.723527i	$-0.257480\pi$
$881$	40.9783	1.38059	0.690296	+	0.723527i	$0.257480\pi$
$882$	0	0
$883$	−21.2554	−0.715302	−0.357651	−	0.933855i	$-0.616422\pi$
$883$	−21.2554	−0.715302	−0.357651	+	0.933855i	$0.616422\pi$
$884$	6.51087	0.218984
$885$	0	0
$886$	−8.62772	−0.289854
$887$	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
$889$	11.3723	0.381414
$890$	−10.2772	−0.344492
$891$	0	0
$892$	−9.25544	−0.309895
$893$	−18.5109	−0.619443
$894$	0	0
$895$	−28.6277	−0.956919
$896$	1.00000	0.0334077
$897$	0	0
$898$	28.9783	0.967017
$899$	−63.2119	−2.10824
$900$	0	0
$901$	−22.5109	−0.749946
$902$	32.4674	1.08105
$903$	0	0
$904$	−16.1168	−0.536038
$905$	28.7881	0.956948
$906$	0	0
$907$	−41.0951	−1.36454	−0.682270	−	0.731100i	$-0.739007\pi$
$907$	−41.0951	−1.36454	−0.682270	+	0.731100i	$0.739007\pi$
$908$	−6.11684	−0.202995
$909$	0	0
$910$	−1.88316	−0.0624260
$911$	−49.3288	−1.63434	−0.817168	−	0.576400i	$-0.804457\pi$
$911$	−49.3288	−1.63434	−0.817168	+	0.576400i	$0.804457\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	10.2337	0.338500
$915$	0	0
$916$	−0.744563	−0.0246010
$917$	18.7446	0.619000
$918$	0	0
$919$	−45.9565	−1.51597	−0.757983	−	0.652275i	$-0.773815\pi$
$919$	−45.9565	−1.51597	−0.757983	+	0.652275i	$0.773815\pi$
$920$	1.37228	0.0452428
$921$	0	0
$922$	−12.5109	−0.412024
$923$	−20.2337	−0.666000
$924$	0	0
$925$	−8.19019	−0.269292
$926$	2.11684	0.0695638
$927$	0	0
$928$	9.37228	0.307660
$929$	−15.8832	−0.521109	−0.260555	−	0.965459i	$-0.583905\pi$
$929$	−15.8832	−0.521109	−0.260555	+	0.965459i	$0.583905\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	−23.4891	−0.769412
$933$	0	0
$934$	28.8614	0.944374
$935$	26.0435	0.851713
$936$	0	0
$937$	−33.3723	−1.09022	−0.545112	−	0.838363i	$-0.683513\pi$
$937$	−33.3723	−1.09022	−0.545112	+	0.838363i	$0.683513\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	6.35053	0.207132
$941$	54.6277	1.78081	0.890406	−	0.455166i	$-0.150420\pi$
$941$	54.6277	1.78081	0.890406	+	0.455166i	$0.150420\pi$
$942$	0	0
$943$	8.11684	0.264321
$944$	2.74456	0.0893279
$945$	0	0
$946$	24.4674	0.795503
$947$	5.64947	0.183583	0.0917915	−	0.995778i	$-0.470741\pi$
$947$	5.64947	0.183583	0.0917915	+	0.995778i	$0.470741\pi$
$948$	0	0
$949$	−17.4891	−0.567721
$950$	12.4674	0.404495
$951$	0	0
$952$	4.74456	0.153772
$953$	−23.4891	−0.760887	−0.380444	−	0.924804i	$-0.624229\pi$
$953$	−23.4891	−0.760887	−0.380444	+	0.924804i	$0.624229\pi$
$954$	0	0
$955$	3.44563	0.111498
$956$	−13.4891	−0.436269
$957$	0	0
$958$	−6.74456	−0.217907
$959$	−8.11684	−0.262107
$960$	0	0
$961$	14.4891	0.467391
$962$	−3.60597	−0.116261
$963$	0	0
$964$	−2.62772	−0.0846331
$965$	0.160343	0.00516162
$966$	0	0
$967$	−53.4891	−1.72009	−0.860047	−	0.510215i	$-0.829566\pi$
$967$	−53.4891	−1.72009	−0.860047	+	0.510215i	$0.829566\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	−12.8614	−0.412955
$971$	28.4674	0.913562	0.456781	−	0.889579i	$-0.349002\pi$
$971$	28.4674	0.913562	0.456781	+	0.889579i	$0.349002\pi$
$972$	0	0
$973$	−8.62772	−0.276592
$974$	−5.88316	−0.188508
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−8.35053	−0.267157	−0.133579	−	0.991038i	$-0.542647\pi$
$977$	−8.35053	−0.267157	−0.133579	+	0.991038i	$0.542647\pi$
$978$	0	0
$979$	29.9565	0.957414
$980$	−1.37228	−0.0438359
$981$	0	0
$982$	−30.9783	−0.988556
$983$	3.76631	0.120127	0.0600633	−	0.998195i	$-0.480870\pi$
$983$	3.76631	0.120127	0.0600633	+	0.998195i	$0.480870\pi$
$984$	0	0
$985$	−29.3288	−0.934493
$986$	44.4674	1.41613
$987$	0	0
$988$	5.48913	0.174632
$989$	6.11684	0.194504
$990$	0	0
$991$	−13.4891	−0.428496	−0.214248	−	0.976779i	$-0.568730\pi$
$991$	−13.4891	−0.428496	−0.214248	+	0.976779i	$0.568730\pi$
$992$	−6.74456	−0.214140
$993$	0	0
$994$	−14.7446	−0.467669
$995$	23.1386	0.733543
$996$	0	0
$997$	−50.0000	−1.58352	−0.791758	−	0.610835i	$-0.790834\pi$
$997$	−50.0000	−1.58352	−0.791758	+	0.610835i	$0.790834\pi$
$998$	−21.7228	−0.687624
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.37228 q^{10} -4.00000 q^{11} +1.37228 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.74456 q^{17} +4.00000 q^{19} -1.37228 q^{20} +4.00000 q^{22} +1.00000 q^{23} -3.11684 q^{25} -1.37228 q^{26} -1.00000 q^{28} -9.37228 q^{29} +6.74456 q^{31} -1.00000 q^{32} -4.74456 q^{34} +1.37228 q^{35} +2.62772 q^{37} -4.00000 q^{38} +1.37228 q^{40} +8.11684 q^{41} +6.11684 q^{43} -4.00000 q^{44} -1.00000 q^{46} -4.62772 q^{47} +1.00000 q^{49} +3.11684 q^{50} +1.37228 q^{52} -4.74456 q^{53} +5.48913 q^{55} +1.00000 q^{56} +9.37228 q^{58} +2.74456 q^{59} -2.00000 q^{61} -6.74456 q^{62} +1.00000 q^{64} -1.88316 q^{65} -4.00000 q^{67} +4.74456 q^{68} -1.37228 q^{70} -14.7446 q^{71} -12.7446 q^{73} -2.62772 q^{74} +4.00000 q^{76} +4.00000 q^{77} -13.4891 q^{79} -1.37228 q^{80} -8.11684 q^{82} -4.00000 q^{83} -6.51087 q^{85} -6.11684 q^{86} +4.00000 q^{88} -7.48913 q^{89} -1.37228 q^{91} +1.00000 q^{92} +4.62772 q^{94} -5.48913 q^{95} -9.37228 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 3 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 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

Modular forms

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Database

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