LMFDB - Embedded newform 4842.2.a.p.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4842,2,Mod(1,4842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4842, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4842 = 2 \cdot 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4842.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:	$38.6635646587$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 21x^{6} + 3x^{5} + 135x^{4} + 76x^{3} - 180x^{2} - 110x + 41 $ x^8 - x^7 - 21x^6 + 3x^5 + 135x^4 + 76x^3 - 180x^2 - 110x + 41
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1614)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.21132$ of defining polynomial
Character	$\chi$	$=$	4842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +3.38195 q^{5} -0.211316 q^{7} -1.00000 q^{8} -3.38195 q^{10} +2.61950 q^{11} +5.18926 q^{13} +0.211316 q^{14} +1.00000 q^{16} -2.02697 q^{17} +6.25886 q^{19} +3.38195 q^{20} -2.61950 q^{22} -1.76118 q^{23} +6.43755 q^{25} -5.18926 q^{26} -0.211316 q^{28} +1.11008 q^{29} +2.08264 q^{31} -1.00000 q^{32} +2.02697 q^{34} -0.714658 q^{35} +8.28583 q^{37} -6.25886 q^{38} -3.38195 q^{40} +5.45377 q^{41} +4.73911 q^{43} +2.61950 q^{44} +1.76118 q^{46} -8.87857 q^{47} -6.95535 q^{49} -6.43755 q^{50} +5.18926 q^{52} -12.4763 q^{53} +8.85899 q^{55} +0.211316 q^{56} -1.11008 q^{58} +7.55598 q^{59} -13.6395 q^{61} -2.08264 q^{62} +1.00000 q^{64} +17.5498 q^{65} +9.62462 q^{67} -2.02697 q^{68} +0.714658 q^{70} -2.61981 q^{71} +6.98186 q^{73} -8.28583 q^{74} +6.25886 q^{76} -0.553540 q^{77} -9.85401 q^{79} +3.38195 q^{80} -5.45377 q^{82} +10.8941 q^{83} -6.85509 q^{85} -4.73911 q^{86} -2.61950 q^{88} -10.9577 q^{89} -1.09657 q^{91} -1.76118 q^{92} +8.87857 q^{94} +21.1671 q^{95} +0.644025 q^{97} +6.95535 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 4 q^{5} + 9 q^{7} - 8 q^{8} + 4 q^{10} - 2 q^{11} + 9 q^{13} - 9 q^{14} + 8 q^{16} + 2 q^{17} + 12 q^{19} - 4 q^{20} + 2 q^{22} - 6 q^{23} + 10 q^{25} - 9 q^{26} + 9 q^{28} - q^{29}+ \cdots + 3 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 4 * q^5 + 9 * q^7 - 8 * q^8 + 4 * q^10 - 2 * q^11 + 9 * q^13 - 9 * q^14 + 8 * q^16 + 2 * q^17 + 12 * q^19 - 4 * q^20 + 2 * q^22 - 6 * q^23 + 10 * q^25 - 9 * q^26 + 9 * q^28 - q^29 + 8 * q^31 - 8 * q^32 - 2 * q^34 + 7 * q^35 + 10 * q^37 - 12 * q^38 + 4 * q^40 + q^41 + 8 * q^43 - 2 * q^44 + 6 * q^46 - 2 * q^47 - 3 * q^49 - 10 * q^50 + 9 * q^52 + 9 * q^53 - q^55 - 9 * q^56 + q^58 + 3 * q^59 - 8 * q^62 + 8 * q^64 + 18 * q^65 + 6 * q^67 + 2 * q^68 - 7 * q^70 + 17 * q^71 - q^73 - 10 * q^74 + 12 * q^76 + 4 * q^77 + q^79 - 4 * q^80 - q^82 + 17 * q^83 - 20 * q^85 - 8 * q^86 + 2 * q^88 + 4 * q^89 + 15 * q^91 - 6 * q^92 + 2 * q^94 + 7 * q^95 + 13 * q^97 + 3 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.38195	1.51245	0.756226	−	0.654311i	$-0.227041\pi$
$5$	3.38195	1.51245	0.756226	+	0.654311i	$0.227041\pi$
$6$	0	0
$7$	−0.211316	−0.0798698	−0.0399349	−	0.999202i	$-0.512715\pi$
$7$	−0.211316	−0.0798698	−0.0399349	+	0.999202i	$0.512715\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.38195	−1.06946
$11$	2.61950	0.789807	0.394904	−	0.918723i	$-0.370778\pi$
$11$	2.61950	0.789807	0.394904	+	0.918723i	$0.370778\pi$
$12$	0	0
$13$	5.18926	1.43924	0.719620	−	0.694368i	$-0.244316\pi$
$13$	5.18926	1.43924	0.719620	+	0.694368i	$0.244316\pi$
$14$	0.211316	0.0564765
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.02697	−0.491612	−0.245806	−	0.969319i	$-0.579053\pi$
$17$	−2.02697	−0.491612	−0.245806	+	0.969319i	$0.579053\pi$
$18$	0	0
$19$	6.25886	1.43588	0.717940	−	0.696105i	$-0.245085\pi$
$19$	6.25886	1.43588	0.717940	+	0.696105i	$0.245085\pi$
$20$	3.38195	0.756226
$21$	0	0
$22$	−2.61950	−0.558478
$23$	−1.76118	−0.367230	−0.183615	−	0.982998i	$-0.558780\pi$
$23$	−1.76118	−0.367230	−0.183615	+	0.982998i	$0.558780\pi$
$24$	0	0
$25$	6.43755	1.28751
$26$	−5.18926	−1.01770
$27$	0	0
$28$	−0.211316	−0.0399349
$29$	1.11008	0.206137	0.103069	−	0.994674i	$-0.467134\pi$
$29$	1.11008	0.206137	0.103069	+	0.994674i	$0.467134\pi$
$30$	0	0
$31$	2.08264	0.374053	0.187026	−	0.982355i	$-0.440115\pi$
$31$	2.08264	0.374053	0.187026	+	0.982355i	$0.440115\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.02697	0.347622
$35$	−0.714658	−0.120799
$36$	0	0
$37$	8.28583	1.36218	0.681091	−	0.732199i	$-0.261506\pi$
$37$	8.28583	1.36218	0.681091	+	0.732199i	$0.261506\pi$
$38$	−6.25886	−1.01532
$39$	0	0
$40$	−3.38195	−0.534732
$41$	5.45377	0.851736	0.425868	−	0.904785i	$-0.359969\pi$
$41$	5.45377	0.851736	0.425868	+	0.904785i	$0.359969\pi$
$42$	0	0
$43$	4.73911	0.722708	0.361354	−	0.932429i	$-0.382315\pi$
$43$	4.73911	0.722708	0.361354	+	0.932429i	$0.382315\pi$
$44$	2.61950	0.394904
$45$	0	0
$46$	1.76118	0.259671
$47$	−8.87857	−1.29507	−0.647536	−	0.762035i	$-0.724201\pi$
$47$	−8.87857	−1.29507	−0.647536	+	0.762035i	$0.724201\pi$
$48$	0	0
$49$	−6.95535	−0.993621
$50$	−6.43755	−0.910407
$51$	0	0
$52$	5.18926	0.719620
$53$	−12.4763	−1.71375	−0.856875	−	0.515524i	$-0.827597\pi$
$53$	−12.4763	−1.71375	−0.856875	+	0.515524i	$0.827597\pi$
$54$	0	0
$55$	8.85899	1.19455
$56$	0.211316	0.0282382
$57$	0	0
$58$	−1.11008	−0.145761
$59$	7.55598	0.983704	0.491852	−	0.870679i	$-0.336320\pi$
$59$	7.55598	0.983704	0.491852	+	0.870679i	$0.336320\pi$
$60$	0	0
$61$	−13.6395	−1.74636	−0.873179	−	0.487399i	$-0.837946\pi$
$61$	−13.6395	−1.74636	−0.873179	+	0.487399i	$0.837946\pi$
$62$	−2.08264	−0.264495
$63$	0	0
$64$	1.00000	0.125000
$65$	17.5498	2.17678
$66$	0	0
$67$	9.62462	1.17583	0.587917	−	0.808921i	$-0.299948\pi$
$67$	9.62462	1.17583	0.587917	+	0.808921i	$0.299948\pi$
$68$	−2.02697	−0.245806
$69$	0	0
$70$	0.714658	0.0854179
$71$	−2.61981	−0.310915	−0.155457	−	0.987843i	$-0.549685\pi$
$71$	−2.61981	−0.310915	−0.155457	+	0.987843i	$0.549685\pi$
$72$	0	0
$73$	6.98186	0.817165	0.408582	−	0.912721i	$-0.366023\pi$
$73$	6.98186	0.817165	0.408582	+	0.912721i	$0.366023\pi$
$74$	−8.28583	−0.963208
$75$	0	0
$76$	6.25886	0.717940
$77$	−0.553540	−0.0630817
$78$	0	0
$79$	−9.85401	−1.10866	−0.554331	−	0.832296i	$-0.687026\pi$
$79$	−9.85401	−1.10866	−0.554331	+	0.832296i	$0.687026\pi$
$80$	3.38195	0.378113
$81$	0	0
$82$	−5.45377	−0.602268
$83$	10.8941	1.19578	0.597890	−	0.801578i	$-0.296006\pi$
$83$	10.8941	1.19578	0.597890	+	0.801578i	$0.296006\pi$
$84$	0	0
$85$	−6.85509	−0.743539
$86$	−4.73911	−0.511032
$87$	0	0
$88$	−2.61950	−0.279239
$89$	−10.9577	−1.16152	−0.580759	−	0.814075i	$-0.697244\pi$
$89$	−10.9577	−1.16152	−0.580759	+	0.814075i	$0.697244\pi$
$90$	0	0
$91$	−1.09657	−0.114952
$92$	−1.76118	−0.183615
$93$	0	0
$94$	8.87857	0.915755
$95$	21.1671	2.17170
$96$	0	0
$97$	0.644025	0.0653908	0.0326954	−	0.999465i	$-0.489591\pi$
$97$	0.644025	0.0653908	0.0326954	+	0.999465i	$0.489591\pi$
$98$	6.95535	0.702596
$99$	0	0
$100$	6.43755	0.643755
$101$	3.84144	0.382238	0.191119	−	0.981567i	$-0.438788\pi$
$101$	3.84144	0.382238	0.191119	+	0.981567i	$0.438788\pi$
$102$	0	0
$103$	13.1362	1.29435	0.647174	−	0.762342i	$-0.275951\pi$
$103$	13.1362	1.29435	0.647174	+	0.762342i	$0.275951\pi$
$104$	−5.18926	−0.508848
$105$	0	0
$106$	12.4763	1.21180
$107$	5.52143	0.533777	0.266889	−	0.963727i	$-0.414004\pi$
$107$	5.52143	0.533777	0.266889	+	0.963727i	$0.414004\pi$
$108$	0	0
$109$	−12.2900	−1.17717	−0.588583	−	0.808437i	$-0.700314\pi$
$109$	−12.2900	−1.17717	−0.588583	+	0.808437i	$0.700314\pi$
$110$	−8.85899	−0.844671
$111$	0	0
$112$	−0.211316	−0.0199674
$113$	−11.9287	−1.12215	−0.561077	−	0.827764i	$-0.689613\pi$
$113$	−11.9287	−1.12215	−0.561077	+	0.827764i	$0.689613\pi$
$114$	0	0
$115$	−5.95620	−0.555418
$116$	1.11008	0.103069
$117$	0	0
$118$	−7.55598	−0.695584
$119$	0.428330	0.0392649
$120$	0	0
$121$	−4.13825	−0.376204
$122$	13.6395	1.23486
$123$	0	0
$124$	2.08264	0.187026
$125$	4.86172	0.434846
$126$	0	0
$127$	−11.7616	−1.04368	−0.521838	−	0.853045i	$-0.674754\pi$
$127$	−11.7616	−1.04368	−0.521838	+	0.853045i	$0.674754\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−17.5498	−1.53922
$131$	13.8987	1.21433	0.607166	−	0.794575i	$-0.292306\pi$
$131$	13.8987	1.21433	0.607166	+	0.794575i	$0.292306\pi$
$132$	0	0
$133$	−1.32259	−0.114683
$134$	−9.62462	−0.831440
$135$	0	0
$136$	2.02697	0.173811
$137$	19.0685	1.62913	0.814567	−	0.580069i	$-0.196975\pi$
$137$	19.0685	1.62913	0.814567	+	0.580069i	$0.196975\pi$
$138$	0	0
$139$	11.4309	0.969559	0.484780	−	0.874636i	$-0.338900\pi$
$139$	11.4309	0.969559	0.484780	+	0.874636i	$0.338900\pi$
$140$	−0.714658	−0.0603996
$141$	0	0
$142$	2.61981	0.219850
$143$	13.5932	1.13672
$144$	0	0
$145$	3.75424	0.311773
$146$	−6.98186	−0.577823
$147$	0	0
$148$	8.28583	0.681091
$149$	−17.0440	−1.39630	−0.698149	−	0.715952i	$-0.745993\pi$
$149$	−17.0440	−1.39630	−0.698149	+	0.715952i	$0.745993\pi$
$150$	0	0
$151$	−8.04111	−0.654376	−0.327188	−	0.944959i	$-0.606101\pi$
$151$	−8.04111	−0.654376	−0.327188	+	0.944959i	$0.606101\pi$
$152$	−6.25886	−0.507660
$153$	0	0
$154$	0.553540	0.0446055
$155$	7.04337	0.565737
$156$	0	0
$157$	6.20006	0.494819	0.247409	−	0.968911i	$-0.420421\pi$
$157$	6.20006	0.494819	0.247409	+	0.968911i	$0.420421\pi$
$158$	9.85401	0.783943
$159$	0	0
$160$	−3.38195	−0.267366
$161$	0.372164	0.0293306
$162$	0	0
$163$	−14.9933	−1.17437	−0.587184	−	0.809453i	$-0.699764\pi$
$163$	−14.9933	−1.17437	−0.587184	+	0.809453i	$0.699764\pi$
$164$	5.45377	0.425868
$165$	0	0
$166$	−10.8941	−0.845545
$167$	−1.94990	−0.150888	−0.0754438	−	0.997150i	$-0.524037\pi$
$167$	−1.94990	−0.150888	−0.0754438	+	0.997150i	$0.524037\pi$
$168$	0	0
$169$	13.9284	1.07141
$170$	6.85509	0.525761
$171$	0	0
$172$	4.73911	0.361354
$173$	1.12775	0.0857414	0.0428707	−	0.999081i	$-0.486350\pi$
$173$	1.12775	0.0857414	0.0428707	+	0.999081i	$0.486350\pi$
$174$	0	0
$175$	−1.36035	−0.102833
$176$	2.61950	0.197452
$177$	0	0
$178$	10.9577	0.821318
$179$	−3.61157	−0.269941	−0.134971	−	0.990850i	$-0.543094\pi$
$179$	−3.61157	−0.269941	−0.134971	+	0.990850i	$0.543094\pi$
$180$	0	0
$181$	−23.8339	−1.77156	−0.885781	−	0.464104i	$-0.846376\pi$
$181$	−23.8339	−1.77156	−0.885781	+	0.464104i	$0.846376\pi$
$182$	1.09657	0.0812832
$183$	0	0
$184$	1.76118	0.129836
$185$	28.0222	2.06023
$186$	0	0
$187$	−5.30963	−0.388279
$188$	−8.87857	−0.647536
$189$	0	0
$190$	−21.1671	−1.53562
$191$	6.39834	0.462968	0.231484	−	0.972839i	$-0.425642\pi$
$191$	6.39834	0.462968	0.231484	+	0.972839i	$0.425642\pi$
$192$	0	0
$193$	−13.5035	−0.972004	−0.486002	−	0.873958i	$-0.661545\pi$
$193$	−13.5035	−0.972004	−0.486002	+	0.873958i	$0.661545\pi$
$194$	−0.644025	−0.0462383
$195$	0	0
$196$	−6.95535	−0.496810
$197$	−5.46878	−0.389634	−0.194817	−	0.980840i	$-0.562411\pi$
$197$	−5.46878	−0.389634	−0.194817	+	0.980840i	$0.562411\pi$
$198$	0	0
$199$	17.1793	1.21781	0.608905	−	0.793243i	$-0.291609\pi$
$199$	17.1793	1.21781	0.608905	+	0.793243i	$0.291609\pi$
$200$	−6.43755	−0.455204
$201$	0	0
$202$	−3.84144	−0.270283
$203$	−0.234578	−0.0164641
$204$	0	0
$205$	18.4444	1.28821
$206$	−13.1362	−0.915243
$207$	0	0
$208$	5.18926	0.359810
$209$	16.3950	1.13407
$210$	0	0
$211$	−8.03629	−0.553241	−0.276620	−	0.960979i	$-0.589214\pi$
$211$	−8.03629	−0.553241	−0.276620	+	0.960979i	$0.589214\pi$
$212$	−12.4763	−0.856875
$213$	0	0
$214$	−5.52143	−0.377437
$215$	16.0274	1.09306
$216$	0	0
$217$	−0.440094	−0.0298755
$218$	12.2900	0.832382
$219$	0	0
$220$	8.85899	0.597273
$221$	−10.5184	−0.707547
$222$	0	0
$223$	6.05563	0.405515	0.202758	−	0.979229i	$-0.435010\pi$
$223$	6.05563	0.405515	0.202758	+	0.979229i	$0.435010\pi$
$224$	0.211316	0.0141191
$225$	0	0
$226$	11.9287	0.793482
$227$	2.97393	0.197386	0.0986932	−	0.995118i	$-0.468534\pi$
$227$	2.97393	0.197386	0.0986932	+	0.995118i	$0.468534\pi$
$228$	0	0
$229$	2.39817	0.158476	0.0792378	−	0.996856i	$-0.474751\pi$
$229$	2.39817	0.158476	0.0792378	+	0.996856i	$0.474751\pi$
$230$	5.95620	0.392740
$231$	0	0
$232$	−1.11008	−0.0728806
$233$	17.4919	1.14593	0.572967	−	0.819578i	$-0.305792\pi$
$233$	17.4919	1.14593	0.572967	+	0.819578i	$0.305792\pi$
$234$	0	0
$235$	−30.0268	−1.95874
$236$	7.55598	0.491852
$237$	0	0
$238$	−0.428330	−0.0277645
$239$	−5.72249	−0.370157	−0.185079	−	0.982724i	$-0.559254\pi$
$239$	−5.72249	−0.370157	−0.185079	+	0.982724i	$0.559254\pi$
$240$	0	0
$241$	13.6485	0.879179	0.439590	−	0.898199i	$-0.355124\pi$
$241$	13.6485	0.879179	0.439590	+	0.898199i	$0.355124\pi$
$242$	4.13825	0.266017
$243$	0	0
$244$	−13.6395	−0.873179
$245$	−23.5226	−1.50280
$246$	0	0
$247$	32.4788	2.06658
$248$	−2.08264	−0.132248
$249$	0	0
$250$	−4.86172	−0.307482
$251$	−16.8931	−1.06628	−0.533142	−	0.846025i	$-0.678989\pi$
$251$	−16.8931	−1.06628	−0.533142	+	0.846025i	$0.678989\pi$
$252$	0	0
$253$	−4.61339	−0.290041
$254$	11.7616	0.737991
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.93163	0.370005	0.185002	−	0.982738i	$-0.440771\pi$
$257$	5.93163	0.370005	0.185002	+	0.982738i	$0.440771\pi$
$258$	0	0
$259$	−1.75092	−0.108797
$260$	17.5498	1.08839
$261$	0	0
$262$	−13.8987	−0.858663
$263$	−6.91162	−0.426189	−0.213094	−	0.977032i	$-0.568354\pi$
$263$	−6.91162	−0.426189	−0.213094	+	0.977032i	$0.568354\pi$
$264$	0	0
$265$	−42.1941	−2.59196
$266$	1.32259	0.0810934
$267$	0	0
$268$	9.62462	0.587917
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	−27.8504	−1.69179	−0.845896	−	0.533348i	$-0.820934\pi$
$271$	−27.8504	−1.69179	−0.845896	+	0.533348i	$0.820934\pi$
$272$	−2.02697	−0.122903
$273$	0	0
$274$	−19.0685	−1.15197
$275$	16.8631	1.01689
$276$	0	0
$277$	21.5773	1.29645	0.648226	−	0.761448i	$-0.275511\pi$
$277$	21.5773	1.29645	0.648226	+	0.761448i	$0.275511\pi$
$278$	−11.4309	−0.685582
$279$	0	0
$280$	0.714658	0.0427090
$281$	1.55635	0.0928438	0.0464219	−	0.998922i	$-0.485218\pi$
$281$	1.55635	0.0928438	0.0464219	+	0.998922i	$0.485218\pi$
$282$	0	0
$283$	−18.9435	−1.12607	−0.563036	−	0.826432i	$-0.690367\pi$
$283$	−18.9435	−1.12607	−0.563036	+	0.826432i	$0.690367\pi$
$284$	−2.61981	−0.155457
$285$	0	0
$286$	−13.5932	−0.803784
$287$	−1.15247	−0.0680280
$288$	0	0
$289$	−12.8914	−0.758318
$290$	−3.75424	−0.220457
$291$	0	0
$292$	6.98186	0.408582
$293$	−19.5482	−1.14202	−0.571008	−	0.820945i	$-0.693447\pi$
$293$	−19.5482	−1.14202	−0.571008	+	0.820945i	$0.693447\pi$
$294$	0	0
$295$	25.5539	1.48781
$296$	−8.28583	−0.481604
$297$	0	0
$298$	17.0440	0.987332
$299$	−9.13919	−0.528533
$300$	0	0
$301$	−1.00145	−0.0577225
$302$	8.04111	0.462714
$303$	0	0
$304$	6.25886	0.358970
$305$	−46.1280	−2.64128
$306$	0	0
$307$	22.7738	1.29977	0.649883	−	0.760034i	$-0.274818\pi$
$307$	22.7738	1.29977	0.649883	+	0.760034i	$0.274818\pi$
$308$	−0.553540	−0.0315409
$309$	0	0
$310$	−7.04337	−0.400036
$311$	−32.0583	−1.81786	−0.908929	−	0.416951i	$-0.863099\pi$
$311$	−32.0583	−1.81786	−0.908929	+	0.416951i	$0.863099\pi$
$312$	0	0
$313$	−1.84320	−0.104184	−0.0520920	−	0.998642i	$-0.516589\pi$
$313$	−1.84320	−0.104184	−0.0520920	+	0.998642i	$0.516589\pi$
$314$	−6.20006	−0.349890
$315$	0	0
$316$	−9.85401	−0.554331
$317$	−21.5496	−1.21034	−0.605172	−	0.796095i	$-0.706896\pi$
$317$	−21.5496	−1.21034	−0.605172	+	0.796095i	$0.706896\pi$
$318$	0	0
$319$	2.90786	0.162809
$320$	3.38195	0.189056
$321$	0	0
$322$	−0.372164	−0.0207399
$323$	−12.6865	−0.705896
$324$	0	0
$325$	33.4061	1.85304
$326$	14.9933	0.830404
$327$	0	0
$328$	−5.45377	−0.301134
$329$	1.87618	0.103437
$330$	0	0
$331$	17.5003	0.961903	0.480952	−	0.876747i	$-0.340291\pi$
$331$	17.5003	0.961903	0.480952	+	0.876747i	$0.340291\pi$
$332$	10.8941	0.597890
$333$	0	0
$334$	1.94990	0.106694
$335$	32.5499	1.77839
$336$	0	0
$337$	0.945178	0.0514871	0.0257436	−	0.999669i	$-0.491805\pi$
$337$	0.945178	0.0514871	0.0257436	+	0.999669i	$0.491805\pi$
$338$	−13.9284	−0.757603
$339$	0	0
$340$	−6.85509	−0.371770
$341$	5.45546	0.295430
$342$	0	0
$343$	2.94898	0.159230
$344$	−4.73911	−0.255516
$345$	0	0
$346$	−1.12775	−0.0606283
$347$	30.5256	1.63870	0.819350	−	0.573293i	$-0.194334\pi$
$347$	30.5256	1.63870	0.819350	+	0.573293i	$0.194334\pi$
$348$	0	0
$349$	−17.0713	−0.913808	−0.456904	−	0.889516i	$-0.651042\pi$
$349$	−17.0713	−0.913808	−0.456904	+	0.889516i	$0.651042\pi$
$350$	1.36035	0.0727140
$351$	0	0
$352$	−2.61950	−0.139620
$353$	11.3088	0.601907	0.300953	−	0.953639i	$-0.402695\pi$
$353$	11.3088	0.601907	0.300953	+	0.953639i	$0.402695\pi$
$354$	0	0
$355$	−8.86006	−0.470243
$356$	−10.9577	−0.580759
$357$	0	0
$358$	3.61157	0.190877
$359$	20.7803	1.09674	0.548371	−	0.836235i	$-0.315248\pi$
$359$	20.7803	1.09674	0.548371	+	0.836235i	$0.315248\pi$
$360$	0	0
$361$	20.1733	1.06175
$362$	23.8339	1.25268
$363$	0	0
$364$	−1.09657	−0.0574759
$365$	23.6123	1.23592
$366$	0	0
$367$	30.5653	1.59550	0.797748	−	0.602990i	$-0.206024\pi$
$367$	30.5653	1.59550	0.797748	+	0.602990i	$0.206024\pi$
$368$	−1.76118	−0.0918076
$369$	0	0
$370$	−28.0222	−1.45681
$371$	2.63643	0.136877
$372$	0	0
$373$	24.0416	1.24483	0.622413	−	0.782689i	$-0.286153\pi$
$373$	24.0416	1.24483	0.622413	+	0.782689i	$0.286153\pi$
$374$	5.30963	0.274554
$375$	0	0
$376$	8.87857	0.457877
$377$	5.76051	0.296681
$378$	0	0
$379$	1.76832	0.0908322	0.0454161	−	0.998968i	$-0.485539\pi$
$379$	1.76832	0.0908322	0.0454161	+	0.998968i	$0.485539\pi$
$380$	21.1671	1.08585
$381$	0	0
$382$	−6.39834	−0.327368
$383$	−3.12837	−0.159852	−0.0799260	−	0.996801i	$-0.525468\pi$
$383$	−3.12837	−0.159852	−0.0799260	+	0.996801i	$0.525468\pi$
$384$	0	0
$385$	−1.87204	−0.0954081
$386$	13.5035	0.687310
$387$	0	0
$388$	0.644025	0.0326954
$389$	12.3873	0.628062	0.314031	−	0.949413i	$-0.398320\pi$
$389$	12.3873	0.628062	0.314031	+	0.949413i	$0.398320\pi$
$390$	0	0
$391$	3.56984	0.180535
$392$	6.95535	0.351298
$393$	0	0
$394$	5.46878	0.275513
$395$	−33.3257	−1.67680
$396$	0	0
$397$	14.6465	0.735088	0.367544	−	0.930006i	$-0.380199\pi$
$397$	14.6465	0.735088	0.367544	+	0.930006i	$0.380199\pi$
$398$	−17.1793	−0.861122
$399$	0	0
$400$	6.43755	0.321878
$401$	25.5604	1.27642	0.638212	−	0.769861i	$-0.279674\pi$
$401$	25.5604	1.27642	0.638212	+	0.769861i	$0.279674\pi$
$402$	0	0
$403$	10.8073	0.538352
$404$	3.84144	0.191119
$405$	0	0
$406$	0.234578	0.0116419
$407$	21.7047	1.07586
$408$	0	0
$409$	−14.0929	−0.696848	−0.348424	−	0.937337i	$-0.613283\pi$
$409$	−14.0929	−0.696848	−0.348424	+	0.937337i	$0.613283\pi$
$410$	−18.4444	−0.910902
$411$	0	0
$412$	13.1362	0.647174
$413$	−1.59670	−0.0785682
$414$	0	0
$415$	36.8432	1.80856
$416$	−5.18926	−0.254424
$417$	0	0
$418$	−16.3950	−0.801908
$419$	−10.3282	−0.504566	−0.252283	−	0.967654i	$-0.581181\pi$
$419$	−10.3282	−0.504566	−0.252283	+	0.967654i	$0.581181\pi$
$420$	0	0
$421$	−21.7560	−1.06032	−0.530160	−	0.847897i	$-0.677868\pi$
$421$	−21.7560	−1.06032	−0.530160	+	0.847897i	$0.677868\pi$
$422$	8.03629	0.391200
$423$	0	0
$424$	12.4763	0.605902
$425$	−13.0487	−0.632955
$426$	0	0
$427$	2.88224	0.139481
$428$	5.52143	0.266889
$429$	0	0
$430$	−16.0274	−0.772911
$431$	−17.2355	−0.830203	−0.415102	−	0.909775i	$-0.636254\pi$
$431$	−17.2355	−0.830203	−0.415102	+	0.909775i	$0.636254\pi$
$432$	0	0
$433$	−24.5562	−1.18009	−0.590047	−	0.807369i	$-0.700891\pi$
$433$	−24.5562	−1.18009	−0.590047	+	0.807369i	$0.700891\pi$
$434$	0.440094	0.0211252
$435$	0	0
$436$	−12.2900	−0.588583
$437$	−11.0229	−0.527299
$438$	0	0
$439$	−21.3132	−1.01722	−0.508611	−	0.860997i	$-0.669841\pi$
$439$	−21.3132	−1.01722	−0.508611	+	0.860997i	$0.669841\pi$
$440$	−8.85899	−0.422336
$441$	0	0
$442$	10.5184	0.500312
$443$	22.6790	1.07751	0.538756	−	0.842462i	$-0.318894\pi$
$443$	22.6790	1.07751	0.538756	+	0.842462i	$0.318894\pi$
$444$	0	0
$445$	−37.0585	−1.75674
$446$	−6.05563	−0.286742
$447$	0	0
$448$	−0.211316	−0.00998372
$449$	32.4566	1.53172	0.765860	−	0.643007i	$-0.222313\pi$
$449$	32.4566	1.53172	0.765860	+	0.643007i	$0.222313\pi$
$450$	0	0
$451$	14.2861	0.672708
$452$	−11.9287	−0.561077
$453$	0	0
$454$	−2.97393	−0.139573
$455$	−3.70854	−0.173859
$456$	0	0
$457$	−28.2009	−1.31918	−0.659591	−	0.751625i	$-0.729271\pi$
$457$	−28.2009	−1.31918	−0.659591	+	0.751625i	$0.729271\pi$
$458$	−2.39817	−0.112059
$459$	0	0
$460$	−5.95620	−0.277709
$461$	34.2926	1.59716	0.798582	−	0.601886i	$-0.205584\pi$
$461$	34.2926	1.59716	0.798582	+	0.601886i	$0.205584\pi$
$462$	0	0
$463$	29.9743	1.39302	0.696512	−	0.717545i	$-0.254734\pi$
$463$	29.9743	1.39302	0.696512	+	0.717545i	$0.254734\pi$
$464$	1.11008	0.0515343
$465$	0	0
$466$	−17.4919	−0.810298
$467$	24.6700	1.14159	0.570795	−	0.821093i	$-0.306635\pi$
$467$	24.6700	1.14159	0.570795	+	0.821093i	$0.306635\pi$
$468$	0	0
$469$	−2.03383	−0.0939136
$470$	30.0268	1.38503
$471$	0	0
$472$	−7.55598	−0.347792
$473$	12.4141	0.570800
$474$	0	0
$475$	40.2917	1.84871
$476$	0.428330	0.0196325
$477$	0	0
$478$	5.72249	0.261741
$479$	10.9908	0.502183	0.251091	−	0.967963i	$-0.419211\pi$
$479$	10.9908	0.502183	0.251091	+	0.967963i	$0.419211\pi$
$480$	0	0
$481$	42.9973	1.96051
$482$	−13.6485	−0.621674
$483$	0	0
$484$	−4.13825	−0.188102
$485$	2.17806	0.0989005
$486$	0	0
$487$	−15.1938	−0.688495	−0.344247	−	0.938879i	$-0.611866\pi$
$487$	−15.1938	−0.688495	−0.344247	+	0.938879i	$0.611866\pi$
$488$	13.6395	0.617431
$489$	0	0
$490$	23.5226	1.06264
$491$	−17.0007	−0.767229	−0.383615	−	0.923493i	$-0.625321\pi$
$491$	−17.0007	−0.767229	−0.383615	+	0.923493i	$0.625321\pi$
$492$	0	0
$493$	−2.25010	−0.101340
$494$	−32.4788	−1.46129
$495$	0	0
$496$	2.08264	0.0935132
$497$	0.553607	0.0248327
$498$	0	0
$499$	−13.2475	−0.593040	−0.296520	−	0.955027i	$-0.595826\pi$
$499$	−13.2475	−0.593040	−0.296520	+	0.955027i	$0.595826\pi$
$500$	4.86172	0.217423
$501$	0	0
$502$	16.8931	0.753977
$503$	35.2526	1.57183	0.785917	−	0.618332i	$-0.212191\pi$
$503$	35.2526	1.57183	0.785917	+	0.618332i	$0.212191\pi$
$504$	0	0
$505$	12.9916	0.578117
$506$	4.61339	0.205090
$507$	0	0
$508$	−11.7616	−0.521838
$509$	6.10249	0.270488	0.135244	−	0.990812i	$-0.456818\pi$
$509$	6.10249	0.270488	0.135244	+	0.990812i	$0.456818\pi$
$510$	0	0
$511$	−1.47537	−0.0652667
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−5.93163	−0.261633
$515$	44.4259	1.95764
$516$	0	0
$517$	−23.2574	−1.02286
$518$	1.75092	0.0769312
$519$	0	0
$520$	−17.5498	−0.769609
$521$	33.2669	1.45745	0.728724	−	0.684808i	$-0.240114\pi$
$521$	33.2669	1.45745	0.728724	+	0.684808i	$0.240114\pi$
$522$	0	0
$523$	16.0834	0.703276	0.351638	−	0.936136i	$-0.385625\pi$
$523$	16.0834	0.703276	0.351638	+	0.936136i	$0.385625\pi$
$524$	13.8987	0.607166
$525$	0	0
$526$	6.91162	0.301361
$527$	−4.22144	−0.183889
$528$	0	0
$529$	−19.8983	−0.865142
$530$	42.1941	1.83280
$531$	0	0
$532$	−1.32259	−0.0573417
$533$	28.3010	1.22585
$534$	0	0
$535$	18.6732	0.807312
$536$	−9.62462	−0.415720
$537$	0	0
$538$	1.00000	0.0431131
$539$	−18.2195	−0.784769
$540$	0	0
$541$	19.6968	0.846833	0.423416	−	0.905935i	$-0.360831\pi$
$541$	19.6968	0.846833	0.423416	+	0.905935i	$0.360831\pi$
$542$	27.8504	1.19628
$543$	0	0
$544$	2.02697	0.0869055
$545$	−41.5640	−1.78041
$546$	0	0
$547$	21.3592	0.913254	0.456627	−	0.889658i	$-0.349057\pi$
$547$	21.3592	0.913254	0.456627	+	0.889658i	$0.349057\pi$
$548$	19.0685	0.814567
$549$	0	0
$550$	−16.8631	−0.719047
$551$	6.94786	0.295989
$552$	0	0
$553$	2.08231	0.0885486
$554$	−21.5773	−0.916730
$555$	0	0
$556$	11.4309	0.484780
$557$	−21.8619	−0.926318	−0.463159	−	0.886275i	$-0.653284\pi$
$557$	−21.8619	−0.926318	−0.463159	+	0.886275i	$0.653284\pi$
$558$	0	0
$559$	24.5925	1.04015
$560$	−0.714658	−0.0301998
$561$	0	0
$562$	−1.55635	−0.0656505
$563$	−39.9978	−1.68571	−0.842853	−	0.538143i	$-0.819126\pi$
$563$	−39.9978	−1.68571	−0.842853	+	0.538143i	$0.819126\pi$
$564$	0	0
$565$	−40.3421	−1.69720
$566$	18.9435	0.796254
$567$	0	0
$568$	2.61981	0.109925
$569$	7.67701	0.321837	0.160918	−	0.986968i	$-0.448554\pi$
$569$	7.67701	0.321837	0.160918	+	0.986968i	$0.448554\pi$
$570$	0	0
$571$	16.7701	0.701805	0.350903	−	0.936412i	$-0.385875\pi$
$571$	16.7701	0.701805	0.350903	+	0.936412i	$0.385875\pi$
$572$	13.5932	0.568361
$573$	0	0
$574$	1.15247	0.0481030
$575$	−11.3377	−0.472813
$576$	0	0
$577$	−4.99672	−0.208016	−0.104008	−	0.994576i	$-0.533167\pi$
$577$	−4.99672	−0.208016	−0.104008	+	0.994576i	$0.533167\pi$
$578$	12.8914	0.536212
$579$	0	0
$580$	3.75424	0.155886
$581$	−2.30209	−0.0955067
$582$	0	0
$583$	−32.6816	−1.35353
$584$	−6.98186	−0.288911
$585$	0	0
$586$	19.5482	0.807527
$587$	−30.4574	−1.25711	−0.628555	−	0.777765i	$-0.716353\pi$
$587$	−30.4574	−1.25711	−0.628555	+	0.777765i	$0.716353\pi$
$588$	0	0
$589$	13.0349	0.537095
$590$	−25.5539	−1.05204
$591$	0	0
$592$	8.28583	0.340545
$593$	9.76939	0.401181	0.200590	−	0.979675i	$-0.435714\pi$
$593$	9.76939	0.401181	0.200590	+	0.979675i	$0.435714\pi$
$594$	0	0
$595$	1.44859	0.0593863
$596$	−17.0440	−0.698149
$597$	0	0
$598$	9.13919	0.373729
$599$	41.8225	1.70882	0.854410	−	0.519599i	$-0.173919\pi$
$599$	41.8225	1.70882	0.854410	+	0.519599i	$0.173919\pi$
$600$	0	0
$601$	9.04220	0.368839	0.184419	−	0.982848i	$-0.440960\pi$
$601$	9.04220	0.368839	0.184419	+	0.982848i	$0.440960\pi$
$602$	1.00145	0.0408160
$603$	0	0
$604$	−8.04111	−0.327188
$605$	−13.9953	−0.568991
$606$	0	0
$607$	36.5577	1.48383	0.741915	−	0.670494i	$-0.233918\pi$
$607$	36.5577	1.48383	0.741915	+	0.670494i	$0.233918\pi$
$608$	−6.25886	−0.253830
$609$	0	0
$610$	46.1280	1.86767
$611$	−46.0732	−1.86392
$612$	0	0
$613$	−17.2863	−0.698185	−0.349093	−	0.937088i	$-0.613510\pi$
$613$	−17.2863	−0.698185	−0.349093	+	0.937088i	$0.613510\pi$
$614$	−22.7738	−0.919074
$615$	0	0
$616$	0.553540	0.0223028
$617$	1.17923	0.0474740	0.0237370	−	0.999718i	$-0.492444\pi$
$617$	1.17923	0.0474740	0.0237370	+	0.999718i	$0.492444\pi$
$618$	0	0
$619$	30.6570	1.23221	0.616103	−	0.787665i	$-0.288710\pi$
$619$	30.6570	1.23221	0.616103	+	0.787665i	$0.288710\pi$
$620$	7.04337	0.282868
$621$	0	0
$622$	32.0583	1.28542
$623$	2.31554	0.0927702
$624$	0	0
$625$	−15.7457	−0.629827
$626$	1.84320	0.0736692
$627$	0	0
$628$	6.20006	0.247409
$629$	−16.7951	−0.669664
$630$	0	0
$631$	34.5328	1.37473	0.687365	−	0.726313i	$-0.258767\pi$
$631$	34.5328	1.37473	0.687365	+	0.726313i	$0.258767\pi$
$632$	9.85401	0.391971
$633$	0	0
$634$	21.5496	0.855842
$635$	−39.7772	−1.57851
$636$	0	0
$637$	−36.0931	−1.43006
$638$	−2.90786	−0.115123
$639$	0	0
$640$	−3.38195	−0.133683
$641$	32.5266	1.28472	0.642362	−	0.766401i	$-0.277955\pi$
$641$	32.5266	1.28472	0.642362	+	0.766401i	$0.277955\pi$
$642$	0	0
$643$	−9.84633	−0.388301	−0.194151	−	0.980972i	$-0.562195\pi$
$643$	−9.84633	−0.388301	−0.194151	+	0.980972i	$0.562195\pi$
$644$	0.372164	0.0146653
$645$	0	0
$646$	12.6865	0.499144
$647$	25.4515	1.00060	0.500300	−	0.865852i	$-0.333223\pi$
$647$	25.4515	1.00060	0.500300	+	0.865852i	$0.333223\pi$
$648$	0	0
$649$	19.7928	0.776937
$650$	−33.4061	−1.31030
$651$	0	0
$652$	−14.9933	−0.587184
$653$	17.3275	0.678078	0.339039	−	0.940772i	$-0.389898\pi$
$653$	17.3275	0.678078	0.339039	+	0.940772i	$0.389898\pi$
$654$	0	0
$655$	47.0045	1.83662
$656$	5.45377	0.212934
$657$	0	0
$658$	−1.87618	−0.0731411
$659$	−7.72466	−0.300910	−0.150455	−	0.988617i	$-0.548074\pi$
$659$	−7.72466	−0.300910	−0.150455	+	0.988617i	$0.548074\pi$
$660$	0	0
$661$	32.8193	1.27652	0.638261	−	0.769820i	$-0.279654\pi$
$661$	32.8193	1.27652	0.638261	+	0.769820i	$0.279654\pi$
$662$	−17.5003	−0.680168
$663$	0	0
$664$	−10.8941	−0.422772
$665$	−4.47294	−0.173453
$666$	0	0
$667$	−1.95505	−0.0756999
$668$	−1.94990	−0.0754438
$669$	0	0
$670$	−32.5499	−1.25751
$671$	−35.7286	−1.37929
$672$	0	0
$673$	−26.0908	−1.00573	−0.502864	−	0.864366i	$-0.667720\pi$
$673$	−26.0908	−1.00573	−0.502864	+	0.864366i	$0.667720\pi$
$674$	−0.945178	−0.0364069
$675$	0	0
$676$	13.9284	0.535707
$677$	11.8070	0.453778	0.226889	−	0.973921i	$-0.427145\pi$
$677$	11.8070	0.453778	0.226889	+	0.973921i	$0.427145\pi$
$678$	0	0
$679$	−0.136093	−0.00522275
$680$	6.85509	0.262881
$681$	0	0
$682$	−5.45546	−0.208900
$683$	−42.5317	−1.62743	−0.813716	−	0.581263i	$-0.802559\pi$
$683$	−42.5317	−1.62743	−0.813716	+	0.581263i	$0.802559\pi$
$684$	0	0
$685$	64.4887	2.46399
$686$	−2.94898	−0.112593
$687$	0	0
$688$	4.73911	0.180677
$689$	−64.7427	−2.46650
$690$	0	0
$691$	19.1908	0.730052	0.365026	−	0.930997i	$-0.381060\pi$
$691$	19.1908	0.730052	0.365026	+	0.930997i	$0.381060\pi$
$692$	1.12775	0.0428707
$693$	0	0
$694$	−30.5256	−1.15874
$695$	38.6588	1.46641
$696$	0	0
$697$	−11.0546	−0.418723
$698$	17.0713	0.646160
$699$	0	0
$700$	−1.36035	−0.0514166
$701$	−10.7604	−0.406413	−0.203207	−	0.979136i	$-0.565136\pi$
$701$	−10.7604	−0.406413	−0.203207	+	0.979136i	$0.565136\pi$
$702$	0	0
$703$	51.8598	1.95593
$704$	2.61950	0.0987259
$705$	0	0
$706$	−11.3088	−0.425612
$707$	−0.811757	−0.0305293
$708$	0	0
$709$	−47.7161	−1.79202	−0.896008	−	0.444038i	$-0.853546\pi$
$709$	−47.7161	−1.79202	−0.896008	+	0.444038i	$0.853546\pi$
$710$	8.86006	0.332512
$711$	0	0
$712$	10.9577	0.410659
$713$	−3.66789	−0.137364
$714$	0	0
$715$	45.9716	1.71924
$716$	−3.61157	−0.134971
$717$	0	0
$718$	−20.7803	−0.775514
$719$	7.84050	0.292401	0.146201	−	0.989255i	$-0.453296\pi$
$719$	7.84050	0.292401	0.146201	+	0.989255i	$0.453296\pi$
$720$	0	0
$721$	−2.77588	−0.103379
$722$	−20.1733	−0.750773
$723$	0	0
$724$	−23.8339	−0.885781
$725$	7.14622	0.265404
$726$	0	0
$727$	−7.18234	−0.266378	−0.133189	−	0.991091i	$-0.542522\pi$
$727$	−7.18234	−0.266378	−0.133189	+	0.991091i	$0.542522\pi$
$728$	1.09657	0.0406416
$729$	0	0
$730$	−23.6123	−0.873929
$731$	−9.60603	−0.355292
$732$	0	0
$733$	−14.5003	−0.535580	−0.267790	−	0.963477i	$-0.586293\pi$
$733$	−14.5003	−0.535580	−0.267790	+	0.963477i	$0.586293\pi$
$734$	−30.5653	−1.12819
$735$	0	0
$736$	1.76118	0.0649178
$737$	25.2116	0.928683
$738$	0	0
$739$	−20.0900	−0.739024	−0.369512	−	0.929226i	$-0.620475\pi$
$739$	−20.0900	−0.739024	−0.369512	+	0.929226i	$0.620475\pi$
$740$	28.0222	1.03012
$741$	0	0
$742$	−2.63643	−0.0967865
$743$	3.89941	0.143055	0.0715277	−	0.997439i	$-0.477213\pi$
$743$	3.89941	0.143055	0.0715277	+	0.997439i	$0.477213\pi$
$744$	0	0
$745$	−57.6419	−2.11183
$746$	−24.0416	−0.880224
$747$	0	0
$748$	−5.30963	−0.194139
$749$	−1.16676	−0.0426327
$750$	0	0
$751$	2.96519	0.108201	0.0541006	−	0.998535i	$-0.482771\pi$
$751$	2.96519	0.108201	0.0541006	+	0.998535i	$0.482771\pi$
$752$	−8.87857	−0.323768
$753$	0	0
$754$	−5.76051	−0.209785
$755$	−27.1946	−0.989712
$756$	0	0
$757$	1.14811	0.0417288	0.0208644	−	0.999782i	$-0.493358\pi$
$757$	1.14811	0.0417288	0.0208644	+	0.999782i	$0.493358\pi$
$758$	−1.76832	−0.0642281
$759$	0	0
$760$	−21.1671	−0.767812
$761$	34.7165	1.25847	0.629236	−	0.777215i	$-0.283368\pi$
$761$	34.7165	1.25847	0.629236	+	0.777215i	$0.283368\pi$
$762$	0	0
$763$	2.59706	0.0940200
$764$	6.39834	0.231484
$765$	0	0
$766$	3.12837	0.113032
$767$	39.2099	1.41579
$768$	0	0
$769$	8.46957	0.305420	0.152710	−	0.988271i	$-0.451200\pi$
$769$	8.46957	0.305420	0.152710	+	0.988271i	$0.451200\pi$
$770$	1.87204	0.0674637
$771$	0	0
$772$	−13.5035	−0.486002
$773$	−48.4852	−1.74389	−0.871945	−	0.489604i	$-0.837141\pi$
$773$	−48.4852	−1.74389	−0.871945	+	0.489604i	$0.837141\pi$
$774$	0	0
$775$	13.4071	0.481597
$776$	−0.644025	−0.0231192
$777$	0	0
$778$	−12.3873	−0.444107
$779$	34.1344	1.22299
$780$	0	0
$781$	−6.86259	−0.245563
$782$	−3.56984	−0.127657
$783$	0	0
$784$	−6.95535	−0.248405
$785$	20.9683	0.748389
$786$	0	0
$787$	−5.21440	−0.185873	−0.0929366	−	0.995672i	$-0.529625\pi$
$787$	−5.21440	−0.185873	−0.0929366	+	0.995672i	$0.529625\pi$
$788$	−5.46878	−0.194817
$789$	0	0
$790$	33.3257	1.18568
$791$	2.52071	0.0896261
$792$	0	0
$793$	−70.7788	−2.51343
$794$	−14.6465	−0.519786
$795$	0	0
$796$	17.1793	0.608905
$797$	−41.1631	−1.45807	−0.729035	−	0.684476i	$-0.760031\pi$
$797$	−41.1631	−1.45807	−0.729035	+	0.684476i	$0.760031\pi$
$798$	0	0
$799$	17.9966	0.636673
$800$	−6.43755	−0.227602
$801$	0	0
$802$	−25.5604	−0.902568
$803$	18.2889	0.645403
$804$	0	0
$805$	1.25864	0.0443611
$806$	−10.8073	−0.380672
$807$	0	0
$808$	−3.84144	−0.135142
$809$	−22.9918	−0.808349	−0.404174	−	0.914682i	$-0.632441\pi$
$809$	−22.9918	−0.808349	−0.404174	+	0.914682i	$0.632441\pi$
$810$	0	0
$811$	12.0503	0.423144	0.211572	−	0.977362i	$-0.432142\pi$
$811$	12.0503	0.423144	0.211572	+	0.977362i	$0.432142\pi$
$812$	−0.234578	−0.00823207
$813$	0	0
$814$	−21.7047	−0.760749
$815$	−50.7066	−1.77618
$816$	0	0
$817$	29.6614	1.03772
$818$	14.0929	0.492746
$819$	0	0
$820$	18.4444	0.644105
$821$	−53.2225	−1.85748	−0.928740	−	0.370732i	$-0.879107\pi$
$821$	−53.2225	−1.85748	−0.928740	+	0.370732i	$0.879107\pi$
$822$	0	0
$823$	−41.2893	−1.43926	−0.719628	−	0.694360i	$-0.755688\pi$
$823$	−41.2893	−1.43926	−0.719628	+	0.694360i	$0.755688\pi$
$824$	−13.1362	−0.457621
$825$	0	0
$826$	1.59670	0.0555561
$827$	−52.4750	−1.82473	−0.912366	−	0.409374i	$-0.865747\pi$
$827$	−52.4750	−1.82473	−0.912366	+	0.409374i	$0.865747\pi$
$828$	0	0
$829$	34.0952	1.18418	0.592088	−	0.805873i	$-0.298304\pi$
$829$	34.0952	1.18418	0.592088	+	0.805873i	$0.298304\pi$
$830$	−36.8432	−1.27885
$831$	0	0
$832$	5.18926	0.179905
$833$	14.0983	0.488476
$834$	0	0
$835$	−6.59445	−0.228210
$836$	16.3950	0.567035
$837$	0	0
$838$	10.3282	0.356782
$839$	−43.8212	−1.51288	−0.756439	−	0.654065i	$-0.773062\pi$
$839$	−43.8212	−1.51288	−0.756439	+	0.654065i	$0.773062\pi$
$840$	0	0
$841$	−27.7677	−0.957507
$842$	21.7560	0.749760
$843$	0	0
$844$	−8.03629	−0.276620
$845$	47.1050	1.62046
$846$	0	0
$847$	0.874476	0.0300473
$848$	−12.4763	−0.428438
$849$	0	0
$850$	13.0487	0.447567
$851$	−14.5928	−0.500234
$852$	0	0
$853$	−46.0183	−1.57564	−0.787818	−	0.615909i	$-0.788789\pi$
$853$	−46.0183	−1.57564	−0.787818	+	0.615909i	$0.788789\pi$
$854$	−2.88224	−0.0986281
$855$	0	0
$856$	−5.52143	−0.188719
$857$	5.48596	0.187397	0.0936985	−	0.995601i	$-0.470131\pi$
$857$	5.48596	0.187397	0.0936985	+	0.995601i	$0.470131\pi$
$858$	0	0
$859$	−21.8117	−0.744207	−0.372103	−	0.928191i	$-0.621363\pi$
$859$	−21.8117	−0.744207	−0.372103	+	0.928191i	$0.621363\pi$
$860$	16.0274	0.546531
$861$	0	0
$862$	17.2355	0.587043
$863$	14.4383	0.491487	0.245743	−	0.969335i	$-0.420968\pi$
$863$	14.4383	0.491487	0.245743	+	0.969335i	$0.420968\pi$
$864$	0	0
$865$	3.81400	0.129680
$866$	24.5562	0.834453
$867$	0	0
$868$	−0.440094	−0.0149378
$869$	−25.8125	−0.875630
$870$	0	0
$871$	49.9446	1.69231
$872$	12.2900	0.416191
$873$	0	0
$874$	11.0229	0.372857
$875$	−1.02736	−0.0347310
$876$	0	0
$877$	−8.02253	−0.270902	−0.135451	−	0.990784i	$-0.543248\pi$
$877$	−8.02253	−0.270902	−0.135451	+	0.990784i	$0.543248\pi$
$878$	21.3132	0.719284
$879$	0	0
$880$	8.85899	0.298636
$881$	−17.5013	−0.589634	−0.294817	−	0.955554i	$-0.595259\pi$
$881$	−17.5013	−0.589634	−0.294817	+	0.955554i	$0.595259\pi$
$882$	0	0
$883$	−17.3465	−0.583757	−0.291878	−	0.956455i	$-0.594280\pi$
$883$	−17.3465	−0.583757	−0.291878	+	0.956455i	$0.594280\pi$
$884$	−10.5184	−0.353774
$885$	0	0
$886$	−22.6790	−0.761917
$887$	52.4553	1.76128	0.880638	−	0.473791i	$-0.157115\pi$
$887$	52.4553	1.76128	0.880638	+	0.473791i	$0.157115\pi$
$888$	0	0
$889$	2.48542	0.0833582
$890$	37.0585	1.24220
$891$	0	0
$892$	6.05563	0.202758
$893$	−55.5697	−1.85957
$894$	0	0
$895$	−12.2141	−0.408273
$896$	0.211316	0.00705956
$897$	0	0
$898$	−32.4566	−1.08309
$899$	2.31190	0.0771063
$900$	0	0
$901$	25.2890	0.842500
$902$	−14.2861	−0.475676
$903$	0	0
$904$	11.9287	0.396741
$905$	−80.6050	−2.67940
$906$	0	0
$907$	−28.7504	−0.954641	−0.477320	−	0.878729i	$-0.658392\pi$
$907$	−28.7504	−0.954641	−0.477320	+	0.878729i	$0.658392\pi$
$908$	2.97393	0.0986932
$909$	0	0
$910$	3.70854	0.122937
$911$	−28.9160	−0.958029	−0.479014	−	0.877807i	$-0.659006\pi$
$911$	−28.9160	−0.958029	−0.479014	+	0.877807i	$0.659006\pi$
$912$	0	0
$913$	28.5370	0.944436
$914$	28.2009	0.932803
$915$	0	0
$916$	2.39817	0.0792378
$917$	−2.93700	−0.0969884
$918$	0	0
$919$	−52.4043	−1.72866	−0.864330	−	0.502925i	$-0.832257\pi$
$919$	−52.4043	−1.72866	−0.864330	+	0.502925i	$0.832257\pi$
$920$	5.95620	0.196370
$921$	0	0
$922$	−34.2926	−1.12937
$923$	−13.5949	−0.447481
$924$	0	0
$925$	53.3404	1.75382
$926$	−29.9743	−0.985017
$927$	0	0
$928$	−1.11008	−0.0364403
$929$	0.985702	0.0323398	0.0161699	−	0.999869i	$-0.494853\pi$
$929$	0.985702	0.0323398	0.0161699	+	0.999869i	$0.494853\pi$
$930$	0	0
$931$	−43.5325	−1.42672
$932$	17.4919	0.572967
$933$	0	0
$934$	−24.6700	−0.807225
$935$	−17.9569	−0.587253
$936$	0	0
$937$	40.1614	1.31202	0.656008	−	0.754754i	$-0.272244\pi$
$937$	40.1614	1.31202	0.656008	+	0.754754i	$0.272244\pi$
$938$	2.03383	0.0664070
$939$	0	0
$940$	−30.0268	−0.979368
$941$	−33.8428	−1.10324	−0.551622	−	0.834094i	$-0.685991\pi$
$941$	−33.8428	−1.10324	−0.551622	+	0.834094i	$0.685991\pi$
$942$	0	0
$943$	−9.60505	−0.312783
$944$	7.55598	0.245926
$945$	0	0
$946$	−12.4141	−0.403617
$947$	−28.8796	−0.938460	−0.469230	−	0.883076i	$-0.655468\pi$
$947$	−28.8796	−0.938460	−0.469230	+	0.883076i	$0.655468\pi$
$948$	0	0
$949$	36.2306	1.17610
$950$	−40.2917	−1.30724
$951$	0	0
$952$	−0.428330	−0.0138822
$953$	13.1799	0.426940	0.213470	−	0.976950i	$-0.431523\pi$
$953$	13.1799	0.426940	0.213470	+	0.976950i	$0.431523\pi$
$954$	0	0
$955$	21.6388	0.700216
$956$	−5.72249	−0.185079
$957$	0	0
$958$	−10.9908	−0.355097
$959$	−4.02948	−0.130119
$960$	0	0
$961$	−26.6626	−0.860084
$962$	−42.9973	−1.38629
$963$	0	0
$964$	13.6485	0.439590
$965$	−45.6681	−1.47011
$966$	0	0
$967$	−22.4858	−0.723094	−0.361547	−	0.932354i	$-0.617751\pi$
$967$	−22.4858	−0.723094	−0.361547	+	0.932354i	$0.617751\pi$
$968$	4.13825	0.133008
$969$	0	0
$970$	−2.17806	−0.0699332
$971$	−1.35975	−0.0436365	−0.0218183	−	0.999762i	$-0.506946\pi$
$971$	−1.35975	−0.0436365	−0.0218183	+	0.999762i	$0.506946\pi$
$972$	0	0
$973$	−2.41553	−0.0774385
$974$	15.1938	0.486839
$975$	0	0
$976$	−13.6395	−0.436590
$977$	23.7667	0.760364	0.380182	−	0.924912i	$-0.375861\pi$
$977$	23.7667	0.760364	0.380182	+	0.924912i	$0.375861\pi$
$978$	0	0
$979$	−28.7038	−0.917376
$980$	−23.5226	−0.751402
$981$	0	0
$982$	17.0007	0.542513
$983$	−36.3850	−1.16050	−0.580251	−	0.814438i	$-0.697046\pi$
$983$	−36.3850	−1.16050	−0.580251	+	0.814438i	$0.697046\pi$
$984$	0	0
$985$	−18.4951	−0.589303
$986$	2.25010	0.0716579
$987$	0	0
$988$	32.4788	1.03329
$989$	−8.34641	−0.265400
$990$	0	0
$991$	−17.8966	−0.568506	−0.284253	−	0.958749i	$-0.591746\pi$
$991$	−17.8966	−0.568506	−0.284253	+	0.958749i	$0.591746\pi$
$992$	−2.08264	−0.0661238
$993$	0	0
$994$	−0.553607	−0.0175594
$995$	58.0995	1.84188
$996$	0	0
$997$	45.9039	1.45379	0.726896	−	0.686747i	$-0.240962\pi$
$997$	45.9039	1.45379	0.726896	+	0.686747i	$0.240962\pi$
$998$	13.2475	0.419343
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.p.1.8		8
3.2	odd	2		1614.2.a.j.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1614.2.a.j.1.1	✓	8	3.2	odd	2
4842.2.a.p.1.8		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.38195 q^{5} -0.211316 q^{7} -1.00000 q^{8} -3.38195 q^{10} +2.61950 q^{11} +5.18926 q^{13} +0.211316 q^{14} +1.00000 q^{16} -2.02697 q^{17} +6.25886 q^{19} +3.38195 q^{20} -2.61950 q^{22} -1.76118 q^{23} +6.43755 q^{25} -5.18926 q^{26} -0.211316 q^{28} +1.11008 q^{29} +2.08264 q^{31} -1.00000 q^{32} +2.02697 q^{34} -0.714658 q^{35} +8.28583 q^{37} -6.25886 q^{38} -3.38195 q^{40} +5.45377 q^{41} +4.73911 q^{43} +2.61950 q^{44} +1.76118 q^{46} -8.87857 q^{47} -6.95535 q^{49} -6.43755 q^{50} +5.18926 q^{52} -12.4763 q^{53} +8.85899 q^{55} +0.211316 q^{56} -1.11008 q^{58} +7.55598 q^{59} -13.6395 q^{61} -2.08264 q^{62} +1.00000 q^{64} +17.5498 q^{65} +9.62462 q^{67} -2.02697 q^{68} +0.714658 q^{70} -2.61981 q^{71} +6.98186 q^{73} -8.28583 q^{74} +6.25886 q^{76} -0.553540 q^{77} -9.85401 q^{79} +3.38195 q^{80} -5.45377 q^{82} +10.8941 q^{83} -6.85509 q^{85} -4.73911 q^{86} -2.61950 q^{88} -10.9577 q^{89} -1.09657 q^{91} -1.76118 q^{92} +8.87857 q^{94} +21.1671 q^{95} +0.644025 q^{97} +6.95535 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 4 q^{5} + 9 q^{7} - 8 q^{8} + 4 q^{10} - 2 q^{11} + 9 q^{13} - 9 q^{14} + 8 q^{16} + 2 q^{17} + 12 q^{19} - 4 q^{20} + 2 q^{22} - 6 q^{23} + 10 q^{25} - 9 q^{26} + 9 q^{28} - q^{29}+ \cdots + 3 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 4 * q^5 + 9 * q^7 - 8 * q^8 + 4 * q^10 - 2 * q^11 + 9 * q^13 - 9 * q^14 + 8 * q^16 + 2 * q^17 + 12 * q^19 - 4 * q^20 + 2 * q^22 - 6 * q^23 + 10 * q^25 - 9 * q^26 + 9 * q^28 - q^29 + 8 * q^31 - 8 * q^32 - 2 * q^34 + 7 * q^35 + 10 * q^37 - 12 * q^38 + 4 * q^40 + q^41 + 8 * q^43 - 2 * q^44 + 6 * q^46 - 2 * q^47 - 3 * q^49 - 10 * q^50 + 9 * q^52 + 9 * q^53 - q^55 - 9 * q^56 + q^58 + 3 * q^59 - 8 * q^62 + 8 * q^64 + 18 * q^65 + 6 * q^67 + 2 * q^68 - 7 * q^70 + 17 * q^71 - q^73 - 10 * q^74 + 12 * q^76 + 4 * q^77 + q^79 - 4 * q^80 - q^82 + 17 * q^83 - 20 * q^85 - 8 * q^86 + 2 * q^88 + 4 * q^89 + 15 * q^91 - 6 * q^92 + 2 * q^94 + 7 * q^95 + 13 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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