LMFDB - Embedded newform 4842.2.a.o.1.5

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 $ x^7 - x^6 - 10x^5 + 7x^4 + 27x^3 - 15x^2 - 20*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.30699$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.40321 q^{5} +1.52017 q^{7} +1.00000 q^{8} +2.40321 q^{10} -5.07638 q^{11} +5.11021 q^{13} +1.52017 q^{14} +1.00000 q^{16} +6.65764 q^{17} -5.71222 q^{19} +2.40321 q^{20} -5.07638 q^{22} +6.36685 q^{23} +0.775401 q^{25} +5.11021 q^{26} +1.52017 q^{28} +6.00511 q^{29} +2.77822 q^{31} +1.00000 q^{32} +6.65764 q^{34} +3.65329 q^{35} -1.59410 q^{37} -5.71222 q^{38} +2.40321 q^{40} -9.69619 q^{41} -9.26120 q^{43} -5.07638 q^{44} +6.36685 q^{46} +7.55235 q^{47} -4.68907 q^{49} +0.775401 q^{50} +5.11021 q^{52} +9.42058 q^{53} -12.1996 q^{55} +1.52017 q^{56} +6.00511 q^{58} +6.31332 q^{59} -7.97918 q^{61} +2.77822 q^{62} +1.00000 q^{64} +12.2809 q^{65} -2.54885 q^{67} +6.65764 q^{68} +3.65329 q^{70} -1.98957 q^{71} +12.7523 q^{73} -1.59410 q^{74} -5.71222 q^{76} -7.71699 q^{77} +3.64283 q^{79} +2.40321 q^{80} -9.69619 q^{82} +0.689504 q^{83} +15.9997 q^{85} -9.26120 q^{86} -5.07638 q^{88} -2.78972 q^{89} +7.76841 q^{91} +6.36685 q^{92} +7.55235 q^{94} -13.7276 q^{95} +11.9363 q^{97} -4.68907 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 7 q^{4} + 6 q^{5} - 3 q^{7} + 7 q^{8} + 6 q^{10} + 12 q^{11} + 3 q^{13} - 3 q^{14} + 7 q^{16} + 8 q^{17} - 7 q^{19} + 6 q^{20} + 12 q^{22} + 22 q^{23} + 9 q^{25} + 3 q^{26} - 3 q^{28} + 7 q^{29}+ \cdots + 6 q^{98}+O(q^{100}) $ 7 * q + 7 * q^2 + 7 * q^4 + 6 * q^5 - 3 * q^7 + 7 * q^8 + 6 * q^10 + 12 * q^11 + 3 * q^13 - 3 * q^14 + 7 * q^16 + 8 * q^17 - 7 * q^19 + 6 * q^20 + 12 * q^22 + 22 * q^23 + 9 * q^25 + 3 * q^26 - 3 * q^28 + 7 * q^29 - 3 * q^31 + 7 * q^32 + 8 * q^34 + 25 * q^35 - 13 * q^37 - 7 * q^38 + 6 * q^40 + 11 * q^41 - 14 * q^43 + 12 * q^44 + 22 * q^46 + 23 * q^47 + 6 * q^49 + 9 * q^50 + 3 * q^52 + 21 * q^53 + 15 * q^55 - 3 * q^56 + 7 * q^58 + 20 * q^59 + 13 * q^61 - 3 * q^62 + 7 * q^64 + 29 * q^65 - 30 * q^67 + 8 * q^68 + 25 * q^70 + 23 * q^71 + 9 * q^73 - 13 * q^74 - 7 * q^76 - 8 * q^77 + 11 * q^79 + 6 * q^80 + 11 * q^82 + 3 * q^83 + 21 * q^85 - 14 * q^86 + 12 * q^88 - 16 * q^89 - 10 * q^91 + 22 * q^92 + 23 * q^94 + 21 * q^97 + 6 * 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
$5$	2.40321	1.07475	0.537373	−	0.843344i	$-0.319417\pi$
$5$	2.40321	1.07475	0.537373	+	0.843344i	$0.319417\pi$
$6$	0	0
$7$	1.52017	0.574572	0.287286	−	0.957845i	$-0.407247\pi$
$7$	1.52017	0.574572	0.287286	+	0.957845i	$0.407247\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.40321	0.759961
$11$	−5.07638	−1.53059	−0.765294	−	0.643681i	$-0.777406\pi$
$11$	−5.07638	−1.53059	−0.765294	+	0.643681i	$0.777406\pi$
$12$	0	0
$13$	5.11021	1.41732	0.708659	−	0.705552i	$-0.249301\pi$
$13$	5.11021	1.41732	0.708659	+	0.705552i	$0.249301\pi$
$14$	1.52017	0.406284
$15$	0	0
$16$	1.00000	0.250000
$17$	6.65764	1.61471	0.807357	−	0.590063i	$-0.200897\pi$
$17$	6.65764	1.61471	0.807357	+	0.590063i	$0.200897\pi$
$18$	0	0
$19$	−5.71222	−1.31047	−0.655236	−	0.755424i	$-0.727431\pi$
$19$	−5.71222	−1.31047	−0.655236	+	0.755424i	$0.727431\pi$
$20$	2.40321	0.537373
$21$	0	0
$22$	−5.07638	−1.08229
$23$	6.36685	1.32758	0.663790	−	0.747919i	$-0.268947\pi$
$23$	6.36685	1.32758	0.663790	+	0.747919i	$0.268947\pi$
$24$	0	0
$25$	0.775401	0.155080
$26$	5.11021	1.00219
$27$	0	0
$28$	1.52017	0.287286
$29$	6.00511	1.11512	0.557560	−	0.830137i	$-0.311738\pi$
$29$	6.00511	1.11512	0.557560	+	0.830137i	$0.311738\pi$
$30$	0	0
$31$	2.77822	0.498984	0.249492	−	0.968377i	$-0.419736\pi$
$31$	2.77822	0.498984	0.249492	+	0.968377i	$0.419736\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.65764	1.14178
$35$	3.65329	0.617519
$36$	0	0
$37$	−1.59410	−0.262069	−0.131034	−	0.991378i	$-0.541830\pi$
$37$	−1.59410	−0.262069	−0.131034	+	0.991378i	$0.541830\pi$
$38$	−5.71222	−0.926644
$39$	0	0
$40$	2.40321	0.379980
$41$	−9.69619	−1.51429	−0.757146	−	0.653246i	$-0.773407\pi$
$41$	−9.69619	−1.51429	−0.757146	+	0.653246i	$0.773407\pi$
$42$	0	0
$43$	−9.26120	−1.41232	−0.706160	−	0.708053i	$-0.749574\pi$
$43$	−9.26120	−1.41232	−0.706160	+	0.708053i	$0.749574\pi$
$44$	−5.07638	−0.765294
$45$	0	0
$46$	6.36685	0.938741
$47$	7.55235	1.10162	0.550811	−	0.834630i	$-0.314318\pi$
$47$	7.55235	1.10162	0.550811	+	0.834630i	$0.314318\pi$
$48$	0	0
$49$	−4.68907	−0.669867
$50$	0.775401	0.109658
$51$	0	0
$52$	5.11021	0.708659
$53$	9.42058	1.29402	0.647008	−	0.762483i	$-0.276020\pi$
$53$	9.42058	1.29402	0.647008	+	0.762483i	$0.276020\pi$
$54$	0	0
$55$	−12.1996	−1.64499
$56$	1.52017	0.203142
$57$	0	0
$58$	6.00511	0.788509
$59$	6.31332	0.821924	0.410962	−	0.911652i	$-0.365193\pi$
$59$	6.31332	0.821924	0.410962	+	0.911652i	$0.365193\pi$
$60$	0	0
$61$	−7.97918	−1.02163	−0.510815	−	0.859691i	$-0.670656\pi$
$61$	−7.97918	−1.02163	−0.510815	+	0.859691i	$0.670656\pi$
$62$	2.77822	0.352835
$63$	0	0
$64$	1.00000	0.125000
$65$	12.2809	1.52326
$66$	0	0
$67$	−2.54885	−0.311392	−0.155696	−	0.987805i	$-0.549762\pi$
$67$	−2.54885	−0.311392	−0.155696	+	0.987805i	$0.549762\pi$
$68$	6.65764	0.807357
$69$	0	0
$70$	3.65329	0.436652
$71$	−1.98957	−0.236118	−0.118059	−	0.993007i	$-0.537667\pi$
$71$	−1.98957	−0.236118	−0.118059	+	0.993007i	$0.537667\pi$
$72$	0	0
$73$	12.7523	1.49254	0.746269	−	0.665644i	$-0.231843\pi$
$73$	12.7523	1.49254	0.746269	+	0.665644i	$0.231843\pi$
$74$	−1.59410	−0.185311
$75$	0	0
$76$	−5.71222	−0.655236
$77$	−7.71699	−0.879432
$78$	0	0
$79$	3.64283	0.409851	0.204925	−	0.978778i	$-0.434305\pi$
$79$	3.64283	0.409851	0.204925	+	0.978778i	$0.434305\pi$
$80$	2.40321	0.268687
$81$	0	0
$82$	−9.69619	−1.07077
$83$	0.689504	0.0756829	0.0378415	−	0.999284i	$-0.487952\pi$
$83$	0.689504	0.0756829	0.0378415	+	0.999284i	$0.487952\pi$
$84$	0	0
$85$	15.9997	1.73541
$86$	−9.26120	−0.998660
$87$	0	0
$88$	−5.07638	−0.541144
$89$	−2.78972	−0.295709	−0.147855	−	0.989009i	$-0.547237\pi$
$89$	−2.78972	−0.295709	−0.147855	+	0.989009i	$0.547237\pi$
$90$	0	0
$91$	7.76841	0.814350
$92$	6.36685	0.663790
$93$	0	0
$94$	7.55235	0.778965
$95$	−13.7276	−1.40843
$96$	0	0
$97$	11.9363	1.21194	0.605972	−	0.795486i	$-0.292784\pi$
$97$	11.9363	1.21194	0.605972	+	0.795486i	$0.292784\pi$
$98$	−4.68907	−0.473668
$99$	0	0
$100$	0.775401	0.0775401
$101$	14.5273	1.44552	0.722759	−	0.691100i	$-0.242873\pi$
$101$	14.5273	1.44552	0.722759	+	0.691100i	$0.242873\pi$
$102$	0	0
$103$	−10.1048	−0.995656	−0.497828	−	0.867276i	$-0.665869\pi$
$103$	−10.1048	−0.995656	−0.497828	+	0.867276i	$0.665869\pi$
$104$	5.11021	0.501097
$105$	0	0
$106$	9.42058	0.915007
$107$	12.2725	1.18643	0.593214	−	0.805045i	$-0.297859\pi$
$107$	12.2725	1.18643	0.593214	+	0.805045i	$0.297859\pi$
$108$	0	0
$109$	14.2163	1.36168	0.680838	−	0.732434i	$-0.261616\pi$
$109$	14.2163	1.36168	0.680838	+	0.732434i	$0.261616\pi$
$110$	−12.1996	−1.16319
$111$	0	0
$112$	1.52017	0.143643
$113$	−12.7024	−1.19494	−0.597471	−	0.801891i	$-0.703828\pi$
$113$	−12.7024	−1.19494	−0.597471	+	0.801891i	$0.703828\pi$
$114$	0	0
$115$	15.3009	1.42681
$116$	6.00511	0.557560
$117$	0	0
$118$	6.31332	0.581188
$119$	10.1208	0.927769
$120$	0	0
$121$	14.7697	1.34270
$122$	−7.97918	−0.722401
$123$	0	0
$124$	2.77822	0.249492
$125$	−10.1526	−0.908075
$126$	0	0
$127$	−17.1800	−1.52448	−0.762238	−	0.647297i	$-0.775899\pi$
$127$	−17.1800	−1.52448	−0.762238	+	0.647297i	$0.775899\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	12.2809	1.07711
$131$	−5.22919	−0.456877	−0.228438	−	0.973558i	$-0.573362\pi$
$131$	−5.22919	−0.456877	−0.228438	+	0.973558i	$0.573362\pi$
$132$	0	0
$133$	−8.68356	−0.752960
$134$	−2.54885	−0.220187
$135$	0	0
$136$	6.65764	0.570888
$137$	12.9664	1.10779	0.553895	−	0.832586i	$-0.313141\pi$
$137$	12.9664	1.10779	0.553895	+	0.832586i	$0.313141\pi$
$138$	0	0
$139$	−0.692646	−0.0587495	−0.0293747	−	0.999568i	$-0.509352\pi$
$139$	−0.692646	−0.0587495	−0.0293747	+	0.999568i	$0.509352\pi$
$140$	3.65329	0.308759
$141$	0	0
$142$	−1.98957	−0.166961
$143$	−25.9414	−2.16933
$144$	0	0
$145$	14.4315	1.19847
$146$	12.7523	1.05538
$147$	0	0
$148$	−1.59410	−0.131034
$149$	−11.5986	−0.950197	−0.475099	−	0.879933i	$-0.657588\pi$
$149$	−11.5986	−0.950197	−0.475099	+	0.879933i	$0.657588\pi$
$150$	0	0
$151$	1.14291	0.0930089	0.0465045	−	0.998918i	$-0.485192\pi$
$151$	1.14291	0.0930089	0.0465045	+	0.998918i	$0.485192\pi$
$152$	−5.71222	−0.463322
$153$	0	0
$154$	−7.71699	−0.621852
$155$	6.67665	0.536281
$156$	0	0
$157$	−12.2350	−0.976460	−0.488230	−	0.872715i	$-0.662357\pi$
$157$	−12.2350	−0.976460	−0.488230	+	0.872715i	$0.662357\pi$
$158$	3.64283	0.289808
$159$	0	0
$160$	2.40321	0.189990
$161$	9.67872	0.762790
$162$	0	0
$163$	−23.6815	−1.85488	−0.927438	−	0.373978i	$-0.877994\pi$
$163$	−23.6815	−1.85488	−0.927438	+	0.373978i	$0.877994\pi$
$164$	−9.69619	−0.757146
$165$	0	0
$166$	0.689504	0.0535159
$167$	3.54694	0.274470	0.137235	−	0.990538i	$-0.456178\pi$
$167$	3.54694	0.274470	0.137235	+	0.990538i	$0.456178\pi$
$168$	0	0
$169$	13.1142	1.00879
$170$	15.9997	1.22712
$171$	0	0
$172$	−9.26120	−0.706160
$173$	−20.8927	−1.58844	−0.794220	−	0.607630i	$-0.792120\pi$
$173$	−20.8927	−1.58844	−0.794220	+	0.607630i	$0.792120\pi$
$174$	0	0
$175$	1.17874	0.0891047
$176$	−5.07638	−0.382647
$177$	0	0
$178$	−2.78972	−0.209098
$179$	−2.13073	−0.159258	−0.0796290	−	0.996825i	$-0.525374\pi$
$179$	−2.13073	−0.159258	−0.0796290	+	0.996825i	$0.525374\pi$
$180$	0	0
$181$	18.5589	1.37948	0.689738	−	0.724059i	$-0.257726\pi$
$181$	18.5589	1.37948	0.689738	+	0.724059i	$0.257726\pi$
$182$	7.76841	0.575833
$183$	0	0
$184$	6.36685	0.469371
$185$	−3.83095	−0.281657
$186$	0	0
$187$	−33.7967	−2.47146
$188$	7.55235	0.550811
$189$	0	0
$190$	−13.7276	−0.995907
$191$	−15.6213	−1.13032	−0.565160	−	0.824982i	$-0.691185\pi$
$191$	−15.6213	−1.13032	−0.565160	+	0.824982i	$0.691185\pi$
$192$	0	0
$193$	−18.5417	−1.33466	−0.667330	−	0.744762i	$-0.732563\pi$
$193$	−18.5417	−1.33466	−0.667330	+	0.744762i	$0.732563\pi$
$194$	11.9363	0.856974
$195$	0	0
$196$	−4.68907	−0.334934
$197$	9.98285	0.711249	0.355624	−	0.934629i	$-0.384268\pi$
$197$	9.98285	0.711249	0.355624	+	0.934629i	$0.384268\pi$
$198$	0	0
$199$	5.33431	0.378139	0.189070	−	0.981964i	$-0.439453\pi$
$199$	5.33431	0.378139	0.189070	+	0.981964i	$0.439453\pi$
$200$	0.775401	0.0548292
$201$	0	0
$202$	14.5273	1.02214
$203$	9.12880	0.640716
$204$	0	0
$205$	−23.3020	−1.62748
$206$	−10.1048	−0.704035
$207$	0	0
$208$	5.11021	0.354329
$209$	28.9974	2.00579
$210$	0	0
$211$	−19.7945	−1.36271	−0.681355	−	0.731953i	$-0.738609\pi$
$211$	−19.7945	−1.36271	−0.681355	+	0.731953i	$0.738609\pi$
$212$	9.42058	0.647008
$213$	0	0
$214$	12.2725	0.838931
$215$	−22.2566	−1.51789
$216$	0	0
$217$	4.22338	0.286702
$218$	14.2163	0.962850
$219$	0	0
$220$	−12.1996	−0.822497
$221$	34.0219	2.28856
$222$	0	0
$223$	29.2824	1.96089	0.980447	−	0.196783i	$-0.0630495\pi$
$223$	29.2824	1.96089	0.980447	+	0.196783i	$0.0630495\pi$
$224$	1.52017	0.101571
$225$	0	0
$226$	−12.7024	−0.844951
$227$	5.90479	0.391915	0.195957	−	0.980612i	$-0.437219\pi$
$227$	5.90479	0.391915	0.195957	+	0.980612i	$0.437219\pi$
$228$	0	0
$229$	14.2761	0.943391	0.471695	−	0.881762i	$-0.343642\pi$
$229$	14.2761	0.943391	0.471695	+	0.881762i	$0.343642\pi$
$230$	15.3009	1.00891
$231$	0	0
$232$	6.00511	0.394254
$233$	27.0850	1.77440	0.887198	−	0.461389i	$-0.152649\pi$
$233$	27.0850	1.77440	0.887198	+	0.461389i	$0.152649\pi$
$234$	0	0
$235$	18.1498	1.18397
$236$	6.31332	0.410962
$237$	0	0
$238$	10.1208	0.656032
$239$	−7.97200	−0.515666	−0.257833	−	0.966190i	$-0.583008\pi$
$239$	−7.97200	−0.515666	−0.257833	+	0.966190i	$0.583008\pi$
$240$	0	0
$241$	−2.36516	−0.152353	−0.0761767	−	0.997094i	$-0.524271\pi$
$241$	−2.36516	−0.152353	−0.0761767	+	0.997094i	$0.524271\pi$
$242$	14.7697	0.949430
$243$	0	0
$244$	−7.97918	−0.510815
$245$	−11.2688	−0.719938
$246$	0	0
$247$	−29.1906	−1.85735
$248$	2.77822	0.176417
$249$	0	0
$250$	−10.1526	−0.642106
$251$	23.0435	1.45449	0.727246	−	0.686376i	$-0.240800\pi$
$251$	23.0435	1.45449	0.727246	+	0.686376i	$0.240800\pi$
$252$	0	0
$253$	−32.3206	−2.03198
$254$	−17.1800	−1.07797
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.2436	−1.44990	−0.724948	−	0.688803i	$-0.758136\pi$
$257$	−23.2436	−1.44990	−0.724948	+	0.688803i	$0.758136\pi$
$258$	0	0
$259$	−2.42331	−0.150577
$260$	12.2809	0.761628
$261$	0	0
$262$	−5.22919	−0.323061
$263$	−4.62308	−0.285072	−0.142536	−	0.989790i	$-0.545526\pi$
$263$	−4.62308	−0.285072	−0.142536	+	0.989790i	$0.545526\pi$
$264$	0	0
$265$	22.6396	1.39074
$266$	−8.68356	−0.532423
$267$	0	0
$268$	−2.54885	−0.155696
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	12.4311	0.755135	0.377568	−	0.925982i	$-0.376761\pi$
$271$	12.4311	0.755135	0.377568	+	0.925982i	$0.376761\pi$
$272$	6.65764	0.403679
$273$	0	0
$274$	12.9664	0.783326
$275$	−3.93623	−0.237364
$276$	0	0
$277$	−4.92762	−0.296072	−0.148036	−	0.988982i	$-0.547295\pi$
$277$	−4.92762	−0.296072	−0.148036	+	0.988982i	$0.547295\pi$
$278$	−0.692646	−0.0415422
$279$	0	0
$280$	3.65329	0.218326
$281$	21.3194	1.27181	0.635905	−	0.771767i	$-0.280627\pi$
$281$	21.3194	1.27181	0.635905	+	0.771767i	$0.280627\pi$
$282$	0	0
$283$	−29.9997	−1.78330	−0.891649	−	0.452728i	$-0.850451\pi$
$283$	−29.9997	−1.78330	−0.891649	+	0.452728i	$0.850451\pi$
$284$	−1.98957	−0.118059
$285$	0	0
$286$	−25.9414	−1.53395
$287$	−14.7399	−0.870069
$288$	0	0
$289$	27.3241	1.60730
$290$	14.4315	0.847447
$291$	0	0
$292$	12.7523	0.746269
$293$	−16.9481	−0.990120	−0.495060	−	0.868859i	$-0.664854\pi$
$293$	−16.9481	−0.990120	−0.495060	+	0.868859i	$0.664854\pi$
$294$	0	0
$295$	15.1722	0.883360
$296$	−1.59410	−0.0926553
$297$	0	0
$298$	−11.5986	−0.671891
$299$	32.5360	1.88160
$300$	0	0
$301$	−14.0786	−0.811479
$302$	1.14291	0.0657672
$303$	0	0
$304$	−5.71222	−0.327618
$305$	−19.1756	−1.09799
$306$	0	0
$307$	18.4715	1.05422	0.527112	−	0.849796i	$-0.323275\pi$
$307$	18.4715	1.05422	0.527112	+	0.849796i	$0.323275\pi$
$308$	−7.71699	−0.439716
$309$	0	0
$310$	6.67665	0.379208
$311$	−13.9141	−0.788995	−0.394497	−	0.918897i	$-0.629081\pi$
$311$	−13.9141	−0.788995	−0.394497	+	0.918897i	$0.629081\pi$
$312$	0	0
$313$	21.9666	1.24162	0.620812	−	0.783959i	$-0.286803\pi$
$313$	21.9666	1.24162	0.620812	+	0.783959i	$0.286803\pi$
$314$	−12.2350	−0.690462
$315$	0	0
$316$	3.64283	0.204925
$317$	3.38070	0.189879	0.0949395	−	0.995483i	$-0.469734\pi$
$317$	3.38070	0.189879	0.0949395	+	0.995483i	$0.469734\pi$
$318$	0	0
$319$	−30.4842	−1.70679
$320$	2.40321	0.134343
$321$	0	0
$322$	9.67872	0.539374
$323$	−38.0299	−2.11604
$324$	0	0
$325$	3.96246	0.219798
$326$	−23.6815	−1.31159
$327$	0	0
$328$	−9.69619	−0.535383
$329$	11.4809	0.632961
$330$	0	0
$331$	21.9585	1.20695	0.603476	−	0.797382i	$-0.293782\pi$
$331$	21.9585	1.20695	0.603476	+	0.797382i	$0.293782\pi$
$332$	0.689504	0.0378415
$333$	0	0
$334$	3.54694	0.194080
$335$	−6.12542	−0.334667
$336$	0	0
$337$	3.21883	0.175341	0.0876705	−	0.996150i	$-0.472058\pi$
$337$	3.21883	0.175341	0.0876705	+	0.996150i	$0.472058\pi$
$338$	13.1142	0.713321
$339$	0	0
$340$	15.9997	0.867704
$341$	−14.1033	−0.763738
$342$	0	0
$343$	−17.7694	−0.959459
$344$	−9.26120	−0.499330
$345$	0	0
$346$	−20.8927	−1.12320
$347$	−15.8474	−0.850734	−0.425367	−	0.905021i	$-0.639855\pi$
$347$	−15.8474	−0.850734	−0.425367	+	0.905021i	$0.639855\pi$
$348$	0	0
$349$	−17.0834	−0.914451	−0.457226	−	0.889351i	$-0.651157\pi$
$349$	−17.0834	−0.914451	−0.457226	+	0.889351i	$0.651157\pi$
$350$	1.17874	0.0630066
$351$	0	0
$352$	−5.07638	−0.270572
$353$	4.73633	0.252090	0.126045	−	0.992025i	$-0.459772\pi$
$353$	4.73633	0.252090	0.126045	+	0.992025i	$0.459772\pi$
$354$	0	0
$355$	−4.78134	−0.253767
$356$	−2.78972	−0.147855
$357$	0	0
$358$	−2.13073	−0.112612
$359$	−18.2662	−0.964051	−0.482025	−	0.876157i	$-0.660099\pi$
$359$	−18.2662	−0.964051	−0.482025	+	0.876157i	$0.660099\pi$
$360$	0	0
$361$	13.6294	0.717338
$362$	18.5589	0.975437
$363$	0	0
$364$	7.76841	0.407175
$365$	30.6463	1.60410
$366$	0	0
$367$	5.99172	0.312765	0.156383	−	0.987697i	$-0.450017\pi$
$367$	5.99172	0.312765	0.156383	+	0.987697i	$0.450017\pi$
$368$	6.36685	0.331895
$369$	0	0
$370$	−3.83095	−0.199162
$371$	14.3209	0.743505
$372$	0	0
$373$	−16.9416	−0.877203	−0.438601	−	0.898682i	$-0.644526\pi$
$373$	−16.9416	−0.877203	−0.438601	+	0.898682i	$0.644526\pi$
$374$	−33.7967	−1.74759
$375$	0	0
$376$	7.55235	0.389483
$377$	30.6873	1.58048
$378$	0	0
$379$	−16.7293	−0.859325	−0.429663	−	0.902989i	$-0.641368\pi$
$379$	−16.7293	−0.859325	−0.429663	+	0.902989i	$0.641368\pi$
$380$	−13.7276	−0.704213
$381$	0	0
$382$	−15.6213	−0.799257
$383$	26.8421	1.37157	0.685785	−	0.727804i	$-0.259459\pi$
$383$	26.8421	1.37157	0.685785	+	0.727804i	$0.259459\pi$
$384$	0	0
$385$	−18.5455	−0.945167
$386$	−18.5417	−0.943747
$387$	0	0
$388$	11.9363	0.605972
$389$	−8.89100	−0.450792	−0.225396	−	0.974267i	$-0.572368\pi$
$389$	−8.89100	−0.450792	−0.225396	+	0.974267i	$0.572368\pi$
$390$	0	0
$391$	42.3882	2.14366
$392$	−4.68907	−0.236834
$393$	0	0
$394$	9.98285	0.502929
$395$	8.75448	0.440486
$396$	0	0
$397$	−15.3011	−0.767939	−0.383970	−	0.923346i	$-0.625443\pi$
$397$	−15.3011	−0.767939	−0.383970	+	0.923346i	$0.625443\pi$
$398$	5.33431	0.267385
$399$	0	0
$400$	0.775401	0.0387701
$401$	9.91869	0.495316	0.247658	−	0.968848i	$-0.420339\pi$
$401$	9.91869	0.495316	0.247658	+	0.968848i	$0.420339\pi$
$402$	0	0
$403$	14.1973	0.707218
$404$	14.5273	0.722759
$405$	0	0
$406$	9.12880	0.453055
$407$	8.09227	0.401119
$408$	0	0
$409$	−13.4737	−0.666233	−0.333117	−	0.942886i	$-0.608100\pi$
$409$	−13.4737	−0.666233	−0.333117	+	0.942886i	$0.608100\pi$
$410$	−23.3020	−1.15080
$411$	0	0
$412$	−10.1048	−0.497828
$413$	9.59734	0.472254
$414$	0	0
$415$	1.65702	0.0813400
$416$	5.11021	0.250549
$417$	0	0
$418$	28.9974	1.41831
$419$	24.5320	1.19847	0.599233	−	0.800574i	$-0.295472\pi$
$419$	24.5320	1.19847	0.599233	+	0.800574i	$0.295472\pi$
$420$	0	0
$421$	−27.8189	−1.35581	−0.677906	−	0.735149i	$-0.737112\pi$
$421$	−27.8189	−1.35581	−0.677906	+	0.735149i	$0.737112\pi$
$422$	−19.7945	−0.963581
$423$	0	0
$424$	9.42058	0.457504
$425$	5.16234	0.250410
$426$	0	0
$427$	−12.1297	−0.586999
$428$	12.2725	0.593214
$429$	0	0
$430$	−22.2566	−1.07331
$431$	−32.5743	−1.56905	−0.784525	−	0.620097i	$-0.787093\pi$
$431$	−32.5743	−1.56905	−0.784525	+	0.620097i	$0.787093\pi$
$432$	0	0
$433$	−5.11693	−0.245904	−0.122952	−	0.992413i	$-0.539236\pi$
$433$	−5.11693	−0.245904	−0.122952	+	0.992413i	$0.539236\pi$
$434$	4.22338	0.202729
$435$	0	0
$436$	14.2163	0.680838
$437$	−36.3688	−1.73976
$438$	0	0
$439$	18.8040	0.897466	0.448733	−	0.893666i	$-0.351875\pi$
$439$	18.8040	0.897466	0.448733	+	0.893666i	$0.351875\pi$
$440$	−12.1996	−0.581593
$441$	0	0
$442$	34.0219	1.61826
$443$	−16.7817	−0.797325	−0.398662	−	0.917098i	$-0.630525\pi$
$443$	−16.7817	−0.797325	−0.398662	+	0.917098i	$0.630525\pi$
$444$	0	0
$445$	−6.70426	−0.317812
$446$	29.2824	1.38656
$447$	0	0
$448$	1.52017	0.0718215
$449$	−29.1983	−1.37795	−0.688977	−	0.724784i	$-0.741940\pi$
$449$	−29.1983	−1.37795	−0.688977	+	0.724784i	$0.741940\pi$
$450$	0	0
$451$	49.2216	2.31775
$452$	−12.7024	−0.597471
$453$	0	0
$454$	5.90479	0.277126
$455$	18.6691	0.875220
$456$	0	0
$457$	6.74219	0.315387	0.157693	−	0.987488i	$-0.449594\pi$
$457$	6.74219	0.315387	0.157693	+	0.987488i	$0.449594\pi$
$458$	14.2761	0.667078
$459$	0	0
$460$	15.3009	0.713406
$461$	−17.8901	−0.833226	−0.416613	−	0.909084i	$-0.636783\pi$
$461$	−17.8901	−0.833226	−0.416613	+	0.909084i	$0.636783\pi$
$462$	0	0
$463$	17.2936	0.803703	0.401851	−	0.915705i	$-0.368367\pi$
$463$	17.2936	0.803703	0.401851	+	0.915705i	$0.368367\pi$
$464$	6.00511	0.278780
$465$	0	0
$466$	27.0850	1.25469
$467$	3.00187	0.138910	0.0694550	−	0.997585i	$-0.477874\pi$
$467$	3.00187	0.138910	0.0694550	+	0.997585i	$0.477874\pi$
$468$	0	0
$469$	−3.87470	−0.178917
$470$	18.1498	0.837190
$471$	0	0
$472$	6.31332	0.290594
$473$	47.0134	2.16168
$474$	0	0
$475$	−4.42926	−0.203228
$476$	10.1208	0.463884
$477$	0	0
$478$	−7.97200	−0.364631
$479$	25.7701	1.17747	0.588734	−	0.808327i	$-0.299627\pi$
$479$	25.7701	1.17747	0.588734	+	0.808327i	$0.299627\pi$
$480$	0	0
$481$	−8.14619	−0.371434
$482$	−2.36516	−0.107730
$483$	0	0
$484$	14.7697	0.671349
$485$	28.6853	1.30253
$486$	0	0
$487$	9.85904	0.446756	0.223378	−	0.974732i	$-0.428292\pi$
$487$	9.85904	0.446756	0.223378	+	0.974732i	$0.428292\pi$
$488$	−7.97918	−0.361200
$489$	0	0
$490$	−11.2688	−0.509073
$491$	−9.75565	−0.440266	−0.220133	−	0.975470i	$-0.570649\pi$
$491$	−9.75565	−0.440266	−0.220133	+	0.975470i	$0.570649\pi$
$492$	0	0
$493$	39.9798	1.80060
$494$	−29.1906	−1.31335
$495$	0	0
$496$	2.77822	0.124746
$497$	−3.02448	−0.135667
$498$	0	0
$499$	−3.06163	−0.137058	−0.0685288	−	0.997649i	$-0.521830\pi$
$499$	−3.06163	−0.137058	−0.0685288	+	0.997649i	$0.521830\pi$
$500$	−10.1526	−0.454037
$501$	0	0
$502$	23.0435	1.02848
$503$	15.0940	0.673010	0.336505	−	0.941682i	$-0.390755\pi$
$503$	15.0940	0.673010	0.336505	+	0.941682i	$0.390755\pi$
$504$	0	0
$505$	34.9121	1.55357
$506$	−32.3206	−1.43683
$507$	0	0
$508$	−17.1800	−0.762238
$509$	−13.2575	−0.587629	−0.293814	−	0.955863i	$-0.594925\pi$
$509$	−13.2575	−0.587629	−0.293814	+	0.955863i	$0.594925\pi$
$510$	0	0
$511$	19.3856	0.857570
$512$	1.00000	0.0441942
$513$	0	0
$514$	−23.2436	−1.02523
$515$	−24.2839	−1.07008
$516$	0	0
$517$	−38.3386	−1.68613
$518$	−2.42331	−0.106474
$519$	0	0
$520$	12.2809	0.538553
$521$	−20.2639	−0.887776	−0.443888	−	0.896082i	$-0.646401\pi$
$521$	−20.2639	−0.887776	−0.443888	+	0.896082i	$0.646401\pi$
$522$	0	0
$523$	−2.18471	−0.0955307	−0.0477654	−	0.998859i	$-0.515210\pi$
$523$	−2.18471	−0.0955307	−0.0477654	+	0.998859i	$0.515210\pi$
$524$	−5.22919	−0.228438
$525$	0	0
$526$	−4.62308	−0.201576
$527$	18.4964	0.805716
$528$	0	0
$529$	17.5368	0.762470
$530$	22.6396	0.983401
$531$	0	0
$532$	−8.68356	−0.376480
$533$	−49.5496	−2.14623
$534$	0	0
$535$	29.4933	1.27511
$536$	−2.54885	−0.110094
$537$	0	0
$538$	1.00000	0.0431131
$539$	23.8035	1.02529
$540$	0	0
$541$	21.9779	0.944903	0.472452	−	0.881357i	$-0.343369\pi$
$541$	21.9779	0.944903	0.472452	+	0.881357i	$0.343369\pi$
$542$	12.4311	0.533961
$543$	0	0
$544$	6.65764	0.285444
$545$	34.1647	1.46346
$546$	0	0
$547$	−2.59714	−0.111046	−0.0555229	−	0.998457i	$-0.517683\pi$
$547$	−2.59714	−0.111046	−0.0555229	+	0.998457i	$0.517683\pi$
$548$	12.9664	0.553895
$549$	0	0
$550$	−3.93623	−0.167842
$551$	−34.3025	−1.46133
$552$	0	0
$553$	5.53774	0.235489
$554$	−4.92762	−0.209354
$555$	0	0
$556$	−0.692646	−0.0293747
$557$	−32.9165	−1.39471	−0.697357	−	0.716724i	$-0.745641\pi$
$557$	−32.9165	−1.39471	−0.697357	+	0.716724i	$0.745641\pi$
$558$	0	0
$559$	−47.3266	−2.00170
$560$	3.65329	0.154380
$561$	0	0
$562$	21.3194	0.899306
$563$	16.1700	0.681485	0.340742	−	0.940157i	$-0.389322\pi$
$563$	16.1700	0.681485	0.340742	+	0.940157i	$0.389322\pi$
$564$	0	0
$565$	−30.5265	−1.28426
$566$	−29.9997	−1.26098
$567$	0	0
$568$	−1.98957	−0.0834803
$569$	11.9495	0.500951	0.250476	−	0.968123i	$-0.419413\pi$
$569$	11.9495	0.500951	0.250476	+	0.968123i	$0.419413\pi$
$570$	0	0
$571$	−18.1326	−0.758824	−0.379412	−	0.925228i	$-0.623874\pi$
$571$	−18.1326	−0.758824	−0.379412	+	0.925228i	$0.623874\pi$
$572$	−25.9414	−1.08466
$573$	0	0
$574$	−14.7399	−0.615232
$575$	4.93687	0.205882
$576$	0	0
$577$	22.5512	0.938817	0.469408	−	0.882981i	$-0.344467\pi$
$577$	22.5512	0.938817	0.469408	+	0.882981i	$0.344467\pi$
$578$	27.3241	1.13653
$579$	0	0
$580$	14.4315	0.599236
$581$	1.04817	0.0434853
$582$	0	0
$583$	−47.8225	−1.98060
$584$	12.7523	0.527692
$585$	0	0
$586$	−16.9481	−0.700121
$587$	−22.1880	−0.915796	−0.457898	−	0.889005i	$-0.651397\pi$
$587$	−22.1880	−0.915796	−0.457898	+	0.889005i	$0.651397\pi$
$588$	0	0
$589$	−15.8698	−0.653905
$590$	15.1722	0.624630
$591$	0	0
$592$	−1.59410	−0.0655172
$593$	−30.4282	−1.24954	−0.624768	−	0.780811i	$-0.714806\pi$
$593$	−30.4282	−1.24954	−0.624768	+	0.780811i	$0.714806\pi$
$594$	0	0
$595$	24.3223	0.997117
$596$	−11.5986	−0.475099
$597$	0	0
$598$	32.5360	1.33049
$599$	−15.6413	−0.639086	−0.319543	−	0.947572i	$-0.603529\pi$
$599$	−15.6413	−0.639086	−0.319543	+	0.947572i	$0.603529\pi$
$600$	0	0
$601$	−19.9435	−0.813512	−0.406756	−	0.913537i	$-0.633340\pi$
$601$	−19.9435	−0.813512	−0.406756	+	0.913537i	$0.633340\pi$
$602$	−14.0786	−0.573802
$603$	0	0
$604$	1.14291	0.0465045
$605$	35.4946	1.44306
$606$	0	0
$607$	−11.4793	−0.465932	−0.232966	−	0.972485i	$-0.574843\pi$
$607$	−11.4793	−0.465932	−0.232966	+	0.972485i	$0.574843\pi$
$608$	−5.71222	−0.231661
$609$	0	0
$610$	−19.1756	−0.776398
$611$	38.5941	1.56135
$612$	0	0
$613$	11.2663	0.455044	0.227522	−	0.973773i	$-0.426938\pi$
$613$	11.2663	0.455044	0.227522	+	0.973773i	$0.426938\pi$
$614$	18.4715	0.745449
$615$	0	0
$616$	−7.71699	−0.310926
$617$	−10.0317	−0.403861	−0.201930	−	0.979400i	$-0.564721\pi$
$617$	−10.0317	−0.403861	−0.201930	+	0.979400i	$0.564721\pi$
$618$	0	0
$619$	10.0658	0.404579	0.202289	−	0.979326i	$-0.435162\pi$
$619$	10.0658	0.404579	0.202289	+	0.979326i	$0.435162\pi$
$620$	6.67665	0.268141
$621$	0	0
$622$	−13.9141	−0.557904
$623$	−4.24085	−0.169906
$624$	0	0
$625$	−28.2758	−1.13103
$626$	21.9666	0.877961
$627$	0	0
$628$	−12.2350	−0.488230
$629$	−10.6129	−0.423166
$630$	0	0
$631$	−35.3530	−1.40738	−0.703691	−	0.710506i	$-0.748466\pi$
$631$	−35.3530	−1.40738	−0.703691	+	0.710506i	$0.748466\pi$
$632$	3.64283	0.144904
$633$	0	0
$634$	3.38070	0.134265
$635$	−41.2870	−1.63842
$636$	0	0
$637$	−23.9621	−0.949414
$638$	−30.4842	−1.20688
$639$	0	0
$640$	2.40321	0.0949951
$641$	−0.166926	−0.00659316	−0.00329658	−	0.999995i	$-0.501049\pi$
$641$	−0.166926	−0.00659316	−0.00329658	+	0.999995i	$0.501049\pi$
$642$	0	0
$643$	−30.9571	−1.22083	−0.610415	−	0.792082i	$-0.708997\pi$
$643$	−30.9571	−1.22083	−0.610415	+	0.792082i	$0.708997\pi$
$644$	9.67872	0.381395
$645$	0	0
$646$	−38.0299	−1.49626
$647$	0.959927	0.0377386	0.0188693	−	0.999822i	$-0.493993\pi$
$647$	0.959927	0.0377386	0.0188693	+	0.999822i	$0.493993\pi$
$648$	0	0
$649$	−32.0488	−1.25803
$650$	3.96246	0.155421
$651$	0	0
$652$	−23.6815	−0.927438
$653$	−27.5173	−1.07684	−0.538418	−	0.842678i	$-0.680978\pi$
$653$	−27.5173	−1.07684	−0.538418	+	0.842678i	$0.680978\pi$
$654$	0	0
$655$	−12.5668	−0.491027
$656$	−9.69619	−0.378573
$657$	0	0
$658$	11.4809	0.447571
$659$	−3.87281	−0.150863	−0.0754316	−	0.997151i	$-0.524033\pi$
$659$	−3.87281	−0.150863	−0.0754316	+	0.997151i	$0.524033\pi$
$660$	0	0
$661$	2.60845	0.101457	0.0507285	−	0.998712i	$-0.483846\pi$
$661$	2.60845	0.101457	0.0507285	+	0.998712i	$0.483846\pi$
$662$	21.9585	0.853443
$663$	0	0
$664$	0.689504	0.0267580
$665$	−20.8684	−0.809242
$666$	0	0
$667$	38.2336	1.48041
$668$	3.54694	0.137235
$669$	0	0
$670$	−6.12542	−0.236645
$671$	40.5054	1.56369
$672$	0	0
$673$	−12.9907	−0.500753	−0.250376	−	0.968149i	$-0.580554\pi$
$673$	−12.9907	−0.500753	−0.250376	+	0.968149i	$0.580554\pi$
$674$	3.21883	0.123985
$675$	0	0
$676$	13.1142	0.504394
$677$	−30.7406	−1.18146	−0.590728	−	0.806871i	$-0.701159\pi$
$677$	−30.7406	−1.18146	−0.590728	+	0.806871i	$0.701159\pi$
$678$	0	0
$679$	18.1452	0.696349
$680$	15.9997	0.613560
$681$	0	0
$682$	−14.1033	−0.540045
$683$	6.20328	0.237362	0.118681	−	0.992932i	$-0.462133\pi$
$683$	6.20328	0.237362	0.118681	+	0.992932i	$0.462133\pi$
$684$	0	0
$685$	31.1608	1.19059
$686$	−17.7694	−0.678440
$687$	0	0
$688$	−9.26120	−0.353080
$689$	48.1411	1.83403
$690$	0	0
$691$	18.1544	0.690626	0.345313	−	0.938488i	$-0.387773\pi$
$691$	18.1544	0.690626	0.345313	+	0.938488i	$0.387773\pi$
$692$	−20.8927	−0.794220
$693$	0	0
$694$	−15.8474	−0.601560
$695$	−1.66457	−0.0631408
$696$	0	0
$697$	−64.5537	−2.44515
$698$	−17.0834	−0.646615
$699$	0	0
$700$	1.17874	0.0445524
$701$	46.2492	1.74681	0.873404	−	0.486996i	$-0.161907\pi$
$701$	46.2492	1.74681	0.873404	+	0.486996i	$0.161907\pi$
$702$	0	0
$703$	9.10585	0.343434
$704$	−5.07638	−0.191323
$705$	0	0
$706$	4.73633	0.178254
$707$	22.0840	0.830554
$708$	0	0
$709$	−35.6219	−1.33781	−0.668904	−	0.743349i	$-0.733236\pi$
$709$	−35.6219	−1.33781	−0.668904	+	0.743349i	$0.733236\pi$
$710$	−4.78134	−0.179440
$711$	0	0
$712$	−2.78972	−0.104549
$713$	17.6885	0.662441
$714$	0	0
$715$	−62.3425	−2.33148
$716$	−2.13073	−0.0796290
$717$	0	0
$718$	−18.2662	−0.681687
$719$	37.4220	1.39561	0.697803	−	0.716290i	$-0.254161\pi$
$719$	37.4220	1.39561	0.697803	+	0.716290i	$0.254161\pi$
$720$	0	0
$721$	−15.3611	−0.572076
$722$	13.6294	0.507234
$723$	0	0
$724$	18.5589	0.689738
$725$	4.65637	0.172933
$726$	0	0
$727$	−24.4331	−0.906172	−0.453086	−	0.891467i	$-0.649677\pi$
$727$	−24.4331	−0.906172	−0.453086	+	0.891467i	$0.649677\pi$
$728$	7.76841	0.287916
$729$	0	0
$730$	30.6463	1.13427
$731$	−61.6577	−2.28049
$732$	0	0
$733$	37.4726	1.38408	0.692042	−	0.721857i	$-0.256711\pi$
$733$	37.4726	1.38408	0.692042	+	0.721857i	$0.256711\pi$
$734$	5.99172	0.221159
$735$	0	0
$736$	6.36685	0.234685
$737$	12.9389	0.476612
$738$	0	0
$739$	−18.6134	−0.684707	−0.342353	−	0.939571i	$-0.611224\pi$
$739$	−18.6134	−0.684707	−0.342353	+	0.939571i	$0.611224\pi$
$740$	−3.83095	−0.140829
$741$	0	0
$742$	14.3209	0.525737
$743$	−0.773612	−0.0283811	−0.0141905	−	0.999899i	$-0.504517\pi$
$743$	−0.773612	−0.0283811	−0.0141905	+	0.999899i	$0.504517\pi$
$744$	0	0
$745$	−27.8739	−1.02122
$746$	−16.9416	−0.620276
$747$	0	0
$748$	−33.7967	−1.23573
$749$	18.6563	0.681688
$750$	0	0
$751$	−11.0543	−0.403377	−0.201689	−	0.979450i	$-0.564643\pi$
$751$	−11.0543	−0.403377	−0.201689	+	0.979450i	$0.564643\pi$
$752$	7.55235	0.275406
$753$	0	0
$754$	30.6873	1.11757
$755$	2.74666	0.0999610
$756$	0	0
$757$	31.9235	1.16028	0.580140	−	0.814517i	$-0.302998\pi$
$757$	31.9235	1.16028	0.580140	+	0.814517i	$0.302998\pi$
$758$	−16.7293	−0.607635
$759$	0	0
$760$	−13.7276	−0.497954
$761$	−14.7166	−0.533477	−0.266739	−	0.963769i	$-0.585946\pi$
$761$	−14.7166	−0.533477	−0.266739	+	0.963769i	$0.585946\pi$
$762$	0	0
$763$	21.6113	0.782380
$764$	−15.6213	−0.565160
$765$	0	0
$766$	26.8421	0.969846
$767$	32.2624	1.16493
$768$	0	0
$769$	−18.0186	−0.649768	−0.324884	−	0.945754i	$-0.605325\pi$
$769$	−18.0186	−0.649768	−0.324884	+	0.945754i	$0.605325\pi$
$770$	−18.5455	−0.668334
$771$	0	0
$772$	−18.5417	−0.667330
$773$	−48.8663	−1.75760	−0.878799	−	0.477193i	$-0.841654\pi$
$773$	−48.8663	−1.75760	−0.878799	+	0.477193i	$0.841654\pi$
$774$	0	0
$775$	2.15424	0.0773825
$776$	11.9363	0.428487
$777$	0	0
$778$	−8.89100	−0.318758
$779$	55.3868	1.98444
$780$	0	0
$781$	10.0998	0.361399
$782$	42.3882	1.51580
$783$	0	0
$784$	−4.68907	−0.167467
$785$	−29.4033	−1.04945
$786$	0	0
$787$	−31.8942	−1.13691	−0.568453	−	0.822715i	$-0.692458\pi$
$787$	−31.8942	−1.13691	−0.568453	+	0.822715i	$0.692458\pi$
$788$	9.98285	0.355624
$789$	0	0
$790$	8.75448	0.311471
$791$	−19.3099	−0.686579
$792$	0	0
$793$	−40.7753	−1.44797
$794$	−15.3011	−0.543015
$795$	0	0
$796$	5.33431	0.189070
$797$	26.0846	0.923964	0.461982	−	0.886889i	$-0.347139\pi$
$797$	26.0846	0.923964	0.461982	+	0.886889i	$0.347139\pi$
$798$	0	0
$799$	50.2808	1.77881
$800$	0.775401	0.0274146
$801$	0	0
$802$	9.91869	0.350241
$803$	−64.7353	−2.28446
$804$	0	0
$805$	23.2600	0.819806
$806$	14.1973	0.500079
$807$	0	0
$808$	14.5273	0.511068
$809$	2.45878	0.0864461	0.0432231	−	0.999065i	$-0.486237\pi$
$809$	2.45878	0.0864461	0.0432231	+	0.999065i	$0.486237\pi$
$810$	0	0
$811$	−20.3841	−0.715783	−0.357891	−	0.933763i	$-0.616504\pi$
$811$	−20.3841	−0.715783	−0.357891	+	0.933763i	$0.616504\pi$
$812$	9.12880	0.320358
$813$	0	0
$814$	8.09227	0.283634
$815$	−56.9114	−1.99352
$816$	0	0
$817$	52.9020	1.85081
$818$	−13.4737	−0.471098
$819$	0	0
$820$	−23.3020	−0.813740
$821$	8.31824	0.290309	0.145154	−	0.989409i	$-0.453632\pi$
$821$	8.31824	0.290309	0.145154	+	0.989409i	$0.453632\pi$
$822$	0	0
$823$	−25.5967	−0.892244	−0.446122	−	0.894972i	$-0.647195\pi$
$823$	−25.5967	−0.892244	−0.446122	+	0.894972i	$0.647195\pi$
$824$	−10.1048	−0.352017
$825$	0	0
$826$	9.59734	0.333934
$827$	−28.4900	−0.990693	−0.495347	−	0.868695i	$-0.664959\pi$
$827$	−28.4900	−0.990693	−0.495347	+	0.868695i	$0.664959\pi$
$828$	0	0
$829$	−15.2875	−0.530956	−0.265478	−	0.964117i	$-0.585530\pi$
$829$	−15.2875	−0.530956	−0.265478	+	0.964117i	$0.585530\pi$
$830$	1.65702	0.0575161
$831$	0	0
$832$	5.11021	0.177165
$833$	−31.2181	−1.08164
$834$	0	0
$835$	8.52403	0.294986
$836$	28.9974	1.00290
$837$	0	0
$838$	24.5320	0.847444
$839$	−33.9640	−1.17257	−0.586283	−	0.810106i	$-0.699409\pi$
$839$	−33.9640	−1.17257	−0.586283	+	0.810106i	$0.699409\pi$
$840$	0	0
$841$	7.06129	0.243493
$842$	−27.8189	−0.958704
$843$	0	0
$844$	−19.7945	−0.681355
$845$	31.5162	1.08419
$846$	0	0
$847$	22.4525	0.771476
$848$	9.42058	0.323504
$849$	0	0
$850$	5.16234	0.177067
$851$	−10.1494	−0.347917
$852$	0	0
$853$	−18.6395	−0.638203	−0.319102	−	0.947720i	$-0.603381\pi$
$853$	−18.6395	−0.638203	−0.319102	+	0.947720i	$0.603381\pi$
$854$	−12.1297	−0.415071
$855$	0	0
$856$	12.2725	0.419465
$857$	−38.5929	−1.31831	−0.659154	−	0.752008i	$-0.729085\pi$
$857$	−38.5929	−1.31831	−0.659154	+	0.752008i	$0.729085\pi$
$858$	0	0
$859$	36.7533	1.25400	0.627002	−	0.779017i	$-0.284282\pi$
$859$	36.7533	1.25400	0.627002	+	0.779017i	$0.284282\pi$
$860$	−22.2566	−0.758943
$861$	0	0
$862$	−32.5743	−1.10949
$863$	−39.5677	−1.34690	−0.673450	−	0.739233i	$-0.735188\pi$
$863$	−39.5677	−1.34690	−0.673450	+	0.739233i	$0.735188\pi$
$864$	0	0
$865$	−50.2094	−1.70717
$866$	−5.11693	−0.173880
$867$	0	0
$868$	4.22338	0.143351
$869$	−18.4924	−0.627313
$870$	0	0
$871$	−13.0252	−0.441341
$872$	14.2163	0.481425
$873$	0	0
$874$	−36.3688	−1.23019
$875$	−15.4337	−0.521754
$876$	0	0
$877$	4.55367	0.153766	0.0768832	−	0.997040i	$-0.475503\pi$
$877$	4.55367	0.153766	0.0768832	+	0.997040i	$0.475503\pi$
$878$	18.8040	0.634604
$879$	0	0
$880$	−12.1996	−0.411248
$881$	16.4979	0.555828	0.277914	−	0.960606i	$-0.410357\pi$
$881$	16.4979	0.555828	0.277914	+	0.960606i	$0.410357\pi$
$882$	0	0
$883$	−43.4286	−1.46149	−0.730744	−	0.682652i	$-0.760827\pi$
$883$	−43.4286	−1.46149	−0.730744	+	0.682652i	$0.760827\pi$
$884$	34.0219	1.14428
$885$	0	0
$886$	−16.7817	−0.563794
$887$	9.36355	0.314397	0.157199	−	0.987567i	$-0.449754\pi$
$887$	9.36355	0.314397	0.157199	+	0.987567i	$0.449754\pi$
$888$	0	0
$889$	−26.1165	−0.875920
$890$	−6.70426	−0.224727
$891$	0	0
$892$	29.2824	0.980447
$893$	−43.1406	−1.44365
$894$	0	0
$895$	−5.12058	−0.171162
$896$	1.52017	0.0507854
$897$	0	0
$898$	−29.1983	−0.974360
$899$	16.6835	0.556427
$900$	0	0
$901$	62.7188	2.08947
$902$	49.2216	1.63890
$903$	0	0
$904$	−12.7024	−0.422476
$905$	44.6010	1.48259
$906$	0	0
$907$	−43.0813	−1.43049	−0.715246	−	0.698873i	$-0.753685\pi$
$907$	−43.0813	−1.43049	−0.715246	+	0.698873i	$0.753685\pi$
$908$	5.90479	0.195957
$909$	0	0
$910$	18.6691	0.618874
$911$	3.70489	0.122749	0.0613743	−	0.998115i	$-0.480452\pi$
$911$	3.70489	0.122749	0.0613743	+	0.998115i	$0.480452\pi$
$912$	0	0
$913$	−3.50019	−0.115839
$914$	6.74219	0.223012
$915$	0	0
$916$	14.2761	0.471695
$917$	−7.94928	−0.262508
$918$	0	0
$919$	42.2046	1.39220	0.696101	−	0.717944i	$-0.254917\pi$
$919$	42.2046	1.39220	0.696101	+	0.717944i	$0.254917\pi$
$920$	15.3009	0.504454
$921$	0	0
$922$	−17.8901	−0.589180
$923$	−10.1671	−0.334654
$924$	0	0
$925$	−1.23607	−0.0406417
$926$	17.2936	0.568304
$927$	0	0
$928$	6.00511	0.197127
$929$	40.9616	1.34391	0.671954	−	0.740593i	$-0.265455\pi$
$929$	40.9616	1.34391	0.671954	+	0.740593i	$0.265455\pi$
$930$	0	0
$931$	26.7850	0.877843
$932$	27.0850	0.887198
$933$	0	0
$934$	3.00187	0.0982243
$935$	−81.2205	−2.65619
$936$	0	0
$937$	−11.2419	−0.367258	−0.183629	−	0.982996i	$-0.558785\pi$
$937$	−11.2419	−0.367258	−0.183629	+	0.982996i	$0.558785\pi$
$938$	−3.87470	−0.126513
$939$	0	0
$940$	18.1498	0.591983
$941$	−1.14007	−0.0371652	−0.0185826	−	0.999827i	$-0.505915\pi$
$941$	−1.14007	−0.0371652	−0.0185826	+	0.999827i	$0.505915\pi$
$942$	0	0
$943$	−61.7342	−2.01034
$944$	6.31332	0.205481
$945$	0	0
$946$	47.0134	1.52854
$947$	20.5389	0.667424	0.333712	−	0.942675i	$-0.391699\pi$
$947$	20.5389	0.667424	0.333712	+	0.942675i	$0.391699\pi$
$948$	0	0
$949$	65.1667	2.11540
$950$	−4.42926	−0.143704
$951$	0	0
$952$	10.1208	0.328016
$953$	18.6016	0.602567	0.301283	−	0.953535i	$-0.402585\pi$
$953$	18.6016	0.602567	0.301283	+	0.953535i	$0.402585\pi$
$954$	0	0
$955$	−37.5413	−1.21481
$956$	−7.97200	−0.257833
$957$	0	0
$958$	25.7701	0.832595
$959$	19.7111	0.636505
$960$	0	0
$961$	−23.2815	−0.751015
$962$	−8.14619	−0.262644
$963$	0	0
$964$	−2.36516	−0.0761767
$965$	−44.5595	−1.43442
$966$	0	0
$967$	−21.3331	−0.686026	−0.343013	−	0.939331i	$-0.611447\pi$
$967$	−21.3331	−0.686026	−0.343013	+	0.939331i	$0.611447\pi$
$968$	14.7697	0.474715
$969$	0	0
$970$	28.6853	0.921030
$971$	−13.0827	−0.419844	−0.209922	−	0.977718i	$-0.567321\pi$
$971$	−13.0827	−0.419844	−0.209922	+	0.977718i	$0.567321\pi$
$972$	0	0
$973$	−1.05294	−0.0337558
$974$	9.85904	0.315904
$975$	0	0
$976$	−7.97918	−0.255407
$977$	22.6902	0.725924	0.362962	−	0.931804i	$-0.381766\pi$
$977$	22.6902	0.725924	0.362962	+	0.931804i	$0.381766\pi$
$978$	0	0
$979$	14.1617	0.452609
$980$	−11.2688	−0.359969
$981$	0	0
$982$	−9.75565	−0.311315
$983$	51.3612	1.63817	0.819084	−	0.573674i	$-0.194482\pi$
$983$	51.3612	1.63817	0.819084	+	0.573674i	$0.194482\pi$
$984$	0	0
$985$	23.9909	0.764412
$986$	39.9798	1.27322
$987$	0	0
$988$	−29.1906	−0.928677
$989$	−58.9647	−1.87497
$990$	0	0
$991$	−11.0015	−0.349475	−0.174737	−	0.984615i	$-0.555908\pi$
$991$	−11.0015	−0.349475	−0.174737	+	0.984615i	$0.555908\pi$
$992$	2.77822	0.0882087
$993$	0	0
$994$	−3.02448	−0.0959308
$995$	12.8195	0.406404
$996$	0	0
$997$	12.2415	0.387693	0.193846	−	0.981032i	$-0.437904\pi$
$997$	12.2415	0.387693	0.193846	+	0.981032i	$0.437904\pi$
$998$	−3.06163	−0.0969143
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.o.1.5		7
3.2	odd	2		538.2.a.d.1.3	✓	7
12.11	even	2		4304.2.a.i.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.d.1.3	✓	7	3.2	odd	2
4304.2.a.i.1.5		7	12.11	even	2
4842.2.a.o.1.5		7	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.40321 q^{5} +1.52017 q^{7} +1.00000 q^{8} +2.40321 q^{10} -5.07638 q^{11} +5.11021 q^{13} +1.52017 q^{14} +1.00000 q^{16} +6.65764 q^{17} -5.71222 q^{19} +2.40321 q^{20} -5.07638 q^{22} +6.36685 q^{23} +0.775401 q^{25} +5.11021 q^{26} +1.52017 q^{28} +6.00511 q^{29} +2.77822 q^{31} +1.00000 q^{32} +6.65764 q^{34} +3.65329 q^{35} -1.59410 q^{37} -5.71222 q^{38} +2.40321 q^{40} -9.69619 q^{41} -9.26120 q^{43} -5.07638 q^{44} +6.36685 q^{46} +7.55235 q^{47} -4.68907 q^{49} +0.775401 q^{50} +5.11021 q^{52} +9.42058 q^{53} -12.1996 q^{55} +1.52017 q^{56} +6.00511 q^{58} +6.31332 q^{59} -7.97918 q^{61} +2.77822 q^{62} +1.00000 q^{64} +12.2809 q^{65} -2.54885 q^{67} +6.65764 q^{68} +3.65329 q^{70} -1.98957 q^{71} +12.7523 q^{73} -1.59410 q^{74} -5.71222 q^{76} -7.71699 q^{77} +3.64283 q^{79} +2.40321 q^{80} -9.69619 q^{82} +0.689504 q^{83} +15.9997 q^{85} -9.26120 q^{86} -5.07638 q^{88} -2.78972 q^{89} +7.76841 q^{91} +6.36685 q^{92} +7.55235 q^{94} -13.7276 q^{95} +11.9363 q^{97} -4.68907 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} + 6 q^{5} - 3 q^{7} + 7 q^{8} + 6 q^{10} + 12 q^{11} + 3 q^{13} - 3 q^{14} + 7 q^{16} + 8 q^{17} - 7 q^{19} + 6 q^{20} + 12 q^{22} + 22 q^{23} + 9 q^{25} + 3 q^{26} - 3 q^{28} + 7 q^{29}+ \cdots + 6 q^{98}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^4 + 6 * q^5 - 3 * q^7 + 7 * q^8 + 6 * q^10 + 12 * q^11 + 3 * q^13 - 3 * q^14 + 7 * q^16 + 8 * q^17 - 7 * q^19 + 6 * q^20 + 12 * q^22 + 22 * q^23 + 9 * q^25 + 3 * q^26 - 3 * q^28 + 7 * q^29 - 3 * q^31 + 7 * q^32 + 8 * q^34 + 25 * q^35 - 13 * q^37 - 7 * q^38 + 6 * q^40 + 11 * q^41 - 14 * q^43 + 12 * q^44 + 22 * q^46 + 23 * q^47 + 6 * q^49 + 9 * q^50 + 3 * q^52 + 21 * q^53 + 15 * q^55 - 3 * q^56 + 7 * q^58 + 20 * q^59 + 13 * q^61 - 3 * q^62 + 7 * q^64 + 29 * q^65 - 30 * q^67 + 8 * q^68 + 25 * q^70 + 23 * q^71 + 9 * q^73 - 13 * q^74 - 7 * q^76 - 8 * q^77 + 11 * q^79 + 6 * q^80 + 11 * q^82 + 3 * q^83 + 21 * q^85 - 14 * q^86 + 12 * q^88 - 16 * q^89 - 10 * q^91 + 22 * q^92 + 23 * q^94 + 21 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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