LMFDB - Embedded newform 4304.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, 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("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.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:	$34.3676130300$
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	$\chi$	$=$	4304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.20937 q^{3} -2.40321 q^{5} -1.52017 q^{7} -1.53742 q^{9} -5.07638 q^{11} +5.11021 q^{13} -2.90637 q^{15} -6.65764 q^{17} +5.71222 q^{19} -1.83846 q^{21} +6.36685 q^{23} +0.775401 q^{25} -5.48743 q^{27} -6.00511 q^{29} -2.77822 q^{31} -6.13924 q^{33} +3.65329 q^{35} -1.59410 q^{37} +6.18015 q^{39} +9.69619 q^{41} +9.26120 q^{43} +3.69473 q^{45} +7.55235 q^{47} -4.68907 q^{49} -8.05157 q^{51} -9.42058 q^{53} +12.1996 q^{55} +6.90820 q^{57} +6.31332 q^{59} -7.97918 q^{61} +2.33714 q^{63} -12.2809 q^{65} +2.54885 q^{67} +7.69990 q^{69} -1.98957 q^{71} +12.7523 q^{73} +0.937750 q^{75} +7.71699 q^{77} -3.64283 q^{79} -2.02411 q^{81} +0.689504 q^{83} +15.9997 q^{85} -7.26242 q^{87} +2.78972 q^{89} -7.76841 q^{91} -3.35991 q^{93} -13.7276 q^{95} +11.9363 q^{97} +7.80451 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{3} - 6 q^{5} + 3 q^{7} + 7 q^{9} + 12 q^{11} + 3 q^{13} + 6 q^{15} - 8 q^{17} + 7 q^{19} - 5 q^{21} + 22 q^{23} + 9 q^{25} + 22 q^{27} - 7 q^{29} + 3 q^{31} - 19 q^{33} + 25 q^{35} - 13 q^{37}+ \cdots + 9 q^{99}+O(q^{100}) $ 7 * q + 4 * q^3 - 6 * q^5 + 3 * q^7 + 7 * q^9 + 12 * q^11 + 3 * q^13 + 6 * q^15 - 8 * q^17 + 7 * q^19 - 5 * q^21 + 22 * q^23 + 9 * q^25 + 22 * q^27 - 7 * q^29 + 3 * q^31 - 19 * q^33 + 25 * q^35 - 13 * q^37 - q^39 - 11 * q^41 + 14 * q^43 - 17 * q^45 + 23 * q^47 + 6 * q^49 + 21 * q^51 - 21 * q^53 - 15 * q^55 - 14 * q^57 + 20 * q^59 + 13 * q^61 + 12 * q^63 - 29 * q^65 + 30 * q^67 + 13 * q^69 + 23 * q^71 + 9 * q^73 - 3 * q^75 + 8 * q^77 - 11 * q^79 + 43 * q^81 + 3 * q^83 + 21 * q^85 - 28 * q^87 + 16 * q^89 + 10 * q^91 - 13 * q^93 + 21 * q^97 + 9 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.20937	0.698232	0.349116	−	0.937079i	$-0.386482\pi$
$3$	1.20937	0.698232	0.349116	+	0.937079i	$0.386482\pi$
$4$	0	0
$5$	−2.40321	−1.07475	−0.537373	−	0.843344i	$-0.680583\pi$
$5$	−2.40321	−1.07475	−0.537373	+	0.843344i	$0.680583\pi$
$6$	0	0
$7$	−1.52017	−0.574572	−0.287286	−	0.957845i	$-0.592753\pi$
$7$	−1.52017	−0.574572	−0.287286	+	0.957845i	$0.592753\pi$
$8$	0	0
$9$	−1.53742	−0.512472
$10$	0	0
$11$	−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	$0.249301\pi$
$14$	0	0
$15$	−2.90637	−0.750423
$16$	0	0
$17$	−6.65764	−1.61471	−0.807357	−	0.590063i	$-0.799103\pi$
$17$	−6.65764	−1.61471	−0.807357	+	0.590063i	$0.799103\pi$
$18$	0	0
$19$	5.71222	1.31047	0.655236	−	0.755424i	$-0.272569\pi$
$19$	5.71222	1.31047	0.655236	+	0.755424i	$0.272569\pi$
$20$	0	0
$21$	−1.83846	−0.401184
$22$	0	0
$23$	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
$26$	0	0
$27$	−5.48743	−1.05606
$28$	0	0
$29$	−6.00511	−1.11512	−0.557560	−	0.830137i	$-0.688262\pi$
$29$	−6.00511	−1.11512	−0.557560	+	0.830137i	$0.688262\pi$
$30$	0	0
$31$	−2.77822	−0.498984	−0.249492	−	0.968377i	$-0.580264\pi$
$31$	−2.77822	−0.498984	−0.249492	+	0.968377i	$0.580264\pi$
$32$	0	0
$33$	−6.13924	−1.06871
$34$	0	0
$35$	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	$0.541830\pi$
$38$	0	0
$39$	6.18015	0.989616
$40$	0	0
$41$	9.69619	1.51429	0.757146	−	0.653246i	$-0.226593\pi$
$41$	9.69619	1.51429	0.757146	+	0.653246i	$0.226593\pi$
$42$	0	0
$43$	9.26120	1.41232	0.706160	−	0.708053i	$-0.250426\pi$
$43$	9.26120	1.41232	0.706160	+	0.708053i	$0.250426\pi$
$44$	0	0
$45$	3.69473	0.550777
$46$	0	0
$47$	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	0
$51$	−8.05157	−1.12745
$52$	0	0
$53$	−9.42058	−1.29402	−0.647008	−	0.762483i	$-0.723980\pi$
$53$	−9.42058	−1.29402	−0.647008	+	0.762483i	$0.723980\pi$
$54$	0	0
$55$	12.1996	1.64499
$56$	0	0
$57$	6.90820	0.915014
$58$	0	0
$59$	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	$0.670656\pi$
$62$	0	0
$63$	2.33714	0.294452
$64$	0	0
$65$	−12.2809	−1.52326
$66$	0	0
$67$	2.54885	0.311392	0.155696	−	0.987805i	$-0.450238\pi$
$67$	2.54885	0.311392	0.155696	+	0.987805i	$0.450238\pi$
$68$	0	0
$69$	7.69990	0.926959
$70$	0	0
$71$	−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	$0.231843\pi$
$74$	0	0
$75$	0.937750	0.108282
$76$	0	0
$77$	7.71699	0.879432
$78$	0	0
$79$	−3.64283	−0.409851	−0.204925	−	0.978778i	$-0.565695\pi$
$79$	−3.64283	−0.409851	−0.204925	+	0.978778i	$0.565695\pi$
$80$	0	0
$81$	−2.02411	−0.224901
$82$	0	0
$83$	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
$86$	0	0
$87$	−7.26242	−0.778613
$88$	0	0
$89$	2.78972	0.295709	0.147855	−	0.989009i	$-0.452763\pi$
$89$	2.78972	0.295709	0.147855	+	0.989009i	$0.452763\pi$
$90$	0	0
$91$	−7.76841	−0.814350
$92$	0	0
$93$	−3.35991	−0.348407
$94$	0	0
$95$	−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	$0.292784\pi$
$98$	0	0
$99$	7.80451	0.784383
$100$	0	0
$101$	−14.5273	−1.44552	−0.722759	−	0.691100i	$-0.757127\pi$
$101$	−14.5273	−1.44552	−0.722759	+	0.691100i	$0.757127\pi$
$102$	0	0
$103$	10.1048	0.995656	0.497828	−	0.867276i	$-0.334131\pi$
$103$	10.1048	0.995656	0.497828	+	0.867276i	$0.334131\pi$
$104$	0	0
$105$	4.41819	0.431172
$106$	0	0
$107$	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	$0.261616\pi$
$110$	0	0
$111$	−1.92786	−0.182985
$112$	0	0
$113$	12.7024	1.19494	0.597471	−	0.801891i	$-0.296172\pi$
$113$	12.7024	1.19494	0.597471	+	0.801891i	$0.296172\pi$
$114$	0	0
$115$	−15.3009	−1.42681
$116$	0	0
$117$	−7.85652	−0.726335
$118$	0	0
$119$	10.1208	0.927769
$120$	0	0
$121$	14.7697	1.34270
$122$	0	0
$123$	11.7263	1.05733
$124$	0	0
$125$	10.1526	0.908075
$126$	0	0
$127$	17.1800	1.52448	0.762238	−	0.647297i	$-0.224101\pi$
$127$	17.1800	1.52448	0.762238	+	0.647297i	$0.224101\pi$
$128$	0	0
$129$	11.2002	0.986127
$130$	0	0
$131$	−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
$134$	0	0
$135$	13.1874	1.13499
$136$	0	0
$137$	−12.9664	−1.10779	−0.553895	−	0.832586i	$-0.686859\pi$
$137$	−12.9664	−1.10779	−0.553895	+	0.832586i	$0.686859\pi$
$138$	0	0
$139$	0.692646	0.0587495	0.0293747	−	0.999568i	$-0.490648\pi$
$139$	0.692646	0.0587495	0.0293747	+	0.999568i	$0.490648\pi$
$140$	0	0
$141$	9.13361	0.769189
$142$	0	0
$143$	−25.9414	−2.16933
$144$	0	0
$145$	14.4315	1.19847
$146$	0	0
$147$	−5.67084	−0.467723
$148$	0	0
$149$	11.5986	0.950197	0.475099	−	0.879933i	$-0.342412\pi$
$149$	11.5986	0.950197	0.475099	+	0.879933i	$0.342412\pi$
$150$	0	0
$151$	−1.14291	−0.0930089	−0.0465045	−	0.998918i	$-0.514808\pi$
$151$	−1.14291	−0.0930089	−0.0465045	+	0.998918i	$0.514808\pi$
$152$	0	0
$153$	10.2356	0.827496
$154$	0	0
$155$	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	$0.662357\pi$
$158$	0	0
$159$	−11.3930	−0.903523
$160$	0	0
$161$	−9.67872	−0.762790
$162$	0	0
$163$	23.6815	1.85488	0.927438	−	0.373978i	$-0.122006\pi$
$163$	23.6815	1.85488	0.927438	+	0.373978i	$0.122006\pi$
$164$	0	0
$165$	14.7539	1.14859
$166$	0	0
$167$	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
$170$	0	0
$171$	−8.78205	−0.671580
$172$	0	0
$173$	20.8927	1.58844	0.794220	−	0.607630i	$-0.207880\pi$
$173$	20.8927	1.58844	0.794220	+	0.607630i	$0.207880\pi$
$174$	0	0
$175$	−1.17874	−0.0891047
$176$	0	0
$177$	7.63516	0.573894
$178$	0	0
$179$	−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	$0.257726\pi$
$182$	0	0
$183$	−9.64981	−0.713334
$184$	0	0
$185$	3.83095	0.281657
$186$	0	0
$187$	33.7967	2.47146
$188$	0	0
$189$	8.34185	0.606780
$190$	0	0
$191$	−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	$0.732563\pi$
$194$	0	0
$195$	−14.8522	−1.06359
$196$	0	0
$197$	−9.98285	−0.711249	−0.355624	−	0.934629i	$-0.615732\pi$
$197$	−9.98285	−0.711249	−0.355624	+	0.934629i	$0.615732\pi$
$198$	0	0
$199$	−5.33431	−0.378139	−0.189070	−	0.981964i	$-0.560547\pi$
$199$	−5.33431	−0.378139	−0.189070	+	0.981964i	$0.560547\pi$
$200$	0	0
$201$	3.08251	0.217424
$202$	0	0
$203$	9.12880	0.640716
$204$	0	0
$205$	−23.3020	−1.62748
$206$	0	0
$207$	−9.78850	−0.680348
$208$	0	0
$209$	−28.9974	−2.00579
$210$	0	0
$211$	19.7945	1.36271	0.681355	−	0.731953i	$-0.261391\pi$
$211$	19.7945	1.36271	0.681355	+	0.731953i	$0.261391\pi$
$212$	0	0
$213$	−2.40613	−0.164865
$214$	0	0
$215$	−22.2566	−1.51789
$216$	0	0
$217$	4.22338	0.286702
$218$	0	0
$219$	15.4222	1.04214
$220$	0	0
$221$	−34.0219	−2.28856
$222$	0	0
$223$	−29.2824	−1.96089	−0.980447	−	0.196783i	$-0.936951\pi$
$223$	−29.2824	−1.96089	−0.980447	+	0.196783i	$0.936951\pi$
$224$	0	0
$225$	−1.19211	−0.0794743
$226$	0	0
$227$	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	$0.343642\pi$
$230$	0	0
$231$	9.33272	0.614048
$232$	0	0
$233$	−27.0850	−1.77440	−0.887198	−	0.461389i	$-0.847351\pi$
$233$	−27.0850	−1.77440	−0.887198	+	0.461389i	$0.847351\pi$
$234$	0	0
$235$	−18.1498	−1.18397
$236$	0	0
$237$	−4.40555	−0.286171
$238$	0	0
$239$	−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	$0.524271\pi$
$242$	0	0
$243$	14.0144	0.899024
$244$	0	0
$245$	11.2688	0.719938
$246$	0	0
$247$	29.1906	1.85735
$248$	0	0
$249$	0.833868	0.0528443
$250$	0	0
$251$	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
$254$	0	0
$255$	19.3496	1.21172
$256$	0	0
$257$	23.2436	1.44990	0.724948	−	0.688803i	$-0.241864\pi$
$257$	23.2436	1.44990	0.724948	+	0.688803i	$0.241864\pi$
$258$	0	0
$259$	2.42331	0.150577
$260$	0	0
$261$	9.23234	0.571468
$262$	0	0
$263$	−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
$266$	0	0
$267$	3.37381	0.206474
$268$	0	0
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	−12.4311	−0.755135	−0.377568	−	0.925982i	$-0.623239\pi$
$271$	−12.4311	−0.755135	−0.377568	+	0.925982i	$0.623239\pi$
$272$	0	0
$273$	−9.39490	−0.568605
$274$	0	0
$275$	−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	0
$279$	4.27129	0.255715
$280$	0	0
$281$	−21.3194	−1.27181	−0.635905	−	0.771767i	$-0.719373\pi$
$281$	−21.3194	−1.27181	−0.635905	+	0.771767i	$0.719373\pi$
$282$	0	0
$283$	29.9997	1.78330	0.891649	−	0.452728i	$-0.149549\pi$
$283$	29.9997	1.78330	0.891649	+	0.452728i	$0.149549\pi$
$284$	0	0
$285$	−16.6018	−0.983408
$286$	0	0
$287$	−14.7399	−0.870069
$288$	0	0
$289$	27.3241	1.60730
$290$	0	0
$291$	14.4354	0.846218
$292$	0	0
$293$	16.9481	0.990120	0.495060	−	0.868859i	$-0.335146\pi$
$293$	16.9481	0.990120	0.495060	+	0.868859i	$0.335146\pi$
$294$	0	0
$295$	−15.1722	−0.883360
$296$	0	0
$297$	27.8563	1.61639
$298$	0	0
$299$	32.5360	1.88160
$300$	0	0
$301$	−14.0786	−0.811479
$302$	0	0
$303$	−17.5689	−1.00931
$304$	0	0
$305$	19.1756	1.09799
$306$	0	0
$307$	−18.4715	−1.05422	−0.527112	−	0.849796i	$-0.676725\pi$
$307$	−18.4715	−1.05422	−0.527112	+	0.849796i	$0.676725\pi$
$308$	0	0
$309$	12.2205	0.695199
$310$	0	0
$311$	−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	$0.286803\pi$
$314$	0	0
$315$	−5.61663	−0.316461
$316$	0	0
$317$	−3.38070	−0.189879	−0.0949395	−	0.995483i	$-0.530266\pi$
$317$	−3.38070	−0.189879	−0.0949395	+	0.995483i	$0.530266\pi$
$318$	0	0
$319$	30.4842	1.70679
$320$	0	0
$321$	14.8420	0.828402
$322$	0	0
$323$	−38.0299	−2.11604
$324$	0	0
$325$	3.96246	0.219798
$326$	0	0
$327$	17.1928	0.950765
$328$	0	0
$329$	−11.4809	−0.632961
$330$	0	0
$331$	−21.9585	−1.20695	−0.603476	−	0.797382i	$-0.706218\pi$
$331$	−21.9585	−1.20695	−0.603476	+	0.797382i	$0.706218\pi$
$332$	0	0
$333$	2.45080	0.134303
$334$	0	0
$335$	−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	$0.472058\pi$
$338$	0	0
$339$	15.3619	0.834346
$340$	0	0
$341$	14.1033	0.763738
$342$	0	0
$343$	17.7694	0.959459
$344$	0	0
$345$	−18.5045	−0.996246
$346$	0	0
$347$	−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	$0.651157\pi$
$350$	0	0
$351$	−28.0419	−1.49677
$352$	0	0
$353$	−4.73633	−0.252090	−0.126045	−	0.992025i	$-0.540228\pi$
$353$	−4.73633	−0.252090	−0.126045	+	0.992025i	$0.540228\pi$
$354$	0	0
$355$	4.78134	0.253767
$356$	0	0
$357$	12.2398	0.647798
$358$	0	0
$359$	−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
$362$	0	0
$363$	17.8620	0.937514
$364$	0	0
$365$	−30.6463	−1.60410
$366$	0	0
$367$	−5.99172	−0.312765	−0.156383	−	0.987697i	$-0.549983\pi$
$367$	−5.99172	−0.312765	−0.156383	+	0.987697i	$0.549983\pi$
$368$	0	0
$369$	−14.9071	−0.776032
$370$	0	0
$371$	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	$0.644526\pi$
$374$	0	0
$375$	12.2783	0.634047
$376$	0	0
$377$	−30.6873	−1.58048
$378$	0	0
$379$	16.7293	0.859325	0.429663	−	0.902989i	$-0.358632\pi$
$379$	16.7293	0.859325	0.429663	+	0.902989i	$0.358632\pi$
$380$	0	0
$381$	20.7770	1.06444
$382$	0	0
$383$	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
$386$	0	0
$387$	−14.2383	−0.723774
$388$	0	0
$389$	8.89100	0.450792	0.225396	−	0.974267i	$-0.427632\pi$
$389$	8.89100	0.450792	0.225396	+	0.974267i	$0.427632\pi$
$390$	0	0
$391$	−42.3882	−2.14366
$392$	0	0
$393$	−6.32405	−0.319006
$394$	0	0
$395$	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	$0.625443\pi$
$398$	0	0
$399$	−10.5017	−0.525741
$400$	0	0
$401$	−9.91869	−0.495316	−0.247658	−	0.968848i	$-0.579661\pi$
$401$	−9.91869	−0.495316	−0.247658	+	0.968848i	$0.579661\pi$
$402$	0	0
$403$	−14.1973	−0.707218
$404$	0	0
$405$	4.86434	0.241711
$406$	0	0
$407$	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	$0.608100\pi$
$410$	0	0
$411$	−15.6812	−0.773495
$412$	0	0
$413$	−9.59734	−0.472254
$414$	0	0
$415$	−1.65702	−0.0813400
$416$	0	0
$417$	0.837668	0.0410208
$418$	0	0
$419$	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	$0.737112\pi$
$422$	0	0
$423$	−11.6111	−0.564551
$424$	0	0
$425$	−5.16234	−0.250410
$426$	0	0
$427$	12.1297	0.586999
$428$	0	0
$429$	−31.3728	−1.51469
$430$	0	0
$431$	−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	$0.539236\pi$
$434$	0	0
$435$	17.4531	0.836811
$436$	0	0
$437$	36.3688	1.73976
$438$	0	0
$439$	−18.8040	−0.897466	−0.448733	−	0.893666i	$-0.648125\pi$
$439$	−18.8040	−0.897466	−0.448733	+	0.893666i	$0.648125\pi$
$440$	0	0
$441$	7.20905	0.343288
$442$	0	0
$443$	−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
$446$	0	0
$447$	14.0271	0.663458
$448$	0	0
$449$	29.1983	1.37795	0.688977	−	0.724784i	$-0.258060\pi$
$449$	29.1983	1.37795	0.688977	+	0.724784i	$0.258060\pi$
$450$	0	0
$451$	−49.2216	−2.31775
$452$	0	0
$453$	−1.38221	−0.0649418
$454$	0	0
$455$	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	$0.449594\pi$
$458$	0	0
$459$	36.5333	1.70523
$460$	0	0
$461$	17.8901	0.833226	0.416613	−	0.909084i	$-0.363217\pi$
$461$	17.8901	0.833226	0.416613	+	0.909084i	$0.363217\pi$
$462$	0	0
$463$	−17.2936	−0.803703	−0.401851	−	0.915705i	$-0.631633\pi$
$463$	−17.2936	−0.803703	−0.401851	+	0.915705i	$0.631633\pi$
$464$	0	0
$465$	8.07456	0.374449
$466$	0	0
$467$	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
$470$	0	0
$471$	−14.7967	−0.681796
$472$	0	0
$473$	−47.0134	−2.16168
$474$	0	0
$475$	4.42926	0.203228
$476$	0	0
$477$	14.4833	0.663147
$478$	0	0
$479$	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
$482$	0	0
$483$	−11.7052	−0.532605
$484$	0	0
$485$	−28.6853	−1.30253
$486$	0	0
$487$	−9.85904	−0.446756	−0.223378	−	0.974732i	$-0.571708\pi$
$487$	−9.85904	−0.446756	−0.223378	+	0.974732i	$0.571708\pi$
$488$	0	0
$489$	28.6397	1.29513
$490$	0	0
$491$	−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
$494$	0	0
$495$	−18.7559	−0.843013
$496$	0	0
$497$	3.02448	0.135667
$498$	0	0
$499$	3.06163	0.137058	0.0685288	−	0.997649i	$-0.478170\pi$
$499$	3.06163	0.137058	0.0685288	+	0.997649i	$0.478170\pi$
$500$	0	0
$501$	4.28957	0.191644
$502$	0	0
$503$	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
$506$	0	0
$507$	15.8600	0.704368
$508$	0	0
$509$	13.2575	0.587629	0.293814	−	0.955863i	$-0.405075\pi$
$509$	13.2575	0.587629	0.293814	+	0.955863i	$0.405075\pi$
$510$	0	0
$511$	−19.3856	−0.857570
$512$	0	0
$513$	−31.3454	−1.38393
$514$	0	0
$515$	−24.2839	−1.07008
$516$	0	0
$517$	−38.3386	−1.68613
$518$	0	0
$519$	25.2670	1.10910
$520$	0	0
$521$	20.2639	0.887776	0.443888	−	0.896082i	$-0.353599\pi$
$521$	20.2639	0.887776	0.443888	+	0.896082i	$0.353599\pi$
$522$	0	0
$523$	2.18471	0.0955307	0.0477654	−	0.998859i	$-0.484790\pi$
$523$	2.18471	0.0955307	0.0477654	+	0.998859i	$0.484790\pi$
$524$	0	0
$525$	−1.42554	−0.0622158
$526$	0	0
$527$	18.4964	0.805716
$528$	0	0
$529$	17.5368	0.762470
$530$	0	0
$531$	−9.70620	−0.421213
$532$	0	0
$533$	49.5496	2.14623
$534$	0	0
$535$	−29.4933	−1.27511
$536$	0	0
$537$	−2.57685	−0.111199
$538$	0	0
$539$	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	$0.343369\pi$
$542$	0	0
$543$	22.4447	0.963194
$544$	0	0
$545$	−34.1647	−1.46346
$546$	0	0
$547$	2.59714	0.111046	0.0555229	−	0.998457i	$-0.482317\pi$
$547$	2.59714	0.111046	0.0555229	+	0.998457i	$0.482317\pi$
$548$	0	0
$549$	12.2673	0.523556
$550$	0	0
$551$	−34.3025	−1.46133
$552$	0	0
$553$	5.53774	0.235489
$554$	0	0
$555$	4.63305	0.196662
$556$	0	0
$557$	32.9165	1.39471	0.697357	−	0.716724i	$-0.254359\pi$
$557$	32.9165	1.39471	0.697357	+	0.716724i	$0.254359\pi$
$558$	0	0
$559$	47.3266	2.00170
$560$	0	0
$561$	40.8729	1.72565
$562$	0	0
$563$	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
$566$	0	0
$567$	3.07699	0.129222
$568$	0	0
$569$	−11.9495	−0.500951	−0.250476	−	0.968123i	$-0.580587\pi$
$569$	−11.9495	−0.500951	−0.250476	+	0.968123i	$0.580587\pi$
$570$	0	0
$571$	18.1326	0.758824	0.379412	−	0.925228i	$-0.376126\pi$
$571$	18.1326	0.758824	0.379412	+	0.925228i	$0.376126\pi$
$572$	0	0
$573$	−18.8920	−0.789225
$574$	0	0
$575$	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	$0.344467\pi$
$578$	0	0
$579$	−22.4238	−0.931903
$580$	0	0
$581$	−1.04817	−0.0434853
$582$	0	0
$583$	47.8225	1.98060
$584$	0	0
$585$	18.8808	0.780626
$586$	0	0
$587$	−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
$590$	0	0
$591$	−12.0730	−0.496617
$592$	0	0
$593$	30.4282	1.24954	0.624768	−	0.780811i	$-0.285194\pi$
$593$	30.4282	1.24954	0.624768	+	0.780811i	$0.285194\pi$
$594$	0	0
$595$	−24.3223	−0.997117
$596$	0	0
$597$	−6.45118	−0.264029
$598$	0	0
$599$	−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	$0.633340\pi$
$602$	0	0
$603$	−3.91864	−0.159580
$604$	0	0
$605$	−35.4946	−1.44306
$606$	0	0
$607$	11.4793	0.465932	0.232966	−	0.972485i	$-0.425157\pi$
$607$	11.4793	0.465932	0.232966	+	0.972485i	$0.425157\pi$
$608$	0	0
$609$	11.0401	0.447369
$610$	0	0
$611$	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	$0.426938\pi$
$614$	0	0
$615$	−28.1808	−1.13636
$616$	0	0
$617$	10.0317	0.403861	0.201930	−	0.979400i	$-0.435279\pi$
$617$	10.0317	0.403861	0.201930	+	0.979400i	$0.435279\pi$
$618$	0	0
$619$	−10.0658	−0.404579	−0.202289	−	0.979326i	$-0.564838\pi$
$619$	−10.0658	−0.404579	−0.202289	+	0.979326i	$0.564838\pi$
$620$	0	0
$621$	−34.9377	−1.40200
$622$	0	0
$623$	−4.24085	−0.169906
$624$	0	0
$625$	−28.2758	−1.13103
$626$	0	0
$627$	−35.0687	−1.40051
$628$	0	0
$629$	10.6129	0.423166
$630$	0	0
$631$	35.3530	1.40738	0.703691	−	0.710506i	$-0.251534\pi$
$631$	35.3530	1.40738	0.703691	+	0.710506i	$0.251534\pi$
$632$	0	0
$633$	23.9389	0.951488
$634$	0	0
$635$	−41.2870	−1.63842
$636$	0	0
$637$	−23.9621	−0.949414
$638$	0	0
$639$	3.05879	0.121004
$640$	0	0
$641$	0.166926	0.00659316	0.00329658	−	0.999995i	$-0.498951\pi$
$641$	0.166926	0.00659316	0.00329658	+	0.999995i	$0.498951\pi$
$642$	0	0
$643$	30.9571	1.22083	0.610415	−	0.792082i	$-0.291003\pi$
$643$	30.9571	1.22083	0.610415	+	0.792082i	$0.291003\pi$
$644$	0	0
$645$	−26.9165	−1.05984
$646$	0	0
$647$	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
$650$	0	0
$651$	5.10765	0.200185
$652$	0	0
$653$	27.5173	1.07684	0.538418	−	0.842678i	$-0.319022\pi$
$653$	27.5173	1.07684	0.538418	+	0.842678i	$0.319022\pi$
$654$	0	0
$655$	12.5668	0.491027
$656$	0	0
$657$	−19.6055	−0.764884
$658$	0	0
$659$	−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	$0.483846\pi$
$662$	0	0
$663$	−41.1452	−1.59795
$664$	0	0
$665$	20.8684	0.809242
$666$	0	0
$667$	−38.2336	−1.48041
$668$	0	0
$669$	−35.4134	−1.36916
$670$	0	0
$671$	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	$0.580554\pi$
$674$	0	0
$675$	−4.25496	−0.163774
$676$	0	0
$677$	30.7406	1.18146	0.590728	−	0.806871i	$-0.298841\pi$
$677$	30.7406	1.18146	0.590728	+	0.806871i	$0.298841\pi$
$678$	0	0
$679$	−18.1452	−0.696349
$680$	0	0
$681$	7.14110	0.273648
$682$	0	0
$683$	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
$686$	0	0
$687$	17.2651	0.658706
$688$	0	0
$689$	−48.1411	−1.83403
$690$	0	0
$691$	−18.1544	−0.690626	−0.345313	−	0.938488i	$-0.612227\pi$
$691$	−18.1544	−0.690626	−0.345313	+	0.938488i	$0.612227\pi$
$692$	0	0
$693$	−11.8642	−0.450684
$694$	0	0
$695$	−1.66457	−0.0631408
$696$	0	0
$697$	−64.5537	−2.44515
$698$	0	0
$699$	−32.7559	−1.23894
$700$	0	0
$701$	−46.2492	−1.74681	−0.873404	−	0.486996i	$-0.838093\pi$
$701$	−46.2492	−1.74681	−0.873404	+	0.486996i	$0.838093\pi$
$702$	0	0
$703$	−9.10585	−0.343434
$704$	0	0
$705$	−21.9499	−0.826683
$706$	0	0
$707$	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	$0.733236\pi$
$710$	0	0
$711$	5.60055	0.210037
$712$	0	0
$713$	−17.6885	−0.662441
$714$	0	0
$715$	62.3425	2.33148
$716$	0	0
$717$	−9.64112	−0.360054
$718$	0	0
$719$	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
$722$	0	0
$723$	−2.86036	−0.106378
$724$	0	0
$725$	−4.65637	−0.172933
$726$	0	0
$727$	24.4331	0.906172	0.453086	−	0.891467i	$-0.350323\pi$
$727$	24.4331	0.906172	0.453086	+	0.891467i	$0.350323\pi$
$728$	0	0
$729$	23.0209	0.852628
$730$	0	0
$731$	−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	$0.256711\pi$
$734$	0	0
$735$	13.6282	0.502684
$736$	0	0
$737$	−12.9389	−0.476612
$738$	0	0
$739$	18.6134	0.684707	0.342353	−	0.939571i	$-0.388776\pi$
$739$	18.6134	0.684707	0.342353	+	0.939571i	$0.388776\pi$
$740$	0	0
$741$	35.3024	1.29686
$742$	0	0
$743$	−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
$746$	0	0
$747$	−1.06005	−0.0387854
$748$	0	0
$749$	−18.6563	−0.681688
$750$	0	0
$751$	11.0543	0.403377	0.201689	−	0.979450i	$-0.435357\pi$
$751$	11.0543	0.403377	0.201689	+	0.979450i	$0.435357\pi$
$752$	0	0
$753$	27.8682	1.01557
$754$	0	0
$755$	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	$0.302998\pi$
$758$	0	0
$759$	−39.0877	−1.41879
$760$	0	0
$761$	14.7166	0.533477	0.266739	−	0.963769i	$-0.414054\pi$
$761$	14.7166	0.533477	0.266739	+	0.963769i	$0.414054\pi$
$762$	0	0
$763$	−21.6113	−0.782380
$764$	0	0
$765$	−24.5982	−0.889348
$766$	0	0
$767$	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	$0.605325\pi$
$770$	0	0
$771$	28.1102	1.01236
$772$	0	0
$773$	48.8663	1.75760	0.878799	−	0.477193i	$-0.158346\pi$
$773$	48.8663	1.75760	0.878799	+	0.477193i	$0.158346\pi$
$774$	0	0
$775$	−2.15424	−0.0773825
$776$	0	0
$777$	2.93069	0.105138
$778$	0	0
$779$	55.3868	1.98444
$780$	0	0
$781$	10.0998	0.361399
$782$	0	0
$783$	32.9526	1.17763
$784$	0	0
$785$	29.4033	1.04945
$786$	0	0
$787$	31.8942	1.13691	0.568453	−	0.822715i	$-0.307542\pi$
$787$	31.8942	1.13691	0.568453	+	0.822715i	$0.307542\pi$
$788$	0	0
$789$	−5.59103	−0.199046
$790$	0	0
$791$	−19.3099	−0.686579
$792$	0	0
$793$	−40.7753	−1.44797
$794$	0	0
$795$	27.3797	0.971059
$796$	0	0
$797$	−26.0846	−0.923964	−0.461982	−	0.886889i	$-0.652861\pi$
$797$	−26.0846	−0.923964	−0.461982	+	0.886889i	$0.652861\pi$
$798$	0	0
$799$	−50.2808	−1.77881
$800$	0	0
$801$	−4.28895	−0.151543
$802$	0	0
$803$	−64.7353	−2.28446
$804$	0	0
$805$	23.2600	0.819806
$806$	0	0
$807$	−1.20937	−0.0425720
$808$	0	0
$809$	−2.45878	−0.0864461	−0.0432231	−	0.999065i	$-0.513763\pi$
$809$	−2.45878	−0.0864461	−0.0432231	+	0.999065i	$0.513763\pi$
$810$	0	0
$811$	20.3841	0.715783	0.357891	−	0.933763i	$-0.383496\pi$
$811$	20.3841	0.715783	0.357891	+	0.933763i	$0.383496\pi$
$812$	0	0
$813$	−15.0338	−0.527260
$814$	0	0
$815$	−56.9114	−1.99352
$816$	0	0
$817$	52.9020	1.85081
$818$	0	0
$819$	11.9433	0.417332
$820$	0	0
$821$	−8.31824	−0.290309	−0.145154	−	0.989409i	$-0.546368\pi$
$821$	−8.31824	−0.290309	−0.145154	+	0.989409i	$0.546368\pi$
$822$	0	0
$823$	25.5967	0.892244	0.446122	−	0.894972i	$-0.352805\pi$
$823$	25.5967	0.892244	0.446122	+	0.894972i	$0.352805\pi$
$824$	0	0
$825$	−4.76038	−0.165735
$826$	0	0
$827$	−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	$0.585530\pi$
$830$	0	0
$831$	−5.95933	−0.206727
$832$	0	0
$833$	31.2181	1.08164
$834$	0	0
$835$	−8.52403	−0.294986
$836$	0	0
$837$	15.2453	0.526955
$838$	0	0
$839$	−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
$842$	0	0
$843$	−25.7832	−0.888019
$844$	0	0
$845$	−31.5162	−1.08419
$846$	0	0
$847$	−22.4525	−0.771476
$848$	0	0
$849$	36.2808	1.24516
$850$	0	0
$851$	−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	$0.603381\pi$
$854$	0	0
$855$	21.1051	0.721779
$856$	0	0
$857$	38.5929	1.31831	0.659154	−	0.752008i	$-0.270915\pi$
$857$	38.5929	1.31831	0.659154	+	0.752008i	$0.270915\pi$
$858$	0	0
$859$	−36.7533	−1.25400	−0.627002	−	0.779017i	$-0.715718\pi$
$859$	−36.7533	−1.25400	−0.627002	+	0.779017i	$0.715718\pi$
$860$	0	0
$861$	−17.8260	−0.607510
$862$	0	0
$863$	−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
$866$	0	0
$867$	33.0451	1.12227
$868$	0	0
$869$	18.4924	0.627313
$870$	0	0
$871$	13.0252	0.441341
$872$	0	0
$873$	−18.3510	−0.621087
$874$	0	0
$875$	−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	$0.475503\pi$
$878$	0	0
$879$	20.4966	0.691334
$880$	0	0
$881$	−16.4979	−0.555828	−0.277914	−	0.960606i	$-0.589643\pi$
$881$	−16.4979	−0.555828	−0.277914	+	0.960606i	$0.589643\pi$
$882$	0	0
$883$	43.4286	1.46149	0.730744	−	0.682652i	$-0.239173\pi$
$883$	43.4286	1.46149	0.730744	+	0.682652i	$0.239173\pi$
$884$	0	0
$885$	−18.3489	−0.616790
$886$	0	0
$887$	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
$890$	0	0
$891$	10.2751	0.344230
$892$	0	0
$893$	43.1406	1.44365
$894$	0	0
$895$	5.12058	0.171162
$896$	0	0
$897$	39.3481	1.31380
$898$	0	0
$899$	16.6835	0.556427
$900$	0	0
$901$	62.7188	2.08947
$902$	0	0
$903$	−17.0263	−0.566600
$904$	0	0
$905$	−44.6010	−1.48259
$906$	0	0
$907$	43.0813	1.43049	0.715246	−	0.698873i	$-0.246315\pi$
$907$	43.0813	1.43049	0.715246	+	0.698873i	$0.246315\pi$
$908$	0	0
$909$	22.3345	0.740788
$910$	0	0
$911$	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
$914$	0	0
$915$	23.1905	0.766654
$916$	0	0
$917$	7.94928	0.262508
$918$	0	0
$919$	−42.2046	−1.39220	−0.696101	−	0.717944i	$-0.745083\pi$
$919$	−42.2046	−1.39220	−0.696101	+	0.717944i	$0.745083\pi$
$920$	0	0
$921$	−22.3390	−0.736093
$922$	0	0
$923$	−10.1671	−0.334654
$924$	0	0
$925$	−1.23607	−0.0406417
$926$	0	0
$927$	−15.5353	−0.510246
$928$	0	0
$929$	−40.9616	−1.34391	−0.671954	−	0.740593i	$-0.734545\pi$
$929$	−40.9616	−1.34391	−0.671954	+	0.740593i	$0.734545\pi$
$930$	0	0
$931$	−26.7850	−0.877843
$932$	0	0
$933$	−16.8273	−0.550901
$934$	0	0
$935$	−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	$0.558785\pi$
$938$	0	0
$939$	26.5658	0.866942
$940$	0	0
$941$	1.14007	0.0371652	0.0185826	−	0.999827i	$-0.494085\pi$
$941$	1.14007	0.0371652	0.0185826	+	0.999827i	$0.494085\pi$
$942$	0	0
$943$	61.7342	2.01034
$944$	0	0
$945$	−20.0472	−0.652135
$946$	0	0
$947$	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
$950$	0	0
$951$	−4.08853	−0.132580
$952$	0	0
$953$	−18.6016	−0.602567	−0.301283	−	0.953535i	$-0.597415\pi$
$953$	−18.6016	−0.602567	−0.301283	+	0.953535i	$0.597415\pi$
$954$	0	0
$955$	37.5413	1.21481
$956$	0	0
$957$	36.8668	1.19173
$958$	0	0
$959$	19.7111	0.636505
$960$	0	0
$961$	−23.2815	−0.751015
$962$	0	0
$963$	−18.8679	−0.608011
$964$	0	0
$965$	44.5595	1.43442
$966$	0	0
$967$	21.3331	0.686026	0.343013	−	0.939331i	$-0.388553\pi$
$967$	21.3331	0.686026	0.343013	+	0.939331i	$0.388553\pi$
$968$	0	0
$969$	−45.9923	−1.47749
$970$	0	0
$971$	−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
$974$	0	0
$975$	4.79210	0.153470
$976$	0	0
$977$	−22.6902	−0.725924	−0.362962	−	0.931804i	$-0.618234\pi$
$977$	−22.6902	−0.725924	−0.362962	+	0.931804i	$0.618234\pi$
$978$	0	0
$979$	−14.1617	−0.452609
$980$	0	0
$981$	−21.8564	−0.697820
$982$	0	0
$983$	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
$986$	0	0
$987$	−13.8847	−0.441954
$988$	0	0
$989$	58.9647	1.87497
$990$	0	0
$991$	11.0015	0.349475	0.174737	−	0.984615i	$-0.444092\pi$
$991$	11.0015	0.349475	0.174737	+	0.984615i	$0.444092\pi$
$992$	0	0
$993$	−26.5561	−0.842732
$994$	0	0
$995$	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	$0.437904\pi$
$998$	0	0
$999$	8.74752	0.276759

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.d.1.3	✓	7	4.3	odd	2
4304.2.a.i.1.5		7	1.1	even	1	trivial
4842.2.a.o.1.5		7	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:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q+1.20937 q^{3} -2.40321 q^{5} -1.52017 q^{7} -1.53742 q^{9} -5.07638 q^{11} +5.11021 q^{13} -2.90637 q^{15} -6.65764 q^{17} +5.71222 q^{19} -1.83846 q^{21} +6.36685 q^{23} +0.775401 q^{25} -5.48743 q^{27} -6.00511 q^{29} -2.77822 q^{31} -6.13924 q^{33} +3.65329 q^{35} -1.59410 q^{37} +6.18015 q^{39} +9.69619 q^{41} +9.26120 q^{43} +3.69473 q^{45} +7.55235 q^{47} -4.68907 q^{49} -8.05157 q^{51} -9.42058 q^{53} +12.1996 q^{55} +6.90820 q^{57} +6.31332 q^{59} -7.97918 q^{61} +2.33714 q^{63} -12.2809 q^{65} +2.54885 q^{67} +7.69990 q^{69} -1.98957 q^{71} +12.7523 q^{73} +0.937750 q^{75} +7.71699 q^{77} -3.64283 q^{79} -2.02411 q^{81} +0.689504 q^{83} +15.9997 q^{85} -7.26242 q^{87} +2.78972 q^{89} -7.76841 q^{91} -3.35991 q^{93} -13.7276 q^{95} +11.9363 q^{97} +7.80451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{3} - 6 q^{5} + 3 q^{7} + 7 q^{9} + 12 q^{11} + 3 q^{13} + 6 q^{15} - 8 q^{17} + 7 q^{19} - 5 q^{21} + 22 q^{23} + 9 q^{25} + 22 q^{27} - 7 q^{29} + 3 q^{31} - 19 q^{33} + 25 q^{35} - 13 q^{37}+ \cdots + 9 q^{99}+O(q^{100}) \) 7 * q + 4 * q^3 - 6 * q^5 + 3 * q^7 + 7 * q^9 + 12 * q^11 + 3 * q^13 + 6 * q^15 - 8 * q^17 + 7 * q^19 - 5 * q^21 + 22 * q^23 + 9 * q^25 + 22 * q^27 - 7 * q^29 + 3 * q^31 - 19 * q^33 + 25 * q^35 - 13 * q^37 - q^39 - 11 * q^41 + 14 * q^43 - 17 * q^45 + 23 * q^47 + 6 * q^49 + 21 * q^51 - 21 * q^53 - 15 * q^55 - 14 * q^57 + 20 * q^59 + 13 * q^61 + 12 * q^63 - 29 * q^65 + 30 * q^67 + 13 * q^69 + 23 * q^71 + 9 * q^73 - 3 * q^75 + 8 * q^77 - 11 * q^79 + 43 * q^81 + 3 * q^83 + 21 * q^85 - 28 * q^87 + 16 * q^89 + 10 * q^91 - 13 * q^93 + 21 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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