LMFDB - Embedded newform 6240.2.a.ca.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -0.508203 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -0.508203 q^{7} +1.00000 q^{9} +5.36266 q^{11} -1.00000 q^{13} -1.00000 q^{15} +7.87086 q^{17} -0.508203 q^{19} -0.508203 q^{21} +1.87086 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.87086 q^{29} -6.37907 q^{31} +5.36266 q^{33} +0.508203 q^{35} -9.74173 q^{37} -1.00000 q^{39} -0.983593 q^{41} +8.00000 q^{43} -1.00000 q^{45} +4.34625 q^{47} -6.74173 q^{49} +7.87086 q^{51} +3.01641 q^{53} -5.36266 q^{55} -0.508203 q^{57} +13.3627 q^{59} +6.00000 q^{61} -0.508203 q^{63} +1.00000 q^{65} -2.37907 q^{67} +1.87086 q^{69} -13.1044 q^{71} -7.87086 q^{73} +1.00000 q^{75} -2.72532 q^{77} +12.7581 q^{79} +1.00000 q^{81} -17.1044 q^{83} -7.87086 q^{85} +3.87086 q^{87} +2.00000 q^{89} +0.508203 q^{91} -6.37907 q^{93} +0.508203 q^{95} +15.6126 q^{97} +5.36266 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 3 q^{15} + 8 q^{17} - 10 q^{23} + 3 q^{25} + 3 q^{27} - 4 q^{29} - 2 q^{31} + 2 q^{33} + 2 q^{37} - 3 q^{39} - 6 q^{41} + 24 q^{43} - 3 q^{45} + 2 q^{47} + 11 q^{49} + 8 q^{51} + 6 q^{53} - 2 q^{55} + 26 q^{59} + 18 q^{61} + 3 q^{65} + 10 q^{67} - 10 q^{69} + 6 q^{71} - 8 q^{73} + 3 q^{75} + 20 q^{77} + 4 q^{79} + 3 q^{81} - 6 q^{83} - 8 q^{85} - 4 q^{87} + 6 q^{89} - 2 q^{93} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 3 * q^15 + 8 * q^17 - 10 * q^23 + 3 * q^25 + 3 * q^27 - 4 * q^29 - 2 * q^31 + 2 * q^33 + 2 * q^37 - 3 * q^39 - 6 * q^41 + 24 * q^43 - 3 * q^45 + 2 * q^47 + 11 * q^49 + 8 * q^51 + 6 * q^53 - 2 * q^55 + 26 * q^59 + 18 * q^61 + 3 * q^65 + 10 * q^67 - 10 * q^69 + 6 * q^71 - 8 * q^73 + 3 * q^75 + 20 * q^77 + 4 * q^79 + 3 * q^81 - 6 * q^83 - 8 * q^85 - 4 * q^87 + 6 * q^89 - 2 * q^93 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.508203	−0.192083	−0.0960414	−	0.995377i	$-0.530618\pi$
$7$	−0.508203	−0.192083	−0.0960414	+	0.995377i	$0.530618\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.36266	1.61690	0.808452	−	0.588563i	$-0.200306\pi$
$11$	5.36266	1.61690	0.808452	+	0.588563i	$0.200306\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	7.87086	1.90897	0.954483	−	0.298267i	$-0.0964085\pi$
$17$	7.87086	1.90897	0.954483	+	0.298267i	$0.0964085\pi$
$18$	0	0
$19$	−0.508203	−0.116590	−0.0582949	−	0.998299i	$-0.518566\pi$
$19$	−0.508203	−0.116590	−0.0582949	+	0.998299i	$0.518566\pi$
$20$	0	0
$21$	−0.508203	−0.110899
$22$	0	0
$23$	1.87086	0.390102	0.195051	−	0.980793i	$-0.437513\pi$
$23$	1.87086	0.390102	0.195051	+	0.980793i	$0.437513\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.87086	0.718802	0.359401	−	0.933183i	$-0.382981\pi$
$29$	3.87086	0.718802	0.359401	+	0.933183i	$0.382981\pi$
$30$	0	0
$31$	−6.37907	−1.14571	−0.572857	−	0.819655i	$-0.694165\pi$
$31$	−6.37907	−1.14571	−0.572857	+	0.819655i	$0.694165\pi$
$32$	0	0
$33$	5.36266	0.933520
$34$	0	0
$35$	0.508203	0.0859020
$36$	0	0
$37$	−9.74173	−1.60153	−0.800765	−	0.598978i	$-0.795574\pi$
$37$	−9.74173	−1.60153	−0.800765	+	0.598978i	$0.795574\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−0.983593	−0.153611	−0.0768057	−	0.997046i	$-0.524472\pi$
$41$	−0.983593	−0.153611	−0.0768057	+	0.997046i	$0.524472\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.34625	0.633966	0.316983	−	0.948431i	$-0.397330\pi$
$47$	4.34625	0.633966	0.316983	+	0.948431i	$0.397330\pi$
$48$	0	0
$49$	−6.74173	−0.963104
$50$	0	0
$51$	7.87086	1.10214
$52$	0	0
$53$	3.01641	0.414335	0.207168	−	0.978305i	$-0.433575\pi$
$53$	3.01641	0.414335	0.207168	+	0.978305i	$0.433575\pi$
$54$	0	0
$55$	−5.36266	−0.723101
$56$	0	0
$57$	−0.508203	−0.0673132
$58$	0	0
$59$	13.3627	1.73967	0.869835	−	0.493342i	$-0.164225\pi$
$59$	13.3627	1.73967	0.869835	+	0.493342i	$0.164225\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−0.508203	−0.0640276
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−2.37907	−0.290649	−0.145325	−	0.989384i	$-0.546423\pi$
$67$	−2.37907	−0.290649	−0.145325	+	0.989384i	$0.546423\pi$
$68$	0	0
$69$	1.87086	0.225226
$70$	0	0
$71$	−13.1044	−1.55521	−0.777603	−	0.628756i	$-0.783564\pi$
$71$	−13.1044	−1.55521	−0.777603	+	0.628756i	$0.783564\pi$
$72$	0	0
$73$	−7.87086	−0.921215	−0.460608	−	0.887604i	$-0.652368\pi$
$73$	−7.87086	−0.921215	−0.460608	+	0.887604i	$0.652368\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.72532	−0.310579
$78$	0	0
$79$	12.7581	1.43540	0.717701	−	0.696351i	$-0.245194\pi$
$79$	12.7581	1.43540	0.717701	+	0.696351i	$0.245194\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−17.1044	−1.87745	−0.938725	−	0.344666i	$-0.887992\pi$
$83$	−17.1044	−1.87745	−0.938725	+	0.344666i	$0.887992\pi$
$84$	0	0
$85$	−7.87086	−0.853715
$86$	0	0
$87$	3.87086	0.415000
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0.508203	0.0532742
$92$	0	0
$93$	−6.37907	−0.661479
$94$	0	0
$95$	0.508203	0.0521406
$96$	0	0
$97$	15.6126	1.58522	0.792609	−	0.609730i	$-0.208722\pi$
$97$	15.6126	1.58522	0.792609	+	0.609730i	$0.208722\pi$
$98$	0	0
$99$	5.36266	0.538968
$100$	0	0
$101$	2.85446	0.284029	0.142015	−	0.989865i	$-0.454642\pi$
$101$	2.85446	0.284029	0.142015	+	0.989865i	$0.454642\pi$
$102$	0	0
$103$	8.75814	0.862965	0.431482	−	0.902121i	$-0.357991\pi$
$103$	8.75814	0.862965	0.431482	+	0.902121i	$0.357991\pi$
$104$	0	0
$105$	0.508203	0.0495956
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−1.14554	−0.109723	−0.0548615	−	0.998494i	$-0.517472\pi$
$109$	−1.14554	−0.109723	−0.0548615	+	0.998494i	$0.517472\pi$
$110$	0	0
$111$	−9.74173	−0.924644
$112$	0	0
$113$	−3.87086	−0.364140	−0.182070	−	0.983286i	$-0.558280\pi$
$113$	−3.87086	−0.364140	−0.182070	+	0.983286i	$0.558280\pi$
$114$	0	0
$115$	−1.87086	−0.174459
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	17.7581	1.61438
$122$	0	0
$123$	−0.983593	−0.0886876
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.72532	−0.596776	−0.298388	−	0.954445i	$-0.596449\pi$
$127$	−6.72532	−0.596776	−0.298388	+	0.954445i	$0.596449\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	−12.5962	−1.10053	−0.550267	−	0.834989i	$-0.685474\pi$
$131$	−12.5962	−1.10053	−0.550267	+	0.834989i	$0.685474\pi$
$132$	0	0
$133$	0.258271	0.0223949
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	12.9836	1.10926	0.554632	−	0.832096i	$-0.312859\pi$
$137$	12.9836	1.10926	0.554632	+	0.832096i	$0.312859\pi$
$138$	0	0
$139$	−7.74173	−0.656645	−0.328322	−	0.944566i	$-0.606483\pi$
$139$	−7.74173	−0.656645	−0.328322	+	0.944566i	$0.606483\pi$
$140$	0	0
$141$	4.34625	0.366021
$142$	0	0
$143$	−5.36266	−0.448448
$144$	0	0
$145$	−3.87086	−0.321458
$146$	0	0
$147$	−6.74173	−0.556048
$148$	0	0
$149$	−1.74173	−0.142688	−0.0713440	−	0.997452i	$-0.522729\pi$
$149$	−1.74173	−0.142688	−0.0713440	+	0.997452i	$0.522729\pi$
$150$	0	0
$151$	−1.62093	−0.131910	−0.0659548	−	0.997823i	$-0.521009\pi$
$151$	−1.62093	−0.131910	−0.0659548	+	0.997823i	$0.521009\pi$
$152$	0	0
$153$	7.87086	0.636322
$154$	0	0
$155$	6.37907	0.512379
$156$	0	0
$157$	20.4671	1.63345	0.816724	−	0.577028i	$-0.195788\pi$
$157$	20.4671	1.63345	0.816724	+	0.577028i	$0.195788\pi$
$158$	0	0
$159$	3.01641	0.239217
$160$	0	0
$161$	−0.950780	−0.0749319
$162$	0	0
$163$	20.0880	1.57341	0.786706	−	0.617328i	$-0.211785\pi$
$163$	20.0880	1.57341	0.786706	+	0.617328i	$0.211785\pi$
$164$	0	0
$165$	−5.36266	−0.417483
$166$	0	0
$167$	8.08798	0.625867	0.312933	−	0.949775i	$-0.398688\pi$
$167$	8.08798	0.625867	0.312933	+	0.949775i	$0.398688\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.508203	−0.0388633
$172$	0	0
$173$	22.4342	1.70564	0.852822	−	0.522202i	$-0.174889\pi$
$173$	22.4342	1.70564	0.852822	+	0.522202i	$0.174889\pi$
$174$	0	0
$175$	−0.508203	−0.0384166
$176$	0	0
$177$	13.3627	1.00440
$178$	0	0
$179$	3.90368	0.291775	0.145887	−	0.989301i	$-0.453396\pi$
$179$	3.90368	0.291775	0.145887	+	0.989301i	$0.453396\pi$
$180$	0	0
$181$	−0.983593	−0.0731099	−0.0365550	−	0.999332i	$-0.511638\pi$
$181$	−0.983593	−0.0731099	−0.0365550	+	0.999332i	$0.511638\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	9.74173	0.716226
$186$	0	0
$187$	42.2088	3.08661
$188$	0	0
$189$	−0.508203	−0.0369664
$190$	0	0
$191$	−5.27468	−0.381662	−0.190831	−	0.981623i	$-0.561118\pi$
$191$	−5.27468	−0.381662	−0.190831	+	0.981623i	$0.561118\pi$
$192$	0	0
$193$	−17.5798	−1.26542	−0.632710	−	0.774389i	$-0.718058\pi$
$193$	−17.5798	−1.26542	−0.632710	+	0.774389i	$0.718058\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	15.0164	1.06987	0.534937	−	0.844892i	$-0.320335\pi$
$197$	15.0164	1.06987	0.534937	+	0.844892i	$0.320335\pi$
$198$	0	0
$199$	−15.4835	−1.09759	−0.548797	−	0.835956i	$-0.684914\pi$
$199$	−15.4835	−1.09759	−0.548797	+	0.835956i	$0.684914\pi$
$200$	0	0
$201$	−2.37907	−0.167807
$202$	0	0
$203$	−1.96719	−0.138069
$204$	0	0
$205$	0.983593	0.0686971
$206$	0	0
$207$	1.87086	0.130034
$208$	0	0
$209$	−2.72532	−0.188515
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	−13.1044	−0.897898
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	3.24186	0.220072
$218$	0	0
$219$	−7.87086	−0.531864
$220$	0	0
$221$	−7.87086	−0.529452
$222$	0	0
$223$	10.5410	0.705879	0.352939	−	0.935646i	$-0.385182\pi$
$223$	10.5410	0.705879	0.352939	+	0.935646i	$0.385182\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−16.3463	−1.08494	−0.542470	−	0.840075i	$-0.682511\pi$
$227$	−16.3463	−1.08494	−0.542470	+	0.840075i	$0.682511\pi$
$228$	0	0
$229$	16.8873	1.11594	0.557971	−	0.829860i	$-0.311580\pi$
$229$	16.8873	1.11594	0.557971	+	0.829860i	$0.311580\pi$
$230$	0	0
$231$	−2.72532	−0.179313
$232$	0	0
$233$	3.11273	0.203922	0.101961	−	0.994788i	$-0.467488\pi$
$233$	3.11273	0.203922	0.101961	+	0.994788i	$0.467488\pi$
$234$	0	0
$235$	−4.34625	−0.283518
$236$	0	0
$237$	12.7581	0.828730
$238$	0	0
$239$	15.0716	0.974899	0.487450	−	0.873151i	$-0.337927\pi$
$239$	15.0716	0.974899	0.487450	+	0.873151i	$0.337927\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.74173	0.430713
$246$	0	0
$247$	0.508203	0.0323362
$248$	0	0
$249$	−17.1044	−1.08395
$250$	0	0
$251$	−15.6454	−0.987529	−0.493765	−	0.869596i	$-0.664380\pi$
$251$	−15.6454	−0.987529	−0.493765	+	0.869596i	$0.664380\pi$
$252$	0	0
$253$	10.0328	0.630758
$254$	0	0
$255$	−7.87086	−0.492893
$256$	0	0
$257$	−14.5962	−0.910485	−0.455243	−	0.890367i	$-0.650447\pi$
$257$	−14.5962	−0.910485	−0.455243	+	0.890367i	$0.650447\pi$
$258$	0	0
$259$	4.95078	0.307626
$260$	0	0
$261$	3.87086	0.239601
$262$	0	0
$263$	−18.5634	−1.14467	−0.572333	−	0.820021i	$-0.693962\pi$
$263$	−18.5634	−1.14467	−0.572333	+	0.820021i	$0.693962\pi$
$264$	0	0
$265$	−3.01641	−0.185296
$266$	0	0
$267$	2.00000	0.122398
$268$	0	0
$269$	7.61259	0.464148	0.232074	−	0.972698i	$-0.425449\pi$
$269$	7.61259	0.464148	0.232074	+	0.972698i	$0.425449\pi$
$270$	0	0
$271$	4.34625	0.264016	0.132008	−	0.991249i	$-0.457858\pi$
$271$	4.34625	0.264016	0.132008	+	0.991249i	$0.457858\pi$
$272$	0	0
$273$	0.508203	0.0307579
$274$	0	0
$275$	5.36266	0.323381
$276$	0	0
$277$	2.25827	0.135686	0.0678432	−	0.997696i	$-0.478388\pi$
$277$	2.25827	0.135686	0.0678432	+	0.997696i	$0.478388\pi$
$278$	0	0
$279$	−6.37907	−0.381905
$280$	0	0
$281$	−0.0328135	−0.00195749	−0.000978745	−	1.00000i	$-0.500312\pi$
$281$	−0.0328135	−0.00195749	−0.000978745		1.00000i	$0.500312\pi$
$282$	0	0
$283$	18.0328	1.07194	0.535970	−	0.844237i	$-0.319946\pi$
$283$	18.0328	1.07194	0.535970	+	0.844237i	$0.319946\pi$
$284$	0	0
$285$	0.508203	0.0301034
$286$	0	0
$287$	0.499865	0.0295061
$288$	0	0
$289$	44.9505	2.64415
$290$	0	0
$291$	15.6126	0.915226
$292$	0	0
$293$	30.4999	1.78182	0.890911	−	0.454179i	$-0.150067\pi$
$293$	30.4999	1.78182	0.890911	+	0.454179i	$0.150067\pi$
$294$	0	0
$295$	−13.3627	−0.778004
$296$	0	0
$297$	5.36266	0.311173
$298$	0	0
$299$	−1.87086	−0.108195
$300$	0	0
$301$	−4.06563	−0.234339
$302$	0	0
$303$	2.85446	0.163984
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	−7.65375	−0.436822	−0.218411	−	0.975857i	$-0.570087\pi$
$307$	−7.65375	−0.436822	−0.218411	+	0.975857i	$0.570087\pi$
$308$	0	0
$309$	8.75814	0.498233
$310$	0	0
$311$	−2.03281	−0.115270	−0.0576351	−	0.998338i	$-0.518356\pi$
$311$	−2.03281	−0.115270	−0.0576351	+	0.998338i	$0.518356\pi$
$312$	0	0
$313$	16.9836	0.959969	0.479985	−	0.877277i	$-0.340642\pi$
$313$	16.9836	0.959969	0.479985	+	0.877277i	$0.340642\pi$
$314$	0	0
$315$	0.508203	0.0286340
$316$	0	0
$317$	−19.4506	−1.09246	−0.546229	−	0.837636i	$-0.683937\pi$
$317$	−19.4506	−1.09246	−0.546229	+	0.837636i	$0.683937\pi$
$318$	0	0
$319$	20.7581	1.16223
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−1.14554	−0.0633486
$328$	0	0
$329$	−2.20878	−0.121774
$330$	0	0
$331$	−15.9917	−0.878981	−0.439491	−	0.898247i	$-0.644841\pi$
$331$	−15.9917	−0.878981	−0.439491	+	0.898247i	$0.644841\pi$
$332$	0	0
$333$	−9.74173	−0.533843
$334$	0	0
$335$	2.37907	0.129982
$336$	0	0
$337$	30.4342	1.65786	0.828929	−	0.559353i	$-0.188951\pi$
$337$	30.4342	1.65786	0.828929	+	0.559353i	$0.188951\pi$
$338$	0	0
$339$	−3.87086	−0.210237
$340$	0	0
$341$	−34.2088	−1.85251
$342$	0	0
$343$	6.98359	0.377079
$344$	0	0
$345$	−1.87086	−0.100724
$346$	0	0
$347$	0.258271	0.0138647	0.00693235	−	0.999976i	$-0.497793\pi$
$347$	0.258271	0.0138647	0.00693235	+	0.999976i	$0.497793\pi$
$348$	0	0
$349$	14.8545	0.795141	0.397570	−	0.917572i	$-0.369853\pi$
$349$	14.8545	0.795141	0.397570	+	0.917572i	$0.369853\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	36.4671	1.94095	0.970473	−	0.241211i	$-0.0775446\pi$
$353$	36.4671	1.94095	0.970473	+	0.241211i	$0.0775446\pi$
$354$	0	0
$355$	13.1044	0.695509
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	11.0716	0.584335	0.292168	−	0.956367i	$-0.405623\pi$
$359$	11.0716	0.584335	0.292168	+	0.956367i	$0.405623\pi$
$360$	0	0
$361$	−18.7417	−0.986407
$362$	0	0
$363$	17.7581	0.932060
$364$	0	0
$365$	7.87086	0.411980
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	−0.983593	−0.0512038
$370$	0	0
$371$	−1.53295	−0.0795867
$372$	0	0
$373$	32.5327	1.68448	0.842239	−	0.539104i	$-0.181237\pi$
$373$	32.5327	1.68448	0.842239	+	0.539104i	$0.181237\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−3.87086	−0.199360
$378$	0	0
$379$	21.2663	1.09238	0.546189	−	0.837662i	$-0.316078\pi$
$379$	21.2663	1.09238	0.546189	+	0.837662i	$0.316078\pi$
$380$	0	0
$381$	−6.72532	−0.344549
$382$	0	0
$383$	−10.6373	−0.543543	−0.271771	−	0.962362i	$-0.587609\pi$
$383$	−10.6373	−0.543543	−0.271771	+	0.962362i	$0.587609\pi$
$384$	0	0
$385$	2.72532	0.138895
$386$	0	0
$387$	8.00000	0.406663
$388$	0	0
$389$	25.6454	1.30027	0.650137	−	0.759817i	$-0.274712\pi$
$389$	25.6454	1.30027	0.650137	+	0.759817i	$0.274712\pi$
$390$	0	0
$391$	14.7253	0.744692
$392$	0	0
$393$	−12.5962	−0.635394
$394$	0	0
$395$	−12.7581	−0.641931
$396$	0	0
$397$	0.291084	0.0146091	0.00730455	−	0.999973i	$-0.497675\pi$
$397$	0.291084	0.0146091	0.00730455	+	0.999973i	$0.497675\pi$
$398$	0	0
$399$	0.258271	0.0129297
$400$	0	0
$401$	−26.4999	−1.32334	−0.661670	−	0.749795i	$-0.730152\pi$
$401$	−26.4999	−1.32334	−0.661670	+	0.749795i	$0.730152\pi$
$402$	0	0
$403$	6.37907	0.317764
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−52.2416	−2.58952
$408$	0	0
$409$	−35.2580	−1.74340	−0.871698	−	0.490043i	$-0.836981\pi$
$409$	−35.2580	−1.74340	−0.871698	+	0.490043i	$0.836981\pi$
$410$	0	0
$411$	12.9836	0.640433
$412$	0	0
$413$	−6.79095	−0.334161
$414$	0	0
$415$	17.1044	0.839622
$416$	0	0
$417$	−7.74173	−0.379114
$418$	0	0
$419$	12.9201	0.631187	0.315594	−	0.948894i	$-0.397796\pi$
$419$	12.9201	0.631187	0.315594	+	0.948894i	$0.397796\pi$
$420$	0	0
$421$	2.59619	0.126530	0.0632652	−	0.997997i	$-0.479849\pi$
$421$	2.59619	0.126530	0.0632652	+	0.997997i	$0.479849\pi$
$422$	0	0
$423$	4.34625	0.211322
$424$	0	0
$425$	7.87086	0.381793
$426$	0	0
$427$	−3.04922	−0.147562
$428$	0	0
$429$	−5.36266	−0.258912
$430$	0	0
$431$	−20.3463	−0.980045	−0.490022	−	0.871710i	$-0.663011\pi$
$431$	−20.3463	−0.980045	−0.490022	+	0.871710i	$0.663011\pi$
$432$	0	0
$433$	−7.70892	−0.370467	−0.185233	−	0.982695i	$-0.559304\pi$
$433$	−7.70892	−0.370467	−0.185233	+	0.982695i	$0.559304\pi$
$434$	0	0
$435$	−3.87086	−0.185594
$436$	0	0
$437$	−0.950780	−0.0454820
$438$	0	0
$439$	−16.7581	−0.799822	−0.399911	−	0.916554i	$-0.630959\pi$
$439$	−16.7581	−0.799822	−0.399911	+	0.916554i	$0.630959\pi$
$440$	0	0
$441$	−6.74173	−0.321035
$442$	0	0
$443$	3.67610	0.174657	0.0873284	−	0.996180i	$-0.472167\pi$
$443$	3.67610	0.174657	0.0873284	+	0.996180i	$0.472167\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	−1.74173	−0.0823810
$448$	0	0
$449$	19.5163	0.921030	0.460515	−	0.887652i	$-0.347665\pi$
$449$	19.5163	0.921030	0.460515	+	0.887652i	$0.347665\pi$
$450$	0	0
$451$	−5.27468	−0.248375
$452$	0	0
$453$	−1.62093	−0.0761580
$454$	0	0
$455$	−0.508203	−0.0238249
$456$	0	0
$457$	0.129135	0.00604070	0.00302035	−	0.999995i	$-0.499039\pi$
$457$	0.129135	0.00604070	0.00302035	+	0.999995i	$0.499039\pi$
$458$	0	0
$459$	7.87086	0.367381
$460$	0	0
$461$	−0.0328135	−0.00152828	−0.000764139	−	1.00000i	$-0.500243\pi$
$461$	−0.0328135	−0.00152828	−0.000764139		1.00000i	$0.500243\pi$
$462$	0	0
$463$	−41.1840	−1.91398	−0.956992	−	0.290113i	$-0.906307\pi$
$463$	−41.1840	−1.91398	−0.956992	+	0.290113i	$0.906307\pi$
$464$	0	0
$465$	6.37907	0.295822
$466$	0	0
$467$	−1.27468	−0.0589850	−0.0294925	−	0.999565i	$-0.509389\pi$
$467$	−1.27468	−0.0589850	−0.0294925	+	0.999565i	$0.509389\pi$
$468$	0	0
$469$	1.20905	0.0558288
$470$	0	0
$471$	20.4671	0.943072
$472$	0	0
$473$	42.9013	1.97260
$474$	0	0
$475$	−0.508203	−0.0233180
$476$	0	0
$477$	3.01641	0.138112
$478$	0	0
$479$	−19.1372	−0.874401	−0.437201	−	0.899364i	$-0.644030\pi$
$479$	−19.1372	−0.874401	−0.437201	+	0.899364i	$0.644030\pi$
$480$	0	0
$481$	9.74173	0.444185
$482$	0	0
$483$	−0.950780	−0.0432620
$484$	0	0
$485$	−15.6126	−0.708931
$486$	0	0
$487$	−1.71725	−0.0778162	−0.0389081	−	0.999243i	$-0.512388\pi$
$487$	−1.71725	−0.0778162	−0.0389081	+	0.999243i	$0.512388\pi$
$488$	0	0
$489$	20.0880	0.908410
$490$	0	0
$491$	31.6454	1.42814	0.714069	−	0.700076i	$-0.246850\pi$
$491$	31.6454	1.42814	0.714069	+	0.700076i	$0.246850\pi$
$492$	0	0
$493$	30.4671	1.37217
$494$	0	0
$495$	−5.36266	−0.241034
$496$	0	0
$497$	6.65970	0.298728
$498$	0	0
$499$	3.75007	0.167876	0.0839380	−	0.996471i	$-0.473250\pi$
$499$	3.75007	0.167876	0.0839380	+	0.996471i	$0.473250\pi$
$500$	0	0
$501$	8.08798	0.361344
$502$	0	0
$503$	−36.5962	−1.63174	−0.815872	−	0.578233i	$-0.803742\pi$
$503$	−36.5962	−1.63174	−0.815872	+	0.578233i	$0.803742\pi$
$504$	0	0
$505$	−2.85446	−0.127022
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−13.4835	−0.597644	−0.298822	−	0.954309i	$-0.596594\pi$
$509$	−13.4835	−0.597644	−0.298822	+	0.954309i	$0.596594\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	−0.508203	−0.0224377
$514$	0	0
$515$	−8.75814	−0.385930
$516$	0	0
$517$	23.3075	1.02506
$518$	0	0
$519$	22.4342	0.984754
$520$	0	0
$521$	−10.9508	−0.479762	−0.239881	−	0.970802i	$-0.577108\pi$
$521$	−10.9508	−0.479762	−0.239881	+	0.970802i	$0.577108\pi$
$522$	0	0
$523$	−32.9341	−1.44011	−0.720054	−	0.693918i	$-0.755883\pi$
$523$	−32.9341	−1.44011	−0.720054	+	0.693918i	$0.755883\pi$
$524$	0	0
$525$	−0.508203	−0.0221798
$526$	0	0
$527$	−50.2088	−2.18713
$528$	0	0
$529$	−19.4999	−0.847820
$530$	0	0
$531$	13.3627	0.579890
$532$	0	0
$533$	0.983593	0.0426042
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	3.90368	0.168456
$538$	0	0
$539$	−36.1536	−1.55725
$540$	0	0
$541$	−38.2723	−1.64545	−0.822727	−	0.568437i	$-0.807548\pi$
$541$	−38.2723	−1.64545	−0.822727	+	0.568437i	$0.807548\pi$
$542$	0	0
$543$	−0.983593	−0.0422100
$544$	0	0
$545$	1.14554	0.0490696
$546$	0	0
$547$	−25.4506	−1.08819	−0.544096	−	0.839023i	$-0.683127\pi$
$547$	−25.4506	−1.08819	−0.544096	+	0.839023i	$0.683127\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−1.96719	−0.0838050
$552$	0	0
$553$	−6.48373	−0.275716
$554$	0	0
$555$	9.74173	0.413513
$556$	0	0
$557$	−19.4506	−0.824150	−0.412075	−	0.911150i	$-0.635196\pi$
$557$	−19.4506	−0.824150	−0.412075	+	0.911150i	$0.635196\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	42.2088	1.78206
$562$	0	0
$563$	16.4342	0.692621	0.346310	−	0.938120i	$-0.387434\pi$
$563$	16.4342	0.692621	0.346310	+	0.938120i	$0.387434\pi$
$564$	0	0
$565$	3.87086	0.162849
$566$	0	0
$567$	−0.508203	−0.0213425
$568$	0	0
$569$	43.1924	1.81072	0.905359	−	0.424646i	$-0.139602\pi$
$569$	43.1924	1.81072	0.905359	+	0.424646i	$0.139602\pi$
$570$	0	0
$571$	−13.0164	−0.544720	−0.272360	−	0.962195i	$-0.587804\pi$
$571$	−13.0164	−0.544720	−0.272360	+	0.962195i	$0.587804\pi$
$572$	0	0
$573$	−5.27468	−0.220353
$574$	0	0
$575$	1.87086	0.0780204
$576$	0	0
$577$	−4.12914	−0.171898	−0.0859491	−	0.996300i	$-0.527392\pi$
$577$	−4.12914	−0.171898	−0.0859491	+	0.996300i	$0.527392\pi$
$578$	0	0
$579$	−17.5798	−0.730591
$580$	0	0
$581$	8.69251	0.360626
$582$	0	0
$583$	16.1760	0.669940
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	9.68656	0.399807	0.199904	−	0.979816i	$-0.435937\pi$
$587$	9.68656	0.399807	0.199904	+	0.979816i	$0.435937\pi$
$588$	0	0
$589$	3.24186	0.133579
$590$	0	0
$591$	15.0164	0.617692
$592$	0	0
$593$	4.65970	0.191351	0.0956754	−	0.995413i	$-0.469499\pi$
$593$	4.65970	0.191351	0.0956754	+	0.995413i	$0.469499\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	−15.4835	−0.633696
$598$	0	0
$599$	−30.1432	−1.23162	−0.615808	−	0.787896i	$-0.711170\pi$
$599$	−30.1432	−1.23162	−0.615808	+	0.787896i	$0.711170\pi$
$600$	0	0
$601$	−35.7089	−1.45660	−0.728299	−	0.685260i	$-0.759689\pi$
$601$	−35.7089	−1.45660	−0.728299	+	0.685260i	$0.759689\pi$
$602$	0	0
$603$	−2.37907	−0.0968831
$604$	0	0
$605$	−17.7581	−0.721971
$606$	0	0
$607$	−34.9669	−1.41926	−0.709632	−	0.704573i	$-0.751139\pi$
$607$	−34.9669	−1.41926	−0.709632	+	0.704573i	$0.751139\pi$
$608$	0	0
$609$	−1.96719	−0.0797144
$610$	0	0
$611$	−4.34625	−0.175831
$612$	0	0
$613$	33.8074	1.36547	0.682733	−	0.730668i	$-0.260791\pi$
$613$	33.8074	1.36547	0.682733	+	0.730668i	$0.260791\pi$
$614$	0	0
$615$	0.983593	0.0396623
$616$	0	0
$617$	44.4671	1.79018	0.895088	−	0.445889i	$-0.147113\pi$
$617$	44.4671	1.79018	0.895088	+	0.445889i	$0.147113\pi$
$618$	0	0
$619$	−25.0081	−1.00516	−0.502580	−	0.864531i	$-0.667616\pi$
$619$	−25.0081	−1.00516	−0.502580	+	0.864531i	$0.667616\pi$
$620$	0	0
$621$	1.87086	0.0750752
$622$	0	0
$623$	−1.01641	−0.0407215
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.72532	−0.108839
$628$	0	0
$629$	−76.6758	−3.05727
$630$	0	0
$631$	25.6043	1.01929	0.509645	−	0.860385i	$-0.329777\pi$
$631$	25.6043	1.01929	0.509645	+	0.860385i	$0.329777\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	6.72532	0.266886
$636$	0	0
$637$	6.74173	0.267117
$638$	0	0
$639$	−13.1044	−0.518402
$640$	0	0
$641$	−34.7581	−1.37286	−0.686432	−	0.727194i	$-0.740824\pi$
$641$	−34.7581	−1.37286	−0.686432	+	0.727194i	$0.740824\pi$
$642$	0	0
$643$	19.0716	0.752110	0.376055	−	0.926597i	$-0.377280\pi$
$643$	19.0716	0.752110	0.376055	+	0.926597i	$0.377280\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	−15.8381	−0.622658	−0.311329	−	0.950302i	$-0.600774\pi$
$647$	−15.8381	−0.622658	−0.311329	+	0.950302i	$0.600774\pi$
$648$	0	0
$649$	71.6594	2.81288
$650$	0	0
$651$	3.24186	0.127059
$652$	0	0
$653$	−24.0328	−0.940477	−0.470238	−	0.882539i	$-0.655832\pi$
$653$	−24.0328	−0.940477	−0.470238	+	0.882539i	$0.655832\pi$
$654$	0	0
$655$	12.5962	0.492174
$656$	0	0
$657$	−7.87086	−0.307072
$658$	0	0
$659$	−5.11273	−0.199164	−0.0995818	−	0.995029i	$-0.531751\pi$
$659$	−5.11273	−0.199164	−0.0995818	+	0.995029i	$0.531751\pi$
$660$	0	0
$661$	34.0796	1.32554	0.662772	−	0.748821i	$-0.269380\pi$
$661$	34.0796	1.32554	0.662772	+	0.748821i	$0.269380\pi$
$662$	0	0
$663$	−7.87086	−0.305679
$664$	0	0
$665$	−0.258271	−0.0100153
$666$	0	0
$667$	7.24186	0.280406
$668$	0	0
$669$	10.5410	0.407539
$670$	0	0
$671$	32.1760	1.24214
$672$	0	0
$673$	−44.1432	−1.70159	−0.850797	−	0.525495i	$-0.823880\pi$
$673$	−44.1432	−1.70159	−0.850797	+	0.525495i	$0.823880\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	26.1760	1.00602	0.503012	−	0.864279i	$-0.332225\pi$
$677$	26.1760	1.00602	0.503012	+	0.864279i	$0.332225\pi$
$678$	0	0
$679$	−7.93437	−0.304493
$680$	0	0
$681$	−16.3463	−0.626390
$682$	0	0
$683$	5.03876	0.192803	0.0964015	−	0.995343i	$-0.469267\pi$
$683$	5.03876	0.192803	0.0964015	+	0.995343i	$0.469267\pi$
$684$	0	0
$685$	−12.9836	−0.496078
$686$	0	0
$687$	16.8873	0.644290
$688$	0	0
$689$	−3.01641	−0.114916
$690$	0	0
$691$	−45.4423	−1.72871	−0.864353	−	0.502885i	$-0.832272\pi$
$691$	−45.4423	−1.72871	−0.864353	+	0.502885i	$0.832272\pi$
$692$	0	0
$693$	−2.72532	−0.103526
$694$	0	0
$695$	7.74173	0.293661
$696$	0	0
$697$	−7.74173	−0.293239
$698$	0	0
$699$	3.11273	0.117734
$700$	0	0
$701$	4.19476	0.158434	0.0792170	−	0.996857i	$-0.474758\pi$
$701$	4.19476	0.158434	0.0792170	+	0.996857i	$0.474758\pi$
$702$	0	0
$703$	4.95078	0.186722
$704$	0	0
$705$	−4.34625	−0.163689
$706$	0	0
$707$	−1.45065	−0.0545571
$708$	0	0
$709$	−28.6946	−1.07765	−0.538825	−	0.842418i	$-0.681131\pi$
$709$	−28.6946	−1.07765	−0.538825	+	0.842418i	$0.681131\pi$
$710$	0	0
$711$	12.7581	0.478467
$712$	0	0
$713$	−11.9344	−0.446946
$714$	0	0
$715$	5.36266	0.200552
$716$	0	0
$717$	15.0716	0.562858
$718$	0	0
$719$	21.7745	0.812053	0.406027	−	0.913861i	$-0.366914\pi$
$719$	21.7745	0.812053	0.406027	+	0.913861i	$0.366914\pi$
$720$	0	0
$721$	−4.45091	−0.165761
$722$	0	0
$723$	2.00000	0.0743808
$724$	0	0
$725$	3.87086	0.143760
$726$	0	0
$727$	−16.4342	−0.609512	−0.304756	−	0.952430i	$-0.598575\pi$
$727$	−16.4342	−0.609512	−0.304756	+	0.952430i	$0.598575\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	62.9669	2.32892
$732$	0	0
$733$	11.7089	0.432479	0.216239	−	0.976340i	$-0.430621\pi$
$733$	11.7089	0.432479	0.216239	+	0.976340i	$0.430621\pi$
$734$	0	0
$735$	6.74173	0.248672
$736$	0	0
$737$	−12.7581	−0.469952
$738$	0	0
$739$	29.4423	1.08305	0.541526	−	0.840684i	$-0.317847\pi$
$739$	29.4423	1.08305	0.541526	+	0.840684i	$0.317847\pi$
$740$	0	0
$741$	0.508203	0.0186693
$742$	0	0
$743$	28.3296	1.03931	0.519656	−	0.854376i	$-0.326060\pi$
$743$	28.3296	1.03931	0.519656	+	0.854376i	$0.326060\pi$
$744$	0	0
$745$	1.74173	0.0638120
$746$	0	0
$747$	−17.1044	−0.625817
$748$	0	0
$749$	2.03281	0.0742774
$750$	0	0
$751$	−7.24186	−0.264259	−0.132130	−	0.991232i	$-0.542182\pi$
$751$	−7.24186	−0.264259	−0.132130	+	0.991232i	$0.542182\pi$
$752$	0	0
$753$	−15.6454	−0.570150
$754$	0	0
$755$	1.62093	0.0589918
$756$	0	0
$757$	32.7909	1.19181	0.595904	−	0.803056i	$-0.296794\pi$
$757$	32.7909	1.19181	0.595904	+	0.803056i	$0.296794\pi$
$758$	0	0
$759$	10.0328	0.364168
$760$	0	0
$761$	−54.4342	−1.97324	−0.986620	−	0.163038i	$-0.947871\pi$
$761$	−54.4342	−1.97324	−0.986620	+	0.163038i	$0.947871\pi$
$762$	0	0
$763$	0.582168	0.0210759
$764$	0	0
$765$	−7.87086	−0.284572
$766$	0	0
$767$	−13.3627	−0.482498
$768$	0	0
$769$	−34.9341	−1.25976	−0.629878	−	0.776694i	$-0.716895\pi$
$769$	−34.9341	−1.25976	−0.629878	+	0.776694i	$0.716895\pi$
$770$	0	0
$771$	−14.5962	−0.525669
$772$	0	0
$773$	−44.5327	−1.60173	−0.800865	−	0.598846i	$-0.795626\pi$
$773$	−44.5327	−1.60173	−0.800865	+	0.598846i	$0.795626\pi$
$774$	0	0
$775$	−6.37907	−0.229143
$776$	0	0
$777$	4.95078	0.177608
$778$	0	0
$779$	0.499865	0.0179095
$780$	0	0
$781$	−70.2744	−2.51462
$782$	0	0
$783$	3.87086	0.138333
$784$	0	0
$785$	−20.4671	−0.730500
$786$	0	0
$787$	40.1536	1.43132	0.715661	−	0.698448i	$-0.246126\pi$
$787$	40.1536	1.43132	0.715661	+	0.698448i	$0.246126\pi$
$788$	0	0
$789$	−18.5634	−0.660874
$790$	0	0
$791$	1.96719	0.0699451
$792$	0	0
$793$	−6.00000	−0.213066
$794$	0	0
$795$	−3.01641	−0.106981
$796$	0	0
$797$	47.1267	1.66932	0.834658	−	0.550769i	$-0.185666\pi$
$797$	47.1267	1.66932	0.834658	+	0.550769i	$0.185666\pi$
$798$	0	0
$799$	34.2088	1.21022
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	−42.2088	−1.48952
$804$	0	0
$805$	0.950780	0.0335106
$806$	0	0
$807$	7.61259	0.267976
$808$	0	0
$809$	−10.2416	−0.360075	−0.180038	−	0.983660i	$-0.557622\pi$
$809$	−10.2416	−0.360075	−0.180038	+	0.983660i	$0.557622\pi$
$810$	0	0
$811$	35.0409	1.23045	0.615226	−	0.788351i	$-0.289065\pi$
$811$	35.0409	1.23045	0.615226	+	0.788351i	$0.289065\pi$
$812$	0	0
$813$	4.34625	0.152430
$814$	0	0
$815$	−20.0880	−0.703651
$816$	0	0
$817$	−4.06563	−0.142238
$818$	0	0
$819$	0.508203	0.0177581
$820$	0	0
$821$	13.7417	0.479590	0.239795	−	0.970824i	$-0.422920\pi$
$821$	13.7417	0.479590	0.239795	+	0.970824i	$0.422920\pi$
$822$	0	0
$823$	50.1432	1.74788	0.873940	−	0.486033i	$-0.161557\pi$
$823$	50.1432	1.74788	0.873940	+	0.486033i	$0.161557\pi$
$824$	0	0
$825$	5.36266	0.186704
$826$	0	0
$827$	9.62093	0.334553	0.167276	−	0.985910i	$-0.446503\pi$
$827$	9.62093	0.334553	0.167276	+	0.985910i	$0.446503\pi$
$828$	0	0
$829$	2.06563	0.0717422	0.0358711	−	0.999356i	$-0.488579\pi$
$829$	2.06563	0.0717422	0.0358711	+	0.999356i	$0.488579\pi$
$830$	0	0
$831$	2.25827	0.0783385
$832$	0	0
$833$	−53.0632	−1.83853
$834$	0	0
$835$	−8.08798	−0.279896
$836$	0	0
$837$	−6.37907	−0.220493
$838$	0	0
$839$	−7.58812	−0.261971	−0.130985	−	0.991384i	$-0.541814\pi$
$839$	−7.58812	−0.261971	−0.130985	+	0.991384i	$0.541814\pi$
$840$	0	0
$841$	−14.0164	−0.483324
$842$	0	0
$843$	−0.0328135	−0.00113016
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−9.02474	−0.310094
$848$	0	0
$849$	18.0328	0.618885
$850$	0	0
$851$	−18.2255	−0.624761
$852$	0	0
$853$	−44.6430	−1.52855	−0.764274	−	0.644892i	$-0.776902\pi$
$853$	−44.6430	−1.52855	−0.764274	+	0.644892i	$0.776902\pi$
$854$	0	0
$855$	0.508203	0.0173802
$856$	0	0
$857$	3.43663	0.117393	0.0586965	−	0.998276i	$-0.481306\pi$
$857$	3.43663	0.117393	0.0586965	+	0.998276i	$0.481306\pi$
$858$	0	0
$859$	41.5819	1.41876	0.709378	−	0.704828i	$-0.248976\pi$
$859$	41.5819	1.41876	0.709378	+	0.704828i	$0.248976\pi$
$860$	0	0
$861$	0.499865	0.0170354
$862$	0	0
$863$	32.9117	1.12033	0.560164	−	0.828381i	$-0.310738\pi$
$863$	32.9117	1.12033	0.560164	+	0.828381i	$0.310738\pi$
$864$	0	0
$865$	−22.4342	−0.762787
$866$	0	0
$867$	44.9505	1.52660
$868$	0	0
$869$	68.4176	2.32091
$870$	0	0
$871$	2.37907	0.0806116
$872$	0	0
$873$	15.6126	0.528406
$874$	0	0
$875$	0.508203	0.0171804
$876$	0	0
$877$	−35.4506	−1.19708	−0.598542	−	0.801092i	$-0.704253\pi$
$877$	−35.4506	−1.19708	−0.598542	+	0.801092i	$0.704253\pi$
$878$	0	0
$879$	30.4999	1.02873
$880$	0	0
$881$	19.1924	0.646608	0.323304	−	0.946295i	$-0.395206\pi$
$881$	19.1924	0.646608	0.323304	+	0.946295i	$0.395206\pi$
$882$	0	0
$883$	−30.2088	−1.01661	−0.508303	−	0.861178i	$-0.669727\pi$
$883$	−30.2088	−1.01661	−0.508303	+	0.861178i	$0.669727\pi$
$884$	0	0
$885$	−13.3627	−0.449181
$886$	0	0
$887$	−12.9201	−0.433814	−0.216907	−	0.976192i	$-0.569597\pi$
$887$	−12.9201	−0.433814	−0.216907	+	0.976192i	$0.569597\pi$
$888$	0	0
$889$	3.41783	0.114630
$890$	0	0
$891$	5.36266	0.179656
$892$	0	0
$893$	−2.20878	−0.0739140
$894$	0	0
$895$	−3.90368	−0.130486
$896$	0	0
$897$	−1.87086	−0.0624664
$898$	0	0
$899$	−24.6925	−0.823541
$900$	0	0
$901$	23.7417	0.790952
$902$	0	0
$903$	−4.06563	−0.135296
$904$	0	0
$905$	0.983593	0.0326957
$906$	0	0
$907$	35.5491	1.18039	0.590194	−	0.807261i	$-0.299051\pi$
$907$	35.5491	1.18039	0.590194	+	0.807261i	$0.299051\pi$
$908$	0	0
$909$	2.85446	0.0946764
$910$	0	0
$911$	1.53295	0.0507888	0.0253944	−	0.999678i	$-0.491916\pi$
$911$	1.53295	0.0507888	0.0253944	+	0.999678i	$0.491916\pi$
$912$	0	0
$913$	−91.7251	−3.03566
$914$	0	0
$915$	−6.00000	−0.198354
$916$	0	0
$917$	6.40142	0.211394
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	−7.65375	−0.252199
$922$	0	0
$923$	13.1044	0.431336
$924$	0	0
$925$	−9.74173	−0.320306
$926$	0	0
$927$	8.75814	0.287655
$928$	0	0
$929$	42.9997	1.41078	0.705388	−	0.708822i	$-0.250773\pi$
$929$	42.9997	1.41078	0.705388	+	0.708822i	$0.250773\pi$
$930$	0	0
$931$	3.42617	0.112288
$932$	0	0
$933$	−2.03281	−0.0665513
$934$	0	0
$935$	−42.2088	−1.38037
$936$	0	0
$937$	−25.4178	−0.830364	−0.415182	−	0.909738i	$-0.636282\pi$
$937$	−25.4178	−0.830364	−0.415182	+	0.909738i	$0.636282\pi$
$938$	0	0
$939$	16.9836	0.554239
$940$	0	0
$941$	−20.2911	−0.661470	−0.330735	−	0.943724i	$-0.607297\pi$
$941$	−20.2911	−0.661470	−0.330735	+	0.943724i	$0.607297\pi$
$942$	0	0
$943$	−1.84017	−0.0599242
$944$	0	0
$945$	0.508203	0.0165319
$946$	0	0
$947$	−24.4119	−0.793280	−0.396640	−	0.917974i	$-0.629824\pi$
$947$	−24.4119	−0.793280	−0.396640	+	0.917974i	$0.629824\pi$
$948$	0	0
$949$	7.87086	0.255499
$950$	0	0
$951$	−19.4506	−0.630730
$952$	0	0
$953$	29.1455	0.944117	0.472058	−	0.881567i	$-0.343511\pi$
$953$	29.1455	0.944117	0.472058	+	0.881567i	$0.343511\pi$
$954$	0	0
$955$	5.27468	0.170685
$956$	0	0
$957$	20.7581	0.671015
$958$	0	0
$959$	−6.59831	−0.213070
$960$	0	0
$961$	9.69251	0.312662
$962$	0	0
$963$	−4.00000	−0.128898
$964$	0	0
$965$	17.5798	0.565913
$966$	0	0
$967$	−25.2007	−0.810400	−0.405200	−	0.914228i	$-0.632798\pi$
$967$	−25.2007	−0.810400	−0.405200	+	0.914228i	$0.632798\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	12.5962	0.404231	0.202115	−	0.979362i	$-0.435218\pi$
$971$	12.5962	0.404231	0.202115	+	0.979362i	$0.435218\pi$
$972$	0	0
$973$	3.93437	0.126130
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	28.7909	0.921104	0.460552	−	0.887633i	$-0.347651\pi$
$977$	28.7909	0.921104	0.460552	+	0.887633i	$0.347651\pi$
$978$	0	0
$979$	10.7253	0.342783
$980$	0	0
$981$	−1.14554	−0.0365743
$982$	0	0
$983$	−50.2968	−1.60422	−0.802109	−	0.597178i	$-0.796289\pi$
$983$	−50.2968	−1.60422	−0.802109	+	0.597178i	$0.796289\pi$
$984$	0	0
$985$	−15.0164	−0.478463
$986$	0	0
$987$	−2.20878	−0.0703063
$988$	0	0
$989$	14.9669	0.475920
$990$	0	0
$991$	−38.2744	−1.21583	−0.607913	−	0.794003i	$-0.707993\pi$
$991$	−38.2744	−1.21583	−0.607913	+	0.794003i	$0.707993\pi$
$992$	0	0
$993$	−15.9917	−0.507480
$994$	0	0
$995$	15.4835	0.490859
$996$	0	0
$997$	43.1267	1.36584	0.682919	−	0.730494i	$-0.260710\pi$
$997$	43.1267	1.36584	0.682919	+	0.730494i	$0.260710\pi$
$998$	0	0
$999$	−9.74173	−0.308215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.ca.1.2	yes	3
4.3	odd	2		6240.2.a.bx.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bx.1.2	✓	3	4.3	odd	2
6240.2.a.ca.1.2	yes	3	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.508203 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.508203 q^{7} +1.00000 q^{9} +5.36266 q^{11} -1.00000 q^{13} -1.00000 q^{15} +7.87086 q^{17} -0.508203 q^{19} -0.508203 q^{21} +1.87086 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.87086 q^{29} -6.37907 q^{31} +5.36266 q^{33} +0.508203 q^{35} -9.74173 q^{37} -1.00000 q^{39} -0.983593 q^{41} +8.00000 q^{43} -1.00000 q^{45} +4.34625 q^{47} -6.74173 q^{49} +7.87086 q^{51} +3.01641 q^{53} -5.36266 q^{55} -0.508203 q^{57} +13.3627 q^{59} +6.00000 q^{61} -0.508203 q^{63} +1.00000 q^{65} -2.37907 q^{67} +1.87086 q^{69} -13.1044 q^{71} -7.87086 q^{73} +1.00000 q^{75} -2.72532 q^{77} +12.7581 q^{79} +1.00000 q^{81} -17.1044 q^{83} -7.87086 q^{85} +3.87086 q^{87} +2.00000 q^{89} +0.508203 q^{91} -6.37907 q^{93} +0.508203 q^{95} +15.6126 q^{97} +5.36266 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 3 q^{15} + 8 q^{17} - 10 q^{23} + 3 q^{25} + 3 q^{27} - 4 q^{29} - 2 q^{31} + 2 q^{33} + 2 q^{37} - 3 q^{39} - 6 q^{41} + 24 q^{43} - 3 q^{45} + 2 q^{47} + 11 q^{49} + 8 q^{51} + 6 q^{53} - 2 q^{55} + 26 q^{59} + 18 q^{61} + 3 q^{65} + 10 q^{67} - 10 q^{69} + 6 q^{71} - 8 q^{73} + 3 q^{75} + 20 q^{77} + 4 q^{79} + 3 q^{81} - 6 q^{83} - 8 q^{85} - 4 q^{87} + 6 q^{89} - 2 q^{93} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 3 * q^15 + 8 * q^17 - 10 * q^23 + 3 * q^25 + 3 * q^27 - 4 * q^29 - 2 * q^31 + 2 * q^33 + 2 * q^37 - 3 * q^39 - 6 * q^41 + 24 * q^43 - 3 * q^45 + 2 * q^47 + 11 * q^49 + 8 * q^51 + 6 * q^53 - 2 * q^55 + 26 * q^59 + 18 * q^61 + 3 * q^65 + 10 * q^67 - 10 * q^69 + 6 * q^71 - 8 * q^73 + 3 * q^75 + 20 * q^77 + 4 * q^79 + 3 * q^81 - 6 * q^83 - 8 * q^85 - 4 * q^87 + 6 * q^89 - 2 * q^93 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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