LMFDB - Embedded newform 495.6.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 143x^{3} + 71x^{2} + 4216x - 3740 $ x^5 - x^4 - 143x^3 + 71x^2 + 4216*x - 3740
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.898099$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.898099 q^{2} -31.1934 q^{4} +25.0000 q^{5} +120.732 q^{7} -56.7540 q^{8} +O(q^{10})$	\(q+0.898099 q^{2} -31.1934 q^{4} +25.0000 q^{5} +120.732 q^{7} -56.7540 q^{8} +22.4525 q^{10} +121.000 q^{11} +667.387 q^{13} +108.430 q^{14} +947.219 q^{16} +74.5586 q^{17} -708.549 q^{19} -779.835 q^{20} +108.670 q^{22} -1238.63 q^{23} +625.000 q^{25} +599.380 q^{26} -3766.06 q^{28} +3855.89 q^{29} -8236.92 q^{31} +2666.82 q^{32} +66.9611 q^{34} +3018.31 q^{35} -5725.33 q^{37} -636.348 q^{38} -1418.85 q^{40} -672.977 q^{41} +16058.1 q^{43} -3774.40 q^{44} -1112.41 q^{46} +7572.96 q^{47} -2230.69 q^{49} +561.312 q^{50} -20818.1 q^{52} +5420.79 q^{53} +3025.00 q^{55} -6852.04 q^{56} +3462.97 q^{58} +33128.4 q^{59} +18038.6 q^{61} -7397.57 q^{62} -27915.9 q^{64} +16684.7 q^{65} -60276.1 q^{67} -2325.74 q^{68} +2710.74 q^{70} -12430.5 q^{71} -1230.79 q^{73} -5141.92 q^{74} +22102.1 q^{76} +14608.6 q^{77} +47015.0 q^{79} +23680.5 q^{80} -604.401 q^{82} +29681.0 q^{83} +1863.97 q^{85} +14421.7 q^{86} -6867.23 q^{88} +125061. q^{89} +80575.2 q^{91} +38637.2 q^{92} +6801.27 q^{94} -17713.7 q^{95} +74465.3 q^{97} -2003.38 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 127 q^{4} + 125 q^{5} + 116 q^{7} + 153 q^{8}+O(q^{10}) $ 5 * q + q^2 + 127 * q^4 + 125 * q^5 + 116 * q^7 + 153 * q^8	$ 5 q + q^{2} + 127 q^{4} + 125 q^{5} + 116 q^{7} + 153 q^{8} + 25 q^{10} + 605 q^{11} - 926 q^{13} - 368 q^{14} + 1891 q^{16} + 246 q^{17} + 3420 q^{19} + 3175 q^{20} + 121 q^{22} + 4244 q^{23} + 3125 q^{25} + 8862 q^{26} - 4904 q^{28} + 2922 q^{29} - 6112 q^{31} + 24757 q^{32} + 10866 q^{34} + 2900 q^{35} + 6654 q^{37} + 45692 q^{38} + 3825 q^{40} + 14934 q^{41} + 10804 q^{43} + 15367 q^{44} - 101500 q^{46} + 41460 q^{47} - 12099 q^{49} + 625 q^{50} - 97742 q^{52} + 62398 q^{53} + 15125 q^{55} + 74368 q^{56} - 27822 q^{58} - 8524 q^{59} + 59010 q^{61} + 142624 q^{62} + 13799 q^{64} - 23150 q^{65} - 15772 q^{67} + 83686 q^{68} - 9200 q^{70} - 88124 q^{71} - 118358 q^{73} - 67194 q^{74} + 100668 q^{76} + 14036 q^{77} + 57324 q^{79} + 47275 q^{80} + 29102 q^{82} + 7268 q^{83} + 6150 q^{85} + 35288 q^{86} + 18513 q^{88} - 72978 q^{89} - 1464 q^{91} - 62148 q^{92} + 344836 q^{94} + 85500 q^{95} - 59174 q^{97} - 272767 q^{98}+O(q^{100}) $ 5 * q + q^2 + 127 * q^4 + 125 * q^5 + 116 * q^7 + 153 * q^8 + 25 * q^10 + 605 * q^11 - 926 * q^13 - 368 * q^14 + 1891 * q^16 + 246 * q^17 + 3420 * q^19 + 3175 * q^20 + 121 * q^22 + 4244 * q^23 + 3125 * q^25 + 8862 * q^26 - 4904 * q^28 + 2922 * q^29 - 6112 * q^31 + 24757 * q^32 + 10866 * q^34 + 2900 * q^35 + 6654 * q^37 + 45692 * q^38 + 3825 * q^40 + 14934 * q^41 + 10804 * q^43 + 15367 * q^44 - 101500 * q^46 + 41460 * q^47 - 12099 * q^49 + 625 * q^50 - 97742 * q^52 + 62398 * q^53 + 15125 * q^55 + 74368 * q^56 - 27822 * q^58 - 8524 * q^59 + 59010 * q^61 + 142624 * q^62 + 13799 * q^64 - 23150 * q^65 - 15772 * q^67 + 83686 * q^68 - 9200 * q^70 - 88124 * q^71 - 118358 * q^73 - 67194 * q^74 + 100668 * q^76 + 14036 * q^77 + 57324 * q^79 + 47275 * q^80 + 29102 * q^82 + 7268 * q^83 + 6150 * q^85 + 35288 * q^86 + 18513 * q^88 - 72978 * q^89 - 1464 * q^91 - 62148 * q^92 + 344836 * q^94 + 85500 * q^95 - 59174 * q^97 - 272767 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.898099	0.158763	0.0793815	−	0.996844i	$-0.474705\pi$
$2$	0.898099	0.158763	0.0793815	+	0.996844i	$0.474705\pi$
$3$	0	0
$3$	0	0
$4$	−31.1934	−0.974794
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	120.732	0.931277	0.465638	−	0.884975i	$-0.345825\pi$
$7$	120.732	0.931277	0.465638	+	0.884975i	$0.345825\pi$
$8$	−56.7540	−0.313524
$9$	0	0
$10$	22.4525	0.0710010
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	667.387	1.09527	0.547633	−	0.836719i	$-0.315529\pi$
$13$	667.387	1.09527	0.547633	+	0.836719i	$0.315529\pi$
$14$	108.430	0.147852
$15$	0	0
$16$	947.219	0.925018
$17$	74.5586	0.0625714	0.0312857	−	0.999510i	$-0.490040\pi$
$17$	74.5586	0.0625714	0.0312857	+	0.999510i	$0.490040\pi$
$18$	0	0
$19$	−708.549	−0.450283	−0.225142	−	0.974326i	$-0.572285\pi$
$19$	−708.549	−0.450283	−0.225142	+	0.974326i	$0.572285\pi$
$20$	−779.835	−0.435941
$21$	0	0
$22$	108.670	0.0478689
$23$	−1238.63	−0.488228	−0.244114	−	0.969747i	$-0.578497\pi$
$23$	−1238.63	−0.488228	−0.244114	+	0.969747i	$0.578497\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	599.380	0.173888
$27$	0	0
$28$	−3766.06	−0.907803
$29$	3855.89	0.851392	0.425696	−	0.904866i	$-0.360029\pi$
$29$	3855.89	0.851392	0.425696	+	0.904866i	$0.360029\pi$
$30$	0	0
$31$	−8236.92	−1.53943	−0.769716	−	0.638387i	$-0.779602\pi$
$31$	−8236.92	−1.53943	−0.769716	+	0.638387i	$0.779602\pi$
$32$	2666.82	0.460383
$33$	0	0
$34$	66.9611	0.00993402
$35$	3018.31	0.416480
$36$	0	0
$37$	−5725.33	−0.687538	−0.343769	−	0.939054i	$-0.611704\pi$
$37$	−5725.33	−0.687538	−0.343769	+	0.939054i	$0.611704\pi$
$38$	−636.348	−0.0714884
$39$	0	0
$40$	−1418.85	−0.140212
$41$	−672.977	−0.0625231	−0.0312616	−	0.999511i	$-0.509952\pi$
$41$	−672.977	−0.0625231	−0.0312616	+	0.999511i	$0.509952\pi$
$42$	0	0
$43$	16058.1	1.32441	0.662204	−	0.749323i	$-0.269621\pi$
$43$	16058.1	1.32441	0.662204	+	0.749323i	$0.269621\pi$
$44$	−3774.40	−0.293912
$45$	0	0
$46$	−1112.41	−0.0775125
$47$	7572.96	0.500059	0.250029	−	0.968238i	$-0.419560\pi$
$47$	7572.96	0.500059	0.250029	+	0.968238i	$0.419560\pi$
$48$	0	0
$49$	−2230.69	−0.132724
$50$	561.312	0.0317526
$51$	0	0
$52$	−20818.1	−1.06766
$53$	5420.79	0.265078	0.132539	−	0.991178i	$-0.457687\pi$
$53$	5420.79	0.265078	0.132539	+	0.991178i	$0.457687\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	−6852.04	−0.291978
$57$	0	0
$58$	3462.97	0.135170
$59$	33128.4	1.23900	0.619499	−	0.784997i	$-0.287336\pi$
$59$	33128.4	1.23900	0.619499	+	0.784997i	$0.287336\pi$
$60$	0	0
$61$	18038.6	0.620695	0.310348	−	0.950623i	$-0.399555\pi$
$61$	18038.6	0.620695	0.310348	+	0.950623i	$0.399555\pi$
$62$	−7397.57	−0.244405
$63$	0	0
$64$	−27915.9	−0.851926
$65$	16684.7	0.489818
$66$	0	0
$67$	−60276.1	−1.64043	−0.820216	−	0.572054i	$-0.806147\pi$
$67$	−60276.1	−1.64043	−0.820216	+	0.572054i	$0.806147\pi$
$68$	−2325.74	−0.0609942
$69$	0	0
$70$	2710.74	0.0661216
$71$	−12430.5	−0.292646	−0.146323	−	0.989237i	$-0.546744\pi$
$71$	−12430.5	−0.292646	−0.146323	+	0.989237i	$0.546744\pi$
$72$	0	0
$73$	−1230.79	−0.0270320	−0.0135160	−	0.999909i	$-0.504302\pi$
$73$	−1230.79	−0.0270320	−0.0135160	+	0.999909i	$0.504302\pi$
$74$	−5141.92	−0.109156
$75$	0	0
$76$	22102.1	0.438934
$77$	14608.6	0.280790
$78$	0	0
$79$	47015.0	0.847557	0.423778	−	0.905766i	$-0.360704\pi$
$79$	47015.0	0.847557	0.423778	+	0.905766i	$0.360704\pi$
$80$	23680.5	0.413681
$81$	0	0
$82$	−604.401	−0.00992636
$83$	29681.0	0.472915	0.236457	−	0.971642i	$-0.424014\pi$
$83$	29681.0	0.472915	0.236457	+	0.971642i	$0.424014\pi$
$84$	0	0
$85$	1863.97	0.0279828
$86$	14421.7	0.210267
$87$	0	0
$88$	−6867.23	−0.0945311
$89$	125061.	1.67358	0.836791	−	0.547522i	$-0.184429\pi$
$89$	125061.	1.67358	0.836791	+	0.547522i	$0.184429\pi$
$90$	0	0
$91$	80575.2	1.02000
$92$	38637.2	0.475922
$93$	0	0
$94$	6801.27	0.0793908
$95$	−17713.7	−0.201373
$96$	0	0
$97$	74465.3	0.803572	0.401786	−	0.915734i	$-0.368390\pi$
$97$	74465.3	0.803572	0.401786	+	0.915734i	$0.368390\pi$
$98$	−2003.38	−0.0210717
$99$	0	0
$100$	−19495.9	−0.194959
$101$	65393.5	0.637868	0.318934	−	0.947777i	$-0.396675\pi$
$101$	65393.5	0.637868	0.318934	+	0.947777i	$0.396675\pi$
$102$	0	0
$103$	−35955.1	−0.333939	−0.166969	−	0.985962i	$-0.553398\pi$
$103$	−35955.1	−0.333939	−0.166969	+	0.985962i	$0.553398\pi$
$104$	−37876.9	−0.343392
$105$	0	0
$106$	4868.41	0.0420845
$107$	118141.	0.997565	0.498783	−	0.866727i	$-0.333781\pi$
$107$	118141.	0.997565	0.498783	+	0.866727i	$0.333781\pi$
$108$	0	0
$109$	216333.	1.74404	0.872019	−	0.489472i	$-0.162811\pi$
$109$	216333.	1.74404	0.872019	+	0.489472i	$0.162811\pi$
$110$	2716.75	0.0214076
$111$	0	0
$112$	114360.	0.861448
$113$	82987.0	0.611384	0.305692	−	0.952130i	$-0.401112\pi$
$113$	82987.0	0.611384	0.305692	+	0.952130i	$0.401112\pi$
$114$	0	0
$115$	−30965.8	−0.218342
$116$	−120278.	−0.829932
$117$	0	0
$118$	29752.6	0.196707
$119$	9001.64	0.0582712
$120$	0	0
$121$	14641.0	0.0909091
$122$	16200.5	0.0985434
$123$	0	0
$124$	256938.	1.50063
$125$	15625.0	0.0894427
$126$	0	0
$127$	−112754.	−0.620330	−0.310165	−	0.950683i	$-0.600384\pi$
$127$	−112754.	−0.620330	−0.310165	+	0.950683i	$0.600384\pi$
$128$	−110410.	−0.595637
$129$	0	0
$130$	14984.5	0.0777650
$131$	−177226.	−0.902297	−0.451149	−	0.892449i	$-0.648986\pi$
$131$	−177226.	−0.902297	−0.451149	+	0.892449i	$0.648986\pi$
$132$	0	0
$133$	−85544.8	−0.419338
$134$	−54133.9	−0.260440
$135$	0	0
$136$	−4231.50	−0.0196176
$137$	179083.	0.815181	0.407590	−	0.913165i	$-0.366369\pi$
$137$	179083.	0.815181	0.407590	+	0.913165i	$0.366369\pi$
$138$	0	0
$139$	140181.	0.615392	0.307696	−	0.951485i	$-0.400442\pi$
$139$	140181.	0.615392	0.307696	+	0.951485i	$0.400442\pi$
$140$	−94151.4	−0.405982
$141$	0	0
$142$	−11163.8	−0.0464614
$143$	80753.9	0.330235
$144$	0	0
$145$	96397.2	0.380754
$146$	−1105.37	−0.00429168
$147$	0	0
$148$	178593.	0.670208
$149$	300863.	1.11020	0.555102	−	0.831783i	$-0.312679\pi$
$149$	300863.	1.11020	0.555102	+	0.831783i	$0.312679\pi$
$150$	0	0
$151$	225360.	0.804330	0.402165	−	0.915567i	$-0.368258\pi$
$151$	225360.	0.804330	0.402165	+	0.915567i	$0.368258\pi$
$152$	40213.0	0.141175
$153$	0	0
$154$	13120.0	0.0445791
$155$	−205923.	−0.688455
$156$	0	0
$157$	−344938.	−1.11684	−0.558421	−	0.829558i	$-0.688593\pi$
$157$	−344938.	−1.11684	−0.558421	+	0.829558i	$0.688593\pi$
$158$	42224.2	0.134561
$159$	0	0
$160$	66670.6	0.205890
$161$	−149543.	−0.454675
$162$	0	0
$163$	−134781.	−0.397337	−0.198669	−	0.980067i	$-0.563662\pi$
$163$	−134781.	−0.397337	−0.198669	+	0.980067i	$0.563662\pi$
$164$	20992.5	0.0609472
$165$	0	0
$166$	26656.5	0.0750814
$167$	−476574.	−1.32233	−0.661163	−	0.750242i	$-0.729937\pi$
$167$	−476574.	−1.32233	−0.661163	+	0.750242i	$0.729937\pi$
$168$	0	0
$169$	74112.7	0.199607
$170$	1674.03	0.00444263
$171$	0	0
$172$	−500906.	−1.29103
$173$	607031.	1.54204	0.771020	−	0.636811i	$-0.219747\pi$
$173$	607031.	1.54204	0.771020	+	0.636811i	$0.219747\pi$
$174$	0	0
$175$	75457.7	0.186255
$176$	114613.	0.278903
$177$	0	0
$178$	112317.	0.265703
$179$	588915.	1.37379	0.686895	−	0.726757i	$-0.258973\pi$
$179$	588915.	1.37379	0.686895	+	0.726757i	$0.258973\pi$
$180$	0	0
$181$	−403724.	−0.915984	−0.457992	−	0.888956i	$-0.651431\pi$
$181$	−403724.	−0.915984	−0.457992	+	0.888956i	$0.651431\pi$
$182$	72364.6	0.161938
$183$	0	0
$184$	70297.3	0.153071
$185$	−143133.	−0.307476
$186$	0	0
$187$	9021.60	0.0188660
$188$	−236226.	−0.487454
$189$	0	0
$190$	−15908.7	−0.0319706
$191$	395279.	0.784007	0.392004	−	0.919964i	$-0.371782\pi$
$191$	395279.	0.784007	0.392004	+	0.919964i	$0.371782\pi$
$192$	0	0
$193$	192334.	0.371675	0.185837	−	0.982581i	$-0.440500\pi$
$193$	192334.	0.371675	0.185837	+	0.982581i	$0.440500\pi$
$194$	66877.3	0.127578
$195$	0	0
$196$	69582.9	0.129379
$197$	537571.	0.986894	0.493447	−	0.869776i	$-0.335737\pi$
$197$	537571.	0.986894	0.493447	+	0.869776i	$0.335737\pi$
$198$	0	0
$199$	−619032.	−1.10810	−0.554052	−	0.832482i	$-0.686919\pi$
$199$	−619032.	−1.10810	−0.554052	+	0.832482i	$0.686919\pi$
$200$	−35471.2	−0.0627049
$201$	0	0
$202$	58729.8	0.101270
$203$	465530.	0.792881
$204$	0	0
$205$	−16824.4	−0.0279612
$206$	−32291.2	−0.0530172
$207$	0	0
$208$	632162.	1.01314
$209$	−85734.5	−0.135766
$210$	0	0
$211$	937975.	1.45039	0.725196	−	0.688543i	$-0.241749\pi$
$211$	937975.	1.45039	0.725196	+	0.688543i	$0.241749\pi$
$212$	−169093.	−0.258396
$213$	0	0
$214$	106102.	0.158376
$215$	401452.	0.592294
$216$	0	0
$217$	−994463.	−1.43364
$218$	194288.	0.276889
$219$	0	0
$220$	−94360.1	−0.131441
$221$	49759.5	0.0685323
$222$	0	0
$223$	−1.05781e6	−1.42445	−0.712224	−	0.701952i	$-0.752312\pi$
$223$	−1.05781e6	−1.42445	−0.712224	+	0.701952i	$0.752312\pi$
$224$	321972.	0.428744
$225$	0	0
$226$	74530.6	0.0970651
$227$	1.38495e6	1.78389	0.891945	−	0.452145i	$-0.149341\pi$
$227$	1.38495e6	1.78389	0.891945	+	0.452145i	$0.149341\pi$
$228$	0	0
$229$	−56996.7	−0.0718225	−0.0359113	−	0.999355i	$-0.511433\pi$
$229$	−56996.7	−0.0718225	−0.0359113	+	0.999355i	$0.511433\pi$
$230$	−27810.4	−0.0346647
$231$	0	0
$232$	−218837.	−0.266932
$233$	27988.8	0.0337750	0.0168875	−	0.999857i	$-0.494624\pi$
$233$	27988.8	0.0337750	0.0168875	+	0.999857i	$0.494624\pi$
$234$	0	0
$235$	189324.	0.223633
$236$	−1.03339e6	−1.20777
$237$	0	0
$238$	8084.37	0.00925132
$239$	1.21406e6	1.37482	0.687408	−	0.726272i	$-0.258748\pi$
$239$	1.21406e6	1.37482	0.687408	+	0.726272i	$0.258748\pi$
$240$	0	0
$241$	1.06213e6	1.17797	0.588987	−	0.808142i	$-0.299527\pi$
$241$	1.06213e6	1.17797	0.588987	+	0.808142i	$0.299527\pi$
$242$	13149.1	0.0144330
$243$	0	0
$244$	−562686.	−0.605050
$245$	−55767.3	−0.0593560
$246$	0	0
$247$	−472877.	−0.493180
$248$	467478.	0.482649
$249$	0	0
$250$	14032.8	0.0142002
$251$	75353.1	0.0754948	0.0377474	−	0.999287i	$-0.487982\pi$
$251$	75353.1	0.0754948	0.0377474	+	0.999287i	$0.487982\pi$
$252$	0	0
$253$	−149874.	−0.147206
$254$	−101264.	−0.0984855
$255$	0	0
$256$	794151.	0.757361
$257$	−1.14147e6	−1.07803	−0.539016	−	0.842296i	$-0.681204\pi$
$257$	−1.14147e6	−1.07803	−0.539016	+	0.842296i	$0.681204\pi$
$258$	0	0
$259$	−691233.	−0.640288
$260$	−520452.	−0.477472
$261$	0	0
$262$	−159167.	−0.143251
$263$	780299.	0.695619	0.347810	−	0.937565i	$-0.386926\pi$
$263$	780299.	0.695619	0.347810	+	0.937565i	$0.386926\pi$
$264$	0	0
$265$	135520.	0.118546
$266$	−76827.8	−0.0665754
$267$	0	0
$268$	1.88022e6	1.59908
$269$	−210163.	−0.177082	−0.0885411	−	0.996073i	$-0.528220\pi$
$269$	−210163.	−0.177082	−0.0885411	+	0.996073i	$0.528220\pi$
$270$	0	0
$271$	−4596.78	−0.00380216	−0.00190108	−	0.999998i	$-0.500605\pi$
$271$	−4596.78	−0.00380216	−0.00190108	+	0.999998i	$0.500605\pi$
$272$	70623.3	0.0578797
$273$	0	0
$274$	160835.	0.129421
$275$	75625.0	0.0603023
$276$	0	0
$277$	−689589.	−0.539996	−0.269998	−	0.962861i	$-0.587023\pi$
$277$	−689589.	−0.539996	−0.269998	+	0.962861i	$0.587023\pi$
$278$	125896.	0.0977016
$279$	0	0
$280$	−171301.	−0.130576
$281$	−1.95020e6	−1.47338	−0.736688	−	0.676233i	$-0.763611\pi$
$281$	−1.95020e6	−1.47338	−0.736688	+	0.676233i	$0.763611\pi$
$282$	0	0
$283$	307845.	0.228489	0.114245	−	0.993453i	$-0.463555\pi$
$283$	307845.	0.228489	0.114245	+	0.993453i	$0.463555\pi$
$284$	387750.	0.285270
$285$	0	0
$286$	72525.0	0.0524291
$287$	−81250.2	−0.0582263
$288$	0	0
$289$	−1.41430e6	−0.996085
$290$	86574.2	0.0604496
$291$	0	0
$292$	38392.7	0.0263506
$293$	−2.19630e6	−1.49459	−0.747297	−	0.664491i	$-0.768649\pi$
$293$	−2.19630e6	−1.49459	−0.747297	+	0.664491i	$0.768649\pi$
$294$	0	0
$295$	828210.	0.554097
$296$	324935.	0.215560
$297$	0	0
$298$	270204.	0.176259
$299$	−826647.	−0.534739
$300$	0	0
$301$	1.93873e6	1.23339
$302$	202396.	0.127698
$303$	0	0
$304$	−671151.	−0.416520
$305$	450965.	0.277583
$306$	0	0
$307$	−905979.	−0.548621	−0.274310	−	0.961641i	$-0.588450\pi$
$307$	−905979.	−0.548621	−0.274310	+	0.961641i	$0.588450\pi$
$308$	−455693.	−0.273713
$309$	0	0
$310$	−184939.	−0.109301
$311$	−1.05496e6	−0.618495	−0.309247	−	0.950982i	$-0.600077\pi$
$311$	−1.05496e6	−0.618495	−0.309247	+	0.950982i	$0.600077\pi$
$312$	0	0
$313$	2.98094e6	1.71985	0.859927	−	0.510417i	$-0.170509\pi$
$313$	2.98094e6	1.71985	0.859927	+	0.510417i	$0.170509\pi$
$314$	−309788.	−0.177313
$315$	0	0
$316$	−1.46656e6	−0.826194
$317$	272200.	0.152139	0.0760694	−	0.997103i	$-0.475763\pi$
$317$	272200.	0.152139	0.0760694	+	0.997103i	$0.475763\pi$
$318$	0	0
$319$	466562.	0.256704
$320$	−697898.	−0.380993
$321$	0	0
$322$	−134304.	−0.0721856
$323$	−52828.5	−0.0281749
$324$	0	0
$325$	417117.	0.219053
$326$	−121047.	−0.0630824
$327$	0	0
$328$	38194.1	0.0196025
$329$	914301.	0.465693
$330$	0	0
$331$	−984627.	−0.493971	−0.246986	−	0.969019i	$-0.579440\pi$
$331$	−984627.	−0.493971	−0.246986	+	0.969019i	$0.579440\pi$
$332$	−925851.	−0.460995
$333$	0	0
$334$	−428010.	−0.209937
$335$	−1.50690e6	−0.733623
$336$	0	0
$337$	−366865.	−0.175967	−0.0879836	−	0.996122i	$-0.528042\pi$
$337$	−366865.	−0.175967	−0.0879836	+	0.996122i	$0.528042\pi$
$338$	66560.6	0.0316902
$339$	0	0
$340$	−58143.5	−0.0272774
$341$	−996667.	−0.464156
$342$	0	0
$343$	−2.29847e6	−1.05488
$344$	−911359.	−0.415234
$345$	0	0
$346$	545174.	0.244819
$347$	558457.	0.248981	0.124490	−	0.992221i	$-0.460270\pi$
$347$	558457.	0.248981	0.124490	+	0.992221i	$0.460270\pi$
$348$	0	0
$349$	1.53340e6	0.673895	0.336947	−	0.941524i	$-0.390606\pi$
$349$	1.53340e6	0.673895	0.336947	+	0.941524i	$0.390606\pi$
$350$	67768.5	0.0295705
$351$	0	0
$352$	322686.	0.138811
$353$	−2.43850e6	−1.04156	−0.520781	−	0.853690i	$-0.674359\pi$
$353$	−2.43850e6	−1.04156	−0.520781	+	0.853690i	$0.674359\pi$
$354$	0	0
$355$	−310762.	−0.130875
$356$	−3.90108e6	−1.63140
$357$	0	0
$358$	528904.	0.218107
$359$	−1.17648e6	−0.481778	−0.240889	−	0.970553i	$-0.577439\pi$
$359$	−1.17648e6	−0.481778	−0.240889	+	0.970553i	$0.577439\pi$
$360$	0	0
$361$	−1.97406e6	−0.797245
$362$	−362584.	−0.145424
$363$	0	0
$364$	−2.51342e6	−0.994286
$365$	−30769.8	−0.0120891
$366$	0	0
$367$	−2.74188e6	−1.06263	−0.531316	−	0.847174i	$-0.678302\pi$
$367$	−2.74188e6	−1.06263	−0.531316	+	0.847174i	$0.678302\pi$
$368$	−1.17325e6	−0.451620
$369$	0	0
$370$	−128548.	−0.0488159
$371$	654465.	0.246861
$372$	0	0
$373$	3.46639e6	1.29005	0.645023	−	0.764163i	$-0.276848\pi$
$373$	3.46639e6	1.29005	0.645023	+	0.764163i	$0.276848\pi$
$374$	8102.29	0.00299522
$375$	0	0
$376$	−429795.	−0.156781
$377$	2.57337e6	0.932500
$378$	0	0
$379$	4.07201e6	1.45617	0.728083	−	0.685489i	$-0.240412\pi$
$379$	4.07201e6	1.45617	0.728083	+	0.685489i	$0.240412\pi$
$380$	552552.	0.196297
$381$	0	0
$382$	355000.	0.124471
$383$	−436616.	−0.152091	−0.0760454	−	0.997104i	$-0.524229\pi$
$383$	−436616.	−0.152091	−0.0760454	+	0.997104i	$0.524229\pi$
$384$	0	0
$385$	365215.	0.125573
$386$	172735.	0.0590082
$387$	0	0
$388$	−2.32283e6	−0.783317
$389$	−4.27642e6	−1.43287	−0.716435	−	0.697654i	$-0.754227\pi$
$389$	−4.27642e6	−1.43287	−0.716435	+	0.697654i	$0.754227\pi$
$390$	0	0
$391$	−92350.7	−0.0305491
$392$	126601.	0.0416122
$393$	0	0
$394$	482792.	0.156682
$395$	1.17538e6	0.379039
$396$	0	0
$397$	−1.61356e6	−0.513817	−0.256909	−	0.966436i	$-0.582704\pi$
$397$	−1.61356e6	−0.513817	−0.256909	+	0.966436i	$0.582704\pi$
$398$	−555952.	−0.175926
$399$	0	0
$400$	592012.	0.185004
$401$	−4.58182e6	−1.42291	−0.711455	−	0.702732i	$-0.751963\pi$
$401$	−4.58182e6	−1.42291	−0.711455	+	0.702732i	$0.751963\pi$
$402$	0	0
$403$	−5.49721e6	−1.68609
$404$	−2.03985e6	−0.621790
$405$	0	0
$406$	418093.	0.125880
$407$	−692766.	−0.207300
$408$	0	0
$409$	2.55861e6	0.756304	0.378152	−	0.925743i	$-0.376560\pi$
$409$	2.55861e6	0.756304	0.378152	+	0.925743i	$0.376560\pi$
$410$	−15110.0	−0.00443920
$411$	0	0
$412$	1.12156e6	0.325522
$413$	3.99967e6	1.15385
$414$	0	0
$415$	742024.	0.211494
$416$	1.77980e6	0.504242
$417$	0	0
$418$	−76998.1	−0.0215546
$419$	4.36111e6	1.21356	0.606781	−	0.794869i	$-0.292460\pi$
$419$	4.36111e6	1.21356	0.606781	+	0.794869i	$0.292460\pi$
$420$	0	0
$421$	−4.39997e6	−1.20989	−0.604943	−	0.796269i	$-0.706804\pi$
$421$	−4.39997e6	−1.20989	−0.604943	+	0.796269i	$0.706804\pi$
$422$	842395.	0.230269
$423$	0	0
$424$	−307651.	−0.0831083
$425$	46599.2	0.0125143
$426$	0	0
$427$	2.17784e6	0.578039
$428$	−3.68522e6	−0.972421
$429$	0	0
$430$	360543.	0.0940343
$431$	−5.08667e6	−1.31899	−0.659493	−	0.751711i	$-0.729229\pi$
$431$	−5.08667e6	−1.31899	−0.659493	+	0.751711i	$0.729229\pi$
$432$	0	0
$433$	1.75095e6	0.448800	0.224400	−	0.974497i	$-0.427958\pi$
$433$	1.75095e6	0.448800	0.224400	+	0.974497i	$0.427958\pi$
$434$	−893126.	−0.227609
$435$	0	0
$436$	−6.74816e6	−1.70008
$437$	877631.	0.219841
$438$	0	0
$439$	−2.66084e6	−0.658957	−0.329479	−	0.944163i	$-0.606873\pi$
$439$	−2.66084e6	−0.658957	−0.329479	+	0.944163i	$0.606873\pi$
$440$	−171681.	−0.0422756
$441$	0	0
$442$	44689.0	0.0108804
$443$	6.98400e6	1.69081	0.845406	−	0.534125i	$-0.179359\pi$
$443$	6.98400e6	1.69081	0.845406	+	0.534125i	$0.179359\pi$
$444$	0	0
$445$	3.12653e6	0.748449
$446$	−950021.	−0.226150
$447$	0	0
$448$	−3.37036e6	−0.793379
$449$	−7.66547e6	−1.79441	−0.897207	−	0.441610i	$-0.854407\pi$
$449$	−7.66547e6	−1.79441	−0.897207	+	0.441610i	$0.854407\pi$
$450$	0	0
$451$	−81430.3	−0.0188514
$452$	−2.58865e6	−0.595973
$453$	0	0
$454$	1.24382e6	0.283216
$455$	2.01438e6	0.456156
$456$	0	0
$457$	3.50188e6	0.784351	0.392175	−	0.919890i	$-0.371723\pi$
$457$	3.50188e6	0.784351	0.392175	+	0.919890i	$0.371723\pi$
$458$	−51188.7	−0.0114028
$459$	0	0
$460$	965929.	0.212839
$461$	6.63603e6	1.45431	0.727153	−	0.686475i	$-0.240843\pi$
$461$	6.63603e6	1.45431	0.727153	+	0.686475i	$0.240843\pi$
$462$	0	0
$463$	1.13310e6	0.245649	0.122825	−	0.992428i	$-0.460805\pi$
$463$	1.13310e6	0.245649	0.122825	+	0.992428i	$0.460805\pi$
$464$	3.65237e6	0.787553
$465$	0	0
$466$	25136.8	0.00536222
$467$	−1.37315e6	−0.291358	−0.145679	−	0.989332i	$-0.546537\pi$
$467$	−1.37315e6	−0.291358	−0.145679	+	0.989332i	$0.546537\pi$
$468$	0	0
$469$	−7.27728e6	−1.52770
$470$	170032.	0.0355047
$471$	0	0
$472$	−1.88017e6	−0.388456
$473$	1.94303e6	0.399324
$474$	0	0
$475$	−442843.	−0.0900567
$476$	−280792.	−0.0568025
$477$	0	0
$478$	1.09034e6	0.218270
$479$	7.71987e6	1.53735	0.768673	−	0.639642i	$-0.220918\pi$
$479$	7.71987e6	1.53735	0.768673	+	0.639642i	$0.220918\pi$
$480$	0	0
$481$	−3.82102e6	−0.753037
$482$	953900.	0.187019
$483$	0	0
$484$	−456703.	−0.0886177
$485$	1.86163e6	0.359368
$486$	0	0
$487$	2.76688e6	0.528650	0.264325	−	0.964434i	$-0.414851\pi$
$487$	2.76688e6	0.528650	0.264325	+	0.964434i	$0.414851\pi$
$488$	−1.02376e6	−0.194603
$489$	0	0
$490$	−50084.6	−0.00942354
$491$	2.58549e6	0.483993	0.241997	−	0.970277i	$-0.422198\pi$
$491$	2.58549e6	0.483993	0.241997	+	0.970277i	$0.422198\pi$
$492$	0	0
$493$	287490.	0.0532727
$494$	−424690.	−0.0782988
$495$	0	0
$496$	−7.80216e6	−1.42400
$497$	−1.50076e6	−0.272534
$498$	0	0
$499$	−580259.	−0.104321	−0.0521604	−	0.998639i	$-0.516611\pi$
$499$	−580259.	−0.104321	−0.0521604	+	0.998639i	$0.516611\pi$
$500$	−487397.	−0.0871883
$501$	0	0
$502$	67674.6	0.0119858
$503$	8.01677e6	1.41280	0.706398	−	0.707815i	$-0.250319\pi$
$503$	8.01677e6	1.41280	0.706398	+	0.707815i	$0.250319\pi$
$504$	0	0
$505$	1.63484e6	0.285263
$506$	−134602.	−0.0233709
$507$	0	0
$508$	3.51718e6	0.604694
$509$	375241.	0.0641971	0.0320986	−	0.999485i	$-0.489781\pi$
$509$	375241.	0.0641971	0.0320986	+	0.999485i	$0.489781\pi$
$510$	0	0
$511$	−148597.	−0.0251743
$512$	4.24633e6	0.715878
$513$	0	0
$514$	−1.02515e6	−0.171152
$515$	−898877.	−0.149342
$516$	0	0
$517$	916328.	0.150773
$518$	−620796.	−0.101654
$519$	0	0
$520$	−946922.	−0.153570
$521$	4.40189e6	0.710469	0.355235	−	0.934777i	$-0.384401\pi$
$521$	4.40189e6	0.710469	0.355235	+	0.934777i	$0.384401\pi$
$522$	0	0
$523$	3.26635e6	0.522166	0.261083	−	0.965316i	$-0.415920\pi$
$523$	3.26635e6	0.522166	0.261083	+	0.965316i	$0.415920\pi$
$524$	5.52829e6	0.879554
$525$	0	0
$526$	700786.	0.110439
$527$	−614133.	−0.0963244
$528$	0	0
$529$	−4.90213e6	−0.761634
$530$	121710.	0.0188208
$531$	0	0
$532$	2.66844e6	0.408769
$533$	−449136.	−0.0684795
$534$	0	0
$535$	2.95353e6	0.446125
$536$	3.42091e6	0.514315
$537$	0	0
$538$	−188747.	−0.0281141
$539$	−269914.	−0.0400178
$540$	0	0
$541$	8.43137e6	1.23853	0.619263	−	0.785184i	$-0.287432\pi$
$541$	8.43137e6	1.23853	0.619263	+	0.785184i	$0.287432\pi$
$542$	−4128.37	−0.000603643 0
$543$	0	0
$544$	198835.	0.0288068
$545$	5.40832e6	0.779957
$546$	0	0
$547$	−4.69753e6	−0.671276	−0.335638	−	0.941991i	$-0.608952\pi$
$547$	−4.69753e6	−0.671276	−0.335638	+	0.941991i	$0.608952\pi$
$548$	−5.58622e6	−0.794633
$549$	0	0
$550$	67918.8	0.00957377
$551$	−2.73209e6	−0.383368
$552$	0	0
$553$	5.67623e6	0.789310
$554$	−619319.	−0.0857315
$555$	0	0
$556$	−4.37272e6	−0.599881
$557$	−1.06786e7	−1.45840	−0.729200	−	0.684300i	$-0.760108\pi$
$557$	−1.06786e7	−1.45840	−0.729200	+	0.684300i	$0.760108\pi$
$558$	0	0
$559$	1.07169e7	1.45058
$560$	2.85900e6	0.385251
$561$	0	0
$562$	−1.75147e6	−0.233917
$563$	−6.73487e6	−0.895485	−0.447742	−	0.894163i	$-0.647772\pi$
$563$	−6.73487e6	−0.895485	−0.447742	+	0.894163i	$0.647772\pi$
$564$	0	0
$565$	2.07467e6	0.273419
$566$	276475.	0.0362756
$567$	0	0
$568$	705480.	0.0917516
$569$	−6.36492e6	−0.824161	−0.412081	−	0.911147i	$-0.635198\pi$
$569$	−6.36492e6	−0.824161	−0.412081	+	0.911147i	$0.635198\pi$
$570$	0	0
$571$	−3.10897e6	−0.399049	−0.199524	−	0.979893i	$-0.563940\pi$
$571$	−3.10897e6	−0.399049	−0.199524	+	0.979893i	$0.563940\pi$
$572$	−2.51899e6	−0.321911
$573$	0	0
$574$	−72970.7	−0.00924419
$575$	−774145.	−0.0976456
$576$	0	0
$577$	1.35678e7	1.69656	0.848280	−	0.529548i	$-0.177639\pi$
$577$	1.35678e7	1.69656	0.848280	+	0.529548i	$0.177639\pi$
$578$	−1.27018e6	−0.158141
$579$	0	0
$580$	−3.00696e6	−0.371157
$581$	3.58345e6	0.440414
$582$	0	0
$583$	655916.	0.0799239
$584$	69852.4	0.00847519
$585$	0	0
$586$	−1.97250e6	−0.237286
$587$	−1.34652e7	−1.61293	−0.806466	−	0.591280i	$-0.798623\pi$
$587$	−1.34652e7	−1.61293	−0.806466	+	0.591280i	$0.798623\pi$
$588$	0	0
$589$	5.83626e6	0.693181
$590$	743815.	0.0879701
$591$	0	0
$592$	−5.42314e6	−0.635985
$593$	−1.49874e7	−1.75021	−0.875105	−	0.483933i	$-0.839208\pi$
$593$	−1.49874e7	−1.75021	−0.875105	+	0.483933i	$0.839208\pi$
$594$	0	0
$595$	225041.	0.0260597
$596$	−9.38493e6	−1.08222
$597$	0	0
$598$	−742411.	−0.0848968
$599$	−4.88452e6	−0.556231	−0.278115	−	0.960548i	$-0.589710\pi$
$599$	−4.88452e6	−0.556231	−0.278115	+	0.960548i	$0.589710\pi$
$600$	0	0
$601$	−801632.	−0.0905293	−0.0452646	−	0.998975i	$-0.514413\pi$
$601$	−801632.	−0.0905293	−0.0452646	+	0.998975i	$0.514413\pi$
$602$	1.74117e6	0.195817
$603$	0	0
$604$	−7.02975e6	−0.784056
$605$	366025.	0.0406558
$606$	0	0
$607$	1.38864e7	1.52974	0.764872	−	0.644182i	$-0.222802\pi$
$607$	1.38864e7	1.52974	0.764872	+	0.644182i	$0.222802\pi$
$608$	−1.88958e6	−0.207303
$609$	0	0
$610$	405011.	0.0440700
$611$	5.05410e6	0.547697
$612$	0	0
$613$	1.48685e6	0.159815	0.0799073	−	0.996802i	$-0.474538\pi$
$613$	1.48685e6	0.159815	0.0799073	+	0.996802i	$0.474538\pi$
$614$	−813659.	−0.0871007
$615$	0	0
$616$	−829097.	−0.0880346
$617$	−2.96802e6	−0.313873	−0.156937	−	0.987609i	$-0.550162\pi$
$617$	−2.96802e6	−0.313873	−0.156937	+	0.987609i	$0.550162\pi$
$618$	0	0
$619$	−1.40553e7	−1.47440	−0.737198	−	0.675677i	$-0.763851\pi$
$619$	−1.40553e7	−1.47440	−0.737198	+	0.675677i	$0.763851\pi$
$620$	6.42344e6	0.671102
$621$	0	0
$622$	−947461.	−0.0981941
$623$	1.50989e7	1.55857
$624$	0	0
$625$	390625.	0.0400000
$626$	2.67718e6	0.273049
$627$	0	0
$628$	1.07598e7	1.08869
$629$	−426873.	−0.0430202
$630$	0	0
$631$	−7.30424e6	−0.730301	−0.365150	−	0.930949i	$-0.618982\pi$
$631$	−7.30424e6	−0.730301	−0.365150	+	0.930949i	$0.618982\pi$
$632$	−2.66829e6	−0.265730
$633$	0	0
$634$	244463.	0.0241540
$635$	−2.81885e6	−0.277420
$636$	0	0
$637$	−1.48874e6	−0.145368
$638$	419019.	0.0407551
$639$	0	0
$640$	−2.76024e6	−0.266377
$641$	7.28278e6	0.700087	0.350043	−	0.936733i	$-0.386167\pi$
$641$	7.28278e6	0.700087	0.350043	+	0.936733i	$0.386167\pi$
$642$	0	0
$643$	1.44225e7	1.37567	0.687833	−	0.725869i	$-0.258562\pi$
$643$	1.44225e7	1.37567	0.687833	+	0.725869i	$0.258562\pi$
$644$	4.66476e6	0.443215
$645$	0	0
$646$	−47445.2	−0.00447313
$647$	1.62591e6	0.152699	0.0763495	−	0.997081i	$-0.475674\pi$
$647$	1.62591e6	0.152699	0.0763495	+	0.997081i	$0.475674\pi$
$648$	0	0
$649$	4.00854e6	0.373572
$650$	374613.	0.0347775
$651$	0	0
$652$	4.20427e6	0.387322
$653$	1.73130e7	1.58887	0.794435	−	0.607349i	$-0.207767\pi$
$653$	1.73130e7	1.58887	0.794435	+	0.607349i	$0.207767\pi$
$654$	0	0
$655$	−4.43065e6	−0.403520
$656$	−637457.	−0.0578350
$657$	0	0
$658$	821133.	0.0739348
$659$	−2.47926e6	−0.222387	−0.111193	−	0.993799i	$-0.535467\pi$
$659$	−2.47926e6	−0.222387	−0.111193	+	0.993799i	$0.535467\pi$
$660$	0	0
$661$	−7.28403e6	−0.648437	−0.324219	−	0.945982i	$-0.605101\pi$
$661$	−7.28403e6	−0.648437	−0.324219	+	0.945982i	$0.605101\pi$
$662$	−884293.	−0.0784244
$663$	0	0
$664$	−1.68451e6	−0.148270
$665$	−2.13862e6	−0.187534
$666$	0	0
$667$	−4.77602e6	−0.415673
$668$	1.48660e7	1.28900
$669$	0	0
$670$	−1.35335e6	−0.116472
$671$	2.18267e6	0.187147
$672$	0	0
$673$	1.73888e7	1.47990	0.739951	−	0.672661i	$-0.234849\pi$
$673$	1.73888e7	1.47990	0.739951	+	0.672661i	$0.234849\pi$
$674$	−329481.	−0.0279371
$675$	0	0
$676$	−2.31183e6	−0.194576
$677$	1.51982e7	1.27444	0.637221	−	0.770681i	$-0.280084\pi$
$677$	1.51982e7	1.27444	0.637221	+	0.770681i	$0.280084\pi$
$678$	0	0
$679$	8.99038e6	0.748348
$680$	−105787.	−0.00877328
$681$	0	0
$682$	−895106.	−0.0736908
$683$	2.02075e7	1.65753	0.828765	−	0.559596i	$-0.189044\pi$
$683$	2.02075e7	1.65753	0.828765	+	0.559596i	$0.189044\pi$
$684$	0	0
$685$	4.47708e6	0.364560
$686$	−2.06425e6	−0.167476
$687$	0	0
$688$	1.52105e7	1.22510
$689$	3.61777e6	0.290330
$690$	0	0
$691$	−1.88904e7	−1.50503	−0.752517	−	0.658573i	$-0.771160\pi$
$691$	−1.88904e7	−1.50503	−0.752517	+	0.658573i	$0.771160\pi$
$692$	−1.89354e7	−1.50317
$693$	0	0
$694$	501550.	0.0395290
$695$	3.50453e6	0.275212
$696$	0	0
$697$	−50176.3	−0.00391216
$698$	1.37715e6	0.106990
$699$	0	0
$700$	−2.35378e6	−0.181561
$701$	1.63467e7	1.25642	0.628210	−	0.778044i	$-0.283788\pi$
$701$	1.63467e7	1.25642	0.628210	+	0.778044i	$0.283788\pi$
$702$	0	0
$703$	4.05668e6	0.309587
$704$	−3.37783e6	−0.256865
$705$	0	0
$706$	−2.19001e6	−0.165362
$707$	7.89511e6	0.594032
$708$	0	0
$709$	2.28062e7	1.70387	0.851936	−	0.523646i	$-0.175429\pi$
$709$	2.28062e7	1.70387	0.851936	+	0.523646i	$0.175429\pi$
$710$	−279095.	−0.0207782
$711$	0	0
$712$	−7.09771e6	−0.524709
$713$	1.02025e7	0.751594
$714$	0	0
$715$	2.01885e6	0.147686
$716$	−1.83703e7	−1.33916
$717$	0	0
$718$	−1.05659e6	−0.0764886
$719$	−1.42246e6	−0.102617	−0.0513083	−	0.998683i	$-0.516339\pi$
$719$	−1.42246e6	−0.102617	−0.0513083	+	0.998683i	$0.516339\pi$
$720$	0	0
$721$	−4.34094e6	−0.310989
$722$	−1.77290e6	−0.126573
$723$	0	0
$724$	1.25935e7	0.892896
$725$	2.40993e6	0.170278
$726$	0	0
$727$	1.58260e7	1.11054	0.555272	−	0.831669i	$-0.312614\pi$
$727$	1.58260e7	1.11054	0.555272	+	0.831669i	$0.312614\pi$
$728$	−4.57297e6	−0.319793
$729$	0	0
$730$	−27634.4	−0.00191930
$731$	1.19727e6	0.0828701
$732$	0	0
$733$	1.04557e7	0.718776	0.359388	−	0.933188i	$-0.382985\pi$
$733$	1.04557e7	0.718776	0.359388	+	0.933188i	$0.382985\pi$
$734$	−2.46248e6	−0.168707
$735$	0	0
$736$	−3.30321e6	−0.224772
$737$	−7.29341e6	−0.494609
$738$	0	0
$739$	−5.11969e6	−0.344852	−0.172426	−	0.985023i	$-0.555160\pi$
$739$	−5.11969e6	−0.344852	−0.172426	+	0.985023i	$0.555160\pi$
$740$	4.46482e6	0.299726
$741$	0	0
$742$	587775.	0.0391923
$743$	4.44877e6	0.295643	0.147822	−	0.989014i	$-0.452774\pi$
$743$	4.44877e6	0.295643	0.147822	+	0.989014i	$0.452774\pi$
$744$	0	0
$745$	7.52156e6	0.496498
$746$	3.11316e6	0.204812
$747$	0	0
$748$	−281414.	−0.0183904
$749$	1.42634e7	0.929009
$750$	0	0
$751$	−1.91736e7	−1.24052	−0.620262	−	0.784395i	$-0.712974\pi$
$751$	−1.91736e7	−1.24052	−0.620262	+	0.784395i	$0.712974\pi$
$752$	7.17325e6	0.462563
$753$	0	0
$754$	2.31114e6	0.148047
$755$	5.63400e6	0.359707
$756$	0	0
$757$	−1.24339e7	−0.788621	−0.394310	−	0.918977i	$-0.629017\pi$
$757$	−1.24339e7	−0.788621	−0.394310	+	0.918977i	$0.629017\pi$
$758$	3.65707e6	0.231185
$759$	0	0
$760$	1.00532e6	0.0631353
$761$	1.41772e7	0.887421	0.443710	−	0.896170i	$-0.353662\pi$
$761$	1.41772e7	0.887421	0.443710	+	0.896170i	$0.353662\pi$
$762$	0	0
$763$	2.61184e7	1.62418
$764$	−1.23301e7	−0.764246
$765$	0	0
$766$	−392125.	−0.0241464
$767$	2.21095e7	1.35703
$768$	0	0
$769$	−4.53694e6	−0.276660	−0.138330	−	0.990386i	$-0.544174\pi$
$769$	−4.53694e6	−0.276660	−0.138330	+	0.990386i	$0.544174\pi$
$770$	328000.	0.0199364
$771$	0	0
$772$	−5.99956e6	−0.362306
$773$	2.11780e7	1.27479	0.637393	−	0.770539i	$-0.280013\pi$
$773$	2.11780e7	1.27479	0.637393	+	0.770539i	$0.280013\pi$
$774$	0	0
$775$	−5.14807e6	−0.307886
$776$	−4.22620e6	−0.251939
$777$	0	0
$778$	−3.84065e6	−0.227487
$779$	476838.	0.0281531
$780$	0	0
$781$	−1.50409e6	−0.0882361
$782$	−82940.1	−0.00485007
$783$	0	0
$784$	−2.11295e6	−0.122772
$785$	−8.62344e6	−0.499467
$786$	0	0
$787$	−2.73190e7	−1.57228	−0.786138	−	0.618051i	$-0.787922\pi$
$787$	−2.73190e7	−1.57228	−0.786138	+	0.618051i	$0.787922\pi$
$788$	−1.67687e7	−0.962018
$789$	0	0
$790$	1.05560e6	0.0601774
$791$	1.00192e7	0.569367
$792$	0	0
$793$	1.20387e7	0.679826
$794$	−1.44914e6	−0.0815752
$795$	0	0
$796$	1.93097e7	1.08017
$797$	8.71204e6	0.485818	0.242909	−	0.970049i	$-0.421898\pi$
$797$	8.71204e6	0.485818	0.242909	+	0.970049i	$0.421898\pi$
$798$	0	0
$799$	564630.	0.0312894
$800$	1.66676e6	0.0920766
$801$	0	0
$802$	−4.11493e6	−0.225905
$803$	−148926.	−0.00815045
$804$	0	0
$805$	−3.73857e6	−0.203337
$806$	−4.93704e6	−0.267688
$807$	0	0
$808$	−3.71134e6	−0.199987
$809$	8.30436e6	0.446103	0.223051	−	0.974807i	$-0.428398\pi$
$809$	8.30436e6	0.446103	0.223051	+	0.974807i	$0.428398\pi$
$810$	0	0
$811$	−2.16579e7	−1.15629	−0.578143	−	0.815936i	$-0.696222\pi$
$811$	−2.16579e7	−1.15629	−0.578143	+	0.815936i	$0.696222\pi$
$812$	−1.45215e7	−0.772896
$813$	0	0
$814$	−622172.	−0.0329116
$815$	−3.36952e6	−0.177695
$816$	0	0
$817$	−1.13779e7	−0.596359
$818$	2.29789e6	0.120073
$819$	0	0
$820$	524812.	0.0272564
$821$	4.66524e6	0.241555	0.120777	−	0.992680i	$-0.461461\pi$
$821$	4.66524e6	0.241555	0.120777	+	0.992680i	$0.461461\pi$
$822$	0	0
$823$	−3.86223e7	−1.98765	−0.993823	−	0.110977i	$-0.964602\pi$
$823$	−3.86223e7	−1.98765	−0.993823	+	0.110977i	$0.964602\pi$
$824$	2.04059e6	0.104698
$825$	0	0
$826$	3.59210e6	0.183189
$827$	−1.64856e7	−0.838187	−0.419094	−	0.907943i	$-0.637652\pi$
$827$	−1.64856e7	−0.838187	−0.419094	+	0.907943i	$0.637652\pi$
$828$	0	0
$829$	1.17447e7	0.593546	0.296773	−	0.954948i	$-0.404089\pi$
$829$	1.17447e7	0.593546	0.296773	+	0.954948i	$0.404089\pi$
$830$	666411.	0.0335774
$831$	0	0
$832$	−1.86307e7	−0.933086
$833$	−166317.	−0.00830473
$834$	0	0
$835$	−1.19143e7	−0.591363
$836$	2.67435e6	0.132344
$837$	0	0
$838$	3.91671e6	0.192669
$839$	−2.08153e7	−1.02089	−0.510444	−	0.859911i	$-0.670519\pi$
$839$	−2.08153e7	−1.02089	−0.510444	+	0.859911i	$0.670519\pi$
$840$	0	0
$841$	−5.64328e6	−0.275132
$842$	−3.95161e6	−0.192085
$843$	0	0
$844$	−2.92587e7	−1.41383
$845$	1.85282e6	0.0892670
$846$	0	0
$847$	1.76764e6	0.0846615
$848$	5.13467e6	0.245202
$849$	0	0
$850$	41850.7	0.00198680
$851$	7.09158e6	0.335675
$852$	0	0
$853$	4.06291e6	0.191190	0.0955950	−	0.995420i	$-0.469525\pi$
$853$	4.06291e6	0.191190	0.0955950	+	0.995420i	$0.469525\pi$
$854$	1.95592e6	0.0917712
$855$	0	0
$856$	−6.70497e6	−0.312761
$857$	−66823.3	−0.00310796	−0.00155398	−	0.999999i	$-0.500495\pi$
$857$	−66823.3	−0.00310796	−0.00155398	+	0.999999i	$0.500495\pi$
$858$	0	0
$859$	9.60662e6	0.444209	0.222105	−	0.975023i	$-0.428707\pi$
$859$	9.60662e6	0.444209	0.222105	+	0.975023i	$0.428707\pi$
$860$	−1.25226e7	−0.577364
$861$	0	0
$862$	−4.56833e6	−0.209406
$863$	3.62555e6	0.165709	0.0828547	−	0.996562i	$-0.473596\pi$
$863$	3.62555e6	0.165709	0.0828547	+	0.996562i	$0.473596\pi$
$864$	0	0
$865$	1.51758e7	0.689621
$866$	1.57252e6	0.0712529
$867$	0	0
$868$	3.10207e7	1.39750
$869$	5.68882e6	0.255548
$870$	0	0
$871$	−4.02275e7	−1.79671
$872$	−1.22777e7	−0.546798
$873$	0	0
$874$	788200.	0.0349026
$875$	1.88644e6	0.0832959
$876$	0	0
$877$	−6.29358e6	−0.276311	−0.138156	−	0.990411i	$-0.544117\pi$
$877$	−6.29358e6	−0.276311	−0.138156	+	0.990411i	$0.544117\pi$
$878$	−2.38970e6	−0.104618
$879$	0	0
$880$	2.86534e6	0.124729
$881$	−2.35769e7	−1.02340	−0.511701	−	0.859163i	$-0.670985\pi$
$881$	−2.35769e7	−1.02340	−0.511701	+	0.859163i	$0.670985\pi$
$882$	0	0
$883$	1.08516e7	0.468372	0.234186	−	0.972192i	$-0.424758\pi$
$883$	1.08516e7	0.468372	0.234186	+	0.972192i	$0.424758\pi$
$884$	−1.55217e6	−0.0668049
$885$	0	0
$886$	6.27233e6	0.268438
$887$	2.44367e7	1.04288	0.521439	−	0.853289i	$-0.325396\pi$
$887$	2.44367e7	1.04288	0.521439	+	0.853289i	$0.325396\pi$
$888$	0	0
$889$	−1.36131e7	−0.577699
$890$	2.80793e6	0.118826
$891$	0	0
$892$	3.29968e7	1.38854
$893$	−5.36581e6	−0.225168
$894$	0	0
$895$	1.47229e7	0.614377
$896$	−1.33300e7	−0.554703
$897$	0	0
$898$	−6.88435e6	−0.284887
$899$	−3.17606e7	−1.31066
$900$	0	0
$901$	404167.	0.0165863
$902$	−73132.5	−0.00299291
$903$	0	0
$904$	−4.70984e6	−0.191684
$905$	−1.00931e7	−0.409641
$906$	0	0
$907$	−4.71119e7	−1.90157	−0.950785	−	0.309850i	$-0.899721\pi$
$907$	−4.71119e7	−1.90157	−0.950785	+	0.309850i	$0.899721\pi$
$908$	−4.32012e7	−1.73893
$909$	0	0
$910$	1.80911e6	0.0724207
$911$	−4.74577e7	−1.89457	−0.947286	−	0.320390i	$-0.896186\pi$
$911$	−4.74577e7	−1.89457	−0.947286	+	0.320390i	$0.896186\pi$
$912$	0	0
$913$	3.59140e6	0.142589
$914$	3.14503e6	0.124526
$915$	0	0
$916$	1.77792e6	0.0700122
$917$	−2.13969e7	−0.840288
$918$	0	0
$919$	1.88222e7	0.735159	0.367580	−	0.929992i	$-0.380187\pi$
$919$	1.88222e7	0.735159	0.367580	+	0.929992i	$0.380187\pi$
$920$	1.75743e6	0.0684556
$921$	0	0
$922$	5.95982e6	0.230890
$923$	−8.29595e6	−0.320525
$924$	0	0
$925$	−3.57833e6	−0.137508
$926$	1.01764e6	0.0390001
$927$	0	0
$928$	1.02830e7	0.391966
$929$	−1.80487e7	−0.686132	−0.343066	−	0.939311i	$-0.611465\pi$
$929$	−1.80487e7	−0.686132	−0.343066	+	0.939311i	$0.611465\pi$
$930$	0	0
$931$	1.58056e6	0.0597634
$932$	−873067.	−0.0329237
$933$	0	0
$934$	−1.23323e6	−0.0462569
$935$	225540.	0.00843712
$936$	0	0
$937$	−5.09381e7	−1.89537	−0.947685	−	0.319208i	$-0.896583\pi$
$937$	−5.09381e7	−1.89537	−0.947685	+	0.319208i	$0.896583\pi$
$938$	−6.53572e6	−0.242542
$939$	0	0
$940$	−5.90566e6	−0.217996
$941$	−4.08876e7	−1.50528	−0.752641	−	0.658431i	$-0.771221\pi$
$941$	−4.08876e7	−1.50528	−0.752641	+	0.658431i	$0.771221\pi$
$942$	0	0
$943$	833571.	0.0305255
$944$	3.13799e7	1.14610
$945$	0	0
$946$	1.74503e6	0.0633979
$947$	1.33924e6	0.0485271	0.0242636	−	0.999706i	$-0.492276\pi$
$947$	1.33924e6	0.0485271	0.0242636	+	0.999706i	$0.492276\pi$
$948$	0	0
$949$	−821416.	−0.0296072
$950$	−397717.	−0.0142977
$951$	0	0
$952$	−510879.	−0.0182695
$953$	−1.17556e6	−0.0419288	−0.0209644	−	0.999780i	$-0.506674\pi$
$953$	−1.17556e6	−0.0419288	−0.0209644	+	0.999780i	$0.506674\pi$
$954$	0	0
$955$	9.88197e6	0.350619
$956$	−3.78706e7	−1.34016
$957$	0	0
$958$	6.93321e6	0.244074
$959$	2.16212e7	0.759159
$960$	0	0
$961$	3.92177e7	1.36985
$962$	−3.43165e6	−0.119554
$963$	0	0
$964$	−3.31315e7	−1.14828
$965$	4.80835e6	0.166218
$966$	0	0
$967$	−1.01207e7	−0.348053	−0.174026	−	0.984741i	$-0.555678\pi$
$967$	−1.01207e7	−0.348053	−0.174026	+	0.984741i	$0.555678\pi$
$968$	−830935.	−0.0285022
$969$	0	0
$970$	1.67193e6	0.0570544
$971$	2.71903e7	0.925476	0.462738	−	0.886495i	$-0.346867\pi$
$971$	2.71903e7	0.925476	0.462738	+	0.886495i	$0.346867\pi$
$972$	0	0
$973$	1.69244e7	0.573101
$974$	2.48493e6	0.0839300
$975$	0	0
$976$	1.70865e7	0.574154
$977$	−2.41360e7	−0.808962	−0.404481	−	0.914546i	$-0.632548\pi$
$977$	−2.41360e7	−0.808962	−0.404481	+	0.914546i	$0.632548\pi$
$978$	0	0
$979$	1.51324e7	0.504604
$980$	1.73957e6	0.0578599
$981$	0	0
$982$	2.32203e6	0.0768402
$983$	2.27328e7	0.750359	0.375180	−	0.926952i	$-0.377581\pi$
$983$	2.27328e7	0.750359	0.375180	+	0.926952i	$0.377581\pi$
$984$	0	0
$985$	1.34393e7	0.441352
$986$	258194.	0.00845774
$987$	0	0
$988$	1.47506e7	0.480749
$989$	−1.98900e7	−0.646613
$990$	0	0
$991$	−1.42957e6	−0.0462403	−0.0231202	−	0.999733i	$-0.507360\pi$
$991$	−1.42957e6	−0.0462403	−0.0231202	+	0.999733i	$0.507360\pi$
$992$	−2.19664e7	−0.708728
$993$	0	0
$994$	−1.34783e6	−0.0432684
$995$	−1.54758e7	−0.495559
$996$	0	0
$997$	−3.51095e6	−0.111863	−0.0559316	−	0.998435i	$-0.517813\pi$
$997$	−3.51095e6	−0.111863	−0.0559316	+	0.998435i	$0.517813\pi$
$998$	−521130.	−0.0165623
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.i.1.3		5
3.2	odd	2		165.6.a.g.1.3	✓	5
15.14	odd	2		825.6.a.k.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.g.1.3	✓	5	3.2	odd	2
495.6.a.i.1.3		5	1.1	even	1	trivial
825.6.a.k.1.3		5	15.14	odd	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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+0.898099 q^{2} -31.1934 q^{4} +25.0000 q^{5} +120.732 q^{7} -56.7540 q^{8} +O(q^{10})\)	\(q+0.898099 q^{2} -31.1934 q^{4} +25.0000 q^{5} +120.732 q^{7} -56.7540 q^{8} +22.4525 q^{10} +121.000 q^{11} +667.387 q^{13} +108.430 q^{14} +947.219 q^{16} +74.5586 q^{17} -708.549 q^{19} -779.835 q^{20} +108.670 q^{22} -1238.63 q^{23} +625.000 q^{25} +599.380 q^{26} -3766.06 q^{28} +3855.89 q^{29} -8236.92 q^{31} +2666.82 q^{32} +66.9611 q^{34} +3018.31 q^{35} -5725.33 q^{37} -636.348 q^{38} -1418.85 q^{40} -672.977 q^{41} +16058.1 q^{43} -3774.40 q^{44} -1112.41 q^{46} +7572.96 q^{47} -2230.69 q^{49} +561.312 q^{50} -20818.1 q^{52} +5420.79 q^{53} +3025.00 q^{55} -6852.04 q^{56} +3462.97 q^{58} +33128.4 q^{59} +18038.6 q^{61} -7397.57 q^{62} -27915.9 q^{64} +16684.7 q^{65} -60276.1 q^{67} -2325.74 q^{68} +2710.74 q^{70} -12430.5 q^{71} -1230.79 q^{73} -5141.92 q^{74} +22102.1 q^{76} +14608.6 q^{77} +47015.0 q^{79} +23680.5 q^{80} -604.401 q^{82} +29681.0 q^{83} +1863.97 q^{85} +14421.7 q^{86} -6867.23 q^{88} +125061. q^{89} +80575.2 q^{91} +38637.2 q^{92} +6801.27 q^{94} -17713.7 q^{95} +74465.3 q^{97} -2003.38 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 127 q^{4} + 125 q^{5} + 116 q^{7} + 153 q^{8}+O(q^{10}) \) 5 * q + q^2 + 127 * q^4 + 125 * q^5 + 116 * q^7 + 153 * q^8	\( 5 q + q^{2} + 127 q^{4} + 125 q^{5} + 116 q^{7} + 153 q^{8} + 25 q^{10} + 605 q^{11} - 926 q^{13} - 368 q^{14} + 1891 q^{16} + 246 q^{17} + 3420 q^{19} + 3175 q^{20} + 121 q^{22} + 4244 q^{23} + 3125 q^{25} + 8862 q^{26} - 4904 q^{28} + 2922 q^{29} - 6112 q^{31} + 24757 q^{32} + 10866 q^{34} + 2900 q^{35} + 6654 q^{37} + 45692 q^{38} + 3825 q^{40} + 14934 q^{41} + 10804 q^{43} + 15367 q^{44} - 101500 q^{46} + 41460 q^{47} - 12099 q^{49} + 625 q^{50} - 97742 q^{52} + 62398 q^{53} + 15125 q^{55} + 74368 q^{56} - 27822 q^{58} - 8524 q^{59} + 59010 q^{61} + 142624 q^{62} + 13799 q^{64} - 23150 q^{65} - 15772 q^{67} + 83686 q^{68} - 9200 q^{70} - 88124 q^{71} - 118358 q^{73} - 67194 q^{74} + 100668 q^{76} + 14036 q^{77} + 57324 q^{79} + 47275 q^{80} + 29102 q^{82} + 7268 q^{83} + 6150 q^{85} + 35288 q^{86} + 18513 q^{88} - 72978 q^{89} - 1464 q^{91} - 62148 q^{92} + 344836 q^{94} + 85500 q^{95} - 59174 q^{97} - 272767 q^{98}+O(q^{100}) \) 5 * q + q^2 + 127 * q^4 + 125 * q^5 + 116 * q^7 + 153 * q^8 + 25 * q^10 + 605 * q^11 - 926 * q^13 - 368 * q^14 + 1891 * q^16 + 246 * q^17 + 3420 * q^19 + 3175 * q^20 + 121 * q^22 + 4244 * q^23 + 3125 * q^25 + 8862 * q^26 - 4904 * q^28 + 2922 * q^29 - 6112 * q^31 + 24757 * q^32 + 10866 * q^34 + 2900 * q^35 + 6654 * q^37 + 45692 * q^38 + 3825 * q^40 + 14934 * q^41 + 10804 * q^43 + 15367 * q^44 - 101500 * q^46 + 41460 * q^47 - 12099 * q^49 + 625 * q^50 - 97742 * q^52 + 62398 * q^53 + 15125 * q^55 + 74368 * q^56 - 27822 * q^58 - 8524 * q^59 + 59010 * q^61 + 142624 * q^62 + 13799 * q^64 - 23150 * q^65 - 15772 * q^67 + 83686 * q^68 - 9200 * q^70 - 88124 * q^71 - 118358 * q^73 - 67194 * q^74 + 100668 * q^76 + 14036 * q^77 + 57324 * q^79 + 47275 * q^80 + 29102 * q^82 + 7268 * q^83 + 6150 * q^85 + 35288 * q^86 + 18513 * q^88 - 72978 * q^89 - 1464 * q^91 - 62148 * q^92 + 344836 * q^94 + 85500 * q^95 - 59174 * q^97 - 272767 * q^98

Introduction

L-functions

Modular forms

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Database

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