LMFDB - Embedded newform 495.6.a.g.1.2

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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 33x^{2} - 8x + 116 $ x^4 - 2x^3 - 33x^2 - 8*x + 116
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.74666$ 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)

$f(q)$	$=$	$q-2.64192 q^{2} -25.0202 q^{4} +25.0000 q^{5} -12.8802 q^{7} +150.643 q^{8} +O(q^{10})$	\(q-2.64192 q^{2} -25.0202 q^{4} +25.0000 q^{5} -12.8802 q^{7} +150.643 q^{8} -66.0481 q^{10} +121.000 q^{11} -485.167 q^{13} +34.0284 q^{14} +402.660 q^{16} -266.661 q^{17} -149.702 q^{19} -625.506 q^{20} -319.673 q^{22} +3213.11 q^{23} +625.000 q^{25} +1281.77 q^{26} +322.265 q^{28} -2948.81 q^{29} +2145.87 q^{31} -5884.38 q^{32} +704.498 q^{34} -322.004 q^{35} -808.357 q^{37} +395.500 q^{38} +3766.08 q^{40} -10105.2 q^{41} +2763.15 q^{43} -3027.45 q^{44} -8488.79 q^{46} -9973.36 q^{47} -16641.1 q^{49} -1651.20 q^{50} +12139.0 q^{52} -7126.92 q^{53} +3025.00 q^{55} -1940.31 q^{56} +7790.52 q^{58} +33337.2 q^{59} -11871.1 q^{61} -5669.22 q^{62} +2660.94 q^{64} -12129.2 q^{65} +4500.58 q^{67} +6671.93 q^{68} +850.710 q^{70} +45977.8 q^{71} -62039.1 q^{73} +2135.62 q^{74} +3745.57 q^{76} -1558.50 q^{77} -57486.6 q^{79} +10066.5 q^{80} +26697.2 q^{82} +90511.7 q^{83} -6666.53 q^{85} -7300.03 q^{86} +18227.8 q^{88} +127861. q^{89} +6249.03 q^{91} -80392.8 q^{92} +26348.9 q^{94} -3742.54 q^{95} +132338. q^{97} +43964.5 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{2} + 61 q^{4} + 100 q^{5} - 90 q^{7} + 135 q^{8}+O(q^{10}) $ 4 * q + 5 * q^2 + 61 * q^4 + 100 * q^5 - 90 * q^7 + 135 * q^8	$ 4 q + 5 q^{2} + 61 q^{4} + 100 q^{5} - 90 q^{7} + 135 q^{8} + 125 q^{10} + 484 q^{11} + 820 q^{13} + 1687 q^{14} - 2671 q^{16} + 3800 q^{17} - 3394 q^{19} + 1525 q^{20} + 605 q^{22} + 3020 q^{23} + 2500 q^{25} + 3650 q^{26} + 2635 q^{28} + 5248 q^{29} + 4732 q^{31} - 11505 q^{32} + 9759 q^{34} - 2250 q^{35} + 10210 q^{37} - 16945 q^{38} + 3375 q^{40} + 21068 q^{41} - 12140 q^{43} + 7381 q^{44} + 58442 q^{46} - 4720 q^{47} + 24550 q^{49} + 3125 q^{50} + 58250 q^{52} + 21670 q^{53} + 12100 q^{55} + 28985 q^{56} - 45435 q^{58} + 69068 q^{59} - 44000 q^{61} - 33375 q^{62} - 68223 q^{64} + 20500 q^{65} - 8720 q^{67} + 107335 q^{68} + 42175 q^{70} + 47516 q^{71} - 2480 q^{73} + 55717 q^{74} - 102461 q^{76} - 10890 q^{77} - 188192 q^{79} - 66775 q^{80} + 279030 q^{82} - 68620 q^{83} + 95000 q^{85} - 115328 q^{86} + 16335 q^{88} + 170266 q^{89} + 97740 q^{91} - 53950 q^{92} - 152926 q^{94} - 84850 q^{95} + 186160 q^{97} + 393590 q^{98}+O(q^{100}) $ 4 * q + 5 * q^2 + 61 * q^4 + 100 * q^5 - 90 * q^7 + 135 * q^8 + 125 * q^10 + 484 * q^11 + 820 * q^13 + 1687 * q^14 - 2671 * q^16 + 3800 * q^17 - 3394 * q^19 + 1525 * q^20 + 605 * q^22 + 3020 * q^23 + 2500 * q^25 + 3650 * q^26 + 2635 * q^28 + 5248 * q^29 + 4732 * q^31 - 11505 * q^32 + 9759 * q^34 - 2250 * q^35 + 10210 * q^37 - 16945 * q^38 + 3375 * q^40 + 21068 * q^41 - 12140 * q^43 + 7381 * q^44 + 58442 * q^46 - 4720 * q^47 + 24550 * q^49 + 3125 * q^50 + 58250 * q^52 + 21670 * q^53 + 12100 * q^55 + 28985 * q^56 - 45435 * q^58 + 69068 * q^59 - 44000 * q^61 - 33375 * q^62 - 68223 * q^64 + 20500 * q^65 - 8720 * q^67 + 107335 * q^68 + 42175 * q^70 + 47516 * q^71 - 2480 * q^73 + 55717 * q^74 - 102461 * q^76 - 10890 * q^77 - 188192 * q^79 - 66775 * q^80 + 279030 * q^82 - 68620 * q^83 + 95000 * q^85 - 115328 * q^86 + 16335 * q^88 + 170266 * q^89 + 97740 * q^91 - 53950 * q^92 - 152926 * q^94 - 84850 * q^95 + 186160 * q^97 + 393590 * 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 $

$2$	−2.64192	−0.467030	−0.233515	−	0.972353i	$-0.575023\pi$
$2$	−2.64192	−0.467030	−0.233515	+	0.972353i	$0.575023\pi$
$3$	0	0
$3$	0	0
$4$	−25.0202	−0.781883
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−12.8802	−0.0993520	−0.0496760	−	0.998765i	$-0.515819\pi$
$7$	−12.8802	−0.0993520	−0.0496760	+	0.998765i	$0.515819\pi$
$8$	150.643	0.832193
$9$	0	0
$10$	−66.0481	−0.208862
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−485.167	−0.796219	−0.398110	−	0.917338i	$-0.630334\pi$
$13$	−485.167	−0.796219	−0.398110	+	0.917338i	$0.630334\pi$
$14$	34.0284	0.0464004
$15$	0	0
$16$	402.660	0.393223
$17$	−266.661	−0.223788	−0.111894	−	0.993720i	$-0.535692\pi$
$17$	−266.661	−0.223788	−0.111894	+	0.993720i	$0.535692\pi$
$18$	0	0
$19$	−149.702	−0.0951356	−0.0475678	−	0.998868i	$-0.515147\pi$
$19$	−149.702	−0.0951356	−0.0475678	+	0.998868i	$0.515147\pi$
$20$	−625.506	−0.349669
$21$	0	0
$22$	−319.673	−0.140815
$23$	3213.11	1.26650	0.633251	−	0.773946i	$-0.281720\pi$
$23$	3213.11	1.26650	0.633251	+	0.773946i	$0.281720\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	1281.77	0.371859
$27$	0	0
$28$	322.265	0.0776816
$29$	−2948.81	−0.651106	−0.325553	−	0.945524i	$-0.605550\pi$
$29$	−2948.81	−0.651106	−0.325553	+	0.945524i	$0.605550\pi$
$30$	0	0
$31$	2145.87	0.401051	0.200525	−	0.979689i	$-0.435735\pi$
$31$	2145.87	0.401051	0.200525	+	0.979689i	$0.435735\pi$
$32$	−5884.38	−1.01584
$33$	0	0
$34$	704.498	0.104516
$35$	−322.004	−0.0444315
$36$	0	0
$37$	−808.357	−0.0970731	−0.0485366	−	0.998821i	$-0.515456\pi$
$37$	−808.357	−0.0970731	−0.0485366	+	0.998821i	$0.515456\pi$
$38$	395.500	0.0444312
$39$	0	0
$40$	3766.08	0.372168
$41$	−10105.2	−0.938829	−0.469414	−	0.882978i	$-0.655535\pi$
$41$	−10105.2	−0.938829	−0.469414	+	0.882978i	$0.655535\pi$
$42$	0	0
$43$	2763.15	0.227894	0.113947	−	0.993487i	$-0.463651\pi$
$43$	2763.15	0.227894	0.113947	+	0.993487i	$0.463651\pi$
$44$	−3027.45	−0.235746
$45$	0	0
$46$	−8488.79	−0.591495
$47$	−9973.36	−0.658562	−0.329281	−	0.944232i	$-0.606806\pi$
$47$	−9973.36	−0.658562	−0.329281	+	0.944232i	$0.606806\pi$
$48$	0	0
$49$	−16641.1	−0.990129
$50$	−1651.20	−0.0934061
$51$	0	0
$52$	12139.0	0.622550
$53$	−7126.92	−0.348508	−0.174254	−	0.984701i	$-0.555751\pi$
$53$	−7126.92	−0.348508	−0.174254	+	0.984701i	$0.555751\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	−1940.31	−0.0826800
$57$	0	0
$58$	7790.52	0.304086
$59$	33337.2	1.24681	0.623403	−	0.781901i	$-0.285750\pi$
$59$	33337.2	1.24681	0.623403	+	0.781901i	$0.285750\pi$
$60$	0	0
$61$	−11871.1	−0.408476	−0.204238	−	0.978921i	$-0.565472\pi$
$61$	−11871.1	−0.408476	−0.204238	+	0.978921i	$0.565472\pi$
$62$	−5669.22	−0.187303
$63$	0	0
$64$	2660.94	0.0812053
$65$	−12129.2	−0.356080
$66$	0	0
$67$	4500.58	0.122485	0.0612423	−	0.998123i	$-0.480494\pi$
$67$	4500.58	0.122485	0.0612423	+	0.998123i	$0.480494\pi$
$68$	6671.93	0.174976
$69$	0	0
$70$	850.710	0.0207509
$71$	45977.8	1.08244	0.541218	−	0.840882i	$-0.317963\pi$
$71$	45977.8	1.08244	0.541218	+	0.840882i	$0.317963\pi$
$72$	0	0
$73$	−62039.1	−1.36257	−0.681284	−	0.732019i	$-0.738578\pi$
$73$	−62039.1	−1.36257	−0.681284	+	0.732019i	$0.738578\pi$
$74$	2135.62	0.0453361
$75$	0	0
$76$	3745.57	0.0743848
$77$	−1558.50	−0.0299557
$78$	0	0
$79$	−57486.6	−1.03633	−0.518166	−	0.855280i	$-0.673385\pi$
$79$	−57486.6	−1.03633	−0.518166	+	0.855280i	$0.673385\pi$
$80$	10066.5	0.175855
$81$	0	0
$82$	26697.2	0.438461
$83$	90511.7	1.44215	0.721074	−	0.692858i	$-0.243649\pi$
$83$	90511.7	1.44215	0.721074	+	0.692858i	$0.243649\pi$
$84$	0	0
$85$	−6666.53	−0.100081
$86$	−7300.03	−0.106434
$87$	0	0
$88$	18227.8	0.250916
$89$	127861.	1.71105	0.855524	−	0.517764i	$-0.173235\pi$
$89$	127861.	1.71105	0.855524	+	0.517764i	$0.173235\pi$
$90$	0	0
$91$	6249.03	0.0791059
$92$	−80392.8	−0.990256
$93$	0	0
$94$	26348.9	0.307569
$95$	−3742.54	−0.0425459
$96$	0	0
$97$	132338.	1.42809	0.714046	−	0.700099i	$-0.246861\pi$
$97$	132338.	1.42809	0.714046	+	0.700099i	$0.246861\pi$
$98$	43964.5	0.462420
$99$	0	0
$100$	−15637.7	−0.156377
$101$	46941.7	0.457884	0.228942	−	0.973440i	$-0.426473\pi$
$101$	46941.7	0.457884	0.228942	+	0.973440i	$0.426473\pi$
$102$	0	0
$103$	94478.2	0.877483	0.438741	−	0.898613i	$-0.355424\pi$
$103$	94478.2	0.877483	0.438741	+	0.898613i	$0.355424\pi$
$104$	−73087.0	−0.662608
$105$	0	0
$106$	18828.8	0.162764
$107$	−133596.	−1.12807	−0.564034	−	0.825752i	$-0.690751\pi$
$107$	−133596.	−1.12807	−0.564034	+	0.825752i	$0.690751\pi$
$108$	0	0
$109$	18537.6	0.149447	0.0747237	−	0.997204i	$-0.476193\pi$
$109$	18537.6	0.149447	0.0747237	+	0.997204i	$0.476193\pi$
$110$	−7991.82	−0.0629744
$111$	0	0
$112$	−5186.33	−0.0390675
$113$	1665.04	0.0122667	0.00613337	−	0.999981i	$-0.498048\pi$
$113$	1665.04	0.0122667	0.00613337	+	0.999981i	$0.498048\pi$
$114$	0	0
$115$	80327.7	0.566397
$116$	73779.9	0.509088
$117$	0	0
$118$	−88074.3	−0.582296
$119$	3434.64	0.0222338
$120$	0	0
$121$	14641.0	0.0909091
$122$	31362.6	0.190771
$123$	0	0
$124$	−53690.2	−0.313574
$125$	15625.0	0.0894427
$126$	0	0
$127$	−108773.	−0.598428	−0.299214	−	0.954186i	$-0.596724\pi$
$127$	−108773.	−0.598428	−0.299214	+	0.954186i	$0.596724\pi$
$128$	181270.	0.977915
$129$	0	0
$130$	32044.3	0.166300
$131$	330399.	1.68214	0.841068	−	0.540930i	$-0.181928\pi$
$131$	330399.	1.68214	0.841068	+	0.540930i	$0.181928\pi$
$132$	0	0
$133$	1928.18	0.00945190
$134$	−11890.2	−0.0572040
$135$	0	0
$136$	−40170.7	−0.186235
$137$	−33498.4	−0.152483	−0.0762417	−	0.997089i	$-0.524292\pi$
$137$	−33498.4	−0.152483	−0.0762417	+	0.997089i	$0.524292\pi$
$138$	0	0
$139$	236204.	1.03693	0.518465	−	0.855099i	$-0.326504\pi$
$139$	236204.	1.03693	0.518465	+	0.855099i	$0.326504\pi$
$140$	8056.62	0.0347403
$141$	0	0
$142$	−121470.	−0.505531
$143$	−58705.2	−0.240069
$144$	0	0
$145$	−73720.2	−0.291183
$146$	163902.	0.636360
$147$	0	0
$148$	20225.3	0.0758998
$149$	340511.	1.25651	0.628255	−	0.778008i	$-0.283770\pi$
$149$	340511.	1.25651	0.628255	+	0.778008i	$0.283770\pi$
$150$	0	0
$151$	478882.	1.70917	0.854587	−	0.519309i	$-0.173811\pi$
$151$	478882.	1.70917	0.854587	+	0.519309i	$0.173811\pi$
$152$	−22551.5	−0.0791712
$153$	0	0
$154$	4117.44	0.0139902
$155$	53646.7	0.179355
$156$	0	0
$157$	−242363.	−0.784726	−0.392363	−	0.919811i	$-0.628342\pi$
$157$	−242363.	−0.784726	−0.392363	+	0.919811i	$0.628342\pi$
$158$	151875.	0.483998
$159$	0	0
$160$	−147109.	−0.454298
$161$	−41385.4	−0.125829
$162$	0	0
$163$	559226.	1.64861	0.824305	−	0.566146i	$-0.191566\pi$
$163$	559226.	1.64861	0.824305	+	0.566146i	$0.191566\pi$
$164$	252835.	0.734054
$165$	0	0
$166$	−239125.	−0.673527
$167$	363650.	1.00900	0.504502	−	0.863411i	$-0.331676\pi$
$167$	363650.	1.00900	0.504502	+	0.863411i	$0.331676\pi$
$168$	0	0
$169$	−135906.	−0.366035
$170$	17612.5	0.0467409
$171$	0	0
$172$	−69134.7	−0.178187
$173$	−283260.	−0.719565	−0.359782	−	0.933036i	$-0.617149\pi$
$173$	−283260.	−0.719565	−0.359782	+	0.933036i	$0.617149\pi$
$174$	0	0
$175$	−8050.10	−0.0198704
$176$	48721.9	0.118561
$177$	0	0
$178$	−337798.	−0.799111
$179$	−458537.	−1.06965	−0.534826	−	0.844963i	$-0.679623\pi$
$179$	−458537.	−1.06965	−0.534826	+	0.844963i	$0.679623\pi$
$180$	0	0
$181$	221669.	0.502932	0.251466	−	0.967866i	$-0.419087\pi$
$181$	221669.	0.502932	0.251466	+	0.967866i	$0.419087\pi$
$182$	−16509.5	−0.0369449
$183$	0	0
$184$	484033.	1.05397
$185$	−20208.9	−0.0434124
$186$	0	0
$187$	−32266.0	−0.0674747
$188$	249536.	0.514919
$189$	0	0
$190$	9887.51	0.0198702
$191$	−422704.	−0.838403	−0.419201	−	0.907893i	$-0.637690\pi$
$191$	−422704.	−0.838403	−0.419201	+	0.907893i	$0.637690\pi$
$192$	0	0
$193$	602369.	1.16404	0.582022	−	0.813173i	$-0.302262\pi$
$193$	602369.	1.16404	0.582022	+	0.813173i	$0.302262\pi$
$194$	−349627.	−0.666962
$195$	0	0
$196$	416364.	0.774165
$197$	954629.	1.75255	0.876273	−	0.481816i	$-0.160023\pi$
$197$	954629.	1.75255	0.876273	+	0.481816i	$0.160023\pi$
$198$	0	0
$199$	163858.	0.293315	0.146658	−	0.989187i	$-0.453148\pi$
$199$	163858.	0.293315	0.146658	+	0.989187i	$0.453148\pi$
$200$	94151.9	0.166439
$201$	0	0
$202$	−124016.	−0.213846
$203$	37981.1	0.0646886
$204$	0	0
$205$	−252631.	−0.419857
$206$	−249604.	−0.409811
$207$	0	0
$208$	−195357.	−0.313092
$209$	−18113.9	−0.0286844
$210$	0	0
$211$	−258014.	−0.398967	−0.199483	−	0.979901i	$-0.563926\pi$
$211$	−258014.	−0.398967	−0.199483	+	0.979901i	$0.563926\pi$
$212$	178317.	0.272492
$213$	0	0
$214$	352951.	0.526842
$215$	69078.8	0.101917
$216$	0	0
$217$	−27639.2	−0.0398452
$218$	−48975.0	−0.0697964
$219$	0	0
$220$	−75686.2	−0.105429
$221$	129375.	0.178185
$222$	0	0
$223$	1.40571e6	1.89293	0.946466	−	0.322805i	$-0.104626\pi$
$223$	1.40571e6	1.89293	0.946466	+	0.322805i	$0.104626\pi$
$224$	75791.8	0.100926
$225$	0	0
$226$	−4398.91	−0.00572894
$227$	14869.8	0.0191532	0.00957661	−	0.999954i	$-0.496952\pi$
$227$	14869.8	0.0191532	0.00957661	+	0.999954i	$0.496952\pi$
$228$	0	0
$229$	1.26453e6	1.59345	0.796727	−	0.604340i	$-0.206563\pi$
$229$	1.26453e6	1.59345	0.796727	+	0.604340i	$0.206563\pi$
$230$	−212220.	−0.264525
$231$	0	0
$232$	−444218.	−0.541846
$233$	1.16476e6	1.40555	0.702775	−	0.711412i	$-0.251944\pi$
$233$	1.16476e6	1.40555	0.702775	+	0.711412i	$0.251944\pi$
$234$	0	0
$235$	−249334.	−0.294518
$236$	−834104.	−0.974856
$237$	0	0
$238$	−9074.05	−0.0103839
$239$	−285003.	−0.322742	−0.161371	−	0.986894i	$-0.551591\pi$
$239$	−285003.	−0.322742	−0.161371	+	0.986894i	$0.551591\pi$
$240$	0	0
$241$	−1.49416e6	−1.65712	−0.828562	−	0.559897i	$-0.810841\pi$
$241$	−1.49416e6	−1.65712	−0.828562	+	0.559897i	$0.810841\pi$
$242$	−38680.4	−0.0424573
$243$	0	0
$244$	297018.	0.319380
$245$	−416028.	−0.442799
$246$	0	0
$247$	72630.3	0.0757488
$248$	323260.	0.333752
$249$	0	0
$250$	−41280.0	−0.0417725
$251$	948891.	0.950675	0.475337	−	0.879804i	$-0.342326\pi$
$251$	948891.	0.950675	0.475337	+	0.879804i	$0.342326\pi$
$252$	0	0
$253$	388786.	0.381865
$254$	287370.	0.279484
$255$	0	0
$256$	−564051.	−0.537921
$257$	314180.	0.296719	0.148360	−	0.988933i	$-0.452601\pi$
$257$	314180.	0.296719	0.148360	+	0.988933i	$0.452601\pi$
$258$	0	0
$259$	10411.8	0.00964440
$260$	303475.	0.278413
$261$	0	0
$262$	−872890.	−0.785608
$263$	−177951.	−0.158639	−0.0793196	−	0.996849i	$-0.525275\pi$
$263$	−177951.	−0.158639	−0.0793196	+	0.996849i	$0.525275\pi$
$264$	0	0
$265$	−178173.	−0.155857
$266$	−5094.11	−0.00441433
$267$	0	0
$268$	−112606.	−0.0957685
$269$	544043.	0.458408	0.229204	−	0.973378i	$-0.426388\pi$
$269$	544043.	0.458408	0.229204	+	0.973378i	$0.426388\pi$
$270$	0	0
$271$	−801819.	−0.663213	−0.331606	−	0.943418i	$-0.607591\pi$
$271$	−801819.	−0.663213	−0.331606	+	0.943418i	$0.607591\pi$
$272$	−107374.	−0.0879987
$273$	0	0
$274$	88500.2	0.0712144
$275$	75625.0	0.0603023
$276$	0	0
$277$	−110444.	−0.0864857	−0.0432429	−	0.999065i	$-0.513769\pi$
$277$	−110444.	−0.0864857	−0.0432429	+	0.999065i	$0.513769\pi$
$278$	−624032.	−0.484278
$279$	0	0
$280$	−48507.7	−0.0369756
$281$	1.76988e6	1.33715	0.668573	−	0.743647i	$-0.266905\pi$
$281$	1.76988e6	1.33715	0.668573	+	0.743647i	$0.266905\pi$
$282$	0	0
$283$	−226526.	−0.168133	−0.0840665	−	0.996460i	$-0.526791\pi$
$283$	−226526.	−0.168133	−0.0840665	+	0.996460i	$0.526791\pi$
$284$	−1.15038e6	−0.846338
$285$	0	0
$286$	155095.	0.112120
$287$	130157.	0.0932744
$288$	0	0
$289$	−1.34875e6	−0.949919
$290$	194763.	0.135991
$291$	0	0
$292$	1.55223e6	1.06537
$293$	311510.	0.211984	0.105992	−	0.994367i	$-0.466198\pi$
$293$	311510.	0.211984	0.105992	+	0.994367i	$0.466198\pi$
$294$	0	0
$295$	833430.	0.557589
$296$	−121773.	−0.0807836
$297$	0	0
$298$	−899604.	−0.586828
$299$	−1.55889e6	−1.00841
$300$	0	0
$301$	−35589.8	−0.0226417
$302$	−1.26517e6	−0.798236
$303$	0	0
$304$	−60279.0	−0.0374095
$305$	−296778.	−0.182676
$306$	0	0
$307$	1.29267e6	0.782785	0.391393	−	0.920224i	$-0.371993\pi$
$307$	1.29267e6	0.782785	0.391393	+	0.920224i	$0.371993\pi$
$308$	38994.1	0.0234219
$309$	0	0
$310$	−141731.	−0.0837644
$311$	−767063.	−0.449707	−0.224854	−	0.974393i	$-0.572190\pi$
$311$	−767063.	−0.449707	−0.224854	+	0.974393i	$0.572190\pi$
$312$	0	0
$313$	766627.	0.442307	0.221153	−	0.975239i	$-0.429018\pi$
$313$	766627.	0.442307	0.221153	+	0.975239i	$0.429018\pi$
$314$	640305.	0.366491
$315$	0	0
$316$	1.43833e6	0.810289
$317$	−1.79729e6	−1.00455	−0.502273	−	0.864709i	$-0.667503\pi$
$317$	−1.79729e6	−1.00455	−0.502273	+	0.864709i	$0.667503\pi$
$318$	0	0
$319$	−356806.	−0.196316
$320$	66523.4	0.0363161
$321$	0	0
$322$	109337.	0.0587662
$323$	39919.6	0.0212902
$324$	0	0
$325$	−303229.	−0.159244
$326$	−1.47743e6	−0.769951
$327$	0	0
$328$	−1.52228e6	−0.781287
$329$	128459.	0.0654295
$330$	0	0
$331$	−1.49238e6	−0.748702	−0.374351	−	0.927287i	$-0.622134\pi$
$331$	−1.49238e6	−0.748702	−0.374351	+	0.927287i	$0.622134\pi$
$332$	−2.26463e6	−1.12759
$333$	0	0
$334$	−960736.	−0.471235
$335$	112514.	0.0547767
$336$	0	0
$337$	−1.30198e6	−0.624498	−0.312249	−	0.950000i	$-0.601082\pi$
$337$	−1.30198e6	−0.624498	−0.312249	+	0.950000i	$0.601082\pi$
$338$	359054.	0.170950
$339$	0	0
$340$	166798.	0.0782517
$341$	259650.	0.120921
$342$	0	0
$343$	430817.	0.197723
$344$	416250.	0.189652
$345$	0	0
$346$	748351.	0.336059
$347$	−2.56747e6	−1.14467	−0.572337	−	0.820019i	$-0.693963\pi$
$347$	−2.56747e6	−1.14467	−0.572337	+	0.820019i	$0.693963\pi$
$348$	0	0
$349$	−4.02595e6	−1.76931	−0.884657	−	0.466243i	$-0.845607\pi$
$349$	−4.02595e6	−1.76931	−0.884657	+	0.466243i	$0.845607\pi$
$350$	21267.8	0.00928008
$351$	0	0
$352$	−712010.	−0.306287
$353$	−1.30479e6	−0.557317	−0.278659	−	0.960390i	$-0.589890\pi$
$353$	−1.30479e6	−0.557317	−0.278659	+	0.960390i	$0.589890\pi$
$354$	0	0
$355$	1.14945e6	0.484080
$356$	−3.19911e6	−1.33784
$357$	0	0
$358$	1.21142e6	0.499560
$359$	−2.41823e6	−0.990289	−0.495145	−	0.868811i	$-0.664885\pi$
$359$	−2.41823e6	−0.990289	−0.495145	+	0.868811i	$0.664885\pi$
$360$	0	0
$361$	−2.45369e6	−0.990949
$362$	−585633.	−0.234884
$363$	0	0
$364$	−156352.	−0.0618515
$365$	−1.55098e6	−0.609359
$366$	0	0
$367$	391486.	0.151723	0.0758614	−	0.997118i	$-0.475829\pi$
$367$	391486.	0.151723	0.0758614	+	0.997118i	$0.475829\pi$
$368$	1.29379e6	0.498018
$369$	0	0
$370$	53390.4	0.0202749
$371$	91795.9	0.0346249
$372$	0	0
$373$	2.56193e6	0.953444	0.476722	−	0.879054i	$-0.341825\pi$
$373$	2.56193e6	0.953444	0.476722	+	0.879054i	$0.341825\pi$
$374$	85244.3	0.0315127
$375$	0	0
$376$	−1.50242e6	−0.548051
$377$	1.43066e6	0.518423
$378$	0	0
$379$	−4.57869e6	−1.63736	−0.818678	−	0.574253i	$-0.805293\pi$
$379$	−4.57869e6	−1.63736	−0.818678	+	0.574253i	$0.805293\pi$
$380$	93639.4	0.0332659
$381$	0	0
$382$	1.11675e6	0.391560
$383$	1.12758e6	0.392782	0.196391	−	0.980526i	$-0.437078\pi$
$383$	1.12758e6	0.392782	0.196391	+	0.980526i	$0.437078\pi$
$384$	0	0
$385$	−38962.5	−0.0133966
$386$	−1.59141e6	−0.543644
$387$	0	0
$388$	−3.31114e6	−1.11660
$389$	329255.	0.110321	0.0551604	−	0.998478i	$-0.482433\pi$
$389$	329255.	0.110321	0.0551604	+	0.998478i	$0.482433\pi$
$390$	0	0
$391$	−856811.	−0.283428
$392$	−2.50687e6	−0.823979
$393$	0	0
$394$	−2.52206e6	−0.818492
$395$	−1.43716e6	−0.463461
$396$	0	0
$397$	−2.54236e6	−0.809582	−0.404791	−	0.914409i	$-0.632656\pi$
$397$	−2.54236e6	−0.809582	−0.404791	+	0.914409i	$0.632656\pi$
$398$	−432900.	−0.136987
$399$	0	0
$400$	251663.	0.0786446
$401$	4.91799e6	1.52731	0.763655	−	0.645625i	$-0.223403\pi$
$401$	4.91799e6	1.52731	0.763655	+	0.645625i	$0.223403\pi$
$402$	0	0
$403$	−1.04110e6	−0.319324
$404$	−1.17449e6	−0.358012
$405$	0	0
$406$	−100343.	−0.0302116
$407$	−97811.2	−0.0292686
$408$	0	0
$409$	−248717.	−0.0735186	−0.0367593	−	0.999324i	$-0.511703\pi$
$409$	−248717.	−0.0735186	−0.0367593	+	0.999324i	$0.511703\pi$
$410$	667431.	0.196086
$411$	0	0
$412$	−2.36387e6	−0.686089
$413$	−429388.	−0.123873
$414$	0	0
$415$	2.26279e6	0.644948
$416$	2.85490e6	0.808832
$417$	0	0
$418$	47855.6	0.0133965
$419$	−4.46344e6	−1.24204	−0.621019	−	0.783795i	$-0.713281\pi$
$419$	−4.46344e6	−1.24204	−0.621019	+	0.783795i	$0.713281\pi$
$420$	0	0
$421$	3.23943e6	0.890766	0.445383	−	0.895340i	$-0.353067\pi$
$421$	3.23943e6	0.890766	0.445383	+	0.895340i	$0.353067\pi$
$422$	681652.	0.186330
$423$	0	0
$424$	−1.07362e6	−0.290026
$425$	−166663.	−0.0447577
$426$	0	0
$427$	152902.	0.0405829
$428$	3.34261e6	0.882017
$429$	0	0
$430$	−182501.	−0.0475985
$431$	534445.	0.138583	0.0692914	−	0.997596i	$-0.477926\pi$
$431$	534445.	0.138583	0.0692914	+	0.997596i	$0.477926\pi$
$432$	0	0
$433$	4.32701e6	1.10909	0.554547	−	0.832152i	$-0.312891\pi$
$433$	4.32701e6	1.10909	0.554547	+	0.832152i	$0.312891\pi$
$434$	73020.5	0.0186089
$435$	0	0
$436$	−463816.	−0.116850
$437$	−481008.	−0.120489
$438$	0	0
$439$	5.48083e6	1.35733	0.678665	−	0.734448i	$-0.262559\pi$
$439$	5.48083e6	1.35733	0.678665	+	0.734448i	$0.262559\pi$
$440$	455695.	0.112213
$441$	0	0
$442$	−341799.	−0.0832176
$443$	1.46278e6	0.354136	0.177068	−	0.984199i	$-0.443339\pi$
$443$	1.46278e6	0.354136	0.177068	+	0.984199i	$0.443339\pi$
$444$	0	0
$445$	3.19652e6	0.765203
$446$	−3.71379e6	−0.884056
$447$	0	0
$448$	−34273.3	−0.00806791
$449$	3.91887e6	0.917371	0.458685	−	0.888599i	$-0.348320\pi$
$449$	3.91887e6	0.917371	0.458685	+	0.888599i	$0.348320\pi$
$450$	0	0
$451$	−1.22273e6	−0.283067
$452$	−41659.7	−0.00959115
$453$	0	0
$454$	−39285.0	−0.00894513
$455$	156226.	0.0353772
$456$	0	0
$457$	8.82515e6	1.97666	0.988329	−	0.152334i	$-0.0486790\pi$
$457$	8.82515e6	1.97666	0.988329	+	0.152334i	$0.0486790\pi$
$458$	−3.34078e6	−0.744191
$459$	0	0
$460$	−2.00982e6	−0.442856
$461$	−4.72413e6	−1.03531	−0.517654	−	0.855590i	$-0.673194\pi$
$461$	−4.72413e6	−1.03531	−0.517654	+	0.855590i	$0.673194\pi$
$462$	0	0
$463$	6.03855e6	1.30912	0.654562	−	0.756009i	$-0.272853\pi$
$463$	6.03855e6	1.30912	0.654562	+	0.756009i	$0.272853\pi$
$464$	−1.18737e6	−0.256030
$465$	0	0
$466$	−3.07720e6	−0.656435
$467$	2.30982e6	0.490101	0.245051	−	0.969510i	$-0.421195\pi$
$467$	2.30982e6	0.490101	0.245051	+	0.969510i	$0.421195\pi$
$468$	0	0
$469$	−57968.2	−0.0121691
$470$	658721.	0.137549
$471$	0	0
$472$	5.02202e6	1.03758
$473$	334341.	0.0687127
$474$	0	0
$475$	−93563.6	−0.0190271
$476$	−85935.5	−0.0173842
$477$	0	0
$478$	752956.	0.150730
$479$	−2.56648e6	−0.511093	−0.255546	−	0.966797i	$-0.582255\pi$
$479$	−2.56648e6	−0.511093	−0.255546	+	0.966797i	$0.582255\pi$
$480$	0	0
$481$	392188.	0.0772915
$482$	3.94746e6	0.773928
$483$	0	0
$484$	−366321.	−0.0710802
$485$	3.30846e6	0.638662
$486$	0	0
$487$	−2.72042e6	−0.519772	−0.259886	−	0.965639i	$-0.583685\pi$
$487$	−2.72042e6	−0.519772	−0.259886	+	0.965639i	$0.583685\pi$
$488$	−1.78830e6	−0.339931
$489$	0	0
$490$	1.09911e6	0.206801
$491$	4.27300e6	0.799888	0.399944	−	0.916540i	$-0.369030\pi$
$491$	4.27300e6	0.799888	0.399944	+	0.916540i	$0.369030\pi$
$492$	0	0
$493$	786332.	0.145710
$494$	−191884.	−0.0353770
$495$	0	0
$496$	864057.	0.157702
$497$	−592202.	−0.107542
$498$	0	0
$499$	−635246.	−0.114206	−0.0571032	−	0.998368i	$-0.518186\pi$
$499$	−635246.	−0.114206	−0.0571032	+	0.998368i	$0.518186\pi$
$500$	−390941.	−0.0699337
$501$	0	0
$502$	−2.50690e6	−0.443994
$503$	6.61468e6	1.16571	0.582853	−	0.812578i	$-0.301936\pi$
$503$	6.61468e6	1.16571	0.582853	+	0.812578i	$0.301936\pi$
$504$	0	0
$505$	1.17354e6	0.204772
$506$	−1.02714e6	−0.178342
$507$	0	0
$508$	2.72153e6	0.467900
$509$	−5.32298e6	−0.910668	−0.455334	−	0.890321i	$-0.650480\pi$
$509$	−5.32298e6	−0.910668	−0.455334	+	0.890321i	$0.650480\pi$
$510$	0	0
$511$	799073.	0.135374
$512$	−4.31046e6	−0.726689
$513$	0	0
$514$	−830039.	−0.138577
$515$	2.36196e6	0.392422
$516$	0	0
$517$	−1.20678e6	−0.198564
$518$	−27507.1	−0.00450423
$519$	0	0
$520$	−1.82718e6	−0.296327
$521$	−2.82964e6	−0.456706	−0.228353	−	0.973578i	$-0.573334\pi$
$521$	−2.82964e6	−0.456706	−0.228353	+	0.973578i	$0.573334\pi$
$522$	0	0
$523$	2.89624e6	0.462999	0.231499	−	0.972835i	$-0.425637\pi$
$523$	2.89624e6	0.462999	0.231499	+	0.972835i	$0.425637\pi$
$524$	−8.26667e6	−1.31523
$525$	0	0
$526$	470132.	0.0740893
$527$	−572220.	−0.0897504
$528$	0	0
$529$	3.88773e6	0.604027
$530$	470719.	0.0727901
$531$	0	0
$532$	−48243.6	−0.00739028
$533$	4.90272e6	0.747513
$534$	0	0
$535$	−3.33991e6	−0.504487
$536$	677981.	0.101931
$537$	0	0
$538$	−1.43732e6	−0.214090
$539$	−2.01357e6	−0.298535
$540$	0	0
$541$	−6.89422e6	−1.01273	−0.506363	−	0.862320i	$-0.669010\pi$
$541$	−6.89422e6	−1.01273	−0.506363	+	0.862320i	$0.669010\pi$
$542$	2.11834e6	0.309741
$543$	0	0
$544$	1.56913e6	0.227333
$545$	463441.	0.0668349
$546$	0	0
$547$	−1.10997e7	−1.58615	−0.793075	−	0.609125i	$-0.791521\pi$
$547$	−1.10997e7	−1.58615	−0.793075	+	0.609125i	$0.791521\pi$
$548$	838138.	0.119224
$549$	0	0
$550$	−199795.	−0.0281630
$551$	441442.	0.0619433
$552$	0	0
$553$	740436.	0.102962
$554$	291786.	0.0403915
$555$	0	0
$556$	−5.90988e6	−0.810758
$557$	1.33678e7	1.82567	0.912836	−	0.408327i	$-0.133888\pi$
$557$	1.33678e7	1.82567	0.912836	+	0.408327i	$0.133888\pi$
$558$	0	0
$559$	−1.34059e6	−0.181454
$560$	−129658.	−0.0174715
$561$	0	0
$562$	−4.67589e6	−0.624488
$563$	−7.33302e6	−0.975017	−0.487508	−	0.873118i	$-0.662094\pi$
$563$	−7.33302e6	−0.975017	−0.487508	+	0.873118i	$0.662094\pi$
$564$	0	0
$565$	41626.0	0.00548585
$566$	598465.	0.0785232
$567$	0	0
$568$	6.92624e6	0.900796
$569$	1.13993e7	1.47603	0.738016	−	0.674784i	$-0.235763\pi$
$569$	1.13993e7	1.47603	0.738016	+	0.674784i	$0.235763\pi$
$570$	0	0
$571$	4.39289e6	0.563845	0.281923	−	0.959437i	$-0.409028\pi$
$571$	4.39289e6	0.563845	0.281923	+	0.959437i	$0.409028\pi$
$572$	1.46882e6	0.187706
$573$	0	0
$574$	−343865.	−0.0435620
$575$	2.00819e6	0.253300
$576$	0	0
$577$	6.17106e6	0.771650	0.385825	−	0.922572i	$-0.373917\pi$
$577$	6.17106e6	0.771650	0.385825	+	0.922572i	$0.373917\pi$
$578$	3.56329e6	0.443641
$579$	0	0
$580$	1.84450e6	0.227671
$581$	−1.16581e6	−0.143280
$582$	0	0
$583$	−862357.	−0.105079
$584$	−9.34576e6	−1.13392
$585$	0	0
$586$	−822986.	−0.0990030
$587$	1.33897e7	1.60389	0.801947	−	0.597395i	$-0.203798\pi$
$587$	1.33897e7	1.60389	0.801947	+	0.597395i	$0.203798\pi$
$588$	0	0
$589$	−321240.	−0.0381542
$590$	−2.20186e6	−0.260411
$591$	0	0
$592$	−325493.	−0.0381714
$593$	5.58517e6	0.652228	0.326114	−	0.945330i	$-0.394261\pi$
$593$	5.58517e6	0.652228	0.326114	+	0.945330i	$0.394261\pi$
$594$	0	0
$595$	85866.0	0.00994326
$596$	−8.51967e6	−0.982443
$597$	0	0
$598$	4.11848e6	0.470960
$599$	1.01847e7	1.15980	0.579899	−	0.814688i	$-0.303092\pi$
$599$	1.01847e7	1.15980	0.579899	+	0.814688i	$0.303092\pi$
$600$	0	0
$601$	−1.25625e7	−1.41870	−0.709350	−	0.704856i	$-0.751012\pi$
$601$	−1.25625e7	−1.41870	−0.709350	+	0.704856i	$0.751012\pi$
$602$	94025.6	0.0105744
$603$	0	0
$604$	−1.19817e7	−1.33637
$605$	366025.	0.0406558
$606$	0	0
$607$	−6.03337e6	−0.664643	−0.332321	−	0.943166i	$-0.607832\pi$
$607$	−6.03337e6	−0.664643	−0.332321	+	0.943166i	$0.607832\pi$
$608$	880901.	0.0966425
$609$	0	0
$610$	784064.	0.0853153
$611$	4.83874e6	0.524360
$612$	0	0
$613$	1.40171e7	1.50663	0.753314	−	0.657661i	$-0.228454\pi$
$613$	1.40171e7	1.50663	0.753314	+	0.657661i	$0.228454\pi$
$614$	−3.41514e6	−0.365585
$615$	0	0
$616$	−234777.	−0.0249290
$617$	1.32807e7	1.40446	0.702228	−	0.711952i	$-0.252189\pi$
$617$	1.32807e7	1.40446	0.702228	+	0.711952i	$0.252189\pi$
$618$	0	0
$619$	9.12137e6	0.956827	0.478414	−	0.878135i	$-0.341212\pi$
$619$	9.12137e6	0.956827	0.478414	+	0.878135i	$0.341212\pi$
$620$	−1.34225e6	−0.140235
$621$	0	0
$622$	2.02652e6	0.210027
$623$	−1.64687e6	−0.169996
$624$	0	0
$625$	390625.	0.0400000
$626$	−2.02537e6	−0.206571
$627$	0	0
$628$	6.06399e6	0.613563
$629$	215557.	0.0217238
$630$	0	0
$631$	−4.93986e6	−0.493902	−0.246951	−	0.969028i	$-0.579429\pi$
$631$	−4.93986e6	−0.493902	−0.246951	+	0.969028i	$0.579429\pi$
$632$	−8.65995e6	−0.862428
$633$	0	0
$634$	4.74830e6	0.469153
$635$	−2.71933e6	−0.267625
$636$	0	0
$637$	8.07371e6	0.788360
$638$	942653.	0.0916854
$639$	0	0
$640$	4.53175e6	0.437337
$641$	−1.61757e7	−1.55496	−0.777480	−	0.628908i	$-0.783502\pi$
$641$	−1.61757e7	−1.55496	−0.777480	+	0.628908i	$0.783502\pi$
$642$	0	0
$643$	3.69641e6	0.352576	0.176288	−	0.984339i	$-0.443591\pi$
$643$	3.69641e6	0.352576	0.176288	+	0.984339i	$0.443591\pi$
$644$	1.03547e6	0.0983839
$645$	0	0
$646$	−105465.	−0.00994318
$647$	−9.88638e6	−0.928489	−0.464244	−	0.885707i	$-0.653674\pi$
$647$	−9.88638e6	−0.928489	−0.464244	+	0.885707i	$0.653674\pi$
$648$	0	0
$649$	4.03380e6	0.375926
$650$	801108.	0.0743717
$651$	0	0
$652$	−1.39920e7	−1.28902
$653$	1.84374e7	1.69207	0.846033	−	0.533130i	$-0.178984\pi$
$653$	1.84374e7	1.69207	0.846033	+	0.533130i	$0.178984\pi$
$654$	0	0
$655$	8.25998e6	0.752274
$656$	−4.06897e6	−0.369169
$657$	0	0
$658$	−339378.	−0.0305575
$659$	1.15450e7	1.03557	0.517787	−	0.855509i	$-0.326756\pi$
$659$	1.15450e7	1.03557	0.517787	+	0.855509i	$0.326756\pi$
$660$	0	0
$661$	9.62805e6	0.857107	0.428553	−	0.903517i	$-0.359023\pi$
$661$	9.62805e6	0.857107	0.428553	+	0.903517i	$0.359023\pi$
$662$	3.94275e6	0.349666
$663$	0	0
$664$	1.36350e7	1.20015
$665$	48204.6	0.00422702
$666$	0	0
$667$	−9.47484e6	−0.824627
$668$	−9.09862e6	−0.788922
$669$	0	0
$670$	−297254.	−0.0255824
$671$	−1.43640e6	−0.123160
$672$	0	0
$673$	−1.38177e7	−1.17597	−0.587986	−	0.808871i	$-0.700079\pi$
$673$	−1.38177e7	−1.17597	−0.587986	+	0.808871i	$0.700079\pi$
$674$	3.43974e6	0.291659
$675$	0	0
$676$	3.40041e6	0.286196
$677$	−1.93156e7	−1.61971	−0.809855	−	0.586630i	$-0.800454\pi$
$677$	−1.93156e7	−1.61971	−0.809855	+	0.586630i	$0.800454\pi$
$678$	0	0
$679$	−1.70454e6	−0.141884
$680$	−1.00427e6	−0.0832869
$681$	0	0
$682$	−685976.	−0.0564739
$683$	8.61517e6	0.706663	0.353331	−	0.935498i	$-0.385049\pi$
$683$	8.61517e6	0.706663	0.353331	+	0.935498i	$0.385049\pi$
$684$	0	0
$685$	−837460.	−0.0681927
$686$	−1.13819e6	−0.0923428
$687$	0	0
$688$	1.11261e6	0.0896133
$689$	3.45774e6	0.277488
$690$	0	0
$691$	−4.86964e6	−0.387973	−0.193987	−	0.981004i	$-0.562142\pi$
$691$	−4.86964e6	−0.387973	−0.193987	+	0.981004i	$0.562142\pi$
$692$	7.08723e6	0.562615
$693$	0	0
$694$	6.78306e6	0.534597
$695$	5.90509e6	0.463730
$696$	0	0
$697$	2.69467e6	0.210099
$698$	1.06362e7	0.826323
$699$	0	0
$700$	201416.	0.0155363
$701$	−7.80927e6	−0.600227	−0.300113	−	0.953904i	$-0.597025\pi$
$701$	−7.80927e6	−0.600227	−0.300113	+	0.953904i	$0.597025\pi$
$702$	0	0
$703$	121012.	0.00923511
$704$	321973.	0.0244843
$705$	0	0
$706$	3.44714e6	0.260284
$707$	−604617.	−0.0454917
$708$	0	0
$709$	−1.31455e7	−0.982116	−0.491058	−	0.871127i	$-0.663390\pi$
$709$	−1.31455e7	−0.982116	−0.491058	+	0.871127i	$0.663390\pi$
$710$	−3.03675e6	−0.226080
$711$	0	0
$712$	1.92613e7	1.42392
$713$	6.89491e6	0.507931
$714$	0	0
$715$	−1.46763e6	−0.107362
$716$	1.14727e7	0.836342
$717$	0	0
$718$	6.38879e6	0.462495
$719$	8.12978e6	0.586485	0.293242	−	0.956038i	$-0.405266\pi$
$719$	8.12978e6	0.586485	0.293242	+	0.956038i	$0.405266\pi$
$720$	0	0
$721$	−1.21690e6	−0.0871796
$722$	6.48246e6	0.462803
$723$	0	0
$724$	−5.54622e6	−0.393233
$725$	−1.84300e6	−0.130221
$726$	0	0
$727$	−2.76931e7	−1.94328	−0.971641	−	0.236461i	$-0.924013\pi$
$727$	−2.76931e7	−1.94328	−0.971641	+	0.236461i	$0.924013\pi$
$728$	941373.	0.0658314
$729$	0	0
$730$	4.09756e6	0.284589
$731$	−736825.	−0.0510001
$732$	0	0
$733$	2.33400e7	1.60451	0.802254	−	0.596983i	$-0.203634\pi$
$733$	2.33400e7	1.60451	0.802254	+	0.596983i	$0.203634\pi$
$734$	−1.03428e6	−0.0708592
$735$	0	0
$736$	−1.89071e7	−1.28656
$737$	544570.	0.0369305
$738$	0	0
$739$	−9.85790e6	−0.664008	−0.332004	−	0.943278i	$-0.607725\pi$
$739$	−9.85790e6	−0.664008	−0.332004	+	0.943278i	$0.607725\pi$
$740$	505632.	0.0339434
$741$	0	0
$742$	−242518.	−0.0161709
$743$	2.47978e7	1.64794	0.823968	−	0.566636i	$-0.191755\pi$
$743$	2.47978e7	1.64794	0.823968	+	0.566636i	$0.191755\pi$
$744$	0	0
$745$	8.51278e6	0.561928
$746$	−6.76842e6	−0.445287
$747$	0	0
$748$	807303.	0.0527573
$749$	1.72074e6	0.112076
$750$	0	0
$751$	−7.05889e6	−0.456706	−0.228353	−	0.973578i	$-0.573334\pi$
$751$	−7.05889e6	−0.456706	−0.228353	+	0.973578i	$0.573334\pi$
$752$	−4.01588e6	−0.258962
$753$	0	0
$754$	−3.77970e6	−0.242119
$755$	1.19720e7	0.764365
$756$	0	0
$757$	2.09440e7	1.32837	0.664187	−	0.747567i	$-0.268778\pi$
$757$	2.09440e7	1.32837	0.664187	+	0.747567i	$0.268778\pi$
$758$	1.20965e7	0.764695
$759$	0	0
$760$	−563788.	−0.0354064
$761$	1.85362e7	1.16027	0.580135	−	0.814521i	$-0.303000\pi$
$761$	1.85362e7	1.16027	0.580135	+	0.814521i	$0.303000\pi$
$762$	0	0
$763$	−238768.	−0.0148479
$764$	1.05762e7	0.655533
$765$	0	0
$766$	−2.97899e6	−0.183441
$767$	−1.61741e7	−0.992731
$768$	0	0
$769$	−2.68455e7	−1.63703	−0.818514	−	0.574487i	$-0.805202\pi$
$769$	−2.68455e7	−1.63703	−0.818514	+	0.574487i	$0.805202\pi$
$770$	102936.	0.00625663
$771$	0	0
$772$	−1.50714e7	−0.910145
$773$	−1.48501e7	−0.893883	−0.446941	−	0.894563i	$-0.647487\pi$
$773$	−1.48501e7	−0.893883	−0.446941	+	0.894563i	$0.647487\pi$
$774$	0	0
$775$	1.34117e6	0.0802101
$776$	1.99358e7	1.18845
$777$	0	0
$778$	−869865.	−0.0515232
$779$	1.51277e6	0.0893160
$780$	0	0
$781$	5.56332e6	0.326367
$782$	2.26363e6	0.132370
$783$	0	0
$784$	−6.70071e6	−0.389342
$785$	−6.05908e6	−0.350940
$786$	0	0
$787$	1.34957e7	0.776709	0.388355	−	0.921510i	$-0.373044\pi$
$787$	1.34957e7	0.776709	0.388355	+	0.921510i	$0.373044\pi$
$788$	−2.38851e7	−1.37028
$789$	0	0
$790$	3.79688e6	0.216451
$791$	−21446.0	−0.00121872
$792$	0	0
$793$	5.75947e6	0.325237
$794$	6.71672e6	0.378099
$795$	0	0
$796$	−4.09976e6	−0.229338
$797$	−2.93737e7	−1.63800	−0.818998	−	0.573796i	$-0.805470\pi$
$797$	−2.93737e7	−1.63800	−0.818998	+	0.573796i	$0.805470\pi$
$798$	0	0
$799$	2.65951e6	0.147379
$800$	−3.67774e6	−0.203168
$801$	0	0
$802$	−1.29930e7	−0.713300
$803$	−7.50673e6	−0.410830
$804$	0	0
$805$	−1.03463e6	−0.0562726
$806$	2.75052e6	0.149134
$807$	0	0
$808$	7.07145e6	0.381048
$809$	2.10996e7	1.13345	0.566727	−	0.823906i	$-0.308209\pi$
$809$	2.10996e7	1.13345	0.566727	+	0.823906i	$0.308209\pi$
$810$	0	0
$811$	461523.	0.0246400	0.0123200	−	0.999924i	$-0.496078\pi$
$811$	461523.	0.0246400	0.0123200	+	0.999924i	$0.496078\pi$
$812$	−950297.	−0.0505789
$813$	0	0
$814$	258410.	0.0136693
$815$	1.39806e7	0.737281
$816$	0	0
$817$	−413648.	−0.0216808
$818$	657091.	0.0343354
$819$	0	0
$820$	6.32088e6	0.328279
$821$	−1.54841e7	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$821$	−1.54841e7	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$822$	0	0
$823$	2.96111e7	1.52389	0.761947	−	0.647640i	$-0.224244\pi$
$823$	2.96111e7	1.52389	0.761947	+	0.647640i	$0.224244\pi$
$824$	1.42325e7	0.730235
$825$	0	0
$826$	1.13441e6	0.0578523
$827$	−2.50220e7	−1.27221	−0.636104	−	0.771604i	$-0.719455\pi$
$827$	−2.50220e7	−1.27221	−0.636104	+	0.771604i	$0.719455\pi$
$828$	0	0
$829$	2.74330e7	1.38640	0.693198	−	0.720747i	$-0.256201\pi$
$829$	2.74330e7	1.38640	0.693198	+	0.720747i	$0.256201\pi$
$830$	−5.97813e6	−0.301210
$831$	0	0
$832$	−1.29100e6	−0.0646572
$833$	4.43753e6	0.221579
$834$	0	0
$835$	9.09125e6	0.451240
$836$	453214.	0.0224279
$837$	0	0
$838$	1.17921e7	0.580070
$839$	−2.22692e7	−1.09219	−0.546097	−	0.837722i	$-0.683887\pi$
$839$	−2.22692e7	−1.09219	−0.546097	+	0.837722i	$0.683887\pi$
$840$	0	0
$841$	−1.18157e7	−0.576061
$842$	−8.55833e6	−0.416015
$843$	0	0
$844$	6.45556e6	0.311945
$845$	−3.39766e6	−0.163696
$846$	0	0
$847$	−188579.	−0.00903200
$848$	−2.86973e6	−0.137041
$849$	0	0
$850$	440311.	0.0209032
$851$	−2.59734e6	−0.122943
$852$	0	0
$853$	−1.78183e7	−0.838483	−0.419242	−	0.907875i	$-0.637704\pi$
$853$	−1.78183e7	−0.838483	−0.419242	+	0.907875i	$0.637704\pi$
$854$	−403955.	−0.0189534
$855$	0	0
$856$	−2.01254e7	−0.938771
$857$	−3.54981e7	−1.65102	−0.825511	−	0.564386i	$-0.809113\pi$
$857$	−3.54981e7	−1.65102	−0.825511	+	0.564386i	$0.809113\pi$
$858$	0	0
$859$	3.07818e7	1.42335	0.711674	−	0.702510i	$-0.247937\pi$
$859$	3.07818e7	1.42335	0.711674	+	0.702510i	$0.247937\pi$
$860$	−1.72837e6	−0.0796875
$861$	0	0
$862$	−1.41196e6	−0.0647224
$863$	−2.39799e6	−0.109602	−0.0548012	−	0.998497i	$-0.517452\pi$
$863$	−2.39799e6	−0.109602	−0.0548012	+	0.998497i	$0.517452\pi$
$864$	0	0
$865$	−7.08150e6	−0.321799
$866$	−1.14316e7	−0.517981
$867$	0	0
$868$	691538.	0.0311542
$869$	−6.95587e6	−0.312466
$870$	0	0
$871$	−2.18353e6	−0.0975245
$872$	2.79257e6	0.124369
$873$	0	0
$874$	1.27079e6	0.0562722
$875$	−201253.	−0.00888631
$876$	0	0
$877$	1.30020e7	0.570837	0.285419	−	0.958403i	$-0.407867\pi$
$877$	1.30020e7	0.570837	0.285419	+	0.958403i	$0.407867\pi$
$878$	−1.44799e7	−0.633914
$879$	0	0
$880$	1.21805e6	0.0530222
$881$	1.63845e7	0.711204	0.355602	−	0.934638i	$-0.384276\pi$
$881$	1.63845e7	0.711204	0.355602	+	0.934638i	$0.384276\pi$
$882$	0	0
$883$	1.48150e7	0.639441	0.319720	−	0.947512i	$-0.396411\pi$
$883$	1.48150e7	0.639441	0.319720	+	0.947512i	$0.396411\pi$
$884$	−3.23700e6	−0.139319
$885$	0	0
$886$	−3.86456e6	−0.165392
$887$	3.12806e7	1.33495	0.667477	−	0.744631i	$-0.267374\pi$
$887$	3.12806e7	1.33495	0.667477	+	0.744631i	$0.267374\pi$
$888$	0	0
$889$	1.40101e6	0.0594550
$890$	−8.44495e6	−0.357373
$891$	0	0
$892$	−3.51713e7	−1.48005
$893$	1.49303e6	0.0626527
$894$	0	0
$895$	−1.14634e7	−0.478362
$896$	−2.33479e6	−0.0971578
$897$	0	0
$898$	−1.03534e7	−0.428440
$899$	−6.32776e6	−0.261126
$900$	0	0
$901$	1.90047e6	0.0779919
$902$	3.23036e6	0.132201
$903$	0	0
$904$	250827.	0.0102083
$905$	5.54173e6	0.224918
$906$	0	0
$907$	1.11714e6	0.0450908	0.0225454	−	0.999746i	$-0.492823\pi$
$907$	1.11714e6	0.0450908	0.0225454	+	0.999746i	$0.492823\pi$
$908$	−372047.	−0.0149756
$909$	0	0
$910$	−412736.	−0.0165222
$911$	−3.00518e7	−1.19970	−0.599852	−	0.800111i	$-0.704774\pi$
$911$	−3.00518e7	−1.19970	−0.599852	+	0.800111i	$0.704774\pi$
$912$	0	0
$913$	1.09519e7	0.434824
$914$	−2.33154e7	−0.923159
$915$	0	0
$916$	−3.16388e7	−1.24589
$917$	−4.25560e6	−0.167123
$918$	0	0
$919$	−4.11066e7	−1.60555	−0.802773	−	0.596285i	$-0.796643\pi$
$919$	−4.11066e7	−1.60555	−0.802773	+	0.596285i	$0.796643\pi$
$920$	1.21008e7	0.471352
$921$	0	0
$922$	1.24808e7	0.483520
$923$	−2.23069e7	−0.861857
$924$	0	0
$925$	−505223.	−0.0194146
$926$	−1.59534e7	−0.611400
$927$	0	0
$928$	1.73519e7	0.661420
$929$	2.06090e6	0.0783463	0.0391732	−	0.999232i	$-0.487528\pi$
$929$	2.06090e6	0.0783463	0.0391732	+	0.999232i	$0.487528\pi$
$930$	0	0
$931$	2.49120e6	0.0941965
$932$	−2.91426e7	−1.09898
$933$	0	0
$934$	−6.10237e6	−0.228892
$935$	−806650.	−0.0301756
$936$	0	0
$937$	−3.48612e7	−1.29716	−0.648580	−	0.761146i	$-0.724637\pi$
$937$	−3.48612e7	−1.29716	−0.648580	+	0.761146i	$0.724637\pi$
$938$	153147.	0.00568333
$939$	0	0
$940$	6.23840e6	0.230279
$941$	−2.80758e7	−1.03361	−0.516806	−	0.856103i	$-0.672879\pi$
$941$	−2.80758e7	−1.03361	−0.516806	+	0.856103i	$0.672879\pi$
$942$	0	0
$943$	−3.24692e7	−1.18903
$944$	1.34236e7	0.490273
$945$	0	0
$946$	−883304.	−0.0320909
$947$	4.71654e7	1.70903	0.854513	−	0.519430i	$-0.173856\pi$
$947$	4.71654e7	1.70903	0.854513	+	0.519430i	$0.173856\pi$
$948$	0	0
$949$	3.00993e7	1.08490
$950$	247188.	0.00888624
$951$	0	0
$952$	517405.	0.0185028
$953$	4.10589e7	1.46445	0.732226	−	0.681062i	$-0.238482\pi$
$953$	4.10589e7	1.46445	0.732226	+	0.681062i	$0.238482\pi$
$954$	0	0
$955$	−1.05676e7	−0.374945
$956$	7.13085e6	0.252346
$957$	0	0
$958$	6.78045e6	0.238696
$959$	431465.	0.0151495
$960$	0	0
$961$	−2.40244e7	−0.839158
$962$	−1.03613e6	−0.0360975
$963$	0	0
$964$	3.73843e7	1.29568
$965$	1.50592e7	0.520576
$966$	0	0
$967$	−1.28295e7	−0.441209	−0.220605	−	0.975363i	$-0.570803\pi$
$967$	−1.28295e7	−0.441209	−0.220605	+	0.975363i	$0.570803\pi$
$968$	2.20557e6	0.0756539
$969$	0	0
$970$	−8.74069e6	−0.298275
$971$	−1.80838e7	−0.615520	−0.307760	−	0.951464i	$-0.599579\pi$
$971$	−1.80838e7	−0.615520	−0.307760	+	0.951464i	$0.599579\pi$
$972$	0	0
$973$	−3.04234e6	−0.103021
$974$	7.18713e6	0.242749
$975$	0	0
$976$	−4.78003e6	−0.160622
$977$	2.30924e7	0.773984	0.386992	−	0.922083i	$-0.373514\pi$
$977$	2.30924e7	0.773984	0.386992	+	0.922083i	$0.373514\pi$
$978$	0	0
$979$	1.54711e7	0.515900
$980$	1.04091e7	0.346217
$981$	0	0
$982$	−1.12889e7	−0.373572
$983$	1.13687e7	0.375257	0.187628	−	0.982240i	$-0.439920\pi$
$983$	1.13687e7	0.375257	0.187628	+	0.982240i	$0.439920\pi$
$984$	0	0
$985$	2.38657e7	0.783762
$986$	−2.07743e6	−0.0680509
$987$	0	0
$988$	−1.81723e6	−0.0592266
$989$	8.87830e6	0.288629
$990$	0	0
$991$	4.29535e7	1.38936	0.694680	−	0.719319i	$-0.255546\pi$
$991$	4.29535e7	1.38936	0.694680	+	0.719319i	$0.255546\pi$
$992$	−1.26271e7	−0.407403
$993$	0	0
$994$	1.56455e6	0.0502255
$995$	4.09645e6	0.131175
$996$	0	0
$997$	−4.64938e7	−1.48135	−0.740674	−	0.671864i	$-0.765494\pi$
$997$	−4.64938e7	−1.48135	−0.740674	+	0.671864i	$0.765494\pi$
$998$	1.67827e6	0.0533378
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.g.1.2		4
3.2	odd	2		55.6.a.b.1.3	✓	4
12.11	even	2		880.6.a.n.1.2		4
15.2	even	4		275.6.b.d.199.5		8
15.8	even	4		275.6.b.d.199.4		8
15.14	odd	2		275.6.a.d.1.2		4
33.32	even	2		605.6.a.c.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.6.a.b.1.3	✓	4	3.2	odd	2
275.6.a.d.1.2		4	15.14	odd	2
275.6.b.d.199.4		8	15.8	even	4
275.6.b.d.199.5		8	15.2	even	4
495.6.a.g.1.2		4	1.1	even	1	trivial
605.6.a.c.1.2		4	33.32	even	2
880.6.a.n.1.2		4	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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64192 q^{2} -25.0202 q^{4} +25.0000 q^{5} -12.8802 q^{7} +150.643 q^{8} +O(q^{10})\)	\(q-2.64192 q^{2} -25.0202 q^{4} +25.0000 q^{5} -12.8802 q^{7} +150.643 q^{8} -66.0481 q^{10} +121.000 q^{11} -485.167 q^{13} +34.0284 q^{14} +402.660 q^{16} -266.661 q^{17} -149.702 q^{19} -625.506 q^{20} -319.673 q^{22} +3213.11 q^{23} +625.000 q^{25} +1281.77 q^{26} +322.265 q^{28} -2948.81 q^{29} +2145.87 q^{31} -5884.38 q^{32} +704.498 q^{34} -322.004 q^{35} -808.357 q^{37} +395.500 q^{38} +3766.08 q^{40} -10105.2 q^{41} +2763.15 q^{43} -3027.45 q^{44} -8488.79 q^{46} -9973.36 q^{47} -16641.1 q^{49} -1651.20 q^{50} +12139.0 q^{52} -7126.92 q^{53} +3025.00 q^{55} -1940.31 q^{56} +7790.52 q^{58} +33337.2 q^{59} -11871.1 q^{61} -5669.22 q^{62} +2660.94 q^{64} -12129.2 q^{65} +4500.58 q^{67} +6671.93 q^{68} +850.710 q^{70} +45977.8 q^{71} -62039.1 q^{73} +2135.62 q^{74} +3745.57 q^{76} -1558.50 q^{77} -57486.6 q^{79} +10066.5 q^{80} +26697.2 q^{82} +90511.7 q^{83} -6666.53 q^{85} -7300.03 q^{86} +18227.8 q^{88} +127861. q^{89} +6249.03 q^{91} -80392.8 q^{92} +26348.9 q^{94} -3742.54 q^{95} +132338. q^{97} +43964.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{2} + 61 q^{4} + 100 q^{5} - 90 q^{7} + 135 q^{8}+O(q^{10}) \) 4 * q + 5 * q^2 + 61 * q^4 + 100 * q^5 - 90 * q^7 + 135 * q^8	\( 4 q + 5 q^{2} + 61 q^{4} + 100 q^{5} - 90 q^{7} + 135 q^{8} + 125 q^{10} + 484 q^{11} + 820 q^{13} + 1687 q^{14} - 2671 q^{16} + 3800 q^{17} - 3394 q^{19} + 1525 q^{20} + 605 q^{22} + 3020 q^{23} + 2500 q^{25} + 3650 q^{26} + 2635 q^{28} + 5248 q^{29} + 4732 q^{31} - 11505 q^{32} + 9759 q^{34} - 2250 q^{35} + 10210 q^{37} - 16945 q^{38} + 3375 q^{40} + 21068 q^{41} - 12140 q^{43} + 7381 q^{44} + 58442 q^{46} - 4720 q^{47} + 24550 q^{49} + 3125 q^{50} + 58250 q^{52} + 21670 q^{53} + 12100 q^{55} + 28985 q^{56} - 45435 q^{58} + 69068 q^{59} - 44000 q^{61} - 33375 q^{62} - 68223 q^{64} + 20500 q^{65} - 8720 q^{67} + 107335 q^{68} + 42175 q^{70} + 47516 q^{71} - 2480 q^{73} + 55717 q^{74} - 102461 q^{76} - 10890 q^{77} - 188192 q^{79} - 66775 q^{80} + 279030 q^{82} - 68620 q^{83} + 95000 q^{85} - 115328 q^{86} + 16335 q^{88} + 170266 q^{89} + 97740 q^{91} - 53950 q^{92} - 152926 q^{94} - 84850 q^{95} + 186160 q^{97} + 393590 q^{98}+O(q^{100}) \) 4 * q + 5 * q^2 + 61 * q^4 + 100 * q^5 - 90 * q^7 + 135 * q^8 + 125 * q^10 + 484 * q^11 + 820 * q^13 + 1687 * q^14 - 2671 * q^16 + 3800 * q^17 - 3394 * q^19 + 1525 * q^20 + 605 * q^22 + 3020 * q^23 + 2500 * q^25 + 3650 * q^26 + 2635 * q^28 + 5248 * q^29 + 4732 * q^31 - 11505 * q^32 + 9759 * q^34 - 2250 * q^35 + 10210 * q^37 - 16945 * q^38 + 3375 * q^40 + 21068 * q^41 - 12140 * q^43 + 7381 * q^44 + 58442 * q^46 - 4720 * q^47 + 24550 * q^49 + 3125 * q^50 + 58250 * q^52 + 21670 * q^53 + 12100 * q^55 + 28985 * q^56 - 45435 * q^58 + 69068 * q^59 - 44000 * q^61 - 33375 * q^62 - 68223 * q^64 + 20500 * q^65 - 8720 * q^67 + 107335 * q^68 + 42175 * q^70 + 47516 * q^71 - 2480 * q^73 + 55717 * q^74 - 102461 * q^76 - 10890 * q^77 - 188192 * q^79 - 66775 * q^80 + 279030 * q^82 - 68620 * q^83 + 95000 * q^85 - 115328 * q^86 + 16335 * q^88 + 170266 * q^89 + 97740 * q^91 - 53950 * q^92 - 152926 * q^94 - 84850 * q^95 + 186160 * q^97 + 393590 * q^98

Introduction

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