LMFDB - Embedded newform 650.6.a.s.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(1,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("650.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.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:	$104.249482878$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 1033x^{4} + 1207x^{3} + 258312x^{2} - 435604x - 11275680 $ x^6 - 3x^5 - 1033x^4 + 1207x^3 + 258312x^2 - 435604*x - 11275680
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-19.7411$ of defining polynomial
Character	$\chi$	$=$	650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +18.7411 q^{3} +16.0000 q^{4} -74.9644 q^{6} +230.695 q^{7} -64.0000 q^{8} +108.229 q^{9} -544.656 q^{11} +299.858 q^{12} -169.000 q^{13} -922.779 q^{14} +256.000 q^{16} +554.057 q^{17} -432.917 q^{18} -2228.18 q^{19} +4323.47 q^{21} +2178.63 q^{22} +2915.01 q^{23} -1199.43 q^{24} +676.000 q^{26} -2525.75 q^{27} +3691.11 q^{28} -6864.22 q^{29} -6257.84 q^{31} -1024.00 q^{32} -10207.5 q^{33} -2216.23 q^{34} +1731.67 q^{36} -5695.55 q^{37} +8912.71 q^{38} -3167.25 q^{39} -11997.2 q^{41} -17293.9 q^{42} -15451.4 q^{43} -8714.50 q^{44} -11660.0 q^{46} +16278.3 q^{47} +4797.72 q^{48} +36413.0 q^{49} +10383.6 q^{51} -2704.00 q^{52} +11052.3 q^{53} +10103.0 q^{54} -14764.5 q^{56} -41758.5 q^{57} +27456.9 q^{58} -24145.9 q^{59} -10681.8 q^{61} +25031.4 q^{62} +24967.9 q^{63} +4096.00 q^{64} +40829.9 q^{66} -52777.4 q^{67} +8864.91 q^{68} +54630.5 q^{69} -44162.3 q^{71} -6926.67 q^{72} +41761.3 q^{73} +22782.2 q^{74} -35650.8 q^{76} -125649. q^{77} +12669.0 q^{78} -59573.9 q^{79} -73635.1 q^{81} +47988.7 q^{82} -22640.6 q^{83} +69175.6 q^{84} +61805.8 q^{86} -128643. q^{87} +34858.0 q^{88} +129864. q^{89} -38987.4 q^{91} +46640.1 q^{92} -117279. q^{93} -65113.1 q^{94} -19190.9 q^{96} -63287.9 q^{97} -145652. q^{98} -58947.7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 24 q^{2} - 9 q^{3} + 96 q^{4} + 36 q^{6} + 63 q^{7} - 384 q^{8} + 629 q^{9} + 370 q^{11} - 144 q^{12} - 1014 q^{13} - 252 q^{14} + 1536 q^{16} - 838 q^{17} - 2516 q^{18} - 1659 q^{19} + 4322 q^{21}+ \cdots + 386119 q^{99}+O(q^{100}) $ 6 * q - 24 * q^2 - 9 * q^3 + 96 * q^4 + 36 * q^6 + 63 * q^7 - 384 * q^8 + 629 * q^9 + 370 * q^11 - 144 * q^12 - 1014 * q^13 - 252 * q^14 + 1536 * q^16 - 838 * q^17 - 2516 * q^18 - 1659 * q^19 + 4322 * q^21 - 1480 * q^22 - 1566 * q^23 + 576 * q^24 + 4056 * q^26 - 7569 * q^27 + 1008 * q^28 - 5911 * q^29 + 1403 * q^31 - 6144 * q^32 - 8993 * q^33 + 3352 * q^34 + 10064 * q^36 - 10598 * q^37 + 6636 * q^38 + 1521 * q^39 + 1487 * q^41 - 17288 * q^42 - 16078 * q^43 + 5920 * q^44 + 6264 * q^46 - 7135 * q^47 - 2304 * q^48 + 28977 * q^49 + 7053 * q^51 - 16224 * q^52 + 7399 * q^53 + 30276 * q^54 - 4032 * q^56 - 51223 * q^57 + 23644 * q^58 + 22367 * q^59 + 6711 * q^61 - 5612 * q^62 + 10699 * q^63 + 24576 * q^64 + 35972 * q^66 - 99008 * q^67 - 13408 * q^68 + 134338 * q^69 + 15220 * q^71 - 40256 * q^72 - 9115 * q^73 + 42392 * q^74 - 26544 * q^76 - 156999 * q^77 - 6084 * q^78 + 95428 * q^79 - 8134 * q^81 - 5948 * q^82 + 14856 * q^83 + 69152 * q^84 + 64312 * q^86 - 204762 * q^87 - 23680 * q^88 + 292311 * q^89 - 10647 * q^91 - 25056 * q^92 - 194512 * q^93 + 28540 * q^94 + 9216 * q^96 - 77054 * q^97 - 115908 * q^98 + 386119 * q^99

Coefficient data

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

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

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

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	18.7411	1.20224	0.601121	−	0.799158i	$-0.294721\pi$
$3$	18.7411	1.20224	0.601121	+	0.799158i	$0.294721\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−74.9644	−0.850114
$7$	230.695	1.77948	0.889739	−	0.456470i	$-0.150887\pi$
$7$	230.695	1.77948	0.889739	+	0.456470i	$0.150887\pi$
$8$	−64.0000	−0.353553
$9$	108.229	0.445388
$10$	0	0
$11$	−544.656	−1.35719	−0.678595	−	0.734512i	$-0.737411\pi$
$11$	−544.656	−1.35719	−0.678595	+	0.734512i	$0.737411\pi$
$12$	299.858	0.601121
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−922.779	−1.25828
$15$	0	0
$16$	256.000	0.250000
$17$	554.057	0.464977	0.232489	−	0.972599i	$-0.425313\pi$
$17$	554.057	0.464977	0.232489	+	0.972599i	$0.425313\pi$
$18$	−432.917	−0.314937
$19$	−2228.18	−1.41601	−0.708004	−	0.706208i	$-0.750404\pi$
$19$	−2228.18	−1.41601	−0.708004	+	0.706208i	$0.750404\pi$
$20$	0	0
$21$	4323.47	2.13936
$22$	2178.63	0.959679
$23$	2915.01	1.14900	0.574500	−	0.818505i	$-0.305197\pi$
$23$	2915.01	1.14900	0.574500	+	0.818505i	$0.305197\pi$
$24$	−1199.43	−0.425057
$25$	0	0
$26$	676.000	0.196116
$27$	−2525.75	−0.666779
$28$	3691.11	0.889739
$29$	−6864.22	−1.51564	−0.757820	−	0.652464i	$-0.773736\pi$
$29$	−6864.22	−1.51564	−0.757820	+	0.652464i	$0.773736\pi$
$30$	0	0
$31$	−6257.84	−1.16955	−0.584777	−	0.811194i	$-0.698818\pi$
$31$	−6257.84	−1.16955	−0.584777	+	0.811194i	$0.698818\pi$
$32$	−1024.00	−0.176777
$33$	−10207.5	−1.63167
$34$	−2216.23	−0.328789
$35$	0	0
$36$	1731.67	0.222694
$37$	−5695.55	−0.683961	−0.341980	−	0.939707i	$-0.611098\pi$
$37$	−5695.55	−0.683961	−0.341980	+	0.939707i	$0.611098\pi$
$38$	8912.71	1.00127
$39$	−3167.25	−0.333442
$40$	0	0
$41$	−11997.2	−1.11460	−0.557300	−	0.830311i	$-0.688163\pi$
$41$	−11997.2	−1.11460	−0.557300	+	0.830311i	$0.688163\pi$
$42$	−17293.9	−1.51276
$43$	−15451.4	−1.27438	−0.637189	−	0.770708i	$-0.719903\pi$
$43$	−15451.4	−1.27438	−0.637189	+	0.770708i	$0.719903\pi$
$44$	−8714.50	−0.678595
$45$	0	0
$46$	−11660.0	−0.812466
$47$	16278.3	1.07489	0.537445	−	0.843299i	$-0.319390\pi$
$47$	16278.3	1.07489	0.537445	+	0.843299i	$0.319390\pi$
$48$	4797.72	0.300561
$49$	36413.0	2.16654
$50$	0	0
$51$	10383.6	0.559016
$52$	−2704.00	−0.138675
$53$	11052.3	0.540462	0.270231	−	0.962796i	$-0.412900\pi$
$53$	11052.3	0.540462	0.270231	+	0.962796i	$0.412900\pi$
$54$	10103.0	0.471484
$55$	0	0
$56$	−14764.5	−0.629140
$57$	−41758.5	−1.70239
$58$	27456.9	1.07172
$59$	−24145.9	−0.903054	−0.451527	−	0.892257i	$-0.649121\pi$
$59$	−24145.9	−0.903054	−0.451527	+	0.892257i	$0.649121\pi$
$60$	0	0
$61$	−10681.8	−0.367552	−0.183776	−	0.982968i	$-0.558832\pi$
$61$	−10681.8	−0.367552	−0.183776	+	0.982968i	$0.558832\pi$
$62$	25031.4	0.827000
$63$	24967.9	0.792557
$64$	4096.00	0.125000
$65$	0	0
$66$	40829.9	1.15377
$67$	−52777.4	−1.43635	−0.718176	−	0.695861i	$-0.755023\pi$
$67$	−52777.4	−1.43635	−0.718176	+	0.695861i	$0.755023\pi$
$68$	8864.91	0.232489
$69$	54630.5	1.38138
$70$	0	0
$71$	−44162.3	−1.03970	−0.519848	−	0.854259i	$-0.674011\pi$
$71$	−44162.3	−1.03970	−0.519848	+	0.854259i	$0.674011\pi$
$72$	−6926.67	−0.157468
$73$	41761.3	0.917205	0.458602	−	0.888642i	$-0.348350\pi$
$73$	41761.3	0.917205	0.458602	+	0.888642i	$0.348350\pi$
$74$	22782.2	0.483633
$75$	0	0
$76$	−35650.8	−0.708004
$77$	−125649.	−2.41509
$78$	12669.0	0.235779
$79$	−59573.9	−1.07396	−0.536980	−	0.843595i	$-0.680435\pi$
$79$	−59573.9	−1.07396	−0.536980	+	0.843595i	$0.680435\pi$
$80$	0	0
$81$	−73635.1	−1.24702
$82$	47988.7	0.788141
$83$	−22640.6	−0.360738	−0.180369	−	0.983599i	$-0.557729\pi$
$83$	−22640.6	−0.360738	−0.180369	+	0.983599i	$0.557729\pi$
$84$	69175.6	1.06968
$85$	0	0
$86$	61805.8	0.901121
$87$	−128643.	−1.82217
$88$	34858.0	0.479839
$89$	129864.	1.73786	0.868930	−	0.494935i	$-0.164808\pi$
$89$	129864.	1.73786	0.868930	+	0.494935i	$0.164808\pi$
$90$	0	0
$91$	−38987.4	−0.493538
$92$	46640.1	0.574500
$93$	−117279.	−1.40609
$94$	−65113.1	−0.760061
$95$	0	0
$96$	−19190.9	−0.212529
$97$	−63287.9	−0.682953	−0.341477	−	0.939890i	$-0.610927\pi$
$97$	−63287.9	−0.682953	−0.341477	+	0.939890i	$0.610927\pi$
$98$	−145652.	−1.53197
$99$	−58947.7	−0.604476
$100$	0	0
$101$	116458.	1.13596	0.567982	−	0.823041i	$-0.307724\pi$
$101$	116458.	1.13596	0.567982	+	0.823041i	$0.307724\pi$
$102$	−41534.6	−0.395284
$103$	48487.1	0.450333	0.225166	−	0.974320i	$-0.427707\pi$
$103$	48487.1	0.450333	0.225166	+	0.974320i	$0.427707\pi$
$104$	10816.0	0.0980581
$105$	0	0
$106$	−44209.4	−0.382164
$107$	189220.	1.59775	0.798874	−	0.601499i	$-0.205430\pi$
$107$	189220.	1.59775	0.798874	+	0.601499i	$0.205430\pi$
$108$	−40412.1	−0.333389
$109$	157628.	1.27077	0.635384	−	0.772197i	$-0.280842\pi$
$109$	157628.	1.27077	0.635384	+	0.772197i	$0.280842\pi$
$110$	0	0
$111$	−106741.	−0.822287
$112$	59057.8	0.444869
$113$	200949.	1.48044	0.740219	−	0.672365i	$-0.234722\pi$
$113$	200949.	1.48044	0.740219	+	0.672365i	$0.234722\pi$
$114$	167034.	1.20377
$115$	0	0
$116$	−109827.	−0.757820
$117$	−18290.7	−0.123528
$118$	96583.7	0.638556
$119$	127818.	0.827417
$120$	0	0
$121$	135600.	0.841966
$122$	42727.1	0.259898
$123$	−224840.	−1.34002
$124$	−100125.	−0.584777
$125$	0	0
$126$	−99871.6	−0.560423
$127$	−283395.	−1.55913	−0.779565	−	0.626321i	$-0.784560\pi$
$127$	−283395.	−1.55913	−0.779565	+	0.626321i	$0.784560\pi$
$128$	−16384.0	−0.0883883
$129$	−289577.	−1.53211
$130$	0	0
$131$	−199352.	−1.01494	−0.507472	−	0.861668i	$-0.669420\pi$
$131$	−199352.	−1.01494	−0.507472	+	0.861668i	$0.669420\pi$
$132$	−163319.	−0.815836
$133$	−514029.	−2.51975
$134$	211110.	1.01565
$135$	0	0
$136$	−35459.6	−0.164394
$137$	265825.	1.21003	0.605013	−	0.796216i	$-0.293168\pi$
$137$	265825.	1.21003	0.605013	+	0.796216i	$0.293168\pi$
$138$	−218522.	−0.976781
$139$	985.292	0.00432541	0.00216271	−	0.999998i	$-0.499312\pi$
$139$	985.292	0.00432541	0.00216271	+	0.999998i	$0.499312\pi$
$140$	0	0
$141$	305073.	1.29228
$142$	176649.	0.735176
$143$	92046.9	0.376417
$144$	27706.7	0.111347
$145$	0	0
$146$	−167045.	−0.648562
$147$	682420.	2.60471
$148$	−91128.7	−0.341980
$149$	−20636.5	−0.0761502	−0.0380751	−	0.999275i	$-0.512123\pi$
$149$	−20636.5	−0.0761502	−0.0380751	+	0.999275i	$0.512123\pi$
$150$	0	0
$151$	31649.1	0.112959	0.0564793	−	0.998404i	$-0.482013\pi$
$151$	31649.1	0.112959	0.0564793	+	0.998404i	$0.482013\pi$
$152$	142603.	0.500635
$153$	59965.1	0.207095
$154$	502597.	1.70773
$155$	0	0
$156$	−50676.0	−0.166721
$157$	−475091.	−1.53825	−0.769126	−	0.639097i	$-0.779308\pi$
$157$	−475091.	−1.53825	−0.769126	+	0.639097i	$0.779308\pi$
$158$	238296.	0.759405
$159$	207133.	0.649766
$160$	0	0
$161$	672477.	2.04462
$162$	294541.	0.881775
$163$	368397.	1.08604	0.543022	−	0.839719i	$-0.317280\pi$
$163$	368397.	1.08604	0.543022	+	0.839719i	$0.317280\pi$
$164$	−191955.	−0.557300
$165$	0	0
$166$	90562.2	0.255080
$167$	−446876.	−1.23993	−0.619963	−	0.784631i	$-0.712852\pi$
$167$	−446876.	−1.23993	−0.619963	+	0.784631i	$0.712852\pi$
$168$	−276702.	−0.756379
$169$	28561.0	0.0769231
$170$	0	0
$171$	−241154.	−0.630673
$172$	−247223.	−0.637189
$173$	−218403.	−0.554810	−0.277405	−	0.960753i	$-0.589474\pi$
$173$	−218403.	−0.554810	−0.277405	+	0.960753i	$0.589474\pi$
$174$	514572.	1.28847
$175$	0	0
$176$	−139432.	−0.339298
$177$	−452521.	−1.08569
$178$	−519457.	−1.22885
$179$	338950.	0.790684	0.395342	−	0.918534i	$-0.370626\pi$
$179$	338950.	0.790684	0.395342	+	0.918534i	$0.370626\pi$
$180$	0	0
$181$	−440716.	−0.999915	−0.499957	−	0.866050i	$-0.666651\pi$
$181$	−440716.	−0.999915	−0.499957	+	0.866050i	$0.666651\pi$
$182$	155950.	0.348984
$183$	−200188.	−0.441887
$184$	−186560.	−0.406233
$185$	0	0
$186$	469116.	0.994255
$187$	−301771.	−0.631063
$188$	260452.	0.537445
$189$	−582678.	−1.18652
$190$	0	0
$191$	643313.	1.27596	0.637982	−	0.770051i	$-0.279769\pi$
$191$	643313.	1.27596	0.637982	+	0.770051i	$0.279769\pi$
$192$	76763.6	0.150280
$193$	−389679.	−0.753033	−0.376517	−	0.926410i	$-0.622878\pi$
$193$	−389679.	−0.753033	−0.376517	+	0.926410i	$0.622878\pi$
$194$	253151.	0.482921
$195$	0	0
$196$	582608.	1.08327
$197$	−326458.	−0.599323	−0.299662	−	0.954046i	$-0.596874\pi$
$197$	−326458.	−0.599323	−0.299662	+	0.954046i	$0.596874\pi$
$198$	235791.	0.427429
$199$	433426.	0.775858	0.387929	−	0.921689i	$-0.373191\pi$
$199$	433426.	0.775858	0.387929	+	0.921689i	$0.373191\pi$
$200$	0	0
$201$	−989107.	−1.72684
$202$	−465830.	−0.803247
$203$	−1.58354e6	−2.69705
$204$	166138.	0.279508
$205$	0	0
$206$	−193949.	−0.318433
$207$	315489.	0.511751
$208$	−43264.0	−0.0693375
$209$	1.21359e6	1.92179
$210$	0	0
$211$	−76110.5	−0.117690	−0.0588448	−	0.998267i	$-0.518742\pi$
$211$	−76110.5	−0.117690	−0.0588448	+	0.998267i	$0.518742\pi$
$212$	176838.	0.270231
$213$	−827651.	−1.24997
$214$	−756881.	−1.12978
$215$	0	0
$216$	161648.	0.235742
$217$	−1.44365e6	−2.08120
$218$	−630510.	−0.898568
$219$	782652.	1.10270
$220$	0	0
$221$	−93635.6	−0.128962
$222$	426963.	0.581444
$223$	141163.	0.190090	0.0950450	−	0.995473i	$-0.469700\pi$
$223$	141163.	0.190090	0.0950450	+	0.995473i	$0.469700\pi$
$224$	−236231.	−0.314570
$225$	0	0
$226$	−803797.	−1.04683
$227$	968458.	1.24743	0.623715	−	0.781652i	$-0.285623\pi$
$227$	968458.	1.24743	0.623715	+	0.781652i	$0.285623\pi$
$228$	−668136.	−0.851193
$229$	72887.3	0.0918466	0.0459233	−	0.998945i	$-0.485377\pi$
$229$	72887.3	0.0918466	0.0459233	+	0.998945i	$0.485377\pi$
$230$	0	0
$231$	−2.35481e6	−2.90352
$232$	439310.	0.535860
$233$	−613692.	−0.740561	−0.370280	−	0.928920i	$-0.620738\pi$
$233$	−613692.	−0.740561	−0.370280	+	0.928920i	$0.620738\pi$
$234$	73163.0	0.0873477
$235$	0	0
$236$	−386335.	−0.451527
$237$	−1.11648e6	−1.29116
$238$	−511272.	−0.585072
$239$	−1.21444e6	−1.37525	−0.687626	−	0.726065i	$-0.741347\pi$
$239$	−1.21444e6	−1.37525	−0.687626	+	0.726065i	$0.741347\pi$
$240$	0	0
$241$	−1.78850e6	−1.98356	−0.991780	−	0.127957i	$-0.959158\pi$
$241$	−1.78850e6	−1.98356	−0.991780	+	0.127957i	$0.959158\pi$
$242$	−542398.	−0.595360
$243$	−766246.	−0.832439
$244$	−170908.	−0.183776
$245$	0	0
$246$	899361.	0.947537
$247$	376562.	0.392730
$248$	400502.	0.413500
$249$	−424309.	−0.433695
$250$	0	0
$251$	427715.	0.428519	0.214259	−	0.976777i	$-0.431266\pi$
$251$	427715.	0.428519	0.214259	+	0.976777i	$0.431266\pi$
$252$	399486.	0.396279
$253$	−1.58768e6	−1.55941
$254$	1.13358e6	1.10247
$255$	0	0
$256$	65536.0	0.0625000
$257$	271022.	0.255960	0.127980	−	0.991777i	$-0.459151\pi$
$257$	271022.	0.255960	0.127980	+	0.991777i	$0.459151\pi$
$258$	1.15831e6	1.08337
$259$	−1.31393e6	−1.21709
$260$	0	0
$261$	−742909.	−0.675048
$262$	797407.	0.717673
$263$	1.31591e6	1.17310	0.586552	−	0.809912i	$-0.300485\pi$
$263$	1.31591e6	1.17310	0.586552	+	0.809912i	$0.300485\pi$
$264$	653278.	0.576883
$265$	0	0
$266$	2.05611e6	1.78174
$267$	2.43380e6	2.08933
$268$	−844438.	−0.718176
$269$	−1.63605e6	−1.37853	−0.689266	−	0.724508i	$-0.742067\pi$
$269$	−1.63605e6	−1.37853	−0.689266	+	0.724508i	$0.742067\pi$
$270$	0	0
$271$	−1.41289e6	−1.16865	−0.584326	−	0.811519i	$-0.698641\pi$
$271$	−1.41289e6	−1.16865	−0.584326	+	0.811519i	$0.698641\pi$
$272$	141839.	0.116244
$273$	−730667.	−0.593353
$274$	−1.06330e6	−0.855617
$275$	0	0
$276$	874088.	0.690689
$277$	−755228.	−0.591397	−0.295698	−	0.955281i	$-0.595552\pi$
$277$	−755228.	−0.591397	−0.295698	+	0.955281i	$0.595552\pi$
$278$	−3941.17	−0.00305853
$279$	−677282.	−0.520905
$280$	0	0
$281$	−550427.	−0.415848	−0.207924	−	0.978145i	$-0.566671\pi$
$281$	−550427.	−0.415848	−0.207924	+	0.978145i	$0.566671\pi$
$282$	−1.22029e6	−0.913778
$283$	1.12011e6	0.831367	0.415683	−	0.909509i	$-0.363542\pi$
$283$	1.12011e6	0.831367	0.415683	+	0.909509i	$0.363542\pi$
$284$	−706597.	−0.519848
$285$	0	0
$286$	−368188.	−0.266167
$287$	−2.76768e6	−1.98341
$288$	−110827.	−0.0787342
$289$	−1.11288e6	−0.783796
$290$	0	0
$291$	−1.18608e6	−0.821076
$292$	668180.	0.458602
$293$	1.47320e6	1.00252	0.501260	−	0.865297i	$-0.332870\pi$
$293$	1.47320e6	1.00252	0.501260	+	0.865297i	$0.332870\pi$
$294$	−2.72968e6	−1.84181
$295$	0	0
$296$	364515.	0.241817
$297$	1.37567e6	0.904946
$298$	82546.1	0.0538463
$299$	−492636.	−0.318675
$300$	0	0
$301$	−3.56457e6	−2.26772
$302$	−126596.	−0.0798737
$303$	2.18254e6	1.36570
$304$	−570414.	−0.354002
$305$	0	0
$306$	−239861.	−0.146438
$307$	−1.42452e6	−0.862623	−0.431312	−	0.902203i	$-0.641949\pi$
$307$	−1.42452e6	−0.862623	−0.431312	+	0.902203i	$0.641949\pi$
$308$	−2.01039e6	−1.20754
$309$	908703.	0.541409
$310$	0	0
$311$	−1.79196e6	−1.05058	−0.525288	−	0.850924i	$-0.676043\pi$
$311$	−1.79196e6	−1.05058	−0.525288	+	0.850924i	$0.676043\pi$
$312$	202704.	0.117890
$313$	2.93685e6	1.69442	0.847209	−	0.531260i	$-0.178281\pi$
$313$	2.93685e6	1.69442	0.847209	+	0.531260i	$0.178281\pi$
$314$	1.90036e6	1.08771
$315$	0	0
$316$	−953182.	−0.536980
$317$	−390977.	−0.218526	−0.109263	−	0.994013i	$-0.534849\pi$
$317$	−390977.	−0.218526	−0.109263	+	0.994013i	$0.534849\pi$
$318$	−828533.	−0.459454
$319$	3.73864e6	2.05701
$320$	0	0
$321$	3.54620e6	1.92088
$322$	−2.68991e6	−1.44576
$323$	−1.23454e6	−0.658412
$324$	−1.17816e6	−0.623509
$325$	0	0
$326$	−1.47359e6	−0.767949
$327$	2.95412e6	1.52777
$328$	767819.	0.394071
$329$	3.75531e6	1.91274
$330$	0	0
$331$	−2.78981e6	−1.39960	−0.699800	−	0.714339i	$-0.746728\pi$
$331$	−2.78981e6	−1.39960	−0.699800	+	0.714339i	$0.746728\pi$
$332$	−362249.	−0.180369
$333$	−616425.	−0.304628
$334$	1.78750e6	0.876760
$335$	0	0
$336$	1.10681e6	0.534841
$337$	−2.51413e6	−1.20591	−0.602953	−	0.797776i	$-0.706009\pi$
$337$	−2.51413e6	−1.20591	−0.602953	+	0.797776i	$0.706009\pi$
$338$	−114244.	−0.0543928
$339$	3.76601e6	1.77985
$340$	0	0
$341$	3.40837e6	1.58731
$342$	964616.	0.445953
$343$	4.52300e6	2.07583
$344$	988893.	0.450560
$345$	0	0
$346$	873614.	0.392310
$347$	−310242.	−0.138317	−0.0691587	−	0.997606i	$-0.522031\pi$
$347$	−310242.	−0.138317	−0.0691587	+	0.997606i	$0.522031\pi$
$348$	−2.05829e6	−0.911084
$349$	−1.17835e6	−0.517859	−0.258929	−	0.965896i	$-0.583370\pi$
$349$	−1.17835e6	−0.517859	−0.258929	+	0.965896i	$0.583370\pi$
$350$	0	0
$351$	426852.	0.184931
$352$	557728.	0.239920
$353$	−1.34357e6	−0.573885	−0.286942	−	0.957948i	$-0.592639\pi$
$353$	−1.34357e6	−0.573885	−0.286942	+	0.957948i	$0.592639\pi$
$354$	1.81009e6	0.767699
$355$	0	0
$356$	2.07783e6	0.868930
$357$	2.39545e6	0.994756
$358$	−1.35580e6	−0.559098
$359$	−2.61851e6	−1.07230	−0.536152	−	0.844122i	$-0.680123\pi$
$359$	−2.61851e6	−1.07230	−0.536152	+	0.844122i	$0.680123\pi$
$360$	0	0
$361$	2.48868e6	1.00508
$362$	1.76287e6	0.707046
$363$	2.54129e6	1.01225
$364$	−623798.	−0.246769
$365$	0	0
$366$	800753.	0.312461
$367$	−119818.	−0.0464363	−0.0232182	−	0.999730i	$-0.507391\pi$
$367$	−119818.	−0.0464363	−0.0232182	+	0.999730i	$0.507391\pi$
$368$	746242.	0.287250
$369$	−1.29844e6	−0.496429
$370$	0	0
$371$	2.54972e6	0.961739
$372$	−1.87646e6	−0.703044
$373$	2.68734e6	1.00012	0.500058	−	0.865992i	$-0.333312\pi$
$373$	2.68734e6	1.00012	0.500058	+	0.865992i	$0.333312\pi$
$374$	1.20708e6	0.446229
$375$	0	0
$376$	−1.04181e6	−0.380031
$377$	1.16005e6	0.420363
$378$	2.33071e6	0.838994
$379$	1.16326e6	0.415988	0.207994	−	0.978130i	$-0.433307\pi$
$379$	1.16326e6	0.415988	0.207994	+	0.978130i	$0.433307\pi$
$380$	0	0
$381$	−5.31113e6	−1.87445
$382$	−2.57325e6	−0.902243
$383$	1.35360e6	0.471514	0.235757	−	0.971812i	$-0.424243\pi$
$383$	1.35360e6	0.471514	0.235757	+	0.971812i	$0.424243\pi$
$384$	−307054.	−0.106264
$385$	0	0
$386$	1.55872e6	0.532475
$387$	−1.67230e6	−0.567592
$388$	−1.01261e6	−0.341477
$389$	1.64100e6	0.549839	0.274920	−	0.961467i	$-0.411349\pi$
$389$	1.64100e6	0.549839	0.274920	+	0.961467i	$0.411349\pi$
$390$	0	0
$391$	1.61508e6	0.534259
$392$	−2.33043e6	−0.765987
$393$	−3.73607e6	−1.22021
$394$	1.30583e6	0.423786
$395$	0	0
$396$	−943164.	−0.302238
$397$	1.51737e6	0.483187	0.241593	−	0.970378i	$-0.422330\pi$
$397$	1.51737e6	0.483187	0.241593	+	0.970378i	$0.422330\pi$
$398$	−1.73370e6	−0.548614
$399$	−9.63347e6	−3.02936
$400$	0	0
$401$	−4.16517e6	−1.29352	−0.646758	−	0.762696i	$-0.723875\pi$
$401$	−4.16517e6	−1.29352	−0.646758	+	0.762696i	$0.723875\pi$
$402$	3.95643e6	1.22106
$403$	1.05758e6	0.324376
$404$	1.86332e6	0.567982
$405$	0	0
$406$	6.33415e6	1.90710
$407$	3.10212e6	0.928265
$408$	−664553.	−0.197642
$409$	2.16258e6	0.639240	0.319620	−	0.947546i	$-0.396445\pi$
$409$	2.16258e6	0.639240	0.319620	+	0.947546i	$0.396445\pi$
$410$	0	0
$411$	4.98186e6	1.45474
$412$	775794.	0.225166
$413$	−5.57034e6	−1.60696
$414$	−1.26196e6	−0.361862
$415$	0	0
$416$	173056.	0.0490290
$417$	18465.5	0.00520020
$418$	−4.85436e6	−1.35891
$419$	−5.87037e6	−1.63354	−0.816772	−	0.576961i	$-0.804238\pi$
$419$	−5.87037e6	−1.63354	−0.816772	+	0.576961i	$0.804238\pi$
$420$	0	0
$421$	−2.14426e6	−0.589621	−0.294810	−	0.955556i	$-0.595256\pi$
$421$	−2.14426e6	−0.589621	−0.294810	+	0.955556i	$0.595256\pi$
$422$	304442.	0.0832192
$423$	1.76178e6	0.478742
$424$	−707350.	−0.191082
$425$	0	0
$426$	3.31061e6	0.883860
$427$	−2.46423e6	−0.654050
$428$	3.02752e6	0.798874
$429$	1.72506e6	0.452545
$430$	0	0
$431$	2.56530e6	0.665189	0.332594	−	0.943070i	$-0.392076\pi$
$431$	2.56530e6	0.665189	0.332594	+	0.943070i	$0.392076\pi$
$432$	−646593.	−0.166695
$433$	6.07499e6	1.55713	0.778567	−	0.627561i	$-0.215947\pi$
$433$	6.07499e6	1.55713	0.778567	+	0.627561i	$0.215947\pi$
$434$	5.77460e6	1.47163
$435$	0	0
$436$	2.52204e6	0.635384
$437$	−6.49515e6	−1.62699
$438$	−3.13061e6	−0.779729
$439$	7.17264e6	1.77631	0.888154	−	0.459547i	$-0.151988\pi$
$439$	7.17264e6	1.77631	0.888154	+	0.459547i	$0.151988\pi$
$440$	0	0
$441$	3.94095e6	0.964950
$442$	374542.	0.0911896
$443$	1.55350e6	0.376099	0.188049	−	0.982160i	$-0.439783\pi$
$443$	1.55350e6	0.376099	0.188049	+	0.982160i	$0.439783\pi$
$444$	−1.70785e6	−0.411143
$445$	0	0
$446$	−564653.	−0.134414
$447$	−386751.	−0.0915510
$448$	944925.	0.222435
$449$	−3.57322e6	−0.836456	−0.418228	−	0.908342i	$-0.637349\pi$
$449$	−3.57322e6	−0.836456	−0.418228	+	0.908342i	$0.637349\pi$
$450$	0	0
$451$	6.53433e6	1.51272
$452$	3.21519e6	0.740219
$453$	593139.	0.135804
$454$	−3.87383e6	−0.882066
$455$	0	0
$456$	2.67255e6	0.601884
$457$	8.13345e6	1.82173	0.910866	−	0.412703i	$-0.135415\pi$
$457$	8.13345e6	1.82173	0.910866	+	0.412703i	$0.135415\pi$
$458$	−291549.	−0.0649454
$459$	−1.39941e6	−0.310037
$460$	0	0
$461$	−3.69923e6	−0.810697	−0.405349	−	0.914162i	$-0.632850\pi$
$461$	−3.69923e6	−0.810697	−0.405349	+	0.914162i	$0.632850\pi$
$462$	9.41923e6	2.05310
$463$	1.34127e6	0.290779	0.145389	−	0.989375i	$-0.453557\pi$
$463$	1.34127e6	0.290779	0.145389	+	0.989375i	$0.453557\pi$
$464$	−1.75724e6	−0.378910
$465$	0	0
$466$	2.45477e6	0.523655
$467$	−2.41068e6	−0.511501	−0.255751	−	0.966743i	$-0.582323\pi$
$467$	−2.41068e6	−0.511501	−0.255751	+	0.966743i	$0.582323\pi$
$468$	−292652.	−0.0617642
$469$	−1.21755e7	−2.55596
$470$	0	0
$471$	−8.90373e6	−1.84935
$472$	1.54534e6	0.319278
$473$	8.41573e6	1.72957
$474$	4.46592e6	0.912989
$475$	0	0
$476$	2.04509e6	0.413708
$477$	1.19619e6	0.240715
$478$	4.85777e6	0.972450
$479$	18669.1	0.00371779	0.00185890	−	0.999998i	$-0.499408\pi$
$479$	18669.1	0.00371779	0.00185890	+	0.999998i	$0.499408\pi$
$480$	0	0
$481$	962547.	0.189697
$482$	7.15398e6	1.40259
$483$	1.26030e7	2.45813
$484$	2.16959e6	0.420983
$485$	0	0
$486$	3.06498e6	0.588623
$487$	4.53165e6	0.865832	0.432916	−	0.901434i	$-0.357485\pi$
$487$	4.53165e6	0.865832	0.432916	+	0.901434i	$0.357485\pi$
$488$	683633.	0.129949
$489$	6.90417e6	1.30569
$490$	0	0
$491$	3.84989e6	0.720684	0.360342	−	0.932820i	$-0.382660\pi$
$491$	3.84989e6	0.720684	0.360342	+	0.932820i	$0.382660\pi$
$492$	−3.59744e6	−0.670010
$493$	−3.80317e6	−0.704739
$494$	−1.50625e6	−0.277702
$495$	0	0
$496$	−1.60201e6	−0.292389
$497$	−1.01880e7	−1.85011
$498$	1.69724e6	0.306668
$499$	2.18845e6	0.393445	0.196723	−	0.980459i	$-0.436970\pi$
$499$	2.18845e6	0.393445	0.196723	+	0.980459i	$0.436970\pi$
$500$	0	0
$501$	−8.37495e6	−1.49069
$502$	−1.71086e6	−0.303009
$503$	−1.37502e6	−0.242319	−0.121160	−	0.992633i	$-0.538661\pi$
$503$	−1.37502e6	−0.242319	−0.121160	+	0.992633i	$0.538661\pi$
$504$	−1.59795e6	−0.280211
$505$	0	0
$506$	6.35071e6	1.10267
$507$	535265.	0.0924802
$508$	−4.53431e6	−0.779565
$509$	7.64261e6	1.30752	0.653759	−	0.756703i	$-0.273191\pi$
$509$	7.64261e6	1.30752	0.653759	+	0.756703i	$0.273191\pi$
$510$	0	0
$511$	9.63410e6	1.63215
$512$	−262144.	−0.0441942
$513$	5.62783e6	0.944164
$514$	−1.08409e6	−0.180991
$515$	0	0
$516$	−4.63324e6	−0.766055
$517$	−8.86606e6	−1.45883
$518$	5.25573e6	0.860614
$519$	−4.09312e6	−0.667016
$520$	0	0
$521$	1.17487e7	1.89625	0.948123	−	0.317903i	$-0.102979\pi$
$521$	1.17487e7	1.89625	0.948123	+	0.317903i	$0.102979\pi$
$522$	2.97164e6	0.477331
$523$	−5.40309e6	−0.863750	−0.431875	−	0.901933i	$-0.642148\pi$
$523$	−5.40309e6	−0.863750	−0.431875	+	0.901933i	$0.642148\pi$
$524$	−3.18963e6	−0.507472
$525$	0	0
$526$	−5.26363e6	−0.829510
$527$	−3.46720e6	−0.543817
$528$	−2.61311e6	−0.407918
$529$	2.06092e6	0.320201
$530$	0	0
$531$	−2.61330e6	−0.402209
$532$	−8.22446e6	−1.25988
$533$	2.02752e6	0.309134
$534$	−9.73520e6	−1.47738
$535$	0	0
$536$	3.37775e6	0.507827
$537$	6.35230e6	0.950594
$538$	6.54422e6	0.974769
$539$	−1.98326e7	−2.94041
$540$	0	0
$541$	−1.33517e7	−1.96129	−0.980646	−	0.195789i	$-0.937273\pi$
$541$	−1.33517e7	−1.96129	−0.980646	+	0.195789i	$0.937273\pi$
$542$	5.65156e6	0.826362
$543$	−8.25952e6	−1.20214
$544$	−567354.	−0.0821972
$545$	0	0
$546$	2.92267e6	0.419564
$547$	−2.40834e6	−0.344152	−0.172076	−	0.985084i	$-0.555047\pi$
$547$	−2.40834e6	−0.344152	−0.172076	+	0.985084i	$0.555047\pi$
$548$	4.25320e6	0.605013
$549$	−1.15608e6	−0.163703
$550$	0	0
$551$	1.52947e7	2.14616
$552$	−3.49635e6	−0.488391
$553$	−1.37434e7	−1.91109
$554$	3.02091e6	0.418180
$555$	0	0
$556$	15764.7	0.00216271
$557$	1.52042e6	0.207647	0.103823	−	0.994596i	$-0.466892\pi$
$557$	1.52042e6	0.207647	0.103823	+	0.994596i	$0.466892\pi$
$558$	2.70913e6	0.368336
$559$	2.61129e6	0.353449
$560$	0	0
$561$	−5.65551e6	−0.758691
$562$	2.20171e6	0.294049
$563$	9.15496e6	1.21727	0.608633	−	0.793452i	$-0.291718\pi$
$563$	9.15496e6	1.21727	0.608633	+	0.793452i	$0.291718\pi$
$564$	4.88117e6	0.646139
$565$	0	0
$566$	−4.48042e6	−0.587865
$567$	−1.69872e7	−2.21904
$568$	2.82639e6	0.367588
$569$	1.18973e7	1.54052	0.770262	−	0.637727i	$-0.220125\pi$
$569$	1.18973e7	1.54052	0.770262	+	0.637727i	$0.220125\pi$
$570$	0	0
$571$	1.20642e7	1.54849	0.774245	−	0.632886i	$-0.218130\pi$
$571$	1.20642e7	1.54849	0.774245	+	0.632886i	$0.218130\pi$
$572$	1.47275e6	0.188208
$573$	1.20564e7	1.53402
$574$	1.10707e7	1.40248
$575$	0	0
$576$	443307.	0.0556735
$577$	1.38845e6	0.173616	0.0868081	−	0.996225i	$-0.472333\pi$
$577$	1.38845e6	0.173616	0.0868081	+	0.996225i	$0.472333\pi$
$578$	4.45151e6	0.554227
$579$	−7.30303e6	−0.905329
$580$	0	0
$581$	−5.22305e6	−0.641925
$582$	4.74434e6	0.580588
$583$	−6.01973e6	−0.733509
$584$	−2.67272e6	−0.324281
$585$	0	0
$586$	−5.89280e6	−0.708888
$587$	−5.80067e6	−0.694836	−0.347418	−	0.937710i	$-0.612942\pi$
$587$	−5.80067e6	−0.694836	−0.347418	+	0.937710i	$0.612942\pi$
$588$	1.09187e7	1.30235
$589$	1.39436e7	1.65610
$590$	0	0
$591$	−6.11818e6	−0.720532
$592$	−1.45806e6	−0.170990
$593$	1.06130e7	1.23937	0.619684	−	0.784852i	$-0.287261\pi$
$593$	1.06130e7	1.23937	0.619684	+	0.784852i	$0.287261\pi$
$594$	−5.50267e6	−0.639893
$595$	0	0
$596$	−330184.	−0.0380751
$597$	8.12288e6	0.932770
$598$	1.97054e6	0.225337
$599$	−4.68069e6	−0.533020	−0.266510	−	0.963832i	$-0.585870\pi$
$599$	−4.68069e6	−0.533020	−0.266510	+	0.963832i	$0.585870\pi$
$600$	0	0
$601$	2.65345e6	0.299657	0.149828	−	0.988712i	$-0.452128\pi$
$601$	2.65345e6	0.299657	0.149828	+	0.988712i	$0.452128\pi$
$602$	1.42583e7	1.60352
$603$	−5.71206e6	−0.639734
$604$	506386.	0.0564793
$605$	0	0
$606$	−8.73018e6	−0.965698
$607$	6.58092e6	0.724962	0.362481	−	0.931991i	$-0.381930\pi$
$607$	6.58092e6	0.724962	0.362481	+	0.931991i	$0.381930\pi$
$608$	2.28165e6	0.250317
$609$	−2.96773e7	−3.24251
$610$	0	0
$611$	−2.75103e6	−0.298121
$612$	959442.	0.103548
$613$	8.28523e6	0.890540	0.445270	−	0.895396i	$-0.353108\pi$
$613$	8.28523e6	0.890540	0.445270	+	0.895396i	$0.353108\pi$
$614$	5.69806e6	0.609967
$615$	0	0
$616$	8.04155e6	0.853863
$617$	−5.42747e6	−0.573964	−0.286982	−	0.957936i	$-0.592652\pi$
$617$	−5.42747e6	−0.573964	−0.286982	+	0.957936i	$0.592652\pi$
$618$	−3.63481e6	−0.382834
$619$	−8.73793e6	−0.916604	−0.458302	−	0.888796i	$-0.651542\pi$
$619$	−8.73793e6	−0.916604	−0.458302	+	0.888796i	$0.651542\pi$
$620$	0	0
$621$	−7.36259e6	−0.766129
$622$	7.16785e6	0.742870
$623$	2.99590e7	3.09248
$624$	−810815.	−0.0833605
$625$	0	0
$626$	−1.17474e7	−1.19813
$627$	2.27440e7	2.31046
$628$	−7.60145e6	−0.769126
$629$	−3.15566e6	−0.318026
$630$	0	0
$631$	−6.93247e6	−0.693130	−0.346565	−	0.938026i	$-0.612652\pi$
$631$	−6.93247e6	−0.693130	−0.346565	+	0.938026i	$0.612652\pi$
$632$	3.81273e6	0.379702
$633$	−1.42639e6	−0.141492
$634$	1.56391e6	0.154521
$635$	0	0
$636$	3.31413e6	0.324883
$637$	−6.15380e6	−0.600890
$638$	−1.49546e7	−1.45453
$639$	−4.77966e6	−0.463068
$640$	0	0
$641$	−9.65973e6	−0.928581	−0.464291	−	0.885683i	$-0.653691\pi$
$641$	−9.65973e6	−0.928581	−0.464291	+	0.885683i	$0.653691\pi$
$642$	−1.41848e7	−1.35827
$643$	−588572.	−0.0561399	−0.0280700	−	0.999606i	$-0.508936\pi$
$643$	−588572.	−0.0561399	−0.0280700	+	0.999606i	$0.508936\pi$
$644$	1.07596e7	1.02231
$645$	0	0
$646$	4.93815e6	0.465568
$647$	7.53056e6	0.707240	0.353620	−	0.935389i	$-0.384951\pi$
$647$	7.53056e6	0.707240	0.353620	+	0.935389i	$0.384951\pi$
$648$	4.71265e6	0.440887
$649$	1.31512e7	1.22562
$650$	0	0
$651$	−2.70556e7	−2.50210
$652$	5.89436e6	0.543022
$653$	−9.16834e6	−0.841410	−0.420705	−	0.907197i	$-0.638217\pi$
$653$	−9.16834e6	−0.841410	−0.420705	+	0.907197i	$0.638217\pi$
$654$	−1.18165e7	−1.08030
$655$	0	0
$656$	−3.07127e6	−0.278650
$657$	4.51979e6	0.408512
$658$	−1.50212e7	−1.35251
$659$	−1.25416e7	−1.12496	−0.562482	−	0.826809i	$-0.690153\pi$
$659$	−1.25416e7	−1.12496	−0.562482	+	0.826809i	$0.690153\pi$
$660$	0	0
$661$	−2.80510e6	−0.249715	−0.124858	−	0.992175i	$-0.539847\pi$
$661$	−2.80510e6	−0.249715	−0.124858	+	0.992175i	$0.539847\pi$
$662$	1.11592e7	0.989667
$663$	−1.75483e6	−0.155043
$664$	1.44900e6	0.127540
$665$	0	0
$666$	2.46570e6	0.215404
$667$	−2.00092e7	−1.74147
$668$	−7.15001e6	−0.619963
$669$	2.64556e6	0.228534
$670$	0	0
$671$	5.81789e6	0.498838
$672$	−4.42724e6	−0.378190
$673$	−1.88930e7	−1.60792	−0.803958	−	0.594686i	$-0.797277\pi$
$673$	−1.88930e7	−1.60792	−0.803958	+	0.594686i	$0.797277\pi$
$674$	1.00565e7	0.852705
$675$	0	0
$676$	456976.	0.0384615
$677$	−7.43678e6	−0.623610	−0.311805	−	0.950146i	$-0.600934\pi$
$677$	−7.43678e6	−0.623610	−0.311805	+	0.950146i	$0.600934\pi$
$678$	−1.50641e7	−1.25854
$679$	−1.46002e7	−1.21530
$680$	0	0
$681$	1.81500e7	1.49971
$682$	−1.36335e7	−1.12240
$683$	−1.01428e7	−0.831967	−0.415984	−	0.909372i	$-0.636563\pi$
$683$	−1.01428e7	−0.831967	−0.415984	+	0.909372i	$0.636563\pi$
$684$	−3.85846e6	−0.315336
$685$	0	0
$686$	−1.80920e7	−1.46783
$687$	1.36599e6	0.110422
$688$	−3.95557e6	−0.318594
$689$	−1.86785e6	−0.149897
$690$	0	0
$691$	−2.15408e6	−0.171620	−0.0858098	−	0.996312i	$-0.527348\pi$
$691$	−2.15408e6	−0.171620	−0.0858098	+	0.996312i	$0.527348\pi$
$692$	−3.49445e6	−0.277405
$693$	−1.35989e7	−1.07565
$694$	1.24097e6	0.0978051
$695$	0	0
$696$	8.23316e6	0.644234
$697$	−6.64711e6	−0.518264
$698$	4.71341e6	0.366181
$699$	−1.15013e7	−0.890334
$700$	0	0
$701$	−6.90173e6	−0.530472	−0.265236	−	0.964183i	$-0.585450\pi$
$701$	−6.90173e6	−0.530472	−0.265236	+	0.964183i	$0.585450\pi$
$702$	−1.70741e6	−0.130766
$703$	1.26907e7	0.968494
$704$	−2.23091e6	−0.169649
$705$	0	0
$706$	5.37429e6	0.405798
$707$	2.68661e7	2.02142
$708$	−7.24034e6	−0.542845
$709$	1.62641e7	1.21511	0.607555	−	0.794278i	$-0.292150\pi$
$709$	1.62641e7	1.21511	0.607555	+	0.794278i	$0.292150\pi$
$710$	0	0
$711$	−6.44764e6	−0.478329
$712$	−8.31131e6	−0.614426
$713$	−1.82417e7	−1.34382
$714$	−9.58180e6	−0.703399
$715$	0	0
$716$	5.42320e6	0.395342
$717$	−2.27600e7	−1.65339
$718$	1.04740e7	0.758233
$719$	−6.59643e6	−0.475868	−0.237934	−	0.971281i	$-0.576470\pi$
$719$	−6.59643e6	−0.475868	−0.237934	+	0.971281i	$0.576470\pi$
$720$	0	0
$721$	1.11857e7	0.801357
$722$	−9.95471e6	−0.710699
$723$	−3.35184e7	−2.38472
$724$	−7.05146e6	−0.499957
$725$	0	0
$726$	−1.01651e7	−0.715767
$727$	2.19325e7	1.53905	0.769523	−	0.638620i	$-0.220494\pi$
$727$	2.19325e7	1.53905	0.769523	+	0.638620i	$0.220494\pi$
$728$	2.49519e6	0.174492
$729$	3.53304e6	0.246223
$730$	0	0
$731$	−8.56098e6	−0.592557
$732$	−3.20301e6	−0.220943
$733$	1.67003e7	1.14806	0.574028	−	0.818836i	$-0.305380\pi$
$733$	1.67003e7	1.14806	0.574028	+	0.818836i	$0.305380\pi$
$734$	479273.	0.0328355
$735$	0	0
$736$	−2.98497e6	−0.203116
$737$	2.87455e7	1.94940
$738$	5.19378e6	0.351028
$739$	−1.69875e6	−0.114425	−0.0572123	−	0.998362i	$-0.518221\pi$
$739$	−1.69875e6	−0.114425	−0.0572123	+	0.998362i	$0.518221\pi$
$740$	0	0
$741$	7.05719e6	0.472157
$742$	−1.01989e7	−0.680052
$743$	−1.82334e7	−1.21170	−0.605851	−	0.795578i	$-0.707167\pi$
$743$	−1.82334e7	−1.21170	−0.605851	+	0.795578i	$0.707167\pi$
$744$	7.50585e6	0.497127
$745$	0	0
$746$	−1.07493e7	−0.707188
$747$	−2.45037e6	−0.160668
$748$	−4.82833e6	−0.315532
$749$	4.36521e7	2.84315
$750$	0	0
$751$	−1.47109e6	−0.0951784	−0.0475892	−	0.998867i	$-0.515154\pi$
$751$	−1.47109e6	−0.0951784	−0.0475892	+	0.998867i	$0.515154\pi$
$752$	4.16724e6	0.268722
$753$	8.01585e6	0.515184
$754$	−4.64021e6	−0.297241
$755$	0	0
$756$	−9.32285e6	−0.593259
$757$	−9.68714e6	−0.614407	−0.307203	−	0.951644i	$-0.599393\pi$
$757$	−9.68714e6	−0.614407	−0.307203	+	0.951644i	$0.599393\pi$
$758$	−4.65306e6	−0.294148
$759$	−2.97548e7	−1.87479
$760$	0	0
$761$	1.09169e7	0.683338	0.341669	−	0.939820i	$-0.389008\pi$
$761$	1.09169e7	0.683338	0.341669	+	0.939820i	$0.389008\pi$
$762$	2.12445e7	1.32544
$763$	3.63638e7	2.26130
$764$	1.02930e7	0.637982
$765$	0	0
$766$	−5.41441e6	−0.333411
$767$	4.08066e6	0.250462
$768$	1.22822e6	0.0751402
$769$	−1.70886e7	−1.04205	−0.521027	−	0.853540i	$-0.674451\pi$
$769$	−1.70886e7	−1.04205	−0.521027	+	0.853540i	$0.674451\pi$
$770$	0	0
$771$	5.07925e6	0.307726
$772$	−6.23487e6	−0.376517
$773$	−5.92288e6	−0.356520	−0.178260	−	0.983983i	$-0.557047\pi$
$773$	−5.92288e6	−0.356520	−0.178260	+	0.983983i	$0.557047\pi$
$774$	6.68919e6	0.401348
$775$	0	0
$776$	4.05042e6	0.241461
$777$	−2.46245e7	−1.46324
$778$	−6.56402e6	−0.388795
$779$	2.67318e7	1.57828
$780$	0	0
$781$	2.40533e7	1.41107
$782$	−6.46032e6	−0.377778
$783$	1.73373e7	1.01060
$784$	9.32173e6	0.541635
$785$	0	0
$786$	1.49443e7	0.862818
$787$	−2.75022e7	−1.58282	−0.791409	−	0.611288i	$-0.790652\pi$
$787$	−2.75022e7	−1.58282	−0.791409	+	0.611288i	$0.790652\pi$
$788$	−5.22332e6	−0.299662
$789$	2.46616e7	1.41036
$790$	0	0
$791$	4.63579e7	2.63441
$792$	3.77265e6	0.213715
$793$	1.80522e6	0.101941
$794$	−6.06947e6	−0.341664
$795$	0	0
$796$	6.93481e6	0.387929
$797$	1.29045e7	0.719606	0.359803	−	0.933028i	$-0.382844\pi$
$797$	1.29045e7	0.719606	0.359803	+	0.933028i	$0.382844\pi$
$798$	3.85339e7	2.14208
$799$	9.01909e6	0.499799
$800$	0	0
$801$	1.40551e7	0.774021
$802$	1.66607e7	0.914653
$803$	−2.27455e7	−1.24482
$804$	−1.58257e7	−0.863422
$805$	0	0
$806$	−4.23030e6	−0.229369
$807$	−3.06615e7	−1.65733
$808$	−7.45329e6	−0.401624
$809$	2.16172e7	1.16126	0.580628	−	0.814169i	$-0.302807\pi$
$809$	2.16172e7	1.16126	0.580628	+	0.814169i	$0.302807\pi$
$810$	0	0
$811$	−1.68933e7	−0.901906	−0.450953	−	0.892548i	$-0.648916\pi$
$811$	−1.68933e7	−0.901906	−0.450953	+	0.892548i	$0.648916\pi$
$812$	−2.53366e7	−1.34852
$813$	−2.64791e7	−1.40500
$814$	−1.24085e7	−0.656382
$815$	0	0
$816$	2.65821e6	0.139754
$817$	3.44286e7	1.80453
$818$	−8.65032e6	−0.452011
$819$	−4.21958e6	−0.219816
$820$	0	0
$821$	−7.06381e6	−0.365747	−0.182874	−	0.983136i	$-0.558540\pi$
$821$	−7.06381e6	−0.365747	−0.182874	+	0.983136i	$0.558540\pi$
$822$	−1.99274e7	−1.02866
$823$	−3.32953e7	−1.71350	−0.856748	−	0.515735i	$-0.827519\pi$
$823$	−3.32953e7	−1.71350	−0.856748	+	0.515735i	$0.827519\pi$
$824$	−3.10318e6	−0.159217
$825$	0	0
$826$	2.22813e7	1.13630
$827$	1.49348e7	0.759341	0.379670	−	0.925122i	$-0.376037\pi$
$827$	1.49348e7	0.759341	0.379670	+	0.925122i	$0.376037\pi$
$828$	5.04782e6	0.255875
$829$	1.10426e7	0.558065	0.279032	−	0.960282i	$-0.409986\pi$
$829$	1.10426e7	0.558065	0.279032	+	0.960282i	$0.409986\pi$
$830$	0	0
$831$	−1.41538e7	−0.711002
$832$	−692224.	−0.0346688
$833$	2.01749e7	1.00739
$834$	−73861.8	−0.00367710
$835$	0	0
$836$	1.94175e7	0.960897
$837$	1.58058e7	0.779834
$838$	2.34815e7	1.15509
$839$	2.86322e7	1.40427	0.702133	−	0.712046i	$-0.252231\pi$
$839$	2.86322e7	1.40427	0.702133	+	0.712046i	$0.252231\pi$
$840$	0	0
$841$	2.66063e7	1.29717
$842$	8.57705e6	0.416925
$843$	−1.03156e7	−0.499950
$844$	−1.21777e6	−0.0588448
$845$	0	0
$846$	−7.04714e6	−0.338522
$847$	3.12821e7	1.49826
$848$	2.82940e6	0.135115
$849$	2.09920e7	0.999505
$850$	0	0
$851$	−1.66026e7	−0.785871
$852$	−1.32424e7	−0.624983
$853$	−1.71429e7	−0.806700	−0.403350	−	0.915046i	$-0.632154\pi$
$853$	−1.71429e7	−0.806700	−0.403350	+	0.915046i	$0.632154\pi$
$854$	9.85691e6	0.462483
$855$	0	0
$856$	−1.21101e7	−0.564889
$857$	−132094.	−0.00614373	−0.00307186	−	0.999995i	$-0.500978\pi$
$857$	−132094.	−0.00614373	−0.00307186	+	0.999995i	$0.500978\pi$
$858$	−6.90025e6	−0.319997
$859$	3.33727e7	1.54315	0.771577	−	0.636136i	$-0.219468\pi$
$859$	3.33727e7	1.54315	0.771577	+	0.636136i	$0.219468\pi$
$860$	0	0
$861$	−5.18694e7	−2.38453
$862$	−1.02612e7	−0.470359
$863$	−1.01919e7	−0.465829	−0.232915	−	0.972497i	$-0.574826\pi$
$863$	−1.01919e7	−0.465829	−0.232915	+	0.972497i	$0.574826\pi$
$864$	2.58637e6	0.117871
$865$	0	0
$866$	−2.43000e7	−1.10106
$867$	−2.08566e7	−0.942313
$868$	−2.30984e7	−1.04060
$869$	3.24473e7	1.45757
$870$	0	0
$871$	8.91938e6	0.398372
$872$	−1.00882e7	−0.449284
$873$	−6.84960e6	−0.304179
$874$	2.59806e7	1.15046
$875$	0	0
$876$	1.25224e7	0.551352
$877$	−1.53857e6	−0.0675490	−0.0337745	−	0.999429i	$-0.510753\pi$
$877$	−1.53857e6	−0.0675490	−0.0337745	+	0.999429i	$0.510753\pi$
$878$	−2.86906e7	−1.25604
$879$	2.76094e7	1.20527
$880$	0	0
$881$	−5.38590e6	−0.233786	−0.116893	−	0.993145i	$-0.537293\pi$
$881$	−5.38590e6	−0.233786	−0.116893	+	0.993145i	$0.537293\pi$
$882$	−1.57638e7	−0.682323
$883$	3.24054e7	1.39867	0.699336	−	0.714793i	$-0.253479\pi$
$883$	3.24054e7	1.39867	0.699336	+	0.714793i	$0.253479\pi$
$884$	−1.49817e6	−0.0644808
$885$	0	0
$886$	−6.21400e6	−0.265942
$887$	3.98506e7	1.70069	0.850346	−	0.526224i	$-0.176393\pi$
$887$	3.98506e7	1.70069	0.850346	+	0.526224i	$0.176393\pi$
$888$	6.83142e6	0.290722
$889$	−6.53776e7	−2.77444
$890$	0	0
$891$	4.01058e7	1.69244
$892$	2.25861e6	0.0950450
$893$	−3.62709e7	−1.52205
$894$	1.54701e6	0.0647363
$895$	0	0
$896$	−3.77970e6	−0.157285
$897$	−9.23255e6	−0.383125
$898$	1.42929e7	0.591464
$899$	4.29552e7	1.77262
$900$	0	0
$901$	6.12363e6	0.251303
$902$	−2.61373e7	−1.06966
$903$	−6.68039e7	−2.72636
$904$	−1.28608e7	−0.523414
$905$	0	0
$906$	−2.37256e6	−0.0960276
$907$	494733.	0.0199688	0.00998441	−	0.999950i	$-0.496822\pi$
$907$	494733.	0.0199688	0.00998441	+	0.999950i	$0.496822\pi$
$908$	1.54953e7	0.623715
$909$	1.26041e7	0.505944
$910$	0	0
$911$	1.03084e7	0.411526	0.205763	−	0.978602i	$-0.434032\pi$
$911$	1.03084e7	0.411526	0.205763	+	0.978602i	$0.434032\pi$
$912$	−1.06902e7	−0.425597
$913$	1.23313e7	0.489590
$914$	−3.25338e7	−1.28816
$915$	0	0
$916$	1.16620e6	0.0459233
$917$	−4.59894e7	−1.80607
$918$	5.59764e6	0.219229
$919$	3.06006e7	1.19520	0.597602	−	0.801793i	$-0.296120\pi$
$919$	3.06006e7	1.19520	0.597602	+	0.801793i	$0.296120\pi$
$920$	0	0
$921$	−2.66970e7	−1.03708
$922$	1.47969e7	0.573250
$923$	7.46344e6	0.288360
$924$	−3.76769e7	−1.45176
$925$	0	0
$926$	−5.36507e6	−0.205612
$927$	5.24772e6	0.200573
$928$	7.02896e6	0.267930
$929$	−4.45717e7	−1.69441	−0.847207	−	0.531263i	$-0.821718\pi$
$929$	−4.45717e7	−1.69441	−0.847207	+	0.531263i	$0.821718\pi$
$930$	0	0
$931$	−8.11347e7	−3.06784
$932$	−9.81907e6	−0.370280
$933$	−3.35833e7	−1.26305
$934$	9.64271e6	0.361686
$935$	0	0
$936$	1.17061e6	0.0436739
$937$	−4.43114e7	−1.64879	−0.824397	−	0.566012i	$-0.808485\pi$
$937$	−4.43114e7	−1.64879	−0.824397	+	0.566012i	$0.808485\pi$
$938$	4.87019e7	1.80733
$939$	5.50398e7	2.03710
$940$	0	0
$941$	1.48172e7	0.545498	0.272749	−	0.962085i	$-0.412067\pi$
$941$	1.48172e7	0.545498	0.272749	+	0.962085i	$0.412067\pi$
$942$	3.56149e7	1.30769
$943$	−3.49718e7	−1.28068
$944$	−6.18136e6	−0.225764
$945$	0	0
$946$	−3.36629e7	−1.22299
$947$	1.14497e7	0.414876	0.207438	−	0.978248i	$-0.433487\pi$
$947$	1.14497e7	0.414876	0.207438	+	0.978248i	$0.433487\pi$
$948$	−1.78637e7	−0.645581
$949$	−7.05765e6	−0.254387
$950$	0	0
$951$	−7.32734e6	−0.262721
$952$	−8.18035e6	−0.292536
$953$	3.21943e7	1.14828	0.574139	−	0.818758i	$-0.305337\pi$
$953$	3.21943e7	1.14828	0.574139	+	0.818758i	$0.305337\pi$
$954$	−4.78475e6	−0.170211
$955$	0	0
$956$	−1.94311e7	−0.687626
$957$	7.00663e7	2.47303
$958$	−74676.5	−0.00262888
$959$	6.13244e7	2.15321
$960$	0	0
$961$	1.05315e7	0.367858
$962$	−3.85019e6	−0.134136
$963$	2.04792e7	0.711617
$964$	−2.86159e7	−0.991780
$965$	0	0
$966$	−5.04118e7	−1.73816
$967$	−4.07255e7	−1.40055	−0.700277	−	0.713871i	$-0.746940\pi$
$967$	−4.07255e7	−1.40055	−0.700277	+	0.713871i	$0.746940\pi$
$968$	−8.67837e6	−0.297680
$969$	−2.31366e7	−0.791571
$970$	0	0
$971$	1.63928e7	0.557961	0.278980	−	0.960297i	$-0.410004\pi$
$971$	1.63928e7	0.557961	0.278980	+	0.960297i	$0.410004\pi$
$972$	−1.22599e7	−0.416220
$973$	227301.	0.00769698
$974$	−1.81266e7	−0.612236
$975$	0	0
$976$	−2.73453e6	−0.0918880
$977$	−4.03891e7	−1.35372	−0.676858	−	0.736114i	$-0.736659\pi$
$977$	−4.03891e7	−1.35372	−0.676858	+	0.736114i	$0.736659\pi$
$978$	−2.76167e7	−0.923261
$979$	−7.07314e7	−2.35861
$980$	0	0
$981$	1.70599e7	0.565984
$982$	−1.53996e7	−0.509601
$983$	−4.46121e7	−1.47254	−0.736272	−	0.676685i	$-0.763416\pi$
$983$	−4.46121e7	−1.47254	−0.736272	+	0.676685i	$0.763416\pi$
$984$	1.43898e7	0.473769
$985$	0	0
$986$	1.52127e7	0.498325
$987$	7.03787e7	2.29958
$988$	6.02499e6	0.196365
$989$	−4.50411e7	−1.46426
$990$	0	0
$991$	−2.35070e7	−0.760351	−0.380175	−	0.924914i	$-0.624136\pi$
$991$	−2.35070e7	−0.760351	−0.380175	+	0.924914i	$0.624136\pi$
$992$	6.40803e6	0.206750
$993$	−5.22841e7	−1.68266
$994$	4.07521e7	1.30823
$995$	0	0
$996$	−6.78895e6	−0.216847
$997$	−1.84154e6	−0.0586737	−0.0293369	−	0.999570i	$-0.509340\pi$
$997$	−1.84154e6	−0.0586737	−0.0293369	+	0.999570i	$0.509340\pi$
$998$	−8.75378e6	−0.278208
$999$	1.43855e7	0.456050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.a.s.1.5	✓	6
5.2	odd	4		650.6.b.o.599.2		12
5.3	odd	4		650.6.b.o.599.11		12
5.4	even	2		650.6.a.t.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.6.a.s.1.5	✓	6	1.1	even	1	trivial
650.6.a.t.1.2	yes	6	5.4	even	2
650.6.b.o.599.2		12	5.2	odd	4
650.6.b.o.599.11		12	5.3	odd	4

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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +18.7411 q^{3} +16.0000 q^{4} -74.9644 q^{6} +230.695 q^{7} -64.0000 q^{8} +108.229 q^{9} -544.656 q^{11} +299.858 q^{12} -169.000 q^{13} -922.779 q^{14} +256.000 q^{16} +554.057 q^{17} -432.917 q^{18} -2228.18 q^{19} +4323.47 q^{21} +2178.63 q^{22} +2915.01 q^{23} -1199.43 q^{24} +676.000 q^{26} -2525.75 q^{27} +3691.11 q^{28} -6864.22 q^{29} -6257.84 q^{31} -1024.00 q^{32} -10207.5 q^{33} -2216.23 q^{34} +1731.67 q^{36} -5695.55 q^{37} +8912.71 q^{38} -3167.25 q^{39} -11997.2 q^{41} -17293.9 q^{42} -15451.4 q^{43} -8714.50 q^{44} -11660.0 q^{46} +16278.3 q^{47} +4797.72 q^{48} +36413.0 q^{49} +10383.6 q^{51} -2704.00 q^{52} +11052.3 q^{53} +10103.0 q^{54} -14764.5 q^{56} -41758.5 q^{57} +27456.9 q^{58} -24145.9 q^{59} -10681.8 q^{61} +25031.4 q^{62} +24967.9 q^{63} +4096.00 q^{64} +40829.9 q^{66} -52777.4 q^{67} +8864.91 q^{68} +54630.5 q^{69} -44162.3 q^{71} -6926.67 q^{72} +41761.3 q^{73} +22782.2 q^{74} -35650.8 q^{76} -125649. q^{77} +12669.0 q^{78} -59573.9 q^{79} -73635.1 q^{81} +47988.7 q^{82} -22640.6 q^{83} +69175.6 q^{84} +61805.8 q^{86} -128643. q^{87} +34858.0 q^{88} +129864. q^{89} -38987.4 q^{91} +46640.1 q^{92} -117279. q^{93} -65113.1 q^{94} -19190.9 q^{96} -63287.9 q^{97} -145652. q^{98} -58947.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 24 q^{2} - 9 q^{3} + 96 q^{4} + 36 q^{6} + 63 q^{7} - 384 q^{8} + 629 q^{9} + 370 q^{11} - 144 q^{12} - 1014 q^{13} - 252 q^{14} + 1536 q^{16} - 838 q^{17} - 2516 q^{18} - 1659 q^{19} + 4322 q^{21}+ \cdots + 386119 q^{99}+O(q^{100}) \) 6 * q - 24 * q^2 - 9 * q^3 + 96 * q^4 + 36 * q^6 + 63 * q^7 - 384 * q^8 + 629 * q^9 + 370 * q^11 - 144 * q^12 - 1014 * q^13 - 252 * q^14 + 1536 * q^16 - 838 * q^17 - 2516 * q^18 - 1659 * q^19 + 4322 * q^21 - 1480 * q^22 - 1566 * q^23 + 576 * q^24 + 4056 * q^26 - 7569 * q^27 + 1008 * q^28 - 5911 * q^29 + 1403 * q^31 - 6144 * q^32 - 8993 * q^33 + 3352 * q^34 + 10064 * q^36 - 10598 * q^37 + 6636 * q^38 + 1521 * q^39 + 1487 * q^41 - 17288 * q^42 - 16078 * q^43 + 5920 * q^44 + 6264 * q^46 - 7135 * q^47 - 2304 * q^48 + 28977 * q^49 + 7053 * q^51 - 16224 * q^52 + 7399 * q^53 + 30276 * q^54 - 4032 * q^56 - 51223 * q^57 + 23644 * q^58 + 22367 * q^59 + 6711 * q^61 - 5612 * q^62 + 10699 * q^63 + 24576 * q^64 + 35972 * q^66 - 99008 * q^67 - 13408 * q^68 + 134338 * q^69 + 15220 * q^71 - 40256 * q^72 - 9115 * q^73 + 42392 * q^74 - 26544 * q^76 - 156999 * q^77 - 6084 * q^78 + 95428 * q^79 - 8134 * q^81 - 5948 * q^82 + 14856 * q^83 + 69152 * q^84 + 64312 * q^86 - 204762 * q^87 - 23680 * q^88 + 292311 * q^89 - 10647 * q^91 - 25056 * q^92 - 194512 * q^93 + 28540 * q^94 + 9216 * q^96 - 77054 * q^97 - 115908 * q^98 + 386119 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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