LMFDB - Embedded newform 819.6.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,6,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, 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("819.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	819.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:	$131.354348427$
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} - 111x^{4} + 75x^{3} + 2750x^{2} + 1800x - 1600 $ x^6 - 3x^5 - 111x^4 + 75x^3 + 2750x^2 + 1800*x - 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.502226$ of defining polynomial
Character	$\chi$	$=$	819.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49777 q^{2} -25.7611 q^{4} +68.1279 q^{5} -49.0000 q^{7} -144.274 q^{8} +O(q^{10})$	\(q+2.49777 q^{2} -25.7611 q^{4} +68.1279 q^{5} -49.0000 q^{7} -144.274 q^{8} +170.168 q^{10} -266.628 q^{11} +169.000 q^{13} -122.391 q^{14} +463.991 q^{16} -995.260 q^{17} +2154.37 q^{19} -1755.05 q^{20} -665.976 q^{22} +2439.72 q^{23} +1516.42 q^{25} +422.124 q^{26} +1262.30 q^{28} +338.189 q^{29} -598.379 q^{31} +5775.72 q^{32} -2485.93 q^{34} -3338.27 q^{35} -10007.9 q^{37} +5381.13 q^{38} -9829.11 q^{40} +11360.1 q^{41} -14664.2 q^{43} +6868.63 q^{44} +6093.86 q^{46} -2509.98 q^{47} +2401.00 q^{49} +3787.66 q^{50} -4353.63 q^{52} -16001.3 q^{53} -18164.8 q^{55} +7069.44 q^{56} +844.721 q^{58} -885.325 q^{59} +21875.3 q^{61} -1494.61 q^{62} -421.278 q^{64} +11513.6 q^{65} -6146.53 q^{67} +25639.0 q^{68} -8338.24 q^{70} -39627.1 q^{71} -29822.5 q^{73} -24997.4 q^{74} -55499.0 q^{76} +13064.8 q^{77} -8448.65 q^{79} +31610.8 q^{80} +28375.0 q^{82} +88251.4 q^{83} -67805.0 q^{85} -36628.0 q^{86} +38467.5 q^{88} -138753. q^{89} -8281.00 q^{91} -62849.8 q^{92} -6269.37 q^{94} +146773. q^{95} -132357. q^{97} +5997.16 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 15 q^{2} + 75 q^{4} - 15 q^{5} - 294 q^{7} + 399 q^{8}+O(q^{10}) $ 6 * q + 15 * q^2 + 75 * q^4 - 15 * q^5 - 294 * q^7 + 399 * q^8	$ 6 q + 15 q^{2} + 75 q^{4} - 15 q^{5} - 294 q^{7} + 399 q^{8} - 265 q^{10} + 90 q^{11} + 1014 q^{13} - 735 q^{14} + 355 q^{16} + 1314 q^{17} - 341 q^{19} - 2085 q^{20} - 6749 q^{22} + 4905 q^{23} - 10231 q^{25} + 2535 q^{26} - 3675 q^{28} + 6771 q^{29} - 14711 q^{31} + 10143 q^{32} + 9595 q^{34} + 735 q^{35} - 13314 q^{37} - 1017 q^{38} - 13195 q^{40} + 5040 q^{41} - 9127 q^{43} - 10041 q^{44} + 23739 q^{46} + 4713 q^{47} + 14406 q^{49} - 18018 q^{50} + 12675 q^{52} + 159 q^{53} - 13590 q^{55} - 19551 q^{56} - 38427 q^{58} - 30288 q^{59} - 78126 q^{61} - 40632 q^{62} - 8461 q^{64} - 2535 q^{65} - 33894 q^{67} + 61863 q^{68} + 12985 q^{70} - 8316 q^{71} - 89861 q^{73} + 16635 q^{74} - 169069 q^{76} - 4410 q^{77} - 239803 q^{79} - 19305 q^{80} - 106624 q^{82} + 42753 q^{83} - 184316 q^{85} - 51609 q^{86} - 205923 q^{88} + 11421 q^{89} - 49686 q^{91} + 252519 q^{92} - 4976 q^{94} + 180807 q^{95} - 199445 q^{97} + 36015 q^{98}+O(q^{100}) $ 6 * q + 15 * q^2 + 75 * q^4 - 15 * q^5 - 294 * q^7 + 399 * q^8 - 265 * q^10 + 90 * q^11 + 1014 * q^13 - 735 * q^14 + 355 * q^16 + 1314 * q^17 - 341 * q^19 - 2085 * q^20 - 6749 * q^22 + 4905 * q^23 - 10231 * q^25 + 2535 * q^26 - 3675 * q^28 + 6771 * q^29 - 14711 * q^31 + 10143 * q^32 + 9595 * q^34 + 735 * q^35 - 13314 * q^37 - 1017 * q^38 - 13195 * q^40 + 5040 * q^41 - 9127 * q^43 - 10041 * q^44 + 23739 * q^46 + 4713 * q^47 + 14406 * q^49 - 18018 * q^50 + 12675 * q^52 + 159 * q^53 - 13590 * q^55 - 19551 * q^56 - 38427 * q^58 - 30288 * q^59 - 78126 * q^61 - 40632 * q^62 - 8461 * q^64 - 2535 * q^65 - 33894 * q^67 + 61863 * q^68 + 12985 * q^70 - 8316 * q^71 - 89861 * q^73 + 16635 * q^74 - 169069 * q^76 - 4410 * q^77 - 239803 * q^79 - 19305 * q^80 - 106624 * q^82 + 42753 * q^83 - 184316 * q^85 - 51609 * q^86 - 205923 * q^88 + 11421 * q^89 - 49686 * q^91 + 252519 * q^92 - 4976 * q^94 + 180807 * q^95 - 199445 * q^97 + 36015 * 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 $

$2$	2.49777	0.441548	0.220774	−	0.975325i	$-0.429142\pi$
$2$	2.49777	0.441548	0.220774	+	0.975325i	$0.429142\pi$
$3$	0	0
$3$	0	0
$4$	−25.7611	−0.805035
$5$	68.1279	1.21871	0.609355	−	0.792898i	$-0.291429\pi$
$5$	68.1279	1.21871	0.609355	+	0.792898i	$0.291429\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−144.274	−0.797010
$9$	0	0
$10$	170.168	0.538119
$11$	−266.628	−0.664391	−0.332196	−	0.943211i	$-0.607789\pi$
$11$	−266.628	−0.664391	−0.332196	+	0.943211i	$0.607789\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−122.391	−0.166890
$15$	0	0
$16$	463.991	0.453117
$17$	−995.260	−0.835246	−0.417623	−	0.908621i	$-0.637137\pi$
$17$	−995.260	−0.835246	−0.417623	+	0.908621i	$0.637137\pi$
$18$	0	0
$19$	2154.37	1.36910	0.684551	−	0.728965i	$-0.259998\pi$
$19$	2154.37	1.36910	0.684551	+	0.728965i	$0.259998\pi$
$20$	−1755.05	−0.981104
$21$	0	0
$22$	−665.976	−0.293361
$23$	2439.72	0.961656	0.480828	−	0.876815i	$-0.340336\pi$
$23$	2439.72	0.961656	0.480828	+	0.876815i	$0.340336\pi$
$24$	0	0
$25$	1516.42	0.485253
$26$	422.124	0.122463
$27$	0	0
$28$	1262.30	0.304275
$29$	338.189	0.0746733	0.0373366	−	0.999303i	$-0.488113\pi$
$29$	338.189	0.0746733	0.0373366	+	0.999303i	$0.488113\pi$
$30$	0	0
$31$	−598.379	−0.111833	−0.0559167	−	0.998435i	$-0.517808\pi$
$31$	−598.379	−0.111833	−0.0559167	+	0.998435i	$0.517808\pi$
$32$	5775.72	0.997083
$33$	0	0
$34$	−2485.93	−0.368801
$35$	−3338.27	−0.460629
$36$	0	0
$37$	−10007.9	−1.20182	−0.600908	−	0.799318i	$-0.705194\pi$
$37$	−10007.9	−1.20182	−0.600908	+	0.799318i	$0.705194\pi$
$38$	5381.13	0.604525
$39$	0	0
$40$	−9829.11	−0.971324
$41$	11360.1	1.05541	0.527707	−	0.849426i	$-0.323052\pi$
$41$	11360.1	1.05541	0.527707	+	0.849426i	$0.323052\pi$
$42$	0	0
$43$	−14664.2	−1.20945	−0.604726	−	0.796434i	$-0.706717\pi$
$43$	−14664.2	−1.20945	−0.604726	+	0.796434i	$0.706717\pi$
$44$	6868.63	0.534858
$45$	0	0
$46$	6093.86	0.424617
$47$	−2509.98	−0.165739	−0.0828697	−	0.996560i	$-0.526409\pi$
$47$	−2509.98	−0.165739	−0.0828697	+	0.996560i	$0.526409\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	3787.66	0.214263
$51$	0	0
$52$	−4353.63	−0.223277
$53$	−16001.3	−0.782468	−0.391234	−	0.920291i	$-0.627952\pi$
$53$	−16001.3	−0.782468	−0.391234	+	0.920291i	$0.627952\pi$
$54$	0	0
$55$	−18164.8	−0.809700
$56$	7069.44	0.301242
$57$	0	0
$58$	844.721	0.0329718
$59$	−885.325	−0.0331110	−0.0165555	−	0.999863i	$-0.505270\pi$
$59$	−885.325	−0.0331110	−0.0165555	+	0.999863i	$0.505270\pi$
$60$	0	0
$61$	21875.3	0.752712	0.376356	−	0.926475i	$-0.377177\pi$
$61$	21875.3	0.752712	0.376356	+	0.926475i	$0.377177\pi$
$62$	−1494.61	−0.0493799
$63$	0	0
$64$	−421.278	−0.0128564
$65$	11513.6	0.338009
$66$	0	0
$67$	−6146.53	−0.167280	−0.0836398	−	0.996496i	$-0.526655\pi$
$67$	−6146.53	−0.167280	−0.0836398	+	0.996496i	$0.526655\pi$
$68$	25639.0	0.672402
$69$	0	0
$70$	−8338.24	−0.203390
$71$	−39627.1	−0.932924	−0.466462	−	0.884541i	$-0.654471\pi$
$71$	−39627.1	−0.932924	−0.466462	+	0.884541i	$0.654471\pi$
$72$	0	0
$73$	−29822.5	−0.654994	−0.327497	−	0.944852i	$-0.606205\pi$
$73$	−29822.5	−0.654994	−0.327497	+	0.944852i	$0.606205\pi$
$74$	−24997.4	−0.530660
$75$	0	0
$76$	−55499.0	−1.10218
$77$	13064.8	0.251116
$78$	0	0
$79$	−8448.65	−0.152307	−0.0761534	−	0.997096i	$-0.524264\pi$
$79$	−8448.65	−0.152307	−0.0761534	+	0.997096i	$0.524264\pi$
$80$	31610.8	0.552218
$81$	0	0
$82$	28375.0	0.466016
$83$	88251.4	1.40613	0.703067	−	0.711124i	$-0.251814\pi$
$83$	88251.4	1.40613	0.703067	+	0.711124i	$0.251814\pi$
$84$	0	0
$85$	−67805.0	−1.01792
$86$	−36628.0	−0.534032
$87$	0	0
$88$	38467.5	0.529526
$89$	−138753.	−1.85680	−0.928402	−	0.371577i	$-0.878817\pi$
$89$	−138753.	−1.85680	−0.928402	+	0.371577i	$0.878817\pi$
$90$	0	0
$91$	−8281.00	−0.104828
$92$	−62849.8	−0.774167
$93$	0	0
$94$	−6269.37	−0.0731820
$95$	146773.	1.66854
$96$	0	0
$97$	−132357.	−1.42829	−0.714147	−	0.699996i	$-0.753185\pi$
$97$	−132357.	−1.42829	−0.714147	+	0.699996i	$0.753185\pi$
$98$	5997.16	0.0630783
$99$	0	0
$100$	−39064.6	−0.390646
$101$	134455.	1.31152	0.655758	−	0.754972i	$-0.272349\pi$
$101$	134455.	1.31152	0.655758	+	0.754972i	$0.272349\pi$
$102$	0	0
$103$	−107230.	−0.995921	−0.497960	−	0.867200i	$-0.665918\pi$
$103$	−107230.	−0.995921	−0.497960	+	0.867200i	$0.665918\pi$
$104$	−24382.3	−0.221051
$105$	0	0
$106$	−39967.7	−0.345497
$107$	97183.5	0.820603	0.410302	−	0.911950i	$-0.365423\pi$
$107$	97183.5	0.820603	0.410302	+	0.911950i	$0.365423\pi$
$108$	0	0
$109$	59828.9	0.482331	0.241165	−	0.970484i	$-0.422470\pi$
$109$	59828.9	0.482331	0.241165	+	0.970484i	$0.422470\pi$
$110$	−45371.6	−0.357522
$111$	0	0
$112$	−22735.6	−0.171262
$113$	116954.	0.861626	0.430813	−	0.902441i	$-0.358227\pi$
$113$	116954.	0.861626	0.430813	+	0.902441i	$0.358227\pi$
$114$	0	0
$115$	166213.	1.17198
$116$	−8712.14	−0.0601146
$117$	0	0
$118$	−2211.34	−0.0146201
$119$	48767.7	0.315693
$120$	0	0
$121$	−89960.6	−0.558584
$122$	54639.5	0.332359
$123$	0	0
$124$	15414.9	0.0900299
$125$	−109590.	−0.627327
$126$	0	0
$127$	−58754.7	−0.323246	−0.161623	−	0.986853i	$-0.551673\pi$
$127$	−58754.7	−0.323246	−0.161623	+	0.986853i	$0.551673\pi$
$128$	−185875.	−1.00276
$129$	0	0
$130$	28758.4	0.149247
$131$	−255806.	−1.30237	−0.651183	−	0.758920i	$-0.725727\pi$
$131$	−255806.	−1.30237	−0.651183	+	0.758920i	$0.725727\pi$
$132$	0	0
$133$	−105564.	−0.517472
$134$	−15352.6	−0.0738620
$135$	0	0
$136$	143590.	0.665699
$137$	121429.	0.552741	0.276371	−	0.961051i	$-0.410868\pi$
$137$	121429.	0.552741	0.276371	+	0.961051i	$0.410868\pi$
$138$	0	0
$139$	−303794.	−1.33365	−0.666825	−	0.745214i	$-0.732347\pi$
$139$	−303794.	−1.33365	−0.666825	+	0.745214i	$0.732347\pi$
$140$	85997.6	0.370822
$141$	0	0
$142$	−98979.5	−0.411931
$143$	−45060.1	−0.184269
$144$	0	0
$145$	23040.2	0.0910050
$146$	−74490.0	−0.289212
$147$	0	0
$148$	257814.	0.967504
$149$	298489.	1.10144	0.550722	−	0.834688i	$-0.314352\pi$
$149$	298489.	1.10144	0.550722	+	0.834688i	$0.314352\pi$
$150$	0	0
$151$	−74579.3	−0.266180	−0.133090	−	0.991104i	$-0.542490\pi$
$151$	−74579.3	−0.266180	−0.133090	+	0.991104i	$0.542490\pi$
$152$	−310820.	−1.09119
$153$	0	0
$154$	32632.8	0.110880
$155$	−40766.3	−0.136293
$156$	0	0
$157$	−199502.	−0.645950	−0.322975	−	0.946408i	$-0.604683\pi$
$157$	−199502.	−0.645950	−0.322975	+	0.946408i	$0.604683\pi$
$158$	−21102.8	−0.0672508
$159$	0	0
$160$	393488.	1.21515
$161$	−119546.	−0.363472
$162$	0	0
$163$	−70986.6	−0.209270	−0.104635	−	0.994511i	$-0.533367\pi$
$163$	−70986.6	−0.209270	−0.104635	+	0.994511i	$0.533367\pi$
$164$	−292649.	−0.849645
$165$	0	0
$166$	220432.	0.620876
$167$	−495866.	−1.37586	−0.687928	−	0.725779i	$-0.741479\pi$
$167$	−495866.	−1.37586	−0.687928	+	0.725779i	$0.741479\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	−169362.	−0.449462
$171$	0	0
$172$	377768.	0.973651
$173$	105536.	0.268092	0.134046	−	0.990975i	$-0.457203\pi$
$173$	105536.	0.268092	0.134046	+	0.990975i	$0.457203\pi$
$174$	0	0
$175$	−74304.4	−0.183408
$176$	−123713.	−0.301047
$177$	0	0
$178$	−346573.	−0.819869
$179$	−645778.	−1.50644	−0.753218	−	0.657771i	$-0.771500\pi$
$179$	−645778.	−1.50644	−0.753218	+	0.657771i	$0.771500\pi$
$180$	0	0
$181$	−651817.	−1.47887	−0.739434	−	0.673229i	$-0.764907\pi$
$181$	−651817.	−1.47887	−0.739434	+	0.673229i	$0.764907\pi$
$182$	−20684.1	−0.0462868
$183$	0	0
$184$	−351988.	−0.766449
$185$	−681817.	−1.46467
$186$	0	0
$187$	265364.	0.554930
$188$	64660.0	0.133426
$189$	0	0
$190$	366605.	0.736740
$191$	−500030.	−0.991774	−0.495887	−	0.868387i	$-0.665157\pi$
$191$	−500030.	−0.991774	−0.495887	+	0.868387i	$0.665157\pi$
$192$	0	0
$193$	290979.	0.562300	0.281150	−	0.959664i	$-0.409284\pi$
$193$	290979.	0.562300	0.281150	+	0.959664i	$0.409284\pi$
$194$	−330598.	−0.630661
$195$	0	0
$196$	−61852.5	−0.115005
$197$	425456.	0.781068	0.390534	−	0.920588i	$-0.372290\pi$
$197$	425456.	0.781068	0.390534	+	0.920588i	$0.372290\pi$
$198$	0	0
$199$	−822265.	−1.47190	−0.735951	−	0.677035i	$-0.763265\pi$
$199$	−822265.	−1.47190	−0.735951	+	0.677035i	$0.763265\pi$
$200$	−218780.	−0.386752
$201$	0	0
$202$	335838.	0.579097
$203$	−16571.3	−0.0282238
$204$	0	0
$205$	773941.	1.28624
$206$	−267837.	−0.439747
$207$	0	0
$208$	78414.6	0.125672
$209$	−574415.	−0.909620
$210$	0	0
$211$	−574429.	−0.888240	−0.444120	−	0.895967i	$-0.646484\pi$
$211$	−574429.	−0.888240	−0.444120	+	0.895967i	$0.646484\pi$
$212$	412212.	0.629914
$213$	0	0
$214$	242743.	0.362336
$215$	−999045.	−1.47397
$216$	0	0
$217$	29320.6	0.0422691
$218$	149439.	0.212972
$219$	0	0
$220$	467946.	0.651837
$221$	−168199.	−0.231655
$222$	0	0
$223$	−1.18160e6	−1.59114	−0.795568	−	0.605864i	$-0.792828\pi$
$223$	−1.18160e6	−1.59114	−0.795568	+	0.605864i	$0.792828\pi$
$224$	−283010.	−0.376862
$225$	0	0
$226$	292124.	0.380449
$227$	−1.37166e6	−1.76677	−0.883387	−	0.468644i	$-0.844743\pi$
$227$	−1.37166e6	−1.76677	−0.883387	+	0.468644i	$0.844743\pi$
$228$	0	0
$229$	−1.47499e6	−1.85866	−0.929328	−	0.369255i	$-0.879613\pi$
$229$	−1.47499e6	−1.85866	−0.929328	+	0.369255i	$0.879613\pi$
$230$	415162.	0.517485
$231$	0	0
$232$	−48792.0	−0.0595153
$233$	−440436.	−0.531487	−0.265744	−	0.964044i	$-0.585617\pi$
$233$	−440436.	−0.531487	−0.265744	+	0.964044i	$0.585617\pi$
$234$	0	0
$235$	−171000.	−0.201988
$236$	22807.0	0.0266555
$237$	0	0
$238$	121811.	0.139394
$239$	566941.	0.642012	0.321006	−	0.947077i	$-0.395979\pi$
$239$	566941.	0.642012	0.321006	+	0.947077i	$0.395979\pi$
$240$	0	0
$241$	1.45248e6	1.61089	0.805446	−	0.592669i	$-0.201926\pi$
$241$	1.45248e6	1.61089	0.805446	+	0.592669i	$0.201926\pi$
$242$	−224701.	−0.246642
$243$	0	0
$244$	−563531.	−0.605960
$245$	163575.	0.174101
$246$	0	0
$247$	364088.	0.379721
$248$	86330.6	0.0891324
$249$	0	0
$250$	−273730.	−0.276995
$251$	109150.	0.109355	0.0546777	−	0.998504i	$-0.482587\pi$
$251$	109150.	0.109355	0.0546777	+	0.998504i	$0.482587\pi$
$252$	0	0
$253$	−650496.	−0.638916
$254$	−146756.	−0.142729
$255$	0	0
$256$	−450794.	−0.429910
$257$	92270.2	0.0871422	0.0435711	−	0.999050i	$-0.486126\pi$
$257$	92270.2	0.0871422	0.0435711	+	0.999050i	$0.486126\pi$
$258$	0	0
$259$	490387.	0.454244
$260$	−296604.	−0.272109
$261$	0	0
$262$	−638947.	−0.575058
$263$	1.46686e6	1.30767	0.653837	−	0.756635i	$-0.273158\pi$
$263$	1.46686e6	1.30767	0.653837	+	0.756635i	$0.273158\pi$
$264$	0	0
$265$	−1.09014e6	−0.953601
$266$	−263675.	−0.228489
$267$	0	0
$268$	158342.	0.134666
$269$	914324.	0.770406	0.385203	−	0.922832i	$-0.374131\pi$
$269$	914324.	0.770406	0.385203	+	0.922832i	$0.374131\pi$
$270$	0	0
$271$	1.06702e6	0.882569	0.441285	−	0.897367i	$-0.354523\pi$
$271$	1.06702e6	0.882569	0.441285	+	0.897367i	$0.354523\pi$
$272$	−461792.	−0.378464
$273$	0	0
$274$	303303.	0.244062
$275$	−404319.	−0.322398
$276$	0	0
$277$	−1.87666e6	−1.46956	−0.734778	−	0.678307i	$-0.762714\pi$
$277$	−1.87666e6	−1.46956	−0.734778	+	0.678307i	$0.762714\pi$
$278$	−758808.	−0.588871
$279$	0	0
$280$	481626.	0.367126
$281$	−928464.	−0.701454	−0.350727	−	0.936478i	$-0.614066\pi$
$281$	−928464.	−0.701454	−0.350727	+	0.936478i	$0.614066\pi$
$282$	0	0
$283$	1.29931e6	0.964374	0.482187	−	0.876068i	$-0.339843\pi$
$283$	1.29931e6	0.964374	0.482187	+	0.876068i	$0.339843\pi$
$284$	1.02084e6	0.751036
$285$	0	0
$286$	−112550.	−0.0813636
$287$	−556645.	−0.398909
$288$	0	0
$289$	−429315.	−0.302365
$290$	57549.1	0.0401831
$291$	0	0
$292$	768262.	0.527294
$293$	333507.	0.226953	0.113476	−	0.993541i	$-0.463801\pi$
$293$	333507.	0.226953	0.113476	+	0.993541i	$0.463801\pi$
$294$	0	0
$295$	−60315.3	−0.0403527
$296$	1.44388e6	0.957860
$297$	0	0
$298$	745558.	0.486341
$299$	412312.	0.266715
$300$	0	0
$301$	718548.	0.457130
$302$	−186282.	−0.117531
$303$	0	0
$304$	999609.	0.620363
$305$	1.49032e6	0.917337
$306$	0	0
$307$	1.76526e6	1.06896	0.534481	−	0.845181i	$-0.320507\pi$
$307$	1.76526e6	1.06896	0.534481	+	0.845181i	$0.320507\pi$
$308$	−336563.	−0.202157
$309$	0	0
$310$	−101825.	−0.0601797
$311$	1.70956e6	1.00227	0.501135	−	0.865369i	$-0.332916\pi$
$311$	1.70956e6	1.00227	0.501135	+	0.865369i	$0.332916\pi$
$312$	0	0
$313$	−1.66406e6	−0.960081	−0.480041	−	0.877246i	$-0.659378\pi$
$313$	−1.66406e6	−0.960081	−0.480041	+	0.877246i	$0.659378\pi$
$314$	−498311.	−0.285218
$315$	0	0
$316$	217647.	0.122612
$317$	−298918.	−0.167072	−0.0835361	−	0.996505i	$-0.526621\pi$
$317$	−298918.	−0.167072	−0.0835361	+	0.996505i	$0.526621\pi$
$318$	0	0
$319$	−90170.7	−0.0496122
$320$	−28700.8	−0.0156682
$321$	0	0
$322$	−298599.	−0.160490
$323$	−2.14416e6	−1.14354
$324$	0	0
$325$	256274.	0.134585
$326$	−177309.	−0.0924029
$327$	0	0
$328$	−1.63897e6	−0.841176
$329$	122989.	0.0626436
$330$	0	0
$331$	3.03880e6	1.52452	0.762259	−	0.647272i	$-0.224090\pi$
$331$	3.03880e6	1.52452	0.762259	+	0.647272i	$0.224090\pi$
$332$	−2.27346e6	−1.13199
$333$	0	0
$334$	−1.23856e6	−0.607507
$335$	−418750.	−0.203865
$336$	0	0
$337$	2.24160e6	1.07519	0.537593	−	0.843204i	$-0.319334\pi$
$337$	2.24160e6	1.07519	0.537593	+	0.843204i	$0.319334\pi$
$338$	71338.9	0.0339653
$339$	0	0
$340$	1.74673e6	0.819463
$341$	159544.	0.0743012
$342$	0	0
$343$	−117649.	−0.0539949
$344$	2.11567e6	0.963946
$345$	0	0
$346$	263604.	0.118376
$347$	2.40273e6	1.07123	0.535614	−	0.844463i	$-0.320080\pi$
$347$	2.40273e6	1.07123	0.535614	+	0.844463i	$0.320080\pi$
$348$	0	0
$349$	−4.25858e6	−1.87155	−0.935775	−	0.352599i	$-0.885298\pi$
$349$	−4.25858e6	−1.87155	−0.935775	+	0.352599i	$0.885298\pi$
$350$	−185596.	−0.0809837
$351$	0	0
$352$	−1.53997e6	−0.662453
$353$	1.59483e6	0.681206	0.340603	−	0.940207i	$-0.389369\pi$
$353$	1.59483e6	0.681206	0.340603	+	0.940207i	$0.389369\pi$
$354$	0	0
$355$	−2.69971e6	−1.13696
$356$	3.57442e6	1.49479
$357$	0	0
$358$	−1.61301e6	−0.665164
$359$	−2.37408e6	−0.972207	−0.486104	−	0.873901i	$-0.661582\pi$
$359$	−2.37408e6	−0.972207	−0.486104	+	0.873901i	$0.661582\pi$
$360$	0	0
$361$	2.16521e6	0.874443
$362$	−1.62809e6	−0.652991
$363$	0	0
$364$	213328.	0.0843906
$365$	−2.03175e6	−0.798248
$366$	0	0
$367$	1.26617e6	0.490711	0.245355	−	0.969433i	$-0.421095\pi$
$367$	1.26617e6	0.490711	0.245355	+	0.969433i	$0.421095\pi$
$368$	1.13201e6	0.435742
$369$	0	0
$370$	−1.70302e6	−0.646720
$371$	784065.	0.295745
$372$	0	0
$373$	188334.	0.0700900	0.0350450	−	0.999386i	$-0.488843\pi$
$373$	188334.	0.0700900	0.0350450	+	0.999386i	$0.488843\pi$
$374$	662819.	0.245028
$375$	0	0
$376$	362126.	0.132096
$377$	57154.0	0.0207106
$378$	0	0
$379$	−3.25232e6	−1.16304	−0.581521	−	0.813531i	$-0.697542\pi$
$379$	−3.25232e6	−1.16304	−0.581521	+	0.813531i	$0.697542\pi$
$380$	−3.78103e6	−1.34323
$381$	0	0
$382$	−1.24896e6	−0.437916
$383$	122177.	0.0425590	0.0212795	−	0.999774i	$-0.493226\pi$
$383$	122177.	0.0425590	0.0212795	+	0.999774i	$0.493226\pi$
$384$	0	0
$385$	890075.	0.306038
$386$	726800.	0.248283
$387$	0	0
$388$	3.40967e6	1.14983
$389$	−787620.	−0.263902	−0.131951	−	0.991256i	$-0.542124\pi$
$389$	−787620.	−0.263902	−0.131951	+	0.991256i	$0.542124\pi$
$390$	0	0
$391$	−2.42815e6	−0.803219
$392$	−346402.	−0.113859
$393$	0	0
$394$	1.06269e6	0.344879
$395$	−575589.	−0.185618
$396$	0	0
$397$	−2.38252e6	−0.758683	−0.379341	−	0.925257i	$-0.623849\pi$
$397$	−2.38252e6	−0.758683	−0.379341	+	0.925257i	$0.623849\pi$
$398$	−2.05383e6	−0.649916
$399$	0	0
$400$	703604.	0.219876
$401$	−2.75440e6	−0.855393	−0.427696	−	0.903923i	$-0.640675\pi$
$401$	−2.75440e6	−0.855393	−0.427696	+	0.903923i	$0.640675\pi$
$402$	0	0
$403$	−101126.	−0.0310170
$404$	−3.46371e6	−1.05582
$405$	0	0
$406$	−41391.3	−0.0124622
$407$	2.66838e6	0.798476
$408$	0	0
$409$	−704205.	−0.208157	−0.104078	−	0.994569i	$-0.533189\pi$
$409$	−704205.	−0.208157	−0.104078	+	0.994569i	$0.533189\pi$
$410$	1.93313e6	0.567938
$411$	0	0
$412$	2.76238e6	0.801751
$413$	43380.9	0.0125148
$414$	0	0
$415$	6.01239e6	1.71367
$416$	976097.	0.276541
$417$	0	0
$418$	−1.43476e6	−0.401641
$419$	285103.	0.0793353	0.0396677	−	0.999213i	$-0.487370\pi$
$419$	285103.	0.0793353	0.0396677	+	0.999213i	$0.487370\pi$
$420$	0	0
$421$	−4.90965e6	−1.35003	−0.675017	−	0.737802i	$-0.735864\pi$
$421$	−4.90965e6	−1.35003	−0.675017	+	0.737802i	$0.735864\pi$
$422$	−1.43479e6	−0.392201
$423$	0	0
$424$	2.30858e6	0.623635
$425$	−1.50923e6	−0.405305
$426$	0	0
$427$	−1.07189e6	−0.284498
$428$	−2.50356e6	−0.660614
$429$	0	0
$430$	−2.49539e6	−0.650829
$431$	−785022.	−0.203558	−0.101779	−	0.994807i	$-0.532454\pi$
$431$	−785022.	−0.203558	−0.101779	+	0.994807i	$0.532454\pi$
$432$	0	0
$433$	−2.45765e6	−0.629942	−0.314971	−	0.949101i	$-0.601995\pi$
$433$	−2.45765e6	−0.629942	−0.314971	+	0.949101i	$0.601995\pi$
$434$	73236.1	0.0186638
$435$	0	0
$436$	−1.54126e6	−0.388293
$437$	5.25605e6	1.31661
$438$	0	0
$439$	−444184.	−0.110002	−0.0550011	−	0.998486i	$-0.517516\pi$
$439$	−444184.	−0.110002	−0.0550011	+	0.998486i	$0.517516\pi$
$440$	2.62071e6	0.645339
$441$	0	0
$442$	−420123.	−0.102287
$443$	2.41197e6	0.583933	0.291967	−	0.956428i	$-0.405690\pi$
$443$	2.41197e6	0.583933	0.291967	+	0.956428i	$0.405690\pi$
$444$	0	0
$445$	−9.45293e6	−2.26290
$446$	−2.95136e6	−0.702564
$447$	0	0
$448$	20642.6	0.00485926
$449$	−1.58394e6	−0.370785	−0.185393	−	0.982665i	$-0.559356\pi$
$449$	−1.58394e6	−0.370785	−0.185393	+	0.982665i	$0.559356\pi$
$450$	0	0
$451$	−3.02892e6	−0.701208
$452$	−3.01286e6	−0.693639
$453$	0	0
$454$	−3.42609e6	−0.780116
$455$	−564167.	−0.127755
$456$	0	0
$457$	1.55855e6	0.349083	0.174542	−	0.984650i	$-0.444156\pi$
$457$	1.55855e6	0.349083	0.174542	+	0.984650i	$0.444156\pi$
$458$	−3.68418e6	−0.820686
$459$	0	0
$460$	−4.28183e6	−0.943484
$461$	1.62909e6	0.357020	0.178510	−	0.983938i	$-0.442872\pi$
$461$	1.62909e6	0.357020	0.178510	+	0.983938i	$0.442872\pi$
$462$	0	0
$463$	−2.58723e6	−0.560896	−0.280448	−	0.959869i	$-0.590483\pi$
$463$	−2.58723e6	−0.560896	−0.280448	+	0.959869i	$0.590483\pi$
$464$	156917.	0.0338357
$465$	0	0
$466$	−1.10011e6	−0.234677
$467$	−144327.	−0.0306236	−0.0153118	−	0.999883i	$-0.504874\pi$
$467$	−144327.	−0.0306236	−0.0153118	+	0.999883i	$0.504874\pi$
$468$	0	0
$469$	301180.	0.0632258
$470$	−427119.	−0.0891876
$471$	0	0
$472$	127730.	0.0263898
$473$	3.90990e6	0.803549
$474$	0	0
$475$	3.26692e6	0.664361
$476$	−1.25631e6	−0.254144
$477$	0	0
$478$	1.41609e6	0.283479
$479$	8.57587e6	1.70781	0.853904	−	0.520430i	$-0.174228\pi$
$479$	8.57587e6	1.70781	0.853904	+	0.520430i	$0.174228\pi$
$480$	0	0
$481$	−1.69133e6	−0.333324
$482$	3.62796e6	0.711287
$483$	0	0
$484$	2.31749e6	0.449680
$485$	−9.01721e6	−1.74068
$486$	0	0
$487$	7.98797e6	1.52621	0.763105	−	0.646275i	$-0.223674\pi$
$487$	7.98797e6	1.52621	0.763105	+	0.646275i	$0.223674\pi$
$488$	−3.15604e6	−0.599919
$489$	0	0
$490$	408574.	0.0768742
$491$	2.37954e6	0.445440	0.222720	−	0.974882i	$-0.428506\pi$
$491$	2.37954e6	0.445440	0.222720	+	0.974882i	$0.428506\pi$
$492$	0	0
$493$	−336586.	−0.0623705
$494$	909410.	0.167665
$495$	0	0
$496$	−277643.	−0.0506736
$497$	1.94173e6	0.352612
$498$	0	0
$499$	5.30806e6	0.954298	0.477149	−	0.878822i	$-0.341670\pi$
$499$	5.30806e6	0.954298	0.477149	+	0.878822i	$0.341670\pi$
$500$	2.82315e6	0.505020
$501$	0	0
$502$	272632.	0.0482857
$503$	2.98917e6	0.526781	0.263391	−	0.964689i	$-0.415159\pi$
$503$	2.98917e6	0.526781	0.263391	+	0.964689i	$0.415159\pi$
$504$	0	0
$505$	9.16014e6	1.59836
$506$	−1.62479e6	−0.282112
$507$	0	0
$508$	1.51359e6	0.260224
$509$	−4.26220e6	−0.729187	−0.364594	−	0.931167i	$-0.618792\pi$
$509$	−4.26220e6	−0.729187	−0.364594	+	0.931167i	$0.618792\pi$
$510$	0	0
$511$	1.46130e6	0.247565
$512$	4.82203e6	0.812934
$513$	0	0
$514$	230470.	0.0384775
$515$	−7.30539e6	−1.21374
$516$	0	0
$517$	669231.	0.110116
$518$	1.22487e6	0.200571
$519$	0	0
$520$	−1.66112e6	−0.269397
$521$	−361723.	−0.0583823	−0.0291912	−	0.999574i	$-0.509293\pi$
$521$	−361723.	−0.0583823	−0.0291912	+	0.999574i	$0.509293\pi$
$522$	0	0
$523$	6.09858e6	0.974932	0.487466	−	0.873142i	$-0.337921\pi$
$523$	6.09858e6	0.974932	0.487466	+	0.873142i	$0.337921\pi$
$524$	6.58986e6	1.04845
$525$	0	0
$526$	3.66389e6	0.577401
$527$	595542.	0.0934084
$528$	0	0
$529$	−484130.	−0.0752183
$530$	−2.72292e6	−0.421061
$531$	0	0
$532$	2.71945e6	0.416583
$533$	1.91986e6	0.292719
$534$	0	0
$535$	6.62091e6	1.00008
$536$	886786.	0.133324
$537$	0	0
$538$	2.28378e6	0.340171
$539$	−640173.	−0.0949130
$540$	0	0
$541$	−3.34366e6	−0.491167	−0.245584	−	0.969375i	$-0.578980\pi$
$541$	−3.34366e6	−0.491167	−0.245584	+	0.969375i	$0.578980\pi$
$542$	2.66517e6	0.389697
$543$	0	0
$544$	−5.74834e6	−0.832809
$545$	4.07602e6	0.587821
$546$	0	0
$547$	4.84327e6	0.692103	0.346051	−	0.938216i	$-0.387522\pi$
$547$	4.84327e6	0.692103	0.346051	+	0.938216i	$0.387522\pi$
$548$	−3.12815e6	−0.444976
$549$	0	0
$550$	−1.00990e6	−0.142354
$551$	728585.	0.102235
$552$	0	0
$553$	413984.	0.0575666
$554$	−4.68747e6	−0.648880
$555$	0	0
$556$	7.82607e6	1.07363
$557$	−2.73808e6	−0.373945	−0.186973	−	0.982365i	$-0.559868\pi$
$557$	−2.73808e6	−0.373945	−0.186973	+	0.982365i	$0.559868\pi$
$558$	0	0
$559$	−2.47826e6	−0.335442
$560$	−1.54893e6	−0.208719
$561$	0	0
$562$	−2.31909e6	−0.309726
$563$	1.19569e7	1.58982	0.794908	−	0.606729i	$-0.207519\pi$
$563$	1.19569e7	1.58982	0.794908	+	0.606729i	$0.207519\pi$
$564$	0	0
$565$	7.96783e6	1.05007
$566$	3.24537e6	0.425818
$567$	0	0
$568$	5.71717e6	0.743550
$569$	−3.69087e6	−0.477912	−0.238956	−	0.971030i	$-0.576805\pi$
$569$	−3.69087e6	−0.477912	−0.238956	+	0.971030i	$0.576805\pi$
$570$	0	0
$571$	1.93668e6	0.248581	0.124290	−	0.992246i	$-0.460335\pi$
$571$	1.93668e6	0.248581	0.124290	+	0.992246i	$0.460335\pi$
$572$	1.16080e6	0.148343
$573$	0	0
$574$	−1.39037e6	−0.176138
$575$	3.69962e6	0.466646
$576$	0	0
$577$	−1.01572e6	−0.127009	−0.0635045	−	0.997982i	$-0.520228\pi$
$577$	−1.01572e6	−0.127009	−0.0635045	+	0.997982i	$0.520228\pi$
$578$	−1.07233e6	−0.133509
$579$	0	0
$580$	−593540.	−0.0732622
$581$	−4.32432e6	−0.531468
$582$	0	0
$583$	4.26640e6	0.519865
$584$	4.30263e6	0.522037
$585$	0	0
$586$	833025.	0.100211
$587$	−1.02927e7	−1.23292	−0.616459	−	0.787387i	$-0.711434\pi$
$587$	−1.02927e7	−1.23292	−0.616459	+	0.787387i	$0.711434\pi$
$588$	0	0
$589$	−1.28913e6	−0.153112
$590$	−150654.	−0.0178177
$591$	0	0
$592$	−4.64357e6	−0.544563
$593$	1.09015e6	0.127306	0.0636532	−	0.997972i	$-0.479725\pi$
$593$	1.09015e6	0.127306	0.0636532	+	0.997972i	$0.479725\pi$
$594$	0	0
$595$	3.32244e6	0.384738
$596$	−7.68941e6	−0.886702
$597$	0	0
$598$	1.02986e6	0.117768
$599$	1.04188e7	1.18645	0.593227	−	0.805035i	$-0.297854\pi$
$599$	1.04188e7	1.18645	0.593227	+	0.805035i	$0.297854\pi$
$600$	0	0
$601$	−1.50549e7	−1.70017	−0.850085	−	0.526645i	$-0.823450\pi$
$601$	−1.50549e7	−1.70017	−0.850085	+	0.526645i	$0.823450\pi$
$602$	1.79477e6	0.201845
$603$	0	0
$604$	1.92125e6	0.214284
$605$	−6.12883e6	−0.680752
$606$	0	0
$607$	1.62983e7	1.79543	0.897717	−	0.440573i	$-0.145225\pi$
$607$	1.62983e7	1.79543	0.897717	+	0.440573i	$0.145225\pi$
$608$	1.24430e7	1.36511
$609$	0	0
$610$	3.72248e6	0.405049
$611$	−424187.	−0.0459679
$612$	0	0
$613$	7.18377e6	0.772149	0.386075	−	0.922468i	$-0.373831\pi$
$613$	7.18377e6	0.772149	0.386075	+	0.922468i	$0.373831\pi$
$614$	4.40922e6	0.471998
$615$	0	0
$616$	−1.88491e6	−0.200142
$617$	783264.	0.0828314	0.0414157	−	0.999142i	$-0.486813\pi$
$617$	783264.	0.0828314	0.0414157	+	0.999142i	$0.486813\pi$
$618$	0	0
$619$	−1.63595e7	−1.71611	−0.858053	−	0.513562i	$-0.828326\pi$
$619$	−1.63595e7	−1.71611	−0.858053	+	0.513562i	$0.828326\pi$
$620$	1.05019e6	0.109720
$621$	0	0
$622$	4.27010e6	0.442550
$623$	6.79888e6	0.701806
$624$	0	0
$625$	−1.22049e7	−1.24978
$626$	−4.15644e6	−0.423922
$627$	0	0
$628$	5.13940e6	0.520012
$629$	9.96045e6	1.00381
$630$	0	0
$631$	1.48240e7	1.48215	0.741076	−	0.671422i	$-0.234316\pi$
$631$	1.48240e7	1.48215	0.741076	+	0.671422i	$0.234316\pi$
$632$	1.21892e6	0.121390
$633$	0	0
$634$	−746630.	−0.0737704
$635$	−4.00283e6	−0.393943
$636$	0	0
$637$	405769.	0.0396214
$638$	−225226.	−0.0219062
$639$	0	0
$640$	−1.26633e7	−1.22207
$641$	1.77892e6	0.171006	0.0855030	−	0.996338i	$-0.472750\pi$
$641$	1.77892e6	0.171006	0.0855030	+	0.996338i	$0.472750\pi$
$642$	0	0
$643$	5.56934e6	0.531222	0.265611	−	0.964080i	$-0.414426\pi$
$643$	5.56934e6	0.531222	0.265611	+	0.964080i	$0.414426\pi$
$644$	3.07964e6	0.292607
$645$	0	0
$646$	−5.35562e6	−0.504927
$647$	7.67738e6	0.721029	0.360514	−	0.932754i	$-0.382601\pi$
$647$	7.67738e6	0.721029	0.360514	+	0.932754i	$0.382601\pi$
$648$	0	0
$649$	236052.	0.0219987
$650$	640115.	0.0594258
$651$	0	0
$652$	1.82870e6	0.168470
$653$	−8.45233e6	−0.775700	−0.387850	−	0.921723i	$-0.626782\pi$
$653$	−8.45233e6	−0.775700	−0.387850	+	0.921723i	$0.626782\pi$
$654$	0	0
$655$	−1.74276e7	−1.58721
$656$	5.27099e6	0.478226
$657$	0	0
$658$	307199.	0.0276602
$659$	1.42018e7	1.27389	0.636943	−	0.770911i	$-0.280198\pi$
$659$	1.42018e7	1.27389	0.636943	+	0.770911i	$0.280198\pi$
$660$	0	0
$661$	6.88535e6	0.612946	0.306473	−	0.951879i	$-0.400851\pi$
$661$	6.88535e6	0.612946	0.306473	+	0.951879i	$0.400851\pi$
$662$	7.59024e6	0.673148
$663$	0	0
$664$	−1.27324e7	−1.12070
$665$	−7.19186e6	−0.630648
$666$	0	0
$667$	825086.	0.0718100
$668$	1.27741e7	1.10761
$669$	0	0
$670$	−1.04594e6	−0.0900164
$671$	−5.83256e6	−0.500095
$672$	0	0
$673$	2.08330e7	1.77302	0.886512	−	0.462705i	$-0.153121\pi$
$673$	2.08330e7	1.77302	0.886512	+	0.462705i	$0.153121\pi$
$674$	5.59902e6	0.474747
$675$	0	0
$676$	−735763.	−0.0619258
$677$	−5.81651e6	−0.487743	−0.243871	−	0.969808i	$-0.578417\pi$
$677$	−5.81651e6	−0.487743	−0.243871	+	0.969808i	$0.578417\pi$
$678$	0	0
$679$	6.48549e6	0.539844
$680$	9.78251e6	0.811294
$681$	0	0
$682$	398506.	0.0328076
$683$	1.11236e7	0.912422	0.456211	−	0.889872i	$-0.349206\pi$
$683$	1.11236e7	0.912422	0.456211	+	0.889872i	$0.349206\pi$
$684$	0	0
$685$	8.27272e6	0.673631
$686$	−293861.	−0.0238414
$687$	0	0
$688$	−6.80409e6	−0.548023
$689$	−2.70423e6	−0.217018
$690$	0	0
$691$	8.85581e6	0.705559	0.352779	−	0.935707i	$-0.385237\pi$
$691$	8.85581e6	0.705559	0.352779	+	0.935707i	$0.385237\pi$
$692$	−2.71872e6	−0.215824
$693$	0	0
$694$	6.00149e6	0.472999
$695$	−2.06968e7	−1.62533
$696$	0	0
$697$	−1.13063e7	−0.881530
$698$	−1.06370e7	−0.826379
$699$	0	0
$700$	1.91416e6	0.147650
$701$	−5.32295e6	−0.409126	−0.204563	−	0.978853i	$-0.565577\pi$
$701$	−5.32295e6	−0.409126	−0.204563	+	0.978853i	$0.565577\pi$
$702$	0	0
$703$	−2.15607e7	−1.64541
$704$	112324.	0.00854167
$705$	0	0
$706$	3.98353e6	0.300785
$707$	−6.58829e6	−0.495706
$708$	0	0
$709$	2.41369e7	1.80329	0.901644	−	0.432479i	$-0.142361\pi$
$709$	2.41369e7	1.80329	0.901644	+	0.432479i	$0.142361\pi$
$710$	−6.74327e6	−0.502024
$711$	0	0
$712$	2.00184e7	1.47989
$713$	−1.45987e6	−0.107545
$714$	0	0
$715$	−3.06985e6	−0.224570
$716$	1.66360e7	1.21273
$717$	0	0
$718$	−5.92991e6	−0.429277
$719$	1.81165e7	1.30693	0.653464	−	0.756958i	$-0.273315\pi$
$719$	1.81165e7	1.30693	0.653464	+	0.756958i	$0.273315\pi$
$720$	0	0
$721$	5.25429e6	0.376423
$722$	5.40820e6	0.386109
$723$	0	0
$724$	1.67915e7	1.19054
$725$	512836.	0.0362354
$726$	0	0
$727$	−2.62393e7	−1.84126	−0.920631	−	0.390434i	$-0.872325\pi$
$727$	−2.62393e7	−1.84126	−0.920631	+	0.390434i	$0.872325\pi$
$728$	1.19474e6	0.0835494
$729$	0	0
$730$	−5.07485e6	−0.352465
$731$	1.45947e7	1.01019
$732$	0	0
$733$	−1.61216e7	−1.10827	−0.554137	−	0.832425i	$-0.686952\pi$
$733$	−1.61216e7	−1.10827	−0.554137	+	0.832425i	$0.686952\pi$
$734$	3.16260e6	0.216673
$735$	0	0
$736$	1.40911e7	0.958851
$737$	1.63884e6	0.111139
$738$	0	0
$739$	−1.38086e7	−0.930117	−0.465059	−	0.885280i	$-0.653967\pi$
$739$	−1.38086e7	−0.930117	−0.465059	+	0.885280i	$0.653967\pi$
$740$	1.75644e7	1.17911
$741$	0	0
$742$	1.95842e6	0.130586
$743$	−2.02443e7	−1.34533	−0.672666	−	0.739946i	$-0.734851\pi$
$743$	−2.02443e7	−1.34533	−0.672666	+	0.739946i	$0.734851\pi$
$744$	0	0
$745$	2.03354e7	1.34234
$746$	470415.	0.0309481
$747$	0	0
$748$	−6.83607e6	−0.446738
$749$	−4.76199e6	−0.310159
$750$	0	0
$751$	−1.14455e7	−0.740517	−0.370259	−	0.928929i	$-0.620731\pi$
$751$	−1.14455e7	−0.740517	−0.370259	+	0.928929i	$0.620731\pi$
$752$	−1.16461e6	−0.0750993
$753$	0	0
$754$	142758.	0.00914474
$755$	−5.08093e6	−0.324396
$756$	0	0
$757$	2.91822e6	0.185088	0.0925440	−	0.995709i	$-0.470500\pi$
$757$	2.91822e6	0.185088	0.0925440	+	0.995709i	$0.470500\pi$
$758$	−8.12356e6	−0.513539
$759$	0	0
$760$	−2.11755e7	−1.32984
$761$	−2.82378e7	−1.76754	−0.883771	−	0.467920i	$-0.845003\pi$
$761$	−2.82378e7	−1.76754	−0.883771	+	0.467920i	$0.845003\pi$
$762$	0	0
$763$	−2.93162e6	−0.182304
$764$	1.28813e7	0.798413
$765$	0	0
$766$	305170.	0.0187919
$767$	−149620.	−0.00918334
$768$	0	0
$769$	−1.43941e7	−0.877743	−0.438872	−	0.898550i	$-0.644622\pi$
$769$	−1.43941e7	−0.877743	−0.438872	+	0.898550i	$0.644622\pi$
$770$	2.22321e6	0.135130
$771$	0	0
$772$	−7.49594e6	−0.452671
$773$	1.08711e7	0.654375	0.327187	−	0.944960i	$-0.393899\pi$
$773$	1.08711e7	0.654375	0.327187	+	0.944960i	$0.393899\pi$
$774$	0	0
$775$	−907391.	−0.0542675
$776$	1.90957e7	1.13836
$777$	0	0
$778$	−1.96730e6	−0.116525
$779$	2.44739e7	1.44497
$780$	0	0
$781$	1.05657e7	0.619826
$782$	−6.06497e6	−0.354660
$783$	0	0
$784$	1.11404e6	0.0647310
$785$	−1.35917e7	−0.787225
$786$	0	0
$787$	1.28559e7	0.739886	0.369943	−	0.929054i	$-0.379377\pi$
$787$	1.28559e7	0.739886	0.369943	+	0.929054i	$0.379377\pi$
$788$	−1.09602e7	−0.628787
$789$	0	0
$790$	−1.43769e6	−0.0819592
$791$	−5.73074e6	−0.325664
$792$	0	0
$793$	3.69692e6	0.208765
$794$	−5.95100e6	−0.334995
$795$	0	0
$796$	2.11825e7	1.18493
$797$	8.48057e6	0.472911	0.236455	−	0.971642i	$-0.424014\pi$
$797$	8.48057e6	0.472911	0.236455	+	0.971642i	$0.424014\pi$
$798$	0	0
$799$	2.49808e6	0.138433
$800$	8.75840e6	0.483838
$801$	0	0
$802$	−6.87986e6	−0.377697
$803$	7.95152e6	0.435172
$804$	0	0
$805$	−8.14443e6	−0.442966
$806$	−252590.	−0.0136955
$807$	0	0
$808$	−1.93984e7	−1.04529
$809$	−2.28514e7	−1.22756	−0.613779	−	0.789478i	$-0.710351\pi$
$809$	−2.28514e7	−1.22756	−0.613779	+	0.789478i	$0.710351\pi$
$810$	0	0
$811$	−1.02688e7	−0.548239	−0.274119	−	0.961696i	$-0.588386\pi$
$811$	−1.02688e7	−0.548239	−0.274119	+	0.961696i	$0.588386\pi$
$812$	426895.	0.0227212
$813$	0	0
$814$	6.66501e6	0.352566
$815$	−4.83617e6	−0.255040
$816$	0	0
$817$	−3.15922e7	−1.65586
$818$	−1.75895e6	−0.0919114
$819$	0	0
$820$	−1.99376e7	−1.03547
$821$	3.09069e6	0.160029	0.0800143	−	0.996794i	$-0.474503\pi$
$821$	3.09069e6	0.160029	0.0800143	+	0.996794i	$0.474503\pi$
$822$	0	0
$823$	−1.65361e7	−0.851010	−0.425505	−	0.904956i	$-0.639903\pi$
$823$	−1.65361e7	−0.851010	−0.425505	+	0.904956i	$0.639903\pi$
$824$	1.54706e7	0.793759
$825$	0	0
$826$	108356.	0.00552588
$827$	617196.	0.0313804	0.0156902	−	0.999877i	$-0.495005\pi$
$827$	617196.	0.0313804	0.0156902	+	0.999877i	$0.495005\pi$
$828$	0	0
$829$	2.20779e7	1.11576	0.557881	−	0.829921i	$-0.311615\pi$
$829$	2.20779e7	1.11576	0.557881	+	0.829921i	$0.311615\pi$
$830$	1.50176e7	0.756667
$831$	0	0
$832$	−71196.0	−0.00356572
$833$	−2.38962e6	−0.119321
$834$	0	0
$835$	−3.37823e7	−1.67677
$836$	1.47976e7	0.732276
$837$	0	0
$838$	712123.	0.0350304
$839$	1.54570e7	0.758089	0.379045	−	0.925378i	$-0.376253\pi$
$839$	1.54570e7	0.758089	0.379045	+	0.925378i	$0.376253\pi$
$840$	0	0
$841$	−2.03968e7	−0.994424
$842$	−1.22632e7	−0.596105
$843$	0	0
$844$	1.47979e7	0.715064
$845$	1.94580e6	0.0937469
$846$	0	0
$847$	4.40807e6	0.211125
$848$	−7.42448e6	−0.354549
$849$	0	0
$850$	−3.76971e6	−0.178962
$851$	−2.44164e7	−1.15573
$852$	0	0
$853$	1.08221e7	0.509260	0.254630	−	0.967039i	$-0.418046\pi$
$853$	1.08221e7	0.509260	0.254630	+	0.967039i	$0.418046\pi$
$854$	−2.67733e6	−0.125620
$855$	0	0
$856$	−1.40211e7	−0.654029
$857$	1.31705e7	0.612562	0.306281	−	0.951941i	$-0.400915\pi$
$857$	1.31705e7	0.612562	0.306281	+	0.951941i	$0.400915\pi$
$858$	0	0
$859$	2.96058e7	1.36897	0.684486	−	0.729026i	$-0.260027\pi$
$859$	2.96058e7	1.36897	0.684486	+	0.729026i	$0.260027\pi$
$860$	2.57365e7	1.18660
$861$	0	0
$862$	−1.96081e6	−0.0898808
$863$	3.32208e7	1.51839	0.759194	−	0.650864i	$-0.225593\pi$
$863$	3.32208e7	1.51839	0.759194	+	0.650864i	$0.225593\pi$
$864$	0	0
$865$	7.18993e6	0.326727
$866$	−6.13866e6	−0.278150
$867$	0	0
$868$	−755330.	−0.0340281
$869$	2.25264e6	0.101191
$870$	0	0
$871$	−1.03876e6	−0.0463950
$872$	−8.63177e6	−0.384422
$873$	0	0
$874$	1.31284e7	0.581345
$875$	5.36989e6	0.237107
$876$	0	0
$877$	1.57291e6	0.0690565	0.0345283	−	0.999404i	$-0.489007\pi$
$877$	1.57291e6	0.0690565	0.0345283	+	0.999404i	$0.489007\pi$
$878$	−1.10947e6	−0.0485713
$879$	0	0
$880$	−8.42832e6	−0.366888
$881$	−1.76041e7	−0.764143	−0.382071	−	0.924133i	$-0.624789\pi$
$881$	−1.76041e7	−0.764143	−0.382071	+	0.924133i	$0.624789\pi$
$882$	0	0
$883$	787748.	0.0340005	0.0170003	−	0.999855i	$-0.494588\pi$
$883$	787748.	0.0340005	0.0170003	+	0.999855i	$0.494588\pi$
$884$	4.33299e6	0.186491
$885$	0	0
$886$	6.02457e6	0.257835
$887$	−501334.	−0.0213953	−0.0106976	−	0.999943i	$-0.503405\pi$
$887$	−501334.	−0.0213953	−0.0106976	+	0.999943i	$0.503405\pi$
$888$	0	0
$889$	2.87898e6	0.122175
$890$	−2.36113e7	−0.999182
$891$	0	0
$892$	3.04393e7	1.28092
$893$	−5.40743e6	−0.226914
$894$	0	0
$895$	−4.39955e7	−1.83591
$896$	9.10789e6	0.379008
$897$	0	0
$898$	−3.95632e6	−0.163720
$899$	−202365.	−0.00835097
$900$	0	0
$901$	1.59255e7	0.653553
$902$	−7.56556e6	−0.309617
$903$	0	0
$904$	−1.68734e7	−0.686724
$905$	−4.44070e7	−1.80231
$906$	0	0
$907$	3.23891e7	1.30731	0.653657	−	0.756791i	$-0.273234\pi$
$907$	3.23891e7	1.30731	0.653657	+	0.756791i	$0.273234\pi$
$908$	3.53354e7	1.42232
$909$	0	0
$910$	−1.40916e6	−0.0564102
$911$	−7.45598e6	−0.297652	−0.148826	−	0.988863i	$-0.547549\pi$
$911$	−7.45598e6	−0.297652	−0.148826	+	0.988863i	$0.547549\pi$
$912$	0	0
$913$	−2.35303e7	−0.934222
$914$	3.89290e6	0.154137
$915$	0	0
$916$	3.79973e7	1.49628
$917$	1.25345e7	0.492248
$918$	0	0
$919$	3.47364e6	0.135674	0.0678369	−	0.997696i	$-0.478390\pi$
$919$	3.47364e6	0.135674	0.0678369	+	0.997696i	$0.478390\pi$
$920$	−2.39802e7	−0.934079
$921$	0	0
$922$	4.06909e6	0.157641
$923$	−6.69697e6	−0.258746
$924$	0	0
$925$	−1.51761e7	−0.583185
$926$	−6.46232e6	−0.247663
$927$	0	0
$928$	1.95329e6	0.0744554
$929$	−3.03330e7	−1.15313	−0.576563	−	0.817053i	$-0.695606\pi$
$929$	−3.03330e7	−1.15313	−0.576563	+	0.817053i	$0.695606\pi$
$930$	0	0
$931$	5.17264e6	0.195586
$932$	1.13461e7	0.427866
$933$	0	0
$934$	−360497.	−0.0135218
$935$	1.80787e7	0.676298
$936$	0	0
$937$	1.79154e7	0.666620	0.333310	−	0.942817i	$-0.391834\pi$
$937$	1.79154e7	0.666620	0.333310	+	0.942817i	$0.391834\pi$
$938$	752280.	0.0279172
$939$	0	0
$940$	4.40515e6	0.162608
$941$	−3.54629e7	−1.30557	−0.652784	−	0.757544i	$-0.726399\pi$
$941$	−3.54629e7	−1.30557	−0.652784	+	0.757544i	$0.726399\pi$
$942$	0	0
$943$	2.77154e7	1.01494
$944$	−410783.	−0.0150032
$945$	0	0
$946$	9.76604e6	0.354806
$947$	−2.97963e7	−1.07966	−0.539830	−	0.841774i	$-0.681512\pi$
$947$	−2.97963e7	−1.07966	−0.539830	+	0.841774i	$0.681512\pi$
$948$	0	0
$949$	−5.04001e6	−0.181663
$950$	8.16003e6	0.293348
$951$	0	0
$952$	−7.03593e6	−0.251611
$953$	1.32959e7	0.474226	0.237113	−	0.971482i	$-0.423799\pi$
$953$	1.32959e7	0.474226	0.237113	+	0.971482i	$0.423799\pi$
$954$	0	0
$955$	−3.40660e7	−1.20868
$956$	−1.46050e7	−0.516842
$957$	0	0
$958$	2.14206e7	0.754080
$959$	−5.95003e6	−0.208917
$960$	0	0
$961$	−2.82711e7	−0.987493
$962$	−4.22457e6	−0.147179
$963$	0	0
$964$	−3.74174e7	−1.29683
$965$	1.98238e7	0.685281
$966$	0	0
$967$	−4.26663e6	−0.146730	−0.0733650	−	0.997305i	$-0.523374\pi$
$967$	−4.26663e6	−0.146730	−0.0733650	+	0.997305i	$0.523374\pi$
$968$	1.29790e7	0.445197
$969$	0	0
$970$	−2.25230e7	−0.768592
$971$	1.69443e7	0.576733	0.288367	−	0.957520i	$-0.406888\pi$
$971$	1.69443e7	0.576733	0.288367	+	0.957520i	$0.406888\pi$
$972$	0	0
$973$	1.48859e7	0.504072
$974$	1.99522e7	0.673895
$975$	0	0
$976$	1.01499e7	0.341066
$977$	−5.72940e7	−1.92032	−0.960158	−	0.279459i	$-0.909845\pi$
$977$	−5.72940e7	−1.92032	−0.960158	+	0.279459i	$0.909845\pi$
$978$	0	0
$979$	3.69953e7	1.23364
$980$	−4.21388e6	−0.140158
$981$	0	0
$982$	5.94355e6	0.196683
$983$	−2.07108e6	−0.0683617	−0.0341808	−	0.999416i	$-0.510882\pi$
$983$	−2.07108e6	−0.0683617	−0.0341808	+	0.999416i	$0.510882\pi$
$984$	0	0
$985$	2.89854e7	0.951895
$986$	−840717.	−0.0275396
$987$	0	0
$988$	−9.37932e6	−0.305689
$989$	−3.57766e7	−1.16308
$990$	0	0
$991$	3.74910e7	1.21267	0.606335	−	0.795209i	$-0.292639\pi$
$991$	3.74910e7	1.21267	0.606335	+	0.795209i	$0.292639\pi$
$992$	−3.45607e6	−0.111507
$993$	0	0
$994$	4.84999e6	0.155695
$995$	−5.60192e7	−1.79382
$996$	0	0
$997$	1.81169e7	0.577225	0.288613	−	0.957446i	$-0.406806\pi$
$997$	1.81169e7	0.577225	0.288613	+	0.957446i	$0.406806\pi$
$998$	1.32583e7	0.421369
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.6.a.f.1.3		6
3.2	odd	2		273.6.a.b.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.6.a.b.1.4	✓	6	3.2	odd	2
819.6.a.f.1.3		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	819.a (trivial)

\(f(q)\)	\(=\)	\(q+2.49777 q^{2} -25.7611 q^{4} +68.1279 q^{5} -49.0000 q^{7} -144.274 q^{8} +O(q^{10})\)	\(q+2.49777 q^{2} -25.7611 q^{4} +68.1279 q^{5} -49.0000 q^{7} -144.274 q^{8} +170.168 q^{10} -266.628 q^{11} +169.000 q^{13} -122.391 q^{14} +463.991 q^{16} -995.260 q^{17} +2154.37 q^{19} -1755.05 q^{20} -665.976 q^{22} +2439.72 q^{23} +1516.42 q^{25} +422.124 q^{26} +1262.30 q^{28} +338.189 q^{29} -598.379 q^{31} +5775.72 q^{32} -2485.93 q^{34} -3338.27 q^{35} -10007.9 q^{37} +5381.13 q^{38} -9829.11 q^{40} +11360.1 q^{41} -14664.2 q^{43} +6868.63 q^{44} +6093.86 q^{46} -2509.98 q^{47} +2401.00 q^{49} +3787.66 q^{50} -4353.63 q^{52} -16001.3 q^{53} -18164.8 q^{55} +7069.44 q^{56} +844.721 q^{58} -885.325 q^{59} +21875.3 q^{61} -1494.61 q^{62} -421.278 q^{64} +11513.6 q^{65} -6146.53 q^{67} +25639.0 q^{68} -8338.24 q^{70} -39627.1 q^{71} -29822.5 q^{73} -24997.4 q^{74} -55499.0 q^{76} +13064.8 q^{77} -8448.65 q^{79} +31610.8 q^{80} +28375.0 q^{82} +88251.4 q^{83} -67805.0 q^{85} -36628.0 q^{86} +38467.5 q^{88} -138753. q^{89} -8281.00 q^{91} -62849.8 q^{92} -6269.37 q^{94} +146773. q^{95} -132357. q^{97} +5997.16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 15 q^{2} + 75 q^{4} - 15 q^{5} - 294 q^{7} + 399 q^{8}+O(q^{10}) \) 6 * q + 15 * q^2 + 75 * q^4 - 15 * q^5 - 294 * q^7 + 399 * q^8	\( 6 q + 15 q^{2} + 75 q^{4} - 15 q^{5} - 294 q^{7} + 399 q^{8} - 265 q^{10} + 90 q^{11} + 1014 q^{13} - 735 q^{14} + 355 q^{16} + 1314 q^{17} - 341 q^{19} - 2085 q^{20} - 6749 q^{22} + 4905 q^{23} - 10231 q^{25} + 2535 q^{26} - 3675 q^{28} + 6771 q^{29} - 14711 q^{31} + 10143 q^{32} + 9595 q^{34} + 735 q^{35} - 13314 q^{37} - 1017 q^{38} - 13195 q^{40} + 5040 q^{41} - 9127 q^{43} - 10041 q^{44} + 23739 q^{46} + 4713 q^{47} + 14406 q^{49} - 18018 q^{50} + 12675 q^{52} + 159 q^{53} - 13590 q^{55} - 19551 q^{56} - 38427 q^{58} - 30288 q^{59} - 78126 q^{61} - 40632 q^{62} - 8461 q^{64} - 2535 q^{65} - 33894 q^{67} + 61863 q^{68} + 12985 q^{70} - 8316 q^{71} - 89861 q^{73} + 16635 q^{74} - 169069 q^{76} - 4410 q^{77} - 239803 q^{79} - 19305 q^{80} - 106624 q^{82} + 42753 q^{83} - 184316 q^{85} - 51609 q^{86} - 205923 q^{88} + 11421 q^{89} - 49686 q^{91} + 252519 q^{92} - 4976 q^{94} + 180807 q^{95} - 199445 q^{97} + 36015 q^{98}+O(q^{100}) \) 6 * q + 15 * q^2 + 75 * q^4 - 15 * q^5 - 294 * q^7 + 399 * q^8 - 265 * q^10 + 90 * q^11 + 1014 * q^13 - 735 * q^14 + 355 * q^16 + 1314 * q^17 - 341 * q^19 - 2085 * q^20 - 6749 * q^22 + 4905 * q^23 - 10231 * q^25 + 2535 * q^26 - 3675 * q^28 + 6771 * q^29 - 14711 * q^31 + 10143 * q^32 + 9595 * q^34 + 735 * q^35 - 13314 * q^37 - 1017 * q^38 - 13195 * q^40 + 5040 * q^41 - 9127 * q^43 - 10041 * q^44 + 23739 * q^46 + 4713 * q^47 + 14406 * q^49 - 18018 * q^50 + 12675 * q^52 + 159 * q^53 - 13590 * q^55 - 19551 * q^56 - 38427 * q^58 - 30288 * q^59 - 78126 * q^61 - 40632 * q^62 - 8461 * q^64 - 2535 * q^65 - 33894 * q^67 + 61863 * q^68 + 12985 * q^70 - 8316 * q^71 - 89861 * q^73 + 16635 * q^74 - 169069 * q^76 - 4410 * q^77 - 239803 * q^79 - 19305 * q^80 - 106624 * q^82 + 42753 * q^83 - 184316 * q^85 - 51609 * q^86 - 205923 * q^88 + 11421 * q^89 - 49686 * q^91 + 252519 * q^92 - 4976 * q^94 + 180807 * q^95 - 199445 * q^97 + 36015 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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