LMFDB - Embedded newform 507.4.b.a.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.a.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +8.00000 q^{4} -5.19615i q^{5} +10.3923i q^{7} +9.00000 q^{9} -51.9615i q^{11} -24.0000 q^{12} +15.5885i q^{15} +64.0000 q^{16} -117.000 q^{17} -24.2487i q^{19} -41.5692i q^{20} -31.1769i q^{21} +18.0000 q^{23} +98.0000 q^{25} -27.0000 q^{27} +83.1384i q^{28} -99.0000 q^{29} -193.990i q^{31} +155.885i q^{33} +54.0000 q^{35} +72.0000 q^{36} -112.583i q^{37} +36.3731i q^{41} -82.0000 q^{43} -415.692i q^{44} -46.7654i q^{45} +72.7461i q^{47} -192.000 q^{48} +235.000 q^{49} +351.000 q^{51} -261.000 q^{53} -270.000 q^{55} +72.7461i q^{57} -789.815i q^{59} +124.708i q^{60} -719.000 q^{61} +93.5307i q^{63} +512.000 q^{64} +703.213i q^{67} -936.000 q^{68} -54.0000 q^{69} -467.654i q^{71} -684.160i q^{73} -294.000 q^{75} -193.990i q^{76} +540.000 q^{77} -440.000 q^{79} -332.554i q^{80} +81.0000 q^{81} -1195.12i q^{83} -249.415i q^{84} +607.950i q^{85} +297.000 q^{87} -1517.28i q^{89} +144.000 q^{92} +581.969i q^{93} -126.000 q^{95} -1157.01i q^{97} -467.654i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 16 q^{4} + 18 q^{9} - 48 q^{12} + 128 q^{16} - 234 q^{17} + 36 q^{23} + 196 q^{25} - 54 q^{27} - 198 q^{29} + 108 q^{35} + 144 q^{36} - 164 q^{43} - 384 q^{48} + 470 q^{49} + 702 q^{51}+ \cdots - 252 q^{95}+O(q^{100}) $ 2 * q - 6 * q^3 + 16 * q^4 + 18 * q^9 - 48 * q^12 + 128 * q^16 - 234 * q^17 + 36 * q^23 + 196 * q^25 - 54 * q^27 - 198 * q^29 + 108 * q^35 + 144 * q^36 - 164 * q^43 - 384 * q^48 + 470 * q^49 + 702 * q^51 - 522 * q^53 - 540 * q^55 - 1438 * q^61 + 1024 * q^64 - 1872 * q^68 - 108 * q^69 - 588 * q^75 + 1080 * q^77 - 880 * q^79 + 162 * q^81 + 594 * q^87 + 288 * q^92 - 252 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	8.00000	1.00000
$5$	− 5.19615i	− 0.464758i	−0.972625	−	0.232379i	$-0.925349\pi$
$5$	− 5.19615i	− 0.464758i	0.972625	−	0.232379i	$-0.0746510\pi$
$6$	0	0
$7$	10.3923i	0.561132i	0.959835	+	0.280566i	$0.0905221\pi$
$7$	10.3923i	0.561132i	−0.959835	+	0.280566i	$0.909478\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	− 51.9615i	− 1.42427i	−0.702042	−	0.712136i	$-0.747728\pi$
$11$	− 51.9615i	− 1.42427i	0.702042	−	0.712136i	$-0.252272\pi$
$12$	−24.0000	−0.577350
$13$	0	0
$13$	0	0
$14$	0	0
$15$	15.5885i	0.268328i
$16$	64.0000	1.00000
$17$	−117.000	−1.66922	−0.834608	−	0.550845i	$-0.814306\pi$
$17$	−117.000	−1.66922	−0.834608	+	0.550845i	$0.814306\pi$
$18$	0	0
$19$	− 24.2487i	− 0.292791i	−0.989226	−	0.146396i	$-0.953233\pi$
$19$	− 24.2487i	− 0.292791i	0.989226	−	0.146396i	$-0.0467673\pi$
$20$	− 41.5692i	− 0.464758i
$21$	− 31.1769i	− 0.323970i
$22$	0	0
$23$	18.0000	0.163185	0.0815926	−	0.996666i	$-0.473999\pi$
$23$	18.0000	0.163185	0.0815926	+	0.996666i	$0.473999\pi$
$24$	0	0
$25$	98.0000	0.784000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	83.1384i	0.561132i
$29$	−99.0000	−0.633925	−0.316963	−	0.948438i	$-0.602663\pi$
$29$	−99.0000	−0.633925	−0.316963	+	0.948438i	$0.602663\pi$
$30$	0	0
$31$	− 193.990i	− 1.12392i	−0.827164	−	0.561961i	$-0.810047\pi$
$31$	− 193.990i	− 1.12392i	0.827164	−	0.561961i	$-0.189953\pi$
$32$	0	0
$33$	155.885i	0.822304i
$34$	0	0
$35$	54.0000	0.260790
$36$	72.0000	0.333333
$37$	− 112.583i	− 0.500232i	−0.968216	−	0.250116i	$-0.919531\pi$
$37$	− 112.583i	− 0.500232i	0.968216	−	0.250116i	$-0.0804687\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	36.3731i	0.138549i	0.997598	+	0.0692746i	$0.0220685\pi$
$41$	36.3731i	0.138549i	−0.997598	+	0.0692746i	$0.977932\pi$
$42$	0	0
$43$	−82.0000	−0.290811	−0.145406	−	0.989372i	$-0.546449\pi$
$43$	−82.0000	−0.290811	−0.145406	+	0.989372i	$0.546449\pi$
$44$	− 415.692i	− 1.42427i
$45$	− 46.7654i	− 0.154919i
$46$	0	0
$47$	72.7461i	0.225768i	0.993608	+	0.112884i	$0.0360089\pi$
$47$	72.7461i	0.225768i	−0.993608	+	0.112884i	$0.963991\pi$
$48$	−192.000	−0.577350
$49$	235.000	0.685131
$50$	0	0
$51$	351.000	0.963722
$52$	0	0
$53$	−261.000	−0.676436	−0.338218	−	0.941068i	$-0.609824\pi$
$53$	−261.000	−0.676436	−0.338218	+	0.941068i	$0.609824\pi$
$54$	0	0
$55$	−270.000	−0.661942
$56$	0	0
$57$	72.7461i	0.169043i
$58$	0	0
$59$	− 789.815i	− 1.74280i	−0.490574	−	0.871400i	$-0.663213\pi$
$59$	− 789.815i	− 1.74280i	0.490574	−	0.871400i	$-0.336787\pi$
$60$	124.708i	0.268328i
$61$	−719.000	−1.50916	−0.754578	−	0.656210i	$-0.772158\pi$
$61$	−719.000	−1.50916	−0.754578	+	0.656210i	$0.772158\pi$
$62$	0	0
$63$	93.5307i	0.187044i
$64$	512.000	1.00000
$65$	0	0
$66$	0	0
$67$	703.213i	1.28226i	0.767434	+	0.641128i	$0.221533\pi$
$67$	703.213i	1.28226i	−0.767434	+	0.641128i	$0.778467\pi$
$68$	−936.000	−1.66922
$69$	−54.0000	−0.0942150
$70$	0	0
$71$	− 467.654i	− 0.781694i	−0.920456	−	0.390847i	$-0.872182\pi$
$71$	− 467.654i	− 0.781694i	0.920456	−	0.390847i	$-0.127818\pi$
$72$	0	0
$73$	− 684.160i	− 1.09692i	−0.836178	−	0.548458i	$-0.815215\pi$
$73$	− 684.160i	− 1.09692i	0.836178	−	0.548458i	$-0.184785\pi$
$74$	0	0
$75$	−294.000	−0.452643
$76$	− 193.990i	− 0.292791i
$77$	540.000	0.799204
$78$	0	0
$79$	−440.000	−0.626631	−0.313316	−	0.949649i	$-0.601440\pi$
$79$	−440.000	−0.626631	−0.313316	+	0.949649i	$0.601440\pi$
$80$	− 332.554i	− 0.464758i
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1195.12i	− 1.58049i	−0.612789	−	0.790247i	$-0.709952\pi$
$83$	− 1195.12i	− 1.58049i	0.612789	−	0.790247i	$-0.290048\pi$
$84$	− 249.415i	− 0.323970i
$85$	607.950i	0.775781i
$86$	0	0
$87$	297.000	0.365997
$88$	0	0
$89$	− 1517.28i	− 1.80709i	−0.428493	−	0.903545i	$-0.640955\pi$
$89$	− 1517.28i	− 1.80709i	0.428493	−	0.903545i	$-0.359045\pi$
$90$	0	0
$91$	0	0
$92$	144.000	0.163185
$93$	581.969i	0.648897i
$94$	0	0
$95$	−126.000	−0.136077
$96$	0	0
$97$	− 1157.01i	− 1.21110i	−0.795808	−	0.605549i	$-0.792953\pi$
$97$	− 1157.01i	− 1.21110i	0.795808	−	0.605549i	$-0.207047\pi$
$98$	0	0
$99$	− 467.654i	− 0.474757i
$100$	784.000	0.784000
$101$	1575.00	1.55167	0.775833	−	0.630938i	$-0.217330\pi$
$101$	1575.00	1.55167	0.775833	+	0.630938i	$0.217330\pi$
$102$	0	0
$103$	794.000	0.759565	0.379782	−	0.925076i	$-0.375999\pi$
$103$	794.000	0.759565	0.379782	+	0.925076i	$0.375999\pi$
$104$	0	0
$105$	−162.000	−0.150567
$106$	0	0
$107$	450.000	0.406571	0.203286	−	0.979119i	$-0.434838\pi$
$107$	450.000	0.406571	0.203286	+	0.979119i	$0.434838\pi$
$108$	−216.000	−0.192450
$109$	− 595.825i	− 0.523576i	−0.965125	−	0.261788i	$-0.915688\pi$
$109$	− 595.825i	− 0.523576i	0.965125	−	0.261788i	$-0.0843120\pi$
$110$	0	0
$111$	337.750i	0.288809i
$112$	665.108i	0.561132i
$113$	−1701.00	−1.41608	−0.708038	−	0.706174i	$-0.750420\pi$
$113$	−1701.00	−1.41608	−0.708038	+	0.706174i	$0.750420\pi$
$114$	0	0
$115$	− 93.5307i	− 0.0758416i
$116$	−792.000	−0.633925
$117$	0	0
$118$	0	0
$119$	− 1215.90i	− 0.936650i
$120$	0	0
$121$	−1369.00	−1.02855
$122$	0	0
$123$	− 109.119i	− 0.0799914i
$124$	− 1551.92i	− 1.12392i
$125$	− 1158.74i	− 0.829128i
$126$	0	0
$127$	−1664.00	−1.16265	−0.581323	−	0.813673i	$-0.697465\pi$
$127$	−1664.00	−1.16265	−0.581323	+	0.813673i	$0.697465\pi$
$128$	0	0
$129$	246.000	0.167900
$130$	0	0
$131$	−1476.00	−0.984418	−0.492209	−	0.870477i	$-0.663810\pi$
$131$	−1476.00	−0.984418	−0.492209	+	0.870477i	$0.663810\pi$
$132$	1247.08i	0.822304i
$133$	252.000	0.164295
$134$	0	0
$135$	140.296i	0.0894427i
$136$	0	0
$137$	1013.25i	0.631882i	0.948779	+	0.315941i	$0.102320\pi$
$137$	1013.25i	0.631882i	−0.948779	+	0.315941i	$0.897680\pi$
$138$	0	0
$139$	1124.00	0.685874	0.342937	−	0.939358i	$-0.388578\pi$
$139$	1124.00	0.685874	0.342937	+	0.939358i	$0.388578\pi$
$140$	432.000	0.260790
$141$	− 218.238i	− 0.130347i
$142$	0	0
$143$	0	0
$144$	576.000	0.333333
$145$	514.419i	0.294622i
$146$	0	0
$147$	−705.000	−0.395561
$148$	− 900.666i	− 0.500232i
$149$	3268.38i	1.79702i	0.438952	+	0.898510i	$0.355350\pi$
$149$	3268.38i	1.79702i	−0.438952	+	0.898510i	$0.644650\pi$
$150$	0	0
$151$	1638.52i	0.883052i	0.897248	+	0.441526i	$0.145563\pi$
$151$	1638.52i	0.883052i	−0.897248	+	0.441526i	$0.854437\pi$
$152$	0	0
$153$	−1053.00	−0.556405
$154$	0	0
$155$	−1008.00	−0.522352
$156$	0	0
$157$	1259.00	0.639995	0.319997	−	0.947418i	$-0.396318\pi$
$157$	1259.00	0.639995	0.319997	+	0.947418i	$0.396318\pi$
$158$	0	0
$159$	783.000	0.390540
$160$	0	0
$161$	187.061i	0.0915684i
$162$	0	0
$163$	2951.41i	1.41824i	0.705089	+	0.709118i	$0.250907\pi$
$163$	2951.41i	1.41824i	−0.705089	+	0.709118i	$0.749093\pi$
$164$	290.985i	0.138549i
$165$	810.000	0.382172
$166$	0	0
$167$	3138.48i	1.45427i	0.686496	+	0.727133i	$0.259148\pi$
$167$	3138.48i	1.45427i	−0.686496	+	0.727133i	$0.740852\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 218.238i	− 0.0975971i
$172$	−656.000	−0.290811
$173$	4266.00	1.87479	0.937393	−	0.348273i	$-0.113232\pi$
$173$	4266.00	1.87479	0.937393	+	0.348273i	$0.113232\pi$
$174$	0	0
$175$	1018.45i	0.439927i
$176$	− 3325.54i	− 1.42427i
$177$	2369.45i	1.00621i
$178$	0	0
$179$	3006.00	1.25519	0.627595	−	0.778540i	$-0.284039\pi$
$179$	3006.00	1.25519	0.627595	+	0.778540i	$0.284039\pi$
$180$	− 374.123i	− 0.154919i
$181$	−1873.00	−0.769166	−0.384583	−	0.923090i	$-0.625655\pi$
$181$	−1873.00	−0.769166	−0.384583	+	0.923090i	$0.625655\pi$
$182$	0	0
$183$	2157.00	0.871312
$184$	0	0
$185$	−585.000	−0.232487
$186$	0	0
$187$	6079.50i	2.37742i
$188$	581.969i	0.225768i
$189$	− 280.592i	− 0.107990i
$190$	0	0
$191$	−2736.00	−1.03649	−0.518246	−	0.855232i	$-0.673415\pi$
$191$	−2736.00	−1.03649	−0.518246	+	0.855232i	$0.673415\pi$
$192$	−1536.00	−0.577350
$193$	2603.27i	0.970920i	0.874259	+	0.485460i	$0.161348\pi$
$193$	2603.27i	0.970920i	−0.874259	+	0.485460i	$0.838652\pi$
$194$	0	0
$195$	0	0
$196$	1880.00	0.685131
$197$	− 3720.45i	− 1.34554i	−0.739853	−	0.672768i	$-0.765105\pi$
$197$	− 3720.45i	− 1.34554i	0.739853	−	0.672768i	$-0.234895\pi$
$198$	0	0
$199$	1198.00	0.426754	0.213377	−	0.976970i	$-0.431554\pi$
$199$	1198.00	0.426754	0.213377	+	0.976970i	$0.431554\pi$
$200$	0	0
$201$	− 2109.64i	− 0.740310i
$202$	0	0
$203$	− 1028.84i	− 0.355716i
$204$	2808.00	0.963722
$205$	189.000	0.0643919
$206$	0	0
$207$	162.000	0.0543951
$208$	0	0
$209$	−1260.00	−0.417014
$210$	0	0
$211$	2392.00	0.780436	0.390218	−	0.920722i	$-0.372400\pi$
$211$	2392.00	0.780436	0.390218	+	0.920722i	$0.372400\pi$
$212$	−2088.00	−0.676436
$213$	1402.96i	0.451311i
$214$	0	0
$215$	426.084i	0.135157i
$216$	0	0
$217$	2016.00	0.630668
$218$	0	0
$219$	2052.48i	0.633305i
$220$	−2160.00	−0.661942
$221$	0	0
$222$	0	0
$223$	2036.89i	0.611661i	0.952086	+	0.305830i	$0.0989341\pi$
$223$	2036.89i	0.611661i	−0.952086	+	0.305830i	$0.901066\pi$
$224$	0	0
$225$	882.000	0.261333
$226$	0	0
$227$	2151.21i	0.628990i	0.949259	+	0.314495i	$0.101835\pi$
$227$	2151.21i	0.628990i	−0.949259	+	0.314495i	$0.898165\pi$
$228$	581.969i	0.169043i
$229$	− 3471.03i	− 1.00162i	−0.865556	−	0.500812i	$-0.833035\pi$
$229$	− 3471.03i	− 1.00162i	0.865556	−	0.500812i	$-0.166965\pi$
$230$	0	0
$231$	−1620.00	−0.461421
$232$	0	0
$233$	−1854.00	−0.521286	−0.260643	−	0.965435i	$-0.583935\pi$
$233$	−1854.00	−0.521286	−0.260643	+	0.965435i	$0.583935\pi$
$234$	0	0
$235$	378.000	0.104928
$236$	− 6318.52i	− 1.74280i
$237$	1320.00	0.361786
$238$	0	0
$239$	4458.30i	1.20662i	0.797505	+	0.603312i	$0.206153\pi$
$239$	4458.30i	1.20662i	−0.797505	+	0.603312i	$0.793847\pi$
$240$	997.661i	0.268328i
$241$	− 417.424i	− 0.111571i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	− 417.424i	− 0.111571i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	−5752.00	−1.50916
$245$	− 1221.10i	− 0.318420i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	3585.35i	0.912498i
$250$	0	0
$251$	4104.00	1.03204	0.516020	−	0.856576i	$-0.327413\pi$
$251$	4104.00	1.03204	0.516020	+	0.856576i	$0.327413\pi$
$252$	748.246i	0.187044i
$253$	− 935.307i	− 0.232420i
$254$	0	0
$255$	− 1823.85i	− 0.447898i
$256$	4096.00	1.00000
$257$	1989.00	0.482764	0.241382	−	0.970430i	$-0.422399\pi$
$257$	1989.00	0.482764	0.241382	+	0.970430i	$0.422399\pi$
$258$	0	0
$259$	1170.00	0.280696
$260$	0	0
$261$	−891.000	−0.211308
$262$	0	0
$263$	738.000	0.173031	0.0865153	−	0.996251i	$-0.472427\pi$
$263$	738.000	0.173031	0.0865153	+	0.996251i	$0.472427\pi$
$264$	0	0
$265$	1356.20i	0.314379i
$266$	0	0
$267$	4551.83i	1.04332i
$268$	5625.70i	1.28226i
$269$	−2106.00	−0.477342	−0.238671	−	0.971100i	$-0.576712\pi$
$269$	−2106.00	−0.477342	−0.238671	+	0.971100i	$0.576712\pi$
$270$	0	0
$271$	− 685.892i	− 0.153745i	−0.997041	−	0.0768727i	$-0.975507\pi$
$271$	− 685.892i	− 0.153745i	0.997041	−	0.0768727i	$-0.0244935\pi$
$272$	−7488.00	−1.66922
$273$	0	0
$274$	0	0
$275$	− 5092.23i	− 1.11663i
$276$	−432.000	−0.0942150
$277$	3665.00	0.794977	0.397488	−	0.917607i	$-0.369882\pi$
$277$	3665.00	0.794977	0.397488	+	0.917607i	$0.369882\pi$
$278$	0	0
$279$	− 1745.91i	− 0.374641i
$280$	0	0
$281$	1719.93i	0.365132i	0.983194	+	0.182566i	$0.0584404\pi$
$281$	1719.93i	0.365132i	−0.983194	+	0.182566i	$0.941560\pi$
$282$	0	0
$283$	−1826.00	−0.383549	−0.191775	−	0.981439i	$-0.561424\pi$
$283$	−1826.00	−0.383549	−0.191775	+	0.981439i	$0.561424\pi$
$284$	− 3741.23i	− 0.781694i
$285$	378.000	0.0785642
$286$	0	0
$287$	−378.000	−0.0777444
$288$	0	0
$289$	8776.00	1.78628
$290$	0	0
$291$	3471.03i	0.699228i
$292$	− 5473.28i	− 1.09692i
$293$	− 504.027i	− 0.100497i	−0.998737	−	0.0502484i	$-0.983999\pi$
$293$	− 504.027i	− 0.100497i	0.998737	−	0.0502484i	$-0.0160013\pi$
$294$	0	0
$295$	−4104.00	−0.809980
$296$	0	0
$297$	1402.96i	0.274101i
$298$	0	0
$299$	0	0
$300$	−2352.00	−0.452643
$301$	− 852.169i	− 0.163183i
$302$	0	0
$303$	−4725.00	−0.895855
$304$	− 1551.92i	− 0.292791i
$305$	3736.03i	0.701392i
$306$	0	0
$307$	− 1950.29i	− 0.362570i	−0.983431	−	0.181285i	$-0.941974\pi$
$307$	− 1950.29i	− 0.362570i	0.983431	−	0.181285i	$-0.0580256\pi$
$308$	4320.00	0.799204
$309$	−2382.00	−0.438535
$310$	0	0
$311$	−3798.00	−0.692491	−0.346246	−	0.938144i	$-0.612544\pi$
$311$	−3798.00	−0.692491	−0.346246	+	0.938144i	$0.612544\pi$
$312$	0	0
$313$	1378.00	0.248847	0.124424	−	0.992229i	$-0.460292\pi$
$313$	1378.00	0.248847	0.124424	+	0.992229i	$0.460292\pi$
$314$	0	0
$315$	486.000	0.0869302
$316$	−3520.00	−0.626631
$317$	− 7103.14i	− 1.25852i	−0.777193	−	0.629262i	$-0.783357\pi$
$317$	− 7103.14i	− 1.25852i	0.777193	−	0.629262i	$-0.216643\pi$
$318$	0	0
$319$	5144.19i	0.902882i
$320$	− 2660.43i	− 0.464758i
$321$	−1350.00	−0.234734
$322$	0	0
$323$	2837.10i	0.488732i
$324$	648.000	0.111111
$325$	0	0
$326$	0	0
$327$	1787.48i	0.302286i
$328$	0	0
$329$	−756.000	−0.126686
$330$	0	0
$331$	10073.6i	1.67280i	0.548122	+	0.836398i	$0.315343\pi$
$331$	10073.6i	1.67280i	−0.548122	+	0.836398i	$0.684657\pi$
$332$	− 9560.92i	− 1.58049i
$333$	− 1013.25i	− 0.166744i
$334$	0	0
$335$	3654.00	0.595938
$336$	− 1995.32i	− 0.323970i
$337$	9001.00	1.45494	0.727471	−	0.686138i	$-0.240695\pi$
$337$	9001.00	1.45494	0.727471	+	0.686138i	$0.240695\pi$
$338$	0	0
$339$	5103.00	0.817572
$340$	4863.60i	0.775781i
$341$	−10080.0	−1.60077
$342$	0	0
$343$	6006.75i	0.945581i
$344$	0	0
$345$	280.592i	0.0437872i
$346$	0	0
$347$	3294.00	0.509600	0.254800	−	0.966994i	$-0.417990\pi$
$347$	3294.00	0.509600	0.254800	+	0.966994i	$0.417990\pi$
$348$	2376.00	0.365997
$349$	10544.7i	1.61732i	0.588273	+	0.808662i	$0.299808\pi$
$349$	10544.7i	1.61732i	−0.588273	+	0.808662i	$0.700192\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2478.56i	0.373713i	0.982387	+	0.186856i	$0.0598299\pi$
$353$	2478.56i	0.373713i	−0.982387	+	0.186856i	$0.940170\pi$
$354$	0	0
$355$	−2430.00	−0.363299
$356$	− 12138.2i	− 1.80709i
$357$	3647.70i	0.540775i
$358$	0	0
$359$	5414.39i	0.795991i	0.917387	+	0.397995i	$0.130294\pi$
$359$	5414.39i	0.795991i	−0.917387	+	0.397995i	$0.869706\pi$
$360$	0	0
$361$	6271.00	0.914273
$362$	0	0
$363$	4107.00	0.593834
$364$	0	0
$365$	−3555.00	−0.509801
$366$	0	0
$367$	−9946.00	−1.41465	−0.707326	−	0.706888i	$-0.750099\pi$
$367$	−9946.00	−1.41465	−0.707326	+	0.706888i	$0.750099\pi$
$368$	1152.00	0.163185
$369$	327.358i	0.0461831i
$370$	0	0
$371$	− 2712.39i	− 0.379570i
$372$	4655.75i	0.648897i
$373$	−7301.00	−1.01349	−0.506745	−	0.862096i	$-0.669151\pi$
$373$	−7301.00	−1.01349	−0.506745	+	0.862096i	$0.669151\pi$
$374$	0	0
$375$	3476.23i	0.478697i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	3422.53i	0.463862i	0.972732	+	0.231931i	$0.0745044\pi$
$379$	3422.53i	0.463862i	−0.972732	+	0.231931i	$0.925496\pi$
$380$	−1008.00	−0.136077
$381$	4992.00	0.671254
$382$	0	0
$383$	− 5778.12i	− 0.770883i	−0.922732	−	0.385442i	$-0.874049\pi$
$383$	− 5778.12i	− 0.770883i	0.922732	−	0.385442i	$-0.125951\pi$
$384$	0	0
$385$	− 2805.92i	− 0.371436i
$386$	0	0
$387$	−738.000	−0.0969371
$388$	− 9256.08i	− 1.21110i
$389$	9153.00	1.19300	0.596498	−	0.802614i	$-0.296558\pi$
$389$	9153.00	1.19300	0.596498	+	0.802614i	$0.296558\pi$
$390$	0	0
$391$	−2106.00	−0.272391
$392$	0	0
$393$	4428.00	0.568354
$394$	0	0
$395$	2286.31i	0.291232i
$396$	− 3741.23i	− 0.474757i
$397$	2023.04i	0.255751i	0.991790	+	0.127876i	$0.0408158\pi$
$397$	2023.04i	0.255751i	−0.991790	+	0.127876i	$0.959184\pi$
$398$	0	0
$399$	−756.000	−0.0948555
$400$	6272.00	0.784000
$401$	− 8308.65i	− 1.03470i	−0.855774	−	0.517349i	$-0.826919\pi$
$401$	− 8308.65i	− 1.03470i	0.855774	−	0.517349i	$-0.173081\pi$
$402$	0	0
$403$	0	0
$404$	12600.0	1.55167
$405$	− 420.888i	− 0.0516398i
$406$	0	0
$407$	−5850.00	−0.712466
$408$	0	0
$409$	− 10418.3i	− 1.25954i	−0.776782	−	0.629769i	$-0.783150\pi$
$409$	− 10418.3i	− 1.25954i	0.776782	−	0.629769i	$-0.216850\pi$
$410$	0	0
$411$	− 3039.75i	− 0.364817i
$412$	6352.00	0.759565
$413$	8208.00	0.977940
$414$	0	0
$415$	−6210.00	−0.734547
$416$	0	0
$417$	−3372.00	−0.395989
$418$	0	0
$419$	4176.00	0.486900	0.243450	−	0.969913i	$-0.421721\pi$
$419$	4176.00	0.486900	0.243450	+	0.969913i	$0.421721\pi$
$420$	−1296.00	−0.150567
$421$	− 14471.3i	− 1.67527i	−0.546233	−	0.837633i	$-0.683939\pi$
$421$	− 14471.3i	− 1.67527i	0.546233	−	0.837633i	$-0.316061\pi$
$422$	0	0
$423$	654.715i	0.0752561i
$424$	0	0
$425$	−11466.0	−1.30867
$426$	0	0
$427$	− 7472.07i	− 0.846835i
$428$	3600.00	0.406571
$429$	0	0
$430$	0	0
$431$	6578.33i	0.735190i	0.929986	+	0.367595i	$0.119819\pi$
$431$	6578.33i	0.735190i	−0.929986	+	0.367595i	$0.880181\pi$
$432$	−1728.00	−0.192450
$433$	−6605.00	−0.733062	−0.366531	−	0.930406i	$-0.619455\pi$
$433$	−6605.00	−0.733062	−0.366531	+	0.930406i	$0.619455\pi$
$434$	0	0
$435$	− 1543.26i	− 0.170100i
$436$	− 4766.60i	− 0.523576i
$437$	− 436.477i	− 0.0477792i
$438$	0	0
$439$	−8542.00	−0.928673	−0.464336	−	0.885659i	$-0.653707\pi$
$439$	−8542.00	−0.928673	−0.464336	+	0.885659i	$0.653707\pi$
$440$	0	0
$441$	2115.00	0.228377
$442$	0	0
$443$	−14328.0	−1.53667	−0.768334	−	0.640049i	$-0.778914\pi$
$443$	−14328.0	−1.53667	−0.768334	+	0.640049i	$0.778914\pi$
$444$	2702.00i	0.288809i
$445$	−7884.00	−0.839859
$446$	0	0
$447$	− 9805.14i	− 1.03751i
$448$	5320.86i	0.561132i
$449$	3013.77i	0.316767i	0.987378	+	0.158384i	$0.0506283\pi$
$449$	3013.77i	0.316767i	−0.987378	+	0.158384i	$0.949372\pi$
$450$	0	0
$451$	1890.00	0.197332
$452$	−13608.0	−1.41608
$453$	− 4915.56i	− 0.509830i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2887.33i	0.295544i	0.989021	+	0.147772i	$0.0472102\pi$
$457$	2887.33i	0.295544i	−0.989021	+	0.147772i	$0.952790\pi$
$458$	0	0
$459$	3159.00	0.321241
$460$	− 748.246i	− 0.0758416i
$461$	3600.93i	0.363801i	0.983317	+	0.181900i	$0.0582249\pi$
$461$	3600.93i	0.363801i	−0.983317	+	0.181900i	$0.941775\pi$
$462$	0	0
$463$	− 2677.75i	− 0.268781i	−0.990928	−	0.134391i	$-0.957092\pi$
$463$	− 2677.75i	− 0.268781i	0.990928	−	0.134391i	$-0.0429077\pi$
$464$	−6336.00	−0.633925
$465$	3024.00	0.301580
$466$	0	0
$467$	13878.0	1.37515	0.687577	−	0.726111i	$-0.258674\pi$
$467$	13878.0	1.37515	0.687577	+	0.726111i	$0.258674\pi$
$468$	0	0
$469$	−7308.00	−0.719514
$470$	0	0
$471$	−3777.00	−0.369501
$472$	0	0
$473$	4260.84i	0.414194i
$474$	0	0
$475$	− 2376.37i	− 0.229548i
$476$	− 9727.20i	− 0.936650i
$477$	−2349.00	−0.225479
$478$	0	0
$479$	− 1101.58i	− 0.105079i	−0.998619	−	0.0525393i	$-0.983269\pi$
$479$	− 1101.58i	− 0.105079i	0.998619	−	0.0525393i	$-0.0167315\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 561.184i	− 0.0528670i
$484$	−10952.0	−1.02855
$485$	−6012.00	−0.562868
$486$	0	0
$487$	− 17123.1i	− 1.59326i	−0.604464	−	0.796632i	$-0.706613\pi$
$487$	− 17123.1i	− 1.59326i	0.604464	−	0.796632i	$-0.293387\pi$
$488$	0	0
$489$	− 8854.24i	− 0.818820i
$490$	0	0
$491$	450.000	0.0413609	0.0206805	−	0.999786i	$-0.493417\pi$
$491$	450.000	0.0413609	0.0206805	+	0.999786i	$0.493417\pi$
$492$	− 872.954i	− 0.0799914i
$493$	11583.0	1.05816
$494$	0	0
$495$	−2430.00	−0.220647
$496$	− 12415.3i	− 1.12392i
$497$	4860.00	0.438633
$498$	0	0
$499$	− 13219.0i	− 1.18590i	−0.805239	−	0.592950i	$-0.797963\pi$
$499$	− 13219.0i	− 1.18590i	0.805239	−	0.592950i	$-0.202037\pi$
$500$	− 9269.94i	− 0.829128i
$501$	− 9415.43i	− 0.839621i
$502$	0	0
$503$	5346.00	0.473889	0.236945	−	0.971523i	$-0.423854\pi$
$503$	5346.00	0.473889	0.236945	+	0.971523i	$0.423854\pi$
$504$	0	0
$505$	− 8183.94i	− 0.721150i
$506$	0	0
$507$	0	0
$508$	−13312.0	−1.16265
$509$	5866.46i	0.510857i	0.966828	+	0.255428i	$0.0822165\pi$
$509$	5866.46i	0.510857i	−0.966828	+	0.255428i	$0.917783\pi$
$510$	0	0
$511$	7110.00	0.615514
$512$	0	0
$513$	654.715i	0.0563477i
$514$	0	0
$515$	− 4125.75i	− 0.353014i
$516$	1968.00	0.167900
$517$	3780.00	0.321556
$518$	0	0
$519$	−12798.0	−1.08241
$520$	0	0
$521$	−9657.00	−0.812055	−0.406028	−	0.913861i	$-0.633086\pi$
$521$	−9657.00	−0.812055	−0.406028	+	0.913861i	$0.633086\pi$
$522$	0	0
$523$	21626.0	1.80811	0.904053	−	0.427421i	$-0.140578\pi$
$523$	21626.0	1.80811	0.904053	+	0.427421i	$0.140578\pi$
$524$	−11808.0	−0.984418
$525$	− 3055.34i	− 0.253992i
$526$	0	0
$527$	22696.8i	1.87607i
$528$	9976.61i	0.822304i
$529$	−11843.0	−0.973371
$530$	0	0
$531$	− 7108.34i	− 0.580933i
$532$	2016.00	0.164295
$533$	0	0
$534$	0	0
$535$	− 2338.27i	− 0.188957i
$536$	0	0
$537$	−9018.00	−0.724684
$538$	0	0
$539$	− 12211.0i	− 0.975813i
$540$	1122.37i	0.0894427i
$541$	5371.09i	0.426841i	0.976960	+	0.213421i	$0.0684605\pi$
$541$	5371.09i	0.426841i	−0.976960	+	0.213421i	$0.931540\pi$
$542$	0	0
$543$	5619.00	0.444078
$544$	0	0
$545$	−3096.00	−0.243336
$546$	0	0
$547$	16946.0	1.32460	0.662302	−	0.749237i	$-0.269579\pi$
$547$	16946.0	1.32460	0.662302	+	0.749237i	$0.269579\pi$
$548$	8106.00i	0.631882i
$549$	−6471.00	−0.503052
$550$	0	0
$551$	2400.62i	0.185608i
$552$	0	0
$553$	− 4572.61i	− 0.351623i
$554$	0	0
$555$	1755.00	0.134226
$556$	8992.00	0.685874
$557$	− 3860.74i	− 0.293689i	−0.989160	−	0.146845i	$-0.953088\pi$
$557$	− 3860.74i	− 0.293689i	0.989160	−	0.146845i	$-0.0469117\pi$
$558$	0	0
$559$	0	0
$560$	3456.00	0.260790
$561$	− 18238.5i	− 1.37260i
$562$	0	0
$563$	21672.0	1.62232	0.811160	−	0.584825i	$-0.198837\pi$
$563$	21672.0	1.62232	0.811160	+	0.584825i	$0.198837\pi$
$564$	− 1745.91i	− 0.130347i
$565$	8838.66i	0.658133i
$566$	0	0
$567$	841.777i	0.0623480i
$568$	0	0
$569$	−1386.00	−0.102116	−0.0510581	−	0.998696i	$-0.516259\pi$
$569$	−1386.00	−0.102116	−0.0510581	+	0.998696i	$0.516259\pi$
$570$	0	0
$571$	1162.00	0.0851632	0.0425816	−	0.999093i	$-0.486442\pi$
$571$	1162.00	0.0851632	0.0425816	+	0.999093i	$0.486442\pi$
$572$	0	0
$573$	8208.00	0.598419
$574$	0	0
$575$	1764.00	0.127937
$576$	4608.00	0.333333
$577$	8045.38i	0.580474i	0.956955	+	0.290237i	$0.0937341\pi$
$577$	8045.38i	0.580474i	−0.956955	+	0.290237i	$0.906266\pi$
$578$	0	0
$579$	− 7809.82i	− 0.560561i
$580$	4115.35i	0.294622i
$581$	12420.0	0.886865
$582$	0	0
$583$	13562.0i	0.963429i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27622.7i	1.94227i	0.238530	+	0.971135i	$0.423335\pi$
$587$	27622.7i	1.94227i	−0.238530	+	0.971135i	$0.576665\pi$
$588$	−5640.00	−0.395561
$589$	−4704.00	−0.329075
$590$	0	0
$591$	11161.3i	0.776846i
$592$	− 7205.33i	− 0.500232i
$593$	275.396i	0.0190711i	0.999955	+	0.00953555i	$0.00303531\pi$
$593$	275.396i	0.0190711i	−0.999955	+	0.00953555i	$0.996965\pi$
$594$	0	0
$595$	−6318.00	−0.435316
$596$	26147.0i	1.79702i
$597$	−3594.00	−0.246386
$598$	0	0
$599$	22356.0	1.52494	0.762472	−	0.647021i	$-0.223986\pi$
$599$	22356.0	1.52494	0.762472	+	0.647021i	$0.223986\pi$
$600$	0	0
$601$	−18083.0	−1.22732	−0.613661	−	0.789569i	$-0.710304\pi$
$601$	−18083.0	−1.22732	−0.613661	+	0.789569i	$0.710304\pi$
$602$	0	0
$603$	6328.91i	0.427418i
$604$	13108.2i	0.883052i
$605$	7113.53i	0.478027i
$606$	0	0
$607$	5480.00	0.366435	0.183218	−	0.983072i	$-0.441349\pi$
$607$	5480.00	0.366435	0.183218	+	0.983072i	$0.441349\pi$
$608$	0	0
$609$	3086.51i	0.205373i
$610$	0	0
$611$	0	0
$612$	−8424.00	−0.556405
$613$	− 17737.9i	− 1.16872i	−0.811493	−	0.584362i	$-0.801345\pi$
$613$	− 17737.9i	− 1.16872i	0.811493	−	0.584362i	$-0.198655\pi$
$614$	0	0
$615$	−567.000	−0.0371767
$616$	0	0
$617$	9867.49i	0.643842i	0.946767	+	0.321921i	$0.104328\pi$
$617$	9867.49i	0.643842i	−0.946767	+	0.321921i	$0.895672\pi$
$618$	0	0
$619$	− 4115.35i	− 0.267221i	−0.991034	−	0.133611i	$-0.957343\pi$
$619$	− 4115.35i	− 0.267221i	0.991034	−	0.133611i	$-0.0426572\pi$
$620$	−8064.00	−0.522352
$621$	−486.000	−0.0314050
$622$	0	0
$623$	15768.0	1.01402
$624$	0	0
$625$	6229.00	0.398656
$626$	0	0
$627$	3780.00	0.240763
$628$	10072.0	0.639995
$629$	13172.2i	0.834995i
$630$	0	0
$631$	− 12664.8i	− 0.799011i	−0.916731	−	0.399506i	$-0.869182\pi$
$631$	− 12664.8i	− 0.799011i	0.916731	−	0.399506i	$-0.130818\pi$
$632$	0	0
$633$	−7176.00	−0.450585
$634$	0	0
$635$	8646.40i	0.540349i
$636$	6264.00	0.390540
$637$	0	0
$638$	0	0
$639$	− 4208.88i	− 0.260565i
$640$	0	0
$641$	−3789.00	−0.233473	−0.116737	−	0.993163i	$-0.537243\pi$
$641$	−3789.00	−0.233473	−0.116737	+	0.993163i	$0.537243\pi$
$642$	0	0
$643$	16911.7i	1.03722i	0.855010	+	0.518611i	$0.173551\pi$
$643$	16911.7i	1.03722i	−0.855010	+	0.518611i	$0.826449\pi$
$644$	1496.49i	0.0915684i
$645$	− 1278.25i	− 0.0780328i
$646$	0	0
$647$	−27792.0	−1.68874	−0.844371	−	0.535759i	$-0.820026\pi$
$647$	−27792.0	−1.68874	−0.844371	+	0.535759i	$0.820026\pi$
$648$	0	0
$649$	−41040.0	−2.48222
$650$	0	0
$651$	−6048.00	−0.364116
$652$	23611.3i	1.41824i
$653$	−594.000	−0.0355973	−0.0177986	−	0.999842i	$-0.505666\pi$
$653$	−594.000	−0.0355973	−0.0177986	+	0.999842i	$0.505666\pi$
$654$	0	0
$655$	7669.52i	0.457516i
$656$	2327.88i	0.138549i
$657$	− 6157.44i	− 0.365639i
$658$	0	0
$659$	17748.0	1.04911	0.524555	−	0.851376i	$-0.324232\pi$
$659$	17748.0	1.04911	0.524555	+	0.851376i	$0.324232\pi$
$660$	6480.00	0.382172
$661$	− 15791.1i	− 0.929203i	−0.885520	−	0.464601i	$-0.846198\pi$
$661$	− 15791.1i	− 0.929203i	0.885520	−	0.464601i	$-0.153802\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1309.43i	− 0.0763572i
$666$	0	0
$667$	−1782.00	−0.103447
$668$	25107.8i	1.45427i
$669$	− 6110.68i	− 0.353143i
$670$	0	0
$671$	37360.3i	2.14945i
$672$	0	0
$673$	20933.0	1.19897	0.599486	−	0.800385i	$-0.295372\pi$
$673$	20933.0	1.19897	0.599486	+	0.800385i	$0.295372\pi$
$674$	0	0
$675$	−2646.00	−0.150881
$676$	0	0
$677$	3402.00	0.193131	0.0965653	−	0.995327i	$-0.469214\pi$
$677$	3402.00	0.193131	0.0965653	+	0.995327i	$0.469214\pi$
$678$	0	0
$679$	12024.0	0.679586
$680$	0	0
$681$	− 6453.62i	− 0.363147i
$682$	0	0
$683$	24983.1i	1.39964i	0.714321	+	0.699818i	$0.246736\pi$
$683$	24983.1i	1.39964i	−0.714321	+	0.699818i	$0.753264\pi$
$684$	− 1745.91i	− 0.0975971i
$685$	5265.00	0.293672
$686$	0	0
$687$	10413.1i	0.578288i
$688$	−5248.00	−0.290811
$689$	0	0
$690$	0	0
$691$	− 13866.8i	− 0.763412i	−0.924284	−	0.381706i	$-0.875337\pi$
$691$	− 13866.8i	− 0.763412i	0.924284	−	0.381706i	$-0.124663\pi$
$692$	34128.0	1.87479
$693$	4860.00	0.266401
$694$	0	0
$695$	− 5840.48i	− 0.318765i
$696$	0	0
$697$	− 4255.65i	− 0.231269i
$698$	0	0
$699$	5562.00	0.300964
$700$	8147.57i	0.439927i
$701$	−21906.0	−1.18028	−0.590141	−	0.807300i	$-0.700928\pi$
$701$	−21906.0	−1.18028	−0.590141	+	0.807300i	$0.700928\pi$
$702$	0	0
$703$	−2730.00	−0.146464
$704$	− 26604.3i	− 1.42427i
$705$	−1134.00	−0.0605800
$706$	0	0
$707$	16367.9i	0.870690i
$708$	18955.6i	1.00621i
$709$	− 13057.9i	− 0.691680i	−0.938294	−	0.345840i	$-0.887594\pi$
$709$	− 13057.9i	− 0.691680i	0.938294	−	0.345840i	$-0.112406\pi$
$710$	0	0
$711$	−3960.00	−0.208877
$712$	0	0
$713$	− 3491.81i	− 0.183407i
$714$	0	0
$715$	0	0
$716$	24048.0	1.25519
$717$	− 13374.9i	− 0.696645i
$718$	0	0
$719$	−14220.0	−0.737575	−0.368788	−	0.929514i	$-0.620227\pi$
$719$	−14220.0	−0.737575	−0.368788	+	0.929514i	$0.620227\pi$
$720$	− 2992.98i	− 0.154919i
$721$	8251.49i	0.426216i
$722$	0	0
$723$	1252.27i	0.0644157i
$724$	−14984.0	−0.769166
$725$	−9702.00	−0.496998
$726$	0	0
$727$	−5282.00	−0.269462	−0.134731	−	0.990882i	$-0.543017\pi$
$727$	−5282.00	−0.269462	−0.134731	+	0.990882i	$0.543017\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	9594.00	0.485427
$732$	17256.0	0.871312
$733$	− 11419.4i	− 0.575424i	−0.957717	−	0.287712i	$-0.907105\pi$
$733$	− 11419.4i	− 0.575424i	0.957717	−	0.287712i	$-0.0928945\pi$
$734$	0	0
$735$	3663.29i	0.183840i
$736$	0	0
$737$	36540.0	1.82628
$738$	0	0
$739$	20535.2i	1.02219i	0.859524	+	0.511096i	$0.170760\pi$
$739$	20535.2i	1.02219i	−0.859524	+	0.511096i	$0.829240\pi$
$740$	−4680.00	−0.232487
$741$	0	0
$742$	0	0
$743$	20826.2i	1.02832i	0.857696	+	0.514158i	$0.171895\pi$
$743$	20826.2i	1.02832i	−0.857696	+	0.514158i	$0.828105\pi$
$744$	0	0
$745$	16983.0	0.835180
$746$	0	0
$747$	− 10756.0i	− 0.526831i
$748$	48636.0i	2.37742i
$749$	4676.54i	0.228140i
$750$	0	0
$751$	−4834.00	−0.234880	−0.117440	−	0.993080i	$-0.537469\pi$
$751$	−4834.00	−0.234880	−0.117440	+	0.993080i	$0.537469\pi$
$752$	4655.75i	0.225768i
$753$	−12312.0	−0.595849
$754$	0	0
$755$	8514.00	0.410406
$756$	− 2244.74i	− 0.107990i
$757$	9046.00	0.434323	0.217161	−	0.976136i	$-0.430320\pi$
$757$	9046.00	0.434323	0.217161	+	0.976136i	$0.430320\pi$
$758$	0	0
$759$	2805.92i	0.134188i
$760$	0	0
$761$	− 12034.3i	− 0.573249i	−0.958043	−	0.286625i	$-0.907467\pi$
$761$	− 12034.3i	− 0.573249i	0.958043	−	0.286625i	$-0.0925332\pi$
$762$	0	0
$763$	6192.00	0.293795
$764$	−21888.0	−1.03649
$765$	5471.55i	0.258594i
$766$	0	0
$767$	0	0
$768$	−12288.0	−0.577350
$769$	− 37543.9i	− 1.76056i	−0.474457	−	0.880279i	$-0.657355\pi$
$769$	− 37543.9i	− 1.76056i	0.474457	−	0.880279i	$-0.342645\pi$
$770$	0	0
$771$	−5967.00	−0.278724
$772$	20826.2i	0.970920i
$773$	− 15713.2i	− 0.731130i	−0.930786	−	0.365565i	$-0.880876\pi$
$773$	− 15713.2i	− 0.731130i	0.930786	−	0.365565i	$-0.119124\pi$
$774$	0	0
$775$	− 19011.0i	− 0.881155i
$776$	0	0
$777$	−3510.00	−0.162060
$778$	0	0
$779$	882.000	0.0405660
$780$	0	0
$781$	−24300.0	−1.11334
$782$	0	0
$783$	2673.00	0.121999
$784$	15040.0	0.685131
$785$	− 6541.96i	− 0.297443i
$786$	0	0
$787$	− 3755.09i	− 0.170082i	−0.996377	−	0.0850409i	$-0.972898\pi$
$787$	− 3755.09i	− 0.170082i	0.996377	−	0.0850409i	$-0.0271021\pi$
$788$	− 29763.6i	− 1.34554i
$789$	−2214.00	−0.0998992
$790$	0	0
$791$	− 17677.3i	− 0.794605i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 4068.59i	− 0.181507i
$796$	9584.00	0.426754
$797$	−7830.00	−0.347996	−0.173998	−	0.984746i	$-0.555669\pi$
$797$	−7830.00	−0.347996	−0.173998	+	0.984746i	$0.555669\pi$
$798$	0	0
$799$	− 8511.30i	− 0.376856i
$800$	0	0
$801$	− 13655.5i	− 0.602363i
$802$	0	0
$803$	−35550.0	−1.56231
$804$	− 16877.1i	− 0.740310i
$805$	972.000	0.0425571
$806$	0	0
$807$	6318.00	0.275594
$808$	0	0
$809$	−6165.00	−0.267923	−0.133962	−	0.990987i	$-0.542770\pi$
$809$	−6165.00	−0.267923	−0.133962	+	0.990987i	$0.542770\pi$
$810$	0	0
$811$	29839.8i	1.29201i	0.763335	+	0.646003i	$0.223560\pi$
$811$	29839.8i	1.29201i	−0.763335	+	0.646003i	$0.776440\pi$
$812$	− 8230.71i	− 0.355716i
$813$	2057.68i	0.0887649i
$814$	0	0
$815$	15336.0	0.659137
$816$	22464.0	0.963722
$817$	1988.39i	0.0851470i
$818$	0	0
$819$	0	0
$820$	1512.00	0.0643919
$821$	29763.6i	1.26523i	0.774466	+	0.632616i	$0.218019\pi$
$821$	29763.6i	1.26523i	−0.774466	+	0.632616i	$0.781981\pi$
$822$	0	0
$823$	−8920.00	−0.377803	−0.188901	−	0.981996i	$-0.560493\pi$
$823$	−8920.00	−0.377803	−0.188901	+	0.981996i	$0.560493\pi$
$824$	0	0
$825$	15276.7i	0.644686i
$826$	0	0
$827$	18041.0i	0.758583i	0.925277	+	0.379292i	$0.123832\pi$
$827$	18041.0i	0.758583i	−0.925277	+	0.379292i	$0.876168\pi$
$828$	1296.00	0.0543951
$829$	−21023.0	−0.880771	−0.440385	−	0.897809i	$-0.645158\pi$
$829$	−21023.0	−0.880771	−0.440385	+	0.897809i	$0.645158\pi$
$830$	0	0
$831$	−10995.0	−0.458980
$832$	0	0
$833$	−27495.0	−1.14363
$834$	0	0
$835$	16308.0	0.675882
$836$	−10080.0	−0.417014
$837$	5237.72i	0.216299i
$838$	0	0
$839$	− 22073.3i	− 0.908288i	−0.890928	−	0.454144i	$-0.849945\pi$
$839$	− 22073.3i	− 0.908288i	0.890928	−	0.454144i	$-0.150055\pi$
$840$	0	0
$841$	−14588.0	−0.598139
$842$	0	0
$843$	− 5159.78i	− 0.210809i
$844$	19136.0	0.780436
$845$	0	0
$846$	0	0
$847$	− 14227.1i	− 0.577152i
$848$	−16704.0	−0.676436
$849$	5478.00	0.221442
$850$	0	0
$851$	− 2026.50i	− 0.0816304i
$852$	11223.7i	0.451311i
$853$	26609.5i	1.06810i	0.845452	+	0.534051i	$0.179331\pi$
$853$	26609.5i	1.06810i	−0.845452	+	0.534051i	$0.820669\pi$
$854$	0	0
$855$	−1134.00	−0.0453590
$856$	0	0
$857$	−12771.0	−0.509042	−0.254521	−	0.967067i	$-0.581918\pi$
$857$	−12771.0	−0.509042	−0.254521	+	0.967067i	$0.581918\pi$
$858$	0	0
$859$	17134.0	0.680564	0.340282	−	0.940323i	$-0.389477\pi$
$859$	17134.0	0.680564	0.340282	+	0.940323i	$0.389477\pi$
$860$	3408.68i	0.135157i
$861$	1134.00	0.0448857
$862$	0	0
$863$	7929.33i	0.312766i	0.987696	+	0.156383i	$0.0499835\pi$
$863$	7929.33i	0.312766i	−0.987696	+	0.156383i	$0.950017\pi$
$864$	0	0
$865$	− 22166.8i	− 0.871322i
$866$	0	0
$867$	−26328.0	−1.03131
$868$	16128.0	0.630668
$869$	22863.1i	0.892493i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 10413.1i	− 0.403700i
$874$	0	0
$875$	12042.0	0.465250
$876$	16419.8i	0.633305i
$877$	9864.03i	0.379800i	0.981803	+	0.189900i	$0.0608164\pi$
$877$	9864.03i	0.379800i	−0.981803	+	0.189900i	$0.939184\pi$
$878$	0	0
$879$	1512.08i	0.0580218i
$880$	−17280.0	−0.661942
$881$	29169.0	1.11547	0.557735	−	0.830019i	$-0.311671\pi$
$881$	29169.0	1.11547	0.557735	+	0.830019i	$0.311671\pi$
$882$	0	0
$883$	928.000	0.0353677	0.0176839	−	0.999844i	$-0.494371\pi$
$883$	928.000	0.0353677	0.0176839	+	0.999844i	$0.494371\pi$
$884$	0	0
$885$	12312.0	0.467642
$886$	0	0
$887$	−14400.0	−0.545101	−0.272551	−	0.962141i	$-0.587867\pi$
$887$	−14400.0	−0.545101	−0.272551	+	0.962141i	$0.587867\pi$
$888$	0	0
$889$	− 17292.8i	− 0.652398i
$890$	0	0
$891$	− 4208.88i	− 0.158252i
$892$	16295.1i	0.611661i
$893$	1764.00	0.0661030
$894$	0	0
$895$	− 15619.6i	− 0.583360i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19205.0i	0.712483i
$900$	7056.00	0.261333
$901$	30537.0	1.12912
$902$	0	0
$903$	2556.51i	0.0942140i
$904$	0	0
$905$	9732.39i	0.357476i
$906$	0	0
$907$	19684.0	0.720614	0.360307	−	0.932834i	$-0.382672\pi$
$907$	19684.0	0.720614	0.360307	+	0.932834i	$0.382672\pi$
$908$	17209.7i	0.628990i
$909$	14175.0	0.517222
$910$	0	0
$911$	24480.0	0.890295	0.445147	−	0.895457i	$-0.353151\pi$
$911$	24480.0	0.890295	0.445147	+	0.895457i	$0.353151\pi$
$912$	4655.75i	0.169043i
$913$	−62100.0	−2.25105
$914$	0	0
$915$	− 11208.1i	− 0.404949i
$916$	− 27768.2i	− 1.00162i
$917$	− 15339.0i	− 0.552388i
$918$	0	0
$919$	38608.0	1.38581	0.692906	−	0.721028i	$-0.256330\pi$
$919$	38608.0	1.38581	0.692906	+	0.721028i	$0.256330\pi$
$920$	0	0
$921$	5850.87i	0.209330i
$922$	0	0
$923$	0	0
$924$	−12960.0	−0.461421
$925$	− 11033.2i	− 0.392182i
$926$	0	0
$927$	7146.00	0.253188
$928$	0	0
$929$	10210.4i	0.360596i	0.983612	+	0.180298i	$0.0577062\pi$
$929$	10210.4i	0.360596i	−0.983612	+	0.180298i	$0.942294\pi$
$930$	0	0
$931$	− 5698.45i	− 0.200600i
$932$	−14832.0	−0.521286
$933$	11394.0	0.399810
$934$	0	0
$935$	31590.0	1.10492
$936$	0	0
$937$	28495.0	0.993480	0.496740	−	0.867899i	$-0.334530\pi$
$937$	28495.0	0.993480	0.496740	+	0.867899i	$0.334530\pi$
$938$	0	0
$939$	−4134.00	−0.143672
$940$	3024.00	0.104928
$941$	− 10724.9i	− 0.371541i	−0.982593	−	0.185771i	$-0.940522\pi$
$941$	− 10724.9i	− 0.371541i	0.982593	−	0.185771i	$-0.0594782\pi$
$942$	0	0
$943$	654.715i	0.0226092i
$944$	− 50548.2i	− 1.74280i
$945$	−1458.00	−0.0501891
$946$	0	0
$947$	− 33962.1i	− 1.16538i	−0.812693	−	0.582692i	$-0.801999\pi$
$947$	− 33962.1i	− 1.16538i	0.812693	−	0.582692i	$-0.198001\pi$
$948$	10560.0	0.361786
$949$	0	0
$950$	0	0
$951$	21309.4i	0.726609i
$952$	0	0
$953$	23814.0	0.809456	0.404728	−	0.914437i	$-0.367366\pi$
$953$	23814.0	0.809456	0.404728	+	0.914437i	$0.367366\pi$
$954$	0	0
$955$	14216.7i	0.481718i
$956$	35666.4i	1.20662i
$957$	− 15432.6i	− 0.521279i
$958$	0	0
$959$	−10530.0	−0.354569
$960$	7981.29i	0.268328i
$961$	−7841.00	−0.263200
$962$	0	0
$963$	4050.00	0.135524
$964$	− 3339.39i	− 0.111571i
$965$	13527.0	0.451243
$966$	0	0
$967$	− 51549.3i	− 1.71429i	−0.515079	−	0.857143i	$-0.672238\pi$
$967$	− 51549.3i	− 1.71429i	0.515079	−	0.857143i	$-0.327762\pi$
$968$	0	0
$969$	− 8511.30i	− 0.282170i
$970$	0	0
$971$	−12312.0	−0.406911	−0.203456	−	0.979084i	$-0.565217\pi$
$971$	−12312.0	−0.406911	−0.203456	+	0.979084i	$0.565217\pi$
$972$	−1944.00	−0.0641500
$973$	11681.0i	0.384865i
$974$	0	0
$975$	0	0
$976$	−46016.0	−1.50916
$977$	− 21538.1i	− 0.705285i	−0.935758	−	0.352642i	$-0.885283\pi$
$977$	− 21538.1i	− 0.705285i	0.935758	−	0.352642i	$-0.114717\pi$
$978$	0	0
$979$	−78840.0	−2.57379
$980$	− 9768.77i	− 0.318420i
$981$	− 5362.43i	− 0.174525i
$982$	0	0
$983$	32611.1i	1.05812i	0.848585	+	0.529060i	$0.177455\pi$
$983$	32611.1i	1.05812i	−0.848585	+	0.529060i	$0.822545\pi$
$984$	0	0
$985$	−19332.0	−0.625349
$986$	0	0
$987$	2268.00	0.0731421
$988$	0	0
$989$	−1476.00	−0.0474561
$990$	0	0
$991$	−22330.0	−0.715778	−0.357889	−	0.933764i	$-0.616503\pi$
$991$	−22330.0	−0.715778	−0.357889	+	0.933764i	$0.616503\pi$
$992$	0	0
$993$	− 30220.8i	− 0.965789i
$994$	0	0
$995$	− 6224.99i	− 0.198337i
$996$	28682.8i	0.912498i
$997$	−24931.0	−0.791949	−0.395974	−	0.918262i	$-0.629593\pi$
$997$	−24931.0	−0.791949	−0.395974	+	0.918262i	$0.629593\pi$
$998$	0	0
$999$	3039.75i	0.0962697i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.a.337.1		2
13.3	even	3		39.4.j.a.4.1	✓	2
13.4	even	6		39.4.j.a.10.1	yes	2
13.5	odd	4		507.4.a.g.1.1		2
13.8	odd	4		507.4.a.g.1.2		2
13.12	even	2	inner	507.4.b.a.337.2		2
39.5	even	4		1521.4.a.m.1.2		2
39.8	even	4		1521.4.a.m.1.1		2
39.17	odd	6		117.4.q.b.10.1		2
39.29	odd	6		117.4.q.b.82.1		2
52.3	odd	6		624.4.bv.a.433.1		2
52.43	odd	6		624.4.bv.a.49.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.a.4.1	✓	2	13.3	even	3
39.4.j.a.10.1	yes	2	13.4	even	6
117.4.q.b.10.1		2	39.17	odd	6
117.4.q.b.82.1		2	39.29	odd	6
507.4.a.g.1.1		2	13.5	odd	4
507.4.a.g.1.2		2	13.8	odd	4
507.4.b.a.337.1		2	1.1	even	1	trivial
507.4.b.a.337.2		2	13.12	even	2	inner
624.4.bv.a.49.1		2	52.43	odd	6
624.4.bv.a.433.1		2	52.3	odd	6
1521.4.a.m.1.1		2	39.8	even	4
1521.4.a.m.1.2		2	39.5	even	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +8.00000 q^{4} -5.19615i q^{5} +10.3923i q^{7} +9.00000 q^{9} -51.9615i q^{11} -24.0000 q^{12} +15.5885i q^{15} +64.0000 q^{16} -117.000 q^{17} -24.2487i q^{19} -41.5692i q^{20} -31.1769i q^{21} +18.0000 q^{23} +98.0000 q^{25} -27.0000 q^{27} +83.1384i q^{28} -99.0000 q^{29} -193.990i q^{31} +155.885i q^{33} +54.0000 q^{35} +72.0000 q^{36} -112.583i q^{37} +36.3731i q^{41} -82.0000 q^{43} -415.692i q^{44} -46.7654i q^{45} +72.7461i q^{47} -192.000 q^{48} +235.000 q^{49} +351.000 q^{51} -261.000 q^{53} -270.000 q^{55} +72.7461i q^{57} -789.815i q^{59} +124.708i q^{60} -719.000 q^{61} +93.5307i q^{63} +512.000 q^{64} +703.213i q^{67} -936.000 q^{68} -54.0000 q^{69} -467.654i q^{71} -684.160i q^{73} -294.000 q^{75} -193.990i q^{76} +540.000 q^{77} -440.000 q^{79} -332.554i q^{80} +81.0000 q^{81} -1195.12i q^{83} -249.415i q^{84} +607.950i q^{85} +297.000 q^{87} -1517.28i q^{89} +144.000 q^{92} +581.969i q^{93} -126.000 q^{95} -1157.01i q^{97} -467.654i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 16 q^{4} + 18 q^{9} - 48 q^{12} + 128 q^{16} - 234 q^{17} + 36 q^{23} + 196 q^{25} - 54 q^{27} - 198 q^{29} + 108 q^{35} + 144 q^{36} - 164 q^{43} - 384 q^{48} + 470 q^{49} + 702 q^{51}+ \cdots - 252 q^{95}+O(q^{100}) \) 2 * q - 6 * q^3 + 16 * q^4 + 18 * q^9 - 48 * q^12 + 128 * q^16 - 234 * q^17 + 36 * q^23 + 196 * q^25 - 54 * q^27 - 198 * q^29 + 108 * q^35 + 144 * q^36 - 164 * q^43 - 384 * q^48 + 470 * q^49 + 702 * q^51 - 522 * q^53 - 540 * q^55 - 1438 * q^61 + 1024 * q^64 - 1872 * q^68 - 108 * q^69 - 588 * q^75 + 1080 * q^77 - 880 * q^79 + 162 * q^81 + 594 * q^87 + 288 * q^92 - 252 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more