LMFDB - Embedded newform 800.6.f.b.49.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [800,6,Mod(49,800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("800.49"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$128.307055850$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 $ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{93}\cdot 3^{4}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.15
Root			$3.18502 + 2.41984i$ of defining polynomial
Character	$\chi$	$=$	800.49
Dual form			800.6.f.b.49.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+17.3148 q^{3} -9.19080i q^{7} +56.8021 q^{9} -160.480i q^{11} +368.546 q^{13} +1261.09i q^{17} +2486.75i q^{19} -159.137i q^{21} +422.882i q^{23} -3223.98 q^{27} -5666.05i q^{29} -9387.13 q^{31} -2778.67i q^{33} +3566.43 q^{37} +6381.30 q^{39} -5949.95 q^{41} -10658.5 q^{43} +9243.94i q^{47} +16722.5 q^{49} +21835.5i q^{51} +8976.82 q^{53} +43057.5i q^{57} +27435.6i q^{59} -50515.3i q^{61} -522.057i q^{63} -5964.39 q^{67} +7322.11i q^{69} -67286.3 q^{71} +85768.5i q^{73} -1474.94 q^{77} +56567.2 q^{79} -69625.4 q^{81} +30208.1 q^{83} -98106.4i q^{87} -113965. q^{89} -3387.24i q^{91} -162536. q^{93} +138806. i q^{97} -9115.59i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 36 q^{3} + 1620 q^{9} - 11664 q^{27} - 7160 q^{31} - 3608 q^{37} - 44904 q^{39} + 11608 q^{41} + 51772 q^{43} - 18756 q^{49} + 928 q^{53} + 161604 q^{67} + 200312 q^{71} + 26008 q^{77} + 282080 q^{79}+ \cdots + 293472 q^{93}+O(q^{100}) $ 20 * q - 36 * q^3 + 1620 * q^9 - 11664 * q^27 - 7160 * q^31 - 3608 * q^37 - 44904 * q^39 + 11608 * q^41 + 51772 * q^43 - 18756 * q^49 + 928 * q^53 + 161604 * q^67 + 200312 * q^71 + 26008 * q^77 + 282080 * q^79 + 65172 * q^81 + 99092 * q^83 + 3160 * q^89 + 293472 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$-1$

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
$2$	0	0
$3$	17.3148	1.11074	0.555372	−	0.831602i	$-0.312576\pi$
$3$	17.3148	1.11074	0.555372	+	0.831602i	$0.312576\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 9.19080i	− 0.0708938i	−0.999372	−	0.0354469i	$-0.988715\pi$
$7$	− 9.19080i	− 0.0708938i	0.999372	−	0.0354469i	$-0.0112855\pi$
$8$	0	0
$9$	56.8021	0.233754
$10$	0	0
$11$	− 160.480i	− 0.399888i	−0.979807	−	0.199944i	$-0.935924\pi$
$11$	− 160.480i	− 0.399888i	0.979807	−	0.199944i	$-0.0640760\pi$
$12$	0	0
$13$	368.546	0.604831	0.302415	−	0.953176i	$-0.402207\pi$
$13$	368.546	0.604831	0.302415	+	0.953176i	$0.402207\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1261.09i	1.05833i	0.848517	+	0.529167i	$0.177496\pi$
$17$	1261.09i	1.05833i	−0.848517	+	0.529167i	$0.822504\pi$
$18$	0	0
$19$	2486.75i	1.58033i	0.612894	+	0.790165i	$0.290005\pi$
$19$	2486.75i	1.58033i	−0.612894	+	0.790165i	$0.709995\pi$
$20$	0	0
$21$	− 159.137i	− 0.0787449i
$22$	0	0
$23$	422.882i	0.166686i	0.996521	+	0.0833431i	$0.0265597\pi$
$23$	422.882i	0.166686i	−0.996521	+	0.0833431i	$0.973440\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3223.98	−0.851104
$28$	0	0
$29$	− 5666.05i	− 1.25108i	−0.780192	−	0.625540i	$-0.784879\pi$
$29$	− 5666.05i	− 1.25108i	0.780192	−	0.625540i	$-0.215121\pi$
$30$	0	0
$31$	−9387.13	−1.75440	−0.877200	−	0.480125i	$-0.840591\pi$
$31$	−9387.13	−1.75440	−0.877200	+	0.480125i	$0.840591\pi$
$32$	0	0
$33$	− 2778.67i	− 0.444174i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3566.43	0.428281	0.214141	−	0.976803i	$-0.431305\pi$
$37$	3566.43	0.428281	0.214141	+	0.976803i	$0.431305\pi$
$38$	0	0
$39$	6381.30	0.671812
$40$	0	0
$41$	−5949.95	−0.552782	−0.276391	−	0.961045i	$-0.589138\pi$
$41$	−5949.95	−0.552782	−0.276391	+	0.961045i	$0.589138\pi$
$42$	0	0
$43$	−10658.5	−0.879077	−0.439538	−	0.898224i	$-0.644858\pi$
$43$	−10658.5	−0.879077	−0.439538	+	0.898224i	$0.644858\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9243.94i	0.610397i	0.952289	+	0.305198i	$0.0987228\pi$
$47$	9243.94i	0.610397i	−0.952289	+	0.305198i	$0.901277\pi$
$48$	0	0
$49$	16722.5	0.994974
$50$	0	0
$51$	21835.5i	1.17554i
$52$	0	0
$53$	8976.82	0.438968	0.219484	−	0.975616i	$-0.429563\pi$
$53$	8976.82	0.438968	0.219484	+	0.975616i	$0.429563\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	43057.5i	1.75534i
$58$	0	0
$59$	27435.6i	1.02609i	0.858363	+	0.513044i	$0.171482\pi$
$59$	27435.6i	1.02609i	−0.858363	+	0.513044i	$0.828518\pi$
$60$	0	0
$61$	− 50515.3i	− 1.73819i	−0.494642	−	0.869097i	$-0.664701\pi$
$61$	− 50515.3i	− 1.73819i	0.494642	−	0.869097i	$-0.335299\pi$
$62$	0	0
$63$	− 522.057i	− 0.0165717i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5964.39	−0.162323	−0.0811613	−	0.996701i	$-0.525863\pi$
$67$	−5964.39	−0.162323	−0.0811613	+	0.996701i	$0.525863\pi$
$68$	0	0
$69$	7322.11i	0.185146i
$70$	0	0
$71$	−67286.3	−1.58409	−0.792046	−	0.610461i	$-0.790984\pi$
$71$	−67286.3	−1.58409	−0.792046	+	0.610461i	$0.790984\pi$
$72$	0	0
$73$	85768.5i	1.88374i	0.335979	+	0.941869i	$0.390933\pi$
$73$	85768.5i	1.88374i	−0.335979	+	0.941869i	$0.609067\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1474.94	−0.0283496
$78$	0	0
$79$	56567.2	1.01976	0.509879	−	0.860246i	$-0.329690\pi$
$79$	56567.2	1.01976	0.509879	+	0.860246i	$0.329690\pi$
$80$	0	0
$81$	−69625.4	−1.17911
$82$	0	0
$83$	30208.1	0.481314	0.240657	−	0.970610i	$-0.422637\pi$
$83$	30208.1	0.481314	0.240657	+	0.970610i	$0.422637\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 98106.4i	− 1.38963i
$88$	0	0
$89$	−113965.	−1.52509	−0.762544	−	0.646936i	$-0.776050\pi$
$89$	−113965.	−1.52509	−0.762544	+	0.646936i	$0.776050\pi$
$90$	0	0
$91$	− 3387.24i	− 0.0428787i
$92$	0	0
$93$	−162536.	−1.94869
$94$	0	0
$95$	0	0
$96$	0	0
$97$	138806.i	1.49788i	0.662635	+	0.748942i	$0.269438\pi$
$97$	138806.i	1.49788i	−0.662635	+	0.748942i	$0.730562\pi$
$98$	0	0
$99$	− 9115.59i	− 0.0934753i
$100$	0	0
$101$	128473.i	1.25317i	0.779354	+	0.626584i	$0.215547\pi$
$101$	128473.i	1.25317i	−0.779354	+	0.626584i	$0.784453\pi$
$102$	0	0
$103$	117231.i	1.08880i	0.838825	+	0.544401i	$0.183243\pi$
$103$	117231.i	1.08880i	−0.838825	+	0.544401i	$0.816757\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17654.6	0.149073	0.0745366	−	0.997218i	$-0.476252\pi$
$107$	17654.6	0.149073	0.0745366	+	0.997218i	$0.476252\pi$
$108$	0	0
$109$	75398.4i	0.607849i	0.952696	+	0.303925i	$0.0982971\pi$
$109$	75398.4i	0.607849i	−0.952696	+	0.303925i	$0.901703\pi$
$110$	0	0
$111$	61752.0	0.475711
$112$	0	0
$113$	69211.7i	0.509898i	0.966954	+	0.254949i	$0.0820587\pi$
$113$	69211.7i	0.509898i	−0.966954	+	0.254949i	$0.917941\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	20934.2	0.141381
$118$	0	0
$119$	11590.4	0.0750294
$120$	0	0
$121$	135297.	0.840089
$122$	0	0
$123$	−103022.	−0.613999
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 118525.i	− 0.652078i	−0.945356	−	0.326039i	$-0.894286\pi$
$127$	− 118525.i	− 0.652078i	0.945356	−	0.326039i	$-0.105714\pi$
$128$	0	0
$129$	−184551.	−0.976430
$130$	0	0
$131$	− 83761.8i	− 0.426450i	−0.977003	−	0.213225i	$-0.931603\pi$
$131$	− 83761.8i	− 0.426450i	0.977003	−	0.213225i	$-0.0683967\pi$
$132$	0	0
$133$	22855.2	0.112036
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 174055.i	− 0.792291i	−0.918188	−	0.396145i	$-0.870348\pi$
$137$	− 174055.i	− 0.792291i	0.918188	−	0.396145i	$-0.129652\pi$
$138$	0	0
$139$	310792.i	1.36437i	0.731179	+	0.682186i	$0.238971\pi$
$139$	310792.i	1.36437i	−0.731179	+	0.682186i	$0.761029\pi$
$140$	0	0
$141$	160057.i	0.677995i
$142$	0	0
$143$	− 59144.2i	− 0.241865i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	289547.	1.10516
$148$	0	0
$149$	293656.i	1.08361i	0.840504	+	0.541805i	$0.182259\pi$
$149$	293656.i	1.08361i	−0.840504	+	0.541805i	$0.817741\pi$
$150$	0	0
$151$	−26058.5	−0.0930052	−0.0465026	−	0.998918i	$-0.514808\pi$
$151$	−26058.5	−0.0930052	−0.0465026	+	0.998918i	$0.514808\pi$
$152$	0	0
$153$	71632.5i	0.247390i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−427448.	−1.38399	−0.691997	−	0.721900i	$-0.743269\pi$
$157$	−427448.	−1.38399	−0.691997	+	0.721900i	$0.743269\pi$
$158$	0	0
$159$	155432.	0.487581
$160$	0	0
$161$	3886.62	0.0118170
$162$	0	0
$163$	−547292.	−1.61343	−0.806715	−	0.590941i	$-0.798757\pi$
$163$	−547292.	−1.61343	−0.806715	+	0.590941i	$0.798757\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 203146.i	− 0.563659i	−0.959465	−	0.281829i	$-0.909059\pi$
$167$	− 203146.i	− 0.563659i	0.959465	−	0.281829i	$-0.0909412\pi$
$168$	0	0
$169$	−235467.	−0.634180
$170$	0	0
$171$	141253.i	0.369408i
$172$	0	0
$173$	−94689.2	−0.240539	−0.120269	−	0.992741i	$-0.538376\pi$
$173$	−94689.2	−0.240539	−0.120269	+	0.992741i	$0.538376\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	475042.i	1.13972i
$178$	0	0
$179$	290099.i	0.676727i	0.941016	+	0.338363i	$0.109873\pi$
$179$	290099.i	0.676727i	−0.941016	+	0.338363i	$0.890127\pi$
$180$	0	0
$181$	63999.0i	0.145203i	0.997361	+	0.0726017i	$0.0231302\pi$
$181$	63999.0i	0.145203i	−0.997361	+	0.0726017i	$0.976870\pi$
$182$	0	0
$183$	− 874662.i	− 1.93069i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	202379.	0.423216
$188$	0	0
$189$	29630.9i	0.0603380i
$190$	0	0
$191$	391768.	0.777044	0.388522	−	0.921439i	$-0.372986\pi$
$191$	391768.	0.777044	0.388522	+	0.921439i	$0.372986\pi$
$192$	0	0
$193$	− 641122.i	− 1.23893i	−0.785024	−	0.619466i	$-0.787349\pi$
$193$	− 641122.i	− 1.23893i	0.785024	−	0.619466i	$-0.212651\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	312130.	0.573019	0.286510	−	0.958077i	$-0.407505\pi$
$197$	312130.	0.573019	0.286510	+	0.958077i	$0.407505\pi$
$198$	0	0
$199$	171473.	0.306946	0.153473	−	0.988153i	$-0.450954\pi$
$199$	171473.	0.306946	0.153473	+	0.988153i	$0.450954\pi$
$200$	0	0
$201$	−103272.	−0.180299
$202$	0	0
$203$	−52075.5	−0.0886938
$204$	0	0
$205$	0	0
$206$	0	0
$207$	24020.6i	0.0389635i
$208$	0	0
$209$	399073.	0.631955
$210$	0	0
$211$	679022.i	1.04997i	0.851111	+	0.524986i	$0.175930\pi$
$211$	679022.i	1.04997i	−0.851111	+	0.524986i	$0.824070\pi$
$212$	0	0
$213$	−1.16505e6	−1.75952
$214$	0	0
$215$	0	0
$216$	0	0
$217$	86275.3i	0.124376i
$218$	0	0
$219$	1.48506e6i	2.09235i
$220$	0	0
$221$	464769.i	0.640113i
$222$	0	0
$223$	− 976043.i	− 1.31434i	−0.753743	−	0.657169i	$-0.771754\pi$
$223$	− 976043.i	− 1.31434i	0.753743	−	0.657169i	$-0.228246\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−243426.	−0.313547	−0.156773	−	0.987635i	$-0.550109\pi$
$227$	−243426.	−0.313547	−0.156773	+	0.987635i	$0.550109\pi$
$228$	0	0
$229$	1.33345e6i	1.68030i	0.542354	+	0.840150i	$0.317533\pi$
$229$	1.33345e6i	1.68030i	−0.542354	+	0.840150i	$0.682467\pi$
$230$	0	0
$231$	−25538.2	−0.0314892
$232$	0	0
$233$	− 1.01947e6i	− 1.23023i	−0.788438	−	0.615114i	$-0.789110\pi$
$233$	− 1.01947e6i	− 1.23023i	0.788438	−	0.615114i	$-0.210890\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	979449.	1.13269
$238$	0	0
$239$	−975796.	−1.10501	−0.552503	−	0.833511i	$-0.686327\pi$
$239$	−975796.	−1.10501	−0.552503	+	0.833511i	$0.686327\pi$
$240$	0	0
$241$	359018.	0.398175	0.199087	−	0.979982i	$-0.436202\pi$
$241$	359018.	0.398175	0.199087	+	0.979982i	$0.436202\pi$
$242$	0	0
$243$	−422124.	−0.458589
$244$	0	0
$245$	0	0
$246$	0	0
$247$	916482.i	0.955832i
$248$	0	0
$249$	523047.	0.534616
$250$	0	0
$251$	− 208180.i	− 0.208572i	−0.994547	−	0.104286i	$-0.966744\pi$
$251$	− 208180.i	− 0.208572i	0.994547	−	0.104286i	$-0.0332557\pi$
$252$	0	0
$253$	67864.0	0.0666558
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.82101e6i	− 1.71981i	−0.510456	−	0.859904i	$-0.670523\pi$
$257$	− 1.82101e6i	− 1.71981i	0.510456	−	0.859904i	$-0.329477\pi$
$258$	0	0
$259$	− 32778.3i	− 0.0303625i
$260$	0	0
$261$	− 321844.i	− 0.292445i
$262$	0	0
$263$	1.74901e6i	1.55921i	0.626274	+	0.779603i	$0.284579\pi$
$263$	1.74901e6i	1.55921i	−0.626274	+	0.779603i	$0.715421\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.97327e6	−1.69398
$268$	0	0
$269$	1.48042e6i	1.24740i	0.781665	+	0.623699i	$0.214371\pi$
$269$	1.48042e6i	1.24740i	−0.781665	+	0.623699i	$0.785629\pi$
$270$	0	0
$271$	−1.04418e6	−0.863680	−0.431840	−	0.901950i	$-0.642136\pi$
$271$	−1.04418e6	−0.863680	−0.431840	+	0.901950i	$0.642136\pi$
$272$	0	0
$273$	− 58649.3i	− 0.0476273i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22792.5	−0.0178481	−0.00892407	−	0.999960i	$-0.502841\pi$
$277$	−22792.5	−0.0178481	−0.00892407	+	0.999960i	$0.502841\pi$
$278$	0	0
$279$	−533209.	−0.410098
$280$	0	0
$281$	181092.	0.136815	0.0684074	−	0.997657i	$-0.478208\pi$
$281$	181092.	0.136815	0.0684074	+	0.997657i	$0.478208\pi$
$282$	0	0
$283$	−1.00940e6	−0.749200	−0.374600	−	0.927187i	$-0.622220\pi$
$283$	−1.00940e6	−0.749200	−0.374600	+	0.927187i	$0.622220\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	54684.8i	0.0391888i
$288$	0	0
$289$	−170486.	−0.120073
$290$	0	0
$291$	2.40339e6i	1.66377i
$292$	0	0
$293$	−2.25558e6	−1.53493	−0.767465	−	0.641090i	$-0.778482\pi$
$293$	−2.25558e6	−1.53493	−0.767465	+	0.641090i	$0.778482\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	517383.i	0.340346i
$298$	0	0
$299$	155852.i	0.100817i
$300$	0	0
$301$	97960.6i	0.0623211i
$302$	0	0
$303$	2.22449e6i	1.39195i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.42181e6	1.46654	0.733270	−	0.679938i	$-0.237993\pi$
$307$	2.42181e6	1.46654	0.733270	+	0.679938i	$0.237993\pi$
$308$	0	0
$309$	2.02983e6i	1.20938i
$310$	0	0
$311$	1.04918e6	0.615107	0.307554	−	0.951531i	$-0.400490\pi$
$311$	1.04918e6	0.615107	0.307554	+	0.951531i	$0.400490\pi$
$312$	0	0
$313$	919355.i	0.530423i	0.964190	+	0.265212i	$0.0854418\pi$
$313$	919355.i	0.530423i	−0.964190	+	0.265212i	$0.914558\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	761677.	0.425719	0.212859	−	0.977083i	$-0.431722\pi$
$317$	761677.	0.425719	0.212859	+	0.977083i	$0.431722\pi$
$318$	0	0
$319$	−909285.	−0.500292
$320$	0	0
$321$	305686.	0.165582
$322$	0	0
$323$	−3.13601e6	−1.67252
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.30551e6i	0.675165i
$328$	0	0
$329$	84959.2	0.0432734
$330$	0	0
$331$	− 1.27292e6i	− 0.638605i	−0.947653	−	0.319302i	$-0.896551\pi$
$331$	− 1.27292e6i	− 0.638605i	0.947653	−	0.319302i	$-0.103449\pi$
$332$	0	0
$333$	202581.	0.100112
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.61635e6i	− 0.775286i	−0.921810	−	0.387643i	$-0.873289\pi$
$337$	− 1.61635e6i	− 0.775286i	0.921810	−	0.387643i	$-0.126711\pi$
$338$	0	0
$339$	1.19839e6i	0.566367i
$340$	0	0
$341$	1.50644e6i	0.701564i
$342$	0	0
$343$	− 308163.i	− 0.141431i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−349978.	−0.156033	−0.0780167	−	0.996952i	$-0.524859\pi$
$347$	−349978.	−0.156033	−0.0780167	+	0.996952i	$0.524859\pi$
$348$	0	0
$349$	− 2.38580e6i	− 1.04850i	−0.851563	−	0.524252i	$-0.824345\pi$
$349$	− 2.38580e6i	− 1.04850i	0.851563	−	0.524252i	$-0.175655\pi$
$350$	0	0
$351$	−1.18819e6	−0.514774
$352$	0	0
$353$	3.84764e6i	1.64345i	0.569882	+	0.821727i	$0.306989\pi$
$353$	3.84764e6i	1.64345i	−0.569882	+	0.821727i	$0.693011\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	200686.	0.0833385
$358$	0	0
$359$	−2.69844e6	−1.10503	−0.552517	−	0.833501i	$-0.686333\pi$
$359$	−2.69844e6	−1.10503	−0.552517	+	0.833501i	$0.686333\pi$
$360$	0	0
$361$	−3.70781e6	−1.49744
$362$	0	0
$363$	2.34264e6	0.933125
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.14096e6i	0.442186i	0.975253	+	0.221093i	$0.0709625\pi$
$367$	1.14096e6i	0.442186i	−0.975253	+	0.221093i	$0.929038\pi$
$368$	0	0
$369$	−337970.	−0.129215
$370$	0	0
$371$	− 82504.2i	− 0.0311201i
$372$	0	0
$373$	461958.	0.171922	0.0859608	−	0.996299i	$-0.472604\pi$
$373$	461958.	0.171922	0.0859608	+	0.996299i	$0.472604\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.08820e6i	− 0.756691i
$378$	0	0
$379$	4.36174e6i	1.55977i	0.625920	+	0.779887i	$0.284723\pi$
$379$	4.36174e6i	1.55977i	−0.625920	+	0.779887i	$0.715277\pi$
$380$	0	0
$381$	− 2.05223e6i	− 0.724292i
$382$	0	0
$383$	417638.i	0.145480i	0.997351	+	0.0727400i	$0.0231743\pi$
$383$	417638.i	0.145480i	−0.997351	+	0.0727400i	$0.976826\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−605428.	−0.205487
$388$	0	0
$389$	1.92396e6i	0.644647i	0.946630	+	0.322323i	$0.104464\pi$
$389$	1.92396e6i	0.644647i	−0.946630	+	0.322323i	$0.895536\pi$
$390$	0	0
$391$	−533291.	−0.176410
$392$	0	0
$393$	− 1.45032e6i	− 0.473677i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.95436e6	−0.940778	−0.470389	−	0.882459i	$-0.655886\pi$
$397$	−2.95436e6	−0.940778	−0.470389	+	0.882459i	$0.655886\pi$
$398$	0	0
$399$	395733.	0.124443
$400$	0	0
$401$	−1.57105e6	−0.487898	−0.243949	−	0.969788i	$-0.578443\pi$
$401$	−1.57105e6	−0.487898	−0.243949	+	0.969788i	$0.578443\pi$
$402$	0	0
$403$	−3.45959e6	−1.06112
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 572339.i	− 0.171265i
$408$	0	0
$409$	5.18294e6	1.53203	0.766016	−	0.642821i	$-0.222236\pi$
$409$	5.18294e6	1.53203	0.766016	+	0.642821i	$0.222236\pi$
$410$	0	0
$411$	− 3.01372e6i	− 0.880033i
$412$	0	0
$413$	252155.	0.0727432
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.38130e6i	1.51547i
$418$	0	0
$419$	− 4.81218e6i	− 1.33908i	−0.742776	−	0.669540i	$-0.766491\pi$
$419$	− 4.81218e6i	− 1.33908i	0.742776	−	0.669540i	$-0.233509\pi$
$420$	0	0
$421$	4.66983e6i	1.28409i	0.766667	+	0.642045i	$0.221914\pi$
$421$	4.66983e6i	1.28409i	−0.766667	+	0.642045i	$0.778086\pi$
$422$	0	0
$423$	525075.i	0.142683i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−464276.	−0.123227
$428$	0	0
$429$	− 1.02407e6i	− 0.268650i
$430$	0	0
$431$	3.07756e6	0.798020	0.399010	−	0.916947i	$-0.369354\pi$
$431$	3.07756e6	0.798020	0.399010	+	0.916947i	$0.369354\pi$
$432$	0	0
$433$	− 2.60694e6i	− 0.668206i	−0.942537	−	0.334103i	$-0.891567\pi$
$433$	− 2.60694e6i	− 0.668206i	0.942537	−	0.334103i	$-0.108433\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.05160e6	−0.263419
$438$	0	0
$439$	7.17040e6	1.77575	0.887876	−	0.460083i	$-0.152180\pi$
$439$	7.17040e6	1.77575	0.887876	+	0.460083i	$0.152180\pi$
$440$	0	0
$441$	949875.	0.232579
$442$	0	0
$443$	5.08067e6	1.23002	0.615009	−	0.788520i	$-0.289152\pi$
$443$	5.08067e6	1.23002	0.615009	+	0.788520i	$0.289152\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.08459e6i	1.20361i
$448$	0	0
$449$	−3.18246e6	−0.744984	−0.372492	−	0.928035i	$-0.621497\pi$
$449$	−3.18246e6	−0.744984	−0.372492	+	0.928035i	$0.621497\pi$
$450$	0	0
$451$	954846.i	0.221051i
$452$	0	0
$453$	−451198.	−0.103305
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.54454e6i	0.345946i	0.984927	+	0.172973i	$0.0553373\pi$
$457$	1.54454e6i	0.345946i	−0.984927	+	0.172973i	$0.944663\pi$
$458$	0	0
$459$	− 4.06572e6i	− 0.900753i
$460$	0	0
$461$	− 888536.i	− 0.194725i	−0.995249	−	0.0973627i	$-0.968959\pi$
$461$	− 888536.i	− 0.194725i	0.995249	−	0.0973627i	$-0.0310407\pi$
$462$	0	0
$463$	− 3.53164e6i	− 0.765640i	−0.923823	−	0.382820i	$-0.874953\pi$
$463$	− 3.53164e6i	− 0.765640i	0.923823	−	0.382820i	$-0.125047\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.13335e6	1.51357	0.756783	−	0.653666i	$-0.226770\pi$
$467$	7.13335e6	1.51357	0.756783	+	0.653666i	$0.226770\pi$
$468$	0	0
$469$	54817.5i	0.0115077i
$470$	0	0
$471$	−7.40118e6	−1.53726
$472$	0	0
$473$	1.71048e6i	0.351532i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	509903.	0.102610
$478$	0	0
$479$	5.02213e6	1.00011	0.500057	−	0.865992i	$-0.333312\pi$
$479$	5.02213e6	1.00011	0.500057	+	0.865992i	$0.333312\pi$
$480$	0	0
$481$	1.31439e6	0.259038
$482$	0	0
$483$	67296.1	0.0131257
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 9.04360e6i	− 1.72790i	−0.503576	−	0.863951i	$-0.667983\pi$
$487$	− 9.04360e6i	− 1.72790i	0.503576	−	0.863951i	$-0.332017\pi$
$488$	0	0
$489$	−9.47625e6	−1.79211
$490$	0	0
$491$	− 1.91922e6i	− 0.359270i	−0.983733	−	0.179635i	$-0.942508\pi$
$491$	− 1.91922e6i	− 0.359270i	0.983733	−	0.179635i	$-0.0574916\pi$
$492$	0	0
$493$	7.14538e6	1.32406
$494$	0	0
$495$	0	0
$496$	0	0
$497$	618415.i	0.112302i
$498$	0	0
$499$	− 1.22004e6i	− 0.219342i	−0.993968	−	0.109671i	$-0.965020\pi$
$499$	− 1.22004e6i	− 0.219342i	0.993968	−	0.109671i	$-0.0349797\pi$
$500$	0	0
$501$	− 3.51742e6i	− 0.626081i
$502$	0	0
$503$	− 7.08742e6i	− 1.24902i	−0.781018	−	0.624508i	$-0.785300\pi$
$503$	− 7.08742e6i	− 1.24902i	0.781018	−	0.624508i	$-0.214700\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4.07706e6	−0.704412
$508$	0	0
$509$	− 5.58984e6i	− 0.956324i	−0.878272	−	0.478162i	$-0.841303\pi$
$509$	− 5.58984e6i	− 0.956324i	0.878272	−	0.478162i	$-0.158697\pi$
$510$	0	0
$511$	788281.	0.133545
$512$	0	0
$513$	− 8.01722e6i	− 1.34502i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.48346e6	0.244090
$518$	0	0
$519$	−1.63952e6	−0.267177
$520$	0	0
$521$	8.47810e6	1.36837	0.684186	−	0.729308i	$-0.260158\pi$
$521$	8.47810e6	1.36837	0.684186	+	0.729308i	$0.260158\pi$
$522$	0	0
$523$	6.21299e6	0.993223	0.496611	−	0.867973i	$-0.334577\pi$
$523$	6.21299e6	0.993223	0.496611	+	0.867973i	$0.334577\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.18380e7i	− 1.85674i
$528$	0	0
$529$	6.25751e6	0.972216
$530$	0	0
$531$	1.55840e6i	0.239852i
$532$	0	0
$533$	−2.19283e6	−0.334339
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.02300e6i	0.751670i
$538$	0	0
$539$	− 2.68363e6i	− 0.397878i
$540$	0	0
$541$	− 7.13892e6i	− 1.04867i	−0.851512	−	0.524335i	$-0.824314\pi$
$541$	− 7.13892e6i	− 1.04867i	0.851512	−	0.524335i	$-0.175686\pi$
$542$	0	0
$543$	1.10813e6i	0.161284i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.31142e6	−1.04480	−0.522400	−	0.852700i	$-0.674963\pi$
$547$	−7.31142e6	−1.04480	−0.522400	+	0.852700i	$0.674963\pi$
$548$	0	0
$549$	− 2.86938e6i	− 0.406309i
$550$	0	0
$551$	1.40900e7	1.97712
$552$	0	0
$553$	− 519898.i	− 0.0722945i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.92783e6	−0.263288	−0.131644	−	0.991297i	$-0.542026\pi$
$557$	−1.92783e6	−0.263288	−0.131644	+	0.991297i	$0.542026\pi$
$558$	0	0
$559$	−3.92817e6	−0.531692
$560$	0	0
$561$	3.50415e6	0.470084
$562$	0	0
$563$	1.29516e7	1.72208	0.861039	−	0.508539i	$-0.169814\pi$
$563$	1.29516e7	1.72208	0.861039	+	0.508539i	$0.169814\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	639914.i	0.0835918i
$568$	0	0
$569$	−7.26039e6	−0.940112	−0.470056	−	0.882637i	$-0.655766\pi$
$569$	−7.26039e6	−0.940112	−0.470056	+	0.882637i	$0.655766\pi$
$570$	0	0
$571$	− 1.83619e6i	− 0.235682i	−0.993032	−	0.117841i	$-0.962403\pi$
$571$	− 1.83619e6i	− 0.235682i	0.993032	−	0.117841i	$-0.0375973\pi$
$572$	0	0
$573$	6.78339e6	0.863098
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.04937e6i	− 0.256260i	−0.991757	−	0.128130i	$-0.959102\pi$
$577$	− 2.04937e6i	− 0.256260i	0.991757	−	0.128130i	$-0.0408975\pi$
$578$	0	0
$579$	− 1.11009e7i	− 1.37614i
$580$	0	0
$581$	− 277637.i	− 0.0341222i
$582$	0	0
$583$	− 1.44060e6i	− 0.175538i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.38716e6	−0.285947	−0.142974	−	0.989726i	$-0.545666\pi$
$587$	−2.38716e6	−0.285947	−0.142974	+	0.989726i	$0.545666\pi$
$588$	0	0
$589$	− 2.33434e7i	− 2.77253i
$590$	0	0
$591$	5.40446e6	0.636478
$592$	0	0
$593$	1.01237e7i	1.18223i	0.806587	+	0.591116i	$0.201312\pi$
$593$	1.01237e7i	1.18223i	−0.806587	+	0.591116i	$0.798688\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.96901e6	0.340939
$598$	0	0
$599$	3.96697e6	0.451743	0.225872	−	0.974157i	$-0.427477\pi$
$599$	3.96697e6	0.451743	0.225872	+	0.974157i	$0.427477\pi$
$600$	0	0
$601$	5.94578e6	0.671464	0.335732	−	0.941958i	$-0.391016\pi$
$601$	5.94578e6	0.671464	0.335732	+	0.941958i	$0.391016\pi$
$602$	0	0
$603$	−338790.	−0.0379435
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 900508.i	− 0.0992010i	−0.998769	−	0.0496005i	$-0.984205\pi$
$607$	− 900508.i	− 0.0992010i	0.998769	−	0.0496005i	$-0.0157948\pi$
$608$	0	0
$609$	−901677.	−0.0985162
$610$	0	0
$611$	3.40682e6i	0.369187i
$612$	0	0
$613$	−4.07379e6	−0.437872	−0.218936	−	0.975739i	$-0.570259\pi$
$613$	−4.07379e6	−0.437872	−0.218936	+	0.975739i	$0.570259\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.83021e6i	− 0.405051i	−0.979277	−	0.202525i	$-0.935085\pi$
$617$	− 3.83021e6i	− 0.405051i	0.979277	−	0.202525i	$-0.0649148\pi$
$618$	0	0
$619$	9.65601e6i	1.01291i	0.862266	+	0.506455i	$0.169045\pi$
$619$	9.65601e6i	1.01291i	−0.862266	+	0.506455i	$0.830955\pi$
$620$	0	0
$621$	− 1.36336e6i	− 0.141867i
$622$	0	0
$623$	1.04743e6i	0.108119i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.90986e6	0.701941
$628$	0	0
$629$	4.49758e6i	0.453265i
$630$	0	0
$631$	−8.80367e6	−0.880218	−0.440109	−	0.897944i	$-0.645060\pi$
$631$	−8.80367e6	−0.880218	−0.440109	+	0.897944i	$0.645060\pi$
$632$	0	0
$633$	1.17571e7i	1.16625i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.16303e6	0.601791
$638$	0	0
$639$	−3.82201e6	−0.370288
$640$	0	0
$641$	−8.42581e6	−0.809966	−0.404983	−	0.914324i	$-0.632722\pi$
$641$	−8.42581e6	−0.809966	−0.404983	+	0.914324i	$0.632722\pi$
$642$	0	0
$643$	−559011.	−0.0533204	−0.0266602	−	0.999645i	$-0.508487\pi$
$643$	−559011.	−0.0533204	−0.0266602	+	0.999645i	$0.508487\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.60950e7i	− 1.51158i	−0.654814	−	0.755790i	$-0.727253\pi$
$647$	− 1.60950e7i	− 1.51158i	0.654814	−	0.755790i	$-0.272747\pi$
$648$	0	0
$649$	4.40286e6	0.410320
$650$	0	0
$651$	1.49384e6i	0.138150i
$652$	0	0
$653$	2.03856e7	1.87085	0.935427	−	0.353520i	$-0.115015\pi$
$653$	2.03856e7	1.87085	0.935427	+	0.353520i	$0.115015\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.87184e6i	0.440331i
$658$	0	0
$659$	− 6.70475e6i	− 0.601408i	−0.953718	−	0.300704i	$-0.902778\pi$
$659$	− 6.70475e6i	− 0.601408i	0.953718	−	0.300704i	$-0.0972216\pi$
$660$	0	0
$661$	− 5.17558e6i	− 0.460739i	−0.973103	−	0.230370i	$-0.926006\pi$
$661$	− 5.17558e6i	− 0.460739i	0.973103	−	0.230370i	$-0.0739935\pi$
$662$	0	0
$663$	8.04739e6i	0.711002i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.39607e6	0.208538
$668$	0	0
$669$	− 1.69000e7i	− 1.45989i
$670$	0	0
$671$	−8.10668e6	−0.695083
$672$	0	0
$673$	− 2.31161e6i	− 0.196733i	−0.995150	−	0.0983664i	$-0.968638\pi$
$673$	− 2.31161e6i	− 0.196733i	0.995150	−	0.0983664i	$-0.0313617\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.01425e7	−0.850496	−0.425248	−	0.905077i	$-0.639813\pi$
$677$	−1.01425e7	−0.850496	−0.425248	+	0.905077i	$0.639813\pi$
$678$	0	0
$679$	1.27574e6	0.106191
$680$	0	0
$681$	−4.21487e6	−0.348270
$682$	0	0
$683$	−5.34488e6	−0.438416	−0.219208	−	0.975678i	$-0.570347\pi$
$683$	−5.34488e6	−0.438416	−0.219208	+	0.975678i	$0.570347\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.30883e7i	1.86638i
$688$	0	0
$689$	3.30837e6	0.265501
$690$	0	0
$691$	− 5.80594e6i	− 0.462570i	−0.972886	−	0.231285i	$-0.925707\pi$
$691$	− 5.80594e6i	− 0.462570i	0.972886	−	0.231285i	$-0.0742930\pi$
$692$	0	0
$693$	−83779.6	−0.00662682
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 7.50341e6i	− 0.585028i
$698$	0	0
$699$	− 1.76520e7i	− 1.36647i
$700$	0	0
$701$	− 6.84282e6i	− 0.525945i	−0.964803	−	0.262972i	$-0.915297\pi$
$701$	− 6.84282e6i	− 0.525945i	0.964803	−	0.262972i	$-0.0847028\pi$
$702$	0	0
$703$	8.86881e6i	0.676826i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.18077e6	0.0888418
$708$	0	0
$709$	1.09625e7i	0.819019i	0.912306	+	0.409509i	$0.134300\pi$
$709$	1.09625e7i	0.819019i	−0.912306	+	0.409509i	$0.865700\pi$
$710$	0	0
$711$	3.21314e6	0.238372
$712$	0	0
$713$	− 3.96965e6i	− 0.292434i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.68957e7	−1.22738
$718$	0	0
$719$	4.16072e6	0.300156	0.150078	−	0.988674i	$-0.452048\pi$
$719$	4.16072e6	0.300156	0.150078	+	0.988674i	$0.452048\pi$
$720$	0	0
$721$	1.07744e6	0.0771893
$722$	0	0
$723$	6.21633e6	0.442271
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.27412e7i	1.59580i	0.602791	+	0.797899i	$0.294055\pi$
$727$	2.27412e7i	1.59580i	−0.602791	+	0.797899i	$0.705945\pi$
$728$	0	0
$729$	9.61000e6	0.669737
$730$	0	0
$731$	− 1.34414e7i	− 0.930358i
$732$	0	0
$733$	2.60697e7	1.79216	0.896078	−	0.443898i	$-0.146405\pi$
$733$	2.60697e7	1.79216	0.896078	+	0.443898i	$0.146405\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	957163.i	0.0649109i
$738$	0	0
$739$	− 467568.i	− 0.0314944i	−0.999876	−	0.0157472i	$-0.994987\pi$
$739$	− 467568.i	− 0.0314944i	0.999876	−	0.0157472i	$-0.00501270\pi$
$740$	0	0
$741$	1.58687e7i	1.06168i
$742$	0	0
$743$	− 1.98379e6i	− 0.131833i	−0.997825	−	0.0659165i	$-0.979003\pi$
$743$	− 1.98379e6i	− 0.131833i	0.997825	−	0.0659165i	$-0.0209971\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.71588e6	0.112509
$748$	0	0
$749$	− 162260.i	− 0.0105684i
$750$	0	0
$751$	−1.55338e7	−1.00503	−0.502515	−	0.864568i	$-0.667592\pi$
$751$	−1.55338e7	−1.00503	−0.502515	+	0.864568i	$0.667592\pi$
$752$	0	0
$753$	− 3.60460e6i	− 0.231670i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.40224e6	−0.152362	−0.0761811	−	0.997094i	$-0.524273\pi$
$757$	−2.40224e6	−0.152362	−0.0761811	+	0.997094i	$0.524273\pi$
$758$	0	0
$759$	1.17505e6	0.0740376
$760$	0	0
$761$	−2.53929e7	−1.58946	−0.794732	−	0.606960i	$-0.792389\pi$
$761$	−2.53929e7	−1.58946	−0.794732	+	0.606960i	$0.792389\pi$
$762$	0	0
$763$	692972.	0.0430928
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.01113e7i	0.620609i
$768$	0	0
$769$	1.96036e6	0.119542	0.0597709	−	0.998212i	$-0.480963\pi$
$769$	1.96036e6	0.119542	0.0597709	+	0.998212i	$0.480963\pi$
$770$	0	0
$771$	− 3.15305e7i	− 1.91027i
$772$	0	0
$773$	−3.14540e7	−1.89334	−0.946668	−	0.322211i	$-0.895574\pi$
$773$	−3.14540e7	−1.89334	−0.946668	+	0.322211i	$0.895574\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 567550.i	− 0.0337250i
$778$	0	0
$779$	− 1.47960e7i	− 0.873577i
$780$	0	0
$781$	1.07981e7i	0.633460i
$782$	0	0
$783$	1.82672e7i	1.06480i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.21797e7	−1.27650	−0.638248	−	0.769831i	$-0.720340\pi$
$787$	−2.21797e7	−1.27650	−0.638248	+	0.769831i	$0.720340\pi$
$788$	0	0
$789$	3.02838e7i	1.73188i
$790$	0	0
$791$	636111.	0.0361486
$792$	0	0
$793$	− 1.86172e7i	− 1.05131i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.05722e7	1.14719	0.573595	−	0.819139i	$-0.305548\pi$
$797$	2.05722e7	1.14719	0.573595	+	0.819139i	$0.305548\pi$
$798$	0	0
$799$	−1.16574e7	−0.646004
$800$	0	0
$801$	−6.47344e6	−0.356495
$802$	0	0
$803$	1.37641e7	0.753285
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.56332e7i	1.38554i
$808$	0	0
$809$	3.45753e7	1.85735	0.928676	−	0.370892i	$-0.120948\pi$
$809$	3.45753e7	1.85735	0.928676	+	0.370892i	$0.120948\pi$
$810$	0	0
$811$	2.43908e7i	1.30219i	0.758997	+	0.651094i	$0.225690\pi$
$811$	2.43908e7i	1.30219i	−0.758997	+	0.651094i	$0.774310\pi$
$812$	0	0
$813$	−1.80798e7	−0.959328
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2.65051e7i	− 1.38923i
$818$	0	0
$819$	− 192402.i	− 0.0100231i
$820$	0	0
$821$	− 5.77124e6i	− 0.298821i	−0.988775	−	0.149410i	$-0.952262\pi$
$821$	− 5.77124e6i	− 0.298821i	0.988775	−	0.149410i	$-0.0477376\pi$
$822$	0	0
$823$	9.22421e6i	0.474711i	0.971423	+	0.237356i	$0.0762807\pi$
$823$	9.22421e6i	0.474711i	−0.971423	+	0.237356i	$0.923719\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.36273e7	0.692859	0.346430	−	0.938076i	$-0.387394\pi$
$827$	1.36273e7	0.692859	0.346430	+	0.938076i	$0.387394\pi$
$828$	0	0
$829$	− 5.64154e6i	− 0.285109i	−0.989787	−	0.142555i	$-0.954468\pi$
$829$	− 5.64154e6i	− 0.285109i	0.989787	−	0.142555i	$-0.0455317\pi$
$830$	0	0
$831$	−394648.	−0.0198247
$832$	0	0
$833$	2.10886e7i	1.05302i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.02639e7	1.49318
$838$	0	0
$839$	1.01209e7	0.496379	0.248189	−	0.968712i	$-0.420164\pi$
$839$	1.01209e7	0.496379	0.248189	+	0.968712i	$0.420164\pi$
$840$	0	0
$841$	−1.15929e7	−0.565201
$842$	0	0
$843$	3.13557e6	0.151966
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.24349e6i	− 0.0595572i
$848$	0	0
$849$	−1.74776e7	−0.832170
$850$	0	0
$851$	1.50818e6i	0.0713886i
$852$	0	0
$853$	8.59170e6	0.404302	0.202151	−	0.979354i	$-0.435207\pi$
$853$	8.59170e6	0.404302	0.202151	+	0.979354i	$0.435207\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.17834e7i	0.548049i	0.961723	+	0.274024i	$0.0883549\pi$
$857$	1.17834e7i	0.548049i	−0.961723	+	0.274024i	$0.911645\pi$
$858$	0	0
$859$	− 2.83473e6i	− 0.131078i	−0.997850	−	0.0655389i	$-0.979123\pi$
$859$	− 2.83473e6i	− 0.131078i	0.997850	−	0.0655389i	$-0.0208767\pi$
$860$	0	0
$861$	946856.i	0.0435287i
$862$	0	0
$863$	− 2.08041e7i	− 0.950870i	−0.879751	−	0.475435i	$-0.842291\pi$
$863$	− 2.08041e7i	− 0.950870i	0.879751	−	0.475435i	$-0.157709\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.95193e6	−0.133370
$868$	0	0
$869$	− 9.07789e6i	− 0.407789i
$870$	0	0
$871$	−2.19815e6	−0.0981776
$872$	0	0
$873$	7.88447e6i	0.350136i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.82780e7	−1.24151	−0.620754	−	0.784006i	$-0.713173\pi$
$877$	−2.82780e7	−1.24151	−0.620754	+	0.784006i	$0.713173\pi$
$878$	0	0
$879$	−3.90549e7	−1.70492
$880$	0	0
$881$	−1.63898e6	−0.0711432	−0.0355716	−	0.999367i	$-0.511325\pi$
$881$	−1.63898e6	−0.0711432	−0.0355716	+	0.999367i	$0.511325\pi$
$882$	0	0
$883$	−2.72729e7	−1.17714	−0.588572	−	0.808445i	$-0.700310\pi$
$883$	−2.72729e7	−1.17714	−0.588572	+	0.808445i	$0.700310\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 3.68277e7i	− 1.57168i	−0.618427	−	0.785842i	$-0.712230\pi$
$887$	− 3.68277e7i	− 1.57168i	0.618427	−	0.785842i	$-0.287770\pi$
$888$	0	0
$889$	−1.08934e6	−0.0462283
$890$	0	0
$891$	1.11735e7i	0.471513i
$892$	0	0
$893$	−2.29873e7	−0.964628
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.69854e6i	0.111982i
$898$	0	0
$899$	5.31879e7i	2.19490i
$900$	0	0
$901$	1.13206e7i	0.464575i
$902$	0	0
$903$	1.69617e6i	0.0692228i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.18940e7	0.480077	0.240038	−	0.970763i	$-0.422840\pi$
$907$	1.18940e7	0.480077	0.240038	+	0.970763i	$0.422840\pi$
$908$	0	0
$909$	7.29756e6i	0.292933i
$910$	0	0
$911$	3.24936e7	1.29719	0.648593	−	0.761135i	$-0.275358\pi$
$911$	3.24936e7	1.29719	0.648593	+	0.761135i	$0.275358\pi$
$912$	0	0
$913$	− 4.84779e6i	− 0.192472i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−769839.	−0.0302327
$918$	0	0
$919$	6.30861e6	0.246402	0.123201	−	0.992382i	$-0.460684\pi$
$919$	6.30861e6	0.246402	0.123201	+	0.992382i	$0.460684\pi$
$920$	0	0
$921$	4.19331e7	1.62895
$922$	0	0
$923$	−2.47981e7	−0.958108
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.65896e6i	0.254511i
$928$	0	0
$929$	9.87673e6	0.375469	0.187734	−	0.982220i	$-0.439886\pi$
$929$	9.87673e6	0.375469	0.187734	+	0.982220i	$0.439886\pi$
$930$	0	0
$931$	4.15847e7i	1.57239i
$932$	0	0
$933$	1.81664e7	0.683227
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.06437e7i	− 0.396046i	−0.980197	−	0.198023i	$-0.936548\pi$
$937$	− 1.06437e7i	− 0.396046i	0.980197	−	0.198023i	$-0.0634520\pi$
$938$	0	0
$939$	1.59184e7i	0.589165i
$940$	0	0
$941$	− 4.65943e7i	− 1.71537i	−0.514174	−	0.857686i	$-0.671901\pi$
$941$	− 4.65943e7i	− 1.71537i	0.514174	−	0.857686i	$-0.328099\pi$
$942$	0	0
$943$	− 2.51613e6i	− 0.0921410i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.99997e7	−1.08703	−0.543516	−	0.839399i	$-0.682907\pi$
$947$	−2.99997e7	−1.08703	−0.543516	+	0.839399i	$0.682907\pi$
$948$	0	0
$949$	3.16097e7i	1.13934i
$950$	0	0
$951$	1.31883e7	0.472865
$952$	0	0
$953$	− 1.52026e7i	− 0.542233i	−0.962547	−	0.271116i	$-0.912607\pi$
$953$	− 1.52026e7i	− 0.542233i	0.962547	−	0.271116i	$-0.0873928\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.57441e7	−0.555697
$958$	0	0
$959$	−1.59970e6	−0.0561685
$960$	0	0
$961$	5.94891e7	2.07792
$962$	0	0
$963$	1.00282e6	0.0348464
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.59930e7i	− 1.58171i	−0.612006	−	0.790853i	$-0.709637\pi$
$967$	− 4.59930e7i	− 1.58171i	0.612006	−	0.790853i	$-0.290363\pi$
$968$	0	0
$969$	−5.42993e7	−1.85774
$970$	0	0
$971$	3.27040e7i	1.11315i	0.830799	+	0.556573i	$0.187884\pi$
$971$	3.27040e7i	1.11315i	−0.830799	+	0.556573i	$0.812116\pi$
$972$	0	0
$973$	2.85643e6	0.0967255
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.60025e7i	− 0.536353i	−0.963370	−	0.268177i	$-0.913579\pi$
$977$	− 1.60025e7i	− 0.536353i	0.963370	−	0.268177i	$-0.0864210\pi$
$978$	0	0
$979$	1.82890e7i	0.609865i
$980$	0	0
$981$	4.28279e6i	0.142087i
$982$	0	0
$983$	− 5.81035e7i	− 1.91787i	−0.283634	−	0.958933i	$-0.591540\pi$
$983$	− 5.81035e7i	− 1.91787i	0.283634	−	0.958933i	$-0.408460\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.47105e6	0.0480657
$988$	0	0
$989$	− 4.50731e6i	− 0.146530i
$990$	0	0
$991$	1.47014e7	0.475527	0.237764	−	0.971323i	$-0.423586\pi$
$991$	1.47014e7	0.475527	0.237764	+	0.971323i	$0.423586\pi$
$992$	0	0
$993$	− 2.20404e7i	− 0.709327i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.26322e7	1.03970	0.519850	−	0.854257i	$-0.325988\pi$
$997$	3.26322e7	1.03970	0.519850	+	0.854257i	$0.325988\pi$
$998$	0	0
$999$	−1.14981e7	−0.364512

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.f.b.49.15		20
4.3	odd	2		200.6.f.c.149.19		20
5.2	odd	4		160.6.d.a.81.5		20
5.3	odd	4		800.6.d.c.401.16		20
5.4	even	2		800.6.f.c.49.6		20
8.3	odd	2		200.6.f.b.149.1		20
8.5	even	2		800.6.f.c.49.5		20
20.3	even	4		200.6.d.b.101.9		20
20.7	even	4		40.6.d.a.21.12	yes	20
20.19	odd	2		200.6.f.b.149.2		20
40.3	even	4		200.6.d.b.101.10		20
40.13	odd	4		800.6.d.c.401.5		20
40.19	odd	2		200.6.f.c.149.20		20
40.27	even	4		40.6.d.a.21.11	✓	20
40.29	even	2	inner	800.6.f.b.49.16		20
40.37	odd	4		160.6.d.a.81.16		20
60.47	odd	4		360.6.k.b.181.9		20
120.107	odd	4		360.6.k.b.181.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.11	✓	20	40.27	even	4
40.6.d.a.21.12	yes	20	20.7	even	4
160.6.d.a.81.5		20	5.2	odd	4
160.6.d.a.81.16		20	40.37	odd	4
200.6.d.b.101.9		20	20.3	even	4
200.6.d.b.101.10		20	40.3	even	4
200.6.f.b.149.1		20	8.3	odd	2
200.6.f.b.149.2		20	20.19	odd	2
200.6.f.c.149.19		20	4.3	odd	2
200.6.f.c.149.20		20	40.19	odd	2
360.6.k.b.181.9		20	60.47	odd	4
360.6.k.b.181.10		20	120.107	odd	4
800.6.d.c.401.5		20	40.13	odd	4
800.6.d.c.401.16		20	5.3	odd	4
800.6.f.b.49.15		20	1.1	even	1	trivial
800.6.f.b.49.16		20	40.29	even	2	inner
800.6.f.c.49.5		20	8.5	even	2
800.6.f.c.49.6		20	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+17.3148 q^{3} -9.19080i q^{7} +56.8021 q^{9} -160.480i q^{11} +368.546 q^{13} +1261.09i q^{17} +2486.75i q^{19} -159.137i q^{21} +422.882i q^{23} -3223.98 q^{27} -5666.05i q^{29} -9387.13 q^{31} -2778.67i q^{33} +3566.43 q^{37} +6381.30 q^{39} -5949.95 q^{41} -10658.5 q^{43} +9243.94i q^{47} +16722.5 q^{49} +21835.5i q^{51} +8976.82 q^{53} +43057.5i q^{57} +27435.6i q^{59} -50515.3i q^{61} -522.057i q^{63} -5964.39 q^{67} +7322.11i q^{69} -67286.3 q^{71} +85768.5i q^{73} -1474.94 q^{77} +56567.2 q^{79} -69625.4 q^{81} +30208.1 q^{83} -98106.4i q^{87} -113965. q^{89} -3387.24i q^{91} -162536. q^{93} +138806. i q^{97} -9115.59i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 36 q^{3} + 1620 q^{9} - 11664 q^{27} - 7160 q^{31} - 3608 q^{37} - 44904 q^{39} + 11608 q^{41} + 51772 q^{43} - 18756 q^{49} + 928 q^{53} + 161604 q^{67} + 200312 q^{71} + 26008 q^{77} + 282080 q^{79}+ \cdots + 293472 q^{93}+O(q^{100}) \) 20 * q - 36 * q^3 + 1620 * q^9 - 11664 * q^27 - 7160 * q^31 - 3608 * q^37 - 44904 * q^39 + 11608 * q^41 + 51772 * q^43 - 18756 * q^49 + 928 * q^53 + 161604 * q^67 + 200312 * q^71 + 26008 * q^77 + 282080 * q^79 + 65172 * q^81 + 99092 * q^83 + 3160 * q^89 + 293472 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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