LMFDB - Embedded newform 800.6.c.a.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.c (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:	$128.307055850$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.6.c.a.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000i q^{3} -208.000i q^{7} +179.000 q^{9} +O(q^{10})$	$q+8.00000i q^{3} -208.000i q^{7} +179.000 q^{9} -536.000 q^{11} -694.000i q^{13} -1278.00i q^{17} -1112.00 q^{19} +1664.00 q^{21} -3216.00i q^{23} +3376.00i q^{27} -2918.00 q^{29} -2624.00 q^{31} -4288.00i q^{33} -9458.00i q^{37} +5552.00 q^{39} +170.000 q^{41} +19928.0i q^{43} +32.0000i q^{47} -26457.0 q^{49} +10224.0 q^{51} +22178.0i q^{53} -8896.00i q^{57} -41480.0 q^{59} +15462.0 q^{61} -37232.0i q^{63} -20744.0i q^{67} +25728.0 q^{69} +28592.0 q^{71} +53670.0i q^{73} +111488. i q^{77} +69152.0 q^{79} +16489.0 q^{81} +37800.0i q^{83} -23344.0i q^{87} +126806. q^{89} -144352. q^{91} -20992.0i q^{93} +62290.0i q^{97} -95944.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 358 q^{9}+O(q^{10}) $ 2 * q + 358 * q^9	$ 2 q + 358 q^{9} - 1072 q^{11} - 2224 q^{19} + 3328 q^{21} - 5836 q^{29} - 5248 q^{31} + 11104 q^{39} + 340 q^{41} - 52914 q^{49} + 20448 q^{51} - 82960 q^{59} + 30924 q^{61} + 51456 q^{69} + 57184 q^{71} + 138304 q^{79} + 32978 q^{81} + 253612 q^{89} - 288704 q^{91} - 191888 q^{99}+O(q^{100}) $ 2 * q + 358 * q^9 - 1072 * q^11 - 2224 * q^19 + 3328 * q^21 - 5836 * q^29 - 5248 * q^31 + 11104 * q^39 + 340 * q^41 - 52914 * q^49 + 20448 * q^51 - 82960 * q^59 + 30924 * q^61 + 51456 * q^69 + 57184 * q^71 + 138304 * q^79 + 32978 * q^81 + 253612 * q^89 - 288704 * q^91 - 191888 * q^99

Display 100 coefficients 10 coefficients

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
$3$	8.00000i	0.513200i	0.966518	+	0.256600i	$0.0826023\pi$
$3$	8.00000i	0.513200i	−0.966518	+	0.256600i	$0.917398\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 208.000i	− 1.60442i	−0.597042	−	0.802210i	$-0.703657\pi$
$7$	− 208.000i	− 1.60442i	0.597042	−	0.802210i	$-0.296343\pi$
$8$	0	0
$9$	179.000	0.736626
$10$	0	0
$11$	−536.000	−1.33562	−0.667810	−	0.744332i	$-0.732768\pi$
$11$	−536.000	−1.33562	−0.667810	+	0.744332i	$0.732768\pi$
$12$	0	0
$13$	− 694.000i	− 1.13894i	−0.822012	−	0.569470i	$-0.807148\pi$
$13$	− 694.000i	− 1.13894i	0.822012	−	0.569470i	$-0.192852\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1278.00i	− 1.07253i	−0.844050	−	0.536264i	$-0.819835\pi$
$17$	− 1278.00i	− 1.07253i	0.844050	−	0.536264i	$-0.180165\pi$
$18$	0	0
$19$	−1112.00	−0.706677	−0.353338	−	0.935496i	$-0.614954\pi$
$19$	−1112.00	−0.706677	−0.353338	+	0.935496i	$0.614954\pi$
$20$	0	0
$21$	1664.00	0.823389
$22$	0	0
$23$	− 3216.00i	− 1.26764i	−0.773480	−	0.633821i	$-0.781486\pi$
$23$	− 3216.00i	− 1.26764i	0.773480	−	0.633821i	$-0.218514\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3376.00i	0.891237i
$28$	0	0
$29$	−2918.00	−0.644303	−0.322152	−	0.946688i	$-0.604406\pi$
$29$	−2918.00	−0.644303	−0.322152	+	0.946688i	$0.604406\pi$
$30$	0	0
$31$	−2624.00	−0.490410	−0.245205	−	0.969471i	$-0.578855\pi$
$31$	−2624.00	−0.490410	−0.245205	+	0.969471i	$0.578855\pi$
$32$	0	0
$33$	− 4288.00i	− 0.685441i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9458.00i	− 1.13578i	−0.823104	−	0.567891i	$-0.807759\pi$
$37$	− 9458.00i	− 1.13578i	0.823104	−	0.567891i	$-0.192241\pi$
$38$	0	0
$39$	5552.00	0.584505
$40$	0	0
$41$	170.000	0.0157939	0.00789695	−	0.999969i	$-0.497486\pi$
$41$	170.000	0.0157939	0.00789695	+	0.999969i	$0.497486\pi$
$42$	0	0
$43$	19928.0i	1.64359i	0.569786	+	0.821793i	$0.307026\pi$
$43$	19928.0i	1.64359i	−0.569786	+	0.821793i	$0.692974\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	32.0000i	0.00211303i	0.999999	+	0.00105651i	$0.000336299\pi$
$47$	32.0000i	0.00211303i	−0.999999	+	0.00105651i	$0.999664\pi$
$48$	0	0
$49$	−26457.0	−1.57417
$50$	0	0
$51$	10224.0	0.550422
$52$	0	0
$53$	22178.0i	1.08451i	0.840215	+	0.542254i	$0.182429\pi$
$53$	22178.0i	1.08451i	−0.840215	+	0.542254i	$0.817571\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 8896.00i	− 0.362667i
$58$	0	0
$59$	−41480.0	−1.55135	−0.775673	−	0.631135i	$-0.782589\pi$
$59$	−41480.0	−1.55135	−0.775673	+	0.631135i	$0.782589\pi$
$60$	0	0
$61$	15462.0	0.532036	0.266018	−	0.963968i	$-0.414292\pi$
$61$	15462.0	0.532036	0.266018	+	0.963968i	$0.414292\pi$
$62$	0	0
$63$	− 37232.0i	− 1.18186i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 20744.0i	− 0.564554i	−0.959333	−	0.282277i	$-0.908910\pi$
$67$	− 20744.0i	− 0.564554i	0.959333	−	0.282277i	$-0.0910897\pi$
$68$	0	0
$69$	25728.0	0.650554
$70$	0	0
$71$	28592.0	0.673130	0.336565	−	0.941660i	$-0.390735\pi$
$71$	28592.0	0.673130	0.336565	+	0.941660i	$0.390735\pi$
$72$	0	0
$73$	53670.0i	1.17876i	0.807857	+	0.589379i	$0.200627\pi$
$73$	53670.0i	1.17876i	−0.807857	+	0.589379i	$0.799373\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	111488.i	2.14290i
$78$	0	0
$79$	69152.0	1.24663	0.623314	−	0.781971i	$-0.285786\pi$
$79$	69152.0	1.24663	0.623314	+	0.781971i	$0.285786\pi$
$80$	0	0
$81$	16489.0	0.279243
$82$	0	0
$83$	37800.0i	0.602277i	0.953580	+	0.301139i	$0.0973667\pi$
$83$	37800.0i	0.602277i	−0.953580	+	0.301139i	$0.902633\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 23344.0i	− 0.330657i
$88$	0	0
$89$	126806.	1.69693	0.848467	−	0.529249i	$-0.177526\pi$
$89$	126806.	1.69693	0.848467	+	0.529249i	$0.177526\pi$
$90$	0	0
$91$	−144352.	−1.82734
$92$	0	0
$93$	− 20992.0i	− 0.251679i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	62290.0i	0.672185i	0.941829	+	0.336093i	$0.109106\pi$
$97$	62290.0i	0.672185i	−0.941829	+	0.336093i	$0.890894\pi$
$98$	0	0
$99$	−95944.0	−0.983852
$100$	0	0
$101$	6414.00	0.0625641	0.0312821	−	0.999511i	$-0.490041\pi$
$101$	6414.00	0.0625641	0.0312821	+	0.999511i	$0.490041\pi$
$102$	0	0
$103$	108432.i	1.00708i	0.863972	+	0.503541i	$0.167970\pi$
$103$	108432.i	1.00708i	−0.863972	+	0.503541i	$0.832030\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	103976.i	0.877958i	0.898497	+	0.438979i	$0.144660\pi$
$107$	103976.i	0.877958i	−0.898497	+	0.438979i	$0.855340\pi$
$108$	0	0
$109$	−2486.00	−0.0200417	−0.0100209	−	0.999950i	$-0.503190\pi$
$109$	−2486.00	−0.0200417	−0.0100209	+	0.999950i	$0.503190\pi$
$110$	0	0
$111$	75664.0	0.582884
$112$	0	0
$113$	− 15794.0i	− 0.116358i	−0.998306	−	0.0581790i	$-0.981471\pi$
$113$	− 15794.0i	− 0.116358i	0.998306	−	0.0581790i	$-0.0185294\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 124226.i	− 0.838973i
$118$	0	0
$119$	−265824.	−1.72079
$120$	0	0
$121$	126245.	0.783882
$122$	0	0
$123$	1360.00i	0.00810543i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1024.00i	0.00563366i	0.999996	+	0.00281683i	$0.000896626\pi$
$127$	1024.00i	0.00563366i	−0.999996	+	0.00281683i	$0.999103\pi$
$128$	0	0
$129$	−159424.	−0.843489
$130$	0	0
$131$	−22664.0	−0.115387	−0.0576937	−	0.998334i	$-0.518375\pi$
$131$	−22664.0	−0.115387	−0.0576937	+	0.998334i	$0.518375\pi$
$132$	0	0
$133$	231296.i	1.13381i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 53238.0i	− 0.242337i	−0.992632	−	0.121169i	$-0.961336\pi$
$137$	− 53238.0i	− 0.242337i	0.992632	−	0.121169i	$-0.0386642\pi$
$138$	0	0
$139$	−19816.0	−0.0869919	−0.0434960	−	0.999054i	$-0.513850\pi$
$139$	−19816.0	−0.0869919	−0.0434960	+	0.999054i	$0.513850\pi$
$140$	0	0
$141$	−256.000	−0.00108441
$142$	0	0
$143$	371984.i	1.52119i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 211656.i	− 0.807862i
$148$	0	0
$149$	−452190.	−1.66861	−0.834306	−	0.551302i	$-0.814131\pi$
$149$	−452190.	−1.66861	−0.834306	+	0.551302i	$0.814131\pi$
$150$	0	0
$151$	−263280.	−0.939670	−0.469835	−	0.882754i	$-0.655687\pi$
$151$	−263280.	−0.939670	−0.469835	+	0.882754i	$0.655687\pi$
$152$	0	0
$153$	− 228762.i	− 0.790051i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 353530.i	− 1.14466i	−0.820023	−	0.572331i	$-0.806039\pi$
$157$	− 353530.i	− 1.14466i	0.820023	−	0.572331i	$-0.193961\pi$
$158$	0	0
$159$	−177424.	−0.556570
$160$	0	0
$161$	−668928.	−2.03383
$162$	0	0
$163$	100936.i	0.297562i	0.988870	+	0.148781i	$0.0475349\pi$
$163$	100936.i	0.297562i	−0.988870	+	0.148781i	$0.952465\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 284944.i	− 0.790621i	−0.918548	−	0.395310i	$-0.870637\pi$
$167$	− 284944.i	− 0.790621i	0.918548	−	0.395310i	$-0.129363\pi$
$168$	0	0
$169$	−110343.	−0.297186
$170$	0	0
$171$	−199048.	−0.520556
$172$	0	0
$173$	− 484374.i	− 1.23045i	−0.788350	−	0.615227i	$-0.789064\pi$
$173$	− 484374.i	− 1.23045i	0.788350	−	0.615227i	$-0.210936\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 331840.i	− 0.796151i
$178$	0	0
$179$	−406680.	−0.948681	−0.474341	−	0.880341i	$-0.657313\pi$
$179$	−406680.	−0.948681	−0.474341	+	0.880341i	$0.657313\pi$
$180$	0	0
$181$	570302.	1.29392	0.646962	−	0.762523i	$-0.276039\pi$
$181$	570302.	1.29392	0.646962	+	0.762523i	$0.276039\pi$
$182$	0	0
$183$	123696.i	0.273041i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	685008.i	1.43249i
$188$	0	0
$189$	702208.	1.42992
$190$	0	0
$191$	138624.	0.274951	0.137475	−	0.990505i	$-0.456101\pi$
$191$	138624.	0.274951	0.137475	+	0.990505i	$0.456101\pi$
$192$	0	0
$193$	− 34482.0i	− 0.0666345i	−0.999445	−	0.0333173i	$-0.989393\pi$
$193$	− 34482.0i	− 0.0666345i	0.999445	−	0.0333173i	$-0.0106072\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	643598.i	1.18154i	0.806839	+	0.590771i	$0.201176\pi$
$197$	643598.i	1.18154i	−0.806839	+	0.590771i	$0.798824\pi$
$198$	0	0
$199$	−1.10738e6	−1.98227	−0.991134	−	0.132865i	$-0.957582\pi$
$199$	−1.10738e6	−1.98227	−0.991134	+	0.132865i	$0.957582\pi$
$200$	0	0
$201$	165952.	0.289729
$202$	0	0
$203$	606944.i	1.03373i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 575664.i	− 0.933777i
$208$	0	0
$209$	596032.	0.943852
$210$	0	0
$211$	229976.	0.355612	0.177806	−	0.984066i	$-0.443100\pi$
$211$	229976.	0.355612	0.177806	+	0.984066i	$0.443100\pi$
$212$	0	0
$213$	228736.i	0.345450i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	545792.i	0.786824i
$218$	0	0
$219$	−429360.	−0.604939
$220$	0	0
$221$	−886932.	−1.22155
$222$	0	0
$223$	1.08947e6i	1.46708i	0.679646	+	0.733540i	$0.262133\pi$
$223$	1.08947e6i	1.46708i	−0.679646	+	0.733540i	$0.737867\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 687048.i	− 0.884958i	−0.896779	−	0.442479i	$-0.854099\pi$
$227$	− 687048.i	− 0.884958i	0.896779	−	0.442479i	$-0.145901\pi$
$228$	0	0
$229$	699730.	0.881743	0.440871	−	0.897570i	$-0.354670\pi$
$229$	699730.	0.881743	0.440871	+	0.897570i	$0.354670\pi$
$230$	0	0
$231$	−891904.	−1.09974
$232$	0	0
$233$	− 937722.i	− 1.13158i	−0.824550	−	0.565789i	$-0.808572\pi$
$233$	− 937722.i	− 1.13158i	0.824550	−	0.565789i	$-0.191428\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	553216.i	0.639770i
$238$	0	0
$239$	−643488.	−0.728695	−0.364347	−	0.931263i	$-0.618708\pi$
$239$	−643488.	−0.728695	−0.364347	+	0.931263i	$0.618708\pi$
$240$	0	0
$241$	157282.	0.174436	0.0872181	−	0.996189i	$-0.472202\pi$
$241$	157282.	0.174436	0.0872181	+	0.996189i	$0.472202\pi$
$242$	0	0
$243$	952280.i	1.03454i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	771728.i	0.804863i
$248$	0	0
$249$	−302400.	−0.309089
$250$	0	0
$251$	−1.58604e6	−1.58902	−0.794511	−	0.607250i	$-0.792273\pi$
$251$	−1.58604e6	−1.58902	−0.794511	+	0.607250i	$0.792273\pi$
$252$	0	0
$253$	1.72378e6i	1.69309i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 654334.i	− 0.617969i	−0.951067	−	0.308984i	$-0.900011\pi$
$257$	− 654334.i	− 0.617969i	0.951067	−	0.308984i	$-0.0999891\pi$
$258$	0	0
$259$	−1.96726e6	−1.82227
$260$	0	0
$261$	−522322.	−0.474610
$262$	0	0
$263$	330192.i	0.294359i	0.989110	+	0.147179i	$0.0470195\pi$
$263$	330192.i	0.294359i	−0.989110	+	0.147179i	$0.952981\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.01445e6i	0.870867i
$268$	0	0
$269$	−1.56956e6	−1.32250	−0.661252	−	0.750164i	$-0.729974\pi$
$269$	−1.56956e6	−1.32250	−0.661252	+	0.750164i	$0.729974\pi$
$270$	0	0
$271$	957792.	0.792224	0.396112	−	0.918202i	$-0.370359\pi$
$271$	957792.	0.792224	0.396112	+	0.918202i	$0.370359\pi$
$272$	0	0
$273$	− 1.15482e6i	− 0.937791i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	565438.i	0.442778i	0.975186	+	0.221389i	$0.0710590\pi$
$277$	565438.i	0.442778i	−0.975186	+	0.221389i	$0.928941\pi$
$278$	0	0
$279$	−469696.	−0.361249
$280$	0	0
$281$	−1.34127e6	−1.01333	−0.506664	−	0.862143i	$-0.669122\pi$
$281$	−1.34127e6	−1.01333	−0.506664	+	0.862143i	$0.669122\pi$
$282$	0	0
$283$	734264.i	0.544987i	0.962158	+	0.272494i	$0.0878483\pi$
$283$	734264.i	0.544987i	−0.962158	+	0.272494i	$0.912152\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 35360.0i	− 0.0253401i
$288$	0	0
$289$	−213427.	−0.150316
$290$	0	0
$291$	−498320.	−0.344966
$292$	0	0
$293$	1.13320e6i	0.771149i	0.922677	+	0.385574i	$0.125997\pi$
$293$	1.13320e6i	0.771149i	−0.922677	+	0.385574i	$0.874003\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.80954e6i	− 1.19035i
$298$	0	0
$299$	−2.23190e6	−1.44377
$300$	0	0
$301$	4.14502e6	2.63700
$302$	0	0
$303$	51312.0i	0.0321079i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.91377e6i	− 1.76445i	−0.470829	−	0.882224i	$-0.656045\pi$
$307$	− 2.91377e6i	− 1.76445i	0.470829	−	0.882224i	$-0.343955\pi$
$308$	0	0
$309$	−867456.	−0.516834
$310$	0	0
$311$	−1.43813e6	−0.843134	−0.421567	−	0.906797i	$-0.638520\pi$
$311$	−1.43813e6	−0.843134	−0.421567	+	0.906797i	$0.638520\pi$
$312$	0	0
$313$	1.37601e6i	0.793888i	0.917843	+	0.396944i	$0.129929\pi$
$313$	1.37601e6i	0.793888i	−0.917843	+	0.396944i	$0.870071\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.23494e6i	− 0.690235i	−0.938559	−	0.345118i	$-0.887839\pi$
$317$	− 1.23494e6i	− 0.690235i	0.938559	−	0.345118i	$-0.112161\pi$
$318$	0	0
$319$	1.56405e6	0.860545
$320$	0	0
$321$	−831808.	−0.450568
$322$	0	0
$323$	1.42114e6i	0.757930i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 19888.0i	− 0.0102854i
$328$	0	0
$329$	6656.00	0.00339019
$330$	0	0
$331$	−1.48930e6	−0.747160	−0.373580	−	0.927598i	$-0.621870\pi$
$331$	−1.48930e6	−0.747160	−0.373580	+	0.927598i	$0.621870\pi$
$332$	0	0
$333$	− 1.69298e6i	− 0.836646i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	838226.i	0.402056i	0.979586	+	0.201028i	$0.0644282\pi$
$337$	838226.i	0.402056i	−0.979586	+	0.201028i	$0.935572\pi$
$338$	0	0
$339$	126352.	0.0597149
$340$	0	0
$341$	1.40646e6	0.655002
$342$	0	0
$343$	2.00720e6i	0.921203i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 350008.i	− 0.156047i	−0.996952	−	0.0780233i	$-0.975139\pi$
$347$	− 350008.i	− 0.156047i	0.996952	−	0.0780233i	$-0.0248609\pi$
$348$	0	0
$349$	383642.	0.168602	0.0843010	−	0.996440i	$-0.473134\pi$
$349$	383642.	0.168602	0.0843010	+	0.996440i	$0.473134\pi$
$350$	0	0
$351$	2.34294e6	1.01507
$352$	0	0
$353$	− 4.09309e6i	− 1.74829i	−0.485661	−	0.874147i	$-0.661421\pi$
$353$	− 4.09309e6i	− 1.74829i	0.485661	−	0.874147i	$-0.338579\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.12659e6i	− 0.883108i
$358$	0	0
$359$	−3.14430e6	−1.28762	−0.643811	−	0.765185i	$-0.722648\pi$
$359$	−3.14430e6	−1.28762	−0.643811	+	0.765185i	$0.722648\pi$
$360$	0	0
$361$	−1.23955e6	−0.500608
$362$	0	0
$363$	1.00996e6i	0.402288i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.47619e6i	− 0.572108i	−0.958214	−	0.286054i	$-0.907656\pi$
$367$	− 1.47619e6i	− 0.572108i	0.958214	−	0.286054i	$-0.0923436\pi$
$368$	0	0
$369$	30430.0	0.0116342
$370$	0	0
$371$	4.61302e6	1.74001
$372$	0	0
$373$	3.73981e6i	1.39180i	0.718138	+	0.695901i	$0.244995\pi$
$373$	3.73981e6i	1.39180i	−0.718138	+	0.695901i	$0.755005\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.02509e6i	0.733823i
$378$	0	0
$379$	−1.89966e6	−0.679324	−0.339662	−	0.940548i	$-0.610313\pi$
$379$	−1.89966e6	−0.679324	−0.339662	+	0.940548i	$0.610313\pi$
$380$	0	0
$381$	−8192.00	−0.00289120
$382$	0	0
$383$	− 1.74310e6i	− 0.607192i	−0.952801	−	0.303596i	$-0.901813\pi$
$383$	− 1.74310e6i	− 0.607192i	0.952801	−	0.303596i	$-0.0981874\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.56711e6i	1.21071i
$388$	0	0
$389$	2.69147e6	0.901812	0.450906	−	0.892571i	$-0.351101\pi$
$389$	2.69147e6	0.901812	0.450906	+	0.892571i	$0.351101\pi$
$390$	0	0
$391$	−4.11005e6	−1.35958
$392$	0	0
$393$	− 181312.i	− 0.0592168i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.37353e6i	1.71113i	0.517695	+	0.855565i	$0.326790\pi$
$397$	5.37353e6i	1.71113i	−0.517695	+	0.855565i	$0.673210\pi$
$398$	0	0
$399$	−1.85037e6	−0.581870
$400$	0	0
$401$	156418.	0.0485765	0.0242882	−	0.999705i	$-0.492268\pi$
$401$	156418.	0.0485765	0.0242882	+	0.999705i	$0.492268\pi$
$402$	0	0
$403$	1.82106e6i	0.558548i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.06949e6i	1.51697i
$408$	0	0
$409$	306086.	0.0904764	0.0452382	−	0.998976i	$-0.485595\pi$
$409$	306086.	0.0904764	0.0452382	+	0.998976i	$0.485595\pi$
$410$	0	0
$411$	425904.	0.124368
$412$	0	0
$413$	8.62784e6i	2.48901i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 158528.i	− 0.0446443i
$418$	0	0
$419$	6.70868e6	1.86682	0.933409	−	0.358814i	$-0.116819\pi$
$419$	6.70868e6	1.86682	0.933409	+	0.358814i	$0.116819\pi$
$420$	0	0
$421$	4.02347e6	1.10636	0.553179	−	0.833063i	$-0.313415\pi$
$421$	4.02347e6	1.10636	0.553179	+	0.833063i	$0.313415\pi$
$422$	0	0
$423$	5728.00i	0.00155651i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3.21610e6i	− 0.853610i
$428$	0	0
$429$	−2.97587e6	−0.780676
$430$	0	0
$431$	−7.04304e6	−1.82628	−0.913139	−	0.407648i	$-0.866349\pi$
$431$	−7.04304e6	−1.82628	−0.913139	+	0.407648i	$0.866349\pi$
$432$	0	0
$433$	1.25142e6i	0.320763i	0.987055	+	0.160381i	$0.0512724\pi$
$433$	1.25142e6i	0.320763i	−0.987055	+	0.160381i	$0.948728\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.57619e6i	0.895813i
$438$	0	0
$439$	1.25406e6	0.310569	0.155285	−	0.987870i	$-0.450371\pi$
$439$	1.25406e6	0.310569	0.155285	+	0.987870i	$0.450371\pi$
$440$	0	0
$441$	−4.73580e6	−1.15957
$442$	0	0
$443$	− 5.18081e6i	− 1.25426i	−0.778914	−	0.627131i	$-0.784229\pi$
$443$	− 5.18081e6i	− 1.25426i	0.778914	−	0.627131i	$-0.215771\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3.61752e6i	− 0.856332i
$448$	0	0
$449$	5.73064e6	1.34149	0.670745	−	0.741688i	$-0.265975\pi$
$449$	5.73064e6	1.34149	0.670745	+	0.741688i	$0.265975\pi$
$450$	0	0
$451$	−91120.0	−0.0210947
$452$	0	0
$453$	− 2.10624e6i	− 0.482239i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5.24153e6i	− 1.17400i	−0.809588	−	0.586999i	$-0.800309\pi$
$457$	− 5.24153e6i	− 1.17400i	0.809588	−	0.586999i	$-0.199691\pi$
$458$	0	0
$459$	4.31453e6	0.955876
$460$	0	0
$461$	−173994.	−0.0381313	−0.0190657	−	0.999818i	$-0.506069\pi$
$461$	−173994.	−0.0381313	−0.0190657	+	0.999818i	$0.506069\pi$
$462$	0	0
$463$	− 4.01277e6i	− 0.869945i	−0.900444	−	0.434972i	$-0.856758\pi$
$463$	− 4.01277e6i	− 0.869945i	0.900444	−	0.434972i	$-0.143242\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	774616.i	0.164359i	0.996618	+	0.0821796i	$0.0261881\pi$
$467$	774616.i	0.164359i	−0.996618	+	0.0821796i	$0.973812\pi$
$468$	0	0
$469$	−4.31475e6	−0.905782
$470$	0	0
$471$	2.82824e6	0.587441
$472$	0	0
$473$	− 1.06814e7i	− 2.19521i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.96986e6i	0.798876i
$478$	0	0
$479$	−2.33530e6	−0.465054	−0.232527	−	0.972590i	$-0.574699\pi$
$479$	−2.33530e6	−0.465054	−0.232527	+	0.972590i	$0.574699\pi$
$480$	0	0
$481$	−6.56385e6	−1.29359
$482$	0	0
$483$	− 5.35142e6i	− 1.04376i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.03947e6i	0.198605i	0.995057	+	0.0993025i	$0.0316612\pi$
$487$	1.03947e6i	0.198605i	−0.995057	+	0.0993025i	$0.968339\pi$
$488$	0	0
$489$	−807488.	−0.152709
$490$	0	0
$491$	7.85092e6	1.46966	0.734830	−	0.678251i	$-0.237262\pi$
$491$	7.85092e6	1.46966	0.734830	+	0.678251i	$0.237262\pi$
$492$	0	0
$493$	3.72920e6i	0.691033i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.94714e6i	− 1.07998i
$498$	0	0
$499$	−2.71644e6	−0.488370	−0.244185	−	0.969729i	$-0.578520\pi$
$499$	−2.71644e6	−0.488370	−0.244185	+	0.969729i	$0.578520\pi$
$500$	0	0
$501$	2.27955e6	0.405747
$502$	0	0
$503$	4.62034e6i	0.814242i	0.913374	+	0.407121i	$0.133467\pi$
$503$	4.62034e6i	0.814242i	−0.913374	+	0.407121i	$0.866533\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 882744.i	− 0.152516i
$508$	0	0
$509$	4.46198e6	0.763366	0.381683	−	0.924293i	$-0.375345\pi$
$509$	4.46198e6	0.763366	0.381683	+	0.924293i	$0.375345\pi$
$510$	0	0
$511$	1.11634e7	1.89122
$512$	0	0
$513$	− 3.75411e6i	− 0.629816i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 17152.0i	− 0.00282220i
$518$	0	0
$519$	3.87499e6	0.631470
$520$	0	0
$521$	3.74375e6	0.604245	0.302122	−	0.953269i	$-0.402305\pi$
$521$	3.74375e6	0.604245	0.302122	+	0.953269i	$0.402305\pi$
$522$	0	0
$523$	− 9.28433e6i	− 1.48421i	−0.670282	−	0.742107i	$-0.733827\pi$
$523$	− 9.28433e6i	− 1.48421i	0.670282	−	0.742107i	$-0.266173\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.35347e6i	0.525979i
$528$	0	0
$529$	−3.90631e6	−0.606915
$530$	0	0
$531$	−7.42492e6	−1.14276
$532$	0	0
$533$	− 117980.i	− 0.0179883i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 3.25344e6i	− 0.486863i
$538$	0	0
$539$	1.41810e7	2.10249
$540$	0	0
$541$	862150.	0.126645	0.0633227	−	0.997993i	$-0.479830\pi$
$541$	862150.	0.126645	0.0633227	+	0.997993i	$0.479830\pi$
$542$	0	0
$543$	4.56242e6i	0.664042i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.75442e6i	0.250707i	0.992112	+	0.125353i	$0.0400065\pi$
$547$	1.75442e6i	0.250707i	−0.992112	+	0.125353i	$0.959994\pi$
$548$	0	0
$549$	2.76770e6	0.391911
$550$	0	0
$551$	3.24482e6	0.455314
$552$	0	0
$553$	− 1.43836e7i	− 2.00012i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.00292e7i	− 1.36971i	−0.728678	−	0.684856i	$-0.759865\pi$
$557$	− 1.00292e7i	− 1.36971i	0.728678	−	0.684856i	$-0.240135\pi$
$558$	0	0
$559$	1.38300e7	1.87195
$560$	0	0
$561$	−5.48006e6	−0.735154
$562$	0	0
$563$	5.27460e6i	0.701324i	0.936502	+	0.350662i	$0.114043\pi$
$563$	5.27460e6i	0.701324i	−0.936502	+	0.350662i	$0.885957\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 3.42971e6i	− 0.448023i
$568$	0	0
$569$	−8.36940e6	−1.08371	−0.541856	−	0.840471i	$-0.682278\pi$
$569$	−8.36940e6	−1.08371	−0.541856	+	0.840471i	$0.682278\pi$
$570$	0	0
$571$	4.02702e6	0.516884	0.258442	−	0.966027i	$-0.416791\pi$
$571$	4.02702e6	0.516884	0.258442	+	0.966027i	$0.416791\pi$
$572$	0	0
$573$	1.10899e6i	0.141105i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.37568e6i	0.297063i	0.988908	+	0.148532i	$0.0474547\pi$
$577$	2.37568e6i	0.297063i	−0.988908	+	0.148532i	$0.952545\pi$
$578$	0	0
$579$	275856.	0.0341968
$580$	0	0
$581$	7.86240e6	0.966306
$582$	0	0
$583$	− 1.18874e7i	− 1.44849i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.44028e6i	− 0.412096i	−0.978542	−	0.206048i	$-0.933940\pi$
$587$	− 3.44028e6i	− 0.412096i	0.978542	−	0.206048i	$-0.0660603\pi$
$588$	0	0
$589$	2.91789e6	0.346562
$590$	0	0
$591$	−5.14878e6	−0.606368
$592$	0	0
$593$	3.22942e6i	0.377127i	0.982061	+	0.188564i	$0.0603832\pi$
$593$	3.22942e6i	0.377127i	−0.982061	+	0.188564i	$0.939617\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8.85901e6i	− 1.01730i
$598$	0	0
$599$	9.29714e6	1.05872	0.529361	−	0.848397i	$-0.322432\pi$
$599$	9.29714e6	1.05872	0.529361	+	0.848397i	$0.322432\pi$
$600$	0	0
$601$	−1.12782e7	−1.27366	−0.636828	−	0.771006i	$-0.719754\pi$
$601$	−1.12782e7	−1.27366	−0.636828	+	0.771006i	$0.719754\pi$
$602$	0	0
$603$	− 3.71318e6i	− 0.415865i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.00115e6i	0.771255i	0.922655	+	0.385627i	$0.126015\pi$
$607$	7.00115e6i	0.771255i	−0.922655	+	0.385627i	$0.873985\pi$
$608$	0	0
$609$	−4.85555e6	−0.530512
$610$	0	0
$611$	22208.0	0.00240661
$612$	0	0
$613$	6.19432e6i	0.665798i	0.942962	+	0.332899i	$0.108027\pi$
$613$	6.19432e6i	0.665798i	−0.942962	+	0.332899i	$0.891973\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.62407e6i	− 0.277500i	−0.990327	−	0.138750i	$-0.955692\pi$
$617$	− 2.62407e6i	− 0.277500i	0.990327	−	0.138750i	$-0.0443084\pi$
$618$	0	0
$619$	2.83721e6	0.297622	0.148811	−	0.988866i	$-0.452455\pi$
$619$	2.83721e6	0.297622	0.148811	+	0.988866i	$0.452455\pi$
$620$	0	0
$621$	1.08572e7	1.12977
$622$	0	0
$623$	− 2.63756e7i	− 2.72259i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.76826e6i	0.484385i
$628$	0	0
$629$	−1.20873e7	−1.21816
$630$	0	0
$631$	1.29656e7	1.29634	0.648170	−	0.761496i	$-0.275535\pi$
$631$	1.29656e7	1.29634	0.648170	+	0.761496i	$0.275535\pi$
$632$	0	0
$633$	1.83981e6i	0.182500i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.83612e7i	1.79288i
$638$	0	0
$639$	5.11797e6	0.495844
$640$	0	0
$641$	−1.16798e7	−1.12276	−0.561382	−	0.827557i	$-0.689730\pi$
$641$	−1.16798e7	−1.12276	−0.561382	+	0.827557i	$0.689730\pi$
$642$	0	0
$643$	− 7.02732e6i	− 0.670289i	−0.942167	−	0.335145i	$-0.891215\pi$
$643$	− 7.02732e6i	− 0.670289i	0.942167	−	0.335145i	$-0.108785\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.72821e6i	− 0.913634i	−0.889561	−	0.456817i	$-0.848989\pi$
$647$	− 9.72821e6i	− 0.913634i	0.889561	−	0.456817i	$-0.151011\pi$
$648$	0	0
$649$	2.22333e7	2.07201
$650$	0	0
$651$	−4.36634e6	−0.403798
$652$	0	0
$653$	9.81425e6i	0.900688i	0.892855	+	0.450344i	$0.148699\pi$
$653$	9.81425e6i	0.900688i	−0.892855	+	0.450344i	$0.851301\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.60693e6i	0.868303i
$658$	0	0
$659$	−1.46652e7	−1.31545	−0.657724	−	0.753259i	$-0.728481\pi$
$659$	−1.46652e7	−1.31545	−0.657724	+	0.753259i	$0.728481\pi$
$660$	0	0
$661$	−1.41836e7	−1.26265	−0.631327	−	0.775517i	$-0.717489\pi$
$661$	−1.41836e7	−1.26265	−0.631327	+	0.775517i	$0.717489\pi$
$662$	0	0
$663$	− 7.09546e6i	− 0.626897i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.38429e6i	0.816745i
$668$	0	0
$669$	−8.71578e6	−0.752906
$670$	0	0
$671$	−8.28763e6	−0.710598
$672$	0	0
$673$	− 5.49941e6i	− 0.468035i	−0.972232	−	0.234018i	$-0.924813\pi$
$673$	− 5.49941e6i	− 0.468035i	0.972232	−	0.234018i	$-0.0751873\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.77375e7i	1.48737i	0.668528	+	0.743687i	$0.266925\pi$
$677$	1.77375e7i	1.48737i	−0.668528	+	0.743687i	$0.733075\pi$
$678$	0	0
$679$	1.29563e7	1.07847
$680$	0	0
$681$	5.49638e6	0.454160
$682$	0	0
$683$	9.39670e6i	0.770768i	0.922756	+	0.385384i	$0.125931\pi$
$683$	9.39670e6i	0.770768i	−0.922756	+	0.385384i	$0.874069\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.59784e6i	0.452510i
$688$	0	0
$689$	1.53915e7	1.23519
$690$	0	0
$691$	−1.34767e7	−1.07371	−0.536857	−	0.843673i	$-0.680389\pi$
$691$	−1.34767e7	−1.07371	−0.536857	+	0.843673i	$0.680389\pi$
$692$	0	0
$693$	1.99564e7i	1.57851i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 217260.i	− 0.0169394i
$698$	0	0
$699$	7.50178e6	0.580726
$700$	0	0
$701$	2.15594e7	1.65707	0.828536	−	0.559935i	$-0.189174\pi$
$701$	2.15594e7	1.65707	0.828536	+	0.559935i	$0.189174\pi$
$702$	0	0
$703$	1.05173e7i	0.802631i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.33411e6i	− 0.100379i
$708$	0	0
$709$	6.38165e6	0.476779	0.238390	−	0.971170i	$-0.423380\pi$
$709$	6.38165e6	0.476779	0.238390	+	0.971170i	$0.423380\pi$
$710$	0	0
$711$	1.23782e7	0.918298
$712$	0	0
$713$	8.43878e6i	0.621664i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 5.14790e6i	− 0.373966i
$718$	0	0
$719$	1.63566e7	1.17997	0.589986	−	0.807413i	$-0.299133\pi$
$719$	1.63566e7	1.17997	0.589986	+	0.807413i	$0.299133\pi$
$720$	0	0
$721$	2.25539e7	1.61578
$722$	0	0
$723$	1.25826e6i	0.0895207i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 2.13130e7i	− 1.49558i	−0.663937	−	0.747788i	$-0.731116\pi$
$727$	− 2.13130e7i	− 1.49558i	0.663937	−	0.747788i	$-0.268884\pi$
$728$	0	0
$729$	−3.61141e6	−0.251686
$730$	0	0
$731$	2.54680e7	1.76279
$732$	0	0
$733$	− 1.21571e7i	− 0.835737i	−0.908507	−	0.417869i	$-0.862777\pi$
$733$	− 1.21571e7i	− 0.835737i	0.908507	−	0.417869i	$-0.137223\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.11188e7i	0.754030i
$738$	0	0
$739$	1.92337e7	1.29555	0.647773	−	0.761834i	$-0.275701\pi$
$739$	1.92337e7	1.29555	0.647773	+	0.761834i	$0.275701\pi$
$740$	0	0
$741$	−6.17382e6	−0.413056
$742$	0	0
$743$	− 1.66565e6i	− 0.110691i	−0.998467	−	0.0553454i	$-0.982374\pi$
$743$	− 1.66565e6i	− 0.110691i	0.998467	−	0.0553454i	$-0.0176260\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.76620e6i	0.443653i
$748$	0	0
$749$	2.16270e7	1.40861
$750$	0	0
$751$	9.81290e6	0.634888	0.317444	−	0.948277i	$-0.397175\pi$
$751$	9.81290e6	0.634888	0.317444	+	0.948277i	$0.397175\pi$
$752$	0	0
$753$	− 1.26883e7i	− 0.815486i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.92753e7i	− 1.22254i	−0.791423	−	0.611269i	$-0.790659\pi$
$757$	− 1.92753e7i	− 1.22254i	0.791423	−	0.611269i	$-0.209341\pi$
$758$	0	0
$759$	−1.37902e7	−0.868893
$760$	0	0
$761$	−1.17863e7	−0.737762	−0.368881	−	0.929477i	$-0.620259\pi$
$761$	−1.17863e7	−0.737762	−0.368881	+	0.929477i	$0.620259\pi$
$762$	0	0
$763$	517088.i	0.0321553i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.87871e7i	1.76689i
$768$	0	0
$769$	−1.22941e7	−0.749690	−0.374845	−	0.927087i	$-0.622304\pi$
$769$	−1.22941e7	−0.749690	−0.374845	+	0.927087i	$0.622304\pi$
$770$	0	0
$771$	5.23467e6	0.317142
$772$	0	0
$773$	− 2.57086e6i	− 0.154750i	−0.997002	−	0.0773749i	$-0.975346\pi$
$773$	− 2.57086e6i	− 0.154750i	0.997002	−	0.0773749i	$-0.0246538\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.57381e7i	− 0.935190i
$778$	0	0
$779$	−189040.	−0.0111612
$780$	0	0
$781$	−1.53253e7	−0.899046
$782$	0	0
$783$	− 9.85117e6i	− 0.574227i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.54594e6i	0.319182i	0.987183	+	0.159591i	$0.0510176\pi$
$787$	5.54594e6i	0.319182i	−0.987183	+	0.159591i	$0.948982\pi$
$788$	0	0
$789$	−2.64154e6	−0.151065
$790$	0	0
$791$	−3.28515e6	−0.186687
$792$	0	0
$793$	− 1.07306e7i	− 0.605958i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.09610e7i	1.16887i	0.811440	+	0.584436i	$0.198684\pi$
$797$	2.09610e7i	1.16887i	−0.811440	+	0.584436i	$0.801316\pi$
$798$	0	0
$799$	40896.0	0.00226628
$800$	0	0
$801$	2.26983e7	1.25000
$802$	0	0
$803$	− 2.87671e7i	− 1.57437i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.25565e7i	− 0.678709i
$808$	0	0
$809$	−2.70297e7	−1.45201	−0.726005	−	0.687690i	$-0.758625\pi$
$809$	−2.70297e7	−1.45201	−0.726005	+	0.687690i	$0.758625\pi$
$810$	0	0
$811$	−2.13052e6	−0.113745	−0.0568727	−	0.998381i	$-0.518113\pi$
$811$	−2.13052e6	−0.113745	−0.0568727	+	0.998381i	$0.518113\pi$
$812$	0	0
$813$	7.66234e6i	0.406570i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2.21599e7i	− 1.16148i
$818$	0	0
$819$	−2.58390e7	−1.34607
$820$	0	0
$821$	1.58060e7	0.818396	0.409198	−	0.912446i	$-0.365809\pi$
$821$	1.58060e7	0.818396	0.409198	+	0.912446i	$0.365809\pi$
$822$	0	0
$823$	2.28848e7i	1.17773i	0.808230	+	0.588867i	$0.200426\pi$
$823$	2.28848e7i	1.17773i	−0.808230	+	0.588867i	$0.799574\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.55336e7i	− 1.29822i	−0.760695	−	0.649109i	$-0.775142\pi$
$827$	− 2.55336e7i	− 1.29822i	0.760695	−	0.649109i	$-0.224858\pi$
$828$	0	0
$829$	−8.31786e6	−0.420364	−0.210182	−	0.977662i	$-0.567406\pi$
$829$	−8.31786e6	−0.420364	−0.210182	+	0.977662i	$0.567406\pi$
$830$	0	0
$831$	−4.52350e6	−0.227234
$832$	0	0
$833$	3.38120e7i	1.68834i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 8.85862e6i	− 0.437072i
$838$	0	0
$839$	−3.66261e7	−1.79633	−0.898164	−	0.439660i	$-0.855099\pi$
$839$	−3.66261e7	−1.79633	−0.898164	+	0.439660i	$0.855099\pi$
$840$	0	0
$841$	−1.19964e7	−0.584873
$842$	0	0
$843$	− 1.07302e7i	− 0.520041i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.62590e7i	− 1.25768i
$848$	0	0
$849$	−5.87411e6	−0.279687
$850$	0	0
$851$	−3.04169e7	−1.43976
$852$	0	0
$853$	− 1.74802e7i	− 0.822571i	−0.911507	−	0.411286i	$-0.865080\pi$
$853$	− 1.74802e7i	− 0.822571i	0.911507	−	0.411286i	$-0.134920\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.31062e6i	0.433038i	0.976278	+	0.216519i	$0.0694704\pi$
$857$	9.31062e6i	0.433038i	−0.976278	+	0.216519i	$0.930530\pi$
$858$	0	0
$859$	3.49525e7	1.61620	0.808101	−	0.589045i	$-0.200496\pi$
$859$	3.49525e7	1.61620	0.808101	+	0.589045i	$0.200496\pi$
$860$	0	0
$861$	282880.	0.0130045
$862$	0	0
$863$	2.02349e7i	0.924858i	0.886656	+	0.462429i	$0.153022\pi$
$863$	2.02349e7i	0.924858i	−0.886656	+	0.462429i	$0.846978\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.70742e6i	− 0.0771421i
$868$	0	0
$869$	−3.70655e7	−1.66502
$870$	0	0
$871$	−1.43963e7	−0.642994
$872$	0	0
$873$	1.11499e7i	0.495149i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.98342e7i	− 0.870797i	−0.900238	−	0.435398i	$-0.856608\pi$
$877$	− 1.98342e7i	− 0.870797i	0.900238	−	0.435398i	$-0.143392\pi$
$878$	0	0
$879$	−9.06562e6	−0.395754
$880$	0	0
$881$	−3.81023e7	−1.65391	−0.826954	−	0.562270i	$-0.809928\pi$
$881$	−3.81023e7	−1.65391	−0.826954	+	0.562270i	$0.809928\pi$
$882$	0	0
$883$	2.41560e7i	1.04261i	0.853369	+	0.521307i	$0.174555\pi$
$883$	2.41560e7i	1.04261i	−0.853369	+	0.521307i	$0.825445\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.02368e7i	0.863638i	0.901960	+	0.431819i	$0.142128\pi$
$887$	2.02368e7i	0.863638i	−0.901960	+	0.431819i	$0.857872\pi$
$888$	0	0
$889$	212992.	0.00903876
$890$	0	0
$891$	−8.83810e6	−0.372962
$892$	0	0
$893$	− 35584.0i	− 0.00149323i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.78552e7i	− 0.740942i
$898$	0	0
$899$	7.65683e6	0.315973
$900$	0	0
$901$	2.83435e7	1.16316
$902$	0	0
$903$	3.31602e7i	1.35331i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.97976e7i	0.799088i	0.916714	+	0.399544i	$0.130831\pi$
$907$	1.97976e7i	0.799088i	−0.916714	+	0.399544i	$0.869169\pi$
$908$	0	0
$909$	1.14811e6	0.0460863
$910$	0	0
$911$	−2.13242e7	−0.851288	−0.425644	−	0.904891i	$-0.639952\pi$
$911$	−2.13242e7	−0.851288	−0.425644	+	0.904891i	$0.639952\pi$
$912$	0	0
$913$	− 2.02608e7i	− 0.804414i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.71411e6i	0.185130i
$918$	0	0
$919$	−3.49941e7	−1.36680	−0.683401	−	0.730043i	$-0.739500\pi$
$919$	−3.49941e7	−1.36680	−0.683401	+	0.730043i	$0.739500\pi$
$920$	0	0
$921$	2.33101e7	0.905515
$922$	0	0
$923$	− 1.98428e7i	− 0.766655i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.94093e7i	0.741842i
$928$	0	0
$929$	2.88107e7	1.09525	0.547627	−	0.836722i	$-0.315531\pi$
$929$	2.88107e7	1.09525	0.547627	+	0.836722i	$0.315531\pi$
$930$	0	0
$931$	2.94202e7	1.11243
$932$	0	0
$933$	− 1.15050e7i	− 0.432697i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.28854e7i	0.479457i	0.970840	+	0.239729i	$0.0770585\pi$
$937$	1.28854e7i	0.479457i	−0.970840	+	0.239729i	$0.922941\pi$
$938$	0	0
$939$	−1.10080e7	−0.407424
$940$	0	0
$941$	−5.27615e7	−1.94242	−0.971210	−	0.238227i	$-0.923434\pi$
$941$	−5.27615e7	−1.94242	−0.971210	+	0.238227i	$0.923434\pi$
$942$	0	0
$943$	− 546720.i	− 0.0200210i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.53961e6i	0.200726i	0.994951	+	0.100363i	$0.0320004\pi$
$947$	5.53961e6i	0.200726i	−0.994951	+	0.100363i	$0.968000\pi$
$948$	0	0
$949$	3.72470e7	1.34253
$950$	0	0
$951$	9.87950e6	0.354229
$952$	0	0
$953$	228102.i	0.00813574i	0.999992	+	0.00406787i	$0.00129485\pi$
$953$	228102.i	0.00813574i	−0.999992	+	0.00406787i	$0.998705\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.25124e7i	0.441632i
$958$	0	0
$959$	−1.10735e7	−0.388811
$960$	0	0
$961$	−2.17438e7	−0.759498
$962$	0	0
$963$	1.86117e7i	0.646726i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	7.36709e6i	0.253355i	0.991944	+	0.126678i	$0.0404313\pi$
$967$	7.36709e6i	0.253355i	−0.991944	+	0.126678i	$0.959569\pi$
$968$	0	0
$969$	−1.13691e7	−0.388970
$970$	0	0
$971$	−1.91161e7	−0.650654	−0.325327	−	0.945602i	$-0.605474\pi$
$971$	−1.91161e7	−0.650654	−0.325327	+	0.945602i	$0.605474\pi$
$972$	0	0
$973$	4.12173e6i	0.139572i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.57040e7i	1.86702i	0.358546	+	0.933512i	$0.383273\pi$
$977$	5.57040e7i	1.86702i	−0.358546	+	0.933512i	$0.616727\pi$
$978$	0	0
$979$	−6.79680e7	−2.26646
$980$	0	0
$981$	−444994.	−0.0147632
$982$	0	0
$983$	1.55469e7i	0.513167i	0.966522	+	0.256584i	$0.0825969\pi$
$983$	1.55469e7i	0.513167i	−0.966522	+	0.256584i	$0.917403\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	53248.0i	0.00173984i
$988$	0	0
$989$	6.40884e7	2.08348
$990$	0	0
$991$	2.36890e7	0.766237	0.383118	−	0.923699i	$-0.374850\pi$
$991$	2.36890e7	0.766237	0.383118	+	0.923699i	$0.374850\pi$
$992$	0	0
$993$	− 1.19144e7i	− 0.383442i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.71720e6i	0.118434i	0.998245	+	0.0592172i	$0.0188605\pi$
$997$	3.71720e6i	0.118434i	−0.998245	+	0.0592172i	$0.981140\pi$
$998$	0	0
$999$	3.19302e7	1.01225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.c.a.449.2		2
4.3	odd	2		800.6.c.b.449.1		2
5.2	odd	4		800.6.a.e.1.1		1
5.3	odd	4		32.6.a.a.1.1	✓	1
5.4	even	2	inner	800.6.c.a.449.1		2
15.8	even	4		288.6.a.d.1.1		1
20.3	even	4		32.6.a.c.1.1	yes	1
20.7	even	4		800.6.a.a.1.1		1
20.19	odd	2		800.6.c.b.449.2		2
40.3	even	4		64.6.a.c.1.1		1
40.13	odd	4		64.6.a.e.1.1		1
60.23	odd	4		288.6.a.e.1.1		1
80.3	even	4		256.6.b.b.129.2		2
80.13	odd	4		256.6.b.h.129.1		2
80.43	even	4		256.6.b.b.129.1		2
80.53	odd	4		256.6.b.h.129.2		2
120.53	even	4		576.6.a.u.1.1		1
120.83	odd	4		576.6.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.6.a.a.1.1	✓	1	5.3	odd	4
32.6.a.c.1.1	yes	1	20.3	even	4
64.6.a.c.1.1		1	40.3	even	4
64.6.a.e.1.1		1	40.13	odd	4
256.6.b.b.129.1		2	80.43	even	4
256.6.b.b.129.2		2	80.3	even	4
256.6.b.h.129.1		2	80.13	odd	4
256.6.b.h.129.2		2	80.53	odd	4
288.6.a.d.1.1		1	15.8	even	4
288.6.a.e.1.1		1	60.23	odd	4
576.6.a.u.1.1		1	120.53	even	4
576.6.a.v.1.1		1	120.83	odd	4
800.6.a.a.1.1		1	20.7	even	4
800.6.a.e.1.1		1	5.2	odd	4
800.6.c.a.449.1		2	5.4	even	2	inner
800.6.c.a.449.2		2	1.1	even	1	trivial
800.6.c.b.449.1		2	4.3	odd	2
800.6.c.b.449.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+8.00000i q^{3} -208.000i q^{7} +179.000 q^{9} +O(q^{10})\)	\(q+8.00000i q^{3} -208.000i q^{7} +179.000 q^{9} -536.000 q^{11} -694.000i q^{13} -1278.00i q^{17} -1112.00 q^{19} +1664.00 q^{21} -3216.00i q^{23} +3376.00i q^{27} -2918.00 q^{29} -2624.00 q^{31} -4288.00i q^{33} -9458.00i q^{37} +5552.00 q^{39} +170.000 q^{41} +19928.0i q^{43} +32.0000i q^{47} -26457.0 q^{49} +10224.0 q^{51} +22178.0i q^{53} -8896.00i q^{57} -41480.0 q^{59} +15462.0 q^{61} -37232.0i q^{63} -20744.0i q^{67} +25728.0 q^{69} +28592.0 q^{71} +53670.0i q^{73} +111488. i q^{77} +69152.0 q^{79} +16489.0 q^{81} +37800.0i q^{83} -23344.0i q^{87} +126806. q^{89} -144352. q^{91} -20992.0i q^{93} +62290.0i q^{97} -95944.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 358 q^{9}+O(q^{10}) \) 2 * q + 358 * q^9	\( 2 q + 358 q^{9} - 1072 q^{11} - 2224 q^{19} + 3328 q^{21} - 5836 q^{29} - 5248 q^{31} + 11104 q^{39} + 340 q^{41} - 52914 q^{49} + 20448 q^{51} - 82960 q^{59} + 30924 q^{61} + 51456 q^{69} + 57184 q^{71} + 138304 q^{79} + 32978 q^{81} + 253612 q^{89} - 288704 q^{91} - 191888 q^{99}+O(q^{100}) \) 2 * q + 358 * q^9 - 1072 * q^11 - 2224 * q^19 + 3328 * q^21 - 5836 * q^29 - 5248 * q^31 + 11104 * q^39 + 340 * q^41 - 52914 * q^49 + 20448 * q^51 - 82960 * q^59 + 30924 * q^61 + 51456 * q^69 + 57184 * q^71 + 138304 * q^79 + 32978 * q^81 + 253612 * q^89 - 288704 * q^91 - 191888 * q^99

Introduction

L-functions

Modular forms

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Database

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