LMFDB - Embedded newform 800.6.d.e.401.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(401,800)]

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, 1, 0]))

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

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

chi := DirichletCharacter("800.401");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.d (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:	$28$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.22
Character	$\chi$	$=$	800.401
Dual form			800.6.d.e.401.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.17847i q^{3} +146.905 q^{7} +191.470 q^{9} +O(q^{10})$	$q+7.17847i q^{3} +146.905 q^{7} +191.470 q^{9} -42.4351i q^{11} -605.383i q^{13} -409.138 q^{17} +2096.86i q^{19} +1054.55i q^{21} +3048.59 q^{23} +3118.83i q^{27} -4594.96i q^{29} +5118.83 q^{31} +304.619 q^{33} +11248.5i q^{37} +4345.73 q^{39} +11002.0 q^{41} -8413.16i q^{43} +3975.72 q^{47} +4774.02 q^{49} -2936.99i q^{51} -38273.5i q^{53} -15052.2 q^{57} -36611.0i q^{59} -3181.32i q^{61} +28127.8 q^{63} +41894.6i q^{67} +21884.2i q^{69} -72161.6 q^{71} -48275.5 q^{73} -6233.92i q^{77} -39428.9 q^{79} +24138.7 q^{81} +46504.5i q^{83} +32984.8 q^{87} +38217.3 q^{89} -88933.7i q^{91} +36745.3i q^{93} +95979.0 q^{97} -8125.03i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 1940 q^{9}+O(q^{10}) $ 28 * q - 1940 * q^9	$ 28 q - 1940 q^{9} - 4368 q^{31} + 23360 q^{39} - 2480 q^{41} + 38420 q^{49} + 69232 q^{71} - 35984 q^{79} + 122596 q^{81} + 178744 q^{89}+O(q^{100}) $ 28 * q - 1940 * q^9 - 4368 * q^31 + 23360 * q^39 - 2480 * q^41 + 38420 * q^49 + 69232 * q^71 - 35984 * q^79 + 122596 * q^81 + 178744 * q^89

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
$3$	7.17847i	0.460499i	0.973132	+	0.230250i	$0.0739542\pi$
$3$	7.17847i	0.460499i	−0.973132	+	0.230250i	$0.926046\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	146.905	1.13316	0.566580	−	0.824007i	$-0.308266\pi$
$7$	146.905	1.13316	0.566580	+	0.824007i	$0.308266\pi$
$8$	0	0
$9$	191.470	0.787941
$10$	0	0
$11$	− 42.4351i	− 0.105741i	−0.998601	−	0.0528705i	$-0.983163\pi$
$11$	− 42.4351i	− 0.105741i	0.998601	−	0.0528705i	$-0.0168371\pi$
$12$	0	0
$13$	− 605.383i	− 0.993510i	−0.867891	−	0.496755i	$-0.834525\pi$
$13$	− 605.383i	− 0.993510i	0.867891	−	0.496755i	$-0.165475\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−409.138	−0.343359	−0.171679	−	0.985153i	$-0.554919\pi$
$17$	−409.138	−0.343359	−0.171679	+	0.985153i	$0.554919\pi$
$18$	0	0
$19$	2096.86i	1.33256i	0.745704	+	0.666278i	$0.232114\pi$
$19$	2096.86i	1.33256i	−0.745704	+	0.666278i	$0.767886\pi$
$20$	0	0
$21$	1054.55i	0.521819i
$22$	0	0
$23$	3048.59	1.20166	0.600828	−	0.799379i	$-0.294838\pi$
$23$	3048.59	1.20166	0.600828	+	0.799379i	$0.294838\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3118.83i	0.823345i
$28$	0	0
$29$	− 4594.96i	− 1.01458i	−0.861775	−	0.507290i	$-0.830647\pi$
$29$	− 4594.96i	− 1.01458i	0.861775	−	0.507290i	$-0.169353\pi$
$30$	0	0
$31$	5118.83	0.956679	0.478339	−	0.878175i	$-0.341239\pi$
$31$	5118.83	0.956679	0.478339	+	0.878175i	$0.341239\pi$
$32$	0	0
$33$	304.619	0.0486937
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11248.5i	1.35080i	0.737454	+	0.675398i	$0.236028\pi$
$37$	11248.5i	1.35080i	−0.737454	+	0.675398i	$0.763972\pi$
$38$	0	0
$39$	4345.73	0.457510
$40$	0	0
$41$	11002.0	1.02215	0.511073	−	0.859537i	$-0.329248\pi$
$41$	11002.0	1.02215	0.511073	+	0.859537i	$0.329248\pi$
$42$	0	0
$43$	− 8413.16i	− 0.693886i	−0.937886	−	0.346943i	$-0.887220\pi$
$43$	− 8413.16i	− 0.693886i	0.937886	−	0.346943i	$-0.112780\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3975.72	0.262525	0.131263	−	0.991348i	$-0.458097\pi$
$47$	3975.72	0.262525	0.131263	+	0.991348i	$0.458097\pi$
$48$	0	0
$49$	4774.02	0.284049
$50$	0	0
$51$	− 2936.99i	− 0.158116i
$52$	0	0
$53$	− 38273.5i	− 1.87158i	−0.352558	−	0.935790i	$-0.614688\pi$
$53$	− 38273.5i	− 1.87158i	0.352558	−	0.935790i	$-0.385312\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−15052.2	−0.613641
$58$	0	0
$59$	− 36611.0i	− 1.36924i	−0.728898	−	0.684622i	$-0.759967\pi$
$59$	− 36611.0i	− 1.36924i	0.728898	−	0.684622i	$-0.240033\pi$
$60$	0	0
$61$	− 3181.32i	− 0.109467i	−0.998501	−	0.0547335i	$-0.982569\pi$
$61$	− 3181.32i	− 0.109467i	0.998501	−	0.0547335i	$-0.0174309\pi$
$62$	0	0
$63$	28127.8	0.892862
$64$	0	0
$65$	0	0
$66$	0	0
$67$	41894.6i	1.14017i	0.821584	+	0.570087i	$0.193091\pi$
$67$	41894.6i	1.14017i	−0.821584	+	0.570087i	$0.806909\pi$
$68$	0	0
$69$	21884.2i	0.553361i
$70$	0	0
$71$	−72161.6	−1.69887	−0.849435	−	0.527693i	$-0.823057\pi$
$71$	−72161.6	−1.69887	−0.849435	+	0.527693i	$0.823057\pi$
$72$	0	0
$73$	−48275.5	−1.06028	−0.530138	−	0.847911i	$-0.677860\pi$
$73$	−48275.5	−1.06028	−0.530138	+	0.847911i	$0.677860\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6233.92i	− 0.119821i
$78$	0	0
$79$	−39428.9	−0.710798	−0.355399	−	0.934715i	$-0.615655\pi$
$79$	−39428.9	−0.710798	−0.355399	+	0.934715i	$0.615655\pi$
$80$	0	0
$81$	24138.7	0.408791
$82$	0	0
$83$	46504.5i	0.740968i	0.928839	+	0.370484i	$0.120808\pi$
$83$	46504.5i	0.740968i	−0.928839	+	0.370484i	$0.879192\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	32984.8	0.467213
$88$	0	0
$89$	38217.3	0.511428	0.255714	−	0.966752i	$-0.417689\pi$
$89$	38217.3	0.511428	0.255714	+	0.966752i	$0.417689\pi$
$90$	0	0
$91$	− 88933.7i	− 1.12580i
$92$	0	0
$93$	36745.3i	0.440550i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	95979.0	1.03573	0.517865	−	0.855462i	$-0.326727\pi$
$97$	95979.0	1.03573	0.517865	+	0.855462i	$0.326727\pi$
$98$	0	0
$99$	− 8125.03i	− 0.0833177i
$100$	0	0
$101$	83900.0i	0.818386i	0.912448	+	0.409193i	$0.134190\pi$
$101$	83900.0i	0.818386i	−0.912448	+	0.409193i	$0.865810\pi$
$102$	0	0
$103$	−10693.4	−0.0993165	−0.0496583	−	0.998766i	$-0.515813\pi$
$103$	−10693.4	−0.0993165	−0.0496583	+	0.998766i	$0.515813\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 109794.i	− 0.927081i	−0.886076	−	0.463540i	$-0.846579\pi$
$107$	− 109794.i	− 0.927081i	0.886076	−	0.463540i	$-0.153421\pi$
$108$	0	0
$109$	− 135615.i	− 1.09330i	−0.837361	−	0.546651i	$-0.815903\pi$
$109$	− 135615.i	− 1.09330i	0.837361	−	0.546651i	$-0.184097\pi$
$110$	0	0
$111$	−80746.9	−0.622040
$112$	0	0
$113$	195440.	1.43985	0.719927	−	0.694050i	$-0.244175\pi$
$113$	195440.	1.43985	0.719927	+	0.694050i	$0.244175\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 115913.i	− 0.782827i
$118$	0	0
$119$	−60104.4	−0.389080
$120$	0	0
$121$	159250.	0.988819
$122$	0	0
$123$	78977.7i	0.470697i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	171245.	0.942126	0.471063	−	0.882100i	$-0.343870\pi$
$127$	171245.	0.942126	0.471063	+	0.882100i	$0.343870\pi$
$128$	0	0
$129$	60393.6	0.319534
$130$	0	0
$131$	− 111420.i	− 0.567265i	−0.958933	−	0.283632i	$-0.908460\pi$
$131$	− 111420.i	− 0.567265i	0.958933	−	0.283632i	$-0.0915395\pi$
$132$	0	0
$133$	308039.i	1.51000i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	282243.	1.28476	0.642379	−	0.766387i	$-0.277948\pi$
$137$	282243.	1.28476	0.642379	+	0.766387i	$0.277948\pi$
$138$	0	0
$139$	− 44060.1i	− 0.193423i	−0.995312	−	0.0967117i	$-0.969168\pi$
$139$	− 44060.1i	− 0.193423i	0.995312	−	0.0967117i	$-0.0308325\pi$
$140$	0	0
$141$	28539.6i	0.120893i
$142$	0	0
$143$	−25689.5	−0.105055
$144$	0	0
$145$	0	0
$146$	0	0
$147$	34270.1i	0.130804i
$148$	0	0
$149$	251552.i	0.928244i	0.885771	+	0.464122i	$0.153630\pi$
$149$	251552.i	0.928244i	−0.885771	+	0.464122i	$0.846370\pi$
$150$	0	0
$151$	−95790.0	−0.341883	−0.170942	−	0.985281i	$-0.554681\pi$
$151$	−95790.0	−0.341883	−0.170942	+	0.985281i	$0.554681\pi$
$152$	0	0
$153$	−78337.6	−0.270546
$154$	0	0
$155$	0	0
$156$	0	0
$157$	178462.i	0.577824i	0.957356	+	0.288912i	$0.0932935\pi$
$157$	178462.i	0.577824i	−0.957356	+	0.288912i	$0.906706\pi$
$158$	0	0
$159$	274745.	0.861861
$160$	0	0
$161$	447853.	1.36167
$162$	0	0
$163$	− 147330.i	− 0.434332i	−0.976135	−	0.217166i	$-0.930319\pi$
$163$	− 147330.i	− 0.434332i	0.976135	−	0.217166i	$-0.0696812\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	310989.	0.862888	0.431444	−	0.902140i	$-0.358004\pi$
$167$	310989.	0.862888	0.431444	+	0.902140i	$0.358004\pi$
$168$	0	0
$169$	4803.84	0.0129381
$170$	0	0
$171$	401485.i	1.04997i
$172$	0	0
$173$	− 120278.i	− 0.305542i	−0.988262	−	0.152771i	$-0.951180\pi$
$173$	− 120278.i	− 0.305542i	0.988262	−	0.152771i	$-0.0488196\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	262811.	0.630536
$178$	0	0
$179$	180544.i	0.421162i	0.977576	+	0.210581i	$0.0675356\pi$
$179$	180544.i	0.421162i	−0.977576	+	0.210581i	$0.932464\pi$
$180$	0	0
$181$	149699.i	0.339643i	0.985475	+	0.169821i	$0.0543191\pi$
$181$	149699.i	0.339643i	−0.985475	+	0.169821i	$0.945681\pi$
$182$	0	0
$183$	22837.0	0.0504095
$184$	0	0
$185$	0	0
$186$	0	0
$187$	17361.8i	0.0363071i
$188$	0	0
$189$	458171.i	0.932981i
$190$	0	0
$191$	−213909.	−0.424273	−0.212136	−	0.977240i	$-0.568042\pi$
$191$	−213909.	−0.424273	−0.212136	+	0.977240i	$0.568042\pi$
$192$	0	0
$193$	729587.	1.40988	0.704942	−	0.709265i	$-0.250973\pi$
$193$	729587.	1.40988	0.704942	+	0.709265i	$0.250973\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	922110.i	1.69285i	0.532512	+	0.846423i	$0.321248\pi$
$197$	922110.i	1.69285i	−0.532512	+	0.846423i	$0.678752\pi$
$198$	0	0
$199$	−985328.	−1.76379	−0.881897	−	0.471442i	$-0.843734\pi$
$199$	−985328.	−1.76379	−0.881897	+	0.471442i	$0.843734\pi$
$200$	0	0
$201$	−300739.	−0.525049
$202$	0	0
$203$	− 675021.i	− 1.14968i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	583713.	0.946833
$208$	0	0
$209$	88980.5	0.140906
$210$	0	0
$211$	− 932526.i	− 1.44197i	−0.692953	−	0.720983i	$-0.743691\pi$
$211$	− 932526.i	− 1.44197i	0.692953	−	0.720983i	$-0.256309\pi$
$212$	0	0
$213$	− 518010.i	− 0.782328i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	751980.	1.08407
$218$	0	0
$219$	− 346544.i	− 0.488257i
$220$	0	0
$221$	247686.i	0.341130i
$222$	0	0
$223$	748564.	1.00801	0.504007	−	0.863699i	$-0.331858\pi$
$223$	748564.	1.00801	0.504007	+	0.863699i	$0.331858\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.47117e6i	1.89495i	0.319829	+	0.947475i	$0.396374\pi$
$227$	1.47117e6i	1.89495i	−0.319829	+	0.947475i	$0.603626\pi$
$228$	0	0
$229$	− 835029.i	− 1.05223i	−0.850412	−	0.526117i	$-0.823647\pi$
$229$	− 835029.i	− 1.05223i	0.850412	−	0.526117i	$-0.176353\pi$
$230$	0	0
$231$	44750.0	0.0551777
$232$	0	0
$233$	−284718.	−0.343577	−0.171789	−	0.985134i	$-0.554955\pi$
$233$	−284718.	−0.343577	−0.171789	+	0.985134i	$0.554955\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 283039.i	− 0.327322i
$238$	0	0
$239$	800635.	0.906650	0.453325	−	0.891345i	$-0.350238\pi$
$239$	800635.	0.906650	0.453325	+	0.891345i	$0.350238\pi$
$240$	0	0
$241$	−1.50596e6	−1.67021	−0.835107	−	0.550088i	$-0.814594\pi$
$241$	−1.50596e6	−1.67021	−0.835107	+	0.550088i	$0.814594\pi$
$242$	0	0
$243$	931154.i	1.01159i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.26940e6	1.32391
$248$	0	0
$249$	−333831.	−0.341215
$250$	0	0
$251$	1.10255e6i	1.10463i	0.833637	+	0.552313i	$0.186255\pi$
$251$	1.10255e6i	1.10463i	−0.833637	+	0.552313i	$0.813745\pi$
$252$	0	0
$253$	− 129367.i	− 0.127064i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.41038e6	1.33200	0.665999	−	0.745953i	$-0.268006\pi$
$257$	1.41038e6	1.33200	0.665999	+	0.745953i	$0.268006\pi$
$258$	0	0
$259$	1.65246e6i	1.53067i
$260$	0	0
$261$	− 879794.i	− 0.799429i
$262$	0	0
$263$	1.68879e6	1.50552	0.752758	−	0.658298i	$-0.228723\pi$
$263$	1.68879e6	1.50552	0.752758	+	0.658298i	$0.228723\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	274342.i	0.235512i
$268$	0	0
$269$	305848.i	0.257706i	0.991664	+	0.128853i	$0.0411295\pi$
$269$	305848.i	0.257706i	−0.991664	+	0.128853i	$0.958870\pi$
$270$	0	0
$271$	1.25325e6	1.03661	0.518306	−	0.855195i	$-0.326563\pi$
$271$	1.25325e6	1.03661	0.518306	+	0.855195i	$0.326563\pi$
$272$	0	0
$273$	638408.	0.518432
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 483396.i	− 0.378533i	−0.981926	−	0.189267i	$-0.939389\pi$
$277$	− 483396.i	− 0.378533i	0.981926	−	0.189267i	$-0.0606110\pi$
$278$	0	0
$279$	980100.	0.753806
$280$	0	0
$281$	88721.8	0.0670293	0.0335147	−	0.999438i	$-0.489330\pi$
$281$	88721.8	0.0670293	0.0335147	+	0.999438i	$0.489330\pi$
$282$	0	0
$283$	1.76445e6i	1.30961i	0.755796	+	0.654807i	$0.227250\pi$
$283$	1.76445e6i	1.30961i	−0.755796	+	0.654807i	$0.772750\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.61625e6	1.15825
$288$	0	0
$289$	−1.25246e6	−0.882105
$290$	0	0
$291$	688982.i	0.476953i
$292$	0	0
$293$	− 9124.96i	− 0.00620957i	−0.999995	−	0.00310479i	$-0.999012\pi$
$293$	− 9124.96i	− 0.00620957i	0.999995	−	0.00310479i	$-0.000988286\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	132348.	0.0870614
$298$	0	0
$299$	− 1.84557e6i	− 1.19386i
$300$	0	0
$301$	− 1.23593e6i	− 0.786283i
$302$	0	0
$303$	−602273.	−0.376866
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.64214e6i	− 1.59996i	−0.600026	−	0.799980i	$-0.704843\pi$
$307$	− 2.64214e6i	− 1.59996i	0.600026	−	0.799980i	$-0.295157\pi$
$308$	0	0
$309$	− 76762.0i	− 0.0457352i
$310$	0	0
$311$	1.40253e6	0.822262	0.411131	−	0.911576i	$-0.365134\pi$
$311$	1.40253e6	0.822262	0.411131	+	0.911576i	$0.365134\pi$
$312$	0	0
$313$	−1.35907e6	−0.784115	−0.392058	−	0.919941i	$-0.628237\pi$
$313$	−1.35907e6	−0.784115	−0.392058	+	0.919941i	$0.628237\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 300344.i	− 0.167869i	−0.996471	−	0.0839344i	$-0.973251\pi$
$317$	− 300344.i	− 0.167869i	0.996471	−	0.0839344i	$-0.0267486\pi$
$318$	0	0
$319$	−194987.	−0.107283
$320$	0	0
$321$	788150.	0.426920
$322$	0	0
$323$	− 857906.i	− 0.457545i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	973505.	0.503464
$328$	0	0
$329$	584052.	0.297483
$330$	0	0
$331$	2.37543e6i	1.19172i	0.803090	+	0.595858i	$0.203188\pi$
$331$	2.37543e6i	1.19172i	−0.803090	+	0.595858i	$0.796812\pi$
$332$	0	0
$333$	2.15374e6i	1.06435i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.69929e6	1.77437	0.887184	−	0.461416i	$-0.152659\pi$
$337$	3.69929e6	1.77437	0.887184	+	0.461416i	$0.152659\pi$
$338$	0	0
$339$	1.40296e6i	0.663051i
$340$	0	0
$341$	− 217218.i	− 0.101160i
$342$	0	0
$343$	−1.76770e6	−0.811286
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.98355e6i	0.884339i	0.896931	+	0.442170i	$0.145791\pi$
$347$	1.98355e6i	0.884339i	−0.896931	+	0.442170i	$0.854209\pi$
$348$	0	0
$349$	− 1.51829e6i	− 0.667256i	−0.942705	−	0.333628i	$-0.891727\pi$
$349$	− 1.51829e6i	− 0.667256i	0.942705	−	0.333628i	$-0.108273\pi$
$350$	0	0
$351$	1.88809e6	0.818001
$352$	0	0
$353$	2.06194e6	0.880724	0.440362	−	0.897820i	$-0.354850\pi$
$353$	2.06194e6	0.880724	0.440362	+	0.897820i	$0.354850\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 431458.i	− 0.179171i
$358$	0	0
$359$	1.33255e6	0.545691	0.272846	−	0.962058i	$-0.412035\pi$
$359$	1.33255e6	0.545691	0.272846	+	0.962058i	$0.412035\pi$
$360$	0	0
$361$	−1.92072e6	−0.775705
$362$	0	0
$363$	1.14317e6i	0.455350i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−313330.	−0.121433	−0.0607166	−	0.998155i	$-0.519339\pi$
$367$	−313330.	−0.121433	−0.0607166	+	0.998155i	$0.519339\pi$
$368$	0	0
$369$	2.10655e6	0.805390
$370$	0	0
$371$	− 5.62256e6i	− 2.12080i
$372$	0	0
$373$	2.93069e6i	1.09068i	0.838215	+	0.545341i	$0.183600\pi$
$373$	2.93069e6i	1.09068i	−0.838215	+	0.545341i	$0.816400\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.78171e6	−1.00800
$378$	0	0
$379$	− 4.10811e6i	− 1.46907i	−0.678569	−	0.734537i	$-0.737399\pi$
$379$	− 4.10811e6i	− 1.46907i	0.678569	−	0.734537i	$-0.262601\pi$
$380$	0	0
$381$	1.22928e6i	0.433848i
$382$	0	0
$383$	958991.	0.334055	0.167027	−	0.985952i	$-0.446583\pi$
$383$	958991.	0.334055	0.167027	+	0.985952i	$0.446583\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.61086e6i	− 0.546741i
$388$	0	0
$389$	− 647634.i	− 0.216998i	−0.994097	−	0.108499i	$-0.965396\pi$
$389$	− 647634.i	− 0.216998i	0.994097	−	0.108499i	$-0.0346044\pi$
$390$	0	0
$391$	−1.24730e6	−0.412599
$392$	0	0
$393$	799827.	0.261225
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.06284e6i	0.656885i	0.944524	+	0.328443i	$0.106524\pi$
$397$	2.06284e6i	0.656885i	−0.944524	+	0.328443i	$0.893476\pi$
$398$	0	0
$399$	−2.21125e6	−0.695352
$400$	0	0
$401$	−1.92225e6	−0.596964	−0.298482	−	0.954415i	$-0.596480\pi$
$401$	−1.92225e6	−0.596964	−0.298482	+	0.954415i	$0.596480\pi$
$402$	0	0
$403$	− 3.09885e6i	− 0.950470i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	477330.	0.142834
$408$	0	0
$409$	−1.40471e6	−0.415221	−0.207610	−	0.978212i	$-0.566569\pi$
$409$	−1.40471e6	−0.415221	−0.207610	+	0.978212i	$0.566569\pi$
$410$	0	0
$411$	2.02607e6i	0.591630i
$412$	0	0
$413$	− 5.37833e6i	− 1.55157i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	316284.	0.0890712
$418$	0	0
$419$	4.03594e6i	1.12308i	0.827451	+	0.561538i	$0.189790\pi$
$419$	4.03594e6i	1.12308i	−0.827451	+	0.561538i	$0.810210\pi$
$420$	0	0
$421$	594910.i	0.163586i	0.996649	+	0.0817929i	$0.0260646\pi$
$421$	594910.i	0.163586i	−0.996649	+	0.0817929i	$0.973935\pi$
$422$	0	0
$423$	761229.	0.206854
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 467352.i	− 0.124044i
$428$	0	0
$429$	− 184411.i	− 0.0483776i
$430$	0	0
$431$	−4.15801e6	−1.07818	−0.539091	−	0.842247i	$-0.681232\pi$
$431$	−4.15801e6	−1.07818	−0.539091	+	0.842247i	$0.681232\pi$
$432$	0	0
$433$	2.27636e6	0.583474	0.291737	−	0.956499i	$-0.405767\pi$
$433$	2.27636e6	0.583474	0.291737	+	0.956499i	$0.405767\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.39247e6i	1.60127i
$438$	0	0
$439$	−1.28126e6	−0.317305	−0.158653	−	0.987334i	$-0.550715\pi$
$439$	−1.28126e6	−0.317305	−0.158653	+	0.987334i	$0.550715\pi$
$440$	0	0
$441$	914079.	0.223814
$442$	0	0
$443$	− 6.01266e6i	− 1.45565i	−0.685763	−	0.727825i	$-0.740531\pi$
$443$	− 6.01266e6i	− 1.45565i	0.685763	−	0.727825i	$-0.259469\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.80576e6	−0.427455
$448$	0	0
$449$	−2.11543e6	−0.495202	−0.247601	−	0.968862i	$-0.579642\pi$
$449$	−2.11543e6	−0.495202	−0.247601	+	0.968862i	$0.579642\pi$
$450$	0	0
$451$	− 466872.i	− 0.108083i
$452$	0	0
$453$	− 687625.i	− 0.157437i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.35964e6	0.304533	0.152267	−	0.988339i	$-0.451343\pi$
$457$	1.35964e6	0.304533	0.152267	+	0.988339i	$0.451343\pi$
$458$	0	0
$459$	− 1.27603e6i	− 0.282703i
$460$	0	0
$461$	− 2.06438e6i	− 0.452415i	−0.974079	−	0.226208i	$-0.927367\pi$
$461$	− 2.06438e6i	− 0.452415i	0.974079	−	0.226208i	$-0.0726328\pi$
$462$	0	0
$463$	−1.48917e6	−0.322843	−0.161421	−	0.986886i	$-0.551608\pi$
$463$	−1.48917e6	−0.322843	−0.161421	+	0.986886i	$0.551608\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.05856e6i	1.49770i	0.662741	+	0.748849i	$0.269393\pi$
$467$	7.05856e6i	1.49770i	−0.662741	+	0.748849i	$0.730607\pi$
$468$	0	0
$469$	6.15452e6i	1.29200i
$470$	0	0
$471$	−1.28108e6	−0.266087
$472$	0	0
$473$	−357014.	−0.0733722
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 7.32821e6i	− 1.47469i
$478$	0	0
$479$	1.47482e6	0.293697	0.146849	−	0.989159i	$-0.453087\pi$
$479$	1.47482e6	0.293697	0.146849	+	0.989159i	$0.453087\pi$
$480$	0	0
$481$	6.80964e6	1.34203
$482$	0	0
$483$	3.21490e6i	0.627046i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9.13416e6	−1.74520	−0.872602	−	0.488433i	$-0.837569\pi$
$487$	−9.13416e6	−1.74520	−0.872602	+	0.488433i	$0.837569\pi$
$488$	0	0
$489$	1.05760e6	0.200009
$490$	0	0
$491$	4.74188e6i	0.887660i	0.896111	+	0.443830i	$0.146381\pi$
$491$	4.74188e6i	0.887660i	−0.896111	+	0.443830i	$0.853619\pi$
$492$	0	0
$493$	1.87997e6i	0.348365i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.06009e7	−1.92509
$498$	0	0
$499$	539009.i	0.0969046i	0.998825	+	0.0484523i	$0.0154289\pi$
$499$	539009.i	0.0969046i	−0.998825	+	0.0484523i	$0.984571\pi$
$500$	0	0
$501$	2.23243e6i	0.397359i
$502$	0	0
$503$	−2.21657e6	−0.390627	−0.195313	−	0.980741i	$-0.562572\pi$
$503$	−2.21657e6	−0.390627	−0.195313	+	0.980741i	$0.562572\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	34484.2i	0.00595800i
$508$	0	0
$509$	− 3.35132e6i	− 0.573352i	−0.958028	−	0.286676i	$-0.907450\pi$
$509$	− 3.35132e6i	− 0.573352i	0.958028	−	0.286676i	$-0.0925503\pi$
$510$	0	0
$511$	−7.09190e6	−1.20146
$512$	0	0
$513$	−6.53974e6	−1.09715
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 168710.i	− 0.0277597i
$518$	0	0
$519$	863411.	0.140702
$520$	0	0
$521$	5.01924e6	0.810109	0.405054	−	0.914293i	$-0.367253\pi$
$521$	5.01924e6	0.810109	0.405054	+	0.914293i	$0.367253\pi$
$522$	0	0
$523$	3.44581e6i	0.550855i	0.961322	+	0.275428i	$0.0888194\pi$
$523$	3.44581e6i	0.550855i	−0.961322	+	0.275428i	$0.911181\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.09431e6	−0.328484
$528$	0	0
$529$	2.85758e6	0.443975
$530$	0	0
$531$	− 7.00988e6i	− 1.07888i
$532$	0	0
$533$	− 6.66044e6i	− 1.01551i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.29603e6	−0.193945
$538$	0	0
$539$	− 202586.i	− 0.0300357i
$540$	0	0
$541$	3.97667e6i	0.584153i	0.956395	+	0.292076i	$0.0943461\pi$
$541$	3.97667e6i	0.584153i	−0.956395	+	0.292076i	$0.905654\pi$
$542$	0	0
$543$	−1.07461e6	−0.156405
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 9.20664e6i	− 1.31563i	−0.753181	−	0.657814i	$-0.771481\pi$
$547$	− 9.20664e6i	− 1.31563i	0.753181	−	0.657814i	$-0.228519\pi$
$548$	0	0
$549$	− 609127.i	− 0.0862535i
$550$	0	0
$551$	9.63498e6	1.35198
$552$	0	0
$553$	−5.79229e6	−0.805448
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 6.00805e6i	− 0.820531i	−0.911966	−	0.410266i	$-0.865436\pi$
$557$	− 6.00805e6i	− 0.820531i	0.911966	−	0.410266i	$-0.134564\pi$
$558$	0	0
$559$	−5.09319e6	−0.689383
$560$	0	0
$561$	−124631.	−0.0167194
$562$	0	0
$563$	3.22469e6i	0.428763i	0.976750	+	0.214381i	$0.0687736\pi$
$563$	3.22469e6i	0.428763i	−0.976750	+	0.214381i	$0.931226\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.54609e6	0.463225
$568$	0	0
$569$	3.74802e6	0.485312	0.242656	−	0.970112i	$-0.421981\pi$
$569$	3.74802e6	0.485312	0.242656	+	0.970112i	$0.421981\pi$
$570$	0	0
$571$	− 4.37966e6i	− 0.562148i	−0.959686	−	0.281074i	$-0.909309\pi$
$571$	− 4.37966e6i	− 0.562148i	0.959686	−	0.281074i	$-0.0906906\pi$
$572$	0	0
$573$	− 1.53554e6i	− 0.195377i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.03145e7	−1.28976	−0.644880	−	0.764284i	$-0.723093\pi$
$577$	−1.03145e7	−1.28976	−0.644880	+	0.764284i	$0.723093\pi$
$578$	0	0
$579$	5.23732e6i	0.649251i
$580$	0	0
$581$	6.83173e6i	0.839635i
$582$	0	0
$583$	−1.62414e6	−0.197903
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.88489e6i	− 0.704925i	−0.935826	−	0.352463i	$-0.885344\pi$
$587$	− 5.88489e6i	− 0.704925i	0.935826	−	0.352463i	$-0.114656\pi$
$588$	0	0
$589$	1.07335e7i	1.27483i
$590$	0	0
$591$	−6.61934e6	−0.779554
$592$	0	0
$593$	−1.07458e7	−1.25488	−0.627442	−	0.778664i	$-0.715898\pi$
$593$	−1.07458e7	−1.25488	−0.627442	+	0.778664i	$0.715898\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 7.07314e6i	− 0.812226i
$598$	0	0
$599$	1.01249e6	0.115299	0.0576493	−	0.998337i	$-0.481639\pi$
$599$	1.01249e6	0.115299	0.0576493	+	0.998337i	$0.481639\pi$
$600$	0	0
$601$	7.16049e6	0.808643	0.404321	−	0.914617i	$-0.367508\pi$
$601$	7.16049e6	0.808643	0.404321	+	0.914617i	$0.367508\pi$
$602$	0	0
$603$	8.02154e6i	0.898390i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.22036e7	−1.34437	−0.672183	−	0.740385i	$-0.734643\pi$
$607$	−1.22036e7	−1.34437	−0.672183	+	0.740385i	$0.734643\pi$
$608$	0	0
$609$	4.84562e6	0.529427
$610$	0	0
$611$	− 2.40683e6i	− 0.260821i
$612$	0	0
$613$	1.15888e7i	1.24563i	0.782369	+	0.622815i	$0.214011\pi$
$613$	1.15888e7i	1.24563i	−0.782369	+	0.622815i	$0.785989\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.47499e6	−0.155983	−0.0779915	−	0.996954i	$-0.524851\pi$
$617$	−1.47499e6	−0.155983	−0.0779915	+	0.996954i	$0.524851\pi$
$618$	0	0
$619$	− 6.94510e6i	− 0.728537i	−0.931294	−	0.364269i	$-0.881319\pi$
$619$	− 6.94510e6i	− 0.728537i	0.931294	−	0.364269i	$-0.118681\pi$
$620$	0	0
$621$	9.50803e6i	0.989377i
$622$	0	0
$623$	5.61430e6	0.579530
$624$	0	0
$625$	0	0
$626$	0	0
$627$	638744.i	0.0648870i
$628$	0	0
$629$	− 4.60218e6i	− 0.463807i
$630$	0	0
$631$	−3.38551e6	−0.338494	−0.169247	−	0.985574i	$-0.554133\pi$
$631$	−3.38551e6	−0.338494	−0.169247	+	0.985574i	$0.554133\pi$
$632$	0	0
$633$	6.69411e6	0.664024
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.89011e6i	− 0.282206i
$638$	0	0
$639$	−1.38167e7	−1.33861
$640$	0	0
$641$	−6.43478e6	−0.618570	−0.309285	−	0.950969i	$-0.600090\pi$
$641$	−6.43478e6	−0.618570	−0.309285	+	0.950969i	$0.600090\pi$
$642$	0	0
$643$	− 819506.i	− 0.0781672i	−0.999236	−	0.0390836i	$-0.987556\pi$
$643$	− 819506.i	− 0.0781672i	0.999236	−	0.0390836i	$-0.0124439\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.00673e7	−0.945482	−0.472741	−	0.881201i	$-0.656735\pi$
$647$	−1.00673e7	−0.945482	−0.472741	+	0.881201i	$0.656735\pi$
$648$	0	0
$649$	−1.55359e6	−0.144785
$650$	0	0
$651$	5.39807e6i	0.499213i
$652$	0	0
$653$	3.73524e6i	0.342796i	0.985202	+	0.171398i	$0.0548284\pi$
$653$	3.73524e6i	0.342796i	−0.985202	+	0.171398i	$0.945172\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.24328e6	−0.835435
$658$	0	0
$659$	2.11585e6i	0.189789i	0.995487	+	0.0948947i	$0.0302514\pi$
$659$	2.11585e6i	0.189789i	−0.995487	+	0.0948947i	$0.969749\pi$
$660$	0	0
$661$	− 1.89096e7i	− 1.68337i	−0.539970	−	0.841684i	$-0.681565\pi$
$661$	− 1.89096e7i	− 1.68337i	0.539970	−	0.841684i	$-0.318435\pi$
$662$	0	0
$663$	−1.77800e6	−0.157090
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.40082e7i	− 1.21918i
$668$	0	0
$669$	5.37355e6i	0.464190i
$670$	0	0
$671$	−135000.	−0.0115752
$672$	0	0
$673$	−2.10683e6	−0.179304	−0.0896522	−	0.995973i	$-0.528576\pi$
$673$	−2.10683e6	−0.179304	−0.0896522	+	0.995973i	$0.528576\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.17934e6i	0.434313i	0.976137	+	0.217156i	$0.0696782\pi$
$677$	5.17934e6i	0.434313i	−0.976137	+	0.217156i	$0.930322\pi$
$678$	0	0
$679$	1.40998e7	1.17365
$680$	0	0
$681$	−1.05607e7	−0.872623
$682$	0	0
$683$	− 8.80956e6i	− 0.722608i	−0.932448	−	0.361304i	$-0.882332\pi$
$683$	− 8.80956e6i	− 0.722608i	0.932448	−	0.361304i	$-0.117668\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.99423e6	0.484553
$688$	0	0
$689$	−2.31701e7	−1.85943
$690$	0	0
$691$	5.78539e6i	0.460933i	0.973080	+	0.230466i	$0.0740252\pi$
$691$	5.78539e6i	0.460933i	−0.973080	+	0.230466i	$0.925975\pi$
$692$	0	0
$693$	− 1.19361e6i	− 0.0944122i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.50135e6	−0.350963
$698$	0	0
$699$	− 2.04384e6i	− 0.158217i
$700$	0	0
$701$	− 2.01716e7i	− 1.55041i	−0.631711	−	0.775204i	$-0.717647\pi$
$701$	− 2.01716e7i	− 1.55041i	0.631711	−	0.775204i	$-0.282353\pi$
$702$	0	0
$703$	−2.35865e7	−1.80001
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.23253e7i	0.927362i
$708$	0	0
$709$	− 1.71661e7i	− 1.28250i	−0.767334	−	0.641248i	$-0.778417\pi$
$709$	− 1.71661e7i	− 1.28250i	0.767334	−	0.641248i	$-0.221583\pi$
$710$	0	0
$711$	−7.54943e6	−0.560067
$712$	0	0
$713$	1.56052e7	1.14960
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.74733e6i	0.417511i
$718$	0	0
$719$	1.37927e7	0.995009	0.497505	−	0.867461i	$-0.334250\pi$
$719$	1.37927e7	0.995009	0.497505	+	0.867461i	$0.334250\pi$
$720$	0	0
$721$	−1.57091e6	−0.112541
$722$	0	0
$723$	− 1.08105e7i	− 0.769132i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−9.10175e6	−0.638688	−0.319344	−	0.947639i	$-0.603463\pi$
$727$	−9.10175e6	−0.638688	−0.319344	+	0.947639i	$0.603463\pi$
$728$	0	0
$729$	−818554.	−0.0570465
$730$	0	0
$731$	3.44215e6i	0.238252i
$732$	0	0
$733$	− 1.79637e7i	− 1.23491i	−0.786605	−	0.617456i	$-0.788163\pi$
$733$	− 1.79637e7i	− 1.23491i	0.786605	−	0.617456i	$-0.211837\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.77780e6	0.120563
$738$	0	0
$739$	− 1.38839e7i	− 0.935192i	−0.883942	−	0.467596i	$-0.845120\pi$
$739$	− 1.38839e7i	− 0.935192i	0.883942	−	0.467596i	$-0.154880\pi$
$740$	0	0
$741$	9.11238e6i	0.609658i
$742$	0	0
$743$	−2.37521e6	−0.157845	−0.0789225	−	0.996881i	$-0.525148\pi$
$743$	−2.37521e6	−0.157845	−0.0789225	+	0.996881i	$0.525148\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.90419e6i	0.583839i
$748$	0	0
$749$	− 1.61292e7i	− 1.05053i
$750$	0	0
$751$	1.99260e7	1.28920	0.644600	−	0.764520i	$-0.277024\pi$
$751$	1.99260e7	1.28920	0.644600	+	0.764520i	$0.277024\pi$
$752$	0	0
$753$	−7.91465e6	−0.508679
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.30771e7i	0.829413i	0.909955	+	0.414707i	$0.136116\pi$
$757$	1.30771e7i	0.829413i	−0.909955	+	0.414707i	$0.863884\pi$
$758$	0	0
$759$	928660.	0.0585130
$760$	0	0
$761$	−1.75251e7	−1.09698	−0.548492	−	0.836156i	$-0.684798\pi$
$761$	−1.75251e7	−1.09698	−0.548492	+	0.836156i	$0.684798\pi$
$762$	0	0
$763$	− 1.99224e7i	− 1.23888i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.21637e7	−1.36036
$768$	0	0
$769$	3.13820e7	1.91366	0.956830	−	0.290647i	$-0.0938706\pi$
$769$	3.13820e7	1.91366	0.956830	+	0.290647i	$0.0938706\pi$
$770$	0	0
$771$	1.01244e7i	0.613384i
$772$	0	0
$773$	103265.i	0.00621590i	0.999995	+	0.00310795i	$0.000989293\pi$
$773$	103265.i	0.00621590i	−0.999995	+	0.00310795i	$0.999011\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.18621e7	−0.704870
$778$	0	0
$779$	2.30697e7i	1.36207i
$780$	0	0
$781$	3.06218e6i	0.179640i
$782$	0	0
$783$	1.43309e7	0.835350
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.01823e6i	0.288811i	0.989519	+	0.144405i	$0.0461270\pi$
$787$	5.01823e6i	0.288811i	−0.989519	+	0.144405i	$0.953873\pi$
$788$	0	0
$789$	1.21229e7i	0.693288i
$790$	0	0
$791$	2.87111e7	1.63158
$792$	0	0
$793$	−1.92592e6	−0.108757
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 9.61578e6i	− 0.536215i	−0.963389	−	0.268107i	$-0.913602\pi$
$797$	− 9.61578e6i	− 0.536215i	0.963389	−	0.268107i	$-0.0863982\pi$
$798$	0	0
$799$	−1.62662e6	−0.0901403
$800$	0	0
$801$	7.31745e6	0.402975
$802$	0	0
$803$	2.04857e6i	0.112115i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.19552e6	−0.118673
$808$	0	0
$809$	−2.06325e7	−1.10836	−0.554181	−	0.832396i	$-0.686968\pi$
$809$	−2.06325e7	−1.10836	−0.554181	+	0.832396i	$0.686968\pi$
$810$	0	0
$811$	− 3.18795e7i	− 1.70200i	−0.525164	−	0.851001i	$-0.675996\pi$
$811$	− 3.18795e7i	− 1.70200i	0.525164	−	0.851001i	$-0.324004\pi$
$812$	0	0
$813$	8.99645e6i	0.477359i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.76412e7	0.924642
$818$	0	0
$819$	− 1.70281e7i	− 0.887067i
$820$	0	0
$821$	− 7.88522e6i	− 0.408278i	−0.978942	−	0.204139i	$-0.934561\pi$
$821$	− 7.88522e6i	− 0.408278i	0.978942	−	0.204139i	$-0.0654394\pi$
$822$	0	0
$823$	−51764.9	−0.00266401	−0.00133200	−	0.999999i	$-0.500424\pi$
$823$	−51764.9	−0.00266401	−0.00133200	+	0.999999i	$0.500424\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.24793e6i	0.266824i	0.991061	+	0.133412i	$0.0425933\pi$
$827$	5.24793e6i	0.266824i	−0.991061	+	0.133412i	$0.957407\pi$
$828$	0	0
$829$	− 6.76109e6i	− 0.341689i	−0.985298	−	0.170844i	$-0.945350\pi$
$829$	− 6.76109e6i	− 0.341689i	0.985298	−	0.170844i	$-0.0546495\pi$
$830$	0	0
$831$	3.47004e6	0.174314
$832$	0	0
$833$	−1.95323e6	−0.0975308
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.59647e7i	0.787677i
$838$	0	0
$839$	−5.32681e6	−0.261254	−0.130627	−	0.991432i	$-0.541699\pi$
$839$	−5.32681e6	−0.261254	−0.130627	+	0.991432i	$0.541699\pi$
$840$	0	0
$841$	−602476.	−0.0293731
$842$	0	0
$843$	636887.i	0.0308669i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.33946e7	1.12049
$848$	0	0
$849$	−1.26660e7	−0.603076
$850$	0	0
$851$	3.42920e7i	1.62319i
$852$	0	0
$853$	− 8.20548e6i	− 0.386128i	−0.981186	−	0.193064i	$-0.938157\pi$
$853$	− 8.20548e6i	− 0.386128i	0.981186	−	0.193064i	$-0.0618425\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.45868e6	0.253884	0.126942	−	0.991910i	$-0.459484\pi$
$857$	5.45868e6	0.253884	0.126942	+	0.991910i	$0.459484\pi$
$858$	0	0
$859$	3.18230e7i	1.47149i	0.677257	+	0.735747i	$0.263169\pi$
$859$	3.18230e7i	1.47149i	−0.677257	+	0.735747i	$0.736831\pi$
$860$	0	0
$861$	1.16022e7i	0.533375i
$862$	0	0
$863$	−9.28914e6	−0.424569	−0.212285	−	0.977208i	$-0.568090\pi$
$863$	−9.28914e6	−0.424569	−0.212285	+	0.977208i	$0.568090\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8.99077e6i	− 0.406208i
$868$	0	0
$869$	1.67317e6i	0.0751606i
$870$	0	0
$871$	2.53623e7	1.13277
$872$	0	0
$873$	1.83771e7	0.816094
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.60806e7i	− 0.705997i	−0.935624	−	0.352999i	$-0.885162\pi$
$877$	− 1.60806e7i	− 0.705997i	0.935624	−	0.352999i	$-0.114838\pi$
$878$	0	0
$879$	65503.2	0.00285950
$880$	0	0
$881$	−1.74471e7	−0.757327	−0.378663	−	0.925535i	$-0.623616\pi$
$881$	−1.74471e7	−0.757327	−0.378663	+	0.925535i	$0.623616\pi$
$882$	0	0
$883$	2.18862e7i	0.944645i	0.881426	+	0.472323i	$0.156584\pi$
$883$	2.18862e7i	0.944645i	−0.881426	+	0.472323i	$0.843416\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.00556e7	1.70944	0.854721	−	0.519088i	$-0.173728\pi$
$887$	4.00556e7	1.70944	0.854721	+	0.519088i	$0.173728\pi$
$888$	0	0
$889$	2.51567e7	1.06758
$890$	0	0
$891$	− 1.02433e6i	− 0.0432260i
$892$	0	0
$893$	8.33653e6i	0.349830i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.32484e7	0.549770
$898$	0	0
$899$	− 2.35208e7i	− 0.970628i
$900$	0	0
$901$	1.56592e7i	0.642623i
$902$	0	0
$903$	8.87211e6	0.362083
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.61797e7i	0.653060i	0.945187	+	0.326530i	$0.105879\pi$
$907$	1.61797e7i	0.653060i	−0.945187	+	0.326530i	$0.894121\pi$
$908$	0	0
$909$	1.60643e7i	0.644840i
$910$	0	0
$911$	−3.81429e7	−1.52271	−0.761356	−	0.648334i	$-0.775466\pi$
$911$	−3.81429e7	−1.52271	−0.761356	+	0.648334i	$0.775466\pi$
$912$	0	0
$913$	1.97342e6	0.0783508
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.63682e7i	− 0.642801i
$918$	0	0
$919$	−5.75041e6	−0.224600	−0.112300	−	0.993674i	$-0.535822\pi$
$919$	−5.75041e6	−0.224600	−0.112300	+	0.993674i	$0.535822\pi$
$920$	0	0
$921$	1.89665e7	0.736780
$922$	0	0
$923$	4.36854e7i	1.68784i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.04745e6	−0.0782555
$928$	0	0
$929$	−1.58033e7	−0.600771	−0.300386	−	0.953818i	$-0.597115\pi$
$929$	−1.58033e7	−0.600771	−0.300386	+	0.953818i	$0.597115\pi$
$930$	0	0
$931$	1.00104e7i	0.378512i
$932$	0	0
$933$	1.00680e7i	0.378651i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.61564e7	0.601167	0.300583	−	0.953756i	$-0.402819\pi$
$937$	1.61564e7	0.601167	0.300583	+	0.953756i	$0.402819\pi$
$938$	0	0
$939$	− 9.75602e6i	− 0.361084i
$940$	0	0
$941$	− 4.39614e7i	− 1.61844i	−0.587504	−	0.809221i	$-0.699889\pi$
$941$	− 4.39614e7i	− 1.61844i	0.587504	−	0.809221i	$-0.300111\pi$
$942$	0	0
$943$	3.35407e7	1.22827
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.52665e7i	− 1.64022i	−0.572205	−	0.820111i	$-0.693912\pi$
$947$	− 4.52665e7i	− 1.64022i	0.572205	−	0.820111i	$-0.306088\pi$
$948$	0	0
$949$	2.92252e7i	1.05340i
$950$	0	0
$951$	2.15601e6	0.0773035
$952$	0	0
$953$	−3.21216e6	−0.114569	−0.0572843	−	0.998358i	$-0.518244\pi$
$953$	−3.21216e6	−0.114569	−0.0572843	+	0.998358i	$0.518244\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.39971e6i	− 0.0494036i
$958$	0	0
$959$	4.14628e7	1.45583
$960$	0	0
$961$	−2.42676e6	−0.0847654
$962$	0	0
$963$	− 2.10221e7i	− 0.730485i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.06241e6	0.139707	0.0698533	−	0.997557i	$-0.477747\pi$
$967$	4.06241e6	0.139707	0.0698533	+	0.997557i	$0.477747\pi$
$968$	0	0
$969$	6.15845e6	0.210699
$970$	0	0
$971$	− 2.41409e7i	− 0.821685i	−0.911706	−	0.410842i	$-0.865235\pi$
$971$	− 2.41409e7i	− 0.821685i	0.911706	−	0.410842i	$-0.134765\pi$
$972$	0	0
$973$	− 6.47264e6i	− 0.219179i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.48416e7	0.497443	0.248722	−	0.968575i	$-0.419990\pi$
$977$	1.48416e7	0.497443	0.248722	+	0.968575i	$0.419990\pi$
$978$	0	0
$979$	− 1.62175e6i	− 0.0540790i
$980$	0	0
$981$	− 2.59661e7i	− 0.861457i
$982$	0	0
$983$	3.34107e7	1.10281	0.551406	−	0.834237i	$-0.314092\pi$
$983$	3.34107e7	1.10281	0.551406	+	0.834237i	$0.314092\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.19260e6i	0.136991i
$988$	0	0
$989$	− 2.56483e7i	− 0.833812i
$990$	0	0
$991$	−5.15057e7	−1.66599	−0.832993	−	0.553284i	$-0.813374\pi$
$991$	−5.15057e7	−1.66599	−0.832993	+	0.553284i	$0.813374\pi$
$992$	0	0
$993$	−1.70520e7	−0.548784
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.18985e7i	0.379102i	0.981871	+	0.189551i	$0.0607032\pi$
$997$	1.18985e7i	0.379102i	−0.981871	+	0.189551i	$0.939297\pi$
$998$	0	0
$999$	−3.50821e7	−1.11217

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.d.e.401.22		28
4.3	odd	2		200.6.d.e.101.25		28
5.2	odd	4		160.6.f.a.49.18		28
5.3	odd	4		160.6.f.a.49.11		28
5.4	even	2	inner	800.6.d.e.401.7		28
8.3	odd	2		200.6.d.e.101.26		28
8.5	even	2	inner	800.6.d.e.401.21		28
20.3	even	4		40.6.f.a.29.11	✓	28
20.7	even	4		40.6.f.a.29.18	yes	28
20.19	odd	2		200.6.d.e.101.4		28
40.3	even	4		40.6.f.a.29.17	yes	28
40.13	odd	4		160.6.f.a.49.17		28
40.19	odd	2		200.6.d.e.101.3		28
40.27	even	4		40.6.f.a.29.12	yes	28
40.29	even	2	inner	800.6.d.e.401.8		28
40.37	odd	4		160.6.f.a.49.12		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.f.a.29.11	✓	28	20.3	even	4
40.6.f.a.29.12	yes	28	40.27	even	4
40.6.f.a.29.17	yes	28	40.3	even	4
40.6.f.a.29.18	yes	28	20.7	even	4
160.6.f.a.49.11		28	5.3	odd	4
160.6.f.a.49.12		28	40.37	odd	4
160.6.f.a.49.17		28	40.13	odd	4
160.6.f.a.49.18		28	5.2	odd	4
200.6.d.e.101.3		28	40.19	odd	2
200.6.d.e.101.4		28	20.19	odd	2
200.6.d.e.101.25		28	4.3	odd	2
200.6.d.e.101.26		28	8.3	odd	2
800.6.d.e.401.7		28	5.4	even	2	inner
800.6.d.e.401.8		28	40.29	even	2	inner
800.6.d.e.401.21		28	8.5	even	2	inner
800.6.d.e.401.22		28	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+7.17847i q^{3} +146.905 q^{7} +191.470 q^{9} +O(q^{10})\)	\(q+7.17847i q^{3} +146.905 q^{7} +191.470 q^{9} -42.4351i q^{11} -605.383i q^{13} -409.138 q^{17} +2096.86i q^{19} +1054.55i q^{21} +3048.59 q^{23} +3118.83i q^{27} -4594.96i q^{29} +5118.83 q^{31} +304.619 q^{33} +11248.5i q^{37} +4345.73 q^{39} +11002.0 q^{41} -8413.16i q^{43} +3975.72 q^{47} +4774.02 q^{49} -2936.99i q^{51} -38273.5i q^{53} -15052.2 q^{57} -36611.0i q^{59} -3181.32i q^{61} +28127.8 q^{63} +41894.6i q^{67} +21884.2i q^{69} -72161.6 q^{71} -48275.5 q^{73} -6233.92i q^{77} -39428.9 q^{79} +24138.7 q^{81} +46504.5i q^{83} +32984.8 q^{87} +38217.3 q^{89} -88933.7i q^{91} +36745.3i q^{93} +95979.0 q^{97} -8125.03i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 1940 q^{9}+O(q^{10}) \) 28 * q - 1940 * q^9	\( 28 q - 1940 q^{9} - 4368 q^{31} + 23360 q^{39} - 2480 q^{41} + 38420 q^{49} + 69232 q^{71} - 35984 q^{79} + 122596 q^{81} + 178744 q^{89}+O(q^{100}) \) 28 * q - 1940 * q^9 - 4368 * q^31 + 23360 * q^39 - 2480 * q^41 + 38420 * q^49 + 69232 * q^71 - 35984 * q^79 + 122596 * q^81 + 178744 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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