LMFDB - Embedded newform 800.6.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(1,800)]

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, 0]))

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

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

chi := DirichletCharacter("800.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$128.307055850$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{70}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 70 $ x^2 - 70
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.36660$ of defining polynomial
Character	$\chi$	$=$	800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.7332 q^{3} +68.7332 q^{7} -80.8656 q^{9} +O(q^{10})$	$q-12.7332 q^{3} +68.7332 q^{7} -80.8656 q^{9} -327.332 q^{11} +719.328 q^{13} +379.328 q^{17} -1029.33 q^{19} -875.194 q^{21} -779.120 q^{23} +4123.85 q^{27} +1392.66 q^{29} -2744.68 q^{31} +4167.98 q^{33} -12640.6 q^{37} -9159.35 q^{39} +8210.43 q^{41} +22524.5 q^{43} +7739.18 q^{47} -12082.7 q^{49} -4830.06 q^{51} +2401.86 q^{53} +13106.6 q^{57} -15734.7 q^{59} +32082.1 q^{61} -5558.15 q^{63} +9009.07 q^{67} +9920.70 q^{69} -43832.3 q^{71} +65837.5 q^{73} -22498.6 q^{77} -39601.3 q^{79} -32859.4 q^{81} -63101.4 q^{83} -17733.0 q^{87} +34510.9 q^{89} +49441.7 q^{91} +34948.6 q^{93} +14081.3 q^{97} +26469.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{3} + 104 q^{7} + 106 q^{9}+O(q^{10}) $ 2 * q + 8 * q^3 + 104 * q^7 + 106 * q^9	$ 2 q + 8 q^{3} + 104 q^{7} + 106 q^{9} - 320 q^{11} + 100 q^{13} - 580 q^{17} - 720 q^{19} - 144 q^{21} + 1688 q^{23} + 2960 q^{27} + 108 q^{29} - 9840 q^{31} + 4320 q^{33} - 6540 q^{37} - 22000 q^{39} - 10620 q^{41} + 25672 q^{43} + 28296 q^{47} - 27646 q^{49} - 24720 q^{51} - 31340 q^{53} + 19520 q^{57} - 30800 q^{59} + 24540 q^{61} + 1032 q^{63} + 34584 q^{67} + 61072 q^{69} + 12400 q^{71} + 7180 q^{73} - 22240 q^{77} - 71840 q^{79} - 102398 q^{81} - 31928 q^{83} - 44368 q^{87} - 40748 q^{89} + 27600 q^{91} - 112160 q^{93} + 190140 q^{97} + 27840 q^{99}+O(q^{100}) $ 2 * q + 8 * q^3 + 104 * q^7 + 106 * q^9 - 320 * q^11 + 100 * q^13 - 580 * q^17 - 720 * q^19 - 144 * q^21 + 1688 * q^23 + 2960 * q^27 + 108 * q^29 - 9840 * q^31 + 4320 * q^33 - 6540 * q^37 - 22000 * q^39 - 10620 * q^41 + 25672 * q^43 + 28296 * q^47 - 27646 * q^49 - 24720 * q^51 - 31340 * q^53 + 19520 * q^57 - 30800 * q^59 + 24540 * q^61 + 1032 * q^63 + 34584 * q^67 + 61072 * q^69 + 12400 * q^71 + 7180 * q^73 - 22240 * q^77 - 71840 * q^79 - 102398 * q^81 - 31928 * q^83 - 44368 * q^87 - 40748 * q^89 + 27600 * q^91 - 112160 * q^93 + 190140 * q^97 + 27840 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−12.7332	−0.816835	−0.408418	−	0.912795i	$-0.633919\pi$
$3$	−12.7332	−0.816835	−0.408418	+	0.912795i	$0.633919\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	68.7332	0.530178	0.265089	−	0.964224i	$-0.414599\pi$
$7$	68.7332	0.530178	0.265089	+	0.964224i	$0.414599\pi$
$8$	0	0
$9$	−80.8656	−0.332780
$10$	0	0
$11$	−327.332	−0.815655	−0.407828	−	0.913059i	$-0.633714\pi$
$11$	−327.332	−0.815655	−0.407828	+	0.913059i	$0.633714\pi$
$12$	0	0
$13$	719.328	1.18051	0.590254	−	0.807218i	$-0.299028\pi$
$13$	719.328	1.18051	0.590254	+	0.807218i	$0.299028\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	379.328	0.318341	0.159171	−	0.987251i	$-0.449118\pi$
$17$	379.328	0.318341	0.159171	+	0.987251i	$0.449118\pi$
$18$	0	0
$19$	−1029.33	−0.654139	−0.327069	−	0.945000i	$-0.606061\pi$
$19$	−1029.33	−0.654139	−0.327069	+	0.945000i	$0.606061\pi$
$20$	0	0
$21$	−875.194	−0.433068
$22$	0	0
$23$	−779.120	−0.307104	−0.153552	−	0.988141i	$-0.549071\pi$
$23$	−779.120	−0.307104	−0.153552	+	0.988141i	$0.549071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4123.85	1.08866
$28$	0	0
$29$	1392.66	0.307503	0.153751	−	0.988110i	$-0.450865\pi$
$29$	1392.66	0.307503	0.153751	+	0.988110i	$0.450865\pi$
$30$	0	0
$31$	−2744.68	−0.512965	−0.256483	−	0.966549i	$-0.582564\pi$
$31$	−2744.68	−0.512965	−0.256483	+	0.966549i	$0.582564\pi$
$32$	0	0
$33$	4167.98	0.666256
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−12640.6	−1.51797	−0.758985	−	0.651108i	$-0.774304\pi$
$37$	−12640.6	−1.51797	−0.758985	+	0.651108i	$0.774304\pi$
$38$	0	0
$39$	−9159.35	−0.964280
$40$	0	0
$41$	8210.43	0.762792	0.381396	−	0.924412i	$-0.375443\pi$
$41$	8210.43	0.762792	0.381396	+	0.924412i	$0.375443\pi$
$42$	0	0
$43$	22524.5	1.85774	0.928869	−	0.370408i	$-0.120782\pi$
$43$	22524.5	1.85774	0.928869	+	0.370408i	$0.120782\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7739.18	0.511035	0.255517	−	0.966804i	$-0.417754\pi$
$47$	7739.18	0.511035	0.255517	+	0.966804i	$0.417754\pi$
$48$	0	0
$49$	−12082.7	−0.718912
$50$	0	0
$51$	−4830.06	−0.260032
$52$	0	0
$53$	2401.86	0.117451	0.0587256	−	0.998274i	$-0.481296\pi$
$53$	2401.86	0.117451	0.0587256	+	0.998274i	$0.481296\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13106.6	0.534323
$58$	0	0
$59$	−15734.7	−0.588474	−0.294237	−	0.955732i	$-0.595066\pi$
$59$	−15734.7	−0.588474	−0.294237	+	0.955732i	$0.595066\pi$
$60$	0	0
$61$	32082.1	1.10392	0.551961	−	0.833870i	$-0.313880\pi$
$61$	32082.1	1.10392	0.551961	+	0.833870i	$0.313880\pi$
$62$	0	0
$63$	−5558.15	−0.176433
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9009.07	0.245184	0.122592	−	0.992457i	$-0.460879\pi$
$67$	9009.07	0.245184	0.122592	+	0.992457i	$0.460879\pi$
$68$	0	0
$69$	9920.70	0.250853
$70$	0	0
$71$	−43832.3	−1.03192	−0.515962	−	0.856611i	$-0.672566\pi$
$71$	−43832.3	−1.03192	−0.515962	+	0.856611i	$0.672566\pi$
$72$	0	0
$73$	65837.5	1.44599	0.722997	−	0.690852i	$-0.242764\pi$
$73$	65837.5	1.44599	0.722997	+	0.690852i	$0.242764\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−22498.6	−0.432442
$78$	0	0
$79$	−39601.3	−0.713907	−0.356954	−	0.934122i	$-0.616185\pi$
$79$	−39601.3	−0.713907	−0.356954	+	0.934122i	$0.616185\pi$
$80$	0	0
$81$	−32859.4	−0.556477
$82$	0	0
$83$	−63101.4	−1.00541	−0.502706	−	0.864458i	$-0.667662\pi$
$83$	−63101.4	−1.00541	−0.502706	+	0.864458i	$0.667662\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−17733.0	−0.251179
$88$	0	0
$89$	34510.9	0.461829	0.230915	−	0.972974i	$-0.425828\pi$
$89$	34510.9	0.461829	0.230915	+	0.972974i	$0.425828\pi$
$90$	0	0
$91$	49441.7	0.625879
$92$	0	0
$93$	34948.6	0.419008
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14081.3	0.151955	0.0759773	−	0.997110i	$-0.475792\pi$
$97$	14081.3	0.151955	0.0759773	+	0.997110i	$0.475792\pi$
$98$	0	0
$99$	26469.9	0.271434
$100$	0	0
$101$	184018.	1.79497	0.897485	−	0.441044i	$-0.145392\pi$
$101$	184018.	1.79497	0.897485	+	0.441044i	$0.145392\pi$
$102$	0	0
$103$	−70168.0	−0.651697	−0.325849	−	0.945422i	$-0.605650\pi$
$103$	−70168.0	−0.651697	−0.325849	+	0.945422i	$0.605650\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7952.89	0.0671530	0.0335765	−	0.999436i	$-0.489310\pi$
$107$	7952.89	0.0671530	0.0335765	+	0.999436i	$0.489310\pi$
$108$	0	0
$109$	168681.	1.35988	0.679939	−	0.733269i	$-0.262006\pi$
$109$	168681.	1.35988	0.679939	+	0.733269i	$0.262006\pi$
$110$	0	0
$111$	160955.	1.23993
$112$	0	0
$113$	61891.3	0.455967	0.227984	−	0.973665i	$-0.426787\pi$
$113$	61891.3	0.455967	0.227984	+	0.973665i	$0.426787\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−58168.9	−0.392849
$118$	0	0
$119$	26072.4	0.168777
$120$	0	0
$121$	−53904.8	−0.334706
$122$	0	0
$123$	−104545.	−0.623075
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−358695.	−1.97340	−0.986702	−	0.162538i	$-0.948032\pi$
$127$	−358695.	−1.97340	−0.986702	+	0.162538i	$0.948032\pi$
$128$	0	0
$129$	−286809.	−1.51747
$130$	0	0
$131$	312592.	1.59148	0.795738	−	0.605641i	$-0.207083\pi$
$131$	312592.	1.59148	0.795738	+	0.605641i	$0.207083\pi$
$132$	0	0
$133$	−70749.0	−0.346810
$134$	0	0
$135$	0	0
$136$	0	0
$137$	33573.9	0.152827	0.0764135	−	0.997076i	$-0.475653\pi$
$137$	33573.9	0.152827	0.0764135	+	0.997076i	$0.475653\pi$
$138$	0	0
$139$	−342175.	−1.50214	−0.751072	−	0.660220i	$-0.770463\pi$
$139$	−342175.	−1.50214	−0.751072	+	0.660220i	$0.770463\pi$
$140$	0	0
$141$	−98544.6	−0.417431
$142$	0	0
$143$	−235459.	−0.962887
$144$	0	0
$145$	0	0
$146$	0	0
$147$	153852.	0.587232
$148$	0	0
$149$	−239318.	−0.883099	−0.441549	−	0.897237i	$-0.645571\pi$
$149$	−239318.	−0.883099	−0.441549	+	0.897237i	$0.645571\pi$
$150$	0	0
$151$	−169513.	−0.605007	−0.302503	−	0.953148i	$-0.597822\pi$
$151$	−169513.	−0.605007	−0.302503	+	0.953148i	$0.597822\pi$
$152$	0	0
$153$	−30674.6	−0.105938
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−186382.	−0.603469	−0.301735	−	0.953392i	$-0.597566\pi$
$157$	−186382.	−0.603469	−0.301735	+	0.953392i	$0.597566\pi$
$158$	0	0
$159$	−30583.3	−0.0959383
$160$	0	0
$161$	−53551.4	−0.162820
$162$	0	0
$163$	−403058.	−1.18822	−0.594112	−	0.804382i	$-0.702496\pi$
$163$	−403058.	−1.18822	−0.594112	+	0.804382i	$0.702496\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	167631.	0.465119	0.232560	−	0.972582i	$-0.425290\pi$
$167$	167631.	0.465119	0.232560	+	0.972582i	$0.425290\pi$
$168$	0	0
$169$	146140.	0.393597
$170$	0	0
$171$	83237.2	0.217684
$172$	0	0
$173$	−84080.4	−0.213589	−0.106795	−	0.994281i	$-0.534059\pi$
$173$	−84080.4	−0.213589	−0.106795	+	0.994281i	$0.534059\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	200353.	0.480686
$178$	0	0
$179$	−741698.	−1.73019	−0.865096	−	0.501607i	$-0.832743\pi$
$179$	−741698.	−1.73019	−0.865096	+	0.501607i	$0.832743\pi$
$180$	0	0
$181$	−343670.	−0.779732	−0.389866	−	0.920872i	$-0.627479\pi$
$181$	−343670.	−0.779732	−0.389866	+	0.920872i	$0.627479\pi$
$182$	0	0
$183$	−408508.	−0.901722
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−124166.	−0.259657
$188$	0	0
$189$	283445.	0.577184
$190$	0	0
$191$	−631035.	−1.25161	−0.625806	−	0.779978i	$-0.715230\pi$
$191$	−631035.	−1.25161	−0.625806	+	0.779978i	$0.715230\pi$
$192$	0	0
$193$	−847791.	−1.63831	−0.819154	−	0.573574i	$-0.805556\pi$
$193$	−847791.	−1.63831	−0.819154	+	0.573574i	$0.805556\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	397282.	0.729346	0.364673	−	0.931136i	$-0.381181\pi$
$197$	397282.	0.729346	0.364673	+	0.931136i	$0.381181\pi$
$198$	0	0
$199$	−896038.	−1.60396	−0.801980	−	0.597350i	$-0.796220\pi$
$199$	−896038.	−1.60396	−0.801980	+	0.597350i	$0.796220\pi$
$200$	0	0
$201$	−114714.	−0.200275
$202$	0	0
$203$	95721.7	0.163031
$204$	0	0
$205$	0	0
$206$	0	0
$207$	63004.0	0.102198
$208$	0	0
$209$	336932.	0.533552
$210$	0	0
$211$	876761.	1.35574	0.677868	−	0.735184i	$-0.262904\pi$
$211$	876761.	1.35574	0.677868	+	0.735184i	$0.262904\pi$
$212$	0	0
$213$	558125.	0.842913
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−188651.	−0.271963
$218$	0	0
$219$	−838322.	−1.18114
$220$	0	0
$221$	272861.	0.375804
$222$	0	0
$223$	54891.1	0.0739162	0.0369581	−	0.999317i	$-0.488233\pi$
$223$	54891.1	0.0739162	0.0369581	+	0.999317i	$0.488233\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	803329.	1.03473	0.517367	−	0.855764i	$-0.326912\pi$
$227$	803329.	1.03473	0.517367	+	0.855764i	$0.326912\pi$
$228$	0	0
$229$	−589546.	−0.742898	−0.371449	−	0.928453i	$-0.621139\pi$
$229$	−589546.	−0.742898	−0.371449	+	0.928453i	$0.621139\pi$
$230$	0	0
$231$	286479.	0.353234
$232$	0	0
$233$	−1.02048e6	−1.23144	−0.615720	−	0.787965i	$-0.711135\pi$
$233$	−1.02048e6	−1.23144	−0.615720	+	0.787965i	$0.711135\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	504251.	0.583145
$238$	0	0
$239$	−1.65512e6	−1.87428	−0.937139	−	0.348956i	$-0.886536\pi$
$239$	−1.65512e6	−1.87428	−0.937139	+	0.348956i	$0.886536\pi$
$240$	0	0
$241$	−1.19028e6	−1.32010	−0.660049	−	0.751223i	$-0.729464\pi$
$241$	−1.19028e6	−1.32010	−0.660049	+	0.751223i	$0.729464\pi$
$242$	0	0
$243$	−583689.	−0.634112
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−740424.	−0.772215
$248$	0	0
$249$	803483.	0.821256
$250$	0	0
$251$	23776.0	0.0238207	0.0119104	−	0.999929i	$-0.496209\pi$
$251$	23776.0	0.0238207	0.0119104	+	0.999929i	$0.496209\pi$
$252$	0	0
$253$	255031.	0.250491
$254$	0	0
$255$	0	0
$256$	0	0
$257$	341681.	0.322692	0.161346	−	0.986898i	$-0.448416\pi$
$257$	341681.	0.322692	0.161346	+	0.986898i	$0.448416\pi$
$258$	0	0
$259$	−868828.	−0.804794
$260$	0	0
$261$	−112618.	−0.102331
$262$	0	0
$263$	−1.09120e6	−0.972782	−0.486391	−	0.873741i	$-0.661687\pi$
$263$	−1.09120e6	−0.972782	−0.486391	+	0.873741i	$0.661687\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−439434.	−0.377238
$268$	0	0
$269$	−922907.	−0.777638	−0.388819	−	0.921314i	$-0.627117\pi$
$269$	−922907.	−0.777638	−0.388819	+	0.921314i	$0.627117\pi$
$270$	0	0
$271$	1.34302e6	1.11086	0.555430	−	0.831564i	$-0.312554\pi$
$271$	1.34302e6	1.11086	0.555430	+	0.831564i	$0.312554\pi$
$272$	0	0
$273$	−629551.	−0.511240
$274$	0	0
$275$	0	0
$276$	0	0
$277$	247543.	0.193843	0.0969217	−	0.995292i	$-0.469100\pi$
$277$	247543.	0.193843	0.0969217	+	0.995292i	$0.469100\pi$
$278$	0	0
$279$	221951.	0.170705
$280$	0	0
$281$	1.03588e6	0.782604	0.391302	−	0.920262i	$-0.372025\pi$
$281$	1.03588e6	0.782604	0.391302	+	0.920262i	$0.372025\pi$
$282$	0	0
$283$	−2.39427e6	−1.77708	−0.888542	−	0.458795i	$-0.848281\pi$
$283$	−2.39427e6	−1.77708	−0.888542	+	0.458795i	$0.848281\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	564329.	0.404415
$288$	0	0
$289$	−1.27597e6	−0.898659
$290$	0	0
$291$	−179300.	−0.124122
$292$	0	0
$293$	2.44817e6	1.66599	0.832995	−	0.553280i	$-0.186624\pi$
$293$	2.44817e6	1.66599	0.832995	+	0.553280i	$0.186624\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.34987e6	−0.887973
$298$	0	0
$299$	−560443.	−0.362538
$300$	0	0
$301$	1.54818e6	0.984931
$302$	0	0
$303$	−2.34314e6	−1.46620
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−939476.	−0.568905	−0.284453	−	0.958690i	$-0.591812\pi$
$307$	−939476.	−0.568905	−0.284453	+	0.958690i	$0.591812\pi$
$308$	0	0
$309$	893463.	0.532329
$310$	0	0
$311$	1.13941e6	0.668004	0.334002	−	0.942572i	$-0.391601\pi$
$311$	1.13941e6	0.668004	0.334002	+	0.942572i	$0.391601\pi$
$312$	0	0
$313$	−1.51692e6	−0.875191	−0.437595	−	0.899172i	$-0.644170\pi$
$313$	−1.51692e6	−0.875191	−0.437595	+	0.899172i	$0.644170\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.73484e6	−1.52857	−0.764284	−	0.644880i	$-0.776907\pi$
$317$	−2.73484e6	−1.52857	−0.764284	+	0.644880i	$0.776907\pi$
$318$	0	0
$319$	−455861.	−0.250816
$320$	0	0
$321$	−101266.	−0.0548530
$322$	0	0
$323$	−390453.	−0.208239
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.14785e6	−1.11080
$328$	0	0
$329$	531939.	0.270939
$330$	0	0
$331$	−122807.	−0.0616104	−0.0308052	−	0.999525i	$-0.509807\pi$
$331$	−122807.	−0.0616104	−0.0308052	+	0.999525i	$0.509807\pi$
$332$	0	0
$333$	1.02219e6	0.505150
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.65582e6	0.794217	0.397108	−	0.917772i	$-0.370014\pi$
$337$	1.65582e6	0.794217	0.397108	+	0.917772i	$0.370014\pi$
$338$	0	0
$339$	−788075.	−0.372450
$340$	0	0
$341$	898423.	0.418403
$342$	0	0
$343$	−1.98568e6	−0.911329
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.63896e6	−0.730711	−0.365355	−	0.930868i	$-0.619052\pi$
$347$	−1.63896e6	−0.730711	−0.365355	+	0.930868i	$0.619052\pi$
$348$	0	0
$349$	2.07756e6	0.913040	0.456520	−	0.889713i	$-0.349096\pi$
$349$	2.07756e6	0.913040	0.456520	+	0.889713i	$0.349096\pi$
$350$	0	0
$351$	2.96640e6	1.28517
$352$	0	0
$353$	3.87344e6	1.65447	0.827236	−	0.561854i	$-0.189912\pi$
$353$	3.87344e6	1.65447	0.827236	+	0.561854i	$0.189912\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−331985.	−0.137863
$358$	0	0
$359$	−3.16016e6	−1.29411	−0.647057	−	0.762441i	$-0.724001\pi$
$359$	−3.16016e6	−1.29411	−0.647057	+	0.762441i	$0.724001\pi$
$360$	0	0
$361$	−1.41658e6	−0.572103
$362$	0	0
$363$	686380.	0.273400
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.76875e6	0.685491	0.342745	−	0.939428i	$-0.388643\pi$
$367$	1.76875e6	0.685491	0.342745	+	0.939428i	$0.388643\pi$
$368$	0	0
$369$	−663941.	−0.253842
$370$	0	0
$371$	165087.	0.0622700
$372$	0	0
$373$	3.86744e6	1.43930	0.719651	−	0.694336i	$-0.244302\pi$
$373$	3.86744e6	1.43930	0.719651	+	0.694336i	$0.244302\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.00178e6	0.363009
$378$	0	0
$379$	5.06193e6	1.81016	0.905082	−	0.425236i	$-0.139809\pi$
$379$	5.06193e6	1.81016	0.905082	+	0.425236i	$0.139809\pi$
$380$	0	0
$381$	4.56734e6	1.61195
$382$	0	0
$383$	4.23524e6	1.47530	0.737652	−	0.675181i	$-0.235935\pi$
$383$	4.23524e6	1.47530	0.737652	+	0.675181i	$0.235935\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.82146e6	−0.618219
$388$	0	0
$389$	−390940.	−0.130989	−0.0654947	−	0.997853i	$-0.520863\pi$
$389$	−390940.	−0.130989	−0.0654947	+	0.997853i	$0.520863\pi$
$390$	0	0
$391$	−295542.	−0.0977637
$392$	0	0
$393$	−3.98030e6	−1.29997
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.36908e6	−1.07284	−0.536421	−	0.843951i	$-0.680224\pi$
$397$	−3.36908e6	−1.07284	−0.536421	+	0.843951i	$0.680224\pi$
$398$	0	0
$399$	900861.	0.283286
$400$	0	0
$401$	−5.51542e6	−1.71284	−0.856422	−	0.516277i	$-0.827318\pi$
$401$	−5.51542e6	−1.71284	−0.856422	+	0.516277i	$0.827318\pi$
$402$	0	0
$403$	−1.97433e6	−0.605559
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.13767e6	1.23814
$408$	0	0
$409$	−3.29662e6	−0.974453	−0.487227	−	0.873276i	$-0.661991\pi$
$409$	−3.29662e6	−0.974453	−0.487227	+	0.873276i	$0.661991\pi$
$410$	0	0
$411$	−427503.	−0.124834
$412$	0	0
$413$	−1.08149e6	−0.311996
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.35699e6	1.22700
$418$	0	0
$419$	−6.88088e6	−1.91474	−0.957368	−	0.288870i	$-0.906721\pi$
$419$	−6.88088e6	−1.91474	−0.957368	+	0.288870i	$0.906721\pi$
$420$	0	0
$421$	3.04971e6	0.838596	0.419298	−	0.907849i	$-0.362276\pi$
$421$	3.04971e6	0.838596	0.419298	+	0.907849i	$0.362276\pi$
$422$	0	0
$423$	−625834.	−0.170062
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.20511e6	0.585275
$428$	0	0
$429$	2.99815e6	0.786520
$430$	0	0
$431$	−6.20632e6	−1.60931	−0.804657	−	0.593740i	$-0.797651\pi$
$431$	−6.20632e6	−1.60931	−0.804657	+	0.593740i	$0.797651\pi$
$432$	0	0
$433$	−1.87723e6	−0.481169	−0.240585	−	0.970628i	$-0.577339\pi$
$433$	−1.87723e6	−0.481169	−0.240585	+	0.970628i	$0.577339\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	801971.	0.200888
$438$	0	0
$439$	−1.85883e6	−0.460341	−0.230170	−	0.973150i	$-0.573928\pi$
$439$	−1.85883e6	−0.460341	−0.230170	+	0.973150i	$0.573928\pi$
$440$	0	0
$441$	977079.	0.239240
$442$	0	0
$443$	4.39604e6	1.06427	0.532136	−	0.846659i	$-0.321390\pi$
$443$	4.39604e6	1.06427	0.532136	+	0.846659i	$0.321390\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.04728e6	0.721346
$448$	0	0
$449$	−984885.	−0.230552	−0.115276	−	0.993333i	$-0.536775\pi$
$449$	−984885.	−0.230552	−0.115276	+	0.993333i	$0.536775\pi$
$450$	0	0
$451$	−2.68754e6	−0.622175
$452$	0	0
$453$	2.15844e6	0.494191
$454$	0	0
$455$	0	0
$456$	0	0
$457$	696014.	0.155893	0.0779466	−	0.996958i	$-0.475164\pi$
$457$	696014.	0.155893	0.0779466	+	0.996958i	$0.475164\pi$
$458$	0	0
$459$	1.56429e6	0.346566
$460$	0	0
$461$	6.03623e6	1.32286	0.661430	−	0.750007i	$-0.269950\pi$
$461$	6.03623e6	1.32286	0.661430	+	0.750007i	$0.269950\pi$
$462$	0	0
$463$	2.07793e6	0.450483	0.225241	−	0.974303i	$-0.427683\pi$
$463$	2.07793e6	0.450483	0.225241	+	0.974303i	$0.427683\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.81361e6	0.384816	0.192408	−	0.981315i	$-0.438370\pi$
$467$	1.81361e6	0.384816	0.192408	+	0.981315i	$0.438370\pi$
$468$	0	0
$469$	619222.	0.129991
$470$	0	0
$471$	2.37324e6	0.492935
$472$	0	0
$473$	−7.37300e6	−1.51527
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−194228.	−0.0390854
$478$	0	0
$479$	6.10687e6	1.21613	0.608065	−	0.793887i	$-0.291946\pi$
$479$	6.10687e6	1.21613	0.608065	+	0.793887i	$0.291946\pi$
$480$	0	0
$481$	−9.09273e6	−1.79197
$482$	0	0
$483$	681881.	0.132997
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.32848e6	−0.253823	−0.126912	−	0.991914i	$-0.540506\pi$
$487$	−1.32848e6	−0.253823	−0.126912	+	0.991914i	$0.540506\pi$
$488$	0	0
$489$	5.13222e6	0.970583
$490$	0	0
$491$	−8.72480e6	−1.63325	−0.816623	−	0.577171i	$-0.804157\pi$
$491$	−8.72480e6	−1.63325	−0.816623	+	0.577171i	$0.804157\pi$
$492$	0	0
$493$	528273.	0.0978907
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.01273e6	−0.547104
$498$	0	0
$499$	5.20143e6	0.935129	0.467564	−	0.883959i	$-0.345132\pi$
$499$	5.20143e6	0.935129	0.467564	+	0.883959i	$0.345132\pi$
$500$	0	0
$501$	−2.13448e6	−0.379926
$502$	0	0
$503$	6.44189e6	1.13525	0.567627	−	0.823286i	$-0.307861\pi$
$503$	6.44189e6	1.13525	0.567627	+	0.823286i	$0.307861\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.86083e6	−0.321504
$508$	0	0
$509$	−2.31511e6	−0.396075	−0.198038	−	0.980194i	$-0.563457\pi$
$509$	−2.31511e6	−0.396075	−0.198038	+	0.980194i	$0.563457\pi$
$510$	0	0
$511$	4.52522e6	0.766633
$512$	0	0
$513$	−4.24479e6	−0.712136
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−2.53328e6	−0.416828
$518$	0	0
$519$	1.07061e6	0.174467
$520$	0	0
$521$	−9.65617e6	−1.55851	−0.779257	−	0.626705i	$-0.784403\pi$
$521$	−9.65617e6	−1.55851	−0.779257	+	0.626705i	$0.784403\pi$
$522$	0	0
$523$	6.40583e6	1.02405	0.512025	−	0.858970i	$-0.328895\pi$
$523$	6.40583e6	1.02405	0.512025	+	0.858970i	$0.328895\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.04114e6	−0.163298
$528$	0	0
$529$	−5.82931e6	−0.905687
$530$	0	0
$531$	1.27239e6	0.195833
$532$	0	0
$533$	5.90599e6	0.900481
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.44418e6	1.41328
$538$	0	0
$539$	3.95507e6	0.586384
$540$	0	0
$541$	1.32300e6	0.194342	0.0971709	−	0.995268i	$-0.469021\pi$
$541$	1.32300e6	0.194342	0.0971709	+	0.995268i	$0.469021\pi$
$542$	0	0
$543$	4.37602e6	0.636912
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−4.68044e6	−0.668834	−0.334417	−	0.942425i	$-0.608539\pi$
$547$	−4.68044e6	−0.668834	−0.334417	+	0.942425i	$0.608539\pi$
$548$	0	0
$549$	−2.59434e6	−0.367363
$550$	0	0
$551$	−1.43350e6	−0.201149
$552$	0	0
$553$	−2.72192e6	−0.378498
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.58860e6	−1.03639	−0.518195	−	0.855262i	$-0.673396\pi$
$557$	−7.58860e6	−1.03639	−0.518195	+	0.855262i	$0.673396\pi$
$558$	0	0
$559$	1.62025e7	2.19307
$560$	0	0
$561$	1.58103e6	0.212097
$562$	0	0
$563$	399946.	0.0531777	0.0265889	−	0.999646i	$-0.491536\pi$
$563$	399946.	0.0531777	0.0265889	+	0.999646i	$0.491536\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.25853e6	−0.295032
$568$	0	0
$569$	1.57419e6	0.203834	0.101917	−	0.994793i	$-0.467502\pi$
$569$	1.57419e6	0.203834	0.101917	+	0.994793i	$0.467502\pi$
$570$	0	0
$571$	3.11290e6	0.399554	0.199777	−	0.979841i	$-0.435978\pi$
$571$	3.11290e6	0.399554	0.199777	+	0.979841i	$0.435978\pi$
$572$	0	0
$573$	8.03509e6	1.02236
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.57621e6	0.697267	0.348634	−	0.937259i	$-0.386646\pi$
$577$	5.57621e6	0.697267	0.348634	+	0.937259i	$0.386646\pi$
$578$	0	0
$579$	1.07951e7	1.33823
$580$	0	0
$581$	−4.33716e6	−0.533047
$582$	0	0
$583$	−786205.	−0.0957997
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.11890e7	1.34028	0.670138	−	0.742236i	$-0.266235\pi$
$587$	1.11890e7	1.34028	0.670138	+	0.742236i	$0.266235\pi$
$588$	0	0
$589$	2.82518e6	0.335551
$590$	0	0
$591$	−5.05868e6	−0.595756
$592$	0	0
$593$	−1.44707e7	−1.68987	−0.844934	−	0.534871i	$-0.820360\pi$
$593$	−1.44707e7	−1.68987	−0.844934	+	0.534871i	$0.820360\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.14094e7	1.31017
$598$	0	0
$599$	2.32734e6	0.265028	0.132514	−	0.991181i	$-0.457695\pi$
$599$	2.32734e6	0.265028	0.132514	+	0.991181i	$0.457695\pi$
$600$	0	0
$601$	4.28568e6	0.483987	0.241993	−	0.970278i	$-0.422199\pi$
$601$	4.28568e6	0.483987	0.241993	+	0.970278i	$0.422199\pi$
$602$	0	0
$603$	−728524.	−0.0815925
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.04310e7	−1.14909	−0.574547	−	0.818471i	$-0.694822\pi$
$607$	−1.04310e7	−1.14909	−0.574547	+	0.818471i	$0.694822\pi$
$608$	0	0
$609$	−1.21884e6	−0.133170
$610$	0	0
$611$	5.56701e6	0.603280
$612$	0	0
$613$	−3.52495e6	−0.378880	−0.189440	−	0.981892i	$-0.560667\pi$
$613$	−3.52495e6	−0.378880	−0.189440	+	0.981892i	$0.560667\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.16178e7	−1.22861	−0.614303	−	0.789070i	$-0.710563\pi$
$617$	−1.16178e7	−1.22861	−0.614303	+	0.789070i	$0.710563\pi$
$618$	0	0
$619$	1.04132e7	1.09234	0.546171	−	0.837673i	$-0.316085\pi$
$619$	1.04132e7	1.09234	0.546171	+	0.837673i	$0.316085\pi$
$620$	0	0
$621$	−3.21297e6	−0.334332
$622$	0	0
$623$	2.37204e6	0.244851
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.29022e6	−0.435824
$628$	0	0
$629$	−4.79493e6	−0.483232
$630$	0	0
$631$	−6.57325e6	−0.657214	−0.328607	−	0.944467i	$-0.606579\pi$
$631$	−6.57325e6	−0.657214	−0.328607	+	0.944467i	$0.606579\pi$
$632$	0	0
$633$	−1.11640e7	−1.10741
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−8.69146e6	−0.848680
$638$	0	0
$639$	3.54452e6	0.343404
$640$	0	0
$641$	7.21768e6	0.693829	0.346914	−	0.937897i	$-0.387229\pi$
$641$	7.21768e6	0.693829	0.346914	+	0.937897i	$0.387229\pi$
$642$	0	0
$643$	−989729.	−0.0944036	−0.0472018	−	0.998885i	$-0.515030\pi$
$643$	−989729.	−0.0944036	−0.0472018	+	0.998885i	$0.515030\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.31383e6	0.405138	0.202569	−	0.979268i	$-0.435071\pi$
$647$	4.31383e6	0.405138	0.202569	+	0.979268i	$0.435071\pi$
$648$	0	0
$649$	5.15046e6	0.479992
$650$	0	0
$651$	2.40213e6	0.222149
$652$	0	0
$653$	1.49637e7	1.37327	0.686633	−	0.727004i	$-0.259088\pi$
$653$	1.49637e7	1.37327	0.686633	+	0.727004i	$0.259088\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.32399e6	−0.481198
$658$	0	0
$659$	4.42210e6	0.396657	0.198328	−	0.980136i	$-0.436449\pi$
$659$	4.42210e6	0.396657	0.198328	+	0.980136i	$0.436449\pi$
$660$	0	0
$661$	1.57925e7	1.40587	0.702937	−	0.711252i	$-0.251872\pi$
$661$	1.57925e7	1.40587	0.702937	+	0.711252i	$0.251872\pi$
$662$	0	0
$663$	−3.47440e6	−0.306970
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.08505e6	−0.0944352
$668$	0	0
$669$	−698939.	−0.0603773
$670$	0	0
$671$	−1.05015e7	−0.900420
$672$	0	0
$673$	5.60799e6	0.477276	0.238638	−	0.971109i	$-0.423299\pi$
$673$	5.60799e6	0.477276	0.238638	+	0.971109i	$0.423299\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.01232e6	0.588017	0.294009	−	0.955803i	$-0.405011\pi$
$677$	7.01232e6	0.588017	0.294009	+	0.955803i	$0.405011\pi$
$678$	0	0
$679$	967853.	0.0805629
$680$	0	0
$681$	−1.02289e7	−0.845207
$682$	0	0
$683$	2.16662e7	1.77718	0.888590	−	0.458703i	$-0.151686\pi$
$683$	2.16662e7	1.77718	0.888590	+	0.458703i	$0.151686\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.50681e6	0.606825
$688$	0	0
$689$	1.72772e6	0.138652
$690$	0	0
$691$	−4.20276e6	−0.334842	−0.167421	−	0.985885i	$-0.553544\pi$
$691$	−4.20276e6	−0.334842	−0.167421	+	0.985885i	$0.553544\pi$
$692$	0	0
$693$	1.81936e6	0.143908
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.11444e6	0.242828
$698$	0	0
$699$	1.29939e7	1.00588
$700$	0	0
$701$	−1.50989e6	−0.116051	−0.0580256	−	0.998315i	$-0.518481\pi$
$701$	−1.50989e6	−0.116051	−0.0580256	+	0.998315i	$0.518481\pi$
$702$	0	0
$703$	1.30113e7	0.992963
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.26482e7	0.951653
$708$	0	0
$709$	2.08359e6	0.155667	0.0778336	−	0.996966i	$-0.475200\pi$
$709$	2.08359e6	0.155667	0.0778336	+	0.996966i	$0.475200\pi$
$710$	0	0
$711$	3.20238e6	0.237574
$712$	0	0
$713$	2.13844e6	0.157534
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.10749e7	1.53098
$718$	0	0
$719$	1.03357e7	0.745618	0.372809	−	0.927908i	$-0.378395\pi$
$719$	1.03357e7	0.745618	0.372809	+	0.927908i	$0.378395\pi$
$720$	0	0
$721$	−4.82287e6	−0.345515
$722$	0	0
$723$	1.51561e7	1.07830
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.39227e6	−0.238042	−0.119021	−	0.992892i	$-0.537976\pi$
$727$	−3.39227e6	−0.238042	−0.119021	+	0.992892i	$0.537976\pi$
$728$	0	0
$729$	1.54171e7	1.07444
$730$	0	0
$731$	8.54418e6	0.591394
$732$	0	0
$733$	1.99922e7	1.37436	0.687180	−	0.726487i	$-0.258848\pi$
$733$	1.99922e7	1.37436	0.687180	+	0.726487i	$0.258848\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.94896e6	−0.199986
$738$	0	0
$739$	−2.37515e7	−1.59986	−0.799928	−	0.600096i	$-0.795129\pi$
$739$	−2.37515e7	−1.59986	−0.799928	+	0.600096i	$0.795129\pi$
$740$	0	0
$741$	9.42797e6	0.630773
$742$	0	0
$743$	−768079.	−0.0510427	−0.0255214	−	0.999674i	$-0.508125\pi$
$743$	−768079.	−0.0510427	−0.0255214	+	0.999674i	$0.508125\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.10273e6	0.334581
$748$	0	0
$749$	546628.	0.0356030
$750$	0	0
$751$	−2.34656e7	−1.51821	−0.759105	−	0.650968i	$-0.774363\pi$
$751$	−2.34656e7	−1.51821	−0.759105	+	0.650968i	$0.774363\pi$
$752$	0	0
$753$	−302745.	−0.0194576
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.58118e7	−1.63711	−0.818557	−	0.574425i	$-0.805226\pi$
$757$	−2.58118e7	−1.63711	−0.818557	+	0.574425i	$0.805226\pi$
$758$	0	0
$759$	−3.24736e6	−0.204610
$760$	0	0
$761$	1.19501e7	0.748013	0.374006	−	0.927426i	$-0.377984\pi$
$761$	1.19501e7	0.748013	0.374006	+	0.927426i	$0.377984\pi$
$762$	0	0
$763$	1.15940e7	0.720977
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.13184e7	−0.694698
$768$	0	0
$769$	−1.61907e7	−0.987302	−0.493651	−	0.869660i	$-0.664338\pi$
$769$	−1.61907e7	−0.987302	−0.493651	+	0.869660i	$0.664338\pi$
$770$	0	0
$771$	−4.35070e6	−0.263586
$772$	0	0
$773$	1.40818e7	0.847637	0.423818	−	0.905747i	$-0.360689\pi$
$773$	1.40818e7	0.847637	0.423818	+	0.905747i	$0.360689\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.10630e7	0.657384
$778$	0	0
$779$	−8.45122e6	−0.498972
$780$	0	0
$781$	1.43477e7	0.841695
$782$	0	0
$783$	5.74310e6	0.334766
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.41819e6	0.254277	0.127139	−	0.991885i	$-0.459421\pi$
$787$	4.41819e6	0.254277	0.127139	+	0.991885i	$0.459421\pi$
$788$	0	0
$789$	1.38945e7	0.794602
$790$	0	0
$791$	4.25399e6	0.241744
$792$	0	0
$793$	2.30776e7	1.30319
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.04472e6	−0.337078	−0.168539	−	0.985695i	$-0.553905\pi$
$797$	−6.04472e6	−0.337078	−0.168539	+	0.985695i	$0.553905\pi$
$798$	0	0
$799$	2.93569e6	0.162683
$800$	0	0
$801$	−2.79074e6	−0.153688
$802$	0	0
$803$	−2.15507e7	−1.17943
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.17516e7	0.635202
$808$	0	0
$809$	−1.71556e7	−0.921583	−0.460791	−	0.887509i	$-0.652434\pi$
$809$	−1.71556e7	−0.921583	−0.460791	+	0.887509i	$0.652434\pi$
$810$	0	0
$811$	−1.83020e7	−0.977114	−0.488557	−	0.872532i	$-0.662477\pi$
$811$	−1.83020e7	−0.977114	−0.488557	+	0.872532i	$0.662477\pi$
$812$	0	0
$813$	−1.71009e7	−0.907389
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.31851e7	−1.21522
$818$	0	0
$819$	−3.99813e6	−0.208280
$820$	0	0
$821$	−77887.9	−0.00403285	−0.00201643	−	0.999998i	$-0.500642\pi$
$821$	−77887.9	−0.00403285	−0.00201643	+	0.999998i	$0.500642\pi$
$822$	0	0
$823$	−423648.	−0.0218025	−0.0109012	−	0.999941i	$-0.503470\pi$
$823$	−423648.	−0.0218025	−0.0109012	+	0.999941i	$0.503470\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.18956e6	0.263856	0.131928	−	0.991259i	$-0.457883\pi$
$827$	5.18956e6	0.263856	0.131928	+	0.991259i	$0.457883\pi$
$828$	0	0
$829$	−2.08613e7	−1.05428	−0.527139	−	0.849779i	$-0.676735\pi$
$829$	−2.08613e7	−1.05428	−0.527139	+	0.849779i	$0.676735\pi$
$830$	0	0
$831$	−3.15201e6	−0.158338
$832$	0	0
$833$	−4.58332e6	−0.228859
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.13187e7	−0.558446
$838$	0	0
$839$	8.18798e6	0.401580	0.200790	−	0.979634i	$-0.435649\pi$
$839$	8.18798e6	0.401580	0.200790	+	0.979634i	$0.435649\pi$
$840$	0	0
$841$	−1.85717e7	−0.905442
$842$	0	0
$843$	−1.31900e7	−0.639258
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.70505e6	−0.177454
$848$	0	0
$849$	3.04868e7	1.45158
$850$	0	0
$851$	9.84854e6	0.466174
$852$	0	0
$853$	−3.11924e7	−1.46783	−0.733915	−	0.679241i	$-0.762309\pi$
$853$	−3.11924e7	−1.46783	−0.733915	+	0.679241i	$0.762309\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.88654e7	0.877433	0.438716	−	0.898626i	$-0.355433\pi$
$857$	1.88654e7	0.877433	0.438716	+	0.898626i	$0.355433\pi$
$858$	0	0
$859$	−1.30703e7	−0.604369	−0.302184	−	0.953249i	$-0.597716\pi$
$859$	−1.30703e7	−0.604369	−0.302184	+	0.953249i	$0.597716\pi$
$860$	0	0
$861$	−7.18571e6	−0.330341
$862$	0	0
$863$	5.58115e6	0.255092	0.127546	−	0.991833i	$-0.459290\pi$
$863$	5.58115e6	0.255092	0.127546	+	0.991833i	$0.459290\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.62471e7	0.734056
$868$	0	0
$869$	1.29628e7	0.582302
$870$	0	0
$871$	6.48047e6	0.289442
$872$	0	0
$873$	−1.13869e6	−0.0505675
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.17437e7	−0.954629	−0.477315	−	0.878732i	$-0.658390\pi$
$877$	−2.17437e7	−0.954629	−0.477315	+	0.878732i	$0.658390\pi$
$878$	0	0
$879$	−3.11730e7	−1.36084
$880$	0	0
$881$	2.26789e7	0.984425	0.492212	−	0.870475i	$-0.336188\pi$
$881$	2.26789e7	0.984425	0.492212	+	0.870475i	$0.336188\pi$
$882$	0	0
$883$	−2.34145e7	−1.01061	−0.505304	−	0.862942i	$-0.668619\pi$
$883$	−2.34145e7	−1.01061	−0.505304	+	0.862942i	$0.668619\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.56731e7	−0.668875	−0.334437	−	0.942418i	$-0.608546\pi$
$887$	−1.56731e7	−0.668875	−0.334437	+	0.942418i	$0.608546\pi$
$888$	0	0
$889$	−2.46543e7	−1.04626
$890$	0	0
$891$	1.07559e7	0.453894
$892$	0	0
$893$	−7.96616e6	−0.334288
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.13624e6	0.296134
$898$	0	0
$899$	−3.82240e6	−0.157738
$900$	0	0
$901$	911092.	0.0373895
$902$	0	0
$903$	−1.97133e7	−0.804527
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−3.67553e7	−1.48355	−0.741775	−	0.670649i	$-0.766016\pi$
$907$	−3.67553e7	−1.48355	−0.741775	+	0.670649i	$0.766016\pi$
$908$	0	0
$909$	−1.48807e7	−0.597331
$910$	0	0
$911$	2.11183e7	0.843067	0.421534	−	0.906813i	$-0.361492\pi$
$911$	2.11183e7	0.843067	0.421534	+	0.906813i	$0.361492\pi$
$912$	0	0
$913$	2.06551e7	0.820070
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.14855e7	0.843765
$918$	0	0
$919$	−7.27922e6	−0.284312	−0.142156	−	0.989844i	$-0.545404\pi$
$919$	−7.27922e6	−0.284312	−0.142156	+	0.989844i	$0.545404\pi$
$920$	0	0
$921$	1.19625e7	0.464702
$922$	0	0
$923$	−3.15298e7	−1.21819
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.67418e6	0.216872
$928$	0	0
$929$	−3.41165e7	−1.29696	−0.648478	−	0.761234i	$-0.724594\pi$
$929$	−3.41165e7	−1.29696	−0.648478	+	0.761234i	$0.724594\pi$
$930$	0	0
$931$	1.24371e7	0.470268
$932$	0	0
$933$	−1.45083e7	−0.545649
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.73880e7	1.76327	0.881637	−	0.471929i	$-0.156442\pi$
$937$	4.73880e7	1.76327	0.881637	+	0.471929i	$0.156442\pi$
$938$	0	0
$939$	1.93153e7	0.714887
$940$	0	0
$941$	−3.97476e7	−1.46331	−0.731656	−	0.681674i	$-0.761252\pi$
$941$	−3.97476e7	−1.46331	−0.731656	+	0.681674i	$0.761252\pi$
$942$	0	0
$943$	−6.39691e6	−0.234256
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.76410e7	−1.00156	−0.500782	−	0.865573i	$-0.666954\pi$
$947$	−2.76410e7	−1.00156	−0.500782	+	0.865573i	$0.666954\pi$
$948$	0	0
$949$	4.73588e7	1.70701
$950$	0	0
$951$	3.48233e7	1.24859
$952$	0	0
$953$	5.22977e6	0.186531	0.0932654	−	0.995641i	$-0.470269\pi$
$953$	5.22977e6	0.186531	0.0932654	+	0.995641i	$0.470269\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.80457e6	0.204876
$958$	0	0
$959$	2.30764e6	0.0810255
$960$	0	0
$961$	−2.10959e7	−0.736866
$962$	0	0
$963$	−643115.	−0.0223472
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.76477e7	0.606905	0.303453	−	0.952847i	$-0.401861\pi$
$967$	1.76477e7	0.606905	0.303453	+	0.952847i	$0.401861\pi$
$968$	0	0
$969$	4.97172e6	0.170097
$970$	0	0
$971$	3.66934e7	1.24893	0.624467	−	0.781051i	$-0.285316\pi$
$971$	3.66934e7	1.24893	0.624467	+	0.781051i	$0.285316\pi$
$972$	0	0
$973$	−2.35188e7	−0.796404
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.16023e7	−1.05921	−0.529605	−	0.848244i	$-0.677660\pi$
$977$	−3.16023e7	−1.05921	−0.529605	+	0.848244i	$0.677660\pi$
$978$	0	0
$979$	−1.12965e7	−0.376693
$980$	0	0
$981$	−1.36405e7	−0.452540
$982$	0	0
$983$	−3.73829e7	−1.23393	−0.616963	−	0.786992i	$-0.711637\pi$
$983$	−3.73829e7	−1.23393	−0.616963	+	0.786992i	$0.711637\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.77328e6	−0.221313
$988$	0	0
$989$	−1.75493e7	−0.570518
$990$	0	0
$991$	2.84243e7	0.919401	0.459701	−	0.888074i	$-0.347957\pi$
$991$	2.84243e7	0.919401	0.459701	+	0.888074i	$0.347957\pi$
$992$	0	0
$993$	1.56373e6	0.0503256
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.70600e7	0.543553	0.271776	−	0.962360i	$-0.412389\pi$
$997$	1.70600e7	0.543553	0.271776	+	0.962360i	$0.412389\pi$
$998$	0	0
$999$	−5.21279e7	−1.65256

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.a.l.1.1		2
4.3	odd	2		800.6.a.g.1.2		2
5.2	odd	4		800.6.c.f.449.3		4
5.3	odd	4		800.6.c.f.449.2		4
5.4	even	2		160.6.a.a.1.2	✓	2
20.3	even	4		800.6.c.g.449.3		4
20.7	even	4		800.6.c.g.449.2		4
20.19	odd	2		160.6.a.e.1.1	yes	2
40.19	odd	2		320.6.a.r.1.2		2
40.29	even	2		320.6.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.a.1.2	✓	2	5.4	even	2
160.6.a.e.1.1	yes	2	20.19	odd	2
320.6.a.r.1.2		2	40.19	odd	2
320.6.a.v.1.1		2	40.29	even	2
800.6.a.g.1.2		2	4.3	odd	2
800.6.a.l.1.1		2	1.1	even	1	trivial
800.6.c.f.449.2		4	5.3	odd	4
800.6.c.f.449.3		4	5.2	odd	4
800.6.c.g.449.2		4	20.7	even	4
800.6.c.g.449.3		4	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q-12.7332 q^{3} +68.7332 q^{7} -80.8656 q^{9} +O(q^{10})\)	\(q-12.7332 q^{3} +68.7332 q^{7} -80.8656 q^{9} -327.332 q^{11} +719.328 q^{13} +379.328 q^{17} -1029.33 q^{19} -875.194 q^{21} -779.120 q^{23} +4123.85 q^{27} +1392.66 q^{29} -2744.68 q^{31} +4167.98 q^{33} -12640.6 q^{37} -9159.35 q^{39} +8210.43 q^{41} +22524.5 q^{43} +7739.18 q^{47} -12082.7 q^{49} -4830.06 q^{51} +2401.86 q^{53} +13106.6 q^{57} -15734.7 q^{59} +32082.1 q^{61} -5558.15 q^{63} +9009.07 q^{67} +9920.70 q^{69} -43832.3 q^{71} +65837.5 q^{73} -22498.6 q^{77} -39601.3 q^{79} -32859.4 q^{81} -63101.4 q^{83} -17733.0 q^{87} +34510.9 q^{89} +49441.7 q^{91} +34948.6 q^{93} +14081.3 q^{97} +26469.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{3} + 104 q^{7} + 106 q^{9}+O(q^{10}) \) 2 * q + 8 * q^3 + 104 * q^7 + 106 * q^9	\( 2 q + 8 q^{3} + 104 q^{7} + 106 q^{9} - 320 q^{11} + 100 q^{13} - 580 q^{17} - 720 q^{19} - 144 q^{21} + 1688 q^{23} + 2960 q^{27} + 108 q^{29} - 9840 q^{31} + 4320 q^{33} - 6540 q^{37} - 22000 q^{39} - 10620 q^{41} + 25672 q^{43} + 28296 q^{47} - 27646 q^{49} - 24720 q^{51} - 31340 q^{53} + 19520 q^{57} - 30800 q^{59} + 24540 q^{61} + 1032 q^{63} + 34584 q^{67} + 61072 q^{69} + 12400 q^{71} + 7180 q^{73} - 22240 q^{77} - 71840 q^{79} - 102398 q^{81} - 31928 q^{83} - 44368 q^{87} - 40748 q^{89} + 27600 q^{91} - 112160 q^{93} + 190140 q^{97} + 27840 q^{99}+O(q^{100}) \) 2 * q + 8 * q^3 + 104 * q^7 + 106 * q^9 - 320 * q^11 + 100 * q^13 - 580 * q^17 - 720 * q^19 - 144 * q^21 + 1688 * q^23 + 2960 * q^27 + 108 * q^29 - 9840 * q^31 + 4320 * q^33 - 6540 * q^37 - 22000 * q^39 - 10620 * q^41 + 25672 * q^43 + 28296 * q^47 - 27646 * q^49 - 24720 * q^51 - 31340 * q^53 + 19520 * q^57 - 30800 * q^59 + 24540 * q^61 + 1032 * q^63 + 34584 * q^67 + 61072 * q^69 + 12400 * q^71 + 7180 * q^73 - 22240 * q^77 - 71840 * q^79 - 102398 * q^81 - 31928 * q^83 - 44368 * q^87 - 40748 * q^89 + 27600 * q^91 - 112160 * q^93 + 190140 * q^97 + 27840 * q^99

Introduction

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