LMFDB - Embedded newform 320.6.a.z.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(1,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, 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("320.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.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:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.81998080.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 100x^{2} - 376x - 367 $ x^4 - 100x^2 - 376x - 367
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.92659$ of defining polynomial
Character	$\chi$	$=$	320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0971 q^{3} -25.0000 q^{5} -250.932 q^{7} +491.252 q^{9} +O(q^{10})$	$q+27.0971 q^{3} -25.0000 q^{5} -250.932 q^{7} +491.252 q^{9} +318.107 q^{11} +1076.75 q^{13} -677.427 q^{15} -1120.75 q^{17} +749.388 q^{19} -6799.52 q^{21} +250.932 q^{23} +625.000 q^{25} +6726.89 q^{27} +5626.53 q^{29} +7570.79 q^{31} +8619.76 q^{33} +6273.30 q^{35} -3072.49 q^{37} +29176.9 q^{39} +5390.72 q^{41} -2088.26 q^{43} -12281.3 q^{45} +4076.00 q^{47} +46159.8 q^{49} -30369.2 q^{51} -9027.77 q^{53} -7952.67 q^{55} +20306.2 q^{57} +27481.5 q^{59} +41567.1 q^{61} -123271. q^{63} -26918.9 q^{65} -27782.1 q^{67} +6799.52 q^{69} -14284.9 q^{71} +32642.3 q^{73} +16935.7 q^{75} -79823.1 q^{77} -19864.3 q^{79} +62904.9 q^{81} +76615.8 q^{83} +28018.9 q^{85} +152462. q^{87} +24407.4 q^{89} -270192. q^{91} +205146. q^{93} -18734.7 q^{95} -65973.5 q^{97} +156270. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 100 q^{5} + 724 q^{9}+O(q^{10}) $ 4 * q - 100 * q^5 + 724 * q^9	$ 4 q - 100 q^{5} + 724 q^{9} + 584 q^{13} - 760 q^{17} - 9824 q^{21} + 2500 q^{25} + 168 q^{29} + 25792 q^{33} - 19736 q^{37} + 47624 q^{41} - 18100 q^{45} + 121348 q^{49} - 17496 q^{53} + 99840 q^{57} + 62024 q^{61} - 14600 q^{65} + 9824 q^{69} + 119400 q^{73} - 17728 q^{77} + 103940 q^{81} + 19000 q^{85} + 209320 q^{89} + 532672 q^{93} - 59128 q^{97}+O(q^{100}) $ 4 * q - 100 * q^5 + 724 * q^9 + 584 * q^13 - 760 * q^17 - 9824 * q^21 + 2500 * q^25 + 168 * q^29 + 25792 * q^33 - 19736 * q^37 + 47624 * q^41 - 18100 * q^45 + 121348 * q^49 - 17496 * q^53 + 99840 * q^57 + 62024 * q^61 - 14600 * q^65 + 9824 * q^69 + 119400 * q^73 - 17728 * q^77 + 103940 * q^81 + 19000 * q^85 + 209320 * q^89 + 532672 * q^93 - 59128 * q^97

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
$3$	27.0971	1.73828	0.869139	−	0.494568i	$-0.164674\pi$
$3$	27.0971	1.73828	0.869139	+	0.494568i	$0.164674\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−250.932	−1.93558	−0.967789	−	0.251762i	$-0.918990\pi$
$7$	−250.932	−1.93558	−0.967789	+	0.251762i	$0.918990\pi$
$8$	0	0
$9$	491.252	2.02161
$10$	0	0
$11$	318.107	0.792668	0.396334	−	0.918107i	$-0.370282\pi$
$11$	318.107	0.792668	0.396334	+	0.918107i	$0.370282\pi$
$12$	0	0
$13$	1076.75	1.76709	0.883544	−	0.468348i	$-0.155150\pi$
$13$	1076.75	1.76709	0.883544	+	0.468348i	$0.155150\pi$
$14$	0	0
$15$	−677.427	−0.777382
$16$	0	0
$17$	−1120.75	−0.940564	−0.470282	−	0.882516i	$-0.655848\pi$
$17$	−1120.75	−0.940564	−0.470282	+	0.882516i	$0.655848\pi$
$18$	0	0
$19$	749.388	0.476237	0.238118	−	0.971236i	$-0.423469\pi$
$19$	749.388	0.476237	0.238118	+	0.971236i	$0.423469\pi$
$20$	0	0
$21$	−6799.52	−3.36457
$22$	0	0
$23$	250.932	0.0989091	0.0494546	−	0.998776i	$-0.484252\pi$
$23$	250.932	0.0989091	0.0494546	+	0.998776i	$0.484252\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	6726.89	1.77584
$28$	0	0
$29$	5626.53	1.24235	0.621177	−	0.783670i	$-0.286655\pi$
$29$	5626.53	1.24235	0.621177	+	0.783670i	$0.286655\pi$
$30$	0	0
$31$	7570.79	1.41494	0.707469	−	0.706745i	$-0.249837\pi$
$31$	7570.79	1.41494	0.707469	+	0.706745i	$0.249837\pi$
$32$	0	0
$33$	8619.76	1.37788
$34$	0	0
$35$	6273.30	0.865617
$36$	0	0
$37$	−3072.49	−0.368966	−0.184483	−	0.982836i	$-0.559061\pi$
$37$	−3072.49	−0.368966	−0.184483	+	0.982836i	$0.559061\pi$
$38$	0	0
$39$	29176.9	3.07169
$40$	0	0
$41$	5390.72	0.500826	0.250413	−	0.968139i	$-0.419434\pi$
$41$	5390.72	0.500826	0.250413	+	0.968139i	$0.419434\pi$
$42$	0	0
$43$	−2088.26	−0.172232	−0.0861159	−	0.996285i	$-0.527446\pi$
$43$	−2088.26	−0.172232	−0.0861159	+	0.996285i	$0.527446\pi$
$44$	0	0
$45$	−12281.3	−0.904092
$46$	0	0
$47$	4076.00	0.269147	0.134573	−	0.990904i	$-0.457034\pi$
$47$	4076.00	0.269147	0.134573	+	0.990904i	$0.457034\pi$
$48$	0	0
$49$	46159.8	2.74646
$50$	0	0
$51$	−30369.2	−1.63496
$52$	0	0
$53$	−9027.77	−0.441460	−0.220730	−	0.975335i	$-0.570844\pi$
$53$	−9027.77	−0.441460	−0.220730	+	0.975335i	$0.570844\pi$
$54$	0	0
$55$	−7952.67	−0.354492
$56$	0	0
$57$	20306.2	0.827832
$58$	0	0
$59$	27481.5	1.02781	0.513903	−	0.857848i	$-0.328199\pi$
$59$	27481.5	1.02781	0.513903	+	0.857848i	$0.328199\pi$
$60$	0	0
$61$	41567.1	1.43029	0.715147	−	0.698974i	$-0.246360\pi$
$61$	41567.1	1.43029	0.715147	+	0.698974i	$0.246360\pi$
$62$	0	0
$63$	−123271.	−3.91299
$64$	0	0
$65$	−26918.9	−0.790266
$66$	0	0
$67$	−27782.1	−0.756099	−0.378049	−	0.925785i	$-0.623405\pi$
$67$	−27782.1	−0.756099	−0.378049	+	0.925785i	$0.623405\pi$
$68$	0	0
$69$	6799.52	0.171932
$70$	0	0
$71$	−14284.9	−0.336303	−0.168151	−	0.985761i	$-0.553780\pi$
$71$	−14284.9	−0.336303	−0.168151	+	0.985761i	$0.553780\pi$
$72$	0	0
$73$	32642.3	0.716924	0.358462	−	0.933544i	$-0.383301\pi$
$73$	32642.3	0.716924	0.358462	+	0.933544i	$0.383301\pi$
$74$	0	0
$75$	16935.7	0.347656
$76$	0	0
$77$	−79823.1	−1.53427
$78$	0	0
$79$	−19864.3	−0.358100	−0.179050	−	0.983840i	$-0.557302\pi$
$79$	−19864.3	−0.358100	−0.179050	+	0.983840i	$0.557302\pi$
$80$	0	0
$81$	62904.9	1.06530
$82$	0	0
$83$	76615.8	1.22074	0.610370	−	0.792117i	$-0.291021\pi$
$83$	76615.8	1.22074	0.610370	+	0.792117i	$0.291021\pi$
$84$	0	0
$85$	28018.9	0.420633
$86$	0	0
$87$	152462.	2.15956
$88$	0	0
$89$	24407.4	0.326622	0.163311	−	0.986575i	$-0.447783\pi$
$89$	24407.4	0.326622	0.163311	+	0.986575i	$0.447783\pi$
$90$	0	0
$91$	−270192.	−3.42034
$92$	0	0
$93$	205146.	2.45955
$94$	0	0
$95$	−18734.7	−0.212980
$96$	0	0
$97$	−65973.5	−0.711935	−0.355967	−	0.934498i	$-0.615849\pi$
$97$	−65973.5	−0.711935	−0.355967	+	0.934498i	$0.615849\pi$
$98$	0	0
$99$	156270.	1.60247
$100$	0	0
$101$	89919.4	0.877101	0.438551	−	0.898706i	$-0.355492\pi$
$101$	89919.4	0.877101	0.438551	+	0.898706i	$0.355492\pi$
$102$	0	0
$103$	−64846.3	−0.602271	−0.301136	−	0.953581i	$-0.597366\pi$
$103$	−64846.3	−0.602271	−0.301136	+	0.953581i	$0.597366\pi$
$104$	0	0
$105$	169988.	1.50468
$106$	0	0
$107$	−127808.	−1.07919	−0.539595	−	0.841925i	$-0.681423\pi$
$107$	−127808.	−1.07919	−0.539595	+	0.841925i	$0.681423\pi$
$108$	0	0
$109$	−23209.8	−0.187114	−0.0935569	−	0.995614i	$-0.529824\pi$
$109$	−23209.8	−0.187114	−0.0935569	+	0.995614i	$0.529824\pi$
$110$	0	0
$111$	−83255.5	−0.641365
$112$	0	0
$113$	−106124.	−0.781842	−0.390921	−	0.920424i	$-0.627844\pi$
$113$	−106124.	−0.781842	−0.390921	+	0.920424i	$0.627844\pi$
$114$	0	0
$115$	−6273.30	−0.0442335
$116$	0	0
$117$	528957.	3.57237
$118$	0	0
$119$	281233.	1.82053
$120$	0	0
$121$	−59859.1	−0.371678
$122$	0	0
$123$	146073.	0.870575
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	55034.4	0.302778	0.151389	−	0.988474i	$-0.451625\pi$
$127$	55034.4	0.302778	0.151389	+	0.988474i	$0.451625\pi$
$128$	0	0
$129$	−56585.8	−0.299387
$130$	0	0
$131$	−123085.	−0.626655	−0.313327	−	0.949645i	$-0.601444\pi$
$131$	−123085.	−0.626655	−0.313327	+	0.949645i	$0.601444\pi$
$132$	0	0
$133$	−188045.	−0.921793
$134$	0	0
$135$	−168172.	−0.794182
$136$	0	0
$137$	227197.	1.03419	0.517095	−	0.855928i	$-0.327013\pi$
$137$	227197.	1.03419	0.517095	+	0.855928i	$0.327013\pi$
$138$	0	0
$139$	−223187.	−0.979786	−0.489893	−	0.871783i	$-0.662964\pi$
$139$	−223187.	−0.979786	−0.489893	+	0.871783i	$0.662964\pi$
$140$	0	0
$141$	110448.	0.467852
$142$	0	0
$143$	342523.	1.40071
$144$	0	0
$145$	−140663.	−0.555598
$146$	0	0
$147$	1.25080e6	4.77412
$148$	0	0
$149$	−31091.8	−0.114731	−0.0573655	−	0.998353i	$-0.518270\pi$
$149$	−31091.8	−0.114731	−0.0573655	+	0.998353i	$0.518270\pi$
$150$	0	0
$151$	−325683.	−1.16239	−0.581196	−	0.813764i	$-0.697415\pi$
$151$	−325683.	−1.16239	−0.581196	+	0.813764i	$0.697415\pi$
$152$	0	0
$153$	−550572.	−1.90145
$154$	0	0
$155$	−189270.	−0.632779
$156$	0	0
$157$	67159.8	0.217450	0.108725	−	0.994072i	$-0.465323\pi$
$157$	67159.8	0.217450	0.108725	+	0.994072i	$0.465323\pi$
$158$	0	0
$159$	−244626.	−0.767380
$160$	0	0
$161$	−62966.8	−0.191446
$162$	0	0
$163$	138531.	0.408394	0.204197	−	0.978930i	$-0.434542\pi$
$163$	138531.	0.408394	0.204197	+	0.978930i	$0.434542\pi$
$164$	0	0
$165$	−215494.	−0.616205
$166$	0	0
$167$	314527.	0.872704	0.436352	−	0.899776i	$-0.356270\pi$
$167$	314527.	0.872704	0.436352	+	0.899776i	$0.356270\pi$
$168$	0	0
$169$	788107.	2.12260
$170$	0	0
$171$	368138.	0.962765
$172$	0	0
$173$	578808.	1.47035	0.735173	−	0.677880i	$-0.237101\pi$
$173$	578808.	1.47035	0.735173	+	0.677880i	$0.237101\pi$
$174$	0	0
$175$	−156832.	−0.387116
$176$	0	0
$177$	744670.	1.78661
$178$	0	0
$179$	−279606.	−0.652250	−0.326125	−	0.945327i	$-0.605743\pi$
$179$	−279606.	−0.652250	−0.326125	+	0.945327i	$0.605743\pi$
$180$	0	0
$181$	474917.	1.07751	0.538755	−	0.842463i	$-0.318895\pi$
$181$	474917.	1.07751	0.538755	+	0.842463i	$0.318895\pi$
$182$	0	0
$183$	1.12635e6	2.48625
$184$	0	0
$185$	76812.3	0.165007
$186$	0	0
$187$	−356519.	−0.745554
$188$	0	0
$189$	−1.68799e6	−3.43729
$190$	0	0
$191$	−680793.	−1.35030	−0.675152	−	0.737679i	$-0.735922\pi$
$191$	−680793.	−1.35030	−0.675152	+	0.737679i	$0.735922\pi$
$192$	0	0
$193$	−480051.	−0.927671	−0.463835	−	0.885921i	$-0.653527\pi$
$193$	−480051.	−0.927671	−0.463835	+	0.885921i	$0.653527\pi$
$194$	0	0
$195$	−729422.	−1.37370
$196$	0	0
$197$	−928980.	−1.70546	−0.852728	−	0.522355i	$-0.825054\pi$
$197$	−928980.	−1.70546	−0.852728	+	0.522355i	$0.825054\pi$
$198$	0	0
$199$	−141359.	−0.253040	−0.126520	−	0.991964i	$-0.540381\pi$
$199$	−141359.	−0.253040	−0.126520	+	0.991964i	$0.540381\pi$
$200$	0	0
$201$	−752815.	−1.31431
$202$	0	0
$203$	−1.41188e6	−2.40467
$204$	0	0
$205$	−134768.	−0.223976
$206$	0	0
$207$	123271.	0.199956
$208$	0	0
$209$	238385.	0.377497
$210$	0	0
$211$	−1.08461e6	−1.67713	−0.838567	−	0.544799i	$-0.816606\pi$
$211$	−1.08461e6	−1.67713	−0.838567	+	0.544799i	$0.816606\pi$
$212$	0	0
$213$	−387078.	−0.584588
$214$	0	0
$215$	52206.5	0.0770244
$216$	0	0
$217$	−1.89975e6	−2.73872
$218$	0	0
$219$	884510.	1.24621
$220$	0	0
$221$	−1.20678e6	−1.66206
$222$	0	0
$223$	−430800.	−0.580114	−0.290057	−	0.957009i	$-0.593674\pi$
$223$	−430800.	−0.580114	−0.290057	+	0.957009i	$0.593674\pi$
$224$	0	0
$225$	307032.	0.404322
$226$	0	0
$227$	524.734	0.000675888 0	0.000337944	−	1.00000i	$-0.499892\pi$
$227$	524.734	0.000675888 0	0.000337944		1.00000i	$0.499892\pi$
$228$	0	0
$229$	563646.	0.710260	0.355130	−	0.934817i	$-0.384437\pi$
$229$	563646.	0.710260	0.355130	+	0.934817i	$0.384437\pi$
$230$	0	0
$231$	−2.16297e6	−2.66699
$232$	0	0
$233$	1.17201e6	1.41431	0.707153	−	0.707061i	$-0.249979\pi$
$233$	1.17201e6	1.41431	0.707153	+	0.707061i	$0.249979\pi$
$234$	0	0
$235$	−101900.	−0.120366
$236$	0	0
$237$	−538263.	−0.622478
$238$	0	0
$239$	−1.34048e6	−1.51798	−0.758990	−	0.651102i	$-0.774307\pi$
$239$	−1.34048e6	−1.51798	−0.758990	+	0.651102i	$0.774307\pi$
$240$	0	0
$241$	1.44553e6	1.60318	0.801592	−	0.597871i	$-0.203987\pi$
$241$	1.44553e6	1.60318	0.801592	+	0.597871i	$0.203987\pi$
$242$	0	0
$243$	69905.4	0.0759443
$244$	0	0
$245$	−1.15400e6	−1.22826
$246$	0	0
$247$	806907.	0.841552
$248$	0	0
$249$	2.07606e6	2.12198
$250$	0	0
$251$	1.73066e6	1.73391	0.866957	−	0.498383i	$-0.166073\pi$
$251$	1.73066e6	1.73391	0.866957	+	0.498383i	$0.166073\pi$
$252$	0	0
$253$	79823.1	0.0784020
$254$	0	0
$255$	759229.	0.731177
$256$	0	0
$257$	−1.46437e6	−1.38298	−0.691492	−	0.722384i	$-0.743046\pi$
$257$	−1.46437e6	−1.38298	−0.691492	+	0.722384i	$0.743046\pi$
$258$	0	0
$259$	770986.	0.714163
$260$	0	0
$261$	2.76404e6	2.51156
$262$	0	0
$263$	−1.06316e6	−0.947787	−0.473894	−	0.880582i	$-0.657152\pi$
$263$	−1.06316e6	−0.947787	−0.473894	+	0.880582i	$0.657152\pi$
$264$	0	0
$265$	225694.	0.197427
$266$	0	0
$267$	661368.	0.567760
$268$	0	0
$269$	−1.46767e6	−1.23665	−0.618327	−	0.785921i	$-0.712189\pi$
$269$	−1.46767e6	−1.23665	−0.618327	+	0.785921i	$0.712189\pi$
$270$	0	0
$271$	2.27316e6	1.88021	0.940104	−	0.340887i	$-0.110727\pi$
$271$	2.27316e6	1.88021	0.940104	+	0.340887i	$0.110727\pi$
$272$	0	0
$273$	−7.32142e6	−5.94550
$274$	0	0
$275$	198817.	0.158534
$276$	0	0
$277$	499874.	0.391436	0.195718	−	0.980660i	$-0.437296\pi$
$277$	499874.	0.391436	0.195718	+	0.980660i	$0.437296\pi$
$278$	0	0
$279$	3.71916e6	2.86045
$280$	0	0
$281$	782548.	0.591215	0.295607	−	0.955310i	$-0.404478\pi$
$281$	782548.	0.591215	0.295607	+	0.955310i	$0.404478\pi$
$282$	0	0
$283$	1.77627e6	1.31839	0.659194	−	0.751973i	$-0.270897\pi$
$283$	1.77627e6	1.31839	0.659194	+	0.751973i	$0.270897\pi$
$284$	0	0
$285$	−507656.	−0.370218
$286$	0	0
$287$	−1.35270e6	−0.969388
$288$	0	0
$289$	−163766.	−0.115340
$290$	0	0
$291$	−1.78769e6	−1.23754
$292$	0	0
$293$	37135.6	0.0252709	0.0126355	−	0.999920i	$-0.495978\pi$
$293$	37135.6	0.0252709	0.0126355	+	0.999920i	$0.495978\pi$
$294$	0	0
$295$	−687039.	−0.459649
$296$	0	0
$297$	2.13987e6	1.40765
$298$	0	0
$299$	270192.	0.174781
$300$	0	0
$301$	524011.	0.333368
$302$	0	0
$303$	2.43655e6	1.52465
$304$	0	0
$305$	−1.03918e6	−0.639647
$306$	0	0
$307$	−442910.	−0.268206	−0.134103	−	0.990967i	$-0.542815\pi$
$307$	−442910.	−0.268206	−0.134103	+	0.990967i	$0.542815\pi$
$308$	0	0
$309$	−1.75715e6	−1.04692
$310$	0	0
$311$	−261003.	−0.153019	−0.0765095	−	0.997069i	$-0.524378\pi$
$311$	−261003.	−0.153019	−0.0765095	+	0.997069i	$0.524378\pi$
$312$	0	0
$313$	−673617.	−0.388644	−0.194322	−	0.980938i	$-0.562251\pi$
$313$	−673617.	−0.388644	−0.194322	+	0.980938i	$0.562251\pi$
$314$	0	0
$315$	3.08177e6	1.74994
$316$	0	0
$317$	965919.	0.539874	0.269937	−	0.962878i	$-0.412997\pi$
$317$	965919.	0.539874	0.269937	+	0.962878i	$0.412997\pi$
$318$	0	0
$319$	1.78984e6	0.984774
$320$	0	0
$321$	−3.46322e6	−1.87593
$322$	0	0
$323$	−839880.	−0.447931
$324$	0	0
$325$	672972.	0.353418
$326$	0	0
$327$	−628919.	−0.325256
$328$	0	0
$329$	−1.02280e6	−0.520955
$330$	0	0
$331$	−1.93523e6	−0.970875	−0.485438	−	0.874271i	$-0.661340\pi$
$331$	−1.93523e6	−0.970875	−0.485438	+	0.874271i	$0.661340\pi$
$332$	0	0
$333$	−1.50937e6	−0.745906
$334$	0	0
$335$	694553.	0.338138
$336$	0	0
$337$	1.83525e6	0.880279	0.440139	−	0.897930i	$-0.354929\pi$
$337$	1.83525e6	0.880279	0.440139	+	0.897930i	$0.354929\pi$
$338$	0	0
$339$	−2.87566e6	−1.35906
$340$	0	0
$341$	2.40832e6	1.12157
$342$	0	0
$343$	−7.36556e6	−3.38042
$344$	0	0
$345$	−169988.	−0.0768901
$346$	0	0
$347$	−2.82770e6	−1.26069	−0.630346	−	0.776314i	$-0.717087\pi$
$347$	−2.82770e6	−1.26069	−0.630346	+	0.776314i	$0.717087\pi$
$348$	0	0
$349$	1.02951e6	0.452445	0.226223	−	0.974076i	$-0.427362\pi$
$349$	1.02951e6	0.452445	0.226223	+	0.974076i	$0.427362\pi$
$350$	0	0
$351$	7.24321e6	3.13807
$352$	0	0
$353$	2.36405e6	1.00977	0.504883	−	0.863188i	$-0.331536\pi$
$353$	2.36405e6	1.00977	0.504883	+	0.863188i	$0.331536\pi$
$354$	0	0
$355$	357122.	0.150399
$356$	0	0
$357$	7.62059e6	3.16460
$358$	0	0
$359$	−2.56931e6	−1.05216	−0.526078	−	0.850436i	$-0.676338\pi$
$359$	−2.56931e6	−1.05216	−0.526078	+	0.850436i	$0.676338\pi$
$360$	0	0
$361$	−1.91452e6	−0.773199
$362$	0	0
$363$	−1.62201e6	−0.646080
$364$	0	0
$365$	−816057.	−0.320618
$366$	0	0
$367$	−1.66700e6	−0.646055	−0.323027	−	0.946390i	$-0.604701\pi$
$367$	−1.66700e6	−0.646055	−0.323027	+	0.946390i	$0.604701\pi$
$368$	0	0
$369$	2.64820e6	1.01248
$370$	0	0
$371$	2.26536e6	0.854480
$372$	0	0
$373$	4.20579e6	1.56522	0.782611	−	0.622511i	$-0.213887\pi$
$373$	4.20579e6	1.56522	0.782611	+	0.622511i	$0.213887\pi$
$374$	0	0
$375$	−423392.	−0.155476
$376$	0	0
$377$	6.05839e6	2.19535
$378$	0	0
$379$	2.96494e6	1.06027	0.530136	−	0.847913i	$-0.322141\pi$
$379$	2.96494e6	1.06027	0.530136	+	0.847913i	$0.322141\pi$
$380$	0	0
$381$	1.49127e6	0.526313
$382$	0	0
$383$	−2.78837e6	−0.971301	−0.485650	−	0.874153i	$-0.661417\pi$
$383$	−2.78837e6	−0.971301	−0.485650	+	0.874153i	$0.661417\pi$
$384$	0	0
$385$	1.99558e6	0.686147
$386$	0	0
$387$	−1.02586e6	−0.348186
$388$	0	0
$389$	−4.84640e6	−1.62385	−0.811924	−	0.583763i	$-0.801580\pi$
$389$	−4.84640e6	−1.62385	−0.811924	+	0.583763i	$0.801580\pi$
$390$	0	0
$391$	−281233.	−0.0930303
$392$	0	0
$393$	−3.33525e6	−1.08930
$394$	0	0
$395$	496607.	0.160147
$396$	0	0
$397$	−2.08924e6	−0.665291	−0.332646	−	0.943052i	$-0.607941\pi$
$397$	−2.08924e6	−0.665291	−0.332646	+	0.943052i	$0.607941\pi$
$398$	0	0
$399$	−5.09548e6	−1.60233
$400$	0	0
$401$	5.72071e6	1.77660	0.888298	−	0.459268i	$-0.151888\pi$
$401$	5.72071e6	1.77660	0.888298	+	0.459268i	$0.151888\pi$
$402$	0	0
$403$	8.15189e6	2.50032
$404$	0	0
$405$	−1.57262e6	−0.476417
$406$	0	0
$407$	−977380.	−0.292467
$408$	0	0
$409$	−437627.	−0.129359	−0.0646794	−	0.997906i	$-0.520602\pi$
$409$	−437627.	−0.129359	−0.0646794	+	0.997906i	$0.520602\pi$
$410$	0	0
$411$	6.15637e6	1.79771
$412$	0	0
$413$	−6.89600e6	−1.98940
$414$	0	0
$415$	−1.91539e6	−0.545931
$416$	0	0
$417$	−6.04770e6	−1.70314
$418$	0	0
$419$	−5.01226e6	−1.39476	−0.697379	−	0.716703i	$-0.745651\pi$
$419$	−5.01226e6	−1.39476	−0.697379	+	0.716703i	$0.745651\pi$
$420$	0	0
$421$	948769.	0.260889	0.130444	−	0.991456i	$-0.458360\pi$
$421$	948769.	0.260889	0.130444	+	0.991456i	$0.458360\pi$
$422$	0	0
$423$	2.00234e6	0.544110
$424$	0	0
$425$	−700472.	−0.188113
$426$	0	0
$427$	−1.04305e7	−2.76845
$428$	0	0
$429$	9.28137e6	2.43483
$430$	0	0
$431$	5.33719e6	1.38395	0.691974	−	0.721923i	$-0.256741\pi$
$431$	5.33719e6	1.38395	0.691974	+	0.721923i	$0.256741\pi$
$432$	0	0
$433$	−3.68308e6	−0.944041	−0.472021	−	0.881588i	$-0.656475\pi$
$433$	−3.68308e6	−0.944041	−0.472021	+	0.881588i	$0.656475\pi$
$434$	0	0
$435$	−3.81156e6	−0.965783
$436$	0	0
$437$	188045.	0.0471041
$438$	0	0
$439$	−3.20149e6	−0.792849	−0.396424	−	0.918067i	$-0.629749\pi$
$439$	−3.20149e6	−0.792849	−0.396424	+	0.918067i	$0.629749\pi$
$440$	0	0
$441$	2.26761e7	5.55228
$442$	0	0
$443$	−2.19675e6	−0.531828	−0.265914	−	0.963997i	$-0.585674\pi$
$443$	−2.19675e6	−0.531828	−0.265914	+	0.963997i	$0.585674\pi$
$444$	0	0
$445$	−610184.	−0.146070
$446$	0	0
$447$	−842498.	−0.199434
$448$	0	0
$449$	2.79059e6	0.653251	0.326626	−	0.945154i	$-0.394088\pi$
$449$	2.79059e6	0.653251	0.326626	+	0.945154i	$0.394088\pi$
$450$	0	0
$451$	1.71482e6	0.396989
$452$	0	0
$453$	−8.82505e6	−2.02056
$454$	0	0
$455$	6.75480e6	1.52962
$456$	0	0
$457$	4.73567e6	1.06070	0.530348	−	0.847780i	$-0.322061\pi$
$457$	4.73567e6	1.06070	0.530348	+	0.847780i	$0.322061\pi$
$458$	0	0
$459$	−7.53919e6	−1.67029
$460$	0	0
$461$	2.07704e6	0.455190	0.227595	−	0.973756i	$-0.426914\pi$
$461$	2.07704e6	0.455190	0.227595	+	0.973756i	$0.426914\pi$
$462$	0	0
$463$	7.07901e6	1.53469	0.767344	−	0.641236i	$-0.221578\pi$
$463$	7.07901e6	1.53469	0.767344	+	0.641236i	$0.221578\pi$
$464$	0	0
$465$	−5.12866e6	−1.09995
$466$	0	0
$467$	−2.56203e6	−0.543615	−0.271808	−	0.962352i	$-0.587621\pi$
$467$	−2.56203e6	−0.543615	−0.271808	+	0.962352i	$0.587621\pi$
$468$	0	0
$469$	6.97142e6	1.46349
$470$	0	0
$471$	1.81983e6	0.377989
$472$	0	0
$473$	−664290.	−0.136523
$474$	0	0
$475$	468368.	0.0952473
$476$	0	0
$477$	−4.43491e6	−0.892460
$478$	0	0
$479$	−9.23584e6	−1.83924	−0.919619	−	0.392812i	$-0.871502\pi$
$479$	−9.23584e6	−1.83924	−0.919619	+	0.392812i	$0.871502\pi$
$480$	0	0
$481$	−3.30832e6	−0.651996
$482$	0	0
$483$	−1.70622e6	−0.332787
$484$	0	0
$485$	1.64934e6	0.318387
$486$	0	0
$487$	3.58613e6	0.685178	0.342589	−	0.939485i	$-0.388696\pi$
$487$	3.58613e6	0.685178	0.342589	+	0.939485i	$0.388696\pi$
$488$	0	0
$489$	3.75380e6	0.709903
$490$	0	0
$491$	4.76716e6	0.892393	0.446197	−	0.894935i	$-0.352778\pi$
$491$	4.76716e6	0.892393	0.446197	+	0.894935i	$0.352778\pi$
$492$	0	0
$493$	−6.30596e6	−1.16851
$494$	0	0
$495$	−3.90676e6	−0.716644
$496$	0	0
$497$	3.58453e6	0.650940
$498$	0	0
$499$	1.13248e6	0.203601	0.101801	−	0.994805i	$-0.467540\pi$
$499$	1.13248e6	0.203601	0.101801	+	0.994805i	$0.467540\pi$
$500$	0	0
$501$	8.52277e6	1.51700
$502$	0	0
$503$	6.63918e6	1.17002	0.585012	−	0.811025i	$-0.301090\pi$
$503$	6.63918e6	1.17002	0.585012	+	0.811025i	$0.301090\pi$
$504$	0	0
$505$	−2.24798e6	−0.392252
$506$	0	0
$507$	2.13554e7	3.68967
$508$	0	0
$509$	−9.86315e6	−1.68741	−0.843706	−	0.536805i	$-0.819631\pi$
$509$	−9.86315e6	−1.68741	−0.843706	+	0.536805i	$0.819631\pi$
$510$	0	0
$511$	−8.19099e6	−1.38766
$512$	0	0
$513$	5.04105e6	0.845722
$514$	0	0
$515$	1.62116e6	0.269344
$516$	0	0
$517$	1.29660e6	0.213344
$518$	0	0
$519$	1.56840e7	2.55587
$520$	0	0
$521$	−5.38917e6	−0.869817	−0.434908	−	0.900475i	$-0.643219\pi$
$521$	−5.38917e6	−0.869817	−0.434908	+	0.900475i	$0.643219\pi$
$522$	0	0
$523$	−1.18840e7	−1.89981	−0.949905	−	0.312539i	$-0.898821\pi$
$523$	−1.18840e7	−1.89981	−0.949905	+	0.312539i	$0.898821\pi$
$524$	0	0
$525$	−4.24970e6	−0.672915
$526$	0	0
$527$	−8.48500e6	−1.33084
$528$	0	0
$529$	−6.37338e6	−0.990217
$530$	0	0
$531$	1.35004e7	2.07782
$532$	0	0
$533$	5.80448e6	0.885004
$534$	0	0
$535$	3.19520e6	0.482629
$536$	0	0
$537$	−7.57651e6	−1.13379
$538$	0	0
$539$	1.46837e7	2.17703
$540$	0	0
$541$	8.38145e6	1.23119	0.615596	−	0.788062i	$-0.288915\pi$
$541$	8.38145e6	1.23119	0.615596	+	0.788062i	$0.288915\pi$
$542$	0	0
$543$	1.28689e7	1.87301
$544$	0	0
$545$	580246.	0.0836798
$546$	0	0
$547$	5.58605e6	0.798246	0.399123	−	0.916897i	$-0.369315\pi$
$547$	5.58605e6	0.798246	0.399123	+	0.916897i	$0.369315\pi$
$548$	0	0
$549$	2.04199e7	2.89150
$550$	0	0
$551$	4.21645e6	0.591655
$552$	0	0
$553$	4.98458e6	0.693131
$554$	0	0
$555$	2.08139e6	0.286827
$556$	0	0
$557$	−6.08231e6	−0.830674	−0.415337	−	0.909668i	$-0.636336\pi$
$557$	−6.08231e6	−0.830674	−0.415337	+	0.909668i	$0.636336\pi$
$558$	0	0
$559$	−2.24854e6	−0.304349
$560$	0	0
$561$	−9.66064e6	−1.29598
$562$	0	0
$563$	7.97762e6	1.06072	0.530362	−	0.847771i	$-0.322056\pi$
$563$	7.97762e6	1.06072	0.530362	+	0.847771i	$0.322056\pi$
$564$	0	0
$565$	2.65311e6	0.349651
$566$	0	0
$567$	−1.57849e7	−2.06197
$568$	0	0
$569$	−3.20355e6	−0.414812	−0.207406	−	0.978255i	$-0.566502\pi$
$569$	−3.20355e6	−0.414812	−0.207406	+	0.978255i	$0.566502\pi$
$570$	0	0
$571$	−1.44485e7	−1.85453	−0.927264	−	0.374408i	$-0.877846\pi$
$571$	−1.44485e7	−1.85453	−0.927264	+	0.374408i	$0.877846\pi$
$572$	0	0
$573$	−1.84475e7	−2.34720
$574$	0	0
$575$	156832.	0.0197818
$576$	0	0
$577$	4.96262e6	0.620542	0.310271	−	0.950648i	$-0.399580\pi$
$577$	4.96262e6	0.620542	0.310271	+	0.950648i	$0.399580\pi$
$578$	0	0
$579$	−1.30080e7	−1.61255
$580$	0	0
$581$	−1.92253e7	−2.36284
$582$	0	0
$583$	−2.87179e6	−0.349931
$584$	0	0
$585$	−1.32239e7	−1.59761
$586$	0	0
$587$	−1.25528e7	−1.50365	−0.751824	−	0.659363i	$-0.770826\pi$
$587$	−1.25528e7	−1.50365	−0.751824	+	0.659363i	$0.770826\pi$
$588$	0	0
$589$	5.67346e6	0.673845
$590$	0	0
$591$	−2.51726e7	−2.96456
$592$	0	0
$593$	−871194.	−0.101737	−0.0508684	−	0.998705i	$-0.516199\pi$
$593$	−871194.	−0.101737	−0.0508684	+	0.998705i	$0.516199\pi$
$594$	0	0
$595$	−7.03083e6	−0.814168
$596$	0	0
$597$	−3.83041e6	−0.439854
$598$	0	0
$599$	9.77951e6	1.11365	0.556827	−	0.830629i	$-0.312019\pi$
$599$	9.77951e6	1.11365	0.556827	+	0.830629i	$0.312019\pi$
$600$	0	0
$601$	1.24588e7	1.40698	0.703492	−	0.710704i	$-0.251623\pi$
$601$	1.24588e7	1.40698	0.703492	+	0.710704i	$0.251623\pi$
$602$	0	0
$603$	−1.36480e7	−1.52854
$604$	0	0
$605$	1.49648e6	0.166220
$606$	0	0
$607$	1.66989e7	1.83957	0.919783	−	0.392427i	$-0.128365\pi$
$607$	1.66989e7	1.83957	0.919783	+	0.392427i	$0.128365\pi$
$608$	0	0
$609$	−3.82577e7	−4.17999
$610$	0	0
$611$	4.38885e6	0.475607
$612$	0	0
$613$	−1.26676e6	−0.136159	−0.0680793	−	0.997680i	$-0.521687\pi$
$613$	−1.26676e6	−0.136159	−0.0680793	+	0.997680i	$0.521687\pi$
$614$	0	0
$615$	−3.65182e6	−0.389333
$616$	0	0
$617$	−1.70474e6	−0.180279	−0.0901393	−	0.995929i	$-0.528731\pi$
$617$	−1.70474e6	−0.180279	−0.0901393	+	0.995929i	$0.528731\pi$
$618$	0	0
$619$	1.01627e7	1.06606	0.533029	−	0.846097i	$-0.321054\pi$
$619$	1.01627e7	1.06606	0.533029	+	0.846097i	$0.321054\pi$
$620$	0	0
$621$	1.68799e6	0.175647
$622$	0	0
$623$	−6.12459e6	−0.632203
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	6.45955e6	0.656195
$628$	0	0
$629$	3.44351e6	0.347036
$630$	0	0
$631$	−1.02073e7	−1.02056	−0.510279	−	0.860009i	$-0.670458\pi$
$631$	−1.02073e7	−1.02056	−0.510279	+	0.860009i	$0.670458\pi$
$632$	0	0
$633$	−2.93898e7	−2.91533
$634$	0	0
$635$	−1.37586e6	−0.135407
$636$	0	0
$637$	4.97028e7	4.85325
$638$	0	0
$639$	−7.01746e6	−0.679873
$640$	0	0
$641$	1.45144e7	1.39526	0.697630	−	0.716458i	$-0.254238\pi$
$641$	1.45144e7	1.39526	0.697630	+	0.716458i	$0.254238\pi$
$642$	0	0
$643$	−6.90150e6	−0.658288	−0.329144	−	0.944280i	$-0.606760\pi$
$643$	−6.90150e6	−0.658288	−0.329144	+	0.944280i	$0.606760\pi$
$644$	0	0
$645$	1.41464e6	0.133890
$646$	0	0
$647$	1.22382e7	1.14937	0.574683	−	0.818376i	$-0.305125\pi$
$647$	1.22382e7	1.14937	0.574683	+	0.818376i	$0.305125\pi$
$648$	0	0
$649$	8.74206e6	0.814708
$650$	0	0
$651$	−5.14778e7	−4.76066
$652$	0	0
$653$	−1.96282e7	−1.80134	−0.900672	−	0.434499i	$-0.856926\pi$
$653$	−1.96282e7	−1.80134	−0.900672	+	0.434499i	$0.856926\pi$
$654$	0	0
$655$	3.07713e6	0.280248
$656$	0	0
$657$	1.60356e7	1.44934
$658$	0	0
$659$	−2.18259e6	−0.195776	−0.0978880	−	0.995197i	$-0.531209\pi$
$659$	−2.18259e6	−0.195776	−0.0978880	+	0.995197i	$0.531209\pi$
$660$	0	0
$661$	−7.44654e6	−0.662904	−0.331452	−	0.943472i	$-0.607538\pi$
$661$	−7.44654e6	−0.662904	−0.331452	+	0.943472i	$0.607538\pi$
$662$	0	0
$663$	−3.27001e7	−2.88912
$664$	0	0
$665$	4.70114e6	0.412239
$666$	0	0
$667$	1.41188e6	0.122880
$668$	0	0
$669$	−1.16734e7	−1.00840
$670$	0	0
$671$	1.32228e7	1.13375
$672$	0	0
$673$	−1.34404e7	−1.14387	−0.571933	−	0.820300i	$-0.693806\pi$
$673$	−1.34404e7	−1.14387	−0.571933	+	0.820300i	$0.693806\pi$
$674$	0	0
$675$	4.20431e6	0.355169
$676$	0	0
$677$	−6.23014e6	−0.522428	−0.261214	−	0.965281i	$-0.584123\pi$
$677$	−6.23014e6	−0.522428	−0.261214	+	0.965281i	$0.584123\pi$
$678$	0	0
$679$	1.65549e7	1.37801
$680$	0	0
$681$	14218.8	0.00117488
$682$	0	0
$683$	1.02607e6	0.0841635	0.0420817	−	0.999114i	$-0.486601\pi$
$683$	1.02607e6	0.0841635	0.0420817	+	0.999114i	$0.486601\pi$
$684$	0	0
$685$	−5.67992e6	−0.462504
$686$	0	0
$687$	1.52732e7	1.23463
$688$	0	0
$689$	−9.72070e6	−0.780098
$690$	0	0
$691$	1.46823e7	1.16976	0.584881	−	0.811119i	$-0.301141\pi$
$691$	1.46823e7	1.16976	0.584881	+	0.811119i	$0.301141\pi$
$692$	0	0
$693$	−3.92132e7	−3.10170
$694$	0	0
$695$	5.57966e6	0.438173
$696$	0	0
$697$	−6.04167e6	−0.471059
$698$	0	0
$699$	3.17582e7	2.45846
$700$	0	0
$701$	4.25346e6	0.326924	0.163462	−	0.986550i	$-0.447734\pi$
$701$	4.25346e6	0.326924	0.163462	+	0.986550i	$0.447734\pi$
$702$	0	0
$703$	−2.30249e6	−0.175715
$704$	0	0
$705$	−2.76119e6	−0.209230
$706$	0	0
$707$	−2.25636e7	−1.69770
$708$	0	0
$709$	−8.46830e6	−0.632675	−0.316337	−	0.948647i	$-0.602453\pi$
$709$	−8.46830e6	−0.632675	−0.316337	+	0.948647i	$0.602453\pi$
$710$	0	0
$711$	−9.75835e6	−0.723940
$712$	0	0
$713$	1.89975e6	0.139950
$714$	0	0
$715$	−8.56307e6	−0.626418
$716$	0	0
$717$	−3.63231e7	−2.63867
$718$	0	0
$719$	245933.	0.0177417	0.00887083	−	0.999961i	$-0.497176\pi$
$719$	245933.	0.0177417	0.00887083	+	0.999961i	$0.497176\pi$
$720$	0	0
$721$	1.62720e7	1.16574
$722$	0	0
$723$	3.91695e7	2.78678
$724$	0	0
$725$	3.51658e6	0.248471
$726$	0	0
$727$	1.48936e7	1.04511	0.522556	−	0.852605i	$-0.324979\pi$
$727$	1.48936e7	1.04511	0.522556	+	0.852605i	$0.324979\pi$
$728$	0	0
$729$	−1.33917e7	−0.933288
$730$	0	0
$731$	2.34043e6	0.161995
$732$	0	0
$733$	1.11220e7	0.764580	0.382290	−	0.924042i	$-0.375136\pi$
$733$	1.11220e7	0.764580	0.382290	+	0.924042i	$0.375136\pi$
$734$	0	0
$735$	−3.12699e7	−2.13505
$736$	0	0
$737$	−8.83768e6	−0.599335
$738$	0	0
$739$	2.15121e7	1.44901	0.724505	−	0.689270i	$-0.242069\pi$
$739$	2.15121e7	1.44901	0.724505	+	0.689270i	$0.242069\pi$
$740$	0	0
$741$	2.18648e7	1.46285
$742$	0	0
$743$	−2.59551e7	−1.72485	−0.862424	−	0.506187i	$-0.831055\pi$
$743$	−2.59551e7	−1.72485	−0.862424	+	0.506187i	$0.831055\pi$
$744$	0	0
$745$	777296.	0.0513093
$746$	0	0
$747$	3.76376e7	2.46786
$748$	0	0
$749$	3.20711e7	2.08886
$750$	0	0
$751$	−37924.6	−0.00245370	−0.00122685	−	0.999999i	$-0.500391\pi$
$751$	−37924.6	−0.00245370	−0.00122685	+	0.999999i	$0.500391\pi$
$752$	0	0
$753$	4.68958e7	3.01402
$754$	0	0
$755$	8.14207e6	0.519837
$756$	0	0
$757$	−1.38963e7	−0.881374	−0.440687	−	0.897661i	$-0.645265\pi$
$757$	−1.38963e7	−0.881374	−0.440687	+	0.897661i	$0.645265\pi$
$758$	0	0
$759$	2.16297e6	0.136285
$760$	0	0
$761$	−2.34668e7	−1.46890	−0.734449	−	0.678664i	$-0.762559\pi$
$761$	−2.34668e7	−1.46890	−0.734449	+	0.678664i	$0.762559\pi$
$762$	0	0
$763$	5.82409e6	0.362173
$764$	0	0
$765$	1.37643e7	0.850356
$766$	0	0
$767$	2.95909e7	1.81622
$768$	0	0
$769$	1.71429e7	1.04537	0.522683	−	0.852527i	$-0.324931\pi$
$769$	1.71429e7	1.04537	0.522683	+	0.852527i	$0.324931\pi$
$770$	0	0
$771$	−3.96801e7	−2.40401
$772$	0	0
$773$	−1.29619e7	−0.780224	−0.390112	−	0.920767i	$-0.627564\pi$
$773$	−1.29619e7	−0.780224	−0.390112	+	0.920767i	$0.627564\pi$
$774$	0	0
$775$	4.73175e6	0.282987
$776$	0	0
$777$	2.08915e7	1.24141
$778$	0	0
$779$	4.03974e6	0.238512
$780$	0	0
$781$	−4.54411e6	−0.266576
$782$	0	0
$783$	3.78490e7	2.20623
$784$	0	0
$785$	−1.67899e6	−0.0972468
$786$	0	0
$787$	2.28899e7	1.31737	0.658683	−	0.752420i	$-0.271114\pi$
$787$	2.28899e7	1.31737	0.658683	+	0.752420i	$0.271114\pi$
$788$	0	0
$789$	−2.88086e7	−1.64752
$790$	0	0
$791$	2.66300e7	1.51332
$792$	0	0
$793$	4.47576e7	2.52746
$794$	0	0
$795$	6.11566e6	0.343183
$796$	0	0
$797$	−2.72212e7	−1.51796	−0.758981	−	0.651113i	$-0.774302\pi$
$797$	−2.72212e7	−1.51796	−0.758981	+	0.651113i	$0.774302\pi$
$798$	0	0
$799$	−4.56819e6	−0.253150
$800$	0	0
$801$	1.19902e7	0.660303
$802$	0	0
$803$	1.03837e7	0.568282
$804$	0	0
$805$	1.57417e6	0.0856174
$806$	0	0
$807$	−3.97696e7	−2.14965
$808$	0	0
$809$	1.71507e7	0.921323	0.460661	−	0.887576i	$-0.347612\pi$
$809$	1.71507e7	0.921323	0.460661	+	0.887576i	$0.347612\pi$
$810$	0	0
$811$	1.85530e6	0.0990515	0.0495258	−	0.998773i	$-0.484229\pi$
$811$	1.85530e6	0.0990515	0.0495258	+	0.998773i	$0.484229\pi$
$812$	0	0
$813$	6.15959e7	3.26833
$814$	0	0
$815$	−3.46329e6	−0.182639
$816$	0	0
$817$	−1.56492e6	−0.0820231
$818$	0	0
$819$	−1.32732e8	−6.91460
$820$	0	0
$821$	4.55795e6	0.236000	0.118000	−	0.993014i	$-0.462352\pi$
$821$	4.55795e6	0.236000	0.118000	+	0.993014i	$0.462352\pi$
$822$	0	0
$823$	2.53222e7	1.30317	0.651586	−	0.758575i	$-0.274104\pi$
$823$	2.53222e7	1.30317	0.651586	+	0.758575i	$0.274104\pi$
$824$	0	0
$825$	5.38735e6	0.275575
$826$	0	0
$827$	−1.37327e7	−0.698219	−0.349110	−	0.937082i	$-0.613516\pi$
$827$	−1.37327e7	−0.698219	−0.349110	+	0.937082i	$0.613516\pi$
$828$	0	0
$829$	3.94031e6	0.199133	0.0995667	−	0.995031i	$-0.468254\pi$
$829$	3.94031e6	0.199133	0.0995667	+	0.995031i	$0.468254\pi$
$830$	0	0
$831$	1.35451e7	0.680425
$832$	0	0
$833$	−5.17338e7	−2.58322
$834$	0	0
$835$	−7.86318e6	−0.390285
$836$	0	0
$837$	5.09279e7	2.51271
$838$	0	0
$839$	−635610.	−0.0311735	−0.0155868	−	0.999879i	$-0.504962\pi$
$839$	−635610.	−0.0311735	−0.0155868	+	0.999879i	$0.504962\pi$
$840$	0	0
$841$	1.11467e7	0.543444
$842$	0	0
$843$	2.12048e7	1.02770
$844$	0	0
$845$	−1.97027e7	−0.949257
$846$	0	0
$847$	1.50206e7	0.719412
$848$	0	0
$849$	4.81318e7	2.29173
$850$	0	0
$851$	−770986.	−0.0364941
$852$	0	0
$853$	1.49876e7	0.705275	0.352638	−	0.935760i	$-0.385285\pi$
$853$	1.49876e7	0.705275	0.352638	+	0.935760i	$0.385285\pi$
$854$	0	0
$855$	−9.20345e6	−0.430562
$856$	0	0
$857$	−1.93680e7	−0.900811	−0.450405	−	0.892824i	$-0.648721\pi$
$857$	−1.93680e7	−0.900811	−0.450405	+	0.892824i	$0.648721\pi$
$858$	0	0
$859$	3.26651e7	1.51043	0.755215	−	0.655477i	$-0.227532\pi$
$859$	3.26651e7	1.51043	0.755215	+	0.655477i	$0.227532\pi$
$860$	0	0
$861$	−3.66543e7	−1.68507
$862$	0	0
$863$	−1.58577e6	−0.0724790	−0.0362395	−	0.999343i	$-0.511538\pi$
$863$	−1.58577e6	−0.0724790	−0.0362395	+	0.999343i	$0.511538\pi$
$864$	0	0
$865$	−1.44702e7	−0.657559
$866$	0	0
$867$	−4.43759e6	−0.200493
$868$	0	0
$869$	−6.31896e6	−0.283855
$870$	0	0
$871$	−2.99145e7	−1.33609
$872$	0	0
$873$	−3.24096e7	−1.43926
$874$	0	0
$875$	3.92081e6	0.173123
$876$	0	0
$877$	−1.76484e7	−0.774828	−0.387414	−	0.921906i	$-0.626632\pi$
$877$	−1.76484e7	−0.774828	−0.387414	+	0.921906i	$0.626632\pi$
$878$	0	0
$879$	1.00627e6	0.0439279
$880$	0	0
$881$	−8.89390e6	−0.386058	−0.193029	−	0.981193i	$-0.561831\pi$
$881$	−8.89390e6	−0.386058	−0.193029	+	0.981193i	$0.561831\pi$
$882$	0	0
$883$	−6.37374e6	−0.275101	−0.137551	−	0.990495i	$-0.543923\pi$
$883$	−6.37374e6	−0.275101	−0.137551	+	0.990495i	$0.543923\pi$
$884$	0	0
$885$	−1.86167e7	−0.798998
$886$	0	0
$887$	−2.99742e7	−1.27920	−0.639601	−	0.768707i	$-0.720900\pi$
$887$	−2.99742e7	−1.27920	−0.639601	+	0.768707i	$0.720900\pi$
$888$	0	0
$889$	−1.38099e7	−0.586051
$890$	0	0
$891$	2.00105e7	0.844429
$892$	0	0
$893$	3.05451e6	0.128178
$894$	0	0
$895$	6.99015e6	0.291695
$896$	0	0
$897$	7.32142e6	0.303818
$898$	0	0
$899$	4.25973e7	1.75785
$900$	0	0
$901$	1.01179e7	0.415221
$902$	0	0
$903$	1.41992e7	0.579487
$904$	0	0
$905$	−1.18729e7	−0.481877
$906$	0	0
$907$	−1.99329e6	−0.0804547	−0.0402273	−	0.999191i	$-0.512808\pi$
$907$	−1.99329e6	−0.0804547	−0.0402273	+	0.999191i	$0.512808\pi$
$908$	0	0
$909$	4.41730e7	1.77316
$910$	0	0
$911$	−1.64815e7	−0.657963	−0.328982	−	0.944336i	$-0.606705\pi$
$911$	−1.64815e7	−0.657963	−0.328982	+	0.944336i	$0.606705\pi$
$912$	0	0
$913$	2.43720e7	0.967640
$914$	0	0
$915$	−2.81587e7	−1.11188
$916$	0	0
$917$	3.08861e7	1.21294
$918$	0	0
$919$	−8.45655e6	−0.330297	−0.165148	−	0.986269i	$-0.552810\pi$
$919$	−8.45655e6	−0.330297	−0.165148	+	0.986269i	$0.552810\pi$
$920$	0	0
$921$	−1.20016e7	−0.466217
$922$	0	0
$923$	−1.53813e7	−0.594277
$924$	0	0
$925$	−1.92031e6	−0.0737932
$926$	0	0
$927$	−3.18558e7	−1.21756
$928$	0	0
$929$	1.70844e7	0.649472	0.324736	−	0.945805i	$-0.394725\pi$
$929$	1.70844e7	0.649472	0.324736	+	0.945805i	$0.394725\pi$
$930$	0	0
$931$	3.45916e7	1.30797
$932$	0	0
$933$	−7.07243e6	−0.265989
$934$	0	0
$935$	8.91299e6	0.333422
$936$	0	0
$937$	−2.35711e7	−0.877064	−0.438532	−	0.898716i	$-0.644501\pi$
$937$	−2.35711e7	−0.877064	−0.438532	+	0.898716i	$0.644501\pi$
$938$	0	0
$939$	−1.82530e7	−0.675572
$940$	0	0
$941$	4.44798e7	1.63753	0.818763	−	0.574131i	$-0.194660\pi$
$941$	4.44798e7	1.63753	0.818763	+	0.574131i	$0.194660\pi$
$942$	0	0
$943$	1.35270e6	0.0495363
$944$	0	0
$945$	4.21998e7	1.53720
$946$	0	0
$947$	−1.90955e7	−0.691921	−0.345961	−	0.938249i	$-0.612447\pi$
$947$	−1.90955e7	−0.691921	−0.345961	+	0.938249i	$0.612447\pi$
$948$	0	0
$949$	3.51477e7	1.26687
$950$	0	0
$951$	2.61736e7	0.938452
$952$	0	0
$953$	2.90401e7	1.03578	0.517888	−	0.855448i	$-0.326718\pi$
$953$	2.90401e7	1.03578	0.517888	+	0.855448i	$0.326718\pi$
$954$	0	0
$955$	1.70198e7	0.603874
$956$	0	0
$957$	4.84993e7	1.71181
$958$	0	0
$959$	−5.70109e7	−2.00176
$960$	0	0
$961$	2.86878e7	1.00205
$962$	0	0
$963$	−6.27858e7	−2.18170
$964$	0	0
$965$	1.20013e7	0.414867
$966$	0	0
$967$	3.46040e6	0.119004	0.0595018	−	0.998228i	$-0.481049\pi$
$967$	3.46040e6	0.119004	0.0595018	+	0.998228i	$0.481049\pi$
$968$	0	0
$969$	−2.27583e7	−0.778629
$970$	0	0
$971$	4.43875e7	1.51082	0.755410	−	0.655252i	$-0.227438\pi$
$971$	4.43875e7	1.51082	0.755410	+	0.655252i	$0.227438\pi$
$972$	0	0
$973$	5.60046e7	1.89645
$974$	0	0
$975$	1.82356e7	0.614338
$976$	0	0
$977$	1.55597e7	0.521512	0.260756	−	0.965405i	$-0.416028\pi$
$977$	1.55597e7	0.521512	0.260756	+	0.965405i	$0.416028\pi$
$978$	0	0
$979$	7.76415e6	0.258903
$980$	0	0
$981$	−1.14019e7	−0.378271
$982$	0	0
$983$	−3.93263e7	−1.29807	−0.649036	−	0.760758i	$-0.724828\pi$
$983$	−3.93263e7	−1.29807	−0.649036	+	0.760758i	$0.724828\pi$
$984$	0	0
$985$	2.32245e7	0.762703
$986$	0	0
$987$	−2.77148e7	−0.905565
$988$	0	0
$989$	−524011.	−0.0170353
$990$	0	0
$991$	−2.63955e7	−0.853780	−0.426890	−	0.904303i	$-0.640391\pi$
$991$	−2.63955e7	−0.853780	−0.426890	+	0.904303i	$0.640391\pi$
$992$	0	0
$993$	−5.24392e7	−1.68765
$994$	0	0
$995$	3.53397e6	0.113163
$996$	0	0
$997$	−3.49308e6	−0.111294	−0.0556468	−	0.998451i	$-0.517722\pi$
$997$	−3.49308e6	−0.111294	−0.0556468	+	0.998451i	$0.517722\pi$
$998$	0	0
$999$	−2.06683e7	−0.655226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.a.z.1.4		4
4.3	odd	2	inner	320.6.a.z.1.1		4
8.3	odd	2		160.6.a.h.1.4	yes	4
8.5	even	2		160.6.a.h.1.1	✓	4
40.3	even	4		800.6.c.l.449.8		8
40.13	odd	4		800.6.c.l.449.1		8
40.19	odd	2		800.6.a.r.1.1		4
40.27	even	4		800.6.c.l.449.2		8
40.29	even	2		800.6.a.r.1.4		4
40.37	odd	4		800.6.c.l.449.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.h.1.1	✓	4	8.5	even	2
160.6.a.h.1.4	yes	4	8.3	odd	2
320.6.a.z.1.1		4	4.3	odd	2	inner
320.6.a.z.1.4		4	1.1	even	1	trivial
800.6.a.r.1.1		4	40.19	odd	2
800.6.a.r.1.4		4	40.29	even	2
800.6.c.l.449.1		8	40.13	odd	4
800.6.c.l.449.2		8	40.27	even	4
800.6.c.l.449.7		8	40.37	odd	4
800.6.c.l.449.8		8	40.3	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0971 q^{3} -25.0000 q^{5} -250.932 q^{7} +491.252 q^{9} +O(q^{10})\)	\(q+27.0971 q^{3} -25.0000 q^{5} -250.932 q^{7} +491.252 q^{9} +318.107 q^{11} +1076.75 q^{13} -677.427 q^{15} -1120.75 q^{17} +749.388 q^{19} -6799.52 q^{21} +250.932 q^{23} +625.000 q^{25} +6726.89 q^{27} +5626.53 q^{29} +7570.79 q^{31} +8619.76 q^{33} +6273.30 q^{35} -3072.49 q^{37} +29176.9 q^{39} +5390.72 q^{41} -2088.26 q^{43} -12281.3 q^{45} +4076.00 q^{47} +46159.8 q^{49} -30369.2 q^{51} -9027.77 q^{53} -7952.67 q^{55} +20306.2 q^{57} +27481.5 q^{59} +41567.1 q^{61} -123271. q^{63} -26918.9 q^{65} -27782.1 q^{67} +6799.52 q^{69} -14284.9 q^{71} +32642.3 q^{73} +16935.7 q^{75} -79823.1 q^{77} -19864.3 q^{79} +62904.9 q^{81} +76615.8 q^{83} +28018.9 q^{85} +152462. q^{87} +24407.4 q^{89} -270192. q^{91} +205146. q^{93} -18734.7 q^{95} -65973.5 q^{97} +156270. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 100 q^{5} + 724 q^{9}+O(q^{10}) \) 4 * q - 100 * q^5 + 724 * q^9	\( 4 q - 100 q^{5} + 724 q^{9} + 584 q^{13} - 760 q^{17} - 9824 q^{21} + 2500 q^{25} + 168 q^{29} + 25792 q^{33} - 19736 q^{37} + 47624 q^{41} - 18100 q^{45} + 121348 q^{49} - 17496 q^{53} + 99840 q^{57} + 62024 q^{61} - 14600 q^{65} + 9824 q^{69} + 119400 q^{73} - 17728 q^{77} + 103940 q^{81} + 19000 q^{85} + 209320 q^{89} + 532672 q^{93} - 59128 q^{97}+O(q^{100}) \) 4 * q - 100 * q^5 + 724 * q^9 + 584 * q^13 - 760 * q^17 - 9824 * q^21 + 2500 * q^25 + 168 * q^29 + 25792 * q^33 - 19736 * q^37 + 47624 * q^41 - 18100 * q^45 + 121348 * q^49 - 17496 * q^53 + 99840 * q^57 + 62024 * q^61 - 14600 * q^65 + 9824 * q^69 + 119400 * q^73 - 17728 * q^77 + 103940 * q^81 + 19000 * q^85 + 209320 * q^89 + 532672 * q^93 - 59128 * q^97

Introduction

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