LMFDB - Embedded newform 160.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(1,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, 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("160.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	160.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:	$25.6614111701$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.18136$ of defining polynomial
Character	$\chi$	$=$	160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.6653 q^{3} +25.0000 q^{5} +176.978 q^{7} -129.252 q^{9} +O(q^{10})$	$q-10.6653 q^{3} +25.0000 q^{5} +176.978 q^{7} -129.252 q^{9} -400.949 q^{11} +784.755 q^{13} -266.632 q^{15} +740.755 q^{17} -2776.65 q^{19} -1887.52 q^{21} -176.978 q^{23} +625.000 q^{25} +3970.17 q^{27} +5542.53 q^{29} +5737.27 q^{31} +4276.24 q^{33} +4424.45 q^{35} +6795.51 q^{37} -8369.64 q^{39} +18421.3 q^{41} -14068.2 q^{43} -3231.29 q^{45} +23388.0 q^{47} +14514.2 q^{49} -7900.36 q^{51} -279.773 q^{53} -10023.7 q^{55} +29613.8 q^{57} +40195.9 q^{59} +10555.1 q^{61} -22874.7 q^{63} +19618.9 q^{65} -35941.5 q^{67} +1887.52 q^{69} +25260.8 q^{71} +27057.7 q^{73} -6665.81 q^{75} -70959.1 q^{77} -32083.9 q^{79} -10934.9 q^{81} -44494.3 q^{83} +18518.9 q^{85} -59112.7 q^{87} +80252.6 q^{89} +138884. q^{91} -61189.6 q^{93} -69416.2 q^{95} +36409.5 q^{97} +51823.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−10.6653	−0.684179	−0.342089	−	0.939667i	$-0.611135\pi$
$3$	−10.6653	−0.684179	−0.342089	+	0.939667i	$0.611135\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	176.978	1.36513	0.682565	−	0.730825i	$-0.260864\pi$
$7$	176.978	1.36513	0.682565	+	0.730825i	$0.260864\pi$
$8$	0	0
$9$	−129.252	−0.531899
$10$	0	0
$11$	−400.949	−0.999097	−0.499548	−	0.866286i	$-0.666501\pi$
$11$	−400.949	−0.999097	−0.499548	+	0.866286i	$0.666501\pi$
$12$	0	0
$13$	784.755	1.28788	0.643940	−	0.765076i	$-0.277299\pi$
$13$	784.755	1.28788	0.643940	+	0.765076i	$0.277299\pi$
$14$	0	0
$15$	−266.632	−0.305974
$16$	0	0
$17$	740.755	0.621659	0.310829	−	0.950466i	$-0.399393\pi$
$17$	740.755	0.621659	0.310829	+	0.950466i	$0.399393\pi$
$18$	0	0
$19$	−2776.65	−1.76456	−0.882281	−	0.470723i	$-0.843993\pi$
$19$	−2776.65	−1.76456	−0.882281	+	0.470723i	$0.843993\pi$
$20$	0	0
$21$	−1887.52	−0.933993
$22$	0	0
$23$	−176.978	−0.0697589	−0.0348794	−	0.999392i	$-0.511105\pi$
$23$	−176.978	−0.0697589	−0.0348794	+	0.999392i	$0.511105\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	3970.17	1.04809
$28$	0	0
$29$	5542.53	1.22381	0.611903	−	0.790933i	$-0.290404\pi$
$29$	5542.53	1.22381	0.611903	+	0.790933i	$0.290404\pi$
$30$	0	0
$31$	5737.27	1.07226	0.536131	−	0.844135i	$-0.319885\pi$
$31$	5737.27	1.07226	0.536131	+	0.844135i	$0.319885\pi$
$32$	0	0
$33$	4276.24	0.683561
$34$	0	0
$35$	4424.45	0.610505
$36$	0	0
$37$	6795.51	0.816052	0.408026	−	0.912970i	$-0.366217\pi$
$37$	6795.51	0.816052	0.408026	+	0.912970i	$0.366217\pi$
$38$	0	0
$39$	−8369.64	−0.881140
$40$	0	0
$41$	18421.3	1.71143	0.855717	−	0.517444i	$-0.173116\pi$
$41$	18421.3	1.71143	0.855717	+	0.517444i	$0.173116\pi$
$42$	0	0
$43$	−14068.2	−1.16029	−0.580147	−	0.814512i	$-0.697005\pi$
$43$	−14068.2	−1.16029	−0.580147	+	0.814512i	$0.697005\pi$
$44$	0	0
$45$	−3231.29	−0.237873
$46$	0	0
$47$	23388.0	1.54436	0.772181	−	0.635402i	$-0.219166\pi$
$47$	23388.0	1.54436	0.772181	+	0.635402i	$0.219166\pi$
$48$	0	0
$49$	14514.2	0.863579
$50$	0	0
$51$	−7900.36	−0.425326
$52$	0	0
$53$	−279.773	−0.0136809	−0.00684046	−	0.999977i	$-0.502177\pi$
$53$	−279.773	−0.0136809	−0.00684046	+	0.999977i	$0.502177\pi$
$54$	0	0
$55$	−10023.7	−0.446810
$56$	0	0
$57$	29613.8	1.20728
$58$	0	0
$59$	40195.9	1.50332	0.751661	−	0.659549i	$-0.229253\pi$
$59$	40195.9	1.50332	0.751661	+	0.659549i	$0.229253\pi$
$60$	0	0
$61$	10555.1	0.363194	0.181597	−	0.983373i	$-0.441873\pi$
$61$	10555.1	0.363194	0.181597	+	0.983373i	$0.441873\pi$
$62$	0	0
$63$	−22874.7	−0.726111
$64$	0	0
$65$	19618.9	0.575958
$66$	0	0
$67$	−35941.5	−0.978158	−0.489079	−	0.872239i	$-0.662667\pi$
$67$	−35941.5	−0.978158	−0.489079	+	0.872239i	$0.662667\pi$
$68$	0	0
$69$	1887.52	0.0477275
$70$	0	0
$71$	25260.8	0.594705	0.297353	−	0.954768i	$-0.403896\pi$
$71$	25260.8	0.594705	0.297353	+	0.954768i	$0.403896\pi$
$72$	0	0
$73$	27057.7	0.594271	0.297135	−	0.954835i	$-0.403969\pi$
$73$	27057.7	0.594271	0.297135	+	0.954835i	$0.403969\pi$
$74$	0	0
$75$	−6665.81	−0.136836
$76$	0	0
$77$	−70959.1	−1.36390
$78$	0	0
$79$	−32083.9	−0.578389	−0.289194	−	0.957270i	$-0.593387\pi$
$79$	−32083.9	−0.578389	−0.289194	+	0.957270i	$0.593387\pi$
$80$	0	0
$81$	−10934.9	−0.185184
$82$	0	0
$83$	−44494.3	−0.708940	−0.354470	−	0.935067i	$-0.615339\pi$
$83$	−44494.3	−0.708940	−0.354470	+	0.935067i	$0.615339\pi$
$84$	0	0
$85$	18518.9	0.278014
$86$	0	0
$87$	−59112.7	−0.837303
$88$	0	0
$89$	80252.6	1.07395	0.536975	−	0.843598i	$-0.319567\pi$
$89$	80252.6	1.07395	0.536975	+	0.843598i	$0.319567\pi$
$90$	0	0
$91$	138884.	1.75812
$92$	0	0
$93$	−61189.6	−0.733619
$94$	0	0
$95$	−69416.2	−0.789136
$96$	0	0
$97$	36409.5	0.392903	0.196452	−	0.980514i	$-0.437058\pi$
$97$	36409.5	0.392903	0.196452	+	0.980514i	$0.437058\pi$
$98$	0	0
$99$	51823.3	0.531419
$100$	0	0
$101$	−145765.	−1.42183	−0.710917	−	0.703276i	$-0.751720\pi$
$101$	−145765.	−1.42183	−0.710917	+	0.703276i	$0.751720\pi$
$102$	0	0
$103$	141853.	1.31749	0.658743	−	0.752368i	$-0.271089\pi$
$103$	141853.	1.31749	0.658743	+	0.752368i	$0.271089\pi$
$104$	0	0
$105$	−47188.0	−0.417694
$106$	0	0
$107$	59128.4	0.499271	0.249636	−	0.968340i	$-0.419689\pi$
$107$	59128.4	0.499271	0.249636	+	0.968340i	$0.419689\pi$
$108$	0	0
$109$	−114542.	−0.923417	−0.461709	−	0.887032i	$-0.652763\pi$
$109$	−114542.	−0.923417	−0.461709	+	0.887032i	$0.652763\pi$
$110$	0	0
$111$	−72476.1	−0.558325
$112$	0	0
$113$	−139632.	−1.02870	−0.514349	−	0.857581i	$-0.671966\pi$
$113$	−139632.	−1.02870	−0.514349	+	0.857581i	$0.671966\pi$
$114$	0	0
$115$	−4424.45	−0.0311971
$116$	0	0
$117$	−101431.	−0.685022
$118$	0	0
$119$	131097.	0.848645
$120$	0	0
$121$	−290.855	−0.00180598
$122$	0	0
$123$	−196468.	−1.17093
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	212200.	1.16744	0.583722	−	0.811953i	$-0.301596\pi$
$127$	212200.	1.16744	0.583722	+	0.811953i	$0.301596\pi$
$128$	0	0
$129$	150042.	0.793849
$130$	0	0
$131$	−47237.2	−0.240495	−0.120247	−	0.992744i	$-0.538369\pi$
$131$	−47237.2	−0.240495	−0.120247	+	0.992744i	$0.538369\pi$
$132$	0	0
$133$	−491405.	−2.40886
$134$	0	0
$135$	99254.3	0.468722
$136$	0	0
$137$	−55752.7	−0.253784	−0.126892	−	0.991917i	$-0.540500\pi$
$137$	−55752.7	−0.253784	−0.126892	+	0.991917i	$0.540500\pi$
$138$	0	0
$139$	−380643.	−1.67102	−0.835509	−	0.549477i	$-0.814827\pi$
$139$	−380643.	−1.67102	−0.835509	+	0.549477i	$0.814827\pi$
$140$	0	0
$141$	−249440.	−1.05662
$142$	0	0
$143$	−314647.	−1.28672
$144$	0	0
$145$	138563.	0.547303
$146$	0	0
$147$	−154798.	−0.590843
$148$	0	0
$149$	302872.	1.11762	0.558809	−	0.829296i	$-0.311258\pi$
$149$	302872.	1.11762	0.558809	+	0.829296i	$0.311258\pi$
$150$	0	0
$151$	−177821.	−0.634660	−0.317330	−	0.948315i	$-0.602786\pi$
$151$	−177821.	−0.634660	−0.317330	+	0.948315i	$0.602786\pi$
$152$	0	0
$153$	−95743.6	−0.330660
$154$	0	0
$155$	143432.	0.479530
$156$	0	0
$157$	55699.8	0.180345	0.0901726	−	0.995926i	$-0.471258\pi$
$157$	55699.8	0.180345	0.0901726	+	0.995926i	$0.471258\pi$
$158$	0	0
$159$	2983.86	0.00936020
$160$	0	0
$161$	−31321.2	−0.0952299
$162$	0	0
$163$	482623.	1.42278	0.711392	−	0.702795i	$-0.248065\pi$
$163$	482623.	1.42278	0.711392	+	0.702795i	$0.248065\pi$
$164$	0	0
$165$	106906.	0.305698
$166$	0	0
$167$	237679.	0.659476	0.329738	−	0.944072i	$-0.393040\pi$
$167$	237679.	0.659476	0.329738	+	0.944072i	$0.393040\pi$
$168$	0	0
$169$	244547.	0.658635
$170$	0	0
$171$	358886.	0.938569
$172$	0	0
$173$	571604.	1.45205	0.726023	−	0.687671i	$-0.241367\pi$
$173$	571604.	1.45205	0.726023	+	0.687671i	$0.241367\pi$
$174$	0	0
$175$	110611.	0.273026
$176$	0	0
$177$	−428702.	−1.02854
$178$	0	0
$179$	−85450.5	−0.199334	−0.0996671	−	0.995021i	$-0.531778\pi$
$179$	−85450.5	−0.199334	−0.0996671	+	0.995021i	$0.531778\pi$
$180$	0	0
$181$	−191967.	−0.435543	−0.217771	−	0.976000i	$-0.569879\pi$
$181$	−191967.	−0.435543	−0.217771	+	0.976000i	$0.569879\pi$
$182$	0	0
$183$	−112574.	−0.248490
$184$	0	0
$185$	169888.	0.364949
$186$	0	0
$187$	−297005.	−0.621097
$188$	0	0
$189$	702633.	1.43078
$190$	0	0
$191$	−722934.	−1.43389	−0.716944	−	0.697131i	$-0.754460\pi$
$191$	−722934.	−1.43389	−0.716944	+	0.697131i	$0.754460\pi$
$192$	0	0
$193$	−887721.	−1.71547	−0.857735	−	0.514091i	$-0.828129\pi$
$193$	−887721.	−1.71547	−0.857735	+	0.514091i	$0.828129\pi$
$194$	0	0
$195$	−209241.	−0.394058
$196$	0	0
$197$	130392.	0.239379	0.119690	−	0.992811i	$-0.461810\pi$
$197$	130392.	0.239379	0.119690	+	0.992811i	$0.461810\pi$
$198$	0	0
$199$	398236.	0.712866	0.356433	−	0.934321i	$-0.383993\pi$
$199$	398236.	0.712866	0.356433	+	0.934321i	$0.383993\pi$
$200$	0	0
$201$	383327.	0.669235
$202$	0	0
$203$	980905.	1.67065
$204$	0	0
$205$	460532.	0.765377
$206$	0	0
$207$	22874.7	0.0371047
$208$	0	0
$209$	1.11329e6	1.76297
$210$	0	0
$211$	−444176.	−0.686829	−0.343415	−	0.939184i	$-0.611584\pi$
$211$	−444176.	−0.686829	−0.343415	+	0.939184i	$0.611584\pi$
$212$	0	0
$213$	−269414.	−0.406885
$214$	0	0
$215$	−351706.	−0.518899
$216$	0	0
$217$	1.01537e6	1.46378
$218$	0	0
$219$	−288579.	−0.406587
$220$	0	0
$221$	581310.	0.800622
$222$	0	0
$223$	1.07046e6	1.44148	0.720738	−	0.693207i	$-0.243803\pi$
$223$	1.07046e6	1.44148	0.720738	+	0.693207i	$0.243803\pi$
$224$	0	0
$225$	−80782.2	−0.106380
$226$	0	0
$227$	−135321.	−0.174301	−0.0871506	−	0.996195i	$-0.527776\pi$
$227$	−135321.	−0.174301	−0.0871506	+	0.996195i	$0.527776\pi$
$228$	0	0
$229$	1.37977e6	1.73867	0.869336	−	0.494221i	$-0.164547\pi$
$229$	1.37977e6	1.73867	0.869336	+	0.494221i	$0.164547\pi$
$230$	0	0
$231$	756800.	0.933149
$232$	0	0
$233$	−1.36150e6	−1.64296	−0.821481	−	0.570236i	$-0.806852\pi$
$233$	−1.36150e6	−1.64296	−0.821481	+	0.570236i	$0.806852\pi$
$234$	0	0
$235$	584701.	0.690660
$236$	0	0
$237$	342185.	0.395721
$238$	0	0
$239$	304035.	0.344293	0.172147	−	0.985071i	$-0.444930\pi$
$239$	304035.	0.344293	0.172147	+	0.985071i	$0.444930\pi$
$240$	0	0
$241$	−1.36349e6	−1.51220	−0.756101	−	0.654455i	$-0.772898\pi$
$241$	−1.36349e6	−1.51220	−0.756101	+	0.654455i	$0.772898\pi$
$242$	0	0
$243$	−848127.	−0.921394
$244$	0	0
$245$	362854.	0.386204
$246$	0	0
$247$	−2.17899e6	−2.27254
$248$	0	0
$249$	474545.	0.485041
$250$	0	0
$251$	479976.	0.480878	0.240439	−	0.970664i	$-0.422709\pi$
$251$	479976.	0.480878	0.240439	+	0.970664i	$0.422709\pi$
$252$	0	0
$253$	70959.1	0.0696958
$254$	0	0
$255$	−197509.	−0.190211
$256$	0	0
$257$	28563.1	0.0269757	0.0134879	−	0.999909i	$-0.495707\pi$
$257$	28563.1	0.0269757	0.0134879	+	0.999909i	$0.495707\pi$
$258$	0	0
$259$	1.20265e6	1.11402
$260$	0	0
$261$	−716380.	−0.650942
$262$	0	0
$263$	−744110.	−0.663358	−0.331679	−	0.943392i	$-0.607615\pi$
$263$	−744110.	−0.663358	−0.331679	+	0.943392i	$0.607615\pi$
$264$	0	0
$265$	−6994.32	−0.00611830
$266$	0	0
$267$	−855918.	−0.734774
$268$	0	0
$269$	2.07825e6	1.75112	0.875560	−	0.483109i	$-0.160492\pi$
$269$	2.07825e6	1.75112	0.875560	+	0.483109i	$0.160492\pi$
$270$	0	0
$271$	−1.52246e6	−1.25928	−0.629642	−	0.776885i	$-0.716798\pi$
$271$	−1.52246e6	−1.25928	−0.629642	+	0.776885i	$0.716798\pi$
$272$	0	0
$273$	−1.48124e6	−1.20287
$274$	0	0
$275$	−250593.	−0.199819
$276$	0	0
$277$	−2.03003e6	−1.58966	−0.794829	−	0.606833i	$-0.792440\pi$
$277$	−2.03003e6	−1.58966	−0.794829	+	0.606833i	$0.792440\pi$
$278$	0	0
$279$	−741551.	−0.570335
$280$	0	0
$281$	121712.	0.0919535	0.0459767	−	0.998943i	$-0.485360\pi$
$281$	121712.	0.0919535	0.0459767	+	0.998943i	$0.485360\pi$
$282$	0	0
$283$	−1.23352e6	−0.915549	−0.457775	−	0.889068i	$-0.651353\pi$
$283$	−1.23352e6	−0.915549	−0.457775	+	0.889068i	$0.651353\pi$
$284$	0	0
$285$	740344.	0.539910
$286$	0	0
$287$	3.26016e6	2.33633
$288$	0	0
$289$	−871140.	−0.613540
$290$	0	0
$291$	−388318.	−0.268816
$292$	0	0
$293$	193692.	0.131808	0.0659040	−	0.997826i	$-0.479007\pi$
$293$	193692.	0.131808	0.0659040	+	0.997826i	$0.479007\pi$
$294$	0	0
$295$	1.00490e6	0.672306
$296$	0	0
$297$	−1.59184e6	−1.04715
$298$	0	0
$299$	−138884.	−0.0898411
$300$	0	0
$301$	−2.48976e6	−1.58395
$302$	0	0
$303$	1.55462e6	0.972788
$304$	0	0
$305$	263878.	0.162425
$306$	0	0
$307$	2.83746e6	1.71824	0.859120	−	0.511775i	$-0.171012\pi$
$307$	2.83746e6	1.71824	0.859120	+	0.511775i	$0.171012\pi$
$308$	0	0
$309$	−1.51291e6	−0.901396
$310$	0	0
$311$	2.05910e6	1.20719	0.603596	−	0.797290i	$-0.293734\pi$
$311$	2.05910e6	1.20719	0.603596	+	0.797290i	$0.293734\pi$
$312$	0	0
$313$	−2.59097e6	−1.49486	−0.747432	−	0.664338i	$-0.768713\pi$
$313$	−2.59097e6	−1.49486	−0.747432	+	0.664338i	$0.768713\pi$
$314$	0	0
$315$	−571866.	−0.324727
$316$	0	0
$317$	1.29209e6	0.722179	0.361090	−	0.932531i	$-0.382405\pi$
$317$	1.29209e6	0.722179	0.361090	+	0.932531i	$0.382405\pi$
$318$	0	0
$319$	−2.22227e6	−1.22270
$320$	0	0
$321$	−630622.	−0.341591
$322$	0	0
$323$	−2.05682e6	−1.09696
$324$	0	0
$325$	490472.	0.257576
$326$	0	0
$327$	1.22162e6	0.631782
$328$	0	0
$329$	4.13917e6	2.10825
$330$	0	0
$331$	−325006.	−0.163050	−0.0815251	−	0.996671i	$-0.525979\pi$
$331$	−325006.	−0.163050	−0.0815251	+	0.996671i	$0.525979\pi$
$332$	0	0
$333$	−878330.	−0.434057
$334$	0	0
$335$	−898537.	−0.437446
$336$	0	0
$337$	−1.86171e6	−0.892970	−0.446485	−	0.894791i	$-0.647324\pi$
$337$	−1.86171e6	−0.892970	−0.446485	+	0.894791i	$0.647324\pi$
$338$	0	0
$339$	1.48921e6	0.703813
$340$	0	0
$341$	−2.30035e6	−1.07129
$342$	0	0
$343$	−405780.	−0.186232
$344$	0	0
$345$	47188.0	0.0213444
$346$	0	0
$347$	−1.00143e6	−0.446476	−0.223238	−	0.974764i	$-0.571663\pi$
$347$	−1.00143e6	−0.446476	−0.223238	+	0.974764i	$0.571663\pi$
$348$	0	0
$349$	415024.	0.182393	0.0911967	−	0.995833i	$-0.470931\pi$
$349$	415024.	0.182393	0.0911967	+	0.995833i	$0.470931\pi$
$350$	0	0
$351$	3.11561e6	1.34982
$352$	0	0
$353$	−4.02837e6	−1.72065	−0.860325	−	0.509746i	$-0.829739\pi$
$353$	−4.02837e6	−1.72065	−0.860325	+	0.509746i	$0.829739\pi$
$354$	0	0
$355$	631521.	0.265960
$356$	0	0
$357$	−1.39819e6	−0.580625
$358$	0	0
$359$	−2.40157e6	−0.983465	−0.491732	−	0.870746i	$-0.663636\pi$
$359$	−2.40157e6	−0.983465	−0.491732	+	0.870746i	$0.663636\pi$
$360$	0	0
$361$	5.23368e6	2.11368
$362$	0	0
$363$	3102.05	0.00123561
$364$	0	0
$365$	676443.	0.265766
$366$	0	0
$367$	−3.40170e6	−1.31835	−0.659176	−	0.751989i	$-0.729095\pi$
$367$	−3.40170e6	−1.31835	−0.659176	+	0.751989i	$0.729095\pi$
$368$	0	0
$369$	−2.38098e6	−0.910311
$370$	0	0
$371$	−49513.6	−0.0186762
$372$	0	0
$373$	−307794.	−0.114548	−0.0572741	−	0.998358i	$-0.518241\pi$
$373$	−307794.	−0.114548	−0.0572741	+	0.998358i	$0.518241\pi$
$374$	0	0
$375$	−166645.	−0.0611948
$376$	0	0
$377$	4.34952e6	1.57612
$378$	0	0
$379$	−2.22446e6	−0.795476	−0.397738	−	0.917499i	$-0.630205\pi$
$379$	−2.22446e6	−0.795476	−0.397738	+	0.917499i	$0.630205\pi$
$380$	0	0
$381$	−2.26318e6	−0.798741
$382$	0	0
$383$	−3.89215e6	−1.35579	−0.677896	−	0.735158i	$-0.737108\pi$
$383$	−3.89215e6	−1.35579	−0.677896	+	0.735158i	$0.737108\pi$
$384$	0	0
$385$	−1.77398e6	−0.609953
$386$	0	0
$387$	1.81834e6	0.617160
$388$	0	0
$389$	−2.45071e6	−0.821142	−0.410571	−	0.911829i	$-0.634671\pi$
$389$	−2.45071e6	−0.821142	−0.410571	+	0.911829i	$0.634671\pi$
$390$	0	0
$391$	−131097.	−0.0433662
$392$	0	0
$393$	503798.	0.164541
$394$	0	0
$395$	−802098.	−0.258663
$396$	0	0
$397$	−3.34451e6	−1.06502	−0.532508	−	0.846425i	$-0.678750\pi$
$397$	−3.34451e6	−1.06502	−0.532508	+	0.846425i	$0.678750\pi$
$398$	0	0
$399$	5.24098e6	1.64809
$400$	0	0
$401$	4.38042e6	1.36036	0.680181	−	0.733044i	$-0.261901\pi$
$401$	4.38042e6	1.36036	0.680181	+	0.733044i	$0.261901\pi$
$402$	0	0
$403$	4.50235e6	1.38094
$404$	0	0
$405$	−273373.	−0.0828168
$406$	0	0
$407$	−2.72465e6	−0.815315
$408$	0	0
$409$	5.76678e6	1.70461	0.852306	−	0.523044i	$-0.175204\pi$
$409$	5.76678e6	1.70461	0.852306	+	0.523044i	$0.175204\pi$
$410$	0	0
$411$	594619.	0.173634
$412$	0	0
$413$	7.11379e6	2.05223
$414$	0	0
$415$	−1.11236e6	−0.317047
$416$	0	0
$417$	4.05967e6	1.14327
$418$	0	0
$419$	1.28648e6	0.357988	0.178994	−	0.983850i	$-0.442716\pi$
$419$	1.28648e6	0.357988	0.178994	+	0.983850i	$0.442716\pi$
$420$	0	0
$421$	−5.08504e6	−1.39826	−0.699132	−	0.714992i	$-0.746430\pi$
$421$	−5.08504e6	−1.39826	−0.699132	+	0.714992i	$0.746430\pi$
$422$	0	0
$423$	−3.02294e6	−0.821445
$424$	0	0
$425$	462972.	0.124332
$426$	0	0
$427$	1.86802e6	0.495807
$428$	0	0
$429$	3.35580e6	0.880344
$430$	0	0
$431$	−532656.	−0.138119	−0.0690595	−	0.997613i	$-0.522000\pi$
$431$	−532656.	−0.138119	−0.0690595	+	0.997613i	$0.522000\pi$
$432$	0	0
$433$	2.04106e6	0.523163	0.261581	−	0.965181i	$-0.415756\pi$
$433$	2.04106e6	0.523163	0.261581	+	0.965181i	$0.415756\pi$
$434$	0	0
$435$	−1.47782e6	−0.374453
$436$	0	0
$437$	491405.	0.123094
$438$	0	0
$439$	6.16789e6	1.52748	0.763740	−	0.645524i	$-0.223361\pi$
$439$	6.16789e6	1.52748	0.763740	+	0.645524i	$0.223361\pi$
$440$	0	0
$441$	−1.87598e6	−0.459337
$442$	0	0
$443$	1.14975e6	0.278352	0.139176	−	0.990268i	$-0.455555\pi$
$443$	1.14975e6	0.278352	0.139176	+	0.990268i	$0.455555\pi$
$444$	0	0
$445$	2.00632e6	0.480285
$446$	0	0
$447$	−3.23022e6	−0.764651
$448$	0	0
$449$	733622.	0.171734	0.0858671	−	0.996307i	$-0.472634\pi$
$449$	733622.	0.171734	0.0858671	+	0.996307i	$0.472634\pi$
$450$	0	0
$451$	−7.38600e6	−1.70989
$452$	0	0
$453$	1.89652e6	0.434221
$454$	0	0
$455$	3.47211e6	0.786257
$456$	0	0
$457$	3.58153e6	0.802193	0.401096	−	0.916036i	$-0.368629\pi$
$457$	3.58153e6	0.802193	0.401096	+	0.916036i	$0.368629\pi$
$458$	0	0
$459$	2.94092e6	0.651556
$460$	0	0
$461$	5.62588e6	1.23293	0.616465	−	0.787382i	$-0.288564\pi$
$461$	5.62588e6	1.23293	0.616465	+	0.787382i	$0.288564\pi$
$462$	0	0
$463$	3.19575e6	0.692819	0.346410	−	0.938083i	$-0.387401\pi$
$463$	3.19575e6	0.692819	0.346410	+	0.938083i	$0.387401\pi$
$464$	0	0
$465$	−1.52974e6	−0.328084
$466$	0	0
$467$	−3.95112e6	−0.838355	−0.419178	−	0.907904i	$-0.637682\pi$
$467$	−3.95112e6	−0.838355	−0.419178	+	0.907904i	$0.637682\pi$
$468$	0	0
$469$	−6.36085e6	−1.33531
$470$	0	0
$471$	−594055.	−0.123388
$472$	0	0
$473$	5.64064e6	1.15925
$474$	0	0
$475$	−1.73541e6	−0.352912
$476$	0	0
$477$	36161.0	0.00727688
$478$	0	0
$479$	−2.58626e6	−0.515031	−0.257516	−	0.966274i	$-0.582904\pi$
$479$	−2.58626e6	−0.515031	−0.257516	+	0.966274i	$0.582904\pi$
$480$	0	0
$481$	5.33281e6	1.05098
$482$	0	0
$483$	334050.	0.0651543
$484$	0	0
$485$	910237.	0.175712
$486$	0	0
$487$	4.20118e6	0.802692	0.401346	−	0.915926i	$-0.368542\pi$
$487$	4.20118e6	0.802692	0.401346	+	0.915926i	$0.368542\pi$
$488$	0	0
$489$	−5.14732e6	−0.973439
$490$	0	0
$491$	8.89283e6	1.66470	0.832351	−	0.554249i	$-0.186995\pi$
$491$	8.89283e6	1.66470	0.832351	+	0.554249i	$0.186995\pi$
$492$	0	0
$493$	4.10565e6	0.760790
$494$	0	0
$495$	1.29558e6	0.237658
$496$	0	0
$497$	4.47061e6	0.811850
$498$	0	0
$499$	−7.55579e6	−1.35840	−0.679201	−	0.733952i	$-0.737674\pi$
$499$	−7.55579e6	−1.35840	−0.679201	+	0.733952i	$0.737674\pi$
$500$	0	0
$501$	−2.53491e6	−0.451200
$502$	0	0
$503$	2.86546e6	0.504980	0.252490	−	0.967600i	$-0.418751\pi$
$503$	2.86546e6	0.504980	0.252490	+	0.967600i	$0.418751\pi$
$504$	0	0
$505$	−3.64412e6	−0.635863
$506$	0	0
$507$	−2.60816e6	−0.450624
$508$	0	0
$509$	−4.27687e6	−0.731697	−0.365849	−	0.930674i	$-0.619221\pi$
$509$	−4.27687e6	−0.731697	−0.365849	+	0.930674i	$0.619221\pi$
$510$	0	0
$511$	4.78862e6	0.811257
$512$	0	0
$513$	−1.10238e7	−1.84943
$514$	0	0
$515$	3.54633e6	0.589198
$516$	0	0
$517$	−9.37741e6	−1.54297
$518$	0	0
$519$	−6.09633e6	−0.993459
$520$	0	0
$521$	1.14984e7	1.85586	0.927929	−	0.372758i	$-0.121588\pi$
$521$	1.14984e7	1.85586	0.927929	+	0.372758i	$0.121588\pi$
$522$	0	0
$523$	5.66192e6	0.905126	0.452563	−	0.891732i	$-0.350510\pi$
$523$	5.66192e6	0.905126	0.452563	+	0.891732i	$0.350510\pi$
$524$	0	0
$525$	−1.17970e6	−0.186799
$526$	0	0
$527$	4.24991e6	0.666581
$528$	0	0
$529$	−6.40502e6	−0.995134
$530$	0	0
$531$	−5.19539e6	−0.799616
$532$	0	0
$533$	1.44562e7	2.20412
$534$	0	0
$535$	1.47821e6	0.223281
$536$	0	0
$537$	911354.	0.136380
$538$	0	0
$539$	−5.81944e6	−0.862799
$540$	0	0
$541$	−3.05381e6	−0.448589	−0.224294	−	0.974521i	$-0.572008\pi$
$541$	−3.05381e6	−0.448589	−0.224294	+	0.974521i	$0.572008\pi$
$542$	0	0
$543$	2.04739e6	0.297989
$544$	0	0
$545$	−2.86355e6	−0.412965
$546$	0	0
$547$	6.09517e6	0.870999	0.435499	−	0.900189i	$-0.356572\pi$
$547$	6.09517e6	0.870999	0.435499	+	0.900189i	$0.356572\pi$
$548$	0	0
$549$	−1.36427e6	−0.193183
$550$	0	0
$551$	−1.53897e7	−2.15948
$552$	0	0
$553$	−5.67815e6	−0.789576
$554$	0	0
$555$	−1.81190e6	−0.249691
$556$	0	0
$557$	5.81797e6	0.794573	0.397287	−	0.917695i	$-0.369952\pi$
$557$	5.81797e6	0.794573	0.397287	+	0.917695i	$0.369952\pi$
$558$	0	0
$559$	−1.10401e7	−1.49432
$560$	0	0
$561$	3.16764e6	0.424942
$562$	0	0
$563$	−9.85597e6	−1.31047	−0.655237	−	0.755424i	$-0.727431\pi$
$563$	−9.85597e6	−1.31047	−0.655237	+	0.755424i	$0.727431\pi$
$564$	0	0
$565$	−3.49079e6	−0.460047
$566$	0	0
$567$	−1.93524e6	−0.252800
$568$	0	0
$569$	−7.46083e6	−0.966065	−0.483032	−	0.875603i	$-0.660465\pi$
$569$	−7.46083e6	−0.966065	−0.483032	+	0.875603i	$0.660465\pi$
$570$	0	0
$571$	−9.61386e6	−1.23398	−0.616989	−	0.786972i	$-0.711648\pi$
$571$	−9.61386e6	−1.23398	−0.616989	+	0.786972i	$0.711648\pi$
$572$	0	0
$573$	7.71030e6	0.981036
$574$	0	0
$575$	−110611.	−0.0139518
$576$	0	0
$577$	−7.67330e6	−0.959495	−0.479748	−	0.877407i	$-0.659272\pi$
$577$	−7.67330e6	−0.959495	−0.479748	+	0.877407i	$0.659272\pi$
$578$	0	0
$579$	9.46781e6	1.17369
$580$	0	0
$581$	−7.87451e6	−0.967794
$582$	0	0
$583$	112175.	0.0136686
$584$	0	0
$585$	−2.53577e6	−0.306351
$586$	0	0
$587$	7.30662e6	0.875228	0.437614	−	0.899163i	$-0.355824\pi$
$587$	7.30662e6	0.875228	0.437614	+	0.899163i	$0.355824\pi$
$588$	0	0
$589$	−1.59304e7	−1.89207
$590$	0	0
$591$	−1.39067e6	−0.163778
$592$	0	0
$593$	−4.69101e6	−0.547810	−0.273905	−	0.961757i	$-0.588315\pi$
$593$	−4.69101e6	−0.547810	−0.273905	+	0.961757i	$0.588315\pi$
$594$	0	0
$595$	3.27743e6	0.379525
$596$	0	0
$597$	−4.24730e6	−0.487728
$598$	0	0
$599$	1.57453e7	1.79302	0.896508	−	0.443028i	$-0.146096\pi$
$599$	1.57453e7	1.79302	0.896508	+	0.443028i	$0.146096\pi$
$600$	0	0
$601$	1.91704e6	0.216494	0.108247	−	0.994124i	$-0.465476\pi$
$601$	1.91704e6	0.216494	0.108247	+	0.994124i	$0.465476\pi$
$602$	0	0
$603$	4.64549e6	0.520282
$604$	0	0
$605$	−7271.37	−0.000807659 0
$606$	0	0
$607$	−4.23572e6	−0.466611	−0.233306	−	0.972403i	$-0.574954\pi$
$607$	−4.23572e6	−0.466611	−0.233306	+	0.972403i	$0.574954\pi$
$608$	0	0
$609$	−1.04616e7	−1.14303
$610$	0	0
$611$	1.83539e7	1.98895
$612$	0	0
$613$	6.37661e6	0.685391	0.342696	−	0.939447i	$-0.388660\pi$
$613$	6.37661e6	0.685391	0.342696	+	0.939447i	$0.388660\pi$
$614$	0	0
$615$	−4.91171e6	−0.523655
$616$	0	0
$617$	−7.99105e6	−0.845067	−0.422533	−	0.906347i	$-0.638859\pi$
$617$	−7.99105e6	−0.845067	−0.422533	+	0.906347i	$0.638859\pi$
$618$	0	0
$619$	1.94974e6	0.204527	0.102263	−	0.994757i	$-0.467392\pi$
$619$	1.94974e6	0.204527	0.102263	+	0.994757i	$0.467392\pi$
$620$	0	0
$621$	−702633.	−0.0731138
$622$	0	0
$623$	1.42029e7	1.46608
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−1.18736e7	−1.20619
$628$	0	0
$629$	5.03380e6	0.507306
$630$	0	0
$631$	1.07899e7	1.07881	0.539403	−	0.842048i	$-0.318650\pi$
$631$	1.07899e7	1.07881	0.539403	+	0.842048i	$0.318650\pi$
$632$	0	0
$633$	4.73727e6	0.469914
$634$	0	0
$635$	5.30500e6	0.522097
$636$	0	0
$637$	1.13901e7	1.11219
$638$	0	0
$639$	−3.26500e6	−0.316323
$640$	0	0
$641$	−1.17794e7	−1.13234	−0.566171	−	0.824288i	$-0.691576\pi$
$641$	−1.17794e7	−1.13234	−0.566171	+	0.824288i	$0.691576\pi$
$642$	0	0
$643$	−1.71052e7	−1.63155	−0.815773	−	0.578372i	$-0.803688\pi$
$643$	−1.71052e7	−1.63155	−0.815773	+	0.578372i	$0.803688\pi$
$644$	0	0
$645$	3.75104e6	0.355020
$646$	0	0
$647$	6.58527e6	0.618461	0.309231	−	0.950987i	$-0.399928\pi$
$647$	6.58527e6	0.618461	0.309231	+	0.950987i	$0.399928\pi$
$648$	0	0
$649$	−1.61165e7	−1.50196
$650$	0	0
$651$	−1.08292e7	−1.00149
$652$	0	0
$653$	1.53597e7	1.40962	0.704808	−	0.709398i	$-0.251033\pi$
$653$	1.53597e7	1.40962	0.704808	+	0.709398i	$0.251033\pi$
$654$	0	0
$655$	−1.18093e6	−0.107552
$656$	0	0
$657$	−3.49725e6	−0.316092
$658$	0	0
$659$	912830.	0.0818797	0.0409398	−	0.999162i	$-0.486965\pi$
$659$	912830.	0.0818797	0.0409398	+	0.999162i	$0.486965\pi$
$660$	0	0
$661$	−275004.	−0.0244813	−0.0122407	−	0.999925i	$-0.503896\pi$
$661$	−275004.	−0.0244813	−0.0122407	+	0.999925i	$0.503896\pi$
$662$	0	0
$663$	−6.19985e6	−0.547769
$664$	0	0
$665$	−1.22851e7	−1.07727
$666$	0	0
$667$	−980905.	−0.0853714
$668$	0	0
$669$	−1.14168e7	−0.986228
$670$	0	0
$671$	−4.23207e6	−0.362866
$672$	0	0
$673$	4.59575e6	0.391128	0.195564	−	0.980691i	$-0.437346\pi$
$673$	4.59575e6	0.391128	0.195564	+	0.980691i	$0.437346\pi$
$674$	0	0
$675$	2.48136e6	0.209619
$676$	0	0
$677$	5.72568e6	0.480126	0.240063	−	0.970757i	$-0.422832\pi$
$677$	5.72568e6	0.480126	0.240063	+	0.970757i	$0.422832\pi$
$678$	0	0
$679$	6.44368e6	0.536364
$680$	0	0
$681$	1.44324e6	0.119253
$682$	0	0
$683$	1.03529e7	0.849201	0.424600	−	0.905381i	$-0.360415\pi$
$683$	1.03529e7	0.849201	0.424600	+	0.905381i	$0.360415\pi$
$684$	0	0
$685$	−1.39382e6	−0.113496
$686$	0	0
$687$	−1.47156e7	−1.18956
$688$	0	0
$689$	−219553.	−0.0176194
$690$	0	0
$691$	1.85254e7	1.47596	0.737978	−	0.674824i	$-0.235781\pi$
$691$	1.85254e7	1.47596	0.737978	+	0.674824i	$0.235781\pi$
$692$	0	0
$693$	9.17157e6	0.725455
$694$	0	0
$695$	−9.51608e6	−0.747302
$696$	0	0
$697$	1.36456e7	1.06393
$698$	0	0
$699$	1.45208e7	1.12408
$700$	0	0
$701$	9.89029e6	0.760175	0.380088	−	0.924950i	$-0.375894\pi$
$701$	9.89029e6	0.760175	0.380088	+	0.924950i	$0.375894\pi$
$702$	0	0
$703$	−1.88687e7	−1.43997
$704$	0	0
$705$	−6.23601e6	−0.472535
$706$	0	0
$707$	−2.57971e7	−1.94099
$708$	0	0
$709$	−2.52577e6	−0.188703	−0.0943515	−	0.995539i	$-0.530078\pi$
$709$	−2.52577e6	−0.188703	−0.0943515	+	0.995539i	$0.530078\pi$
$710$	0	0
$711$	4.14690e6	0.307645
$712$	0	0
$713$	−1.01537e6	−0.0747998
$714$	0	0
$715$	−7.86616e6	−0.575437
$716$	0	0
$717$	−3.24262e6	−0.235558
$718$	0	0
$719$	−1.81111e7	−1.30654	−0.653270	−	0.757125i	$-0.726603\pi$
$719$	−1.81111e7	−1.30654	−0.653270	+	0.757125i	$0.726603\pi$
$720$	0	0
$721$	2.51049e7	1.79854
$722$	0	0
$723$	1.45420e7	1.03462
$724$	0	0
$725$	3.46408e6	0.244761
$726$	0	0
$727$	2.18698e7	1.53465	0.767324	−	0.641260i	$-0.221588\pi$
$727$	2.18698e7	1.53465	0.767324	+	0.641260i	$0.221588\pi$
$728$	0	0
$729$	1.17027e7	0.815582
$730$	0	0
$731$	−1.04211e7	−0.721307
$732$	0	0
$733$	−1.92614e6	−0.132412	−0.0662060	−	0.997806i	$-0.521089\pi$
$733$	−1.92614e6	−0.132412	−0.0662060	+	0.997806i	$0.521089\pi$
$734$	0	0
$735$	−3.86995e6	−0.264233
$736$	0	0
$737$	1.44107e7	0.977274
$738$	0	0
$739$	1.64047e7	1.10499	0.552495	−	0.833516i	$-0.313676\pi$
$739$	1.64047e7	1.10499	0.552495	+	0.833516i	$0.313676\pi$
$740$	0	0
$741$	2.32395e7	1.55483
$742$	0	0
$743$	−1.38568e6	−0.0920855	−0.0460428	−	0.998939i	$-0.514661\pi$
$743$	−1.38568e6	−0.0920855	−0.0460428	+	0.998939i	$0.514661\pi$
$744$	0	0
$745$	7.57180e6	0.499814
$746$	0	0
$747$	5.75096e6	0.377084
$748$	0	0
$749$	1.04644e7	0.681570
$750$	0	0
$751$	8.82817e6	0.571177	0.285589	−	0.958352i	$-0.407811\pi$
$751$	8.82817e6	0.571177	0.285589	+	0.958352i	$0.407811\pi$
$752$	0	0
$753$	−5.11909e6	−0.329007
$754$	0	0
$755$	−4.44553e6	−0.283829
$756$	0	0
$757$	−1.03629e7	−0.657265	−0.328632	−	0.944458i	$-0.606588\pi$
$757$	−1.03629e7	−0.657265	−0.328632	+	0.944458i	$0.606588\pi$
$758$	0	0
$759$	−756800.	−0.0476844
$760$	0	0
$761$	4.64947e6	0.291033	0.145516	−	0.989356i	$-0.453516\pi$
$761$	4.64947e6	0.291033	0.145516	+	0.989356i	$0.453516\pi$
$762$	0	0
$763$	−2.02714e7	−1.26058
$764$	0	0
$765$	−2.39359e6	−0.147876
$766$	0	0
$767$	3.15440e7	1.93610
$768$	0	0
$769$	9.17190e6	0.559298	0.279649	−	0.960102i	$-0.409782\pi$
$769$	9.17190e6	0.559298	0.279649	+	0.960102i	$0.409782\pi$
$770$	0	0
$771$	−304634.	−0.0184562
$772$	0	0
$773$	2.98978e6	0.179966	0.0899829	−	0.995943i	$-0.471319\pi$
$773$	2.98978e6	0.179966	0.0899829	+	0.995943i	$0.471319\pi$
$774$	0	0
$775$	3.58579e6	0.214452
$776$	0	0
$777$	−1.28267e7	−0.762187
$778$	0	0
$779$	−5.11494e7	−3.01993
$780$	0	0
$781$	−1.01283e7	−0.594168
$782$	0	0
$783$	2.20048e7	1.28266
$784$	0	0
$785$	1.39249e6	0.0806528
$786$	0	0
$787$	9.40314e6	0.541173	0.270587	−	0.962696i	$-0.412782\pi$
$787$	9.40314e6	0.541173	0.270587	+	0.962696i	$0.412782\pi$
$788$	0	0
$789$	7.93615e6	0.453855
$790$	0	0
$791$	−2.47117e7	−1.40431
$792$	0	0
$793$	8.28318e6	0.467751
$794$	0	0
$795$	74596.4	0.00418601
$796$	0	0
$797$	−8.06747e6	−0.449875	−0.224937	−	0.974373i	$-0.572218\pi$
$797$	−8.06747e6	−0.449875	−0.224937	+	0.974373i	$0.572218\pi$
$798$	0	0
$799$	1.73248e7	0.960066
$800$	0	0
$801$	−1.03728e7	−0.571233
$802$	0	0
$803$	−1.08488e7	−0.593734
$804$	0	0
$805$	−783029.	−0.0425881
$806$	0	0
$807$	−2.21651e7	−1.19808
$808$	0	0
$809$	−9.89326e6	−0.531457	−0.265728	−	0.964048i	$-0.585612\pi$
$809$	−9.89326e6	−0.531457	−0.265728	+	0.964048i	$0.585612\pi$
$810$	0	0
$811$	3.43904e7	1.83605	0.918026	−	0.396520i	$-0.129782\pi$
$811$	3.43904e7	1.83605	0.918026	+	0.396520i	$0.129782\pi$
$812$	0	0
$813$	1.62375e7	0.861576
$814$	0	0
$815$	1.20656e7	0.636289
$816$	0	0
$817$	3.90625e7	2.04741
$818$	0	0
$819$	−1.79510e7	−0.935144
$820$	0	0
$821$	−1.78431e6	−0.0923871	−0.0461936	−	0.998933i	$-0.514709\pi$
$821$	−1.78431e6	−0.0923871	−0.0461936	+	0.998933i	$0.514709\pi$
$822$	0	0
$823$	−3.28202e7	−1.68904	−0.844522	−	0.535521i	$-0.820115\pi$
$823$	−3.28202e7	−1.68904	−0.844522	+	0.535521i	$0.820115\pi$
$824$	0	0
$825$	2.67265e6	0.136712
$826$	0	0
$827$	1.96677e7	0.999976	0.499988	−	0.866032i	$-0.333338\pi$
$827$	1.96677e7	0.999976	0.499988	+	0.866032i	$0.333338\pi$
$828$	0	0
$829$	−2.66433e7	−1.34648	−0.673242	−	0.739422i	$-0.735099\pi$
$829$	−2.66433e7	−1.34648	−0.673242	+	0.739422i	$0.735099\pi$
$830$	0	0
$831$	2.16509e7	1.08761
$832$	0	0
$833$	1.07514e7	0.536851
$834$	0	0
$835$	5.94197e6	0.294927
$836$	0	0
$837$	2.27779e7	1.12383
$838$	0	0
$839$	3.37529e7	1.65541	0.827706	−	0.561163i	$-0.189646\pi$
$839$	3.37529e7	1.65541	0.827706	+	0.561163i	$0.189646\pi$
$840$	0	0
$841$	1.02085e7	0.497703
$842$	0	0
$843$	−1.29810e6	−0.0629126
$844$	0	0
$845$	6.11367e6	0.294551
$846$	0	0
$847$	−51474.9	−0.00246540
$848$	0	0
$849$	1.31559e7	0.626399
$850$	0	0
$851$	−1.20265e6	−0.0569268
$852$	0	0
$853$	−6.29991e6	−0.296457	−0.148228	−	0.988953i	$-0.547357\pi$
$853$	−6.29991e6	−0.296457	−0.148228	+	0.988953i	$0.547357\pi$
$854$	0	0
$855$	8.97215e6	0.419741
$856$	0	0
$857$	−2.84410e7	−1.32280	−0.661399	−	0.750035i	$-0.730037\pi$
$857$	−2.84410e7	−1.32280	−0.661399	+	0.750035i	$0.730037\pi$
$858$	0	0
$859$	−1.11999e7	−0.517882	−0.258941	−	0.965893i	$-0.583374\pi$
$859$	−1.11999e7	−0.517882	−0.258941	+	0.965893i	$0.583374\pi$
$860$	0	0
$861$	−3.47706e7	−1.59847
$862$	0	0
$863$	−1.17959e7	−0.539143	−0.269572	−	0.962980i	$-0.586882\pi$
$863$	−1.17959e7	−0.539143	−0.269572	+	0.962980i	$0.586882\pi$
$864$	0	0
$865$	1.42901e7	0.649374
$866$	0	0
$867$	9.29096e6	0.419771
$868$	0	0
$869$	1.28640e7	0.577866
$870$	0	0
$871$	−2.82052e7	−1.25975
$872$	0	0
$873$	−4.70598e6	−0.208985
$874$	0	0
$875$	2.76528e6	0.122101
$876$	0	0
$877$	−2.54270e7	−1.11634	−0.558169	−	0.829728i	$-0.688496\pi$
$877$	−2.54270e7	−1.11634	−0.558169	+	0.829728i	$0.688496\pi$
$878$	0	0
$879$	−2.06578e6	−0.0901802
$880$	0	0
$881$	184676.	0.00801623	0.00400812	−	0.999992i	$-0.498724\pi$
$881$	184676.	0.00801623	0.00400812	+	0.999992i	$0.498724\pi$
$882$	0	0
$883$	1.39631e7	0.602669	0.301335	−	0.953518i	$-0.402568\pi$
$883$	1.39631e7	0.602669	0.301335	+	0.953518i	$0.402568\pi$
$884$	0	0
$885$	−1.07175e7	−0.459978
$886$	0	0
$887$	−9.88158e6	−0.421713	−0.210857	−	0.977517i	$-0.567625\pi$
$887$	−9.88158e6	−0.421713	−0.210857	+	0.977517i	$0.567625\pi$
$888$	0	0
$889$	3.75547e7	1.59371
$890$	0	0
$891$	4.38435e6	0.185017
$892$	0	0
$893$	−6.49404e7	−2.72512
$894$	0	0
$895$	−2.13626e6	−0.0891450
$896$	0	0
$897$	1.48124e6	0.0614674
$898$	0	0
$899$	3.17990e7	1.31224
$900$	0	0
$901$	−207243.	−0.00850487
$902$	0	0
$903$	2.65541e7	1.08371
$904$	0	0
$905$	−4.79918e6	−0.194781
$906$	0	0
$907$	6.12846e6	0.247362	0.123681	−	0.992322i	$-0.460530\pi$
$907$	6.12846e6	0.247362	0.123681	+	0.992322i	$0.460530\pi$
$908$	0	0
$909$	1.88403e7	0.756272
$910$	0	0
$911$	−5.46846e6	−0.218308	−0.109154	−	0.994025i	$-0.534814\pi$
$911$	−5.46846e6	−0.218308	−0.109154	+	0.994025i	$0.534814\pi$
$912$	0	0
$913$	1.78400e7	0.708299
$914$	0	0
$915$	−2.81434e6	−0.111128
$916$	0	0
$917$	−8.35993e6	−0.328306
$918$	0	0
$919$	−3.54300e7	−1.38383	−0.691914	−	0.721980i	$-0.743232\pi$
$919$	−3.54300e7	−1.38383	−0.691914	+	0.721980i	$0.743232\pi$
$920$	0	0
$921$	−3.02623e7	−1.17558
$922$	0	0
$923$	1.98235e7	0.765909
$924$	0	0
$925$	4.24719e6	0.163210
$926$	0	0
$927$	−1.83347e7	−0.700770
$928$	0	0
$929$	−3.46525e7	−1.31733	−0.658667	−	0.752435i	$-0.728879\pi$
$929$	−3.46525e7	−1.31733	−0.658667	+	0.752435i	$0.728879\pi$
$930$	0	0
$931$	−4.03008e7	−1.52384
$932$	0	0
$933$	−2.19609e7	−0.825936
$934$	0	0
$935$	−7.42512e6	−0.277763
$936$	0	0
$937$	−3.30034e7	−1.22803	−0.614016	−	0.789294i	$-0.710447\pi$
$937$	−3.30034e7	−1.22803	−0.614016	+	0.789294i	$0.710447\pi$
$938$	0	0
$939$	2.76335e7	1.02275
$940$	0	0
$941$	−2.74358e7	−1.01005	−0.505025	−	0.863104i	$-0.668517\pi$
$941$	−2.74358e7	−1.01005	−0.505025	+	0.863104i	$0.668517\pi$
$942$	0	0
$943$	−3.26016e6	−0.119388
$944$	0	0
$945$	1.75658e7	0.639866
$946$	0	0
$947$	631704.	0.0228896	0.0114448	−	0.999935i	$-0.496357\pi$
$947$	631704.	0.0228896	0.0114448	+	0.999935i	$0.496357\pi$
$948$	0	0
$949$	2.12337e7	0.765349
$950$	0	0
$951$	−1.37805e7	−0.494100
$952$	0	0
$953$	−4.38938e7	−1.56556	−0.782782	−	0.622295i	$-0.786200\pi$
$953$	−4.38938e7	−1.56556	−0.782782	+	0.622295i	$0.786200\pi$
$954$	0	0
$955$	−1.80734e7	−0.641254
$956$	0	0
$957$	2.37012e7	0.836546
$958$	0	0
$959$	−9.86699e6	−0.346448
$960$	0	0
$961$	4.28709e6	0.149746
$962$	0	0
$963$	−7.64244e6	−0.265562
$964$	0	0
$965$	−2.21930e7	−0.767182
$966$	0	0
$967$	−1.50345e7	−0.517040	−0.258520	−	0.966006i	$-0.583235\pi$
$967$	−1.50345e7	−0.517040	−0.258520	+	0.966006i	$0.583235\pi$
$968$	0	0
$969$	2.19365e7	0.750514
$970$	0	0
$971$	−2.47407e7	−0.842101	−0.421051	−	0.907037i	$-0.638338\pi$
$971$	−2.47407e7	−0.842101	−0.421051	+	0.907037i	$0.638338\pi$
$972$	0	0
$973$	−6.73654e7	−2.28116
$974$	0	0
$975$	−5.23102e6	−0.176228
$976$	0	0
$977$	1.28586e7	0.430982	0.215491	−	0.976506i	$-0.430865\pi$
$977$	1.28586e7	0.430982	0.215491	+	0.976506i	$0.430865\pi$
$978$	0	0
$979$	−3.21772e7	−1.07298
$980$	0	0
$981$	1.48047e7	0.491165
$982$	0	0
$983$	8.44459e6	0.278737	0.139369	−	0.990241i	$-0.455493\pi$
$983$	8.44459e6	0.278737	0.139369	+	0.990241i	$0.455493\pi$
$984$	0	0
$985$	3.25981e6	0.107054
$986$	0	0
$987$	−4.41454e7	−1.44242
$988$	0	0
$989$	2.48976e6	0.0809408
$990$	0	0
$991$	2.32146e7	0.750892	0.375446	−	0.926844i	$-0.377490\pi$
$991$	2.32146e7	0.750892	0.375446	+	0.926844i	$0.377490\pi$
$992$	0	0
$993$	3.46628e6	0.111555
$994$	0	0
$995$	9.95590e6	0.318803
$996$	0	0
$997$	−4.50681e7	−1.43592	−0.717962	−	0.696082i	$-0.754925\pi$
$997$	−4.50681e7	−1.43592	−0.717962	+	0.696082i	$0.754925\pi$
$998$	0	0
$999$	2.69793e7	0.855298

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.a.h.1.2	✓	4
4.3	odd	2	inner	160.6.a.h.1.3	yes	4
5.2	odd	4		800.6.c.l.449.5		8
5.3	odd	4		800.6.c.l.449.3		8
5.4	even	2		800.6.a.r.1.3		4
8.3	odd	2		320.6.a.z.1.2		4
8.5	even	2		320.6.a.z.1.3		4
20.3	even	4		800.6.c.l.449.6		8
20.7	even	4		800.6.c.l.449.4		8
20.19	odd	2		800.6.a.r.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.h.1.2	✓	4	1.1	even	1	trivial
160.6.a.h.1.3	yes	4	4.3	odd	2	inner
320.6.a.z.1.2		4	8.3	odd	2
320.6.a.z.1.3		4	8.5	even	2
800.6.a.r.1.2		4	20.19	odd	2
800.6.a.r.1.3		4	5.4	even	2
800.6.c.l.449.3		8	5.3	odd	4
800.6.c.l.449.4		8	20.7	even	4
800.6.c.l.449.5		8	5.2	odd	4
800.6.c.l.449.6		8	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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	160.a (trivial)

\(f(q)\)	\(=\)	\(q-10.6653 q^{3} +25.0000 q^{5} +176.978 q^{7} -129.252 q^{9} +O(q^{10})\)	\(q-10.6653 q^{3} +25.0000 q^{5} +176.978 q^{7} -129.252 q^{9} -400.949 q^{11} +784.755 q^{13} -266.632 q^{15} +740.755 q^{17} -2776.65 q^{19} -1887.52 q^{21} -176.978 q^{23} +625.000 q^{25} +3970.17 q^{27} +5542.53 q^{29} +5737.27 q^{31} +4276.24 q^{33} +4424.45 q^{35} +6795.51 q^{37} -8369.64 q^{39} +18421.3 q^{41} -14068.2 q^{43} -3231.29 q^{45} +23388.0 q^{47} +14514.2 q^{49} -7900.36 q^{51} -279.773 q^{53} -10023.7 q^{55} +29613.8 q^{57} +40195.9 q^{59} +10555.1 q^{61} -22874.7 q^{63} +19618.9 q^{65} -35941.5 q^{67} +1887.52 q^{69} +25260.8 q^{71} +27057.7 q^{73} -6665.81 q^{75} -70959.1 q^{77} -32083.9 q^{79} -10934.9 q^{81} -44494.3 q^{83} +18518.9 q^{85} -59112.7 q^{87} +80252.6 q^{89} +138884. q^{91} -61189.6 q^{93} -69416.2 q^{95} +36409.5 q^{97} +51823.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 100 q^{5} + 724 q^{9}+O(q^{10}) \) 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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Modular forms

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