LMFDB - Embedded newform 160.6.f.a.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(49,160)]

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, 1, 1]))

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

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

chi := DirichletCharacter("160.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	160.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Character	$\chi$	$=$	160.49
Dual form			160.6.f.a.49.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.6833 q^{3} +(36.8367 - 42.0483i) q^{5} -236.448i q^{7} +227.167 q^{9} +O(q^{10})$	$q-21.6833 q^{3} +(36.8367 - 42.0483i) q^{5} -236.448i q^{7} +227.167 q^{9} +192.686i q^{11} +975.206 q^{13} +(-798.742 + 911.747i) q^{15} -670.908i q^{17} +456.924i q^{19} +5126.98i q^{21} -2029.03i q^{23} +(-411.118 - 3097.84i) q^{25} +343.317 q^{27} -2782.43i q^{29} +963.333 q^{31} -4178.07i q^{33} +(-9942.24 - 8709.97i) q^{35} -8727.50 q^{37} -21145.7 q^{39} -1812.39 q^{41} -254.225 q^{43} +(8368.07 - 9551.98i) q^{45} -19737.7i q^{47} -39100.7 q^{49} +14547.5i q^{51} -23002.2 q^{53} +(8102.10 + 7097.90i) q^{55} -9907.64i q^{57} +24661.0i q^{59} -189.904i q^{61} -53713.2i q^{63} +(35923.3 - 41005.7i) q^{65} +24912.8 q^{67} +43996.2i q^{69} +38239.7 q^{71} +44775.5i q^{73} +(8914.41 + 67171.5i) q^{75} +45560.2 q^{77} -69761.6 q^{79} -62645.8 q^{81} +4339.15 q^{83} +(-28210.5 - 24714.0i) q^{85} +60332.4i q^{87} -5562.17 q^{89} -230586. i q^{91} -20888.3 q^{93} +(19212.9 + 16831.6i) q^{95} -98102.9i q^{97} +43771.8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 1940 q^{9}+O(q^{10}) $ 28 * q + 1940 * q^9	$ 28 q + 1940 q^{9} + 488 q^{15} + 1556 q^{25} - 4368 q^{31} - 23360 q^{39} - 2480 q^{41} - 38420 q^{49} + 48776 q^{55} + 37200 q^{65} + 69232 q^{71} + 35984 q^{79} + 122596 q^{81} - 178744 q^{89} - 89416 q^{95}+O(q^{100}) $ 28 * q + 1940 * q^9 + 488 * q^15 + 1556 * q^25 - 4368 * q^31 - 23360 * q^39 - 2480 * q^41 - 38420 * q^49 + 48776 * q^55 + 37200 * q^65 + 69232 * q^71 + 35984 * q^79 + 122596 * q^81 - 178744 * q^89 - 89416 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−21.6833			−1.39099			−0.695493	−	0.718533i	$-0.744814\pi$
$3$	−21.6833			−1.39099			−0.695493	+	0.718533i	$0.744814\pi$
$4$		0			0
$5$	36.8367	−	42.0483i	0.658955	−	0.752183i
$5$	36.8367	−	42.0483i	0.658955	−	0.752183i
$6$		0			0
$7$		−	236.448i		−	1.82386i	−0.410349	−	0.911929i	$-0.634593\pi$
$7$		−	236.448i		−	1.82386i	0.410349	−	0.911929i	$-0.365407\pi$
$8$		0			0
$9$	227.167			0.934843
$10$		0			0
$11$			192.686i			0.480140i	0.970756	+	0.240070i	$0.0771704\pi$
$11$			192.686i			0.480140i	−0.970756	+	0.240070i	$0.922830\pi$
$12$		0			0
$13$	975.206			1.60043			0.800217	−	0.599710i	$-0.204718\pi$
$13$	975.206			1.60043			0.800217	+	0.599710i	$0.204718\pi$
$14$		0			0
$15$	−798.742	+	911.747i	−0.916597	+	1.04628i
$16$		0			0
$17$		−	670.908i		−	0.563042i	−0.959555	−	0.281521i	$-0.909161\pi$
$17$		−	670.908i		−	0.563042i	0.959555	−	0.281521i	$-0.0908388\pi$
$18$		0			0
$19$			456.924i			0.290376i	0.989404	+	0.145188i	$0.0463786\pi$
$19$			456.924i			0.290376i	−0.989404	+	0.145188i	$0.953621\pi$
$20$		0			0
$21$			5126.98i			2.53696i
$22$		0			0
$23$		−	2029.03i		−	0.799778i	−0.916564	−	0.399889i	$-0.869049\pi$
$23$		−	2029.03i		−	0.799778i	0.916564	−	0.399889i	$-0.130951\pi$
$24$		0			0
$25$	−411.118	−	3097.84i	−0.131558	−	0.991309i
$26$		0			0
$27$	343.317			0.0906329
$28$		0			0
$29$		−	2782.43i		−	0.614370i	−0.951650	−	0.307185i	$-0.900613\pi$
$29$		−	2782.43i		−	0.614370i	0.951650	−	0.307185i	$-0.0993870\pi$
$30$		0			0
$31$	963.333			0.180041			0.0900207	−	0.995940i	$-0.471307\pi$
$31$	963.333			0.180041			0.0900207	+	0.995940i	$0.471307\pi$
$32$		0			0
$33$		−	4178.07i		−	0.667868i
$34$		0			0
$35$	−9942.24	−	8709.97i	−1.37187	−	1.20184i
$36$		0			0
$37$	−8727.50			−1.04806			−0.524030	−	0.851700i	$-0.675572\pi$
$37$	−8727.50			−1.04806			−0.524030	+	0.851700i	$0.675572\pi$
$38$		0			0
$39$	−21145.7			−2.22618
$40$		0			0
$41$	−1812.39			−0.168381			−0.0841905	−	0.996450i	$-0.526830\pi$
$41$	−1812.39			−0.168381			−0.0841905	+	0.996450i	$0.526830\pi$
$42$		0			0
$43$	−254.225			−0.0209675			−0.0104838	−	0.999945i	$-0.503337\pi$
$43$	−254.225			−0.0209675			−0.0104838	+	0.999945i	$0.503337\pi$
$44$		0			0
$45$	8368.07	−	9551.98i	0.616019	−	0.703173i
$46$		0			0
$47$		−	19737.7i		−	1.30332i	−0.758511	−	0.651660i	$-0.774073\pi$
$47$		−	19737.7i		−	1.30332i	0.758511	−	0.651660i	$-0.225927\pi$
$48$		0			0
$49$	−39100.7			−2.32646
$50$		0			0
$51$			14547.5i			0.783183i
$52$		0			0
$53$	−23002.2			−1.12481			−0.562405	−	0.826862i	$-0.690124\pi$
$53$	−23002.2			−1.12481			−0.562405	+	0.826862i	$0.690124\pi$
$54$		0			0
$55$	8102.10	+	7097.90i	0.361153	+	0.316390i
$56$		0			0
$57$		−	9907.64i		−	0.403908i
$58$		0			0
$59$			24661.0i			0.922319i	0.887317	+	0.461159i	$0.152566\pi$
$59$			24661.0i			0.922319i	−0.887317	+	0.461159i	$0.847434\pi$
$60$		0			0
$61$		−	189.904i		−	0.00653447i	−0.999995	−	0.00326723i	$-0.998960\pi$
$61$		−	189.904i		−	0.00653447i	0.999995	−	0.00326723i	$-0.00103999\pi$
$62$		0			0
$63$		−	53713.2i		−	1.70502i
$64$		0			0
$65$	35923.3	−	41005.7i	1.05461	−	1.20382i
$66$		0			0
$67$	24912.8			0.678010			0.339005	−	0.940785i	$-0.389910\pi$
$67$	24912.8			0.678010			0.339005	+	0.940785i	$0.389910\pi$
$68$		0			0
$69$			43996.2i			1.11248i
$70$		0			0
$71$	38239.7			0.900262			0.450131	−	0.892963i	$-0.351377\pi$
$71$	38239.7			0.900262			0.450131	+	0.892963i	$0.351377\pi$
$72$		0			0
$73$			44775.5i			0.983407i	0.870763	+	0.491704i	$0.163626\pi$
$73$			44775.5i			0.983407i	−0.870763	+	0.491704i	$0.836374\pi$
$74$		0			0
$75$	8914.41	+	67171.5i	0.182995	+	1.37890i
$76$		0			0
$77$	45560.2			0.875707
$78$		0			0
$79$	−69761.6			−1.25762			−0.628809	−	0.777560i	$-0.716457\pi$
$79$	−69761.6			−1.25762			−0.628809	+	0.777560i	$0.716457\pi$
$80$		0			0
$81$	−62645.8			−1.06091
$82$		0			0
$83$	4339.15			0.0691368			0.0345684	−	0.999402i	$-0.488994\pi$
$83$	4339.15			0.0691368			0.0345684	+	0.999402i	$0.488994\pi$
$84$		0			0
$85$	−28210.5	−	24714.0i	−0.423510	−	0.371019i
$86$		0			0
$87$			60332.4i			0.854580i
$88$		0			0
$89$	−5562.17			−0.0744336			−0.0372168	−	0.999307i	$-0.511849\pi$
$89$	−5562.17			−0.0744336			−0.0372168	+	0.999307i	$0.511849\pi$
$90$		0			0
$91$		−	230586.i		−	2.91896i
$92$		0			0
$93$	−20888.3			−0.250435
$94$		0			0
$95$	19212.9	+	16831.6i	0.218415	+	0.191344i
$96$		0			0
$97$		−	98102.9i		−	1.05865i	−0.848419	−	0.529325i	$-0.822445\pi$
$97$		−	98102.9i		−	1.05865i	0.848419	−	0.529325i	$-0.177555\pi$
$98$		0			0
$99$			43771.8i			0.448855i
$100$		0			0
$101$		−	19871.5i		−	0.193833i	−0.995293	−	0.0969163i	$-0.969102\pi$
$101$		−	19871.5i		−	0.193833i	0.995293	−	0.0969163i	$-0.0308979\pi$
$102$		0			0
$103$			120201.i			1.11639i	0.829711	+	0.558194i	$0.188505\pi$
$103$			120201.i			1.11639i	−0.829711	+	0.558194i	$0.811495\pi$
$104$		0			0
$105$	215581.	+	188861.i	1.90826	+	1.67174i
$106$		0			0
$107$	−125007.			−1.05554			−0.527769	−	0.849388i	$-0.676971\pi$
$107$	−125007.			−1.05554			−0.527769	+	0.849388i	$0.676971\pi$
$108$		0			0
$109$		−	31222.3i		−	0.251709i	−0.992049	−	0.125854i	$-0.959833\pi$
$109$		−	31222.3i		−	0.251709i	0.992049	−	0.125854i	$-0.0401672\pi$
$110$		0			0
$111$	189241.			1.45784
$112$		0			0
$113$			18028.4i			0.132819i	0.997792	+	0.0664097i	$0.0211544\pi$
$113$			18028.4i			0.132819i	−0.997792	+	0.0664097i	$0.978846\pi$
$114$		0			0
$115$	−85317.4	−	74742.9i	−0.601579	−	0.527018i
$116$		0			0
$117$	221534.			1.49615
$118$		0			0
$119$	−158635.			−1.02691
$120$		0			0
$121$	123923.			0.769466
$122$		0			0
$123$	39298.7			0.234216
$124$		0			0
$125$	−145403.	−	96827.3i	−0.832336	−	0.554272i
$126$		0			0
$127$			66022.6i			0.363231i	0.983370	+	0.181616i	$0.0581326\pi$
$127$			66022.6i			0.363231i	−0.983370	+	0.181616i	$0.941867\pi$
$128$		0			0
$129$	5512.45			0.0291656
$130$		0			0
$131$			147752.i			0.752238i	0.926571	+	0.376119i	$0.122742\pi$
$131$			147752.i			0.752238i	−0.926571	+	0.376119i	$0.877258\pi$
$132$		0			0
$133$	108039.			0.529604
$134$		0			0
$135$	12646.7	−	14435.9i	0.0597230	−	0.0681725i
$136$		0			0
$137$			357827.i			1.62881i	0.580295	+	0.814406i	$0.302937\pi$
$137$			357827.i			1.62881i	−0.580295	+	0.814406i	$0.697063\pi$
$138$		0			0
$139$			49329.3i			0.216555i	0.994121	+	0.108277i	$0.0345335\pi$
$139$			49329.3i			0.216555i	−0.994121	+	0.108277i	$0.965467\pi$
$140$		0			0
$141$			427979.i			1.81290i
$142$		0			0
$143$			187908.i			0.768432i
$144$		0			0
$145$	−116997.	−	102496.i	−0.462118	−	0.404842i
$146$		0			0
$147$	847834.			3.23607
$148$		0			0
$149$		−	310847.i		−	1.14705i	−0.819189	−	0.573524i	$-0.805576\pi$
$149$		−	310847.i		−	1.14705i	0.819189	−	0.573524i	$-0.194424\pi$
$150$		0			0
$151$	367611.			1.31204			0.656019	−	0.754744i	$-0.272239\pi$
$151$	367611.			1.31204			0.656019	+	0.754744i	$0.272239\pi$
$152$		0			0
$153$		−	152408.i		−	0.526355i
$154$		0			0
$155$	35486.0	−	40506.5i	0.118639	−	0.135424i
$156$		0			0
$157$	−51468.7			−0.166646			−0.0833228	−	0.996523i	$-0.526553\pi$
$157$	−51468.7			−0.166646			−0.0833228	+	0.996523i	$0.526553\pi$
$158$		0			0
$159$	498764.			1.56459
$160$		0			0
$161$	−479761.			−1.45868
$162$		0			0
$163$	−471607.			−1.39031			−0.695155	−	0.718860i	$-0.744664\pi$
$163$	−471607.			−1.39031			−0.695155	+	0.718860i	$0.744664\pi$
$164$		0			0
$165$	−175681.	−	153906.i	−0.502359	−	0.440095i
$166$		0			0
$167$		−	644743.i		−	1.78894i	−0.447130	−	0.894469i	$-0.647554\pi$
$167$		−	644743.i		−	1.78894i	0.447130	−	0.894469i	$-0.352446\pi$
$168$		0			0
$169$	579733.			1.56139
$170$		0			0
$171$			103798.i			0.271455i
$172$		0			0
$173$	−162302.			−0.412295			−0.206147	−	0.978521i	$-0.566093\pi$
$173$	−162302.			−0.412295			−0.206147	+	0.978521i	$0.566093\pi$
$174$		0			0
$175$	−732478.	+	97208.1i	−1.80801	+	0.239943i
$176$		0			0
$177$		−	534733.i		−	1.28293i
$178$		0			0
$179$		−	74777.3i		−	0.174436i	−0.996189	−	0.0872182i	$-0.972202\pi$
$179$		−	74777.3i		−	0.174436i	0.996189	−	0.0872182i	$-0.0277977\pi$
$180$		0			0
$181$			69208.2i			0.157022i	0.996913	+	0.0785111i	$0.0250166\pi$
$181$			69208.2i			0.157022i	−0.996913	+	0.0785111i	$0.974983\pi$
$182$		0			0
$183$			4117.76i			0.00908935i
$184$		0			0
$185$	−321492.	+	366977.i	−0.690623	+	0.788332i
$186$		0			0
$187$	129274.			0.270339
$188$		0			0
$189$		−	81176.7i		−	0.165301i
$190$		0			0
$191$	−712525.			−1.41324			−0.706621	−	0.707592i	$-0.749782\pi$
$191$	−712525.			−1.41324			−0.706621	+	0.707592i	$0.749782\pi$
$192$		0			0
$193$			191610.i			0.370276i	0.982713	+	0.185138i	$0.0592732\pi$
$193$			191610.i			0.370276i	−0.982713	+	0.185138i	$0.940727\pi$
$194$		0			0
$195$	−778938.	+	889141.i	−1.46695	+	1.67450i
$196$		0			0
$197$	447522.			0.821578			0.410789	−	0.911730i	$-0.365253\pi$
$197$	447522.			0.821578			0.410789	+	0.911730i	$0.365253\pi$
$198$		0			0
$199$	−116939.			−0.209328			−0.104664	−	0.994508i	$-0.533377\pi$
$199$	−116939.			−0.209328			−0.104664	+	0.994508i	$0.533377\pi$
$200$		0			0
$201$	−540193.			−0.943103
$202$		0			0
$203$	−657902.			−1.12052
$204$		0			0
$205$	−66762.6	+	76208.1i	−0.110955	+	0.126653i
$206$		0			0
$207$		−	460929.i		−	0.747667i
$208$		0			0
$209$	−88042.7			−0.139421
$210$		0			0
$211$		−	669883.i		−	1.03584i	−0.855429	−	0.517920i	$-0.826706\pi$
$211$		−	669883.i		−	1.03584i	0.855429	−	0.517920i	$-0.173294\pi$
$212$		0			0
$213$	−829164.			−1.25225
$214$		0			0
$215$	−9364.81	+	10689.7i	−0.0138167	+	0.0157714i
$216$		0			0
$217$		−	227778.i		−	0.328370i
$218$		0			0
$219$		−	970882.i		−	1.36791i
$220$		0			0
$221$		−	654273.i		−	0.901111i
$222$		0			0
$223$			1718.93i			0.00231470i	0.999999	+	0.00115735i	$0.000368396\pi$
$223$			1718.93i			0.00231470i	−0.999999	+	0.00115735i	$0.999632\pi$
$224$		0			0
$225$	−93392.3	−	703726.i	−0.122986	−	0.926718i
$226$		0			0
$227$	713312.			0.918787			0.459393	−	0.888233i	$-0.348067\pi$
$227$	713312.			0.918787			0.459393	+	0.888233i	$0.348067\pi$
$228$		0			0
$229$			1.17021e6i			1.47461i	0.675561	+	0.737304i	$0.263902\pi$
$229$			1.17021e6i			1.47461i	−0.675561	+	0.737304i	$0.736098\pi$
$230$		0			0
$231$	−987896.			−1.21810
$232$		0			0
$233$		−	1.52755e6i		−	1.84334i	−0.387976	−	0.921669i	$-0.626826\pi$
$233$		−	1.52755e6i		−	1.84334i	0.387976	−	0.921669i	$-0.373174\pi$
$234$		0			0
$235$	−829936.	−	727070.i	−0.980335	−	0.858829i
$236$		0			0
$237$	1.51266e6			1.74933
$238$		0			0
$239$	567244.			0.642355			0.321178	−	0.947019i	$-0.395921\pi$
$239$	567244.			0.642355			0.321178	+	0.947019i	$0.395921\pi$
$240$		0			0
$241$	688312.			0.763383			0.381692	−	0.924290i	$-0.375342\pi$
$241$	688312.			0.763383			0.381692	+	0.924290i	$0.375342\pi$
$242$		0			0
$243$	1.27494e6			1.38508
$244$		0			0
$245$	−1.44034e6	+	1.64412e6i	−1.53303	+	1.74992i
$246$		0			0
$247$			445595.i			0.464727i
$248$		0			0
$249$	−94087.1			−0.0961683
$250$		0			0
$251$			663210.i			0.664457i	0.943199	+	0.332229i	$0.107801\pi$
$251$			663210.i			0.664457i	−0.943199	+	0.332229i	$0.892199\pi$
$252$		0			0
$253$	390966.			0.384005
$254$		0			0
$255$	611698.	+	535882.i	0.589097	+	0.516082i
$256$		0			0
$257$		−	1.68030e6i		−	1.58691i	−0.608628	−	0.793456i	$-0.708280\pi$
$257$		−	1.68030e6i		−	1.58691i	0.608628	−	0.793456i	$-0.291720\pi$
$258$		0			0
$259$			2.06360e6i			1.91151i
$260$		0			0
$261$		−	632077.i		−	0.574339i
$262$		0			0
$263$		−	1.66101e6i		−	1.48075i	−0.672194	−	0.740375i	$-0.734648\pi$
$263$		−	1.66101e6i		−	1.48075i	0.672194	−	0.740375i	$-0.265352\pi$
$264$		0			0
$265$	−847323.	+	967202.i	−0.741199	+	0.846063i
$266$		0			0
$267$	120606.			0.103536
$268$		0			0
$269$		−	686913.i		−	0.578790i	−0.957210	−	0.289395i	$-0.906546\pi$
$269$		−	686913.i		−	0.578790i	0.957210	−	0.289395i	$-0.0934541\pi$
$270$		0			0
$271$	−237762.			−0.196662			−0.0983309	−	0.995154i	$-0.531350\pi$
$271$	−237762.			−0.196662			−0.0983309	+	0.995154i	$0.531350\pi$
$272$		0			0
$273$			4.99986e6i			4.06024i
$274$		0			0
$275$	596909.	−	79216.5i	0.475967	−	0.0631661i
$276$		0			0
$277$	2.37393e6			1.85895			0.929475	−	0.368886i	$-0.120261\pi$
$277$	2.37393e6			1.85895			0.929475	+	0.368886i	$0.120261\pi$
$278$		0			0
$279$	218837.			0.168310
$280$		0			0
$281$	726368.			0.548771			0.274386	−	0.961620i	$-0.411526\pi$
$281$	726368.			0.548771			0.274386	+	0.961620i	$0.411526\pi$
$282$		0			0
$283$	−1.05205e6			−0.780853			−0.390427	−	0.920634i	$-0.627672\pi$
$283$	−1.05205e6			−0.780853			−0.390427	+	0.920634i	$0.627672\pi$
$284$		0			0
$285$	−416599.	−	364964.i	−0.303813	−	0.266157i
$286$		0			0
$287$			428537.i			0.307103i
$288$		0			0
$289$	969740.			0.682984
$290$		0			0
$291$			2.12720e6i			1.47257i
$292$		0			0
$293$	−1.54163e6			−1.04909			−0.524544	−	0.851383i	$-0.675764\pi$
$293$	−1.54163e6			−1.04909			−0.524544	+	0.851383i	$0.675764\pi$
$294$		0			0
$295$	1.03695e6	+	908430.i	0.693752	+	0.607766i
$296$		0			0
$297$			66152.3i			0.0435165i
$298$		0			0
$299$		−	1.97872e6i		−	1.27999i
$300$		0			0
$301$			60111.1i			0.0382418i
$302$		0			0
$303$			430880.i			0.269619i
$304$		0			0
$305$	−7985.15	−	6995.44i	−0.00491511	−	0.00430592i
$306$		0			0
$307$	2.21416e6			1.34080			0.670398	−	0.742002i	$-0.266123\pi$
$307$	2.21416e6			1.34080			0.670398	+	0.742002i	$0.266123\pi$
$308$		0			0
$309$		−	2.60636e6i		−	1.55288i
$310$		0			0
$311$	1.53824e6			0.901827			0.450913	−	0.892568i	$-0.351098\pi$
$311$	1.53824e6			0.901827			0.450913	+	0.892568i	$0.351098\pi$
$312$		0			0
$313$		−	2.33683e6i		−	1.34824i	−0.738622	−	0.674120i	$-0.764523\pi$
$313$		−	2.33683e6i		−	1.34824i	0.738622	−	0.674120i	$-0.235477\pi$
$314$		0			0
$315$	−2.25855e6	−	1.97862e6i	−1.28249	−	1.12353i
$316$		0			0
$317$	−1.15261e6			−0.644220			−0.322110	−	0.946702i	$-0.604392\pi$
$317$	−1.15261e6			−0.644220			−0.322110	+	0.946702i	$0.604392\pi$
$318$		0			0
$319$	536135.			0.294983
$320$		0			0
$321$	2.71056e6			1.46824
$322$		0			0
$323$	306554.			0.163494
$324$		0			0
$325$	−400925.	−	3.02103e6i	−0.210550	−	1.58652i
$326$		0			0
$327$			677003.i			0.350123i
$328$		0			0
$329$	−4.66694e6			−2.37707
$330$		0			0
$331$		−	2.04520e6i		−	1.02604i	−0.858376	−	0.513022i	$-0.828526\pi$
$331$		−	2.04520e6i		−	1.02604i	0.858376	−	0.513022i	$-0.171474\pi$
$332$		0			0
$333$	−1.98260e6			−0.979770
$334$		0			0
$335$	917706.	−	1.04754e6i	0.446778	−	0.509988i
$336$		0			0
$337$			3.68207e6i			1.76611i	0.469273	+	0.883053i	$0.344516\pi$
$337$			3.68207e6i			1.76611i	−0.469273	+	0.883053i	$0.655484\pi$
$338$		0			0
$339$		−	390916.i		−	0.184750i
$340$		0			0
$341$			185620.i			0.0864450i
$342$		0			0
$343$			5.27132e6i			2.41927i
$344$		0			0
$345$	1.84996e6	+	1.62067e6i	0.836789	+	0.733074i
$346$		0			0
$347$	2.20981e6			0.985215			0.492607	−	0.870252i	$-0.336044\pi$
$347$	2.20981e6			0.985215			0.492607	+	0.870252i	$0.336044\pi$
$348$		0			0
$349$			4.13502e6i			1.81725i	0.417618	+	0.908623i	$0.362865\pi$
$349$			4.13502e6i			1.81725i	−0.417618	+	0.908623i	$0.637135\pi$
$350$		0			0
$351$	334805.			0.145052
$352$		0			0
$353$			376581.i			0.160850i	0.996761	+	0.0804251i	$0.0256278\pi$
$353$			376581.i			0.160850i	−0.996761	+	0.0804251i	$0.974372\pi$
$354$		0			0
$355$	1.40862e6	−	1.60791e6i	0.593232	−	0.677161i
$356$		0			0
$357$	3.43973e6			1.42841
$358$		0			0
$359$	−4.04157e6			−1.65506			−0.827530	−	0.561422i	$-0.810254\pi$
$359$	−4.04157e6			−1.65506			−0.827530	+	0.561422i	$0.810254\pi$
$360$		0			0
$361$	2.26732e6			0.915682
$362$		0			0
$363$	−2.68707e6			−1.07032
$364$		0			0
$365$	1.88273e6	+	1.64938e6i	0.739702	+	0.648021i
$366$		0			0
$367$		−	4.17682e6i		−	1.61875i	−0.587289	−	0.809377i	$-0.699805\pi$
$367$		−	4.17682e6i		−	1.61875i	0.587289	−	0.809377i	$-0.300195\pi$
$368$		0			0
$369$	−411716.			−0.157410
$370$		0			0
$371$			5.43882e6i			2.05149i
$372$		0			0
$373$	3.84926e6			1.43254			0.716268	−	0.697826i	$-0.245849\pi$
$373$	3.84926e6			1.43254			0.716268	+	0.697826i	$0.245849\pi$
$374$		0			0
$375$	3.15282e6	+	2.09954e6i	1.15777	+	0.770984i
$376$		0			0
$377$		−	2.71345e6i		−	0.983259i
$378$		0			0
$379$			771105.i			0.275750i	0.990450	+	0.137875i	$0.0440272\pi$
$379$			771105.i			0.275750i	−0.990450	+	0.137875i	$0.955973\pi$
$380$		0			0
$381$		−	1.43159e6i		−	0.505250i
$382$		0			0
$383$		−	571373.i		−	0.199032i	−0.995036	−	0.0995160i	$-0.968271\pi$
$383$		−	571373.i		−	0.199032i	0.995036	−	0.0995160i	$-0.0317294\pi$
$384$		0			0
$385$	1.67829e6	−	1.91573e6i	0.577051	−	0.658691i
$386$		0			0
$387$	−57751.5			−0.0196014
$388$		0			0
$389$			5.01422e6i			1.68008i	0.542525	+	0.840040i	$0.317468\pi$
$389$			5.01422e6i			1.68008i	−0.542525	+	0.840040i	$0.682532\pi$
$390$		0			0
$391$	−1.36129e6			−0.450308
$392$		0			0
$393$		−	3.20376e6i		−	1.04635i
$394$		0			0
$395$	−2.56979e6	+	2.93336e6i	−0.828713	+	0.945959i
$396$		0			0
$397$	−5.51448e6			−1.75602			−0.878008	−	0.478647i	$-0.841127\pi$
$397$	−5.51448e6			−1.75602			−0.878008	+	0.478647i	$0.841127\pi$
$398$		0			0
$399$	−2.34264e6			−0.736671
$400$		0			0
$401$	646084.			0.200645			0.100322	−	0.994955i	$-0.468013\pi$
$401$	646084.			0.200645			0.100322	+	0.994955i	$0.468013\pi$
$402$		0			0
$403$	939448.			0.288144
$404$		0			0
$405$	−2.30766e6	+	2.63415e6i	−0.699093	+	0.798000i
$406$		0			0
$407$		−	1.68167e6i		−	0.503215i
$408$		0			0
$409$	1.02076e6			0.301726			0.150863	−	0.988555i	$-0.451795\pi$
$409$	1.02076e6			0.301726			0.150863	+	0.988555i	$0.451795\pi$
$410$		0			0
$411$		−	7.75887e6i		−	2.26566i
$412$		0			0
$413$	5.83105e6			1.68218
$414$		0			0
$415$	159840.	−	182454.i	0.0455580	−	0.0520035i
$416$		0			0
$417$		−	1.06962e6i		−	0.301225i
$418$		0			0
$419$			1.63720e6i			0.455581i	0.973710	+	0.227790i	$0.0731501\pi$
$419$			1.63720e6i			0.455581i	−0.973710	+	0.227790i	$0.926850\pi$
$420$		0			0
$421$		−	3.24490e6i		−	0.892270i	−0.894966	−	0.446135i	$-0.852800\pi$
$421$		−	3.24490e6i		−	0.892270i	0.894966	−	0.446135i	$-0.147200\pi$
$422$		0			0
$423$		−	4.48374e6i		−	1.21840i
$424$		0			0
$425$	−2.07836e6	+	275822.i	−0.558148	+	0.0740725i
$426$		0			0
$427$	−44902.5			−0.0119179
$428$		0			0
$429$		−	4.07447e6i		−	1.06888i
$430$		0			0
$431$	−3.39290e6			−0.879789			−0.439894	−	0.898049i	$-0.644984\pi$
$431$	−3.39290e6			−0.879789			−0.439894	+	0.898049i	$0.644984\pi$
$432$		0			0
$433$		−	2.40449e6i		−	0.616315i	−0.951335	−	0.308158i	$-0.900288\pi$
$433$		−	2.40449e6i		−	0.616315i	0.951335	−	0.308158i	$-0.0997124\pi$
$434$		0			0
$435$	2.53688e6	+	2.22245e6i	0.642800	+	0.563129i
$436$		0			0
$437$	927114.			0.232236
$438$		0			0
$439$	5.28610e6			1.30910			0.654552	−	0.756017i	$-0.272857\pi$
$439$	5.28610e6			1.30910			0.654552	+	0.756017i	$0.272857\pi$
$440$		0			0
$441$	−8.88239e6			−2.17487
$442$		0			0
$443$	−4.55556e6			−1.10289			−0.551445	−	0.834211i	$-0.685923\pi$
$443$	−4.55556e6			−1.10289			−0.551445	+	0.834211i	$0.685923\pi$
$444$		0			0
$445$	−204892.	+	233880.i	−0.0490484	+	0.0559877i
$446$		0			0
$447$			6.74020e6i			1.59553i
$448$		0			0
$449$	−3.24132e6			−0.758762			−0.379381	−	0.925241i	$-0.623863\pi$
$449$	−3.24132e6			−0.758762			−0.379381	+	0.925241i	$0.623863\pi$
$450$		0			0
$451$		−	349222.i		−	0.0808464i
$452$		0			0
$453$	−7.97104e6			−1.82503
$454$		0			0
$455$	−9.69573e6	−	8.49401e6i	−2.19559	−	1.92346i
$456$		0			0
$457$			5.12416e6i			1.14771i	0.818957	+	0.573855i	$0.194553\pi$
$457$			5.12416e6i			1.14771i	−0.818957	+	0.573855i	$0.805447\pi$
$458$		0			0
$459$		−	230334.i		−	0.0510301i
$460$		0			0
$461$			789831.i			0.173094i	0.996248	+	0.0865470i	$0.0275833\pi$
$461$			789831.i			0.173094i	−0.996248	+	0.0865470i	$0.972417\pi$
$462$		0			0
$463$		−	73457.7i		−	0.0159252i	−0.999968	−	0.00796260i	$-0.997465\pi$
$463$		−	73457.7i		−	0.0159252i	0.999968	−	0.00796260i	$-0.00253460\pi$
$464$		0			0
$465$	−769454.	+	878316.i	−0.165025	+	0.188373i
$466$		0			0
$467$	−5.37276e6			−1.14000			−0.570001	−	0.821644i	$-0.693057\pi$
$467$	−5.37276e6			−1.14000			−0.570001	+	0.821644i	$0.693057\pi$
$468$		0			0
$469$		−	5.89060e6i		−	1.23659i
$470$		0			0
$471$	1.11601e6			0.231802
$472$		0			0
$473$		−	48985.6i		−	0.0100674i
$474$		0			0
$475$	1.41548e6	−	187850.i	0.287852	−	0.0382011i
$476$		0			0
$477$	−5.22533e6			−1.05152
$478$		0			0
$479$	−2.42372e6			−0.482663			−0.241331	−	0.970443i	$-0.577584\pi$
$479$	−2.42372e6			−0.482663			−0.241331	+	0.970443i	$0.577584\pi$
$480$		0			0
$481$	−8.51111e6			−1.67735
$482$		0			0
$483$	1.04028e7			2.02901
$484$		0			0
$485$	−4.12506e6	−	3.61379e6i	−0.796299	−	0.697603i
$486$		0			0
$487$		−	4.49413e6i		−	0.858663i	−0.903147	−	0.429332i	$-0.858749\pi$
$487$		−	4.49413e6i		−	0.858663i	0.903147	−	0.429332i	$-0.141251\pi$
$488$		0			0
$489$	1.02260e7			1.93390
$490$		0			0
$491$		−	3.33347e6i		−	0.624011i	−0.950080	−	0.312006i	$-0.898999\pi$
$491$		−	3.33347e6i		−	0.624011i	0.950080	−	0.312006i	$-0.101001\pi$
$492$		0			0
$493$	−1.86676e6			−0.345916
$494$		0			0
$495$	1.84053e6	+	1.61241e6i	0.337621	+	0.295775i
$496$		0			0
$497$		−	9.04171e6i		−	1.64195i
$498$		0			0
$499$			6.47383e6i			1.16388i	0.813230	+	0.581942i	$0.197707\pi$
$499$			6.47383e6i			1.16388i	−0.813230	+	0.581942i	$0.802293\pi$
$500$		0			0
$501$			1.39802e7i			2.48839i
$502$		0			0
$503$		−	754370.i		−	0.132943i	−0.997788	−	0.0664713i	$-0.978826\pi$
$503$		−	754370.i		−	0.132943i	0.997788	−	0.0664713i	$-0.0211741\pi$
$504$		0			0
$505$	−835562.	−	732000.i	−0.145798	−	0.127727i
$506$		0			0
$507$	−1.25705e7			−2.17187
$508$		0			0
$509$		−	2.62746e6i		−	0.449513i	−0.974415	−	0.224757i	$-0.927841\pi$
$509$		−	2.62746e6i		−	0.449513i	0.974415	−	0.224757i	$-0.0721587\pi$
$510$		0			0
$511$	1.05871e7			1.79359
$512$		0			0
$513$			156870.i			0.0263176i
$514$		0			0
$515$	5.05424e6	+	4.42780e6i	0.839727	+	0.735648i
$516$		0			0
$517$	3.80317e6			0.625776
$518$		0			0
$519$	3.51924e6			0.573497
$520$		0			0
$521$	5.10889e6			0.824580			0.412290	−	0.911053i	$-0.364729\pi$
$521$	5.10889e6			0.824580			0.412290	+	0.911053i	$0.364729\pi$
$522$		0			0
$523$	2.60300e6			0.416121			0.208060	−	0.978116i	$-0.433285\pi$
$523$	2.60300e6			0.416121			0.208060	+	0.978116i	$0.433285\pi$
$524$		0			0
$525$	1.58826e7	−	2.10780e6i	2.51491	−	0.333757i
$526$		0			0
$527$		−	646308.i		−	0.101371i
$528$		0			0
$529$	2.31937e6			0.360355
$530$		0			0
$531$			5.60216e6i			0.862223i
$532$		0			0
$533$	−1.76746e6			−0.269483
$534$		0			0
$535$	−4.60483e6	+	5.25632e6i	−0.695551	+	0.793957i
$536$		0			0
$537$			1.62142e6i			0.242639i
$538$		0			0
$539$		−	7.53415e6i		−	1.11702i
$540$		0			0
$541$			2.80703e6i			0.412339i	0.978516	+	0.206169i	$0.0660998\pi$
$541$			2.80703e6i			0.412339i	−0.978516	+	0.206169i	$0.933900\pi$
$542$		0			0
$543$		−	1.50066e6i		−	0.218416i
$544$		0			0
$545$	−1.31284e6	−	1.15013e6i	−0.189331	−	0.165865i
$546$		0			0
$547$	6.42557e6			0.918213			0.459106	−	0.888381i	$-0.348170\pi$
$547$	6.42557e6			0.918213			0.459106	+	0.888381i	$0.348170\pi$
$548$		0			0
$549$		−	43139.9i		−	0.00610870i
$550$		0			0
$551$	1.27136e6			0.178398
$552$		0			0
$553$			1.64950e7i			2.29372i
$554$		0			0
$555$	6.97102e6	−	7.95728e6i	0.960647	−	1.09656i
$556$		0			0
$557$	−6.14000e6			−0.838553			−0.419276	−	0.907859i	$-0.637716\pi$
$557$	−6.14000e6			−0.838553			−0.419276	+	0.907859i	$0.637716\pi$
$558$		0			0
$559$	−247922.			−0.0335572
$560$		0			0
$561$	−2.80310e6			−0.376037
$562$		0			0
$563$	−4.04571e6			−0.537927			−0.268964	−	0.963150i	$-0.586681\pi$
$563$	−4.04571e6			−0.537927			−0.268964	+	0.963150i	$0.586681\pi$
$564$		0			0
$565$	758064.	+	664107.i	0.0999045	+	0.0875219i
$566$		0			0
$567$			1.48125e7i			1.93495i
$568$		0			0
$569$	3.29664e6			0.426865			0.213433	−	0.976958i	$-0.431536\pi$
$569$	3.29664e6			0.426865			0.213433	+	0.976958i	$0.431536\pi$
$570$		0			0
$571$			8.76573e6i			1.12512i	0.826758	+	0.562558i	$0.190183\pi$
$571$			8.76573e6i			1.12512i	−0.826758	+	0.562558i	$0.809817\pi$
$572$		0			0
$573$	1.54499e7			1.96580
$574$		0			0
$575$	−6.28562e6	+	834172.i	−0.792827	+	0.105217i
$576$		0			0
$577$			8.81970e6i			1.10284i	0.834226	+	0.551422i	$0.185915\pi$
$577$			8.81970e6i			1.10284i	−0.834226	+	0.551422i	$0.814085\pi$
$578$		0			0
$579$		−	4.15475e6i		−	0.515049i
$580$		0			0
$581$		−	1.02598e6i		−	0.126096i
$582$		0			0
$583$		−	4.43219e6i		−	0.540066i
$584$		0			0
$585$	8.16059e6	−	9.31514e6i	0.985898	−	1.12538i
$586$		0			0
$587$	−1.29013e7			−1.54539			−0.772693	−	0.634780i	$-0.781091\pi$
$587$	−1.29013e7			−1.54539			−0.772693	+	0.634780i	$0.781091\pi$
$588$		0			0
$589$			440170.i			0.0522796i
$590$		0			0
$591$	−9.70377e6			−1.14280
$592$		0			0
$593$		−	1.26142e7i		−	1.47307i	−0.676398	−	0.736536i	$-0.736460\pi$
$593$		−	1.26142e7i		−	1.47307i	0.676398	−	0.736536i	$-0.263540\pi$
$594$		0			0
$595$	−5.84358e6	+	6.67033e6i	−0.676685	+	0.772422i
$596$		0			0
$597$	2.53563e6			0.291172
$598$		0			0
$599$	1.12524e7			1.28138			0.640688	−	0.767802i	$-0.278649\pi$
$599$	1.12524e7			1.28138			0.640688	+	0.767802i	$0.278649\pi$
$600$		0			0
$601$	1.11847e7			1.26310			0.631549	−	0.775336i	$-0.282420\pi$
$601$	1.11847e7			1.26310			0.631549	+	0.775336i	$0.282420\pi$
$602$		0			0
$603$	5.65937e6			0.633833
$604$		0			0
$605$	4.56492e6	−	5.21076e6i	0.507043	−	0.578779i
$606$		0			0
$607$			1.18645e7i			1.30701i	0.756922	+	0.653505i	$0.226702\pi$
$607$			1.18645e7i			1.30701i	−0.756922	+	0.653505i	$0.773298\pi$
$608$		0			0
$609$	1.42655e7			1.55863
$610$		0			0
$611$		−	1.92483e7i		−	2.08588i
$612$		0			0
$613$	1.85839e6			0.199750			0.0998748	−	0.995000i	$-0.468156\pi$
$613$	1.85839e6			0.199750			0.0998748	+	0.995000i	$0.468156\pi$
$614$		0			0
$615$	1.44764e6	−	1.65245e6i	0.154337	−	0.176173i
$616$		0			0
$617$			6.78893e6i			0.717940i	0.933349	+	0.358970i	$0.116872\pi$
$617$			6.78893e6i			0.717940i	−0.933349	+	0.358970i	$0.883128\pi$
$618$		0			0
$619$		−	1.53528e7i		−	1.61050i	−0.592933	−	0.805252i	$-0.702030\pi$
$619$		−	1.53528e7i		−	1.61050i	0.592933	−	0.805252i	$-0.297970\pi$
$620$		0			0
$621$		−	696602.i		−	0.0724862i
$622$		0			0
$623$			1.31517e6i			0.135756i
$624$		0			0
$625$	−9.42759e6	+	2.54715e6i	−0.965385	+	0.260829i
$626$		0			0
$627$	1.90906e6			0.193933
$628$		0			0
$629$			5.85535e6i			0.590101i
$630$		0			0
$631$	1.36354e7			1.36331			0.681657	−	0.731672i	$-0.261260\pi$
$631$	1.36354e7			1.36331			0.681657	+	0.731672i	$0.261260\pi$
$632$		0			0
$633$			1.45253e7i			1.44084i
$634$		0			0
$635$	2.77614e6	+	2.43205e6i	0.273216	+	0.239353i
$636$		0			0
$637$	−3.81313e7			−3.72334
$638$		0			0
$639$	8.68679e6			0.841603
$640$		0			0
$641$	−9.20659e6			−0.885022			−0.442511	−	0.896763i	$-0.645912\pi$
$641$	−9.20659e6			−0.885022			−0.442511	+	0.896763i	$0.645912\pi$
$642$		0			0
$643$	2.77419e6			0.264611			0.132306	−	0.991209i	$-0.457762\pi$
$643$	2.77419e6			0.264611			0.132306	+	0.991209i	$0.457762\pi$
$644$		0			0
$645$	203060.	−	231789.i	0.0192188	−	0.0219378i
$646$		0			0
$647$		−	4.39670e6i		−	0.412921i	−0.978455	−	0.206460i	$-0.933806\pi$
$647$		−	4.39670e6i		−	0.412921i	0.978455	−	0.206460i	$-0.0661944\pi$
$648$		0			0
$649$	−4.75183e6			−0.442842
$650$		0			0
$651$			4.93899e6i			0.456758i
$652$		0			0
$653$	−6.49195e6			−0.595788			−0.297894	−	0.954599i	$-0.596284\pi$
$653$	−6.49195e6			−0.595788			−0.297894	+	0.954599i	$0.596284\pi$
$654$		0			0
$655$	6.21272e6	+	5.44269e6i	0.565820	+	0.495691i
$656$		0			0
$657$			1.01715e7i			0.919331i
$658$		0			0
$659$			1.08903e7i			0.976850i	0.872606	+	0.488425i	$0.162428\pi$
$659$			1.08903e7i			0.976850i	−0.872606	+	0.488425i	$0.837572\pi$
$660$		0			0
$661$		−	1.61309e7i		−	1.43600i	−0.696042	−	0.718001i	$-0.745057\pi$
$661$		−	1.61309e7i		−	1.43600i	0.696042	−	0.718001i	$-0.254943\pi$
$662$		0			0
$663$			1.41868e7i			1.25343i
$664$		0			0
$665$	3.97979e6	−	4.54285e6i	0.348985	−	0.398359i
$666$		0			0
$667$	−5.64565e6			−0.491360
$668$		0			0
$669$		−	37272.0i		−	0.00321972i
$670$		0			0
$671$	36591.8			0.00313746
$672$		0			0
$673$		−	2.20227e7i		−	1.87428i	−0.348957	−	0.937139i	$-0.613464\pi$
$673$		−	2.20227e7i		−	1.87428i	0.348957	−	0.937139i	$-0.386536\pi$
$674$		0			0
$675$	−141144.	−	1.06354e6i	−0.0119235	−	0.0898452i
$676$		0			0
$677$	8.56722e6			0.718403			0.359202	−	0.933260i	$-0.383049\pi$
$677$	8.56722e6			0.718403			0.359202	+	0.933260i	$0.383049\pi$
$678$		0			0
$679$	−2.31963e7			−1.93083
$680$		0			0
$681$	−1.54670e7			−1.27802
$682$		0			0
$683$	1.45813e7			1.19603			0.598017	−	0.801483i	$-0.295955\pi$
$683$	1.45813e7			1.19603			0.598017	+	0.801483i	$0.295955\pi$
$684$		0			0
$685$	1.50460e7	+	1.31811e7i	1.22516	+	1.07331i
$686$		0			0
$687$		−	2.53741e7i		−	2.05116i
$688$		0			0
$689$	−2.24318e7			−1.80018
$690$		0			0
$691$		−	1.29644e7i		−	1.03290i	−0.856318	−	0.516449i	$-0.827254\pi$
$691$		−	1.29644e7i		−	1.03290i	0.856318	−	0.516449i	$-0.172746\pi$
$692$		0			0
$693$	1.03498e7			0.818648
$694$		0			0
$695$	2.07421e6	+	1.81713e6i	0.162889	+	0.142700i
$696$		0			0
$697$			1.21595e6i			0.0948055i
$698$		0			0
$699$			3.31223e7i			2.56406i
$700$		0			0
$701$		−	846059.i		−	0.0650288i	−0.999471	−	0.0325144i	$-0.989649\pi$
$701$		−	846059.i		−	0.0650288i	0.999471	−	0.0325144i	$-0.0103515\pi$
$702$		0			0
$703$		−	3.98781e6i		−	0.304331i
$704$		0			0
$705$	1.79958e7	+	1.57653e7i	1.36363	+	1.19462i
$706$		0			0
$707$	−4.69858e6			−0.353523
$708$		0			0
$709$		−	1.99203e7i		−	1.48827i	−0.668031	−	0.744133i	$-0.732863\pi$
$709$		−	1.99203e7i		−	1.48827i	0.668031	−	0.744133i	$-0.267137\pi$
$710$		0			0
$711$	−1.58475e7			−1.17568
$712$		0			0
$713$		−	1.95463e6i		−	0.143993i
$714$		0			0
$715$	7.90122e6	+	6.92191e6i	0.578001	+	0.506362i
$716$		0			0
$717$	−1.22997e7			−0.893508
$718$		0			0
$719$	−1.24447e7			−0.897767			−0.448883	−	0.893590i	$-0.648178\pi$
$719$	−1.24447e7			−0.897767			−0.448883	+	0.893590i	$0.648178\pi$
$720$		0			0
$721$	2.84213e7			2.03613
$722$		0			0
$723$	−1.49249e7			−1.06186
$724$		0			0
$725$	−8.61953e6	+	1.14391e6i	−0.609030	+	0.0808251i
$726$		0			0
$727$		−	4.45620e6i		−	0.312700i	−0.987702	−	0.156350i	$-0.950027\pi$
$727$		−	4.45620e6i		−	0.312700i	0.987702	−	0.156350i	$-0.0499728\pi$
$728$		0			0
$729$	−1.24221e7			−0.865717
$730$		0			0
$731$			170562.i			0.0118056i
$732$		0			0
$733$	1.16096e7			0.798101			0.399051	−	0.916929i	$-0.369340\pi$
$733$	1.16096e7			0.798101			0.399051	+	0.916929i	$0.369340\pi$
$734$		0			0
$735$	3.12314e7	−	3.56500e7i	2.13242	−	2.43411i
$736$		0			0
$737$			4.80035e6i			0.325540i
$738$		0			0
$739$			2.17540e6i			0.146531i	0.997312	+	0.0732653i	$0.0233420\pi$
$739$			2.17540e6i			0.146531i	−0.997312	+	0.0732653i	$0.976658\pi$
$740$		0			0
$741$		−	9.66198e6i		−	0.646429i
$742$		0			0
$743$		−	1.20942e7i		−	0.803720i	−0.915701	−	0.401860i	$-0.868364\pi$
$743$		−	1.20942e7i		−	0.803720i	0.915701	−	0.401860i	$-0.131636\pi$
$744$		0			0
$745$	−1.30706e7	−	1.14506e7i	−0.862789	−	0.755852i
$746$		0			0
$747$	985710.			0.0646320
$748$		0			0
$749$			2.95576e7i			1.92515i
$750$		0			0
$751$	410935.			0.0265873			0.0132936	−	0.999912i	$-0.495768\pi$
$751$	410935.			0.0265873			0.0132936	+	0.999912i	$0.495768\pi$
$752$		0			0
$753$		−	1.43806e7i		−	0.924251i
$754$		0			0
$755$	1.35416e7	−	1.54574e7i	0.864573	−	0.986892i
$756$		0			0
$757$	2.24651e7			1.42485			0.712424	−	0.701750i	$-0.247597\pi$
$757$	2.24651e7			1.42485			0.712424	+	0.701750i	$0.247597\pi$
$758$		0			0
$759$	−8.47744e6			−0.534146
$760$		0			0
$761$	1.00934e7			0.631793			0.315897	−	0.948794i	$-0.397695\pi$
$761$	1.00934e7			0.631793			0.315897	+	0.948794i	$0.397695\pi$
$762$		0			0
$763$	−7.38245e6			−0.459081
$764$		0			0
$765$	−6.40849e6	−	5.61420e6i	−0.395915	−	0.346844i
$766$		0			0
$767$			2.40496e7i			1.47611i
$768$		0			0
$769$	−2.06066e7			−1.25658			−0.628290	−	0.777980i	$-0.716245\pi$
$769$	−2.06066e7			−1.25658			−0.628290	+	0.777980i	$0.716245\pi$
$770$		0			0
$771$			3.64344e7i			2.20737i
$772$		0			0
$773$	−2.40425e7			−1.44721			−0.723605	−	0.690215i	$-0.757516\pi$
$773$	−2.40425e7			−1.44721			−0.723605	+	0.690215i	$0.757516\pi$
$774$		0			0
$775$	−396043.	−	2.98425e6i	−0.0236858	−	0.178476i
$776$		0			0
$777$		−	4.47458e7i		−	2.65888i
$778$		0			0
$779$		−	828127.i		−	0.0488937i
$780$		0			0
$781$			7.36825e6i			0.432252i
$782$		0			0
$783$		−	955257.i		−	0.0556821i
$784$		0			0
$785$	−1.89594e6	+	2.16417e6i	−0.109812	+	0.125348i
$786$		0			0
$787$	−1.44261e7			−0.830256			−0.415128	−	0.909763i	$-0.636263\pi$
$787$	−1.44261e7			−0.830256			−0.415128	+	0.909763i	$0.636263\pi$
$788$		0			0
$789$			3.60161e7i			2.05970i
$790$		0			0
$791$	4.26279e6			0.242244
$792$		0			0
$793$		−	185196.i		−	0.0104580i
$794$		0			0
$795$	1.83728e7	−	2.09722e7i	1.03100	−	1.17686i
$796$		0			0
$797$	−7.27613e6			−0.405746			−0.202873	−	0.979205i	$-0.565028\pi$
$797$	−7.27613e6			−0.405746			−0.202873	+	0.979205i	$0.565028\pi$
$798$		0			0
$799$	−1.32422e7			−0.733824
$800$		0			0
$801$	−1.26354e6			−0.0695838
$802$		0			0
$803$	−8.62760e6			−0.472173
$804$		0			0
$805$	−1.76728e7	+	2.01731e7i	−0.961205	+	1.09720i
$806$		0			0
$807$			1.48946e7i			0.805089i
$808$		0			0
$809$	−1.86768e7			−1.00330			−0.501650	−	0.865070i	$-0.667274\pi$
$809$	−1.86768e7			−1.00330			−0.501650	+	0.865070i	$0.667274\pi$
$810$		0			0
$811$			7.20855e6i			0.384854i	0.981311	+	0.192427i	$0.0616358\pi$
$811$			7.20855e6i			0.384854i	−0.981311	+	0.192427i	$0.938364\pi$
$812$		0			0
$813$	5.15548e6			0.273554
$814$		0			0
$815$	−1.73724e7	+	1.98303e7i	−0.916151	+	1.04577i
$816$		0			0
$817$		−	116162.i		−	0.00608846i
$818$		0			0
$819$		−	5.23814e7i		−	2.72877i
$820$		0			0
$821$			6.00999e6i			0.311183i	0.987821	+	0.155592i	$0.0497284\pi$
$821$			6.00999e6i			0.311183i	−0.987821	+	0.155592i	$0.950272\pi$
$822$		0			0
$823$			2.98829e6i			0.153788i	0.997039	+	0.0768940i	$0.0245003\pi$
$823$			2.98829e6i			0.153788i	−0.997039	+	0.0768940i	$0.975500\pi$
$824$		0			0
$825$	−1.29430e7	+	1.71768e6i	−0.662063	+	0.0878632i
$826$		0			0
$827$	6.14178e6			0.312270			0.156135	−	0.987736i	$-0.450096\pi$
$827$	6.14178e6			0.312270			0.156135	+	0.987736i	$0.450096\pi$
$828$		0			0
$829$		−	3.62053e7i		−	1.82972i	−0.403768	−	0.914862i	$-0.632300\pi$
$829$		−	3.62053e7i		−	1.82972i	0.403768	−	0.914862i	$-0.367700\pi$
$830$		0			0
$831$	−5.14746e7			−2.58577
$832$		0			0
$833$			2.62330e7i			1.30989i
$834$		0			0
$835$	−2.71103e7	−	2.37502e7i	−1.34561	−	1.17883i
$836$		0			0
$837$	330729.			0.0163177
$838$		0			0
$839$	1.12551e7			0.552005			0.276002	−	0.961157i	$-0.410990\pi$
$839$	1.12551e7			0.552005			0.276002	+	0.961157i	$0.410990\pi$
$840$		0			0
$841$	1.27692e7			0.622550
$842$		0			0
$843$	−1.57501e7			−0.763333
$844$		0			0
$845$	2.13554e7	−	2.43768e7i	1.02889	−	1.17445i
$846$		0			0
$847$		−	2.93014e7i		−	1.40340i
$848$		0			0
$849$	2.28119e7			1.08616
$850$		0			0
$851$			1.77084e7i			0.838215i
$852$		0			0
$853$	−1.79575e6			−0.0845032			−0.0422516	−	0.999107i	$-0.513453\pi$
$853$	−1.79575e6			−0.0845032			−0.0422516	+	0.999107i	$0.513453\pi$
$854$		0			0
$855$	4.36453e6	+	3.82357e6i	0.204184	+	0.178877i
$856$		0			0
$857$		−	2.19588e6i		−	0.102131i	−0.998695	−	0.0510653i	$-0.983738\pi$
$857$		−	2.19588e6i		−	0.102131i	0.998695	−	0.0510653i	$-0.0162617\pi$
$858$		0			0
$859$		−	3.86186e6i		−	0.178572i	−0.996006	−	0.0892860i	$-0.971541\pi$
$859$		−	3.86186e6i		−	0.178572i	0.996006	−	0.0892860i	$-0.0284585\pi$
$860$		0			0
$861$		−	9.29212e6i		−	0.427176i
$862$		0			0
$863$			1.07467e7i			0.491188i	0.969373	+	0.245594i	$0.0789830\pi$
$863$			1.07467e7i			0.491188i	−0.969373	+	0.245594i	$0.921017\pi$
$864$		0			0
$865$	−5.97866e6	+	6.82451e6i	−0.271684	+	0.310121i
$866$		0			0
$867$	−2.10272e7			−0.950022
$868$		0			0
$869$		−	1.34421e7i		−	0.603833i
$870$		0			0
$871$	2.42951e7			1.08511
$872$		0			0
$873$		−	2.22857e7i		−	0.989672i
$874$		0			0
$875$	−2.28946e7	+	3.43803e7i	−1.01091	+	1.51806i
$876$		0			0
$877$	2.12160e7			0.931460			0.465730	−	0.884927i	$-0.345792\pi$
$877$	2.12160e7			0.931460			0.465730	+	0.884927i	$0.345792\pi$
$878$		0			0
$879$	3.34278e7			1.45927
$880$		0			0
$881$	3.72206e7			1.61564			0.807818	−	0.589432i	$-0.200649\pi$
$881$	3.72206e7			1.61564			0.807818	+	0.589432i	$0.200649\pi$
$882$		0			0
$883$	2.67206e7			1.15331			0.576654	−	0.816989i	$-0.304358\pi$
$883$	2.67206e7			1.15331			0.576654	+	0.816989i	$0.304358\pi$
$884$		0			0
$885$	−2.24846e7	−	1.96978e7i	−0.965000	−	0.845394i
$886$		0			0
$887$		−	1.29413e7i		−	0.552290i	−0.961116	−	0.276145i	$-0.910943\pi$
$887$		−	1.29413e7i		−	0.552290i	0.961116	−	0.276145i	$-0.0890570\pi$
$888$		0			0
$889$	1.56109e7			0.662482
$890$		0			0
$891$		−	1.20709e7i		−	0.509386i
$892$		0			0
$893$	9.01862e6			0.378453
$894$		0			0
$895$	−3.14426e6	−	2.75455e6i	−0.131208	−	0.114946i
$896$		0			0
$897$			4.29053e7i			1.78045i
$898$		0			0
$899$		−	2.68041e6i		−	0.110612i
$900$		0			0
$901$			1.54323e7i			0.633315i
$902$		0			0
$903$		−	1.30341e6i		−	0.0531938i
$904$		0			0
$905$	2.91009e6	+	2.54940e6i	0.118109	+	0.103470i
$906$		0			0
$907$	3.82873e7			1.54538			0.772692	−	0.634781i	$-0.218910\pi$
$907$	3.82873e7			1.54538			0.772692	+	0.634781i	$0.218910\pi$
$908$		0			0
$909$		−	4.51414e6i		−	0.181203i
$910$		0			0
$911$	1.48261e7			0.591876			0.295938	−	0.955207i	$-0.404368\pi$
$911$	1.48261e7			0.591876			0.295938	+	0.955207i	$0.404368\pi$
$912$		0			0
$913$			836091.i			0.0331953i
$914$		0			0
$915$	173145.	+	151684.i	0.00683685	+	0.00598947i
$916$		0			0
$917$	3.49357e7			1.37197
$918$		0			0
$919$	−2.75810e7			−1.07726			−0.538631	−	0.842542i	$-0.681058\pi$
$919$	−2.75810e7			−1.07726			−0.538631	+	0.842542i	$0.681058\pi$
$920$		0			0
$921$	−4.80103e7			−1.86503
$922$		0			0
$923$	3.72916e7			1.44081
$924$		0			0
$925$	3.58803e6	+	2.70364e7i	0.137880	+	1.03895i
$926$		0			0
$927$			2.73057e7i			1.04365i
$928$		0			0
$929$	3.07295e7			1.16820			0.584099	−	0.811682i	$-0.301448\pi$
$929$	3.07295e7			1.16820			0.584099	+	0.811682i	$0.301448\pi$
$930$		0			0
$931$		−	1.78661e7i		−	0.675546i
$932$		0			0
$933$	−3.33542e7			−1.25443
$934$		0			0
$935$	4.76204e6	−	5.43576e6i	0.178141	−	0.203344i
$936$		0			0
$937$			9.51349e6i			0.353990i	0.984212	+	0.176995i	$0.0566377\pi$
$937$			9.51349e6i			0.353990i	−0.984212	+	0.176995i	$0.943362\pi$
$938$		0			0
$939$			5.06703e7i			1.87538i
$940$		0			0
$941$		−	2.64149e7i		−	0.972466i	−0.873829	−	0.486233i	$-0.838371\pi$
$941$		−	2.64149e7i		−	0.972466i	0.873829	−	0.486233i	$-0.161629\pi$
$942$		0			0
$943$			3.67741e6i			0.134667i
$944$		0			0
$945$	−3.41334e6	−	2.99028e6i	−0.124337	−	0.108926i
$946$		0			0
$947$	4.05240e7			1.46837			0.734187	−	0.678947i	$-0.237563\pi$
$947$	4.05240e7			1.46837			0.734187	+	0.678947i	$0.237563\pi$
$948$		0			0
$949$			4.36653e7i			1.57388i
$950$		0			0
$951$	2.49924e7			0.896101
$952$		0			0
$953$			3.84770e7i			1.37236i	0.727431	+	0.686181i	$0.240714\pi$
$953$			3.84770e7i			1.37236i	−0.727431	+	0.686181i	$0.759286\pi$
$954$		0			0
$955$	−2.62471e7	+	2.99605e7i	−0.931263	+	1.06302i
$956$		0			0
$957$	−1.16252e7			−0.410318
$958$		0			0
$959$	8.46074e7			2.97072
$960$		0			0
$961$	−2.77011e7			−0.967585
$962$		0			0
$963$	−2.83974e7			−0.986762
$964$		0			0
$965$	8.05688e6	+	7.05828e6i	0.278515	+	0.243995i
$966$		0			0
$967$			9.84732e6i			0.338651i	0.985560	+	0.169325i	$0.0541588\pi$
$967$			9.84732e6i			0.338651i	−0.985560	+	0.169325i	$0.945841\pi$
$968$		0			0
$969$	−6.64711e6			−0.227417
$970$		0			0
$971$			1.96906e7i			0.670211i	0.942181	+	0.335105i	$0.108772\pi$
$971$			1.96906e7i			0.670211i	−0.942181	+	0.335105i	$0.891228\pi$
$972$		0			0
$973$	1.16638e7			0.394965
$974$		0			0
$975$	8.69338e6	+	6.55060e7i	0.292871	+	2.20683i
$976$		0			0
$977$		−	2.36761e7i		−	0.793549i	−0.917916	−	0.396775i	$-0.870129\pi$
$977$		−	2.36761e7i		−	0.793549i	0.917916	−	0.396775i	$-0.129871\pi$
$978$		0			0
$979$		−	1.07175e6i		−	0.0357386i
$980$		0			0
$981$		−	7.09267e6i		−	0.235308i
$982$		0			0
$983$		−	2.18805e7i		−	0.722225i	−0.932522	−	0.361112i	$-0.882397\pi$
$983$		−	2.18805e7i		−	0.722225i	0.932522	−	0.361112i	$-0.117603\pi$
$984$		0			0
$985$	1.64852e7	−	1.88175e7i	0.541383	−	0.617977i
$986$		0			0
$987$	1.01195e8			3.30647
$988$		0			0
$989$			515831.i			0.0167694i
$990$		0			0
$991$	−3.03505e7			−0.981706			−0.490853	−	0.871242i	$-0.663315\pi$
$991$	−3.03505e7			−0.981706			−0.490853	+	0.871242i	$0.663315\pi$
$992$		0			0
$993$			4.43467e7i			1.42721i
$994$		0			0
$995$	−4.30765e6	+	4.91709e6i	−0.137938	+	0.157453i
$996$		0			0
$997$	−4.83451e6			−0.154033			−0.0770167	−	0.997030i	$-0.524539\pi$
$997$	−4.83451e6			−0.154033			−0.0770167	+	0.997030i	$0.524539\pi$
$998$		0			0
$999$	−2.99630e6			−0.0949886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.f.a.49.3		28
4.3	odd	2		40.6.f.a.29.3	✓	28
5.2	odd	4		800.6.d.e.401.28		28
5.3	odd	4		800.6.d.e.401.1		28
5.4	even	2	inner	160.6.f.a.49.26		28
8.3	odd	2		40.6.f.a.29.25	yes	28
8.5	even	2	inner	160.6.f.a.49.25		28
20.3	even	4		200.6.d.e.101.12		28
20.7	even	4		200.6.d.e.101.17		28
20.19	odd	2		40.6.f.a.29.26	yes	28
40.3	even	4		200.6.d.e.101.11		28
40.13	odd	4		800.6.d.e.401.2		28
40.19	odd	2		40.6.f.a.29.4	yes	28
40.27	even	4		200.6.d.e.101.18		28
40.29	even	2	inner	160.6.f.a.49.4		28
40.37	odd	4		800.6.d.e.401.27		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.f.a.29.3	✓	28	4.3	odd	2
40.6.f.a.29.4	yes	28	40.19	odd	2
40.6.f.a.29.25	yes	28	8.3	odd	2
40.6.f.a.29.26	yes	28	20.19	odd	2
160.6.f.a.49.3		28	1.1	even	1	trivial
160.6.f.a.49.4		28	40.29	even	2	inner
160.6.f.a.49.25		28	8.5	even	2	inner
160.6.f.a.49.26		28	5.4	even	2	inner
200.6.d.e.101.11		28	40.3	even	4
200.6.d.e.101.12		28	20.3	even	4
200.6.d.e.101.17		28	20.7	even	4
200.6.d.e.101.18		28	40.27	even	4
800.6.d.e.401.1		28	5.3	odd	4
800.6.d.e.401.2		28	40.13	odd	4
800.6.d.e.401.27		28	40.37	odd	4
800.6.d.e.401.28		28	5.2	odd	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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-21.6833 q^{3} +(36.8367 - 42.0483i) q^{5} -236.448i q^{7} +227.167 q^{9} +O(q^{10})\)	\(q-21.6833 q^{3} +(36.8367 - 42.0483i) q^{5} -236.448i q^{7} +227.167 q^{9} +192.686i q^{11} +975.206 q^{13} +(-798.742 + 911.747i) q^{15} -670.908i q^{17} +456.924i q^{19} +5126.98i q^{21} -2029.03i q^{23} +(-411.118 - 3097.84i) q^{25} +343.317 q^{27} -2782.43i q^{29} +963.333 q^{31} -4178.07i q^{33} +(-9942.24 - 8709.97i) q^{35} -8727.50 q^{37} -21145.7 q^{39} -1812.39 q^{41} -254.225 q^{43} +(8368.07 - 9551.98i) q^{45} -19737.7i q^{47} -39100.7 q^{49} +14547.5i q^{51} -23002.2 q^{53} +(8102.10 + 7097.90i) q^{55} -9907.64i q^{57} +24661.0i q^{59} -189.904i q^{61} -53713.2i q^{63} +(35923.3 - 41005.7i) q^{65} +24912.8 q^{67} +43996.2i q^{69} +38239.7 q^{71} +44775.5i q^{73} +(8914.41 + 67171.5i) q^{75} +45560.2 q^{77} -69761.6 q^{79} -62645.8 q^{81} +4339.15 q^{83} +(-28210.5 - 24714.0i) q^{85} +60332.4i q^{87} -5562.17 q^{89} -230586. i q^{91} -20888.3 q^{93} +(19212.9 + 16831.6i) q^{95} -98102.9i q^{97} +43771.8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 1940 q^{9}+O(q^{10}) \) 28 * q + 1940 * q^9	\( 28 q + 1940 q^{9} + 488 q^{15} + 1556 q^{25} - 4368 q^{31} - 23360 q^{39} - 2480 q^{41} - 38420 q^{49} + 48776 q^{55} + 37200 q^{65} + 69232 q^{71} + 35984 q^{79} + 122596 q^{81} - 178744 q^{89} - 89416 q^{95}+O(q^{100}) \) 28 * q + 1940 * q^9 + 488 * q^15 + 1556 * q^25 - 4368 * q^31 - 23360 * q^39 - 2480 * q^41 - 38420 * q^49 + 48776 * q^55 + 37200 * q^65 + 69232 * q^71 + 35984 * q^79 + 122596 * q^81 - 178744 * q^89 - 89416 * q^95

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