LMFDB - Embedded newform 160.6.d.a.81.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(81,160)]

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, 0]))

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

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

chi := DirichletCharacter("160.81");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 $ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{90}\cdot 3^{4}\cdot 5^{12} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			81.17
Root			$2.93366 - 2.71913i$ of defining polynomial
Character	$\chi$	$=$	160.81
Dual form			160.6.d.a.81.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.7876i q^{3} -25.0000i q^{5} +107.536 q^{7} -109.975 q^{9} +O(q^{10})$	$q+18.7876i q^{3} -25.0000i q^{5} +107.536 q^{7} -109.975 q^{9} +272.206i q^{11} +198.402i q^{13} +469.691 q^{15} +2065.79 q^{17} -1891.04i q^{19} +2020.35i q^{21} +987.677 q^{23} -625.000 q^{25} +2499.22i q^{27} +8015.26i q^{29} -827.342 q^{31} -5114.10 q^{33} -2688.40i q^{35} +9426.30i q^{37} -3727.50 q^{39} -8221.07 q^{41} +9301.63i q^{43} +2749.38i q^{45} -13837.9 q^{47} -5242.97 q^{49} +38811.3i q^{51} -27751.2i q^{53} +6805.14 q^{55} +35528.1 q^{57} +25106.1i q^{59} +26404.6i q^{61} -11826.3 q^{63} +4960.05 q^{65} +38563.9i q^{67} +18556.1i q^{69} +71073.0 q^{71} +18622.0 q^{73} -11742.3i q^{75} +29271.9i q^{77} +75599.4 q^{79} -73678.4 q^{81} -125298. i q^{83} -51644.7i q^{85} -150588. q^{87} +30341.5 q^{89} +21335.4i q^{91} -15543.8i q^{93} -47275.9 q^{95} +15635.2 q^{97} -29935.9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) $ 20 * q + 196 * q^7 - 1620 * q^9	$ 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63} + 200312 q^{71} - 105136 q^{73} - 282080 q^{79} + 65172 q^{81} + 332592 q^{87} - 3160 q^{89} - 144400 q^{95} + 147376 q^{97}+O(q^{100}) $ 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

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
$3$	18.7876i	1.20523i	0.798033	+	0.602614i	$0.205874\pi$
$3$	18.7876i	1.20523i	−0.798033	+	0.602614i	$0.794126\pi$
$4$	0	0
$5$	− 25.0000i	− 0.447214i
$5$	− 25.0000i	− 0.447214i
$6$	0	0
$7$	107.536	0.829487	0.414743	−	0.909938i	$-0.363871\pi$
$7$	107.536	0.829487	0.414743	+	0.909938i	$0.363871\pi$
$8$	0	0
$9$	−109.975	−0.452573
$10$	0	0
$11$	272.206i	0.678290i	0.940734	+	0.339145i	$0.110138\pi$
$11$	272.206i	0.678290i	−0.940734	+	0.339145i	$0.889862\pi$
$12$	0	0
$13$	198.402i	0.325602i	0.986659	+	0.162801i	$0.0520529\pi$
$13$	198.402i	0.325602i	−0.986659	+	0.162801i	$0.947947\pi$
$14$	0	0
$15$	469.691	0.538994
$16$	0	0
$17$	2065.79	1.73366	0.866829	−	0.498606i	$-0.166154\pi$
$17$	2065.79	1.73366	0.866829	+	0.498606i	$0.166154\pi$
$18$	0	0
$19$	− 1891.04i	− 1.20176i	−0.799341	−	0.600878i	$-0.794818\pi$
$19$	− 1891.04i	− 1.20176i	0.799341	−	0.600878i	$-0.205182\pi$
$20$	0	0
$21$	2020.35i	0.999720i
$22$	0	0
$23$	987.677	0.389310	0.194655	−	0.980872i	$-0.437641\pi$
$23$	987.677	0.389310	0.194655	+	0.980872i	$0.437641\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	2499.22i	0.659774i
$28$	0	0
$29$	8015.26i	1.76979i	0.465788	+	0.884896i	$0.345771\pi$
$29$	8015.26i	1.76979i	−0.465788	+	0.884896i	$0.654229\pi$
$30$	0	0
$31$	−827.342	−0.154625	−0.0773127	−	0.997007i	$-0.524634\pi$
$31$	−827.342	−0.154625	−0.0773127	+	0.997007i	$0.524634\pi$
$32$	0	0
$33$	−5114.10	−0.817494
$34$	0	0
$35$	− 2688.40i	− 0.370958i
$36$	0	0
$37$	9426.30i	1.13198i	0.824414	+	0.565988i	$0.191505\pi$
$37$	9426.30i	1.13198i	−0.824414	+	0.565988i	$0.808495\pi$
$38$	0	0
$39$	−3727.50	−0.392425
$40$	0	0
$41$	−8221.07	−0.763780	−0.381890	−	0.924208i	$-0.624727\pi$
$41$	−8221.07	−0.763780	−0.381890	+	0.924208i	$0.624727\pi$
$42$	0	0
$43$	9301.63i	0.767164i	0.923507	+	0.383582i	$0.125310\pi$
$43$	9301.63i	0.767164i	−0.923507	+	0.383582i	$0.874690\pi$
$44$	0	0
$45$	2749.38i	0.202397i
$46$	0	0
$47$	−13837.9	−0.913745	−0.456873	−	0.889532i	$-0.651031\pi$
$47$	−13837.9	−0.913745	−0.456873	+	0.889532i	$0.651031\pi$
$48$	0	0
$49$	−5242.97	−0.311952
$50$	0	0
$51$	38811.3i	2.08945i
$52$	0	0
$53$	− 27751.2i	− 1.35704i	−0.734582	−	0.678520i	$-0.762622\pi$
$53$	− 27751.2i	− 1.35704i	0.734582	−	0.678520i	$-0.237378\pi$
$54$	0	0
$55$	6805.14	0.303340
$56$	0	0
$57$	35528.1	1.44839
$58$	0	0
$59$	25106.1i	0.938965i	0.882942	+	0.469482i	$0.155559\pi$
$59$	25106.1i	0.938965i	−0.882942	+	0.469482i	$0.844441\pi$
$60$	0	0
$61$	26404.6i	0.908563i	0.890858	+	0.454282i	$0.150104\pi$
$61$	26404.6i	0.908563i	−0.890858	+	0.454282i	$0.849896\pi$
$62$	0	0
$63$	−11826.3	−0.375404
$64$	0	0
$65$	4960.05	0.145614
$66$	0	0
$67$	38563.9i	1.04953i	0.851248	+	0.524764i	$0.175846\pi$
$67$	38563.9i	1.04953i	−0.851248	+	0.524764i	$0.824154\pi$
$68$	0	0
$69$	18556.1i	0.469207i
$70$	0	0
$71$	71073.0	1.67324	0.836621	−	0.547782i	$-0.184528\pi$
$71$	71073.0	1.67324	0.836621	+	0.547782i	$0.184528\pi$
$72$	0	0
$73$	18622.0	0.408996	0.204498	−	0.978867i	$-0.434444\pi$
$73$	18622.0	0.408996	0.204498	+	0.978867i	$0.434444\pi$
$74$	0	0
$75$	− 11742.3i	− 0.241045i
$76$	0	0
$77$	29271.9i	0.562632i
$78$	0	0
$79$	75599.4	1.36286	0.681429	−	0.731884i	$-0.261359\pi$
$79$	75599.4	1.36286	0.681429	+	0.731884i	$0.261359\pi$
$80$	0	0
$81$	−73678.4	−1.24775
$82$	0	0
$83$	− 125298.i	− 1.99641i	−0.0599304	−	0.998203i	$-0.519088\pi$
$83$	− 125298.i	− 1.99641i	0.0599304	−	0.998203i	$-0.480912\pi$
$84$	0	0
$85$	− 51644.7i	− 0.775315i
$86$	0	0
$87$	−150588.	−2.13300
$88$	0	0
$89$	30341.5	0.406034	0.203017	−	0.979175i	$-0.434925\pi$
$89$	30341.5	0.406034	0.203017	+	0.979175i	$0.434925\pi$
$90$	0	0
$91$	21335.4i	0.270083i
$92$	0	0
$93$	− 15543.8i	− 0.186359i
$94$	0	0
$95$	−47275.9	−0.537441
$96$	0	0
$97$	15635.2	0.168723	0.0843617	−	0.996435i	$-0.473115\pi$
$97$	15635.2	0.168723	0.0843617	+	0.996435i	$0.473115\pi$
$98$	0	0
$99$	− 29935.9i	− 0.306976i
$100$	0	0
$101$	102692.i	1.00169i	0.865538	+	0.500843i	$0.166976\pi$
$101$	102692.i	1.00169i	−0.865538	+	0.500843i	$0.833024\pi$
$102$	0	0
$103$	35981.8	0.334187	0.167093	−	0.985941i	$-0.446562\pi$
$103$	35981.8	0.334187	0.167093	+	0.985941i	$0.446562\pi$
$104$	0	0
$105$	50508.8	0.447088
$106$	0	0
$107$	− 94984.6i	− 0.802035i	−0.916070	−	0.401018i	$-0.868657\pi$
$107$	− 94984.6i	− 0.802035i	0.916070	−	0.401018i	$-0.131343\pi$
$108$	0	0
$109$	− 173158.i	− 1.39597i	−0.716113	−	0.697984i	$-0.754081\pi$
$109$	− 173158.i	− 1.39597i	0.716113	−	0.697984i	$-0.245919\pi$
$110$	0	0
$111$	−177098.	−1.36429
$112$	0	0
$113$	−237780.	−1.75178	−0.875891	−	0.482510i	$-0.839725\pi$
$113$	−237780.	−1.75178	−0.875891	+	0.482510i	$0.839725\pi$
$114$	0	0
$115$	− 24691.9i	− 0.174105i
$116$	0	0
$117$	− 21819.3i	− 0.147359i
$118$	0	0
$119$	222147.	1.43805
$120$	0	0
$121$	86955.1	0.539923
$122$	0	0
$123$	− 154454.i	− 0.920529i
$124$	0	0
$125$	15625.0i	0.0894427i
$126$	0	0
$127$	63282.9	0.348158	0.174079	−	0.984732i	$-0.444305\pi$
$127$	63282.9	0.348158	0.174079	+	0.984732i	$0.444305\pi$
$128$	0	0
$129$	−174756.	−0.924607
$130$	0	0
$131$	− 180082.i	− 0.916835i	−0.888737	−	0.458417i	$-0.848416\pi$
$131$	− 180082.i	− 0.916835i	0.888737	−	0.458417i	$-0.151584\pi$
$132$	0	0
$133$	− 203355.i	− 0.996840i
$134$	0	0
$135$	62480.5	0.295060
$136$	0	0
$137$	365739.	1.66483	0.832414	−	0.554155i	$-0.186958\pi$
$137$	365739.	1.66483	0.832414	+	0.554155i	$0.186958\pi$
$138$	0	0
$139$	111886.i	0.491180i	0.969374	+	0.245590i	$0.0789817\pi$
$139$	111886.i	0.491180i	−0.969374	+	0.245590i	$0.921018\pi$
$140$	0	0
$141$	− 259981.i	− 1.10127i
$142$	0	0
$143$	−54006.1	−0.220853
$144$	0	0
$145$	200381.	0.791475
$146$	0	0
$147$	− 98503.1i	− 0.375973i
$148$	0	0
$149$	136480.i	0.503621i	0.967777	+	0.251811i	$0.0810260\pi$
$149$	136480.i	0.503621i	−0.967777	+	0.251811i	$0.918974\pi$
$150$	0	0
$151$	−186354.	−0.665115	−0.332557	−	0.943083i	$-0.607912\pi$
$151$	−186354.	−0.665115	−0.332557	+	0.943083i	$0.607912\pi$
$152$	0	0
$153$	−227186.	−0.784607
$154$	0	0
$155$	20683.6i	0.0691506i
$156$	0	0
$157$	− 74689.6i	− 0.241830i	−0.992663	−	0.120915i	$-0.961417\pi$
$157$	− 74689.6i	− 0.241830i	0.992663	−	0.120915i	$-0.0385829\pi$
$158$	0	0
$159$	521380.	1.63554
$160$	0	0
$161$	106211.	0.322927
$162$	0	0
$163$	548779.i	1.61781i	0.587937	+	0.808907i	$0.299940\pi$
$163$	548779.i	1.61781i	−0.587937	+	0.808907i	$0.700060\pi$
$164$	0	0
$165$	127852.i	0.365594i
$166$	0	0
$167$	−224312.	−0.622387	−0.311194	−	0.950347i	$-0.600729\pi$
$167$	−224312.	−0.622387	−0.311194	+	0.950347i	$0.600729\pi$
$168$	0	0
$169$	331930.	0.893983
$170$	0	0
$171$	207967.i	0.543882i
$172$	0	0
$173$	165260.i	0.419809i	0.977722	+	0.209905i	$0.0673154\pi$
$173$	165260.i	0.419809i	−0.977722	+	0.209905i	$0.932685\pi$
$174$	0	0
$175$	−67210.1	−0.165897
$176$	0	0
$177$	−471684.	−1.13167
$178$	0	0
$179$	− 431975.i	− 1.00769i	−0.863795	−	0.503844i	$-0.831919\pi$
$179$	− 431975.i	− 1.00769i	0.863795	−	0.503844i	$-0.168081\pi$
$180$	0	0
$181$	− 216944.i	− 0.492210i	−0.969243	−	0.246105i	$-0.920849\pi$
$181$	− 216944.i	− 0.492210i	0.969243	−	0.246105i	$-0.0791508\pi$
$182$	0	0
$183$	−496080.	−1.09503
$184$	0	0
$185$	235658.	0.506235
$186$	0	0
$187$	562319.i	1.17592i
$188$	0	0
$189$	268756.i	0.547274i
$190$	0	0
$191$	34566.0	0.0685592	0.0342796	−	0.999412i	$-0.489086\pi$
$191$	34566.0	0.0685592	0.0342796	+	0.999412i	$0.489086\pi$
$192$	0	0
$193$	−473601.	−0.915208	−0.457604	−	0.889156i	$-0.651292\pi$
$193$	−473601.	−0.915208	−0.457604	+	0.889156i	$0.651292\pi$
$194$	0	0
$195$	93187.6i	0.175498i
$196$	0	0
$197$	− 394784.i	− 0.724760i	−0.932030	−	0.362380i	$-0.881964\pi$
$197$	− 394784.i	− 0.724760i	0.932030	−	0.362380i	$-0.118036\pi$
$198$	0	0
$199$	477089.	0.854017	0.427009	−	0.904248i	$-0.359567\pi$
$199$	477089.	0.854017	0.427009	+	0.904248i	$0.359567\pi$
$200$	0	0
$201$	−724524.	−1.26492
$202$	0	0
$203$	861930.i	1.46802i
$204$	0	0
$205$	205527.i	0.341573i
$206$	0	0
$207$	−108620.	−0.176191
$208$	0	0
$209$	514751.	0.815139
$210$	0	0
$211$	− 435894.i	− 0.674023i	−0.941500	−	0.337012i	$-0.890584\pi$
$211$	− 435894.i	− 0.674023i	0.941500	−	0.337012i	$-0.109416\pi$
$212$	0	0
$213$	1.33529e6i	2.01664i
$214$	0	0
$215$	232541.	0.343086
$216$	0	0
$217$	−88969.2	−0.128260
$218$	0	0
$219$	349863.i	0.492933i
$220$	0	0
$221$	409856.i	0.564483i
$222$	0	0
$223$	−1.14075e6	−1.53613	−0.768067	−	0.640369i	$-0.778781\pi$
$223$	−1.14075e6	−1.53613	−0.768067	+	0.640369i	$0.778781\pi$
$224$	0	0
$225$	68734.6	0.0905147
$226$	0	0
$227$	− 1.08895e6i	− 1.40262i	−0.712854	−	0.701312i	$-0.752598\pi$
$227$	− 1.08895e6i	− 1.40262i	0.712854	−	0.701312i	$-0.247402\pi$
$228$	0	0
$229$	474386.i	0.597783i	0.954287	+	0.298891i	$0.0966168\pi$
$229$	474386.i	0.597783i	−0.954287	+	0.298891i	$0.903383\pi$
$230$	0	0
$231$	−549951.	−0.678100
$232$	0	0
$233$	−271057.	−0.327093	−0.163547	−	0.986536i	$-0.552293\pi$
$233$	−271057.	−0.327093	−0.163547	+	0.986536i	$0.552293\pi$
$234$	0	0
$235$	345947.i	0.408639i
$236$	0	0
$237$	1.42033e6i	1.64255i
$238$	0	0
$239$	824404.	0.933567	0.466784	−	0.884372i	$-0.345413\pi$
$239$	824404.	0.933567	0.466784	+	0.884372i	$0.345413\pi$
$240$	0	0
$241$	−86717.2	−0.0961750	−0.0480875	−	0.998843i	$-0.515313\pi$
$241$	−86717.2	−0.0961750	−0.0480875	+	0.998843i	$0.515313\pi$
$242$	0	0
$243$	− 776933.i	− 0.844050i
$244$	0	0
$245$	131074.i	0.139509i
$246$	0	0
$247$	375186.	0.391294
$248$	0	0
$249$	2.35405e6	2.40612
$250$	0	0
$251$	− 411977.i	− 0.412751i	−0.978473	−	0.206376i	$-0.933833\pi$
$251$	− 411977.i	− 0.412751i	0.978473	−	0.206376i	$-0.0661669\pi$
$252$	0	0
$253$	268851.i	0.264065i
$254$	0	0
$255$	970282.	0.934431
$256$	0	0
$257$	−88057.5	−0.0831637	−0.0415818	−	0.999135i	$-0.513240\pi$
$257$	−88057.5	−0.0831637	−0.0415818	+	0.999135i	$0.513240\pi$
$258$	0	0
$259$	1.01367e6i	0.938958i
$260$	0	0
$261$	− 881480.i	− 0.800961i
$262$	0	0
$263$	126284.	0.112579	0.0562896	−	0.998414i	$-0.482073\pi$
$263$	126284.	0.112579	0.0562896	+	0.998414i	$0.482073\pi$
$264$	0	0
$265$	−693781.	−0.606886
$266$	0	0
$267$	570045.i	0.489363i
$268$	0	0
$269$	− 1.88178e6i	− 1.58558i	−0.609495	−	0.792790i	$-0.708628\pi$
$269$	− 1.88178e6i	− 1.58558i	0.609495	−	0.792790i	$-0.291372\pi$
$270$	0	0
$271$	1.17095e6	0.968532	0.484266	−	0.874921i	$-0.339087\pi$
$271$	1.17095e6	0.968532	0.484266	+	0.874921i	$0.339087\pi$
$272$	0	0
$273$	−400841.	−0.325511
$274$	0	0
$275$	− 170128.i	− 0.135658i
$276$	0	0
$277$	− 1.11420e6i	− 0.872496i	−0.899827	−	0.436248i	$-0.856307\pi$
$277$	− 1.11420e6i	− 0.872496i	0.899827	−	0.436248i	$-0.143693\pi$
$278$	0	0
$279$	90987.2	0.0699793
$280$	0	0
$281$	−1.87861e6	−1.41929	−0.709646	−	0.704558i	$-0.751145\pi$
$281$	−1.87861e6	−1.41929	−0.709646	+	0.704558i	$0.751145\pi$
$282$	0	0
$283$	3072.89i	0.00228077i	0.999999	+	0.00114038i	$0.000362996\pi$
$283$	3072.89i	0.00228077i	−0.999999	+	0.00114038i	$0.999637\pi$
$284$	0	0
$285$	− 888203.i	− 0.647739i
$286$	0	0
$287$	−884062.	−0.633546
$288$	0	0
$289$	2.84762e6	2.00557
$290$	0	0
$291$	293749.i	0.203350i
$292$	0	0
$293$	− 2.45968e6i	− 1.67382i	−0.547339	−	0.836911i	$-0.684359\pi$
$293$	− 2.45968e6i	− 1.67382i	0.547339	−	0.836911i	$-0.315641\pi$
$294$	0	0
$295$	627653.	0.419918
$296$	0	0
$297$	−680301.	−0.447518
$298$	0	0
$299$	195957.i	0.126760i
$300$	0	0
$301$	1.00026e6i	0.636352i
$302$	0	0
$303$	−1.92933e6	−1.20726
$304$	0	0
$305$	660115.	0.406322
$306$	0	0
$307$	257478.i	0.155917i	0.996957	+	0.0779585i	$0.0248402\pi$
$307$	257478.i	0.155917i	−0.996957	+	0.0779585i	$0.975160\pi$
$308$	0	0
$309$	676012.i	0.402771i
$310$	0	0
$311$	−2.43192e6	−1.42576	−0.712882	−	0.701284i	$-0.752610\pi$
$311$	−2.43192e6	−1.42576	−0.712882	+	0.701284i	$0.752610\pi$
$312$	0	0
$313$	−2.62542e6	−1.51474	−0.757369	−	0.652987i	$-0.773515\pi$
$313$	−2.62542e6	−1.51474	−0.757369	+	0.652987i	$0.773515\pi$
$314$	0	0
$315$	295658.i	0.167886i
$316$	0	0
$317$	− 2.54754e6i	− 1.42388i	−0.702240	−	0.711940i	$-0.747817\pi$
$317$	− 2.54754e6i	− 1.42388i	0.702240	−	0.711940i	$-0.252183\pi$
$318$	0	0
$319$	−2.18180e6	−1.20043
$320$	0	0
$321$	1.78454e6	0.966635
$322$	0	0
$323$	− 3.90648e6i	− 2.08343i
$324$	0	0
$325$	− 124001.i	− 0.0651205i
$326$	0	0
$327$	3.25322e6	1.68246
$328$	0	0
$329$	−1.48807e6	−0.757940
$330$	0	0
$331$	2.11710e6i	1.06211i	0.847336	+	0.531057i	$0.178205\pi$
$331$	2.11710e6i	1.06211i	−0.847336	+	0.531057i	$0.821795\pi$
$332$	0	0
$333$	− 1.03666e6i	− 0.512302i
$334$	0	0
$335$	964097.	0.469363
$336$	0	0
$337$	−1.56954e6	−0.752832	−0.376416	−	0.926451i	$-0.622844\pi$
$337$	−1.56954e6	−0.752832	−0.376416	+	0.926451i	$0.622844\pi$
$338$	0	0
$339$	− 4.46733e6i	− 2.11129i
$340$	0	0
$341$	− 225207.i	− 0.104881i
$342$	0	0
$343$	−2.37117e6	−1.08825
$344$	0	0
$345$	463903.	0.209836
$346$	0	0
$347$	428538.i	0.191058i	0.995427	+	0.0955292i	$0.0304543\pi$
$347$	428538.i	0.191058i	−0.995427	+	0.0955292i	$0.969546\pi$
$348$	0	0
$349$	522898.i	0.229802i	0.993377	+	0.114901i	$0.0366551\pi$
$349$	522898.i	0.229802i	−0.993377	+	0.114901i	$0.963345\pi$
$350$	0	0
$351$	−495850.	−0.214824
$352$	0	0
$353$	1.89389e6	0.808943	0.404472	−	0.914550i	$-0.367455\pi$
$353$	1.89389e6	0.808943	0.404472	+	0.914550i	$0.367455\pi$
$354$	0	0
$355$	− 1.77682e6i	− 0.748297i
$356$	0	0
$357$	4.17361e6i	1.73317i
$358$	0	0
$359$	2.57094e6	1.05282	0.526412	−	0.850229i	$-0.323537\pi$
$359$	2.57094e6	1.05282	0.526412	+	0.850229i	$0.323537\pi$
$360$	0	0
$361$	−1.09992e6	−0.444216
$362$	0	0
$363$	1.63368e6i	0.650730i
$364$	0	0
$365$	− 465550.i	− 0.182909i
$366$	0	0
$367$	2.47263e6	0.958282	0.479141	−	0.877738i	$-0.340948\pi$
$367$	2.47263e6	0.958282	0.479141	+	0.877738i	$0.340948\pi$
$368$	0	0
$369$	904114.	0.345667
$370$	0	0
$371$	− 2.98426e6i	− 1.12565i
$372$	0	0
$373$	− 2.62533e6i	− 0.977038i	−0.872553	−	0.488519i	$-0.837537\pi$
$373$	− 2.62533e6i	− 0.977038i	0.872553	−	0.488519i	$-0.162463\pi$
$374$	0	0
$375$	−293557.	−0.107799
$376$	0	0
$377$	−1.59024e6	−0.576249
$378$	0	0
$379$	2.14283e6i	0.766284i	0.923689	+	0.383142i	$0.125158\pi$
$379$	2.14283e6i	0.766284i	−0.923689	+	0.383142i	$0.874842\pi$
$380$	0	0
$381$	1.18894e6i	0.419610i
$382$	0	0
$383$	641783.	0.223559	0.111779	−	0.993733i	$-0.464345\pi$
$383$	641783.	0.223559	0.111779	+	0.993733i	$0.464345\pi$
$384$	0	0
$385$	731799.	0.251617
$386$	0	0
$387$	− 1.02295e6i	− 0.347198i
$388$	0	0
$389$	2.36321e6i	0.791823i	0.918289	+	0.395911i	$0.129571\pi$
$389$	2.36321e6i	0.791823i	−0.918289	+	0.395911i	$0.870429\pi$
$390$	0	0
$391$	2.04033e6	0.674930
$392$	0	0
$393$	3.38331e6	1.10499
$394$	0	0
$395$	− 1.88998e6i	− 0.609489i
$396$	0	0
$397$	403888.i	0.128613i	0.997930	+	0.0643065i	$0.0204835\pi$
$397$	403888.i	0.128613i	−0.997930	+	0.0643065i	$0.979516\pi$
$398$	0	0
$399$	3.82056e6	1.20142
$400$	0	0
$401$	2.28335e6	0.709108	0.354554	−	0.935036i	$-0.384633\pi$
$401$	2.28335e6	0.709108	0.354554	+	0.935036i	$0.384633\pi$
$402$	0	0
$403$	− 164146.i	− 0.0503464i
$404$	0	0
$405$	1.84196e6i	0.558011i
$406$	0	0
$407$	−2.56589e6	−0.767807
$408$	0	0
$409$	3.27964e6	0.969435	0.484717	−	0.874671i	$-0.338922\pi$
$409$	3.27964e6	0.969435	0.484717	+	0.874671i	$0.338922\pi$
$410$	0	0
$411$	6.87136e6i	2.00650i
$412$	0	0
$413$	2.69981e6i	0.778859i
$414$	0	0
$415$	−3.13245e6	−0.892820
$416$	0	0
$417$	−2.10208e6	−0.591983
$418$	0	0
$419$	604670.i	0.168261i	0.996455	+	0.0841305i	$0.0268113\pi$
$419$	604670.i	0.168261i	−0.996455	+	0.0841305i	$0.973189\pi$
$420$	0	0
$421$	3.46346e6i	0.952367i	0.879346	+	0.476184i	$0.157980\pi$
$421$	3.46346e6i	0.952367i	−0.879346	+	0.476184i	$0.842020\pi$
$422$	0	0
$423$	1.52183e6	0.413537
$424$	0	0
$425$	−1.29112e6	−0.346732
$426$	0	0
$427$	2.83945e6i	0.753641i
$428$	0	0
$429$	− 1.01465e6i	− 0.266178i
$430$	0	0
$431$	−5.00345e6	−1.29741	−0.648704	−	0.761041i	$-0.724689\pi$
$431$	−5.00345e6	−1.29741	−0.648704	+	0.761041i	$0.724689\pi$
$432$	0	0
$433$	−4.11259e6	−1.05413	−0.527067	−	0.849824i	$-0.676708\pi$
$433$	−4.11259e6	−1.05413	−0.527067	+	0.849824i	$0.676708\pi$
$434$	0	0
$435$	3.76469e6i	0.953908i
$436$	0	0
$437$	− 1.86773e6i	− 0.467855i
$438$	0	0
$439$	−4.20313e6	−1.04091	−0.520454	−	0.853890i	$-0.674237\pi$
$439$	−4.20313e6	−1.04091	−0.520454	+	0.853890i	$0.674237\pi$
$440$	0	0
$441$	576598.	0.141181
$442$	0	0
$443$	1.32165e6i	0.319969i	0.987120	+	0.159984i	$0.0511443\pi$
$443$	1.32165e6i	0.319969i	−0.987120	+	0.159984i	$0.948856\pi$
$444$	0	0
$445$	− 758538.i	− 0.181584i
$446$	0	0
$447$	−2.56414e6	−0.606978
$448$	0	0
$449$	−5.52202e6	−1.29265	−0.646327	−	0.763061i	$-0.723696\pi$
$449$	−5.52202e6	−1.29265	−0.646327	+	0.763061i	$0.723696\pi$
$450$	0	0
$451$	− 2.23782e6i	− 0.518064i
$452$	0	0
$453$	− 3.50115e6i	− 0.801615i
$454$	0	0
$455$	533385.	0.120785
$456$	0	0
$457$	3.46838e6	0.776849	0.388424	−	0.921481i	$-0.373019\pi$
$457$	3.46838e6	0.776849	0.388424	+	0.921481i	$0.373019\pi$
$458$	0	0
$459$	5.16286e6i	1.14382i
$460$	0	0
$461$	− 2.42345e6i	− 0.531107i	−0.964096	−	0.265554i	$-0.914445\pi$
$461$	− 2.42345e6i	− 0.531107i	0.964096	−	0.265554i	$-0.0855547\pi$
$462$	0	0
$463$	6.58501e6	1.42759	0.713796	−	0.700354i	$-0.246974\pi$
$463$	6.58501e6	1.42759	0.713796	+	0.700354i	$0.246974\pi$
$464$	0	0
$465$	−388595.	−0.0833422
$466$	0	0
$467$	− 1.15697e6i	− 0.245488i	−0.992438	−	0.122744i	$-0.960831\pi$
$467$	− 1.15697e6i	− 0.245488i	0.992438	−	0.122744i	$-0.0391694\pi$
$468$	0	0
$469$	4.14701e6i	0.870569i
$470$	0	0
$471$	1.40324e6	0.291461
$472$	0	0
$473$	−2.53196e6	−0.520359
$474$	0	0
$475$	1.18190e6i	0.240351i
$476$	0	0
$477$	3.05195e6i	0.614160i
$478$	0	0
$479$	380517.	0.0757767	0.0378883	−	0.999282i	$-0.487937\pi$
$479$	380517.	0.0757767	0.0378883	+	0.999282i	$0.487937\pi$
$480$	0	0
$481$	−1.87020e6	−0.368574
$482$	0	0
$483$	1.99545e6i	0.389201i
$484$	0	0
$485$	− 390881.i	− 0.0754554i
$486$	0	0
$487$	−2.67715e6	−0.511505	−0.255753	−	0.966742i	$-0.582323\pi$
$487$	−2.67715e6	−0.511505	−0.255753	+	0.966742i	$0.582323\pi$
$488$	0	0
$489$	−1.03103e7	−1.94983
$490$	0	0
$491$	− 5.10695e6i	− 0.956000i	−0.878360	−	0.478000i	$-0.841362\pi$
$491$	− 5.10695e6i	− 0.956000i	0.878360	−	0.478000i	$-0.158638\pi$
$492$	0	0
$493$	1.65578e7i	3.06822i
$494$	0	0
$495$	−748397.	−0.137284
$496$	0	0
$497$	7.64292e6	1.38793
$498$	0	0
$499$	259376.i	0.0466313i	0.999728	+	0.0233157i	$0.00742228\pi$
$499$	259376.i	0.0466313i	−0.999728	+	0.0233157i	$0.992578\pi$
$500$	0	0
$501$	− 4.21429e6i	− 0.750118i
$502$	0	0
$503$	324388.	0.0571669	0.0285835	−	0.999591i	$-0.490900\pi$
$503$	324388.	0.0571669	0.0285835	+	0.999591i	$0.490900\pi$
$504$	0	0
$505$	2.56729e6	0.447967
$506$	0	0
$507$	6.23617e6i	1.07745i
$508$	0	0
$509$	− 1.13575e7i	− 1.94307i	−0.236890	−	0.971537i	$-0.576128\pi$
$509$	− 1.13575e7i	− 1.94307i	0.236890	−	0.971537i	$-0.423872\pi$
$510$	0	0
$511$	2.00254e6	0.339257
$512$	0	0
$513$	4.72612e6	0.792887
$514$	0	0
$515$	− 899544.i	− 0.149453i
$516$	0	0
$517$	− 3.76675e6i	− 0.619784i
$518$	0	0
$519$	−3.10484e6	−0.505966
$520$	0	0
$521$	9.15918e6	1.47830	0.739149	−	0.673541i	$-0.235228\pi$
$521$	9.15918e6	1.47830	0.739149	+	0.673541i	$0.235228\pi$
$522$	0	0
$523$	7.08351e6i	1.13238i	0.824273	+	0.566192i	$0.191584\pi$
$523$	7.08351e6i	1.13238i	−0.824273	+	0.566192i	$0.808416\pi$
$524$	0	0
$525$	− 1.26272e6i	− 0.199944i
$526$	0	0
$527$	−1.70911e6	−0.268068
$528$	0	0
$529$	−5.46084e6	−0.848438
$530$	0	0
$531$	− 2.76105e6i	− 0.424950i
$532$	0	0
$533$	− 1.63108e6i	− 0.248689i
$534$	0	0
$535$	−2.37461e6	−0.358681
$536$	0	0
$537$	8.11579e6	1.21449
$538$	0	0
$539$	− 1.42717e6i	− 0.211594i
$540$	0	0
$541$	2.53136e6i	0.371843i	0.982565	+	0.185922i	$0.0595271\pi$
$541$	2.53136e6i	0.371843i	−0.982565	+	0.185922i	$0.940473\pi$
$542$	0	0
$543$	4.07586e6	0.593225
$544$	0	0
$545$	−4.32894e6	−0.624296
$546$	0	0
$547$	− 1.30020e7i	− 1.85798i	−0.370108	−	0.928989i	$-0.620679\pi$
$547$	− 1.30020e7i	− 1.85798i	0.370108	−	0.928989i	$-0.379321\pi$
$548$	0	0
$549$	− 2.90386e6i	− 0.411192i
$550$	0	0
$551$	1.51571e7	2.12686
$552$	0	0
$553$	8.12967e6	1.13047
$554$	0	0
$555$	4.42745e6i	0.610128i
$556$	0	0
$557$	− 5.70438e6i	− 0.779059i	−0.921014	−	0.389530i	$-0.872638\pi$
$557$	− 5.70438e6i	− 0.779059i	0.921014	−	0.389530i	$-0.127362\pi$
$558$	0	0
$559$	−1.84546e6	−0.249790
$560$	0	0
$561$	−1.05646e7	−1.41725
$562$	0	0
$563$	− 1.98939e6i	− 0.264514i	−0.991215	−	0.132257i	$-0.957778\pi$
$563$	− 1.98939e6i	− 0.264514i	0.991215	−	0.132257i	$-0.0422224\pi$
$564$	0	0
$565$	5.94451e6i	0.783420i
$566$	0	0
$567$	−7.92310e6	−1.03499
$568$	0	0
$569$	1.30277e6	0.168689	0.0843447	−	0.996437i	$-0.473120\pi$
$569$	1.30277e6	0.168689	0.0843447	+	0.996437i	$0.473120\pi$
$570$	0	0
$571$	− 2.90628e6i	− 0.373033i	−0.982452	−	0.186517i	$-0.940280\pi$
$571$	− 2.90628e6i	− 0.373033i	0.982452	−	0.186517i	$-0.0597199\pi$
$572$	0	0
$573$	649413.i	0.0826294i
$574$	0	0
$575$	−617298.	−0.0778619
$576$	0	0
$577$	4.01853e6	0.502490	0.251245	−	0.967923i	$-0.419160\pi$
$577$	4.01853e6	0.502490	0.251245	+	0.967923i	$0.419160\pi$
$578$	0	0
$579$	− 8.89785e6i	− 1.10303i
$580$	0	0
$581$	− 1.34741e7i	− 1.65599i
$582$	0	0
$583$	7.55404e6	0.920466
$584$	0	0
$585$	−545483.	−0.0659009
$586$	0	0
$587$	− 1.29654e7i	− 1.55307i	−0.630074	−	0.776535i	$-0.716975\pi$
$587$	− 1.29654e7i	− 1.55307i	0.630074	−	0.776535i	$-0.283025\pi$
$588$	0	0
$589$	1.56454e6i	0.185822i
$590$	0	0
$591$	7.41706e6	0.873501
$592$	0	0
$593$	−107000.	−0.0124953	−0.00624767	−	0.999980i	$-0.501989\pi$
$593$	−107000.	−0.0124953	−0.00624767	+	0.999980i	$0.501989\pi$
$594$	0	0
$595$	− 5.55367e6i	− 0.643114i
$596$	0	0
$597$	8.96337e6i	1.02928i
$598$	0	0
$599$	−1.06394e7	−1.21157	−0.605785	−	0.795629i	$-0.707141\pi$
$599$	−1.06394e7	−1.21157	−0.605785	+	0.795629i	$0.707141\pi$
$600$	0	0
$601$	−6.42705e6	−0.725814	−0.362907	−	0.931825i	$-0.618216\pi$
$601$	−6.42705e6	−0.725814	−0.362907	+	0.931825i	$0.618216\pi$
$602$	0	0
$603$	− 4.24108e6i	− 0.474988i
$604$	0	0
$605$	− 2.17388e6i	− 0.241461i
$606$	0	0
$607$	−1.96807e6	−0.216805	−0.108403	−	0.994107i	$-0.534574\pi$
$607$	−1.96807e6	−0.216805	−0.108403	+	0.994107i	$0.534574\pi$
$608$	0	0
$609$	−1.61936e7	−1.76930
$610$	0	0
$611$	− 2.74546e6i	− 0.297518i
$612$	0	0
$613$	− 8.12485e6i	− 0.873301i	−0.899631	−	0.436651i	$-0.856165\pi$
$613$	− 8.12485e6i	− 0.873301i	0.899631	−	0.436651i	$-0.143835\pi$
$614$	0	0
$615$	−3.86136e6	−0.411673
$616$	0	0
$617$	1.40750e6	0.148846	0.0744229	−	0.997227i	$-0.476289\pi$
$617$	1.40750e6	0.148846	0.0744229	+	0.997227i	$0.476289\pi$
$618$	0	0
$619$	− 5.01827e6i	− 0.526414i	−0.964739	−	0.263207i	$-0.915220\pi$
$619$	− 5.01827e6i	− 0.526414i	0.964739	−	0.263207i	$-0.0847802\pi$
$620$	0	0
$621$	2.46842e6i	0.256856i
$622$	0	0
$623$	3.26281e6	0.336800
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	9.67095e6i	0.982427i
$628$	0	0
$629$	1.94727e7i	1.96246i
$630$	0	0
$631$	−5.59458e6	−0.559363	−0.279682	−	0.960093i	$-0.590229\pi$
$631$	−5.59458e6	−0.559363	−0.279682	+	0.960093i	$0.590229\pi$
$632$	0	0
$633$	8.18942e6	0.812351
$634$	0	0
$635$	− 1.58207e6i	− 0.155701i
$636$	0	0
$637$	− 1.04022e6i	− 0.101572i
$638$	0	0
$639$	−7.81627e6	−0.757265
$640$	0	0
$641$	3.70806e6	0.356453	0.178226	−	0.983990i	$-0.442964\pi$
$641$	3.70806e6	0.356453	0.178226	+	0.983990i	$0.442964\pi$
$642$	0	0
$643$	− 1.36989e6i	− 0.130665i	−0.997864	−	0.0653325i	$-0.979189\pi$
$643$	− 1.36989e6i	− 0.130665i	0.997864	−	0.0653325i	$-0.0208108\pi$
$644$	0	0
$645$	4.36889e6i	0.413497i
$646$	0	0
$647$	1.38641e7	1.30206	0.651028	−	0.759053i	$-0.274338\pi$
$647$	1.38641e7	1.30206	0.651028	+	0.759053i	$0.274338\pi$
$648$	0	0
$649$	−6.83402e6	−0.636890
$650$	0	0
$651$	− 1.67152e6i	− 0.154582i
$652$	0	0
$653$	− 1.00652e7i	− 0.923714i	−0.886954	−	0.461857i	$-0.847183\pi$
$653$	− 1.00652e7i	− 0.923714i	0.886954	−	0.461857i	$-0.152817\pi$
$654$	0	0
$655$	−4.50204e6	−0.410021
$656$	0	0
$657$	−2.04796e6	−0.185101
$658$	0	0
$659$	2.94962e6i	0.264578i	0.991211	+	0.132289i	$0.0422326\pi$
$659$	2.94962e6i	0.264578i	−0.991211	+	0.132289i	$0.957767\pi$
$660$	0	0
$661$	− 2.46213e6i	− 0.219184i	−0.993977	−	0.109592i	$-0.965046\pi$
$661$	− 2.46213e6i	− 0.219184i	0.993977	−	0.109592i	$-0.0349544\pi$
$662$	0	0
$663$	−7.70023e6	−0.680331
$664$	0	0
$665$	−5.08387e6	−0.445801
$666$	0	0
$667$	7.91648e6i	0.688997i
$668$	0	0
$669$	− 2.14320e7i	− 1.85139i
$670$	0	0
$671$	−7.18748e6	−0.616269
$672$	0	0
$673$	8.46100e6	0.720085	0.360043	−	0.932936i	$-0.382762\pi$
$673$	8.46100e6	0.720085	0.360043	+	0.932936i	$0.382762\pi$
$674$	0	0
$675$	− 1.56201e6i	− 0.131955i
$676$	0	0
$677$	− 2.01968e7i	− 1.69360i	−0.531913	−	0.846799i	$-0.678527\pi$
$677$	− 2.01968e7i	− 1.69360i	0.531913	−	0.846799i	$-0.321473\pi$
$678$	0	0
$679$	1.68135e6	0.139954
$680$	0	0
$681$	2.04587e7	1.69048
$682$	0	0
$683$	5.52541e6i	0.453224i	0.973985	+	0.226612i	$0.0727649\pi$
$683$	5.52541e6i	0.453224i	−0.973985	+	0.226612i	$0.927235\pi$
$684$	0	0
$685$	− 9.14346e6i	− 0.744534i
$686$	0	0
$687$	−8.91259e6	−0.720464
$688$	0	0
$689$	5.50590e6	0.441855
$690$	0	0
$691$	1.66737e7i	1.32843i	0.747543	+	0.664214i	$0.231234\pi$
$691$	1.66737e7i	1.32843i	−0.747543	+	0.664214i	$0.768766\pi$
$692$	0	0
$693$	− 3.21919e6i	− 0.254632i
$694$	0	0
$695$	2.79716e6	0.219662
$696$	0	0
$697$	−1.69830e7	−1.32413
$698$	0	0
$699$	− 5.09253e6i	− 0.394222i
$700$	0	0
$701$	1.49989e7i	1.15283i	0.817157	+	0.576415i	$0.195549\pi$
$701$	1.49989e7i	1.15283i	−0.817157	+	0.576415i	$0.804451\pi$
$702$	0	0
$703$	1.78255e7	1.36036
$704$	0	0
$705$	−6.49953e6	−0.492503
$706$	0	0
$707$	1.10431e7i	0.830884i
$708$	0	0
$709$	1.84405e7i	1.37771i	0.724900	+	0.688854i	$0.241886\pi$
$709$	1.84405e7i	1.37771i	−0.724900	+	0.688854i	$0.758114\pi$
$710$	0	0
$711$	−8.31407e6	−0.616793
$712$	0	0
$713$	−817147.	−0.0601972
$714$	0	0
$715$	1.35015e6i	0.0987684i
$716$	0	0
$717$	1.54886e7i	1.12516i
$718$	0	0
$719$	334083.	0.0241009	0.0120504	−	0.999927i	$-0.496164\pi$
$719$	334083.	0.0241009	0.0120504	+	0.999927i	$0.496164\pi$
$720$	0	0
$721$	3.86934e6	0.277204
$722$	0	0
$723$	− 1.62921e6i	− 0.115913i
$724$	0	0
$725$	− 5.00953e6i	− 0.353959i
$726$	0	0
$727$	2.11438e7	1.48370	0.741851	−	0.670564i	$-0.233948\pi$
$727$	2.11438e7	1.48370	0.741851	+	0.670564i	$0.233948\pi$
$728$	0	0
$729$	−3.30712e6	−0.230479
$730$	0	0
$731$	1.92152e7i	1.33000i
$732$	0	0
$733$	1.48236e6i	0.101904i	0.998701	+	0.0509521i	$0.0162256\pi$
$733$	1.48236e6i	0.101904i	−0.998701	+	0.0509521i	$0.983774\pi$
$734$	0	0
$735$	−2.46258e6	−0.168140
$736$	0	0
$737$	−1.04973e7	−0.711884
$738$	0	0
$739$	1.69772e7i	1.14355i	0.820411	+	0.571775i	$0.193745\pi$
$739$	1.69772e7i	1.14355i	−0.820411	+	0.571775i	$0.806255\pi$
$740$	0	0
$741$	7.04885e6i	0.471599i
$742$	0	0
$743$	2.01294e7	1.33770	0.668849	−	0.743399i	$-0.266787\pi$
$743$	2.01294e7	1.33770	0.668849	+	0.743399i	$0.266787\pi$
$744$	0	0
$745$	3.41201e6	0.225226
$746$	0	0
$747$	1.37797e7i	0.903520i
$748$	0	0
$749$	− 1.02143e7i	− 0.665278i
$750$	0	0
$751$	1.16155e7	0.751514	0.375757	−	0.926718i	$-0.377383\pi$
$751$	1.16155e7	0.751514	0.375757	+	0.926718i	$0.377383\pi$
$752$	0	0
$753$	7.74007e6	0.497459
$754$	0	0
$755$	4.65885e6i	0.297448i
$756$	0	0
$757$	9.97313e6i	0.632545i	0.948668	+	0.316273i	$0.102431\pi$
$757$	9.97313e6i	0.632545i	−0.948668	+	0.316273i	$0.897569\pi$
$758$	0	0
$759$	−5.05108e6	−0.318258
$760$	0	0
$761$	2.00566e7	1.25544	0.627719	−	0.778440i	$-0.283989\pi$
$761$	2.00566e7	1.25544	0.627719	+	0.778440i	$0.283989\pi$
$762$	0	0
$763$	− 1.86207e7i	− 1.15794i
$764$	0	0
$765$	5.67964e6i	0.350887i
$766$	0	0
$767$	−4.98110e6	−0.305729
$768$	0	0
$769$	2.63971e7	1.60968	0.804842	−	0.593489i	$-0.202250\pi$
$769$	2.63971e7	1.60968	0.804842	+	0.593489i	$0.202250\pi$
$770$	0	0
$771$	− 1.65439e6i	− 0.100231i
$772$	0	0
$773$	902367.i	0.0543169i	0.999631	+	0.0271584i	$0.00864586\pi$
$773$	902367.i	0.0543169i	−0.999631	+	0.0271584i	$0.991354\pi$
$774$	0	0
$775$	517089.	0.0309251
$776$	0	0
$777$	−1.90444e7	−1.13166
$778$	0	0
$779$	1.55463e7i	0.917877i
$780$	0	0
$781$	1.93465e7i	1.13494i
$782$	0	0
$783$	−2.00319e7	−1.16766
$784$	0	0
$785$	−1.86724e6	−0.108150
$786$	0	0
$787$	− 1.59032e7i	− 0.915269i	−0.889140	−	0.457635i	$-0.848697\pi$
$787$	− 1.59032e7i	− 0.915269i	0.889140	−	0.457635i	$-0.151303\pi$
$788$	0	0
$789$	2.37258e6i	0.135684i
$790$	0	0
$791$	−2.55700e7	−1.45308
$792$	0	0
$793$	−5.23873e6	−0.295830
$794$	0	0
$795$	− 1.30345e7i	− 0.731436i
$796$	0	0
$797$	2.52099e7i	1.40580i	0.711287	+	0.702902i	$0.248113\pi$
$797$	2.52099e7i	1.40580i	−0.711287	+	0.702902i	$0.751887\pi$
$798$	0	0
$799$	−2.85861e7	−1.58412
$800$	0	0
$801$	−3.33682e6	−0.183760
$802$	0	0
$803$	5.06901e6i	0.277418i
$804$	0	0
$805$	− 2.65527e6i	− 0.144417i
$806$	0	0
$807$	3.53542e7	1.91099
$808$	0	0
$809$	−1.43630e7	−0.771570	−0.385785	−	0.922589i	$-0.626069\pi$
$809$	−1.43630e7	−0.771570	−0.385785	+	0.922589i	$0.626069\pi$
$810$	0	0
$811$	− 1.49512e7i	− 0.798222i	−0.916903	−	0.399111i	$-0.869319\pi$
$811$	− 1.49512e7i	− 0.798222i	0.916903	−	0.399111i	$-0.130681\pi$
$812$	0	0
$813$	2.19993e7i	1.16730i
$814$	0	0
$815$	1.37195e7	0.723508
$816$	0	0
$817$	1.75897e7	0.921943
$818$	0	0
$819$	− 2.34637e6i	− 0.122232i
$820$	0	0
$821$	2.49701e7i	1.29289i	0.762960	+	0.646446i	$0.223746\pi$
$821$	2.49701e7i	1.29289i	−0.762960	+	0.646446i	$0.776254\pi$
$822$	0	0
$823$	−2.50946e7	−1.29146	−0.645729	−	0.763567i	$-0.723446\pi$
$823$	−2.50946e7	−1.29146	−0.645729	+	0.763567i	$0.723446\pi$
$824$	0	0
$825$	3.19631e6	0.163499
$826$	0	0
$827$	− 2.47367e7i	− 1.25770i	−0.777526	−	0.628850i	$-0.783526\pi$
$827$	− 2.47367e7i	− 1.25770i	0.777526	−	0.628850i	$-0.216474\pi$
$828$	0	0
$829$	− 1.02078e7i	− 0.515878i	−0.966161	−	0.257939i	$-0.916957\pi$
$829$	− 1.02078e7i	− 0.515878i	0.966161	−	0.257939i	$-0.0830434\pi$
$830$	0	0
$831$	2.09332e7	1.05156
$832$	0	0
$833$	−1.08309e7	−0.540818
$834$	0	0
$835$	5.60779e6i	0.278340i
$836$	0	0
$837$	− 2.06771e6i	− 0.102018i
$838$	0	0
$839$	−2.64451e7	−1.29700	−0.648501	−	0.761214i	$-0.724604\pi$
$839$	−2.64451e7	−1.29700	−0.648501	+	0.761214i	$0.724604\pi$
$840$	0	0
$841$	−4.37332e7	−2.13217
$842$	0	0
$843$	− 3.52947e7i	− 1.71057i
$844$	0	0
$845$	− 8.29824e6i	− 0.399801i
$846$	0	0
$847$	9.35082e6	0.447859
$848$	0	0
$849$	−57732.4	−0.00274885
$850$	0	0
$851$	9.31014e6i	0.440689i
$852$	0	0
$853$	− 2.30070e7i	− 1.08265i	−0.840814	−	0.541324i	$-0.817923\pi$
$853$	− 2.30070e7i	− 1.08265i	0.840814	−	0.541324i	$-0.182077\pi$
$854$	0	0
$855$	5.19919e6	0.243232
$856$	0	0
$857$	−1.81332e7	−0.843379	−0.421689	−	0.906740i	$-0.638563\pi$
$857$	−1.81332e7	−0.843379	−0.421689	+	0.906740i	$0.638563\pi$
$858$	0	0
$859$	1.83891e6i	0.0850310i	0.999096	+	0.0425155i	$0.0135372\pi$
$859$	1.83891e6i	0.0850310i	−0.999096	+	0.0425155i	$0.986463\pi$
$860$	0	0
$861$	− 1.66094e7i	− 0.763567i
$862$	0	0
$863$	−1.14230e7	−0.522097	−0.261049	−	0.965326i	$-0.584068\pi$
$863$	−1.14230e7	−0.522097	−0.261049	+	0.965326i	$0.584068\pi$
$864$	0	0
$865$	4.13150e6	0.187744
$866$	0	0
$867$	5.35001e7i	2.41717i
$868$	0	0
$869$	2.05786e7i	0.924413i
$870$	0	0
$871$	−7.65115e6	−0.341729
$872$	0	0
$873$	−1.71949e6	−0.0763597
$874$	0	0
$875$	1.68025e6i	0.0741915i
$876$	0	0
$877$	− 2.12365e7i	− 0.932362i	−0.884689	−	0.466181i	$-0.845630\pi$
$877$	− 2.12365e7i	− 0.932362i	0.884689	−	0.466181i	$-0.154370\pi$
$878$	0	0
$879$	4.62116e7	2.01734
$880$	0	0
$881$	1.35067e7	0.586287	0.293143	−	0.956069i	$-0.405299\pi$
$881$	1.35067e7	0.586287	0.293143	+	0.956069i	$0.405299\pi$
$882$	0	0
$883$	2.44466e7i	1.05515i	0.849507	+	0.527577i	$0.176900\pi$
$883$	2.44466e7i	1.05515i	−0.849507	+	0.527577i	$0.823100\pi$
$884$	0	0
$885$	1.17921e7i	0.506096i
$886$	0	0
$887$	−8.02242e6	−0.342371	−0.171185	−	0.985239i	$-0.554760\pi$
$887$	−8.02242e6	−0.342371	−0.171185	+	0.985239i	$0.554760\pi$
$888$	0	0
$889$	6.80520e6	0.288793
$890$	0	0
$891$	− 2.00557e7i	− 0.846337i
$892$	0	0
$893$	2.61680e7i	1.09810i
$894$	0	0
$895$	−1.07994e7	−0.450652
$896$	0	0
$897$	−3.68157e6	−0.152775
$898$	0	0
$899$	− 6.63136e6i	− 0.273655i
$900$	0	0
$901$	− 5.73281e7i	− 2.35264i
$902$	0	0
$903$	−1.87926e7	−0.766949
$904$	0	0
$905$	−5.42359e6	−0.220123
$906$	0	0
$907$	− 2.02868e7i	− 0.818831i	−0.912348	−	0.409416i	$-0.865733\pi$
$907$	− 2.02868e7i	− 0.818831i	0.912348	−	0.409416i	$-0.134267\pi$
$908$	0	0
$909$	− 1.12935e7i	− 0.453336i
$910$	0	0
$911$	−1.72893e7	−0.690209	−0.345104	−	0.938564i	$-0.612156\pi$
$911$	−1.72893e7	−0.690209	−0.345104	+	0.938564i	$0.612156\pi$
$912$	0	0
$913$	3.41068e7	1.35414
$914$	0	0
$915$	1.24020e7i	0.489710i
$916$	0	0
$917$	− 1.93653e7i	− 0.760502i
$918$	0	0
$919$	2.14060e7	0.836079	0.418039	−	0.908429i	$-0.362717\pi$
$919$	2.14060e7	0.836079	0.418039	+	0.908429i	$0.362717\pi$
$920$	0	0
$921$	−4.83740e6	−0.187916
$922$	0	0
$923$	1.41010e7i	0.544812i
$924$	0	0
$925$	− 5.89144e6i	− 0.226395i
$926$	0	0
$927$	−3.95710e6	−0.151244
$928$	0	0
$929$	4.14158e7	1.57444	0.787221	−	0.616670i	$-0.211519\pi$
$929$	4.14158e7	1.57444	0.787221	+	0.616670i	$0.211519\pi$
$930$	0	0
$931$	9.91466e6i	0.374890i
$932$	0	0
$933$	− 4.56900e7i	− 1.71837i
$934$	0	0
$935$	1.40580e7	0.525889
$936$	0	0
$937$	1.39335e7	0.518455	0.259228	−	0.965816i	$-0.416532\pi$
$937$	1.39335e7	0.518455	0.259228	+	0.965816i	$0.416532\pi$
$938$	0	0
$939$	− 4.93254e7i	− 1.82560i
$940$	0	0
$941$	− 1.51311e7i	− 0.557051i	−0.960429	−	0.278526i	$-0.910154\pi$
$941$	− 1.51311e7i	− 0.557051i	0.960429	−	0.278526i	$-0.0898457\pi$
$942$	0	0
$943$	−8.11975e6	−0.297347
$944$	0	0
$945$	6.71891e6	0.244748
$946$	0	0
$947$	− 2.72196e7i	− 0.986296i	−0.869945	−	0.493148i	$-0.835846\pi$
$947$	− 2.72196e7i	− 0.986296i	0.869945	−	0.493148i	$-0.164154\pi$
$948$	0	0
$949$	3.69464e6i	0.133170i
$950$	0	0
$951$	4.78623e7	1.71610
$952$	0	0
$953$	−2.79494e7	−0.996873	−0.498437	−	0.866926i	$-0.666092\pi$
$953$	−2.79494e7	−0.996873	−0.498437	+	0.866926i	$0.666092\pi$
$954$	0	0
$955$	− 864150.i	− 0.0306606i
$956$	0	0
$957$	− 4.09908e7i	− 1.44679i
$958$	0	0
$959$	3.93301e7	1.38095
$960$	0	0
$961$	−2.79447e7	−0.976091
$962$	0	0
$963$	1.04460e7i	0.362980i
$964$	0	0
$965$	1.18400e7i	0.409293i
$966$	0	0
$967$	−3.73167e7	−1.28333	−0.641663	−	0.766987i	$-0.721755\pi$
$967$	−3.73167e7	−1.28333	−0.641663	+	0.766987i	$0.721755\pi$
$968$	0	0
$969$	7.33936e7	2.51101
$970$	0	0
$971$	− 1.77774e7i	− 0.605091i	−0.953135	−	0.302545i	$-0.902164\pi$
$971$	− 1.77774e7i	− 0.605091i	0.953135	−	0.302545i	$-0.0978364\pi$
$972$	0	0
$973$	1.20318e7i	0.407427i
$974$	0	0
$975$	2.32969e6	0.0784850
$976$	0	0
$977$	1.67378e7	0.561000	0.280500	−	0.959854i	$-0.409500\pi$
$977$	1.67378e7	0.561000	0.280500	+	0.959854i	$0.409500\pi$
$978$	0	0
$979$	8.25913e6i	0.275409i
$980$	0	0
$981$	1.90431e7i	0.631778i
$982$	0	0
$983$	2.99351e7	0.988091	0.494045	−	0.869436i	$-0.335518\pi$
$983$	2.99351e7	0.988091	0.494045	+	0.869436i	$0.335518\pi$
$984$	0	0
$985$	−9.86961e6	−0.324123
$986$	0	0
$987$	− 2.79574e7i	− 0.913490i
$988$	0	0
$989$	9.18701e6i	0.298664i
$990$	0	0
$991$	−1.30589e7	−0.422398	−0.211199	−	0.977443i	$-0.567737\pi$
$991$	−1.30589e7	−0.422398	−0.211199	+	0.977443i	$0.567737\pi$
$992$	0	0
$993$	−3.97753e7	−1.28009
$994$	0	0
$995$	− 1.19272e7i	− 0.381928i
$996$	0	0
$997$	− 7.99417e6i	− 0.254704i	−0.991858	−	0.127352i	$-0.959352\pi$
$997$	− 7.99417e6i	− 0.254704i	0.991858	−	0.127352i	$-0.0406477\pi$
$998$	0	0
$999$	−2.35584e7	−0.746848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.d.a.81.17		20
4.3	odd	2		40.6.d.a.21.19	✓	20
5.2	odd	4		800.6.f.b.49.18		20
5.3	odd	4		800.6.f.c.49.3		20
5.4	even	2		800.6.d.c.401.4		20
8.3	odd	2		40.6.d.a.21.20	yes	20
8.5	even	2	inner	160.6.d.a.81.4		20
12.11	even	2		360.6.k.b.181.2		20
20.3	even	4		200.6.f.b.149.11		20
20.7	even	4		200.6.f.c.149.10		20
20.19	odd	2		200.6.d.b.101.2		20
24.11	even	2		360.6.k.b.181.1		20
40.3	even	4		200.6.f.c.149.9		20
40.13	odd	4		800.6.f.b.49.17		20
40.19	odd	2		200.6.d.b.101.1		20
40.27	even	4		200.6.f.b.149.12		20
40.29	even	2		800.6.d.c.401.17		20
40.37	odd	4		800.6.f.c.49.4		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.19	✓	20	4.3	odd	2
40.6.d.a.21.20	yes	20	8.3	odd	2
160.6.d.a.81.4		20	8.5	even	2	inner
160.6.d.a.81.17		20	1.1	even	1	trivial
200.6.d.b.101.1		20	40.19	odd	2
200.6.d.b.101.2		20	20.19	odd	2
200.6.f.b.149.11		20	20.3	even	4
200.6.f.b.149.12		20	40.27	even	4
200.6.f.c.149.9		20	40.3	even	4
200.6.f.c.149.10		20	20.7	even	4
360.6.k.b.181.1		20	24.11	even	2
360.6.k.b.181.2		20	12.11	even	2
800.6.d.c.401.4		20	5.4	even	2
800.6.d.c.401.17		20	40.29	even	2
800.6.f.b.49.17		20	40.13	odd	4
800.6.f.b.49.18		20	5.2	odd	4
800.6.f.c.49.3		20	5.3	odd	4
800.6.f.c.49.4		20	40.37	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.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+18.7876i q^{3} -25.0000i q^{5} +107.536 q^{7} -109.975 q^{9} +O(q^{10})\)	\(q+18.7876i q^{3} -25.0000i q^{5} +107.536 q^{7} -109.975 q^{9} +272.206i q^{11} +198.402i q^{13} +469.691 q^{15} +2065.79 q^{17} -1891.04i q^{19} +2020.35i q^{21} +987.677 q^{23} -625.000 q^{25} +2499.22i q^{27} +8015.26i q^{29} -827.342 q^{31} -5114.10 q^{33} -2688.40i q^{35} +9426.30i q^{37} -3727.50 q^{39} -8221.07 q^{41} +9301.63i q^{43} +2749.38i q^{45} -13837.9 q^{47} -5242.97 q^{49} +38811.3i q^{51} -27751.2i q^{53} +6805.14 q^{55} +35528.1 q^{57} +25106.1i q^{59} +26404.6i q^{61} -11826.3 q^{63} +4960.05 q^{65} +38563.9i q^{67} +18556.1i q^{69} +71073.0 q^{71} +18622.0 q^{73} -11742.3i q^{75} +29271.9i q^{77} +75599.4 q^{79} -73678.4 q^{81} -125298. i q^{83} -51644.7i q^{85} -150588. q^{87} +30341.5 q^{89} +21335.4i q^{91} -15543.8i q^{93} -47275.9 q^{95} +15635.2 q^{97} -29935.9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) \) 20 * q + 196 * q^7 - 1620 * q^9	\( 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63} + 200312 q^{71} - 105136 q^{73} - 282080 q^{79} + 65172 q^{81} + 332592 q^{87} - 3160 q^{89} - 144400 q^{95} + 147376 q^{97}+O(q^{100}) \) 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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