LMFDB - Embedded newform 160.6.d.a.81.1

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			81.1
Root			$3.46430 + 1.99965i$ of defining polynomial
Character	$\chi$	$=$	160.81
Dual form			160.6.d.a.81.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-29.2080i q^{3} -25.0000i q^{5} +168.173 q^{7} -610.110 q^{9} -514.493i q^{11} -491.622i q^{13} -730.201 q^{15} +183.094 q^{17} -1250.96i q^{19} -4912.01i q^{21} +423.498 q^{23} -625.000 q^{25} +10722.5i q^{27} +3463.40i q^{29} -2343.92 q^{31} -15027.3 q^{33} -4204.33i q^{35} +7388.25i q^{37} -14359.3 q^{39} +4240.39 q^{41} +15159.4i q^{43} +15252.7i q^{45} +15357.8 q^{47} +11475.3 q^{49} -5347.82i q^{51} -11393.9i q^{53} -12862.3 q^{55} -36538.2 q^{57} -11978.0i q^{59} +41454.0i q^{61} -102604. q^{63} -12290.5 q^{65} -66524.9i q^{67} -12369.6i q^{69} +26214.5 q^{71} -86291.9 q^{73} +18255.0i q^{75} -86524.0i q^{77} +19799.4 q^{79} +164928. q^{81} +8370.24i q^{83} -4577.35i q^{85} +101159. q^{87} -3824.45 q^{89} -82677.7i q^{91} +68461.3i q^{93} -31274.1 q^{95} +35158.5 q^{97} +313897. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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
$3$	− 29.2080i	− 1.87370i	−0.349736	−	0.936848i	$-0.613729\pi$
$3$	− 29.2080i	− 1.87370i	0.349736	−	0.936848i	$-0.386271\pi$
$4$	0	0
$5$	− 25.0000i	− 0.447214i
$5$	− 25.0000i	− 0.447214i
$6$	0	0
$7$	168.173	1.29722	0.648608	−	0.761123i	$-0.275352\pi$
$7$	168.173	1.29722	0.648608	+	0.761123i	$0.275352\pi$
$8$	0	0
$9$	−610.110	−2.51074
$10$	0	0
$11$	− 514.493i	− 1.28203i	−0.767529	−	0.641014i	$-0.778514\pi$
$11$	− 514.493i	− 1.28203i	0.767529	−	0.641014i	$-0.221486\pi$
$12$	0	0
$13$	− 491.622i	− 0.806813i	−0.915021	−	0.403406i	$-0.867826\pi$
$13$	− 491.622i	− 0.806813i	0.915021	−	0.403406i	$-0.132174\pi$
$14$	0	0
$15$	−730.201	−0.837943
$16$	0	0
$17$	183.094	0.153657	0.0768284	−	0.997044i	$-0.475521\pi$
$17$	183.094	0.153657	0.0768284	+	0.997044i	$0.475521\pi$
$18$	0	0
$19$	− 1250.96i	− 0.794988i	−0.917605	−	0.397494i	$-0.869880\pi$
$19$	− 1250.96i	− 0.794988i	0.917605	−	0.397494i	$-0.130120\pi$
$20$	0	0
$21$	− 4912.01i	− 2.43059i
$22$	0	0
$23$	423.498	0.166929	0.0834646	−	0.996511i	$-0.473401\pi$
$23$	423.498	0.166929	0.0834646	+	0.996511i	$0.473401\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	10722.5i	2.83067i
$28$	0	0
$29$	3463.40i	0.764729i	0.924012	+	0.382364i	$0.124890\pi$
$29$	3463.40i	0.764729i	−0.924012	+	0.382364i	$0.875110\pi$
$30$	0	0
$31$	−2343.92	−0.438065	−0.219032	−	0.975718i	$-0.570290\pi$
$31$	−2343.92	−0.438065	−0.219032	+	0.975718i	$0.570290\pi$
$32$	0	0
$33$	−15027.3	−2.40213
$34$	0	0
$35$	− 4204.33i	− 0.580132i
$36$	0	0
$37$	7388.25i	0.887232i	0.896217	+	0.443616i	$0.146305\pi$
$37$	7388.25i	0.887232i	−0.896217	+	0.443616i	$0.853695\pi$
$38$	0	0
$39$	−14359.3	−1.51172
$40$	0	0
$41$	4240.39	0.393954	0.196977	−	0.980408i	$-0.436888\pi$
$41$	4240.39	0.393954	0.196977	+	0.980408i	$0.436888\pi$
$42$	0	0
$43$	15159.4i	1.25029i	0.780510	+	0.625143i	$0.214960\pi$
$43$	15159.4i	1.25029i	−0.780510	+	0.625143i	$0.785040\pi$
$44$	0	0
$45$	15252.7i	1.12284i
$46$	0	0
$47$	15357.8	1.01411	0.507055	−	0.861914i	$-0.330734\pi$
$47$	15357.8	1.01411	0.507055	+	0.861914i	$0.330734\pi$
$48$	0	0
$49$	11475.3	0.682768
$50$	0	0
$51$	− 5347.82i	− 0.287906i
$52$	0	0
$53$	− 11393.9i	− 0.557162i	−0.960413	−	0.278581i	$-0.910136\pi$
$53$	− 11393.9i	− 0.557162i	0.960413	−	0.278581i	$-0.0898640\pi$
$54$	0	0
$55$	−12862.3	−0.573340
$56$	0	0
$57$	−36538.2	−1.48957
$58$	0	0
$59$	− 11978.0i	− 0.447974i	−0.974592	−	0.223987i	$-0.928093\pi$
$59$	− 11978.0i	− 0.447974i	0.974592	−	0.223987i	$-0.0719074\pi$
$60$	0	0
$61$	41454.0i	1.42640i	0.700960	+	0.713200i	$0.252755\pi$
$61$	41454.0i	1.42640i	−0.700960	+	0.713200i	$0.747245\pi$
$62$	0	0
$63$	−102604.	−3.25697
$64$	0	0
$65$	−12290.5	−0.360818
$66$	0	0
$67$	− 66524.9i	− 1.81049i	−0.424885	−	0.905247i	$-0.639685\pi$
$67$	− 66524.9i	− 1.81049i	0.424885	−	0.905247i	$-0.360315\pi$
$68$	0	0
$69$	− 12369.6i	− 0.312775i
$70$	0	0
$71$	26214.5	0.617157	0.308579	−	0.951199i	$-0.400147\pi$
$71$	26214.5	0.617157	0.308579	+	0.951199i	$0.400147\pi$
$72$	0	0
$73$	−86291.9	−1.89523	−0.947617	−	0.319409i	$-0.896516\pi$
$73$	−86291.9	−1.89523	−0.947617	+	0.319409i	$0.896516\pi$
$74$	0	0
$75$	18255.0i	0.374739i
$76$	0	0
$77$	− 86524.0i	− 1.66307i
$78$	0	0
$79$	19799.4	0.356931	0.178466	−	0.983946i	$-0.442887\pi$
$79$	19799.4	0.356931	0.178466	+	0.983946i	$0.442887\pi$
$80$	0	0
$81$	164928.	2.79307
$82$	0	0
$83$	8370.24i	0.133365i	0.997774	+	0.0666826i	$0.0212415\pi$
$83$	8370.24i	0.133365i	−0.997774	+	0.0666826i	$0.978759\pi$
$84$	0	0
$85$	− 4577.35i	− 0.0687174i
$86$	0	0
$87$	101159.	1.43287
$88$	0	0
$89$	−3824.45	−0.0511793	−0.0255896	−	0.999673i	$-0.508146\pi$
$89$	−3824.45	−0.0511793	−0.0255896	+	0.999673i	$0.508146\pi$
$90$	0	0
$91$	− 82677.7i	− 1.04661i
$92$	0	0
$93$	68461.3i	0.820801i
$94$	0	0
$95$	−31274.1	−0.355529
$96$	0	0
$97$	35158.5	0.379403	0.189702	−	0.981842i	$-0.439248\pi$
$97$	35158.5	0.379403	0.189702	+	0.981842i	$0.439248\pi$
$98$	0	0
$99$	313897.i	3.21884i
$100$	0	0
$101$	− 99536.3i	− 0.970908i	−0.874262	−	0.485454i	$-0.838654\pi$
$101$	− 99536.3i	− 0.970908i	0.874262	−	0.485454i	$-0.161346\pi$
$102$	0	0
$103$	−46921.2	−0.435789	−0.217895	−	0.975972i	$-0.569919\pi$
$103$	−46921.2	−0.435789	−0.217895	+	0.975972i	$0.569919\pi$
$104$	0	0
$105$	−122800.	−1.08699
$106$	0	0
$107$	38381.3i	0.324086i	0.986784	+	0.162043i	$0.0518083\pi$
$107$	38381.3i	0.324086i	−0.986784	+	0.162043i	$0.948192\pi$
$108$	0	0
$109$	− 14288.7i	− 0.115193i	−0.998340	−	0.0575966i	$-0.981656\pi$
$109$	− 14288.7i	− 0.115193i	0.998340	−	0.0575966i	$-0.0183437\pi$
$110$	0	0
$111$	215796.	1.66240
$112$	0	0
$113$	130552.	0.961804	0.480902	−	0.876774i	$-0.340309\pi$
$113$	130552.	0.961804	0.480902	+	0.876774i	$0.340309\pi$
$114$	0	0
$115$	− 10587.5i	− 0.0746530i
$116$	0	0
$117$	299943.i	2.02570i
$118$	0	0
$119$	30791.5	0.199326
$120$	0	0
$121$	−103652.	−0.643596
$122$	0	0
$123$	− 123853.i	− 0.738151i
$124$	0	0
$125$	15625.0i	0.0894427i
$126$	0	0
$127$	327594.	1.80230	0.901150	−	0.433508i	$-0.142724\pi$
$127$	327594.	1.80230	0.901150	+	0.433508i	$0.142724\pi$
$128$	0	0
$129$	442775.	2.34266
$130$	0	0
$131$	− 116999.i	− 0.595669i	−0.954618	−	0.297834i	$-0.903736\pi$
$131$	− 116999.i	− 0.595669i	0.954618	−	0.297834i	$-0.0962643\pi$
$132$	0	0
$133$	− 210379.i	− 1.03127i
$134$	0	0
$135$	268064.	1.26591
$136$	0	0
$137$	−74409.1	−0.338707	−0.169354	−	0.985555i	$-0.554168\pi$
$137$	−74409.1	−0.338707	−0.169354	+	0.985555i	$0.554168\pi$
$138$	0	0
$139$	− 80434.5i	− 0.353106i	−0.984291	−	0.176553i	$-0.943505\pi$
$139$	− 80434.5i	− 0.353106i	0.984291	−	0.176553i	$-0.0564947\pi$
$140$	0	0
$141$	− 448572.i	− 1.90013i
$142$	0	0
$143$	−252936.	−1.03436
$144$	0	0
$145$	86585.0	0.341997
$146$	0	0
$147$	− 335171.i	− 1.27930i
$148$	0	0
$149$	− 57917.4i	− 0.213719i	−0.994274	−	0.106860i	$-0.965920\pi$
$149$	− 57917.4i	− 0.213719i	0.994274	−	0.106860i	$-0.0340795\pi$
$150$	0	0
$151$	−450813.	−1.60899	−0.804497	−	0.593957i	$-0.797565\pi$
$151$	−450813.	−1.60899	−0.804497	+	0.593957i	$0.797565\pi$
$152$	0	0
$153$	−111707.	−0.385792
$154$	0	0
$155$	58598.0i	0.195909i
$156$	0	0
$157$	72067.3i	0.233340i	0.993171	+	0.116670i	$0.0372220\pi$
$157$	72067.3i	0.233340i	−0.993171	+	0.116670i	$0.962778\pi$
$158$	0	0
$159$	−332792.	−1.04395
$160$	0	0
$161$	71221.2	0.216543
$162$	0	0
$163$	− 471144.i	− 1.38895i	−0.719519	−	0.694473i	$-0.755638\pi$
$163$	− 471144.i	− 1.38895i	0.719519	−	0.694473i	$-0.244362\pi$
$164$	0	0
$165$	375683.i	1.07427i
$166$	0	0
$167$	−519164.	−1.44050	−0.720250	−	0.693715i	$-0.755973\pi$
$167$	−519164.	−1.44050	−0.720250	+	0.693715i	$0.755973\pi$
$168$	0	0
$169$	129601.	0.349053
$170$	0	0
$171$	763225.i	1.99601i
$172$	0	0
$173$	− 726898.i	− 1.84654i	−0.384153	−	0.923269i	$-0.625506\pi$
$173$	− 726898.i	− 1.84654i	0.384153	−	0.923269i	$-0.374494\pi$
$174$	0	0
$175$	−105108.	−0.259443
$176$	0	0
$177$	−349853.	−0.839368
$178$	0	0
$179$	− 327278.i	− 0.763457i	−0.924274	−	0.381729i	$-0.875329\pi$
$179$	− 327278.i	− 0.763457i	0.924274	−	0.381729i	$-0.124671\pi$
$180$	0	0
$181$	651584.i	1.47834i	0.673519	+	0.739170i	$0.264782\pi$
$181$	651584.i	1.47834i	−0.673519	+	0.739170i	$0.735218\pi$
$182$	0	0
$183$	1.21079e6	2.67264
$184$	0	0
$185$	184706.	0.396782
$186$	0	0
$187$	− 94200.5i	− 0.196992i
$188$	0	0
$189$	1.80325e6i	3.67198i
$190$	0	0
$191$	427987.	0.848882	0.424441	−	0.905456i	$-0.360471\pi$
$191$	427987.	0.848882	0.424441	+	0.905456i	$0.360471\pi$
$192$	0	0
$193$	560696.	1.08351	0.541757	−	0.840535i	$-0.317759\pi$
$193$	560696.	1.08351	0.541757	+	0.840535i	$0.317759\pi$
$194$	0	0
$195$	358983.i	0.676063i
$196$	0	0
$197$	− 268824.i	− 0.493518i	−0.969077	−	0.246759i	$-0.920634\pi$
$197$	− 268824.i	− 0.493518i	0.969077	−	0.246759i	$-0.0793656\pi$
$198$	0	0
$199$	−512715.	−0.917789	−0.458895	−	0.888491i	$-0.651754\pi$
$199$	−512715.	−0.917789	−0.458895	+	0.888491i	$0.651754\pi$
$200$	0	0
$201$	−1.94306e6	−3.39232
$202$	0	0
$203$	582451.i	0.992018i
$204$	0	0
$205$	− 106010.i	− 0.176182i
$206$	0	0
$207$	−258380.	−0.419116
$208$	0	0
$209$	−643612.	−1.01920
$210$	0	0
$211$	− 661872.i	− 1.02345i	−0.859148	−	0.511726i	$-0.829006\pi$
$211$	− 661872.i	− 1.02345i	0.859148	−	0.511726i	$-0.170994\pi$
$212$	0	0
$213$	− 765674.i	− 1.15637i
$214$	0	0
$215$	378984.	0.559145
$216$	0	0
$217$	−394185.	−0.568265
$218$	0	0
$219$	2.52042e6i	3.55109i
$220$	0	0
$221$	− 90013.0i	− 0.123972i
$222$	0	0
$223$	−1.06479e6	−1.43384	−0.716922	−	0.697153i	$-0.754450\pi$
$223$	−1.06479e6	−1.43384	−0.716922	+	0.697153i	$0.754450\pi$
$224$	0	0
$225$	381318.	0.502148
$226$	0	0
$227$	619730.i	0.798248i	0.916897	+	0.399124i	$0.130686\pi$
$227$	619730.i	0.798248i	−0.916897	+	0.399124i	$0.869314\pi$
$228$	0	0
$229$	− 434907.i	− 0.548035i	−0.961725	−	0.274017i	$-0.911647\pi$
$229$	− 434907.i	− 0.548035i	0.961725	−	0.274017i	$-0.0883525\pi$
$230$	0	0
$231$	−2.52720e6	−3.11608
$232$	0	0
$233$	793810.	0.957915	0.478957	−	0.877838i	$-0.341015\pi$
$233$	793810.	0.957915	0.478957	+	0.877838i	$0.341015\pi$
$234$	0	0
$235$	− 383945.i	− 0.453523i
$236$	0	0
$237$	− 578302.i	− 0.668781i
$238$	0	0
$239$	1.64777e6	1.86596	0.932978	−	0.359933i	$-0.117200\pi$
$239$	1.64777e6	1.86596	0.932978	+	0.359933i	$0.117200\pi$
$240$	0	0
$241$	592599.	0.657231	0.328616	−	0.944464i	$-0.393418\pi$
$241$	592599.	0.657231	0.328616	+	0.944464i	$0.393418\pi$
$242$	0	0
$243$	− 2.21164e6i	− 2.40270i
$244$	0	0
$245$	− 286882.i	− 0.305343i
$246$	0	0
$247$	−615001.	−0.641406
$248$	0	0
$249$	244478.	0.249886
$250$	0	0
$251$	− 1.64581e6i	− 1.64890i	−0.565933	−	0.824451i	$-0.691484\pi$
$251$	− 1.64581e6i	− 1.64890i	0.565933	−	0.824451i	$-0.308516\pi$
$252$	0	0
$253$	− 217887.i	− 0.214008i
$254$	0	0
$255$	−133695.	−0.128756
$256$	0	0
$257$	−1.18756e6	−1.12156	−0.560779	−	0.827966i	$-0.689498\pi$
$257$	−1.18756e6	−1.12156	−0.560779	+	0.827966i	$0.689498\pi$
$258$	0	0
$259$	1.24251e6i	1.15093i
$260$	0	0
$261$	− 2.11305e6i	− 1.92003i
$262$	0	0
$263$	1.62916e6	1.45236	0.726180	−	0.687505i	$-0.241294\pi$
$263$	1.62916e6	1.45236	0.726180	+	0.687505i	$0.241294\pi$
$264$	0	0
$265$	−284846.	−0.249170
$266$	0	0
$267$	111705.i	0.0958944i
$268$	0	0
$269$	895226.i	0.754314i	0.926149	+	0.377157i	$0.123098\pi$
$269$	895226.i	0.754314i	−0.926149	+	0.377157i	$0.876902\pi$
$270$	0	0
$271$	−16721.4	−0.0138309	−0.00691545	−	0.999976i	$-0.502201\pi$
$271$	−16721.4	−0.0138309	−0.00691545	+	0.999976i	$0.502201\pi$
$272$	0	0
$273$	−2.41485e6	−1.96103
$274$	0	0
$275$	321558.i	0.256406i
$276$	0	0
$277$	573914.i	0.449415i	0.974426	+	0.224708i	$0.0721427\pi$
$277$	573914.i	0.449415i	−0.974426	+	0.224708i	$0.927857\pi$
$278$	0	0
$279$	1.43005e6	1.09987
$280$	0	0
$281$	1.95965e6	1.48052	0.740258	−	0.672322i	$-0.234703\pi$
$281$	1.95965e6	1.48052	0.740258	+	0.672322i	$0.234703\pi$
$282$	0	0
$283$	− 2.04454e6i	− 1.51751i	−0.651378	−	0.758753i	$-0.725809\pi$
$283$	− 2.04454e6i	− 1.51751i	0.651378	−	0.758753i	$-0.274191\pi$
$284$	0	0
$285$	913455.i	0.666154i
$286$	0	0
$287$	713120.	0.511044
$288$	0	0
$289$	−1.38633e6	−0.976390
$290$	0	0
$291$	− 1.02691e6i	− 0.710886i
$292$	0	0
$293$	2.25113e6i	1.53190i	0.642899	+	0.765951i	$0.277731\pi$
$293$	2.25113e6i	1.53190i	−0.642899	+	0.765951i	$0.722269\pi$
$294$	0	0
$295$	−299449.	−0.200340
$296$	0	0
$297$	5.51667e6	3.62899
$298$	0	0
$299$	− 208201.i	− 0.134681i
$300$	0	0
$301$	2.54940e6i	1.62189i
$302$	0	0
$303$	−2.90726e6	−1.81919
$304$	0	0
$305$	1.03635e6	0.637906
$306$	0	0
$307$	572436.i	0.346642i	0.984865	+	0.173321i	$0.0554498\pi$
$307$	572436.i	0.346642i	−0.984865	+	0.173321i	$0.944550\pi$
$308$	0	0
$309$	1.37048e6i	0.816537i
$310$	0	0
$311$	2.86177e6	1.67778	0.838889	−	0.544303i	$-0.183206\pi$
$311$	2.86177e6	1.67778	0.838889	+	0.544303i	$0.183206\pi$
$312$	0	0
$313$	−345643.	−0.199419	−0.0997095	−	0.995017i	$-0.531791\pi$
$313$	−345643.	−0.199419	−0.0997095	+	0.995017i	$0.531791\pi$
$314$	0	0
$315$	2.56510e6i	1.45656i
$316$	0	0
$317$	− 2.84370e6i	− 1.58941i	−0.606995	−	0.794706i	$-0.707625\pi$
$317$	− 2.84370e6i	− 1.58941i	0.606995	−	0.794706i	$-0.292375\pi$
$318$	0	0
$319$	1.78189e6	0.980404
$320$	0	0
$321$	1.12104e6	0.607238
$322$	0	0
$323$	− 229044.i	− 0.122155i
$324$	0	0
$325$	307264.i	0.161363i
$326$	0	0
$327$	−417345.	−0.215837
$328$	0	0
$329$	2.58278e6	1.31552
$330$	0	0
$331$	2.20630e6i	1.10686i	0.832895	+	0.553431i	$0.186682\pi$
$331$	2.20630e6i	1.10686i	−0.832895	+	0.553431i	$0.813318\pi$
$332$	0	0
$333$	− 4.50764e6i	− 2.22761i
$334$	0	0
$335$	−1.66312e6	−0.809678
$336$	0	0
$337$	−210820.	−0.101120	−0.0505599	−	0.998721i	$-0.516101\pi$
$337$	−210820.	−0.101120	−0.0505599	+	0.998721i	$0.516101\pi$
$338$	0	0
$339$	− 3.81316e6i	− 1.80213i
$340$	0	0
$341$	1.20593e6i	0.561611i
$342$	0	0
$343$	−896652.	−0.411518
$344$	0	0
$345$	−309239.	−0.139877
$346$	0	0
$347$	2.39916e6i	1.06963i	0.844968	+	0.534817i	$0.179619\pi$
$347$	2.39916e6i	1.06963i	−0.844968	+	0.534817i	$0.820381\pi$
$348$	0	0
$349$	1.76207e6i	0.774392i	0.921997	+	0.387196i	$0.126556\pi$
$349$	1.76207e6i	0.774392i	−0.921997	+	0.387196i	$0.873444\pi$
$350$	0	0
$351$	5.27144e6	2.28382
$352$	0	0
$353$	−2.11387e6	−0.902904	−0.451452	−	0.892296i	$-0.649094\pi$
$353$	−2.11387e6	−0.902904	−0.451452	+	0.892296i	$0.649094\pi$
$354$	0	0
$355$	− 655363.i	− 0.276001i
$356$	0	0
$357$	− 899360.i	− 0.373476i
$358$	0	0
$359$	25391.5	0.0103981	0.00519903	−	0.999986i	$-0.498345\pi$
$359$	25391.5	0.0103981	0.00519903	+	0.999986i	$0.498345\pi$
$360$	0	0
$361$	911190.	0.367994
$362$	0	0
$363$	3.02747e6i	1.20590i
$364$	0	0
$365$	2.15730e6i	0.847574i
$366$	0	0
$367$	−1.55872e6	−0.604091	−0.302046	−	0.953293i	$-0.597670\pi$
$367$	−1.55872e6	−0.604091	−0.302046	+	0.953293i	$0.597670\pi$
$368$	0	0
$369$	−2.58710e6	−0.989116
$370$	0	0
$371$	− 1.91614e6i	− 0.722759i
$372$	0	0
$373$	− 741743.i	− 0.276046i	−0.990429	−	0.138023i	$-0.955925\pi$
$373$	− 741743.i	− 0.276046i	0.990429	−	0.138023i	$-0.0440748\pi$
$374$	0	0
$375$	456376.	0.167589
$376$	0	0
$377$	1.70268e6	0.616993
$378$	0	0
$379$	2.61039e6i	0.933487i	0.884393	+	0.466743i	$0.154573\pi$
$379$	2.61039e6i	0.933487i	−0.884393	+	0.466743i	$0.845427\pi$
$380$	0	0
$381$	− 9.56839e6i	− 3.37696i
$382$	0	0
$383$	3.05949e6	1.06574	0.532871	−	0.846197i	$-0.321113\pi$
$383$	3.05949e6	1.06574	0.532871	+	0.846197i	$0.321113\pi$
$384$	0	0
$385$	−2.16310e6	−0.743746
$386$	0	0
$387$	− 9.24887e6i	− 3.13914i
$388$	0	0
$389$	− 498203.i	− 0.166929i	−0.996511	−	0.0834646i	$-0.973401\pi$
$389$	− 498203.i	− 0.166929i	0.996511	−	0.0834646i	$-0.0265985\pi$
$390$	0	0
$391$	77540.0	0.0256498
$392$	0	0
$393$	−3.41732e6	−1.11610
$394$	0	0
$395$	− 494985.i	− 0.159624i
$396$	0	0
$397$	3.95290e6i	1.25875i	0.777101	+	0.629376i	$0.216689\pi$
$397$	3.95290e6i	1.25875i	−0.777101	+	0.629376i	$0.783311\pi$
$398$	0	0
$399$	−6.14475e6	−1.93229
$400$	0	0
$401$	−1.15248e6	−0.357909	−0.178955	−	0.983857i	$-0.557272\pi$
$401$	−1.15248e6	−0.357909	−0.178955	+	0.983857i	$0.557272\pi$
$402$	0	0
$403$	1.15232e6i	0.353436i
$404$	0	0
$405$	− 4.12320e6i	− 1.24910i
$406$	0	0
$407$	3.80120e6	1.13746
$408$	0	0
$409$	2.54459e6	0.752159	0.376079	−	0.926587i	$-0.377272\pi$
$409$	2.54459e6	0.752159	0.376079	+	0.926587i	$0.377272\pi$
$410$	0	0
$411$	2.17334e6i	0.634635i
$412$	0	0
$413$	− 2.01438e6i	− 0.581119i
$414$	0	0
$415$	209256.	0.0596427
$416$	0	0
$417$	−2.34933e6	−0.661614
$418$	0	0
$419$	62820.4i	0.0174810i	0.999962	+	0.00874049i	$0.00278222\pi$
$419$	62820.4i	0.0174810i	−0.999962	+	0.00874049i	$0.997218\pi$
$420$	0	0
$421$	1.89940e6i	0.522289i	0.965300	+	0.261145i	$0.0841000\pi$
$421$	1.89940e6i	0.522289i	−0.965300	+	0.261145i	$0.915900\pi$
$422$	0	0
$423$	−9.36995e6	−2.54616
$424$	0	0
$425$	−114434.	−0.0307314
$426$	0	0
$427$	6.97145e6i	1.85035i
$428$	0	0
$429$	7.38776e6i	1.93807i
$430$	0	0
$431$	−5.91659e6	−1.53419	−0.767094	−	0.641535i	$-0.778298\pi$
$431$	−5.91659e6	−1.53419	−0.767094	+	0.641535i	$0.778298\pi$
$432$	0	0
$433$	−1.64308e6	−0.421153	−0.210577	−	0.977577i	$-0.567534\pi$
$433$	−1.64308e6	−0.421153	−0.210577	+	0.977577i	$0.567534\pi$
$434$	0	0
$435$	− 2.52898e6i	− 0.640799i
$436$	0	0
$437$	− 529781.i	− 0.132707i
$438$	0	0
$439$	4.34976e6	1.07722	0.538610	−	0.842555i	$-0.318950\pi$
$439$	4.34976e6	1.07722	0.538610	+	0.842555i	$0.318950\pi$
$440$	0	0
$441$	−7.00118e6	−1.71425
$442$	0	0
$443$	2.27280e6i	0.550239i	0.961410	+	0.275119i	$0.0887174\pi$
$443$	2.27280e6i	0.550239i	−0.961410	+	0.275119i	$0.911283\pi$
$444$	0	0
$445$	95611.3i	0.0228881i
$446$	0	0
$447$	−1.69165e6	−0.400445
$448$	0	0
$449$	7.36304e6	1.72362	0.861809	−	0.507233i	$-0.169331\pi$
$449$	7.36304e6	1.72362	0.861809	+	0.507233i	$0.169331\pi$
$450$	0	0
$451$	− 2.18165e6i	− 0.505060i
$452$	0	0
$453$	1.31674e7i	3.01477i
$454$	0	0
$455$	−2.06694e6	−0.468058
$456$	0	0
$457$	−6.23984e6	−1.39760	−0.698800	−	0.715317i	$-0.746282\pi$
$457$	−6.23984e6	−1.39760	−0.698800	+	0.715317i	$0.746282\pi$
$458$	0	0
$459$	1.96323e6i	0.434951i
$460$	0	0
$461$	832291.i	0.182399i	0.995833	+	0.0911996i	$0.0290701\pi$
$461$	832291.i	0.182399i	−0.995833	+	0.0911996i	$0.970930\pi$
$462$	0	0
$463$	3.61852e6	0.784474	0.392237	−	0.919864i	$-0.371701\pi$
$463$	3.61852e6	0.784474	0.392237	+	0.919864i	$0.371701\pi$
$464$	0	0
$465$	1.71153e6	0.367073
$466$	0	0
$467$	− 286010.i	− 0.0606861i	−0.999540	−	0.0303431i	$-0.990340\pi$
$467$	− 286010.i	− 0.0606861i	0.999540	−	0.0303431i	$-0.00965998\pi$
$468$	0	0
$469$	− 1.11877e7i	− 2.34860i
$470$	0	0
$471$	2.10494e6	0.437208
$472$	0	0
$473$	7.79938e6	1.60290
$474$	0	0
$475$	781852.i	0.158998i
$476$	0	0
$477$	6.95150e6i	1.39889i
$478$	0	0
$479$	2.28996e6	0.456026	0.228013	−	0.973658i	$-0.426777\pi$
$479$	2.28996e6	0.456026	0.228013	+	0.973658i	$0.426777\pi$
$480$	0	0
$481$	3.63222e6	0.715830
$482$	0	0
$483$	− 2.08023e6i	− 0.405736i
$484$	0	0
$485$	− 878962.i	− 0.169674i
$486$	0	0
$487$	−6.55915e6	−1.25321	−0.626607	−	0.779335i	$-0.715557\pi$
$487$	−6.55915e6	−1.25321	−0.626607	+	0.779335i	$0.715557\pi$
$488$	0	0
$489$	−1.37612e7	−2.60246
$490$	0	0
$491$	3.50573e6i	0.656257i	0.944633	+	0.328129i	$0.106418\pi$
$491$	3.50573e6i	0.656257i	−0.944633	+	0.328129i	$0.893582\pi$
$492$	0	0
$493$	634127.i	0.117506i
$494$	0	0
$495$	7.84742e6	1.43951
$496$	0	0
$497$	4.40858e6	0.800586
$498$	0	0
$499$	1.11789e6i	0.200977i	0.994938	+	0.100489i	$0.0320406\pi$
$499$	1.11789e6i	0.200977i	−0.994938	+	0.100489i	$0.967959\pi$
$500$	0	0
$501$	1.51638e7i	2.69906i
$502$	0	0
$503$	3.97264e6	0.700098	0.350049	−	0.936731i	$-0.386165\pi$
$503$	3.97264e6	0.700098	0.350049	+	0.936731i	$0.386165\pi$
$504$	0	0
$505$	−2.48841e6	−0.434203
$506$	0	0
$507$	− 3.78539e6i	− 0.654020i
$508$	0	0
$509$	− 5.19239e6i	− 0.888327i	−0.895946	−	0.444164i	$-0.853501\pi$
$509$	− 5.19239e6i	− 0.888327i	0.895946	−	0.444164i	$-0.146499\pi$
$510$	0	0
$511$	−1.45120e7	−2.45853
$512$	0	0
$513$	1.34135e7	2.25035
$514$	0	0
$515$	1.17303e6i	0.194891i
$516$	0	0
$517$	− 7.90148e6i	− 1.30012i
$518$	0	0
$519$	−2.12313e7	−3.45985
$520$	0	0
$521$	1.09827e6	0.177262	0.0886311	−	0.996065i	$-0.471751\pi$
$521$	1.09827e6	0.177262	0.0886311	+	0.996065i	$0.471751\pi$
$522$	0	0
$523$	− 8.67189e6i	− 1.38631i	−0.720790	−	0.693154i	$-0.756221\pi$
$523$	− 8.67189e6i	− 1.38631i	0.720790	−	0.693154i	$-0.243779\pi$
$524$	0	0
$525$	3.07001e6i	0.486118i
$526$	0	0
$527$	−429158.	−0.0673116
$528$	0	0
$529$	−6.25699e6	−0.972135
$530$	0	0
$531$	7.30787e6i	1.12475i
$532$	0	0
$533$	− 2.08467e6i	− 0.317847i
$534$	0	0
$535$	959532.	0.144936
$536$	0	0
$537$	−9.55916e6	−1.43049
$538$	0	0
$539$	− 5.90395e6i	− 0.875328i
$540$	0	0
$541$	− 746497.i	− 0.109657i	−0.998496	−	0.0548283i	$-0.982539\pi$
$541$	− 746497.i	− 0.109657i	0.998496	−	0.0548283i	$-0.0174612\pi$
$542$	0	0
$543$	1.90315e7	2.76996
$544$	0	0
$545$	−357218.	−0.0515160
$546$	0	0
$547$	2.52087e6i	0.360231i	0.983645	+	0.180116i	$0.0576472\pi$
$547$	2.52087e6i	0.360231i	−0.983645	+	0.180116i	$0.942353\pi$
$548$	0	0
$549$	− 2.52915e7i	− 3.58132i
$550$	0	0
$551$	4.33258e6	0.607950
$552$	0	0
$553$	3.32973e6	0.463017
$554$	0	0
$555$	− 5.39490e6i	− 0.743449i
$556$	0	0
$557$	186236.i	0.0254347i	0.999919	+	0.0127173i	$0.00404816\pi$
$557$	186236.i	0.0254347i	−0.999919	+	0.0127173i	$0.995952\pi$
$558$	0	0
$559$	7.45267e6	1.00875
$560$	0	0
$561$	−2.75141e6	−0.369104
$562$	0	0
$563$	− 358662.i	− 0.0476886i	−0.999716	−	0.0238443i	$-0.992409\pi$
$563$	− 358662.i	− 0.0476886i	0.999716	−	0.0238443i	$-0.00759059\pi$
$564$	0	0
$565$	− 3.26379e6i	− 0.430132i
$566$	0	0
$567$	2.77365e7	3.62321
$568$	0	0
$569$	1.08964e7	1.41093	0.705463	−	0.708747i	$-0.250739\pi$
$569$	1.08964e7	1.41093	0.705463	+	0.708747i	$0.250739\pi$
$570$	0	0
$571$	− 1.93042e6i	− 0.247778i	−0.992296	−	0.123889i	$-0.960463\pi$
$571$	− 1.93042e6i	− 0.247778i	0.992296	−	0.123889i	$-0.0395366\pi$
$572$	0	0
$573$	− 1.25007e7i	− 1.59055i
$574$	0	0
$575$	−264687.	−0.0333858
$576$	0	0
$577$	−1.33310e7	−1.66695	−0.833477	−	0.552555i	$-0.813653\pi$
$577$	−1.33310e7	−1.66695	−0.833477	+	0.552555i	$0.813653\pi$
$578$	0	0
$579$	− 1.63768e7i	− 2.03018i
$580$	0	0
$581$	1.40765e6i	0.173003i
$582$	0	0
$583$	−5.86206e6	−0.714297
$584$	0	0
$585$	7.49858e6	0.905919
$586$	0	0
$587$	− 7.15350e6i	− 0.856887i	−0.903569	−	0.428443i	$-0.859062\pi$
$587$	− 7.15350e6i	− 0.856887i	0.903569	−	0.428443i	$-0.140938\pi$
$588$	0	0
$589$	2.93216e6i	0.348256i
$590$	0	0
$591$	−7.85183e6	−0.924703
$592$	0	0
$593$	1.46858e7	1.71499	0.857496	−	0.514490i	$-0.172019\pi$
$593$	1.46858e7	1.71499	0.857496	+	0.514490i	$0.172019\pi$
$594$	0	0
$595$	− 769788.i	− 0.0891413i
$596$	0	0
$597$	1.49754e7i	1.71966i
$598$	0	0
$599$	9.83597e6	1.12008	0.560042	−	0.828465i	$-0.310785\pi$
$599$	9.83597e6	1.12008	0.560042	+	0.828465i	$0.310785\pi$
$600$	0	0
$601$	2.68643e6	0.303381	0.151691	−	0.988428i	$-0.451528\pi$
$601$	2.68643e6	0.303381	0.151691	+	0.988428i	$0.451528\pi$
$602$	0	0
$603$	4.05875e7i	4.54568i
$604$	0	0
$605$	2.59130e6i	0.287825i
$606$	0	0
$607$	−387079.	−0.0426411	−0.0213205	−	0.999773i	$-0.506787\pi$
$607$	−387079.	−0.0426411	−0.0213205	+	0.999773i	$0.506787\pi$
$608$	0	0
$609$	1.70123e7	1.85874
$610$	0	0
$611$	− 7.55024e6i	− 0.818196i
$612$	0	0
$613$	1.37500e7i	1.47792i	0.673748	+	0.738962i	$0.264684\pi$
$613$	1.37500e7i	1.47792i	−0.673748	+	0.738962i	$0.735316\pi$
$614$	0	0
$615$	−3.09633e6	−0.330111
$616$	0	0
$617$	−1.13239e7	−1.19752	−0.598758	−	0.800930i	$-0.704339\pi$
$617$	−1.13239e7	−1.19752	−0.598758	+	0.800930i	$0.704339\pi$
$618$	0	0
$619$	6.62505e6i	0.694965i	0.937687	+	0.347482i	$0.112963\pi$
$619$	6.62505e6i	0.694965i	−0.937687	+	0.347482i	$0.887037\pi$
$620$	0	0
$621$	4.54098e6i	0.472521i
$622$	0	0
$623$	−643171.	−0.0663905
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	1.87986e7i	1.90967i
$628$	0	0
$629$	1.35274e6i	0.136329i
$630$	0	0
$631$	1.12049e7	1.12030	0.560152	−	0.828390i	$-0.310743\pi$
$631$	1.12049e7	1.12030	0.560152	+	0.828390i	$0.310743\pi$
$632$	0	0
$633$	−1.93320e7	−1.91764
$634$	0	0
$635$	− 8.18986e6i	− 0.806013i
$636$	0	0
$637$	− 5.64150e6i	− 0.550866i
$638$	0	0
$639$	−1.59937e7	−1.54952
$640$	0	0
$641$	−5.35866e6	−0.515123	−0.257562	−	0.966262i	$-0.582919\pi$
$641$	−5.35866e6	−0.515123	−0.257562	+	0.966262i	$0.582919\pi$
$642$	0	0
$643$	5.64079e6i	0.538038i	0.963135	+	0.269019i	$0.0866994\pi$
$643$	5.64079e6i	0.538038i	−0.963135	+	0.269019i	$0.913301\pi$
$644$	0	0
$645$	− 1.10694e7i	− 1.04767i
$646$	0	0
$647$	−5.67405e6	−0.532884	−0.266442	−	0.963851i	$-0.585848\pi$
$647$	−5.67405e6	−0.532884	−0.266442	+	0.963851i	$0.585848\pi$
$648$	0	0
$649$	−6.16258e6	−0.574316
$650$	0	0
$651$	1.15134e7i	1.06476i
$652$	0	0
$653$	− 1.50064e7i	− 1.37719i	−0.725148	−	0.688593i	$-0.758229\pi$
$653$	− 1.50064e7i	− 1.37719i	0.725148	−	0.688593i	$-0.241771\pi$
$654$	0	0
$655$	−2.92498e6	−0.266391
$656$	0	0
$657$	5.26475e7	4.75844
$658$	0	0
$659$	− 7.48265e6i	− 0.671184i	−0.942007	−	0.335592i	$-0.891064\pi$
$659$	− 7.48265e6i	− 0.671184i	0.942007	−	0.335592i	$-0.108936\pi$
$660$	0	0
$661$	− 1.85142e7i	− 1.64817i	−0.566467	−	0.824084i	$-0.691690\pi$
$661$	− 1.85142e7i	− 1.64817i	0.566467	−	0.824084i	$-0.308310\pi$
$662$	0	0
$663$	−2.62910e6	−0.232286
$664$	0	0
$665$	−5.25947e6	−0.461198
$666$	0	0
$667$	1.46674e6i	0.127656i
$668$	0	0
$669$	3.11004e7i	2.68659i
$670$	0	0
$671$	2.13278e7	1.82869
$672$	0	0
$673$	−4.62662e6	−0.393755	−0.196878	−	0.980428i	$-0.563080\pi$
$673$	−4.62662e6	−0.393755	−0.196878	+	0.980428i	$0.563080\pi$
$674$	0	0
$675$	− 6.70159e6i	− 0.566133i
$676$	0	0
$677$	1.21437e7i	1.01831i	0.860675	+	0.509155i	$0.170042\pi$
$677$	1.21437e7i	1.01831i	−0.860675	+	0.509155i	$0.829958\pi$
$678$	0	0
$679$	5.91272e6	0.492168
$680$	0	0
$681$	1.81011e7	1.49567
$682$	0	0
$683$	− 9.19906e6i	− 0.754557i	−0.926100	−	0.377278i	$-0.876860\pi$
$683$	− 9.19906e6i	− 0.754557i	0.926100	−	0.377278i	$-0.123140\pi$
$684$	0	0
$685$	1.86023e6i	0.151474i
$686$	0	0
$687$	−1.27028e7	−1.02685
$688$	0	0
$689$	−5.60147e6	−0.449525
$690$	0	0
$691$	− 1.02344e7i	− 0.815394i	−0.913117	−	0.407697i	$-0.866332\pi$
$691$	− 1.02344e7i	− 0.815394i	0.913117	−	0.407697i	$-0.133668\pi$
$692$	0	0
$693$	5.27891e7i	4.17553i
$694$	0	0
$695$	−2.01086e6	−0.157914
$696$	0	0
$697$	776389.	0.0605338
$698$	0	0
$699$	− 2.31856e7i	− 1.79484i
$700$	0	0
$701$	− 1.23863e6i	− 0.0952023i	−0.998866	−	0.0476011i	$-0.984842\pi$
$701$	− 1.23863e6i	− 0.0952023i	0.998866	−	0.0476011i	$-0.0151576\pi$
$702$	0	0
$703$	9.24242e6	0.705339
$704$	0	0
$705$	−1.12143e7	−0.849765
$706$	0	0
$707$	− 1.67394e7i	− 1.25948i
$708$	0	0
$709$	− 2.52759e7i	− 1.88839i	−0.329386	−	0.944195i	$-0.606842\pi$
$709$	− 2.52759e7i	− 1.88839i	0.329386	−	0.944195i	$-0.393158\pi$
$710$	0	0
$711$	−1.20798e7	−0.896161
$712$	0	0
$713$	−992646.	−0.0731258
$714$	0	0
$715$	6.32340e6i	0.462578i
$716$	0	0
$717$	− 4.81281e7i	− 3.49624i
$718$	0	0
$719$	5.23203e6	0.377440	0.188720	−	0.982031i	$-0.439566\pi$
$719$	5.23203e6	0.377440	0.188720	+	0.982031i	$0.439566\pi$
$720$	0	0
$721$	−7.89090e6	−0.565313
$722$	0	0
$723$	− 1.73087e7i	− 1.23145i
$724$	0	0
$725$	− 2.16462e6i	− 0.152946i
$726$	0	0
$727$	−1.79530e7	−1.25980	−0.629899	−	0.776677i	$-0.716904\pi$
$727$	−1.79530e7	−1.25980	−0.629899	+	0.776677i	$0.716904\pi$
$728$	0	0
$729$	−2.45203e7	−1.70886
$730$	0	0
$731$	2.77559e6i	0.192115i
$732$	0	0
$733$	2.06019e6i	0.141627i	0.997490	+	0.0708137i	$0.0225596\pi$
$733$	2.06019e6i	0.141627i	−0.997490	+	0.0708137i	$0.977440\pi$
$734$	0	0
$735$	−8.37927e6	−0.572121
$736$	0	0
$737$	−3.42266e7	−2.32111
$738$	0	0
$739$	1.47746e6i	0.0995187i	0.998761	+	0.0497594i	$0.0158454\pi$
$739$	1.47746e6i	0.0995187i	−0.998761	+	0.0497594i	$0.984155\pi$
$740$	0	0
$741$	1.79630e7i	1.20180i
$742$	0	0
$743$	−2.46543e7	−1.63840	−0.819200	−	0.573508i	$-0.805582\pi$
$743$	−2.46543e7	−1.63840	−0.819200	+	0.573508i	$0.805582\pi$
$744$	0	0
$745$	−1.44794e6	−0.0955782
$746$	0	0
$747$	− 5.10676e6i	− 0.334845i
$748$	0	0
$749$	6.45471e6i	0.420409i
$750$	0	0
$751$	6.30342e6	0.407827	0.203914	−	0.978989i	$-0.434634\pi$
$751$	6.30342e6	0.407827	0.203914	+	0.978989i	$0.434634\pi$
$752$	0	0
$753$	−4.80708e7	−3.08954
$754$	0	0
$755$	1.12703e7i	0.719564i
$756$	0	0
$757$	1.44827e7i	0.918567i	0.888290	+	0.459284i	$0.151894\pi$
$757$	1.44827e7i	0.918567i	−0.888290	+	0.459284i	$0.848106\pi$
$758$	0	0
$759$	−6.36405e6	−0.400986
$760$	0	0
$761$	9.50962e6	0.595253	0.297626	−	0.954682i	$-0.403805\pi$
$761$	9.50962e6	0.595253	0.297626	+	0.954682i	$0.403805\pi$
$762$	0	0
$763$	− 2.40298e6i	− 0.149430i
$764$	0	0
$765$	2.79268e6i	0.172531i
$766$	0	0
$767$	−5.88863e6	−0.361431
$768$	0	0
$769$	2.88208e6	0.175748	0.0878741	−	0.996132i	$-0.471993\pi$
$769$	2.88208e6	0.175748	0.0878741	+	0.996132i	$0.471993\pi$
$770$	0	0
$771$	3.46862e7i	2.10146i
$772$	0	0
$773$	− 1.83423e7i	− 1.10409i	−0.833815	−	0.552044i	$-0.813848\pi$
$773$	− 1.83423e7i	− 1.10409i	0.833815	−	0.552044i	$-0.186152\pi$
$774$	0	0
$775$	1.46495e6	0.0876130
$776$	0	0
$777$	3.62912e7	2.15650
$778$	0	0
$779$	− 5.30457e6i	− 0.313189i
$780$	0	0
$781$	− 1.34872e7i	− 0.791213i
$782$	0	0
$783$	−3.71365e7	−2.16469
$784$	0	0
$785$	1.80168e6	0.104353
$786$	0	0
$787$	− 1.35279e7i	− 0.778564i	−0.921119	−	0.389282i	$-0.872723\pi$
$787$	− 1.35279e7i	− 0.778564i	0.921119	−	0.389282i	$-0.127277\pi$
$788$	0	0
$789$	− 4.75846e7i	− 2.72128i
$790$	0	0
$791$	2.19553e7	1.24767
$792$	0	0
$793$	2.03797e7	1.15084
$794$	0	0
$795$	8.31981e6i	0.466869i
$796$	0	0
$797$	3.10994e7i	1.73423i	0.498109	+	0.867115i	$0.334028\pi$
$797$	3.10994e7i	1.73423i	−0.498109	+	0.867115i	$0.665972\pi$
$798$	0	0
$799$	2.81192e6	0.155825
$800$	0	0
$801$	2.33333e6	0.128498
$802$	0	0
$803$	4.43965e7i	2.42974i
$804$	0	0
$805$	− 1.78053e6i	− 0.0968410i
$806$	0	0
$807$	2.61478e7	1.41336
$808$	0	0
$809$	1.19594e6	0.0642449	0.0321225	−	0.999484i	$-0.489773\pi$
$809$	1.19594e6	0.0642449	0.0321225	+	0.999484i	$0.489773\pi$
$810$	0	0
$811$	− 5.59211e6i	− 0.298554i	−0.988795	−	0.149277i	$-0.952305\pi$
$811$	− 5.59211e6i	− 0.298554i	0.988795	−	0.149277i	$-0.0476947\pi$
$812$	0	0
$813$	488400.i	0.0259149i
$814$	0	0
$815$	−1.17786e7	−0.621155
$816$	0	0
$817$	1.89638e7	0.993963
$818$	0	0
$819$	5.04424e7i	2.62776i
$820$	0	0
$821$	2.42817e7i	1.25725i	0.777708	+	0.628625i	$0.216382\pi$
$821$	2.42817e7i	1.25725i	−0.777708	+	0.628625i	$0.783618\pi$
$822$	0	0
$823$	1.21094e7	0.623193	0.311597	−	0.950214i	$-0.399136\pi$
$823$	1.21094e7	0.623193	0.311597	+	0.950214i	$0.399136\pi$
$824$	0	0
$825$	9.39208e6	0.480426
$826$	0	0
$827$	2.18004e7i	1.10841i	0.832380	+	0.554205i	$0.186977\pi$
$827$	2.18004e7i	1.10841i	−0.832380	+	0.554205i	$0.813023\pi$
$828$	0	0
$829$	2.81218e7i	1.42121i	0.703593	+	0.710603i	$0.251578\pi$
$829$	2.81218e7i	1.42121i	−0.703593	+	0.710603i	$0.748422\pi$
$830$	0	0
$831$	1.67629e7	0.842067
$832$	0	0
$833$	2.10106e6	0.104912
$834$	0	0
$835$	1.29791e7i	0.644211i
$836$	0	0
$837$	− 2.51328e7i	− 1.24002i
$838$	0	0
$839$	−1.77035e6	−0.0868270	−0.0434135	−	0.999057i	$-0.513823\pi$
$839$	−1.77035e6	−0.0868270	−0.0434135	+	0.999057i	$0.513823\pi$
$840$	0	0
$841$	8.51602e6	0.415190
$842$	0	0
$843$	− 5.72376e7i	− 2.77404i
$844$	0	0
$845$	− 3.24003e6i	− 0.156101i
$846$	0	0
$847$	−1.74315e7	−0.834883
$848$	0	0
$849$	−5.97171e7	−2.84335
$850$	0	0
$851$	3.12891e6i	0.148105i
$852$	0	0
$853$	2.98606e7i	1.40516i	0.711604	+	0.702581i	$0.247969\pi$
$853$	2.98606e7i	1.40516i	−0.711604	+	0.702581i	$0.752031\pi$
$854$	0	0
$855$	1.90806e7	0.892641
$856$	0	0
$857$	−1.59538e7	−0.742012	−0.371006	−	0.928630i	$-0.620987\pi$
$857$	−1.59538e7	−0.742012	−0.371006	+	0.928630i	$0.620987\pi$
$858$	0	0
$859$	− 2.68382e7i	− 1.24099i	−0.784209	−	0.620497i	$-0.786931\pi$
$859$	− 2.68382e7i	− 1.24099i	0.784209	−	0.620497i	$-0.213069\pi$
$860$	0	0
$861$	− 2.08288e7i	− 0.957541i
$862$	0	0
$863$	−1.99512e7	−0.911891	−0.455946	−	0.890008i	$-0.650699\pi$
$863$	−1.99512e7	−0.911891	−0.455946	+	0.890008i	$0.650699\pi$
$864$	0	0
$865$	−1.81725e7	−0.825797
$866$	0	0
$867$	4.04921e7i	1.82946i
$868$	0	0
$869$	− 1.01867e7i	− 0.457596i
$870$	0	0
$871$	−3.27051e7	−1.46073
$872$	0	0
$873$	−2.14505e7	−0.952582
$874$	0	0
$875$	2.62771e6i	0.116026i
$876$	0	0
$877$	2.73500e7i	1.20077i	0.799712	+	0.600384i	$0.204986\pi$
$877$	2.73500e7i	1.20077i	−0.799712	+	0.600384i	$0.795014\pi$
$878$	0	0
$879$	6.57510e7	2.87032
$880$	0	0
$881$	−3.29524e7	−1.43037	−0.715184	−	0.698937i	$-0.753657\pi$
$881$	−3.29524e7	−1.43037	−0.715184	+	0.698937i	$0.753657\pi$
$882$	0	0
$883$	1.45135e7i	0.626429i	0.949682	+	0.313214i	$0.101406\pi$
$883$	1.45135e7i	0.626429i	−0.949682	+	0.313214i	$0.898594\pi$
$884$	0	0
$885$	8.74633e6i	0.375377i
$886$	0	0
$887$	−2.89058e7	−1.23360	−0.616802	−	0.787118i	$-0.711572\pi$
$887$	−2.89058e7	−1.23360	−0.616802	+	0.787118i	$0.711572\pi$
$888$	0	0
$889$	5.50926e7	2.33797
$890$	0	0
$891$	− 8.48543e7i	− 3.58079i
$892$	0	0
$893$	− 1.92121e7i	− 0.806205i
$894$	0	0
$895$	−8.18196e6	−0.341428
$896$	0	0
$897$	−6.08114e6	−0.252351
$898$	0	0
$899$	− 8.11793e6i	− 0.335001i
$900$	0	0
$901$	− 2.08615e6i	− 0.0856117i
$902$	0	0
$903$	7.44630e7	3.03893
$904$	0	0
$905$	1.62896e7	0.661134
$906$	0	0
$907$	1.75219e7i	0.707233i	0.935390	+	0.353617i	$0.115048\pi$
$907$	1.75219e7i	0.707233i	−0.935390	+	0.353617i	$0.884952\pi$
$908$	0	0
$909$	6.07280e7i	2.43770i
$910$	0	0
$911$	5.49598e6	0.219406	0.109703	−	0.993964i	$-0.465010\pi$
$911$	5.49598e6	0.219406	0.109703	+	0.993964i	$0.465010\pi$
$912$	0	0
$913$	4.30643e6	0.170978
$914$	0	0
$915$	− 3.02697e7i	− 1.19524i
$916$	0	0
$917$	− 1.96762e7i	− 0.772711i
$918$	0	0
$919$	−1.89582e6	−0.0740472	−0.0370236	−	0.999314i	$-0.511788\pi$
$919$	−1.89582e6	−0.0740472	−0.0370236	+	0.999314i	$0.511788\pi$
$920$	0	0
$921$	1.67197e7	0.649502
$922$	0	0
$923$	− 1.28876e7i	− 0.497930i
$924$	0	0
$925$	− 4.61765e6i	− 0.177446i
$926$	0	0
$927$	2.86271e7	1.09415
$928$	0	0
$929$	3.44700e7	1.31040	0.655198	−	0.755457i	$-0.272585\pi$
$929$	3.44700e7	1.31040	0.655198	+	0.755457i	$0.272585\pi$
$930$	0	0
$931$	− 1.43552e7i	− 0.542793i
$932$	0	0
$933$	− 8.35868e7i	− 3.14365i
$934$	0	0
$935$	−2.35501e6	−0.0880977
$936$	0	0
$937$	3.96544e7	1.47551	0.737756	−	0.675068i	$-0.235886\pi$
$937$	3.96544e7	1.47551	0.737756	+	0.675068i	$0.235886\pi$
$938$	0	0
$939$	1.00955e7i	0.373651i
$940$	0	0
$941$	− 1.88367e7i	− 0.693474i	−0.937962	−	0.346737i	$-0.887290\pi$
$941$	− 1.88367e7i	− 0.693474i	0.937962	−	0.346737i	$-0.112710\pi$
$942$	0	0
$943$	1.79580e6	0.0657625
$944$	0	0
$945$	4.50812e7	1.64216
$946$	0	0
$947$	− 4.04648e6i	− 0.146623i	−0.997309	−	0.0733115i	$-0.976643\pi$
$947$	− 4.04648e6i	− 0.146623i	0.997309	−	0.0733115i	$-0.0233567\pi$
$948$	0	0
$949$	4.24230e7i	1.52910i
$950$	0	0
$951$	−8.30590e7	−2.97807
$952$	0	0
$953$	4.74092e7	1.69095	0.845474	−	0.534016i	$-0.179318\pi$
$953$	4.74092e7	1.69095	0.845474	+	0.534016i	$0.179318\pi$
$954$	0	0
$955$	− 1.06997e7i	− 0.379632i
$956$	0	0
$957$	− 5.20456e7i	− 1.83698i
$958$	0	0
$959$	−1.25136e7	−0.439376
$960$	0	0
$961$	−2.31352e7	−0.808099
$962$	0	0
$963$	− 2.34168e7i	− 0.813695i
$964$	0	0
$965$	− 1.40174e7i	− 0.484562i
$966$	0	0
$967$	1.05815e7	0.363898	0.181949	−	0.983308i	$-0.441759\pi$
$967$	1.05815e7	0.363898	0.181949	+	0.983308i	$0.441759\pi$
$968$	0	0
$969$	−6.68992e6	−0.228882
$970$	0	0
$971$	3.45206e7i	1.17498i	0.809232	+	0.587489i	$0.199884\pi$
$971$	3.45206e7i	1.17498i	−0.809232	+	0.587489i	$0.800116\pi$
$972$	0	0
$973$	− 1.35269e7i	− 0.458055i
$974$	0	0
$975$	8.97457e6	0.302344
$976$	0	0
$977$	−1.80039e7	−0.603433	−0.301717	−	0.953398i	$-0.597560\pi$
$977$	−1.80039e7	−0.603433	−0.301717	+	0.953398i	$0.597560\pi$
$978$	0	0
$979$	1.96765e6i	0.0656133i
$980$	0	0
$981$	8.71768e6i	0.289220i
$982$	0	0
$983$	2.71815e7	0.897200	0.448600	−	0.893733i	$-0.351923\pi$
$983$	2.71815e7	0.897200	0.448600	+	0.893733i	$0.351923\pi$
$984$	0	0
$985$	−6.72061e6	−0.220708
$986$	0	0
$987$	− 7.54378e7i	− 2.46488i
$988$	0	0
$989$	6.41996e6i	0.208709i
$990$	0	0
$991$	2.26954e7	0.734097	0.367048	−	0.930202i	$-0.380368\pi$
$991$	2.26954e7	0.734097	0.367048	+	0.930202i	$0.380368\pi$
$992$	0	0
$993$	6.44416e7	2.07393
$994$	0	0
$995$	1.28179e7i	0.410448i
$996$	0	0
$997$	2.28102e7i	0.726762i	0.931641	+	0.363381i	$0.118378\pi$
$997$	2.28102e7i	0.726762i	−0.931641	+	0.363381i	$0.881622\pi$
$998$	0	0
$999$	−7.92208e7	−2.51146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.d.a.81.1		20
4.3	odd	2		40.6.d.a.21.13	✓	20
5.2	odd	4		800.6.f.b.49.2		20
5.3	odd	4		800.6.f.c.49.19		20
5.4	even	2		800.6.d.c.401.20		20
8.3	odd	2		40.6.d.a.21.14	yes	20
8.5	even	2	inner	160.6.d.a.81.20		20
12.11	even	2		360.6.k.b.181.8		20
20.3	even	4		200.6.f.b.149.3		20
20.7	even	4		200.6.f.c.149.18		20
20.19	odd	2		200.6.d.b.101.8		20
24.11	even	2		360.6.k.b.181.7		20
40.3	even	4		200.6.f.c.149.17		20
40.13	odd	4		800.6.f.b.49.1		20
40.19	odd	2		200.6.d.b.101.7		20
40.27	even	4		200.6.f.b.149.4		20
40.29	even	2		800.6.d.c.401.1		20
40.37	odd	4		800.6.f.c.49.20		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.13	✓	20	4.3	odd	2
40.6.d.a.21.14	yes	20	8.3	odd	2
160.6.d.a.81.1		20	1.1	even	1	trivial
160.6.d.a.81.20		20	8.5	even	2	inner
200.6.d.b.101.7		20	40.19	odd	2
200.6.d.b.101.8		20	20.19	odd	2
200.6.f.b.149.3		20	20.3	even	4
200.6.f.b.149.4		20	40.27	even	4
200.6.f.c.149.17		20	40.3	even	4
200.6.f.c.149.18		20	20.7	even	4
360.6.k.b.181.7		20	24.11	even	2
360.6.k.b.181.8		20	12.11	even	2
800.6.d.c.401.1		20	40.29	even	2
800.6.d.c.401.20		20	5.4	even	2
800.6.f.b.49.1		20	40.13	odd	4
800.6.f.b.49.2		20	5.2	odd	4
800.6.f.c.49.19		20	5.3	odd	4
800.6.f.c.49.20		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)

\(f(q)\)	\(=\)	\(q-29.2080i q^{3} -25.0000i q^{5} +168.173 q^{7} -610.110 q^{9} -514.493i q^{11} -491.622i q^{13} -730.201 q^{15} +183.094 q^{17} -1250.96i q^{19} -4912.01i q^{21} +423.498 q^{23} -625.000 q^{25} +10722.5i q^{27} +3463.40i q^{29} -2343.92 q^{31} -15027.3 q^{33} -4204.33i q^{35} +7388.25i q^{37} -14359.3 q^{39} +4240.39 q^{41} +15159.4i q^{43} +15252.7i q^{45} +15357.8 q^{47} +11475.3 q^{49} -5347.82i q^{51} -11393.9i q^{53} -12862.3 q^{55} -36538.2 q^{57} -11978.0i q^{59} +41454.0i q^{61} -102604. q^{63} -12290.5 q^{65} -66524.9i q^{67} -12369.6i q^{69} +26214.5 q^{71} -86291.9 q^{73} +18255.0i q^{75} -86524.0i q^{77} +19799.4 q^{79} +164928. q^{81} +8370.24i q^{83} -4577.35i q^{85} +101159. q^{87} -3824.45 q^{89} -82677.7i q^{91} +68461.3i q^{93} -31274.1 q^{95} +35158.5 q^{97} +313897. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63}+ \cdots + 147376 q^{97}+O(q^{100}) \) 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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