LMFDB - Embedded newform 160.6.d.a.81.15

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			81.15
Root			$-3.80026 + 1.24819i$ of defining polynomial
Character	$\chi$	$=$	160.81
Dual form			160.6.d.a.81.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.5927i q^{3} +25.0000i q^{5} +231.529 q^{7} +108.609 q^{9} +O(q^{10})$	$q+11.5927i q^{3} +25.0000i q^{5} +231.529 q^{7} +108.609 q^{9} -559.335i q^{11} -107.903i q^{13} -289.818 q^{15} -441.735 q^{17} -1873.33i q^{19} +2684.04i q^{21} +3835.50 q^{23} -625.000 q^{25} +4076.10i q^{27} -3369.76i q^{29} +7955.88 q^{31} +6484.21 q^{33} +5788.21i q^{35} +10687.6i q^{37} +1250.89 q^{39} -9963.87 q^{41} +925.409i q^{43} +2715.23i q^{45} -8063.78 q^{47} +36798.5 q^{49} -5120.91i q^{51} +7952.45i q^{53} +13983.4 q^{55} +21717.0 q^{57} +16801.2i q^{59} -11297.4i q^{61} +25146.1 q^{63} +2697.57 q^{65} +33619.4i q^{67} +44463.9i q^{69} +8869.47 q^{71} +55505.3 q^{73} -7245.44i q^{75} -129502. i q^{77} -69413.7 q^{79} -20861.0 q^{81} -10231.5i q^{83} -11043.4i q^{85} +39064.7 q^{87} -92458.4 q^{89} -24982.6i q^{91} +92230.2i q^{93} +46833.2 q^{95} -88657.9 q^{97} -60748.9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	11.5927i	0.743673i	0.928298	+	0.371836i	$0.121272\pi$
$3$	11.5927i	0.743673i	−0.928298	+	0.371836i	$0.878728\pi$
$4$	0	0
$5$	25.0000i	0.447214i
$5$	25.0000i	0.447214i
$6$	0	0
$7$	231.529	1.78591	0.892955	−	0.450146i	$-0.148628\pi$
$7$	231.529	1.78591	0.892955	+	0.450146i	$0.148628\pi$
$8$	0	0
$9$	108.609	0.446951
$10$	0	0
$11$	− 559.335i	− 1.39377i	−0.717184	−	0.696884i	$-0.754569\pi$
$11$	− 559.335i	− 1.39377i	0.717184	−	0.696884i	$-0.245431\pi$
$12$	0	0
$13$	− 107.903i	− 0.177082i	−0.996073	−	0.0885411i	$-0.971780\pi$
$13$	− 107.903i	− 0.177082i	0.996073	−	0.0885411i	$-0.0282205\pi$
$14$	0	0
$15$	−289.818	−0.332580
$16$	0	0
$17$	−441.735	−0.370715	−0.185357	−	0.982671i	$-0.559344\pi$
$17$	−441.735	−0.370715	−0.185357	+	0.982671i	$0.559344\pi$
$18$	0	0
$19$	− 1873.33i	− 1.19050i	−0.803540	−	0.595251i	$-0.797053\pi$
$19$	− 1873.33i	− 1.19050i	0.803540	−	0.595251i	$-0.202947\pi$
$20$	0	0
$21$	2684.04i	1.32813i
$22$	0	0
$23$	3835.50	1.51183	0.755915	−	0.654670i	$-0.227193\pi$
$23$	3835.50	1.51183	0.755915	+	0.654670i	$0.227193\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	4076.10i	1.07606i
$28$	0	0
$29$	− 3369.76i	− 0.744054i	−0.928222	−	0.372027i	$-0.878663\pi$
$29$	− 3369.76i	− 0.744054i	0.928222	−	0.372027i	$-0.121337\pi$
$30$	0	0
$31$	7955.88	1.48691	0.743454	−	0.668787i	$-0.233186\pi$
$31$	7955.88	1.48691	0.743454	+	0.668787i	$0.233186\pi$
$32$	0	0
$33$	6484.21	1.03651
$34$	0	0
$35$	5788.21i	0.798683i
$36$	0	0
$37$	10687.6i	1.28344i	0.766939	+	0.641720i	$0.221779\pi$
$37$	10687.6i	1.28344i	−0.766939	+	0.641720i	$0.778221\pi$
$38$	0	0
$39$	1250.89	0.131691
$40$	0	0
$41$	−9963.87	−0.925696	−0.462848	−	0.886438i	$-0.653172\pi$
$41$	−9963.87	−0.925696	−0.462848	+	0.886438i	$0.653172\pi$
$42$	0	0
$43$	925.409i	0.0763243i	0.999272	+	0.0381621i	$0.0121503\pi$
$43$	925.409i	0.0763243i	−0.999272	+	0.0381621i	$0.987850\pi$
$44$	0	0
$45$	2715.23i	0.199883i
$46$	0	0
$47$	−8063.78	−0.532469	−0.266234	−	0.963908i	$-0.585779\pi$
$47$	−8063.78	−0.532469	−0.266234	+	0.963908i	$0.585779\pi$
$48$	0	0
$49$	36798.5	2.18947
$50$	0	0
$51$	− 5120.91i	− 0.275690i
$52$	0	0
$53$	7952.45i	0.388876i	0.980915	+	0.194438i	$0.0622883\pi$
$53$	7952.45i	0.388876i	−0.980915	+	0.194438i	$0.937712\pi$
$54$	0	0
$55$	13983.4	0.623312
$56$	0	0
$57$	21717.0	0.885344
$58$	0	0
$59$	16801.2i	0.628361i	0.949363	+	0.314181i	$0.101730\pi$
$59$	16801.2i	0.628361i	−0.949363	+	0.314181i	$0.898270\pi$
$60$	0	0
$61$	− 11297.4i	− 0.388734i	−0.980929	−	0.194367i	$-0.937735\pi$
$61$	− 11297.4i	− 0.388734i	0.980929	−	0.194367i	$-0.0622653\pi$
$62$	0	0
$63$	25146.1	0.798214
$64$	0	0
$65$	2697.57	0.0791935
$66$	0	0
$67$	33619.4i	0.914963i	0.889219	+	0.457481i	$0.151248\pi$
$67$	33619.4i	0.914963i	−0.889219	+	0.457481i	$0.848752\pi$
$68$	0	0
$69$	44463.9i	1.12431i
$70$	0	0
$71$	8869.47	0.208810	0.104405	−	0.994535i	$-0.466706\pi$
$71$	8869.47	0.208810	0.104405	+	0.994535i	$0.466706\pi$
$72$	0	0
$73$	55505.3	1.21907	0.609533	−	0.792761i	$-0.291357\pi$
$73$	55505.3	1.21907	0.609533	+	0.792761i	$0.291357\pi$
$74$	0	0
$75$	− 7245.44i	− 0.148735i
$76$	0	0
$77$	− 129502.i	− 2.48914i
$78$	0	0
$79$	−69413.7	−1.25135	−0.625673	−	0.780085i	$-0.715176\pi$
$79$	−69413.7	−1.25135	−0.625673	+	0.780085i	$0.715176\pi$
$80$	0	0
$81$	−20861.0	−0.353284
$82$	0	0
$83$	− 10231.5i	− 0.163021i	−0.996672	−	0.0815106i	$-0.974026\pi$
$83$	− 10231.5i	− 0.163021i	0.996672	−	0.0815106i	$-0.0259744\pi$
$84$	0	0
$85$	− 11043.4i	− 0.165789i
$86$	0	0
$87$	39064.7	0.553333
$88$	0	0
$89$	−92458.4	−1.23729	−0.618645	−	0.785671i	$-0.712318\pi$
$89$	−92458.4	−1.23729	−0.618645	+	0.785671i	$0.712318\pi$
$90$	0	0
$91$	− 24982.6i	− 0.316253i
$92$	0	0
$93$	92230.2i	1.10577i
$94$	0	0
$95$	46833.2	0.532409
$96$	0	0
$97$	−88657.9	−0.956727	−0.478364	−	0.878162i	$-0.658770\pi$
$97$	−88657.9	−0.956727	−0.478364	+	0.878162i	$0.658770\pi$
$98$	0	0
$99$	− 60748.9i	− 0.622946i
$100$	0	0
$101$	− 20080.2i	− 0.195869i	−0.995193	−	0.0979343i	$-0.968776\pi$
$101$	− 20080.2i	− 0.195869i	0.995193	−	0.0979343i	$-0.0312235\pi$
$102$	0	0
$103$	−16926.9	−0.157212	−0.0786059	−	0.996906i	$-0.525047\pi$
$103$	−16926.9	−0.157212	−0.0786059	+	0.996906i	$0.525047\pi$
$104$	0	0
$105$	−67101.1	−0.593959
$106$	0	0
$107$	− 69502.3i	− 0.586867i	−0.955979	−	0.293434i	$-0.905202\pi$
$107$	− 69502.3i	− 0.586867i	0.955979	−	0.293434i	$-0.0947979\pi$
$108$	0	0
$109$	229650.i	1.85140i	0.378260	+	0.925699i	$0.376523\pi$
$109$	229650.i	1.85140i	−0.378260	+	0.925699i	$0.623477\pi$
$110$	0	0
$111$	−123898.	−0.954459
$112$	0	0
$113$	60045.6	0.442369	0.221185	−	0.975232i	$-0.429008\pi$
$113$	60045.6	0.442369	0.221185	+	0.975232i	$0.429008\pi$
$114$	0	0
$115$	95887.6i	0.676111i
$116$	0	0
$117$	− 11719.2i	− 0.0791470i
$118$	0	0
$119$	−102274.	−0.662063
$120$	0	0
$121$	−151805.	−0.942590
$122$	0	0
$123$	− 115508.i	− 0.688415i
$124$	0	0
$125$	− 15625.0i	− 0.0894427i
$126$	0	0
$127$	45815.4	0.252059	0.126030	−	0.992026i	$-0.459777\pi$
$127$	45815.4	0.252059	0.126030	+	0.992026i	$0.459777\pi$
$128$	0	0
$129$	−10728.0	−0.0567603
$130$	0	0
$131$	− 135284.i	− 0.688761i	−0.938830	−	0.344381i	$-0.888089\pi$
$131$	− 135284.i	− 0.688761i	0.938830	−	0.344381i	$-0.111911\pi$
$132$	0	0
$133$	− 433729.i	− 2.12613i
$134$	0	0
$135$	−101903.	−0.481228
$136$	0	0
$137$	−345313.	−1.57185	−0.785927	−	0.618320i	$-0.787814\pi$
$137$	−345313.	−1.57185	−0.785927	+	0.618320i	$0.787814\pi$
$138$	0	0
$139$	− 116846.i	− 0.512951i	−0.966551	−	0.256475i	$-0.917439\pi$
$139$	− 116846.i	− 0.512951i	0.966551	−	0.256475i	$-0.0825612\pi$
$140$	0	0
$141$	− 93481.0i	− 0.395982i
$142$	0	0
$143$	−60353.9	−0.246811
$144$	0	0
$145$	84244.1	0.332751
$146$	0	0
$147$	426594.i	1.62825i
$148$	0	0
$149$	− 87842.0i	− 0.324143i	−0.986779	−	0.162072i	$-0.948182\pi$
$149$	− 87842.0i	− 0.324143i	0.986779	−	0.162072i	$-0.0518175\pi$
$150$	0	0
$151$	−165109.	−0.589288	−0.294644	−	0.955607i	$-0.595201\pi$
$151$	−165109.	−0.589288	−0.294644	+	0.955607i	$0.595201\pi$
$152$	0	0
$153$	−47976.5	−0.165691
$154$	0	0
$155$	198897.i	0.664965i
$156$	0	0
$157$	− 191163.i	− 0.618948i	−0.950908	−	0.309474i	$-0.899847\pi$
$157$	− 191163.i	− 0.618948i	0.950908	−	0.309474i	$-0.100153\pi$
$158$	0	0
$159$	−92190.4	−0.289197
$160$	0	0
$161$	888029.	2.69999
$162$	0	0
$163$	− 478403.i	− 1.41034i	−0.709037	−	0.705172i	$-0.750870\pi$
$163$	− 478403.i	− 1.41034i	0.709037	−	0.705172i	$-0.249130\pi$
$164$	0	0
$165$	162105.i	0.463540i
$166$	0	0
$167$	307544.	0.853328	0.426664	−	0.904410i	$-0.359689\pi$
$167$	307544.	0.853328	0.426664	+	0.904410i	$0.359689\pi$
$168$	0	0
$169$	359650.	0.968642
$170$	0	0
$171$	− 203461.i	− 0.532096i
$172$	0	0
$173$	55343.3i	0.140588i	0.997526	+	0.0702942i	$0.0223938\pi$
$173$	55343.3i	0.140588i	−0.997526	+	0.0702942i	$0.977606\pi$
$174$	0	0
$175$	−144705.	−0.357182
$176$	0	0
$177$	−194771.	−0.467295
$178$	0	0
$179$	703665.i	1.64147i	0.571308	+	0.820736i	$0.306436\pi$
$179$	703665.i	1.64147i	−0.571308	+	0.820736i	$0.693564\pi$
$180$	0	0
$181$	− 96255.4i	− 0.218388i	−0.994020	−	0.109194i	$-0.965173\pi$
$181$	− 96255.4i	− 0.218388i	0.994020	−	0.109194i	$-0.0348270\pi$
$182$	0	0
$183$	130967.	0.289091
$184$	0	0
$185$	−267190.	−0.573972
$186$	0	0
$187$	247078.i	0.516690i
$188$	0	0
$189$	943734.i	1.92174i
$190$	0	0
$191$	504776.	1.00119	0.500594	−	0.865682i	$-0.333115\pi$
$191$	504776.	1.00119	0.500594	+	0.865682i	$0.333115\pi$
$192$	0	0
$193$	233138.	0.450527	0.225263	−	0.974298i	$-0.427676\pi$
$193$	233138.	0.450527	0.225263	+	0.974298i	$0.427676\pi$
$194$	0	0
$195$	31272.2i	0.0588941i
$196$	0	0
$197$	322071.i	0.591270i	0.955301	+	0.295635i	$0.0955313\pi$
$197$	322071.i	0.591270i	−0.955301	+	0.295635i	$0.904469\pi$
$198$	0	0
$199$	−457179.	−0.818377	−0.409189	−	0.912450i	$-0.634188\pi$
$199$	−457179.	−0.818377	−0.409189	+	0.912450i	$0.634188\pi$
$200$	0	0
$201$	−389740.	−0.680433
$202$	0	0
$203$	− 780197.i	− 1.32881i
$204$	0	0
$205$	− 249097.i	− 0.413984i
$206$	0	0
$207$	416571.	0.675714
$208$	0	0
$209$	−1.04782e6	−1.65928
$210$	0	0
$211$	826083.i	1.27737i	0.769468	+	0.638686i	$0.220522\pi$
$211$	826083.i	1.27737i	−0.769468	+	0.638686i	$0.779478\pi$
$212$	0	0
$213$	102821.i	0.155286i
$214$	0	0
$215$	−23135.2	−0.0341333
$216$	0	0
$217$	1.84201e6	2.65548
$218$	0	0
$219$	643457.i	0.906586i
$220$	0	0
$221$	47664.5i	0.0656470i
$222$	0	0
$223$	1.03975e6	1.40013	0.700066	−	0.714079i	$-0.253154\pi$
$223$	1.03975e6	1.40013	0.700066	+	0.714079i	$0.253154\pi$
$224$	0	0
$225$	−67880.7	−0.0893902
$226$	0	0
$227$	− 1.32339e6i	− 1.70460i	−0.523051	−	0.852302i	$-0.675206\pi$
$227$	− 1.32339e6i	− 1.70460i	0.523051	−	0.852302i	$-0.324794\pi$
$228$	0	0
$229$	1.12200e6i	1.41385i	0.707288	+	0.706926i	$0.249918\pi$
$229$	1.12200e6i	1.41385i	−0.707288	+	0.706926i	$0.750082\pi$
$230$	0	0
$231$	1.50128e6	1.85111
$232$	0	0
$233$	−20887.8	−0.0252059	−0.0126030	−	0.999921i	$-0.504012\pi$
$233$	−20887.8	−0.0252059	−0.0126030	+	0.999921i	$0.504012\pi$
$234$	0	0
$235$	− 201594.i	− 0.238127i
$236$	0	0
$237$	− 804693.i	− 0.930592i
$238$	0	0
$239$	−40580.0	−0.0459534	−0.0229767	−	0.999736i	$-0.507314\pi$
$239$	−40580.0	−0.0459534	−0.0229767	+	0.999736i	$0.507314\pi$
$240$	0	0
$241$	841414.	0.933184	0.466592	−	0.884473i	$-0.345482\pi$
$241$	841414.	0.933184	0.466592	+	0.884473i	$0.345482\pi$
$242$	0	0
$243$	748657.i	0.813330i
$244$	0	0
$245$	919962.i	0.979162i
$246$	0	0
$247$	−202138.	−0.210817
$248$	0	0
$249$	118611.	0.121234
$250$	0	0
$251$	− 1.24994e6i	− 1.25229i	−0.779707	−	0.626144i	$-0.784632\pi$
$251$	− 1.24994e6i	− 1.25229i	0.779707	−	0.626144i	$-0.215368\pi$
$252$	0	0
$253$	− 2.14533e6i	− 2.10714i
$254$	0	0
$255$	128023.	0.123292
$256$	0	0
$257$	−766259.	−0.723674	−0.361837	−	0.932241i	$-0.617850\pi$
$257$	−766259.	−0.723674	−0.361837	+	0.932241i	$0.617850\pi$
$258$	0	0
$259$	2.47448e6i	2.29211i
$260$	0	0
$261$	− 365987.i	− 0.332556i
$262$	0	0
$263$	−1.35524e6	−1.20817	−0.604084	−	0.796921i	$-0.706461\pi$
$263$	−1.35524e6	−1.20817	−0.604084	+	0.796921i	$0.706461\pi$
$264$	0	0
$265$	−198811.	−0.173911
$266$	0	0
$267$	− 1.07184e6i	− 0.920139i
$268$	0	0
$269$	− 173063.i	− 0.145822i	−0.997338	−	0.0729110i	$-0.976771\pi$
$269$	− 173063.i	− 0.145822i	0.997338	−	0.0729110i	$-0.0232289\pi$
$270$	0	0
$271$	−802529.	−0.663800	−0.331900	−	0.943315i	$-0.607690\pi$
$271$	−802529.	−0.663800	−0.331900	+	0.943315i	$0.607690\pi$
$272$	0	0
$273$	289616.	0.235188
$274$	0	0
$275$	349585.i	0.278754i
$276$	0	0
$277$	1.01003e6i	0.790924i	0.918482	+	0.395462i	$0.129415\pi$
$277$	1.01003e6i	0.790924i	−0.918482	+	0.395462i	$0.870585\pi$
$278$	0	0
$279$	864081.	0.664575
$280$	0	0
$281$	−2.24480e6	−1.69595	−0.847973	−	0.530039i	$-0.822177\pi$
$281$	−2.24480e6	−1.69595	−0.847973	+	0.530039i	$0.822177\pi$
$282$	0	0
$283$	− 1.71474e6i	− 1.27272i	−0.771392	−	0.636361i	$-0.780439\pi$
$283$	− 1.71474e6i	− 1.27272i	0.771392	−	0.636361i	$-0.219561\pi$
$284$	0	0
$285$	542924.i	0.395938i
$286$	0	0
$287$	−2.30692e6	−1.65321
$288$	0	0
$289$	−1.22473e6	−0.862571
$290$	0	0
$291$	− 1.02779e6i	− 0.711492i
$292$	0	0
$293$	− 2.27442e6i	− 1.54775i	−0.633337	−	0.773876i	$-0.718315\pi$
$293$	− 2.27442e6i	− 1.54775i	0.633337	−	0.773876i	$-0.281685\pi$
$294$	0	0
$295$	−420029.	−0.281012
$296$	0	0
$297$	2.27991e6	1.49978
$298$	0	0
$299$	− 413862.i	− 0.267718i
$300$	0	0
$301$	214259.i	0.136308i
$302$	0	0
$303$	232784.	0.145662
$304$	0	0
$305$	282434.	0.173847
$306$	0	0
$307$	2.36152e6i	1.43003i	0.699109	+	0.715015i	$0.253580\pi$
$307$	2.36152e6i	1.43003i	−0.699109	+	0.715015i	$0.746420\pi$
$308$	0	0
$309$	− 196229.i	− 0.116914i
$310$	0	0
$311$	−797118.	−0.467328	−0.233664	−	0.972317i	$-0.575072\pi$
$311$	−797118.	−0.467328	−0.233664	+	0.972317i	$0.575072\pi$
$312$	0	0
$313$	1.93581e6	1.11687	0.558434	−	0.829549i	$-0.311402\pi$
$313$	1.93581e6	1.11687	0.558434	+	0.829549i	$0.311402\pi$
$314$	0	0
$315$	628653.i	0.356972i
$316$	0	0
$317$	2.34736e6i	1.31200i	0.754763	+	0.655998i	$0.227752\pi$
$317$	2.34736e6i	1.31200i	−0.754763	+	0.655998i	$0.772248\pi$
$318$	0	0
$319$	−1.88483e6	−1.03704
$320$	0	0
$321$	805720.	0.436437
$322$	0	0
$323$	827516.i	0.441337i
$324$	0	0
$325$	67439.3i	0.0354164i
$326$	0	0
$327$	−2.66226e6	−1.37683
$328$	0	0
$329$	−1.86700e6	−0.950941
$330$	0	0
$331$	− 291538.i	− 0.146260i	−0.997322	−	0.0731300i	$-0.976701\pi$
$331$	− 291538.i	− 0.146260i	0.997322	−	0.0731300i	$-0.0232988\pi$
$332$	0	0
$333$	1.16077e6i	0.573635i
$334$	0	0
$335$	−840486.	−0.409184
$336$	0	0
$337$	−3.41829e6	−1.63958	−0.819792	−	0.572661i	$-0.805911\pi$
$337$	−3.41829e6	−1.63958	−0.819792	+	0.572661i	$0.805911\pi$
$338$	0	0
$339$	696091.i	0.328978i
$340$	0	0
$341$	− 4.45001e6i	− 2.07241i
$342$	0	0
$343$	4.62860e6	2.12429
$344$	0	0
$345$	−1.11160e6	−0.502805
$346$	0	0
$347$	− 3.87943e6i	− 1.72960i	−0.502120	−	0.864798i	$-0.667447\pi$
$347$	− 3.87943e6i	− 1.72960i	0.502120	−	0.864798i	$-0.332553\pi$
$348$	0	0
$349$	− 2.29111e6i	− 1.00689i	−0.864027	−	0.503445i	$-0.832066\pi$
$349$	− 2.29111e6i	− 1.00689i	0.864027	−	0.503445i	$-0.167934\pi$
$350$	0	0
$351$	439823.	0.190551
$352$	0	0
$353$	−2.16132e6	−0.923171	−0.461586	−	0.887096i	$-0.652719\pi$
$353$	−2.16132e6	−0.923171	−0.461586	+	0.887096i	$0.652719\pi$
$354$	0	0
$355$	221737.i	0.0933828i
$356$	0	0
$357$	− 1.18564e6i	− 0.492358i
$358$	0	0
$359$	−2.30863e6	−0.945408	−0.472704	−	0.881221i	$-0.656722\pi$
$359$	−2.30863e6	−0.945408	−0.472704	+	0.881221i	$0.656722\pi$
$360$	0	0
$361$	−1.03326e6	−0.417295
$362$	0	0
$363$	− 1.75983e6i	− 0.700979i
$364$	0	0
$365$	1.38763e6i	0.545183i
$366$	0	0
$367$	1.68650e6	0.653614	0.326807	−	0.945091i	$-0.394027\pi$
$367$	1.68650e6	0.653614	0.326807	+	0.945091i	$0.394027\pi$
$368$	0	0
$369$	−1.08217e6	−0.413741
$370$	0	0
$371$	1.84122e6i	0.694498i
$372$	0	0
$373$	171338.i	0.0637647i	0.999492	+	0.0318824i	$0.0101502\pi$
$373$	171338.i	0.0637647i	−0.999492	+	0.0318824i	$0.989850\pi$
$374$	0	0
$375$	181136.	0.0665161
$376$	0	0
$377$	−363607.	−0.131759
$378$	0	0
$379$	1.11665e6i	0.399317i	0.979866	+	0.199658i	$0.0639832\pi$
$379$	1.11665e6i	0.399317i	−0.979866	+	0.199658i	$0.936017\pi$
$380$	0	0
$381$	531125.i	0.187449i
$382$	0	0
$383$	433589.	0.151036	0.0755182	−	0.997144i	$-0.475939\pi$
$383$	433589.	0.151036	0.0755182	+	0.997144i	$0.475939\pi$
$384$	0	0
$385$	3.23755e6	1.11318
$386$	0	0
$387$	100508.i	0.0341132i
$388$	0	0
$389$	3.20076e6i	1.07245i	0.844074	+	0.536227i	$0.180151\pi$
$389$	3.20076e6i	1.07245i	−0.844074	+	0.536227i	$0.819849\pi$
$390$	0	0
$391$	−1.69428e6	−0.560457
$392$	0	0
$393$	1.56831e6	0.512213
$394$	0	0
$395$	− 1.73534e6i	− 0.559619i
$396$	0	0
$397$	1.44525e6i	0.460220i	0.973165	+	0.230110i	$0.0739086\pi$
$397$	1.44525e6i	0.460220i	−0.973165	+	0.230110i	$0.926091\pi$
$398$	0	0
$399$	5.02810e6	1.58114
$400$	0	0
$401$	1.44642e6	0.449194	0.224597	−	0.974452i	$-0.427893\pi$
$401$	1.44642e6	0.449194	0.224597	+	0.974452i	$0.427893\pi$
$402$	0	0
$403$	− 858463.i	− 0.263305i
$404$	0	0
$405$	− 521526.i	− 0.157993i
$406$	0	0
$407$	5.97795e6	1.78882
$408$	0	0
$409$	2.45947e6	0.726997	0.363498	−	0.931595i	$-0.381582\pi$
$409$	2.45947e6	0.726997	0.363498	+	0.931595i	$0.381582\pi$
$410$	0	0
$411$	− 4.00312e6i	− 1.16894i
$412$	0	0
$413$	3.88995e6i	1.12220i
$414$	0	0
$415$	255787.	0.0729053
$416$	0	0
$417$	1.35456e6	0.381467
$418$	0	0
$419$	2.72546e6i	0.758410i	0.925313	+	0.379205i	$0.123802\pi$
$419$	2.72546e6i	0.758410i	−0.925313	+	0.379205i	$0.876198\pi$
$420$	0	0
$421$	6.51391e6i	1.79117i	0.444892	+	0.895584i	$0.353242\pi$
$421$	6.51391e6i	1.79117i	−0.444892	+	0.895584i	$0.646758\pi$
$422$	0	0
$423$	−875800.	−0.237987
$424$	0	0
$425$	276085.	0.0741429
$426$	0	0
$427$	− 2.61567e6i	− 0.694245i
$428$	0	0
$429$	− 699665.i	− 0.183547i
$430$	0	0
$431$	−3.85877e6	−1.00059	−0.500295	−	0.865855i	$-0.666775\pi$
$431$	−3.85877e6	−1.00059	−0.500295	+	0.865855i	$0.666775\pi$
$432$	0	0
$433$	−1.22019e6	−0.312757	−0.156379	−	0.987697i	$-0.549982\pi$
$433$	−1.22019e6	−0.312757	−0.156379	+	0.987697i	$0.549982\pi$
$434$	0	0
$435$	976617.i	0.247458i
$436$	0	0
$437$	− 7.18516e6i	− 1.79984i
$438$	0	0
$439$	1.29701e6	0.321204	0.160602	−	0.987019i	$-0.448656\pi$
$439$	1.29701e6	0.321204	0.160602	+	0.987019i	$0.448656\pi$
$440$	0	0
$441$	3.99665e6	0.978588
$442$	0	0
$443$	2.57412e6i	0.623188i	0.950215	+	0.311594i	$0.100863\pi$
$443$	2.57412e6i	0.623188i	−0.950215	+	0.311594i	$0.899137\pi$
$444$	0	0
$445$	− 2.31146e6i	− 0.553333i
$446$	0	0
$447$	1.01833e6	0.241056
$448$	0	0
$449$	52050.0	0.0121844	0.00609221	−	0.999981i	$-0.498061\pi$
$449$	52050.0	0.0121844	0.00609221	+	0.999981i	$0.498061\pi$
$450$	0	0
$451$	5.57315e6i	1.29021i
$452$	0	0
$453$	− 1.91406e6i	− 0.438238i
$454$	0	0
$455$	624565.	0.141433
$456$	0	0
$457$	446306.	0.0999637	0.0499818	−	0.998750i	$-0.484084\pi$
$457$	446306.	0.0999637	0.0499818	+	0.998750i	$0.484084\pi$
$458$	0	0
$459$	− 1.80056e6i	− 0.398910i
$460$	0	0
$461$	− 3.55954e6i	− 0.780085i	−0.920797	−	0.390043i	$-0.872460\pi$
$461$	− 3.55954e6i	− 0.780085i	0.920797	−	0.390043i	$-0.127540\pi$
$462$	0	0
$463$	−2.31611e6	−0.502120	−0.251060	−	0.967972i	$-0.580779\pi$
$463$	−2.31611e6	−0.502120	−0.251060	+	0.967972i	$0.580779\pi$
$464$	0	0
$465$	−2.30576e6	−0.494517
$466$	0	0
$467$	7.14211e6i	1.51542i	0.652589	+	0.757712i	$0.273683\pi$
$467$	7.14211e6i	1.51542i	−0.652589	+	0.757712i	$0.726317\pi$
$468$	0	0
$469$	7.78386e6i	1.63404i
$470$	0	0
$471$	2.21609e6	0.460295
$472$	0	0
$473$	517614.	0.106378
$474$	0	0
$475$	1.17083e6i	0.238100i
$476$	0	0
$477$	863709.i	0.173809i
$478$	0	0
$479$	−2.43371e6	−0.484651	−0.242326	−	0.970195i	$-0.577910\pi$
$479$	−2.43371e6	−0.484651	−0.242326	+	0.970195i	$0.577910\pi$
$480$	0	0
$481$	1.15322e6	0.227274
$482$	0	0
$483$	1.02947e7i	2.00791i
$484$	0	0
$485$	− 2.21645e6i	− 0.427861i
$486$	0	0
$487$	967697.	0.184892	0.0924458	−	0.995718i	$-0.470532\pi$
$487$	967697.	0.184892	0.0924458	+	0.995718i	$0.470532\pi$
$488$	0	0
$489$	5.54599e6	1.04883
$490$	0	0
$491$	1.94045e6i	0.363245i	0.983368	+	0.181623i	$0.0581349\pi$
$491$	1.94045e6i	0.363245i	−0.983368	+	0.181623i	$0.941865\pi$
$492$	0	0
$493$	1.48854e6i	0.275832i
$494$	0	0
$495$	1.51872e6	0.278590
$496$	0	0
$497$	2.05354e6	0.372916
$498$	0	0
$499$	− 8.97327e6i	− 1.61324i	−0.591069	−	0.806621i	$-0.701294\pi$
$499$	− 8.97327e6i	− 1.61324i	0.591069	−	0.806621i	$-0.298706\pi$
$500$	0	0
$501$	3.56527e6i	0.634597i
$502$	0	0
$503$	−1.02287e7	−1.80261	−0.901306	−	0.433184i	$-0.857390\pi$
$503$	−1.02287e7	−1.80261	−0.901306	+	0.433184i	$0.857390\pi$
$504$	0	0
$505$	502005.	0.0875951
$506$	0	0
$507$	4.16932e6i	0.720352i
$508$	0	0
$509$	− 1.00938e7i	− 1.72687i	−0.504459	−	0.863435i	$-0.668308\pi$
$509$	− 1.00938e7i	− 1.72687i	0.504459	−	0.863435i	$-0.331692\pi$
$510$	0	0
$511$	1.28511e7	2.17714
$512$	0	0
$513$	7.63588e6	1.28105
$514$	0	0
$515$	− 423173.i	− 0.0703073i
$516$	0	0
$517$	4.51036e6i	0.742138i
$518$	0	0
$519$	−641579.	−0.104552
$520$	0	0
$521$	−4.23461e6	−0.683470	−0.341735	−	0.939796i	$-0.611015\pi$
$521$	−4.23461e6	−0.683470	−0.341735	+	0.939796i	$0.611015\pi$
$522$	0	0
$523$	− 5.78898e6i	− 0.925438i	−0.886505	−	0.462719i	$-0.846874\pi$
$523$	− 5.78898e6i	− 0.925438i	0.886505	−	0.462719i	$-0.153126\pi$
$524$	0	0
$525$	− 1.67753e6i	− 0.265626i
$526$	0	0
$527$	−3.51439e6	−0.551219
$528$	0	0
$529$	8.27474e6	1.28563
$530$	0	0
$531$	1.82476e6i	0.280847i
$532$	0	0
$533$	1.07513e6i	0.163924i
$534$	0	0
$535$	1.73756e6	0.262455
$536$	0	0
$537$	−8.15738e6	−1.22072
$538$	0	0
$539$	− 2.05827e7i	− 3.05162i
$540$	0	0
$541$	2.51591e6i	0.369574i	0.982779	+	0.184787i	$0.0591596\pi$
$541$	2.51591e6i	0.369574i	−0.982779	+	0.184787i	$0.940840\pi$
$542$	0	0
$543$	1.11586e6	0.162409
$544$	0	0
$545$	−5.74125e6	−0.827971
$546$	0	0
$547$	5.57811e6i	0.797111i	0.917144	+	0.398556i	$0.130488\pi$
$547$	5.57811e6i	0.797111i	−0.917144	+	0.398556i	$0.869512\pi$
$548$	0	0
$549$	− 1.22700e6i	− 0.173745i
$550$	0	0
$551$	−6.31268e6	−0.885798
$552$	0	0
$553$	−1.60713e7	−2.23479
$554$	0	0
$555$	− 3.09745e6i	− 0.426847i
$556$	0	0
$557$	− 954006.i	− 0.130291i	−0.997876	−	0.0651453i	$-0.979249\pi$
$557$	− 954006.i	− 0.130291i	0.997876	−	0.0651453i	$-0.0207511\pi$
$558$	0	0
$559$	99854.4	0.0135157
$560$	0	0
$561$	−2.86431e6	−0.384249
$562$	0	0
$563$	− 9.17088e6i	− 1.21938i	−0.792639	−	0.609691i	$-0.791293\pi$
$563$	− 9.17088e6i	− 1.21938i	0.792639	−	0.609691i	$-0.208707\pi$
$564$	0	0
$565$	1.50114e6i	0.197834i
$566$	0	0
$567$	−4.82993e6	−0.630933
$568$	0	0
$569$	−8.51511e6	−1.10258	−0.551289	−	0.834314i	$-0.685864\pi$
$569$	−8.51511e6	−1.10258	−0.551289	+	0.834314i	$0.685864\pi$
$570$	0	0
$571$	− 8.35546e6i	− 1.07246i	−0.844073	−	0.536229i	$-0.819848\pi$
$571$	− 8.35546e6i	− 1.07246i	0.844073	−	0.536229i	$-0.180152\pi$
$572$	0	0
$573$	5.85172e6i	0.744556i
$574$	0	0
$575$	−2.39719e6	−0.302366
$576$	0	0
$577$	2.48061e6	0.310184	0.155092	−	0.987900i	$-0.450433\pi$
$577$	2.48061e6	0.310184	0.155092	+	0.987900i	$0.450433\pi$
$578$	0	0
$579$	2.70271e6i	0.335044i
$580$	0	0
$581$	− 2.36888e6i	− 0.291141i
$582$	0	0
$583$	4.44809e6	0.542003
$584$	0	0
$585$	292981.	0.0353956
$586$	0	0
$587$	1.08423e7i	1.29875i	0.760469	+	0.649374i	$0.224969\pi$
$587$	1.08423e7i	1.29875i	−0.760469	+	0.649374i	$0.775031\pi$
$588$	0	0
$589$	− 1.49040e7i	− 1.77017i
$590$	0	0
$591$	−3.73368e6	−0.439712
$592$	0	0
$593$	−1.62349e6	−0.189589	−0.0947943	−	0.995497i	$-0.530219\pi$
$593$	−1.62349e6	−0.189589	−0.0947943	+	0.995497i	$0.530219\pi$
$594$	0	0
$595$	− 2.55686e6i	− 0.296084i
$596$	0	0
$597$	− 5.29994e6i	− 0.608605i
$598$	0	0
$599$	5.09707e6	0.580435	0.290217	−	0.956961i	$-0.406272\pi$
$599$	5.09707e6	0.580435	0.290217	+	0.956961i	$0.406272\pi$
$600$	0	0
$601$	−46987.4	−0.00530634	−0.00265317	−	0.999996i	$-0.500845\pi$
$601$	−46987.4	−0.00530634	−0.00265317	+	0.999996i	$0.500845\pi$
$602$	0	0
$603$	3.65138e6i	0.408944i
$604$	0	0
$605$	− 3.79513e6i	− 0.421539i
$606$	0	0
$607$	5.02427e6	0.553479	0.276739	−	0.960945i	$-0.410746\pi$
$607$	5.02427e6	0.553479	0.276739	+	0.960945i	$0.410746\pi$
$608$	0	0
$609$	9.04459e6	0.988202
$610$	0	0
$611$	870105.i	0.0942907i
$612$	0	0
$613$	9.19740e6i	0.988584i	0.869296	+	0.494292i	$0.164573\pi$
$613$	9.19740e6i	0.988584i	−0.869296	+	0.494292i	$0.835427\pi$
$614$	0	0
$615$	2.88771e6	0.307869
$616$	0	0
$617$	−8.76074e6	−0.926462	−0.463231	−	0.886237i	$-0.653310\pi$
$617$	−8.76074e6	−0.926462	−0.463231	+	0.886237i	$0.653310\pi$
$618$	0	0
$619$	− 750516.i	− 0.0787287i	−0.999225	−	0.0393643i	$-0.987467\pi$
$619$	− 750516.i	− 0.0787287i	0.999225	−	0.0393643i	$-0.0125333\pi$
$620$	0	0
$621$	1.56339e7i	1.62682i
$622$	0	0
$623$	−2.14068e7	−2.20969
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	− 1.21471e7i	− 1.23396i
$628$	0	0
$629$	− 4.72109e6i	− 0.475790i
$630$	0	0
$631$	1.72922e6	0.172893	0.0864463	−	0.996257i	$-0.472449\pi$
$631$	1.72922e6	0.172893	0.0864463	+	0.996257i	$0.472449\pi$
$632$	0	0
$633$	−9.57653e6	−0.949946
$634$	0	0
$635$	1.14539e6i	0.112724i
$636$	0	0
$637$	− 3.97066e6i	− 0.387717i
$638$	0	0
$639$	963305.	0.0933280
$640$	0	0
$641$	−1.99305e7	−1.91590	−0.957950	−	0.286936i	$-0.907363\pi$
$641$	−1.99305e7	−1.91590	−0.957950	+	0.286936i	$0.907363\pi$
$642$	0	0
$643$	1.82551e7i	1.74123i	0.491963	+	0.870616i	$0.336279\pi$
$643$	1.82551e7i	1.74123i	−0.491963	+	0.870616i	$0.663721\pi$
$644$	0	0
$645$	− 268200.i	− 0.0253840i
$646$	0	0
$647$	621345.	0.0583542	0.0291771	−	0.999574i	$-0.490711\pi$
$647$	621345.	0.0583542	0.0291771	+	0.999574i	$0.490711\pi$
$648$	0	0
$649$	9.39748e6	0.875790
$650$	0	0
$651$	2.13539e7i	1.97481i
$652$	0	0
$653$	1.14728e7i	1.05290i	0.850207	+	0.526449i	$0.176477\pi$
$653$	1.14728e7i	1.05290i	−0.850207	+	0.526449i	$0.823523\pi$
$654$	0	0
$655$	3.38211e6	0.308023
$656$	0	0
$657$	6.02838e6	0.544863
$658$	0	0
$659$	1.17996e6i	0.105841i	0.998599	+	0.0529204i	$0.0168530\pi$
$659$	1.17996e6i	0.105841i	−0.998599	+	0.0529204i	$0.983147\pi$
$660$	0	0
$661$	− 1.87825e7i	− 1.67205i	−0.548688	−	0.836027i	$-0.684873\pi$
$661$	− 1.87825e7i	− 1.67205i	0.548688	−	0.836027i	$-0.315127\pi$
$662$	0	0
$663$	−552561.	−0.0488198
$664$	0	0
$665$	1.08432e7	0.950834
$666$	0	0
$667$	− 1.29247e7i	− 1.12488i
$668$	0	0
$669$	1.20536e7i	1.04124i
$670$	0	0
$671$	−6.31902e6	−0.541806
$672$	0	0
$673$	−1.18969e7	−1.01250	−0.506250	−	0.862387i	$-0.668969\pi$
$673$	−1.18969e7	−1.01250	−0.506250	+	0.862387i	$0.668969\pi$
$674$	0	0
$675$	− 2.54756e6i	− 0.215212i
$676$	0	0
$677$	− 2.73577e6i	− 0.229407i	−0.993400	−	0.114704i	$-0.963408\pi$
$677$	− 2.73577e6i	− 0.229407i	0.993400	−	0.114704i	$-0.0365919\pi$
$678$	0	0
$679$	−2.05268e7	−1.70863
$680$	0	0
$681$	1.53417e7	1.26767
$682$	0	0
$683$	− 5.35182e6i	− 0.438985i	−0.975614	−	0.219493i	$-0.929560\pi$
$683$	− 5.35182e6i	− 0.438985i	0.975614	−	0.219493i	$-0.0704402\pi$
$684$	0	0
$685$	− 8.63284e6i	− 0.702954i
$686$	0	0
$687$	−1.30070e7	−1.05144
$688$	0	0
$689$	858092.	0.0688630
$690$	0	0
$691$	− 6.84272e6i	− 0.545172i	−0.962131	−	0.272586i	$-0.912121\pi$
$691$	− 6.84272e6i	− 0.545172i	0.962131	−	0.272586i	$-0.0878790\pi$
$692$	0	0
$693$	− 1.40651e7i	− 1.11253i
$694$	0	0
$695$	2.92114e6	0.229399
$696$	0	0
$697$	4.40139e6	0.343169
$698$	0	0
$699$	− 242146.i	− 0.0187450i
$700$	0	0
$701$	1.33086e7i	1.02291i	0.859310	+	0.511455i	$0.170893\pi$
$701$	1.33086e7i	1.02291i	−0.859310	+	0.511455i	$0.829107\pi$
$702$	0	0
$703$	2.00214e7	1.52794
$704$	0	0
$705$	2.33703e6	0.177089
$706$	0	0
$707$	− 4.64914e6i	− 0.349804i
$708$	0	0
$709$	− 5.66978e6i	− 0.423595i	−0.977314	−	0.211797i	$-0.932068\pi$
$709$	− 5.66978e6i	− 0.423595i	0.977314	−	0.211797i	$-0.0679317\pi$
$710$	0	0
$711$	−7.53896e6	−0.559291
$712$	0	0
$713$	3.05148e7	2.24795
$714$	0	0
$715$	− 1.50885e6i	− 0.110377i
$716$	0	0
$717$	− 470432.i	− 0.0341743i
$718$	0	0
$719$	2.39385e7	1.72693	0.863466	−	0.504408i	$-0.168289\pi$
$719$	2.39385e7	1.72693	0.863466	+	0.504408i	$0.168289\pi$
$720$	0	0
$721$	−3.91907e6	−0.280766
$722$	0	0
$723$	9.75427e6i	0.693983i
$724$	0	0
$725$	2.10610e6i	0.148811i
$726$	0	0
$727$	1.81741e7	1.27531	0.637656	−	0.770321i	$-0.279904\pi$
$727$	1.81741e7	1.27531	0.637656	+	0.770321i	$0.279904\pi$
$728$	0	0
$729$	−1.37482e7	−0.958135
$730$	0	0
$731$	− 408786.i	− 0.0282945i
$732$	0	0
$733$	1.39671e7i	0.960169i	0.877222	+	0.480085i	$0.159394\pi$
$733$	1.39671e7i	0.960169i	−0.877222	+	0.480085i	$0.840606\pi$
$734$	0	0
$735$	−1.06649e7	−0.728176
$736$	0	0
$737$	1.88045e7	1.27525
$738$	0	0
$739$	− 1.00299e7i	− 0.675597i	−0.941219	−	0.337798i	$-0.890318\pi$
$739$	− 1.00299e7i	− 0.675597i	0.941219	−	0.337798i	$-0.109682\pi$
$740$	0	0
$741$	− 2.34332e6i	− 0.156779i
$742$	0	0
$743$	8.29176e6	0.551030	0.275515	−	0.961297i	$-0.411152\pi$
$743$	8.29176e6	0.551030	0.275515	+	0.961297i	$0.411152\pi$
$744$	0	0
$745$	2.19605e6	0.144961
$746$	0	0
$747$	− 1.11123e6i	− 0.0728625i
$748$	0	0
$749$	− 1.60918e7i	− 1.04809i
$750$	0	0
$751$	−1.07708e7	−0.696865	−0.348432	−	0.937334i	$-0.613286\pi$
$751$	−1.07708e7	−0.696865	−0.348432	+	0.937334i	$0.613286\pi$
$752$	0	0
$753$	1.44902e7	0.931292
$754$	0	0
$755$	− 4.12772e6i	− 0.263538i
$756$	0	0
$757$	2.72641e7i	1.72923i	0.502437	+	0.864614i	$0.332437\pi$
$757$	2.72641e7i	1.72923i	−0.502437	+	0.864614i	$0.667563\pi$
$758$	0	0
$759$	2.48702e7	1.56702
$760$	0	0
$761$	2.70646e7	1.69410	0.847052	−	0.531510i	$-0.178375\pi$
$761$	2.70646e7	1.69410	0.847052	+	0.531510i	$0.178375\pi$
$762$	0	0
$763$	5.31705e7i	3.30643i
$764$	0	0
$765$	− 1.19941e6i	− 0.0740994i
$766$	0	0
$767$	1.81289e6	0.111272
$768$	0	0
$769$	−1.10424e7	−0.673361	−0.336681	−	0.941619i	$-0.609304\pi$
$769$	−1.10424e7	−0.673361	−0.336681	+	0.941619i	$0.609304\pi$
$770$	0	0
$771$	− 8.88302e6i	− 0.538176i
$772$	0	0
$773$	1.78160e6i	0.107241i	0.998561	+	0.0536207i	$0.0170762\pi$
$773$	1.78160e6i	0.107241i	−0.998561	+	0.0536207i	$0.982924\pi$
$774$	0	0
$775$	−4.97243e6	−0.297382
$776$	0	0
$777$	−2.86860e7	−1.70458
$778$	0	0
$779$	1.86656e7i	1.10204i
$780$	0	0
$781$	− 4.96101e6i	− 0.291033i
$782$	0	0
$783$	1.37355e7	0.800645
$784$	0	0
$785$	4.77907e6	0.276802
$786$	0	0
$787$	2.97053e6i	0.170961i	0.996340	+	0.0854806i	$0.0272426\pi$
$787$	2.97053e6i	0.170961i	−0.996340	+	0.0854806i	$0.972757\pi$
$788$	0	0
$789$	− 1.57109e7i	− 0.898481i
$790$	0	0
$791$	1.39023e7	0.790032
$792$	0	0
$793$	−1.21902e6	−0.0688379
$794$	0	0
$795$	− 2.30476e6i	− 0.129333i
$796$	0	0
$797$	9.06646e6i	0.505583i	0.967521	+	0.252791i	$0.0813486\pi$
$797$	9.06646e6i	0.505583i	−0.967521	+	0.252791i	$0.918651\pi$
$798$	0	0
$799$	3.56206e6	0.197394
$800$	0	0
$801$	−1.00418e7	−0.553008
$802$	0	0
$803$	− 3.10461e7i	− 1.69910i
$804$	0	0
$805$	2.22007e7i	1.20747i
$806$	0	0
$807$	2.00627e6	0.108444
$808$	0	0
$809$	−1.49840e7	−0.804926	−0.402463	−	0.915436i	$-0.631846\pi$
$809$	−1.49840e7	−0.804926	−0.402463	+	0.915436i	$0.631846\pi$
$810$	0	0
$811$	− 1.54716e7i	− 0.826005i	−0.910730	−	0.413003i	$-0.864480\pi$
$811$	− 1.54716e7i	− 0.826005i	0.910730	−	0.413003i	$-0.135520\pi$
$812$	0	0
$813$	− 9.30348e6i	− 0.493650i
$814$	0	0
$815$	1.19601e7	0.630725
$816$	0	0
$817$	1.73360e6	0.0908642
$818$	0	0
$819$	− 2.71334e6i	− 0.141349i
$820$	0	0
$821$	1.28381e7i	0.664726i	0.943152	+	0.332363i	$0.107846\pi$
$821$	1.28381e7i	0.664726i	−0.943152	+	0.332363i	$0.892154\pi$
$822$	0	0
$823$	−1.52419e7	−0.784402	−0.392201	−	0.919880i	$-0.628286\pi$
$823$	−1.52419e7	−0.784402	−0.392201	+	0.919880i	$0.628286\pi$
$824$	0	0
$825$	−4.05263e6	−0.207301
$826$	0	0
$827$	3.93361e6i	0.199999i	0.994987	+	0.0999994i	$0.0318841\pi$
$827$	3.93361e6i	0.199999i	−0.994987	+	0.0999994i	$0.968116\pi$
$828$	0	0
$829$	− 4.75338e6i	− 0.240224i	−0.992760	−	0.120112i	$-0.961675\pi$
$829$	− 4.75338e6i	− 0.240224i	0.992760	−	0.120112i	$-0.0383253\pi$
$830$	0	0
$831$	−1.17090e7	−0.588188
$832$	0	0
$833$	−1.62552e7	−0.811670
$834$	0	0
$835$	7.68860e6i	0.381620i
$836$	0	0
$837$	3.24290e7i	1.60000i
$838$	0	0
$839$	2.71998e6	0.133401	0.0667007	−	0.997773i	$-0.478753\pi$
$839$	2.71998e6	0.133401	0.0667007	+	0.997773i	$0.478753\pi$
$840$	0	0
$841$	9.15584e6	0.446383
$842$	0	0
$843$	− 2.60233e7i	− 1.26123i
$844$	0	0
$845$	8.99125e6i	0.433190i
$846$	0	0
$847$	−3.51472e7	−1.68338
$848$	0	0
$849$	1.98785e7	0.946488
$850$	0	0
$851$	4.09923e7i	1.94034i
$852$	0	0
$853$	− 2.68253e7i	− 1.26233i	−0.775649	−	0.631164i	$-0.782577\pi$
$853$	− 2.68253e7i	− 1.26233i	0.775649	−	0.631164i	$-0.217423\pi$
$854$	0	0
$855$	5.08652e6	0.237961
$856$	0	0
$857$	8.98268e6	0.417786	0.208893	−	0.977938i	$-0.433014\pi$
$857$	8.98268e6	0.417786	0.208893	+	0.977938i	$0.433014\pi$
$858$	0	0
$859$	2.14669e7i	0.992629i	0.868143	+	0.496315i	$0.165314\pi$
$859$	2.14669e7i	0.992629i	−0.868143	+	0.496315i	$0.834686\pi$
$860$	0	0
$861$	− 2.67435e7i	− 1.22945i
$862$	0	0
$863$	2.02164e7	0.924013	0.462006	−	0.886877i	$-0.347130\pi$
$863$	2.02164e7	0.924013	0.462006	+	0.886877i	$0.347130\pi$
$864$	0	0
$865$	−1.38358e6	−0.0628731
$866$	0	0
$867$	− 1.41979e7i	− 0.641470i
$868$	0	0
$869$	3.88256e7i	1.74409i
$870$	0	0
$871$	3.62764e6	0.162024
$872$	0	0
$873$	−9.62906e6	−0.427610
$874$	0	0
$875$	− 3.61763e6i	− 0.159737i
$876$	0	0
$877$	− 8.97881e6i	− 0.394203i	−0.980383	−	0.197101i	$-0.936847\pi$
$877$	− 8.97881e6i	− 0.394203i	0.980383	−	0.197101i	$-0.0631528\pi$
$878$	0	0
$879$	2.63667e7	1.15102
$880$	0	0
$881$	−2.96009e7	−1.28489	−0.642444	−	0.766333i	$-0.722080\pi$
$881$	−2.96009e7	−1.28489	−0.642444	+	0.766333i	$0.722080\pi$
$882$	0	0
$883$	− 2.24867e7i	− 0.970565i	−0.874357	−	0.485282i	$-0.838717\pi$
$883$	− 2.24867e7i	− 0.970565i	0.874357	−	0.485282i	$-0.161283\pi$
$884$	0	0
$885$	− 4.86927e6i	− 0.208981i
$886$	0	0
$887$	−1.24041e7	−0.529365	−0.264682	−	0.964336i	$-0.585267\pi$
$887$	−1.24041e7	−0.529365	−0.264682	+	0.964336i	$0.585267\pi$
$888$	0	0
$889$	1.06076e7	0.450155
$890$	0	0
$891$	1.16683e7i	0.492396i
$892$	0	0
$893$	1.51061e7i	0.633905i
$894$	0	0
$895$	−1.75916e7	−0.734088
$896$	0	0
$897$	4.79778e6	0.199094
$898$	0	0
$899$	− 2.68094e7i	− 1.10634i
$900$	0	0
$901$	− 3.51288e6i	− 0.144162i
$902$	0	0
$903$	−2.48384e6	−0.101369
$904$	0	0
$905$	2.40639e6	0.0976661
$906$	0	0
$907$	1.05997e7i	0.427834i	0.976852	+	0.213917i	$0.0686222\pi$
$907$	1.05997e7i	0.427834i	−0.976852	+	0.213917i	$0.931378\pi$
$908$	0	0
$909$	− 2.18089e6i	− 0.0875437i
$910$	0	0
$911$	4.24265e7	1.69372	0.846858	−	0.531818i	$-0.178491\pi$
$911$	4.24265e7	1.69372	0.846858	+	0.531818i	$0.178491\pi$
$912$	0	0
$913$	−5.72284e6	−0.227214
$914$	0	0
$915$	3.27418e6i	0.129285i
$916$	0	0
$917$	− 3.13222e7i	− 1.23007i
$918$	0	0
$919$	−4.18031e7	−1.63275	−0.816376	−	0.577521i	$-0.804020\pi$
$919$	−4.18031e7	−1.63275	−0.816376	+	0.577521i	$0.804020\pi$
$920$	0	0
$921$	−2.73764e7	−1.06347
$922$	0	0
$923$	− 957041.i	− 0.0369766i
$924$	0	0
$925$	− 6.67975e6i	− 0.256688i
$926$	0	0
$927$	−1.83842e6	−0.0702660
$928$	0	0
$929$	−1.67378e7	−0.636295	−0.318147	−	0.948041i	$-0.603061\pi$
$929$	−1.67378e7	−0.636295	−0.318147	+	0.948041i	$0.603061\pi$
$930$	0	0
$931$	− 6.89357e7i	− 2.60657i
$932$	0	0
$933$	− 9.24076e6i	− 0.347539i
$934$	0	0
$935$	−6.17696e6	−0.231071
$936$	0	0
$937$	4.81906e7	1.79314	0.896569	−	0.442904i	$-0.146052\pi$
$937$	4.81906e7	1.79314	0.896569	+	0.442904i	$0.146052\pi$
$938$	0	0
$939$	2.24413e7i	0.830584i
$940$	0	0
$941$	− 2.29054e7i	− 0.843263i	−0.906767	−	0.421632i	$-0.861458\pi$
$941$	− 2.29054e7i	− 0.843263i	0.906767	−	0.421632i	$-0.138542\pi$
$942$	0	0
$943$	−3.82165e7	−1.39949
$944$	0	0
$945$	−2.35934e7	−0.859429
$946$	0	0
$947$	− 1.56151e7i	− 0.565810i	−0.959148	−	0.282905i	$-0.908702\pi$
$947$	− 1.56151e7i	− 0.565810i	0.959148	−	0.282905i	$-0.0912981\pi$
$948$	0	0
$949$	− 5.98918e6i	− 0.215875i
$950$	0	0
$951$	−2.72123e7	−0.975695
$952$	0	0
$953$	1.10672e7	0.394733	0.197367	−	0.980330i	$-0.436761\pi$
$953$	1.10672e7	0.394733	0.197367	+	0.980330i	$0.436761\pi$
$954$	0	0
$955$	1.26194e7i	0.447745i
$956$	0	0
$957$	− 2.18503e7i	− 0.771218i
$958$	0	0
$959$	−7.99499e7	−2.80719
$960$	0	0
$961$	3.46669e7	1.21090
$962$	0	0
$963$	− 7.54858e6i	− 0.262301i
$964$	0	0
$965$	5.82846e6i	0.201482i
$966$	0	0
$967$	1.07940e7	0.371206	0.185603	−	0.982625i	$-0.440576\pi$
$967$	1.07940e7	0.371206	0.185603	+	0.982625i	$0.440576\pi$
$968$	0	0
$969$	−9.59315e6	−0.328210
$970$	0	0
$971$	3.92390e7i	1.33558i	0.744349	+	0.667790i	$0.232760\pi$
$971$	3.92390e7i	1.33558i	−0.744349	+	0.667790i	$0.767240\pi$
$972$	0	0
$973$	− 2.70531e7i	− 0.916084i
$974$	0	0
$975$	−781804.	−0.0263382
$976$	0	0
$977$	−2.54755e7	−0.853860	−0.426930	−	0.904285i	$-0.640405\pi$
$977$	−2.54755e7	−0.853860	−0.426930	+	0.904285i	$0.640405\pi$
$978$	0	0
$979$	5.17153e7i	1.72450i
$980$	0	0
$981$	2.49421e7i	0.827485i
$982$	0	0
$983$	−1.56587e7	−0.516858	−0.258429	−	0.966030i	$-0.583205\pi$
$983$	−1.56587e7	−0.516858	−0.258429	+	0.966030i	$0.583205\pi$
$984$	0	0
$985$	−8.05178e6	−0.264424
$986$	0	0
$987$	− 2.16435e7i	− 0.707189i
$988$	0	0
$989$	3.54941e6i	0.115389i
$990$	0	0
$991$	9.68181e6	0.313164	0.156582	−	0.987665i	$-0.449952\pi$
$991$	9.68181e6	0.313164	0.156582	+	0.987665i	$0.449952\pi$
$992$	0	0
$993$	3.37972e6	0.108770
$994$	0	0
$995$	− 1.14295e7i	− 0.365989i
$996$	0	0
$997$	− 3.15560e7i	− 1.00541i	−0.864457	−	0.502707i	$-0.832338\pi$
$997$	− 3.15560e7i	− 1.00541i	0.864457	−	0.502707i	$-0.167662\pi$
$998$	0	0
$999$	−4.35637e7	−1.38106

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.d.a.81.15		20
4.3	odd	2		40.6.d.a.21.7	✓	20
5.2	odd	4		800.6.f.c.49.16		20
5.3	odd	4		800.6.f.b.49.5		20
5.4	even	2		800.6.d.c.401.6		20
8.3	odd	2		40.6.d.a.21.8	yes	20
8.5	even	2	inner	160.6.d.a.81.6		20
12.11	even	2		360.6.k.b.181.14		20
20.3	even	4		200.6.f.c.149.4		20
20.7	even	4		200.6.f.b.149.17		20
20.19	odd	2		200.6.d.b.101.14		20
24.11	even	2		360.6.k.b.181.13		20
40.3	even	4		200.6.f.b.149.18		20
40.13	odd	4		800.6.f.c.49.15		20
40.19	odd	2		200.6.d.b.101.13		20
40.27	even	4		200.6.f.c.149.3		20
40.29	even	2		800.6.d.c.401.15		20
40.37	odd	4		800.6.f.b.49.6		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.7	✓	20	4.3	odd	2
40.6.d.a.21.8	yes	20	8.3	odd	2
160.6.d.a.81.6		20	8.5	even	2	inner
160.6.d.a.81.15		20	1.1	even	1	trivial
200.6.d.b.101.13		20	40.19	odd	2
200.6.d.b.101.14		20	20.19	odd	2
200.6.f.b.149.17		20	20.7	even	4
200.6.f.b.149.18		20	40.3	even	4
200.6.f.c.149.3		20	40.27	even	4
200.6.f.c.149.4		20	20.3	even	4
360.6.k.b.181.13		20	24.11	even	2
360.6.k.b.181.14		20	12.11	even	2
800.6.d.c.401.6		20	5.4	even	2
800.6.d.c.401.15		20	40.29	even	2
800.6.f.b.49.5		20	5.3	odd	4
800.6.f.b.49.6		20	40.37	odd	4
800.6.f.c.49.15		20	40.13	odd	4
800.6.f.c.49.16		20	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.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+11.5927i q^{3} +25.0000i q^{5} +231.529 q^{7} +108.609 q^{9} +O(q^{10})\)	\(q+11.5927i q^{3} +25.0000i q^{5} +231.529 q^{7} +108.609 q^{9} -559.335i q^{11} -107.903i q^{13} -289.818 q^{15} -441.735 q^{17} -1873.33i q^{19} +2684.04i q^{21} +3835.50 q^{23} -625.000 q^{25} +4076.10i q^{27} -3369.76i q^{29} +7955.88 q^{31} +6484.21 q^{33} +5788.21i q^{35} +10687.6i q^{37} +1250.89 q^{39} -9963.87 q^{41} +925.409i q^{43} +2715.23i q^{45} -8063.78 q^{47} +36798.5 q^{49} -5120.91i q^{51} +7952.45i q^{53} +13983.4 q^{55} +21717.0 q^{57} +16801.2i q^{59} -11297.4i q^{61} +25146.1 q^{63} +2697.57 q^{65} +33619.4i q^{67} +44463.9i q^{69} +8869.47 q^{71} +55505.3 q^{73} -7245.44i q^{75} -129502. i q^{77} -69413.7 q^{79} -20861.0 q^{81} -10231.5i q^{83} -11043.4i q^{85} +39064.7 q^{87} -92458.4 q^{89} -24982.6i q^{91} +92230.2i q^{93} +46833.2 q^{95} -88657.9 q^{97} -60748.9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) \) 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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