LMFDB - Embedded newform 400.10.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,10,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$206.014334466$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 652x + 4000 $ x^3 - x^2 - 652*x + 4000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-27.7229$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+268.664 q^{3} -637.237 q^{7} +52497.3 q^{9} +O(q^{10})$	$q+268.664 q^{3} -637.237 q^{7} +52497.3 q^{9} +49042.6 q^{11} +72726.2 q^{13} +67319.3 q^{17} -341136. q^{19} -171203. q^{21} +134355. q^{23} +8.81603e6 q^{27} +4.45784e6 q^{29} -456520. q^{31} +1.31760e7 q^{33} +1.30516e7 q^{37} +1.95389e7 q^{39} -2.56667e7 q^{41} -3.42710e6 q^{43} +3.39814e7 q^{47} -3.99475e7 q^{49} +1.80863e7 q^{51} -8.42456e7 q^{53} -9.16509e7 q^{57} +7.46358e7 q^{59} +1.78017e8 q^{61} -3.34533e7 q^{63} -6.94299e7 q^{67} +3.60963e7 q^{69} +2.07860e8 q^{71} -3.02516e8 q^{73} -3.12518e7 q^{77} -3.72244e8 q^{79} +1.33524e9 q^{81} -4.50079e8 q^{83} +1.19766e9 q^{87} +5.82741e7 q^{89} -4.63438e7 q^{91} -1.22650e8 q^{93} +7.85850e8 q^{97} +2.57461e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9}+O(q^{10}) $ 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9	$ 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9} + 54699 q^{11} - 215884 q^{13} - 334983 q^{17} - 818845 q^{19} - 2375394 q^{21} + 3526854 q^{23} + 6633395 q^{27} + 2175480 q^{29} - 4274066 q^{31} + 22122137 q^{33} + 10305042 q^{37} + 50414092 q^{39} + 5926311 q^{41} - 24429956 q^{43} + 66858708 q^{47} - 6453929 q^{49} + 25634699 q^{51} - 132620514 q^{53} - 252946415 q^{57} - 5670960 q^{59} + 125306926 q^{61} + 284323404 q^{63} + 88829483 q^{67} - 314274942 q^{69} - 297550596 q^{71} - 181321729 q^{73} - 561214086 q^{77} + 310025170 q^{79} + 1398847363 q^{81} - 731088801 q^{83} + 1046385560 q^{87} - 1103860035 q^{89} - 1183187656 q^{91} - 107386758 q^{93} + 332236842 q^{97} + 892234522 q^{99}+O(q^{100}) $ 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9 + 54699 * q^11 - 215884 * q^13 - 334983 * q^17 - 818845 * q^19 - 2375394 * q^21 + 3526854 * q^23 + 6633395 * q^27 + 2175480 * q^29 - 4274066 * q^31 + 22122137 * q^33 + 10305042 * q^37 + 50414092 * q^39 + 5926311 * q^41 - 24429956 * q^43 + 66858708 * q^47 - 6453929 * q^49 + 25634699 * q^51 - 132620514 * q^53 - 252946415 * q^57 - 5670960 * q^59 + 125306926 * q^61 + 284323404 * q^63 + 88829483 * q^67 - 314274942 * q^69 - 297550596 * q^71 - 181321729 * q^73 - 561214086 * q^77 + 310025170 * q^79 + 1398847363 * q^81 - 731088801 * q^83 + 1046385560 * q^87 - 1103860035 * q^89 - 1183187656 * q^91 - 107386758 * q^93 + 332236842 * q^97 + 892234522 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	268.664	1.91498	0.957489	−	0.288470i	$-0.0931465\pi$
$3$	268.664	1.91498	0.957489	+	0.288470i	$0.0931465\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−637.237	−0.100314	−0.0501568	−	0.998741i	$-0.515972\pi$
$7$	−637.237	−0.100314	−0.0501568	+	0.998741i	$0.515972\pi$
$8$	0	0
$9$	52497.3	2.66714
$10$	0	0
$11$	49042.6	1.00996	0.504982	−	0.863130i	$-0.331499\pi$
$11$	49042.6	1.00996	0.504982	+	0.863130i	$0.331499\pi$
$12$	0	0
$13$	72726.2	0.706229	0.353115	−	0.935580i	$-0.385123\pi$
$13$	72726.2	0.706229	0.353115	+	0.935580i	$0.385123\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	67319.3	0.195488	0.0977439	−	0.995212i	$-0.468837\pi$
$17$	67319.3	0.195488	0.0977439	+	0.995212i	$0.468837\pi$
$18$	0	0
$19$	−341136.	−0.600532	−0.300266	−	0.953855i	$-0.597075\pi$
$19$	−341136.	−0.600532	−0.300266	+	0.953855i	$0.597075\pi$
$20$	0	0
$21$	−171203.	−0.192098
$22$	0	0
$23$	134355.	0.100110	0.0500550	−	0.998746i	$-0.484060\pi$
$23$	134355.	0.100110	0.0500550	+	0.998746i	$0.484060\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	8.81603e6	3.19254
$28$	0	0
$29$	4.45784e6	1.17040	0.585199	−	0.810890i	$-0.301016\pi$
$29$	4.45784e6	1.17040	0.585199	+	0.810890i	$0.301016\pi$
$30$	0	0
$31$	−456520.	−0.0887834	−0.0443917	−	0.999014i	$-0.514135\pi$
$31$	−456520.	−0.0887834	−0.0443917	+	0.999014i	$0.514135\pi$
$32$	0	0
$33$	1.31760e7	1.93406
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.30516e7	1.14487	0.572433	−	0.819951i	$-0.305999\pi$
$37$	1.30516e7	1.14487	0.572433	+	0.819951i	$0.305999\pi$
$38$	0	0
$39$	1.95389e7	1.35241
$40$	0	0
$41$	−2.56667e7	−1.41854	−0.709272	−	0.704935i	$-0.750976\pi$
$41$	−2.56667e7	−1.41854	−0.709272	+	0.704935i	$0.750976\pi$
$42$	0	0
$43$	−3.42710e6	−0.152869	−0.0764344	−	0.997075i	$-0.524354\pi$
$43$	−3.42710e6	−0.152869	−0.0764344	+	0.997075i	$0.524354\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.39814e7	1.01578	0.507891	−	0.861421i	$-0.330425\pi$
$47$	3.39814e7	1.01578	0.507891	+	0.861421i	$0.330425\pi$
$48$	0	0
$49$	−3.99475e7	−0.989937
$50$	0	0
$51$	1.80863e7	0.374355
$52$	0	0
$53$	−8.42456e7	−1.46658	−0.733290	−	0.679916i	$-0.762016\pi$
$53$	−8.42456e7	−1.46658	−0.733290	+	0.679916i	$0.762016\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.16509e7	−1.15001
$58$	0	0
$59$	7.46358e7	0.801887	0.400944	−	0.916103i	$-0.368682\pi$
$59$	7.46358e7	0.801887	0.400944	+	0.916103i	$0.368682\pi$
$60$	0	0
$61$	1.78017e8	1.64618	0.823088	−	0.567913i	$-0.192249\pi$
$61$	1.78017e8	1.64618	0.823088	+	0.567913i	$0.192249\pi$
$62$	0	0
$63$	−3.34533e7	−0.267551
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.94299e7	−0.420930	−0.210465	−	0.977601i	$-0.567498\pi$
$67$	−6.94299e7	−0.420930	−0.210465	+	0.977601i	$0.567498\pi$
$68$	0	0
$69$	3.60963e7	0.191708
$70$	0	0
$71$	2.07860e8	0.970753	0.485376	−	0.874305i	$-0.338683\pi$
$71$	2.07860e8	0.970753	0.485376	+	0.874305i	$0.338683\pi$
$72$	0	0
$73$	−3.02516e8	−1.24680	−0.623398	−	0.781905i	$-0.714248\pi$
$73$	−3.02516e8	−1.24680	−0.623398	+	0.781905i	$0.714248\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.12518e7	−0.101313
$78$	0	0
$79$	−3.72244e8	−1.07524	−0.537621	−	0.843187i	$-0.680677\pi$
$79$	−3.72244e8	−1.07524	−0.537621	+	0.843187i	$0.680677\pi$
$80$	0	0
$81$	1.33524e9	3.44650
$82$	0	0
$83$	−4.50079e8	−1.04097	−0.520484	−	0.853872i	$-0.674248\pi$
$83$	−4.50079e8	−1.04097	−0.520484	+	0.853872i	$0.674248\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.19766e9	2.24129
$88$	0	0
$89$	5.82741e7	0.0984511	0.0492255	−	0.998788i	$-0.484325\pi$
$89$	5.82741e7	0.0984511	0.0492255	+	0.998788i	$0.484325\pi$
$90$	0	0
$91$	−4.63438e7	−0.0708444
$92$	0	0
$93$	−1.22650e8	−0.170018
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.85850e8	0.901295	0.450647	−	0.892702i	$-0.351193\pi$
$97$	7.85850e8	0.901295	0.450647	+	0.892702i	$0.351193\pi$
$98$	0	0
$99$	2.57461e9	2.69372
$100$	0	0
$101$	−1.60460e8	−0.153433	−0.0767167	−	0.997053i	$-0.524444\pi$
$101$	−1.60460e8	−0.153433	−0.0767167	+	0.997053i	$0.524444\pi$
$102$	0	0
$103$	1.39454e9	1.22085	0.610425	−	0.792074i	$-0.290999\pi$
$103$	1.39454e9	1.22085	0.610425	+	0.792074i	$0.290999\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.56463e9	1.15394	0.576972	−	0.816764i	$-0.304234\pi$
$107$	1.56463e9	1.15394	0.576972	+	0.816764i	$0.304234\pi$
$108$	0	0
$109$	1.24703e9	0.846172	0.423086	−	0.906089i	$-0.360947\pi$
$109$	1.24703e9	0.846172	0.423086	+	0.906089i	$0.360947\pi$
$110$	0	0
$111$	3.50649e9	2.19240
$112$	0	0
$113$	−1.81056e9	−1.04463	−0.522313	−	0.852754i	$-0.674931\pi$
$113$	−1.81056e9	−1.04463	−0.522313	+	0.852754i	$0.674931\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.81793e9	1.88361
$118$	0	0
$119$	−4.28984e7	−0.0196101
$120$	0	0
$121$	4.72275e7	0.0200291
$122$	0	0
$123$	−6.89572e9	−2.71648
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.06491e9	1.04545	0.522723	−	0.852503i	$-0.324916\pi$
$127$	3.06491e9	1.04545	0.522723	+	0.852503i	$0.324916\pi$
$128$	0	0
$129$	−9.20739e8	−0.292740
$130$	0	0
$131$	1.83508e9	0.544421	0.272211	−	0.962238i	$-0.412245\pi$
$131$	1.83508e9	0.544421	0.272211	+	0.962238i	$0.412245\pi$
$132$	0	0
$133$	2.17385e8	0.0602416
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.62426e9	−1.12150	−0.560751	−	0.827984i	$-0.689488\pi$
$137$	−4.62426e9	−1.12150	−0.560751	+	0.827984i	$0.689488\pi$
$138$	0	0
$139$	−2.57140e8	−0.0584255	−0.0292128	−	0.999573i	$-0.509300\pi$
$139$	−2.57140e8	−0.0584255	−0.0292128	+	0.999573i	$0.509300\pi$
$140$	0	0
$141$	9.12958e9	1.94520
$142$	0	0
$143$	3.56668e9	0.713267
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.07325e10	−1.89571
$148$	0	0
$149$	3.14809e9	0.523250	0.261625	−	0.965170i	$-0.415742\pi$
$149$	3.14809e9	0.523250	0.261625	+	0.965170i	$0.415742\pi$
$150$	0	0
$151$	1.02772e10	1.60871	0.804356	−	0.594147i	$-0.202510\pi$
$151$	1.02772e10	1.60871	0.804356	+	0.594147i	$0.202510\pi$
$152$	0	0
$153$	3.53408e9	0.521393
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.25077e9	0.427009	0.213505	−	0.976942i	$-0.431512\pi$
$157$	3.25077e9	0.427009	0.213505	+	0.976942i	$0.431512\pi$
$158$	0	0
$159$	−2.26338e10	−2.80847
$160$	0	0
$161$	−8.56158e7	−0.0100424
$162$	0	0
$163$	−3.40049e9	−0.377309	−0.188655	−	0.982043i	$-0.560413\pi$
$163$	−3.40049e9	−0.377309	−0.188655	+	0.982043i	$0.560413\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.27785e9	−0.525089	−0.262544	−	0.964920i	$-0.584562\pi$
$167$	−5.27785e9	−0.525089	−0.262544	+	0.964920i	$0.584562\pi$
$168$	0	0
$169$	−5.31540e9	−0.501240
$170$	0	0
$171$	−1.79087e10	−1.60170
$172$	0	0
$173$	6.58861e9	0.559224	0.279612	−	0.960113i	$-0.409794\pi$
$173$	6.58861e9	0.559224	0.279612	+	0.960113i	$0.409794\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.00520e10	1.53560
$178$	0	0
$179$	1.23663e10	0.900326	0.450163	−	0.892946i	$-0.351366\pi$
$179$	1.23663e10	0.900326	0.450163	+	0.892946i	$0.351366\pi$
$180$	0	0
$181$	−2.59914e10	−1.80002	−0.900009	−	0.435872i	$-0.856440\pi$
$181$	−2.59914e10	−1.80002	−0.900009	+	0.435872i	$0.856440\pi$
$182$	0	0
$183$	4.78267e10	3.15239
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.30151e9	0.197436
$188$	0	0
$189$	−5.61790e9	−0.320255
$190$	0	0
$191$	−1.50506e10	−0.818284	−0.409142	−	0.912471i	$-0.634172\pi$
$191$	−1.50506e10	−0.818284	−0.409142	+	0.912471i	$0.634172\pi$
$192$	0	0
$193$	1.40329e10	0.728014	0.364007	−	0.931396i	$-0.381408\pi$
$193$	1.40329e10	0.728014	0.364007	+	0.931396i	$0.381408\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.57285e10	−1.21707	−0.608536	−	0.793526i	$-0.708243\pi$
$197$	−2.57285e10	−1.21707	−0.608536	+	0.793526i	$0.708243\pi$
$198$	0	0
$199$	2.94367e10	1.33061	0.665303	−	0.746573i	$-0.268302\pi$
$199$	2.94367e10	1.33061	0.665303	+	0.746573i	$0.268302\pi$
$200$	0	0
$201$	−1.86533e10	−0.806072
$202$	0	0
$203$	−2.84070e9	−0.117407
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.05326e9	0.267007
$208$	0	0
$209$	−1.67302e10	−0.606516
$210$	0	0
$211$	−1.17275e10	−0.407319	−0.203659	−	0.979042i	$-0.565284\pi$
$211$	−1.17275e10	−0.407319	−0.203659	+	0.979042i	$0.565284\pi$
$212$	0	0
$213$	5.58445e10	1.85897
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.90911e8	0.00890619
$218$	0	0
$219$	−8.12751e10	−2.38759
$220$	0	0
$221$	4.89588e9	0.138059
$222$	0	0
$223$	−2.58340e10	−0.699550	−0.349775	−	0.936834i	$-0.613742\pi$
$223$	−2.58340e10	−0.699550	−0.349775	+	0.936834i	$0.613742\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.50896e10	0.627159	0.313580	−	0.949562i	$-0.398472\pi$
$227$	2.50896e10	0.627159	0.313580	+	0.949562i	$0.398472\pi$
$228$	0	0
$229$	−2.30463e10	−0.553785	−0.276892	−	0.960901i	$-0.589305\pi$
$229$	−2.30463e10	−0.553785	−0.276892	+	0.960901i	$0.589305\pi$
$230$	0	0
$231$	−8.39622e9	−0.194013
$232$	0	0
$233$	3.17197e10	0.705062	0.352531	−	0.935800i	$-0.385321\pi$
$233$	3.17197e10	0.705062	0.352531	+	0.935800i	$0.385321\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1.00009e11	−2.05906
$238$	0	0
$239$	−2.23794e10	−0.443668	−0.221834	−	0.975084i	$-0.571204\pi$
$239$	−2.23794e10	−0.443668	−0.221834	+	0.975084i	$0.571204\pi$
$240$	0	0
$241$	−2.10823e10	−0.402569	−0.201285	−	0.979533i	$-0.564512\pi$
$241$	−2.10823e10	−0.402569	−0.201285	+	0.979533i	$0.564512\pi$
$242$	0	0
$243$	1.85206e11	3.40743
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.48095e10	−0.424113
$248$	0	0
$249$	−1.20920e11	−1.99343
$250$	0	0
$251$	7.96509e10	1.26666	0.633329	−	0.773883i	$-0.281688\pi$
$251$	7.96509e10	1.26666	0.633329	+	0.773883i	$0.281688\pi$
$252$	0	0
$253$	6.58910e9	0.101108
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.30815e10	−0.473027	−0.236514	−	0.971628i	$-0.576005\pi$
$257$	−3.30815e10	−0.473027	−0.236514	+	0.971628i	$0.576005\pi$
$258$	0	0
$259$	−8.31695e9	−0.114846
$260$	0	0
$261$	2.34025e11	3.12162
$262$	0	0
$263$	−6.05028e10	−0.779784	−0.389892	−	0.920861i	$-0.627488\pi$
$263$	−6.05028e10	−0.779784	−0.389892	+	0.920861i	$0.627488\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.56562e10	0.188532
$268$	0	0
$269$	2.61933e10	0.305004	0.152502	−	0.988303i	$-0.451267\pi$
$269$	2.61933e10	0.305004	0.152502	+	0.988303i	$0.451267\pi$
$270$	0	0
$271$	−5.42857e10	−0.611398	−0.305699	−	0.952128i	$-0.598890\pi$
$271$	−5.42857e10	−0.611398	−0.305699	+	0.952128i	$0.598890\pi$
$272$	0	0
$273$	−1.24509e10	−0.135666
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.76561e11	1.80192	0.900961	−	0.433900i	$-0.142863\pi$
$277$	1.76561e11	1.80192	0.900961	+	0.433900i	$0.142863\pi$
$278$	0	0
$279$	−2.39661e10	−0.236798
$280$	0	0
$281$	−7.91126e9	−0.0756950	−0.0378475	−	0.999284i	$-0.512050\pi$
$281$	−7.91126e9	−0.0756950	−0.0378475	+	0.999284i	$0.512050\pi$
$282$	0	0
$283$	1.34806e11	1.24931	0.624657	−	0.780899i	$-0.285239\pi$
$283$	1.34806e11	1.24931	0.624657	+	0.780899i	$0.285239\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.63558e10	0.142299
$288$	0	0
$289$	−1.14056e11	−0.961785
$290$	0	0
$291$	2.11130e11	1.72596
$292$	0	0
$293$	1.02784e11	0.814741	0.407370	−	0.913263i	$-0.366446\pi$
$293$	1.02784e11	0.814741	0.407370	+	0.913263i	$0.366446\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.32361e11	3.22435
$298$	0	0
$299$	9.77110e9	0.0707006
$300$	0	0
$301$	2.18388e9	0.0153348
$302$	0	0
$303$	−4.31098e10	−0.293822
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.61472e11	1.67998	0.839988	−	0.542606i	$-0.182562\pi$
$307$	2.61472e11	1.67998	0.839988	+	0.542606i	$0.182562\pi$
$308$	0	0
$309$	3.74662e11	2.33790
$310$	0	0
$311$	8.24828e10	0.499967	0.249984	−	0.968250i	$-0.419575\pi$
$311$	8.24828e10	0.499967	0.249984	+	0.968250i	$0.419575\pi$
$312$	0	0
$313$	−1.35766e11	−0.799540	−0.399770	−	0.916615i	$-0.630910\pi$
$313$	−1.35766e11	−0.799540	−0.399770	+	0.916615i	$0.630910\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.06853e10	−0.504394	−0.252197	−	0.967676i	$-0.581153\pi$
$317$	−9.06853e10	−0.504394	−0.252197	+	0.967676i	$0.581153\pi$
$318$	0	0
$319$	2.18624e11	1.18206
$320$	0	0
$321$	4.20360e11	2.20978
$322$	0	0
$323$	−2.29650e10	−0.117397
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.35033e11	1.62040
$328$	0	0
$329$	−2.16542e10	−0.101897
$330$	0	0
$331$	3.45169e11	1.58054	0.790271	−	0.612758i	$-0.209940\pi$
$331$	3.45169e11	1.58054	0.790271	+	0.612758i	$0.209940\pi$
$332$	0	0
$333$	6.85173e11	3.05352
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.39418e10	0.354523	0.177261	−	0.984164i	$-0.443276\pi$
$337$	8.39418e10	0.354523	0.177261	+	0.984164i	$0.443276\pi$
$338$	0	0
$339$	−4.86433e11	−2.00043
$340$	0	0
$341$	−2.23889e10	−0.0896681
$342$	0	0
$343$	5.11709e10	0.199618
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.56153e11	−0.578188	−0.289094	−	0.957301i	$-0.593354\pi$
$347$	−1.56153e11	−0.578188	−0.289094	+	0.957301i	$0.593354\pi$
$348$	0	0
$349$	6.31862e9	0.0227986	0.0113993	−	0.999935i	$-0.496371\pi$
$349$	6.31862e9	0.0227986	0.0113993	+	0.999935i	$0.496371\pi$
$350$	0	0
$351$	6.41156e11	2.25466
$352$	0	0
$353$	−2.83875e11	−0.973064	−0.486532	−	0.873663i	$-0.661738\pi$
$353$	−2.83875e11	−0.973064	−0.486532	+	0.873663i	$0.661738\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.15253e10	−0.0375529
$358$	0	0
$359$	−6.00733e11	−1.90878	−0.954391	−	0.298558i	$-0.903494\pi$
$359$	−6.00733e11	−1.90878	−0.954391	+	0.298558i	$0.903494\pi$
$360$	0	0
$361$	−2.06314e11	−0.639361
$362$	0	0
$363$	1.26883e10	0.0383552
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.97260e11	−1.43082	−0.715412	−	0.698702i	$-0.753761\pi$
$367$	−4.97260e11	−1.43082	−0.715412	+	0.698702i	$0.753761\pi$
$368$	0	0
$369$	−1.34743e12	−3.78346
$370$	0	0
$371$	5.36845e10	0.147118
$372$	0	0
$373$	1.39183e11	0.372303	0.186151	−	0.982521i	$-0.440398\pi$
$373$	1.39183e11	0.372303	0.186151	+	0.982521i	$0.440398\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.24202e11	0.826569
$378$	0	0
$379$	5.65247e11	1.40722	0.703610	−	0.710587i	$-0.251570\pi$
$379$	5.65247e11	1.40722	0.703610	+	0.710587i	$0.251570\pi$
$380$	0	0
$381$	8.23432e11	2.00201
$382$	0	0
$383$	5.38864e11	1.27963	0.639816	−	0.768528i	$-0.279011\pi$
$383$	5.38864e11	1.27963	0.639816	+	0.768528i	$0.279011\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.79914e11	−0.407723
$388$	0	0
$389$	7.67552e11	1.69955	0.849776	−	0.527144i	$-0.176737\pi$
$389$	7.67552e11	1.69955	0.849776	+	0.527144i	$0.176737\pi$
$390$	0	0
$391$	9.04466e9	0.0195703
$392$	0	0
$393$	4.93021e11	1.04255
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.25258e11	−1.26329	−0.631643	−	0.775259i	$-0.717619\pi$
$397$	−6.25258e11	−1.26329	−0.631643	+	0.775259i	$0.717619\pi$
$398$	0	0
$399$	5.84034e10	0.115361
$400$	0	0
$401$	−1.61329e11	−0.311574	−0.155787	−	0.987791i	$-0.549791\pi$
$401$	−1.61329e11	−0.311574	−0.155787	+	0.987791i	$0.549791\pi$
$402$	0	0
$403$	−3.32009e10	−0.0627014
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.40083e11	1.15628
$408$	0	0
$409$	−7.24004e11	−1.27934	−0.639670	−	0.768649i	$-0.720929\pi$
$409$	−7.24004e11	−1.27934	−0.639670	+	0.768649i	$0.720929\pi$
$410$	0	0
$411$	−1.24237e12	−2.14765
$412$	0	0
$413$	−4.75607e10	−0.0804403
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.90842e10	−0.111884
$418$	0	0
$419$	−2.80040e11	−0.443871	−0.221936	−	0.975061i	$-0.571237\pi$
$419$	−2.80040e11	−0.443871	−0.221936	+	0.975061i	$0.571237\pi$
$420$	0	0
$421$	−6.55915e10	−0.101760	−0.0508801	−	0.998705i	$-0.516203\pi$
$421$	−6.55915e10	−0.101760	−0.0508801	+	0.998705i	$0.516203\pi$
$422$	0	0
$423$	1.78393e12	2.70924
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.13439e11	−0.165134
$428$	0	0
$429$	9.58239e11	1.36589
$430$	0	0
$431$	−1.00292e12	−1.39997	−0.699987	−	0.714156i	$-0.746811\pi$
$431$	−1.00292e12	−1.39997	−0.699987	+	0.714156i	$0.746811\pi$
$432$	0	0
$433$	−7.98482e11	−1.09162	−0.545808	−	0.837910i	$-0.683777\pi$
$433$	−7.98482e11	−1.09162	−0.545808	+	0.837910i	$0.683777\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.58332e10	−0.0601193
$438$	0	0
$439$	−7.98379e11	−1.02593	−0.512966	−	0.858409i	$-0.671453\pi$
$439$	−7.98379e11	−1.02593	−0.512966	+	0.858409i	$0.671453\pi$
$440$	0	0
$441$	−2.09714e12	−2.64030
$442$	0	0
$443$	−7.56642e11	−0.933413	−0.466707	−	0.884412i	$-0.654560\pi$
$443$	−7.56642e11	−0.933413	−0.466707	+	0.884412i	$0.654560\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.45779e11	1.00201
$448$	0	0
$449$	−6.39419e11	−0.742467	−0.371234	−	0.928540i	$-0.621065\pi$
$449$	−6.39419e11	−0.742467	−0.371234	+	0.928540i	$0.621065\pi$
$450$	0	0
$451$	−1.25876e12	−1.43268
$452$	0	0
$453$	2.76111e12	3.08065
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.34276e12	1.44005	0.720024	−	0.693949i	$-0.244131\pi$
$457$	1.34276e12	1.44005	0.720024	+	0.693949i	$0.244131\pi$
$458$	0	0
$459$	5.93489e11	0.624102
$460$	0	0
$461$	3.47112e11	0.357945	0.178972	−	0.983854i	$-0.442723\pi$
$461$	3.47112e11	0.357945	0.178972	+	0.983854i	$0.442723\pi$
$462$	0	0
$463$	7.20214e11	0.728362	0.364181	−	0.931328i	$-0.381349\pi$
$463$	7.20214e11	0.728362	0.364181	+	0.931328i	$0.381349\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.31553e10	−0.0711737	−0.0355869	−	0.999367i	$-0.511330\pi$
$467$	−7.31553e10	−0.0711737	−0.0355869	+	0.999367i	$0.511330\pi$
$468$	0	0
$469$	4.42433e10	0.0422250
$470$	0	0
$471$	8.73364e11	0.817713
$472$	0	0
$473$	−1.68074e11	−0.154392
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.42267e12	−3.91158
$478$	0	0
$479$	4.80307e10	0.0416878	0.0208439	−	0.999783i	$-0.493365\pi$
$479$	4.80307e10	0.0416878	0.0208439	+	0.999783i	$0.493365\pi$
$480$	0	0
$481$	9.49191e11	0.808539
$482$	0	0
$483$	−2.30019e10	−0.0192310
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.24384e12	−1.00204	−0.501021	−	0.865435i	$-0.667042\pi$
$487$	−1.24384e12	−1.00204	−0.501021	+	0.865435i	$0.667042\pi$
$488$	0	0
$489$	−9.13590e11	−0.722539
$490$	0	0
$491$	2.63145e11	0.204328	0.102164	−	0.994768i	$-0.467423\pi$
$491$	2.63145e11	0.204328	0.102164	+	0.994768i	$0.467423\pi$
$492$	0	0
$493$	3.00099e11	0.228798
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.32456e11	−0.0973798
$498$	0	0
$499$	−3.75452e10	−0.0271083	−0.0135542	−	0.999908i	$-0.504315\pi$
$499$	−3.75452e10	−0.0271083	−0.0135542	+	0.999908i	$0.504315\pi$
$500$	0	0
$501$	−1.41797e12	−1.00553
$502$	0	0
$503$	−2.86189e11	−0.199341	−0.0996707	−	0.995020i	$-0.531779\pi$
$503$	−2.86189e11	−0.199341	−0.0996707	+	0.995020i	$0.531779\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.42806e12	−0.959864
$508$	0	0
$509$	−1.48505e12	−0.980646	−0.490323	−	0.871541i	$-0.663121\pi$
$509$	−1.48505e12	−0.980646	−0.490323	+	0.871541i	$0.663121\pi$
$510$	0	0
$511$	1.92774e11	0.125071
$512$	0	0
$513$	−3.00746e12	−1.91722
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.66654e12	1.02590
$518$	0	0
$519$	1.77012e12	1.07090
$520$	0	0
$521$	−1.77521e12	−1.05556	−0.527778	−	0.849383i	$-0.676975\pi$
$521$	−1.77521e12	−1.05556	−0.527778	+	0.849383i	$0.676975\pi$
$522$	0	0
$523$	−1.20614e12	−0.704922	−0.352461	−	0.935826i	$-0.614655\pi$
$523$	−1.20614e12	−0.704922	−0.352461	+	0.935826i	$0.614655\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.07326e10	−0.0173561
$528$	0	0
$529$	−1.78310e12	−0.989978
$530$	0	0
$531$	3.91818e12	2.13875
$532$	0	0
$533$	−1.86664e12	−1.00182
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.32237e12	1.72410
$538$	0	0
$539$	−1.95913e12	−0.999802
$540$	0	0
$541$	3.24195e12	1.62711	0.813557	−	0.581485i	$-0.197528\pi$
$541$	3.24195e12	1.62711	0.813557	+	0.581485i	$0.197528\pi$
$542$	0	0
$543$	−6.98296e12	−3.44699
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−9.33463e11	−0.445814	−0.222907	−	0.974840i	$-0.571555\pi$
$547$	−9.33463e11	−0.445814	−0.222907	+	0.974840i	$0.571555\pi$
$548$	0	0
$549$	9.34540e12	4.39059
$550$	0	0
$551$	−1.52073e12	−0.702861
$552$	0	0
$553$	2.37208e11	0.107861
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.31760e12	−0.580012	−0.290006	−	0.957025i	$-0.593657\pi$
$557$	−1.31760e12	−0.580012	−0.290006	+	0.957025i	$0.593657\pi$
$558$	0	0
$559$	−2.49240e11	−0.107960
$560$	0	0
$561$	8.86998e11	0.378085
$562$	0	0
$563$	4.33363e12	1.81788	0.908938	−	0.416931i	$-0.136894\pi$
$563$	4.33363e12	1.81788	0.908938	+	0.416931i	$0.136894\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−8.50868e11	−0.345731
$568$	0	0
$569$	3.08940e12	1.23558	0.617788	−	0.786345i	$-0.288029\pi$
$569$	3.08940e12	1.23558	0.617788	+	0.786345i	$0.288029\pi$
$570$	0	0
$571$	7.43095e11	0.292538	0.146269	−	0.989245i	$-0.453274\pi$
$571$	7.43095e11	0.292538	0.146269	+	0.989245i	$0.453274\pi$
$572$	0	0
$573$	−4.04356e12	−1.56700
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.25852e12	−0.848269	−0.424134	−	0.905599i	$-0.639422\pi$
$577$	−2.25852e12	−0.848269	−0.424134	+	0.905599i	$0.639422\pi$
$578$	0	0
$579$	3.77013e12	1.39413
$580$	0	0
$581$	2.86807e11	0.104423
$582$	0	0
$583$	−4.13162e12	−1.48119
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.75794e12	1.65405	0.827023	−	0.562169i	$-0.190033\pi$
$587$	4.75794e12	1.65405	0.827023	+	0.562169i	$0.190033\pi$
$588$	0	0
$589$	1.55735e11	0.0533173
$590$	0	0
$591$	−6.91232e12	−2.33067
$592$	0	0
$593$	−2.01077e12	−0.667755	−0.333878	−	0.942616i	$-0.608357\pi$
$593$	−2.01077e12	−0.667755	−0.333878	+	0.942616i	$0.608357\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.90857e12	2.54808
$598$	0	0
$599$	3.29427e12	1.04554	0.522768	−	0.852475i	$-0.324899\pi$
$599$	3.29427e12	1.04554	0.522768	+	0.852475i	$0.324899\pi$
$600$	0	0
$601$	−1.98887e12	−0.621830	−0.310915	−	0.950438i	$-0.600635\pi$
$601$	−1.98887e12	−0.621830	−0.310915	+	0.950438i	$0.600635\pi$
$602$	0	0
$603$	−3.64488e12	−1.12268
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.11631e12	1.23072	0.615360	−	0.788246i	$-0.289011\pi$
$607$	4.11631e12	1.23072	0.615360	+	0.788246i	$0.289011\pi$
$608$	0	0
$609$	−7.63194e11	−0.224832
$610$	0	0
$611$	2.47134e12	0.717376
$612$	0	0
$613$	2.06864e12	0.591715	0.295858	−	0.955232i	$-0.404395\pi$
$613$	2.06864e12	0.591715	0.295858	+	0.955232i	$0.404395\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.42058e12	1.50578	0.752891	−	0.658145i	$-0.228659\pi$
$617$	5.42058e12	1.50578	0.752891	+	0.658145i	$0.228659\pi$
$618$	0	0
$619$	−3.87240e12	−1.06016	−0.530081	−	0.847947i	$-0.677838\pi$
$619$	−3.87240e12	−1.06016	−0.530081	+	0.847947i	$0.677838\pi$
$620$	0	0
$621$	1.18447e12	0.319605
$622$	0	0
$623$	−3.71344e10	−0.00987599
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.49480e12	−1.16147
$628$	0	0
$629$	8.78623e11	0.223807
$630$	0	0
$631$	3.22937e12	0.810933	0.405467	−	0.914110i	$-0.367109\pi$
$631$	3.22937e12	0.810933	0.405467	+	0.914110i	$0.367109\pi$
$632$	0	0
$633$	−3.15076e12	−0.780006
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.90523e12	−0.699123
$638$	0	0
$639$	1.09121e13	2.58913
$640$	0	0
$641$	1.10935e12	0.259543	0.129771	−	0.991544i	$-0.458576\pi$
$641$	1.10935e12	0.259543	0.129771	+	0.991544i	$0.458576\pi$
$642$	0	0
$643$	−2.83993e12	−0.655177	−0.327588	−	0.944821i	$-0.606236\pi$
$643$	−2.83993e12	−0.655177	−0.327588	+	0.944821i	$0.606236\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.66962e12	−0.598935	−0.299467	−	0.954107i	$-0.596809\pi$
$647$	−2.66962e12	−0.598935	−0.299467	+	0.954107i	$0.596809\pi$
$648$	0	0
$649$	3.66033e12	0.809878
$650$	0	0
$651$	7.81574e10	0.0170552
$652$	0	0
$653$	−4.72156e12	−1.01619	−0.508097	−	0.861300i	$-0.669651\pi$
$653$	−4.72156e12	−1.01619	−0.508097	+	0.861300i	$0.669651\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.58813e13	−3.32538
$658$	0	0
$659$	−4.01448e12	−0.829173	−0.414586	−	0.910010i	$-0.636074\pi$
$659$	−4.01448e12	−0.829173	−0.414586	+	0.910010i	$0.636074\pi$
$660$	0	0
$661$	−8.01591e12	−1.63323	−0.816613	−	0.577186i	$-0.804151\pi$
$661$	−8.01591e12	−1.63323	−0.816613	+	0.577186i	$0.804151\pi$
$662$	0	0
$663$	1.31535e12	0.264380
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.98931e11	0.117168
$668$	0	0
$669$	−6.94065e12	−1.33962
$670$	0	0
$671$	8.73040e12	1.66258
$672$	0	0
$673$	−3.55147e12	−0.667330	−0.333665	−	0.942692i	$-0.608286\pi$
$673$	−3.55147e12	−0.667330	−0.333665	+	0.942692i	$0.608286\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.06872e12	−1.11032	−0.555160	−	0.831744i	$-0.687343\pi$
$677$	−6.06872e12	−1.11032	−0.555160	+	0.831744i	$0.687343\pi$
$678$	0	0
$679$	−5.00773e11	−0.0904122
$680$	0	0
$681$	6.74068e12	1.20100
$682$	0	0
$683$	−9.96514e11	−0.175223	−0.0876113	−	0.996155i	$-0.527923\pi$
$683$	−9.96514e11	−0.175223	−0.0876113	+	0.996155i	$0.527923\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6.19170e12	−1.06049
$688$	0	0
$689$	−6.12686e12	−1.03574
$690$	0	0
$691$	−1.02702e13	−1.71367	−0.856835	−	0.515591i	$-0.827572\pi$
$691$	−1.02702e13	−1.71367	−0.856835	+	0.515591i	$0.827572\pi$
$692$	0	0
$693$	−1.64063e12	−0.270217
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.72786e12	−0.277308
$698$	0	0
$699$	8.52194e12	1.35018
$700$	0	0
$701$	−8.26127e12	−1.29216	−0.646079	−	0.763270i	$-0.723592\pi$
$701$	−8.26127e12	−1.29216	−0.646079	+	0.763270i	$0.723592\pi$
$702$	0	0
$703$	−4.45236e12	−0.687529
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.02251e11	0.0153915
$708$	0	0
$709$	−8.11428e12	−1.20598	−0.602992	−	0.797747i	$-0.706025\pi$
$709$	−8.11428e12	−1.20598	−0.602992	+	0.797747i	$0.706025\pi$
$710$	0	0
$711$	−1.95418e13	−2.86782
$712$	0	0
$713$	−6.13355e10	−0.00888810
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.01254e12	−0.849614
$718$	0	0
$719$	−1.06399e13	−1.48476	−0.742380	−	0.669979i	$-0.766303\pi$
$719$	−1.06399e13	−1.48476	−0.742380	+	0.669979i	$0.766303\pi$
$720$	0	0
$721$	−8.88651e11	−0.122468
$722$	0	0
$723$	−5.66405e12	−0.770911
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.35040e13	1.79290	0.896450	−	0.443145i	$-0.146137\pi$
$727$	1.35040e13	1.79290	0.896450	+	0.443145i	$0.146137\pi$
$728$	0	0
$729$	2.34766e13	3.07866
$730$	0	0
$731$	−2.30710e11	−0.0298840
$732$	0	0
$733$	3.03764e12	0.388659	0.194329	−	0.980936i	$-0.437747\pi$
$733$	3.03764e12	0.388659	0.194329	+	0.980936i	$0.437747\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.40502e12	−0.425124
$738$	0	0
$739$	−7.29483e12	−0.899736	−0.449868	−	0.893095i	$-0.648529\pi$
$739$	−7.29483e12	−0.899736	−0.449868	+	0.893095i	$0.648529\pi$
$740$	0	0
$741$	−6.66542e12	−0.812168
$742$	0	0
$743$	1.18034e13	1.42088	0.710442	−	0.703756i	$-0.248495\pi$
$743$	1.18034e13	1.42088	0.710442	+	0.703756i	$0.248495\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.36279e13	−2.77641
$748$	0	0
$749$	−9.97041e11	−0.115756
$750$	0	0
$751$	−1.20855e13	−1.38639	−0.693197	−	0.720748i	$-0.743798\pi$
$751$	−1.20855e13	−1.38639	−0.693197	+	0.720748i	$0.743798\pi$
$752$	0	0
$753$	2.13993e13	2.42562
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.48077e13	1.63891	0.819455	−	0.573144i	$-0.194276\pi$
$757$	1.48077e13	1.63891	0.819455	+	0.573144i	$0.194276\pi$
$758$	0	0
$759$	1.77025e12	0.193619
$760$	0	0
$761$	−1.75185e13	−1.89350	−0.946749	−	0.321972i	$-0.895654\pi$
$761$	−1.75185e13	−1.89350	−0.946749	+	0.321972i	$0.895654\pi$
$762$	0	0
$763$	−7.94656e11	−0.0848826
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.42798e12	0.566316
$768$	0	0
$769$	4.16729e12	0.429720	0.214860	−	0.976645i	$-0.431070\pi$
$769$	4.16729e12	0.429720	0.214860	+	0.976645i	$0.431070\pi$
$770$	0	0
$771$	−8.88781e12	−0.905837
$772$	0	0
$773$	−1.05896e13	−1.06677	−0.533386	−	0.845872i	$-0.679081\pi$
$773$	−1.05896e13	−1.06677	−0.533386	+	0.845872i	$0.679081\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.23446e12	−0.219927
$778$	0	0
$779$	8.75583e12	0.851881
$780$	0	0
$781$	1.01940e13	0.980426
$782$	0	0
$783$	3.93004e13	3.73654
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.50861e12	−0.233102	−0.116551	−	0.993185i	$-0.537184\pi$
$787$	−2.50861e12	−0.233102	−0.116551	+	0.993185i	$0.537184\pi$
$788$	0	0
$789$	−1.62549e13	−1.49327
$790$	0	0
$791$	1.15376e12	0.104790
$792$	0	0
$793$	1.29465e13	1.16258
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.85822e13	1.63131	0.815653	−	0.578541i	$-0.196378\pi$
$797$	1.85822e13	1.63131	0.815653	+	0.578541i	$0.196378\pi$
$798$	0	0
$799$	2.28760e12	0.198573
$800$	0	0
$801$	3.05923e12	0.262583
$802$	0	0
$803$	−1.48362e13	−1.25922
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.03720e12	0.584075
$808$	0	0
$809$	−4.47333e12	−0.367166	−0.183583	−	0.983004i	$-0.558770\pi$
$809$	−4.47333e12	−0.367166	−0.183583	+	0.983004i	$0.558770\pi$
$810$	0	0
$811$	3.49582e12	0.283763	0.141881	−	0.989884i	$-0.454685\pi$
$811$	3.49582e12	0.283763	0.141881	+	0.989884i	$0.454685\pi$
$812$	0	0
$813$	−1.45846e13	−1.17081
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.16911e12	0.0918026
$818$	0	0
$819$	−2.43293e12	−0.188952
$820$	0	0
$821$	−6.84197e12	−0.525578	−0.262789	−	0.964853i	$-0.584642\pi$
$821$	−6.84197e12	−0.525578	−0.262789	+	0.964853i	$0.584642\pi$
$822$	0	0
$823$	−8.11058e12	−0.616244	−0.308122	−	0.951347i	$-0.599700\pi$
$823$	−8.11058e12	−0.616244	−0.308122	+	0.951347i	$0.599700\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.06349e10	0.00376422	0.00188211	−	0.999998i	$-0.499401\pi$
$827$	5.06349e10	0.00376422	0.00188211	+	0.999998i	$0.499401\pi$
$828$	0	0
$829$	−1.50474e13	−1.10654	−0.553269	−	0.833002i	$-0.686620\pi$
$829$	−1.50474e13	−1.10654	−0.553269	+	0.833002i	$0.686620\pi$
$830$	0	0
$831$	4.74356e13	3.45064
$832$	0	0
$833$	−2.68924e12	−0.193521
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.02469e12	−0.283444
$838$	0	0
$839$	−2.68352e13	−1.86972	−0.934859	−	0.355020i	$-0.884474\pi$
$839$	−2.68352e13	−1.86972	−0.934859	+	0.355020i	$0.884474\pi$
$840$	0	0
$841$	5.36519e12	0.369831
$842$	0	0
$843$	−2.12547e12	−0.144954
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.00951e10	−0.00200919
$848$	0	0
$849$	3.62176e13	2.39241
$850$	0	0
$851$	1.75354e12	0.114613
$852$	0	0
$853$	4.22107e12	0.272993	0.136497	−	0.990641i	$-0.456416\pi$
$853$	4.22107e12	0.272993	0.136497	+	0.990641i	$0.456416\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.57740e12	−0.479851	−0.239926	−	0.970791i	$-0.577123\pi$
$857$	−7.57740e12	−0.479851	−0.239926	+	0.970791i	$0.577123\pi$
$858$	0	0
$859$	−1.41463e13	−0.886486	−0.443243	−	0.896401i	$-0.646172\pi$
$859$	−1.41463e13	−0.886486	−0.443243	+	0.896401i	$0.646172\pi$
$860$	0	0
$861$	4.39421e12	0.272500
$862$	0	0
$863$	−1.52498e13	−0.935868	−0.467934	−	0.883763i	$-0.655002\pi$
$863$	−1.52498e13	−0.935868	−0.467934	+	0.883763i	$0.655002\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.06427e13	−1.84180
$868$	0	0
$869$	−1.82558e13	−1.08596
$870$	0	0
$871$	−5.04937e12	−0.297273
$872$	0	0
$873$	4.12550e13	2.40388
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.30350e12	0.188572	0.0942858	−	0.995545i	$-0.469943\pi$
$877$	3.30350e12	0.188572	0.0942858	+	0.995545i	$0.469943\pi$
$878$	0	0
$879$	2.76142e13	1.56021
$880$	0	0
$881$	3.14728e13	1.76013	0.880063	−	0.474856i	$-0.157500\pi$
$881$	3.14728e13	1.76013	0.880063	+	0.474856i	$0.157500\pi$
$882$	0	0
$883$	−1.10669e13	−0.612637	−0.306318	−	0.951929i	$-0.599097\pi$
$883$	−1.10669e13	−0.612637	−0.306318	+	0.951929i	$0.599097\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.92035e12	0.375381	0.187690	−	0.982228i	$-0.439900\pi$
$887$	6.92035e12	0.375381	0.187690	+	0.982228i	$0.439900\pi$
$888$	0	0
$889$	−1.95308e12	−0.104872
$890$	0	0
$891$	6.54838e13	3.48084
$892$	0	0
$893$	−1.15923e13	−0.610010
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.62514e12	0.135390
$898$	0	0
$899$	−2.03509e12	−0.103912
$900$	0	0
$901$	−5.67136e12	−0.286699
$902$	0	0
$903$	5.86729e11	0.0293659
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.00486e13	0.983676	0.491838	−	0.870687i	$-0.336325\pi$
$907$	2.00486e13	0.983676	0.491838	+	0.870687i	$0.336325\pi$
$908$	0	0
$909$	−8.42371e12	−0.409229
$910$	0	0
$911$	−2.53798e13	−1.22083	−0.610416	−	0.792081i	$-0.708998\pi$
$911$	−2.53798e13	−1.22083	−0.610416	+	0.792081i	$0.708998\pi$
$912$	0	0
$913$	−2.20730e13	−1.05134
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.16938e12	−0.0546129
$918$	0	0
$919$	6.15926e12	0.284845	0.142423	−	0.989806i	$-0.454511\pi$
$919$	6.15926e12	0.284845	0.142423	+	0.989806i	$0.454511\pi$
$920$	0	0
$921$	7.02482e13	3.21712
$922$	0	0
$923$	1.51169e13	0.685574
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.32095e13	3.25618
$928$	0	0
$929$	2.20000e13	0.969061	0.484531	−	0.874774i	$-0.338990\pi$
$929$	2.20000e13	0.969061	0.484531	+	0.874774i	$0.338990\pi$
$930$	0	0
$931$	1.36275e13	0.594489
$932$	0	0
$933$	2.21601e13	0.957426
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.49274e12	0.105645	0.0528225	−	0.998604i	$-0.483178\pi$
$937$	2.49274e12	0.105645	0.0528225	+	0.998604i	$0.483178\pi$
$938$	0	0
$939$	−3.64753e13	−1.53110
$940$	0	0
$941$	−1.26366e13	−0.525382	−0.262691	−	0.964880i	$-0.584610\pi$
$941$	−1.26366e13	−0.525382	−0.262691	+	0.964880i	$0.584610\pi$
$942$	0	0
$943$	−3.44844e12	−0.142010
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.05809e13	0.831551	0.415775	−	0.909467i	$-0.363510\pi$
$947$	2.05809e13	0.831551	0.415775	+	0.909467i	$0.363510\pi$
$948$	0	0
$949$	−2.20008e13	−0.880524
$950$	0	0
$951$	−2.43639e13	−0.965904
$952$	0	0
$953$	−2.97887e13	−1.16986	−0.584929	−	0.811084i	$-0.698878\pi$
$953$	−2.97887e13	−1.16986	−0.584929	+	0.811084i	$0.698878\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.87364e13	2.26362
$958$	0	0
$959$	2.94675e12	0.112502
$960$	0	0
$961$	−2.62312e13	−0.992118
$962$	0	0
$963$	8.21389e13	3.07773
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.74473e13	−0.641664	−0.320832	−	0.947136i	$-0.603963\pi$
$967$	−1.74473e13	−0.641664	−0.320832	+	0.947136i	$0.603963\pi$
$968$	0	0
$969$	−6.16988e12	−0.224812
$970$	0	0
$971$	2.53844e13	0.916390	0.458195	−	0.888852i	$-0.348496\pi$
$971$	2.53844e13	0.916390	0.458195	+	0.888852i	$0.348496\pi$
$972$	0	0
$973$	1.63859e11	0.00586088
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.23029e13	1.13427	0.567135	−	0.823625i	$-0.308052\pi$
$977$	3.23029e13	1.13427	0.567135	+	0.823625i	$0.308052\pi$
$978$	0	0
$979$	2.85791e12	0.0994321
$980$	0	0
$981$	6.54659e13	2.25686
$982$	0	0
$983$	1.78014e13	0.608083	0.304041	−	0.952659i	$-0.401664\pi$
$983$	1.78014e13	0.608083	0.304041	+	0.952659i	$0.401664\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.81771e12	−0.195130
$988$	0	0
$989$	−4.60447e11	−0.0153037
$990$	0	0
$991$	3.38638e12	0.111533	0.0557666	−	0.998444i	$-0.482240\pi$
$991$	3.38638e12	0.111533	0.0557666	+	0.998444i	$0.482240\pi$
$992$	0	0
$993$	9.27345e13	3.02670
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.17766e13	1.33908	0.669538	−	0.742778i	$-0.266492\pi$
$997$	4.17766e13	1.33908	0.669538	+	0.742778i	$0.266492\pi$
$998$	0	0
$999$	1.15063e14	3.65503

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.10.a.y.1.3		3
4.3	odd	2		25.10.a.c.1.1	✓	3
5.2	odd	4		400.10.c.q.49.1		6
5.3	odd	4		400.10.c.q.49.6		6
5.4	even	2		400.10.a.u.1.1		3
12.11	even	2		225.10.a.p.1.3		3
20.3	even	4		25.10.b.c.24.6		6
20.7	even	4		25.10.b.c.24.1		6
20.19	odd	2		25.10.a.d.1.3	yes	3
60.23	odd	4		225.10.b.m.199.1		6
60.47	odd	4		225.10.b.m.199.6		6
60.59	even	2		225.10.a.m.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.10.a.c.1.1	✓	3	4.3	odd	2
25.10.a.d.1.3	yes	3	20.19	odd	2
25.10.b.c.24.1		6	20.7	even	4
25.10.b.c.24.6		6	20.3	even	4
225.10.a.m.1.1		3	60.59	even	2
225.10.a.p.1.3		3	12.11	even	2
225.10.b.m.199.1		6	60.23	odd	4
225.10.b.m.199.6		6	60.47	odd	4
400.10.a.u.1.1		3	5.4	even	2
400.10.a.y.1.3		3	1.1	even	1	trivial
400.10.c.q.49.1		6	5.2	odd	4
400.10.c.q.49.6		6	5.3	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q+268.664 q^{3} -637.237 q^{7} +52497.3 q^{9} +O(q^{10})\)	\(q+268.664 q^{3} -637.237 q^{7} +52497.3 q^{9} +49042.6 q^{11} +72726.2 q^{13} +67319.3 q^{17} -341136. q^{19} -171203. q^{21} +134355. q^{23} +8.81603e6 q^{27} +4.45784e6 q^{29} -456520. q^{31} +1.31760e7 q^{33} +1.30516e7 q^{37} +1.95389e7 q^{39} -2.56667e7 q^{41} -3.42710e6 q^{43} +3.39814e7 q^{47} -3.99475e7 q^{49} +1.80863e7 q^{51} -8.42456e7 q^{53} -9.16509e7 q^{57} +7.46358e7 q^{59} +1.78017e8 q^{61} -3.34533e7 q^{63} -6.94299e7 q^{67} +3.60963e7 q^{69} +2.07860e8 q^{71} -3.02516e8 q^{73} -3.12518e7 q^{77} -3.72244e8 q^{79} +1.33524e9 q^{81} -4.50079e8 q^{83} +1.19766e9 q^{87} +5.82741e7 q^{89} -4.63438e7 q^{91} -1.22650e8 q^{93} +7.85850e8 q^{97} +2.57461e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9}+O(q^{10}) \) 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9	\( 3 q + 89 q^{3} + 5258 q^{7} + 58234 q^{9} + 54699 q^{11} - 215884 q^{13} - 334983 q^{17} - 818845 q^{19} - 2375394 q^{21} + 3526854 q^{23} + 6633395 q^{27} + 2175480 q^{29} - 4274066 q^{31} + 22122137 q^{33} + 10305042 q^{37} + 50414092 q^{39} + 5926311 q^{41} - 24429956 q^{43} + 66858708 q^{47} - 6453929 q^{49} + 25634699 q^{51} - 132620514 q^{53} - 252946415 q^{57} - 5670960 q^{59} + 125306926 q^{61} + 284323404 q^{63} + 88829483 q^{67} - 314274942 q^{69} - 297550596 q^{71} - 181321729 q^{73} - 561214086 q^{77} + 310025170 q^{79} + 1398847363 q^{81} - 731088801 q^{83} + 1046385560 q^{87} - 1103860035 q^{89} - 1183187656 q^{91} - 107386758 q^{93} + 332236842 q^{97} + 892234522 q^{99}+O(q^{100}) \) 3 * q + 89 * q^3 + 5258 * q^7 + 58234 * q^9 + 54699 * q^11 - 215884 * q^13 - 334983 * q^17 - 818845 * q^19 - 2375394 * q^21 + 3526854 * q^23 + 6633395 * q^27 + 2175480 * q^29 - 4274066 * q^31 + 22122137 * q^33 + 10305042 * q^37 + 50414092 * q^39 + 5926311 * q^41 - 24429956 * q^43 + 66858708 * q^47 - 6453929 * q^49 + 25634699 * q^51 - 132620514 * q^53 - 252946415 * q^57 - 5670960 * q^59 + 125306926 * q^61 + 284323404 * q^63 + 88829483 * q^67 - 314274942 * q^69 - 297550596 * q^71 - 181321729 * q^73 - 561214086 * q^77 + 310025170 * q^79 + 1398847363 * q^81 - 731088801 * q^83 + 1046385560 * q^87 - 1103860035 * q^89 - 1183187656 * q^91 - 107386758 * q^93 + 332236842 * q^97 + 892234522 * q^99

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