LMFDB - Embedded newform 4000.2.d.c.2001.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(2001,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4000.2001");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 1000)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2001.20
Character	$\chi$	$=$	4000.2001
Dual form			4000.2.d.c.2001.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.513027i q^{3} -1.12889 q^{7} +2.73680 q^{9} +O(q^{10})$	$q+0.513027i q^{3} -1.12889 q^{7} +2.73680 q^{9} +3.99787i q^{11} +1.12390i q^{13} +2.93982 q^{17} +3.24181i q^{19} -0.579150i q^{21} +5.39053 q^{23} +2.94314i q^{27} +7.35068i q^{29} -7.19613 q^{31} -2.05101 q^{33} -10.6176i q^{37} -0.576592 q^{39} +2.08421 q^{41} -6.26007i q^{43} -2.24328 q^{47} -5.72561 q^{49} +1.50821i q^{51} +1.64494i q^{53} -1.66314 q^{57} +5.55039i q^{59} -6.65010i q^{61} -3.08954 q^{63} +6.79788i q^{67} +2.76549i q^{69} +3.27833 q^{71} +6.97819 q^{73} -4.51314i q^{77} -8.60333 q^{79} +6.70050 q^{81} +13.4761i q^{83} -3.77110 q^{87} -10.9910 q^{89} -1.26876i q^{91} -3.69181i q^{93} +18.6691 q^{97} +10.9414i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 24 q^{9}+O(q^{10}) $ 40 * q - 24 * q^9	$ 40 q - 24 q^{9} - 48 q^{31} - 8 q^{39} + 44 q^{41} + 12 q^{49} - 96 q^{71} - 96 q^{79} - 56 q^{81} - 44 q^{89}+O(q^{100}) $ 40 * q - 24 * q^9 - 48 * q^31 - 8 * q^39 + 44 * q^41 + 12 * q^49 - 96 * q^71 - 96 * q^79 - 56 * q^81 - 44 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	0.513027i	0.296196i	0.988973	+	0.148098i	$0.0473152\pi$
$3$	0.513027i	0.296196i	−0.988973	+	0.148098i	$0.952685\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.12889	−0.426679	−0.213340	−	0.976978i	$-0.568434\pi$
$7$	−1.12889	−0.426679	−0.213340	+	0.976978i	$0.568434\pi$
$8$	0	0
$9$	2.73680	0.912268
$10$	0	0
$11$	3.99787i	1.20540i	0.797967	+	0.602701i	$0.205909\pi$
$11$	3.99787i	1.20540i	−0.797967	+	0.602701i	$0.794091\pi$
$12$	0	0
$13$	1.12390i	0.311714i	0.987780	+	0.155857i	$0.0498140\pi$
$13$	1.12390i	0.311714i	−0.987780	+	0.155857i	$0.950186\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.93982	0.713010	0.356505	−	0.934293i	$-0.383968\pi$
$17$	2.93982	0.713010	0.356505	+	0.934293i	$0.383968\pi$
$18$	0	0
$19$	3.24181i	0.743723i	0.928288	+	0.371862i	$0.121280\pi$
$19$	3.24181i	0.743723i	−0.928288	+	0.371862i	$0.878720\pi$
$20$	0	0
$21$	− 0.579150i	− 0.126381i
$22$	0	0
$23$	5.39053	1.12400	0.562002	−	0.827136i	$-0.310031\pi$
$23$	5.39053	1.12400	0.562002	+	0.827136i	$0.310031\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.94314i	0.566407i
$28$	0	0
$29$	7.35068i	1.36499i	0.730891	+	0.682494i	$0.239105\pi$
$29$	7.35068i	1.36499i	−0.730891	+	0.682494i	$0.760895\pi$
$30$	0	0
$31$	−7.19613	−1.29246	−0.646232	−	0.763141i	$-0.723656\pi$
$31$	−7.19613	−1.29246	−0.646232	+	0.763141i	$0.723656\pi$
$32$	0	0
$33$	−2.05101	−0.357036
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.6176i	− 1.74552i	−0.488152	−	0.872759i	$-0.662329\pi$
$37$	− 10.6176i	− 1.74552i	0.488152	−	0.872759i	$-0.337671\pi$
$38$	0	0
$39$	−0.576592	−0.0923287
$40$	0	0
$41$	2.08421	0.325498	0.162749	−	0.986667i	$-0.447964\pi$
$41$	2.08421	0.325498	0.162749	+	0.986667i	$0.447964\pi$
$42$	0	0
$43$	− 6.26007i	− 0.954652i	−0.878726	−	0.477326i	$-0.841606\pi$
$43$	− 6.26007i	− 0.954652i	0.878726	−	0.477326i	$-0.158394\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.24328	−0.327216	−0.163608	−	0.986525i	$-0.552313\pi$
$47$	−2.24328	−0.327216	−0.163608	+	0.986525i	$0.552313\pi$
$48$	0	0
$49$	−5.72561	−0.817945
$50$	0	0
$51$	1.50821i	0.211191i
$52$	0	0
$53$	1.64494i	0.225950i	0.993598	+	0.112975i	$0.0360380\pi$
$53$	1.64494i	0.225950i	−0.993598	+	0.112975i	$0.963962\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.66314	−0.220288
$58$	0	0
$59$	5.55039i	0.722599i	0.932450	+	0.361299i	$0.117667\pi$
$59$	5.55039i	0.722599i	−0.932450	+	0.361299i	$0.882333\pi$
$60$	0	0
$61$	− 6.65010i	− 0.851458i	−0.904851	−	0.425729i	$-0.860018\pi$
$61$	− 6.65010i	− 0.851458i	0.904851	−	0.425729i	$-0.139982\pi$
$62$	0	0
$63$	−3.08954	−0.389246
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.79788i	0.830493i	0.909709	+	0.415247i	$0.136305\pi$
$67$	6.79788i	0.830493i	−0.909709	+	0.415247i	$0.863695\pi$
$68$	0	0
$69$	2.76549i	0.332926i
$70$	0	0
$71$	3.27833	0.389066	0.194533	−	0.980896i	$-0.437681\pi$
$71$	3.27833	0.389066	0.194533	+	0.980896i	$0.437681\pi$
$72$	0	0
$73$	6.97819	0.816736	0.408368	−	0.912817i	$-0.366098\pi$
$73$	6.97819	0.816736	0.408368	+	0.912817i	$0.366098\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.51314i	− 0.514320i
$78$	0	0
$79$	−8.60333	−0.967950	−0.483975	−	0.875082i	$-0.660807\pi$
$79$	−8.60333	−0.967950	−0.483975	+	0.875082i	$0.660807\pi$
$80$	0	0
$81$	6.70050	0.744500
$82$	0	0
$83$	13.4761i	1.47919i	0.673052	+	0.739596i	$0.264983\pi$
$83$	13.4761i	1.47919i	−0.673052	+	0.739596i	$0.735017\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.77110	−0.404304
$88$	0	0
$89$	−10.9910	−1.16504	−0.582521	−	0.812816i	$-0.697934\pi$
$89$	−10.9910	−1.16504	−0.582521	+	0.812816i	$0.697934\pi$
$90$	0	0
$91$	− 1.26876i	− 0.133002i
$92$	0	0
$93$	− 3.69181i	− 0.382823i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.6691	1.89556	0.947778	−	0.318931i	$-0.103324\pi$
$97$	18.6691	1.89556	0.947778	+	0.318931i	$0.103324\pi$
$98$	0	0
$99$	10.9414i	1.09965i
$100$	0	0
$101$	− 1.76372i	− 0.175497i	−0.996143	−	0.0877485i	$-0.972033\pi$
$101$	− 1.76372i	− 0.175497i	0.996143	−	0.0877485i	$-0.0279672\pi$
$102$	0	0
$103$	−3.89194	−0.383484	−0.191742	−	0.981445i	$-0.561414\pi$
$103$	−3.89194	−0.383484	−0.191742	+	0.981445i	$0.561414\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0146i	1.54819i	0.633071	+	0.774094i	$0.281794\pi$
$107$	16.0146i	1.54819i	−0.633071	+	0.774094i	$0.718206\pi$
$108$	0	0
$109$	15.6094i	1.49511i	0.664199	+	0.747556i	$0.268773\pi$
$109$	15.6094i	1.49511i	−0.664199	+	0.747556i	$0.731227\pi$
$110$	0	0
$111$	5.44710	0.517016
$112$	0	0
$113$	2.97052	0.279443	0.139722	−	0.990191i	$-0.455379\pi$
$113$	2.97052	0.279443	0.139722	+	0.990191i	$0.455379\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.07590i	0.284367i
$118$	0	0
$119$	−3.31872	−0.304227
$120$	0	0
$121$	−4.98293	−0.452994
$122$	0	0
$123$	1.06925i	0.0964114i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.2692	−1.08871	−0.544356	−	0.838855i	$-0.683226\pi$
$127$	−12.2692	−1.08871	−0.544356	+	0.838855i	$0.683226\pi$
$128$	0	0
$129$	3.21158	0.282764
$130$	0	0
$131$	− 13.2639i	− 1.15887i	−0.815019	−	0.579434i	$-0.803274\pi$
$131$	− 13.2639i	− 1.15887i	0.815019	−	0.579434i	$-0.196726\pi$
$132$	0	0
$133$	− 3.65964i	− 0.317331i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.3863	−1.14367	−0.571836	−	0.820368i	$-0.693769\pi$
$137$	−13.3863	−1.14367	−0.571836	+	0.820368i	$0.693769\pi$
$138$	0	0
$139$	20.2594i	1.71838i	0.511657	+	0.859190i	$0.329032\pi$
$139$	20.2594i	1.71838i	−0.511657	+	0.859190i	$0.670968\pi$
$140$	0	0
$141$	− 1.15086i	− 0.0969201i
$142$	0	0
$143$	−4.49321	−0.375741
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.93739i	− 0.242272i
$148$	0	0
$149$	14.8567i	1.21711i	0.793512	+	0.608555i	$0.208251\pi$
$149$	14.8567i	1.21711i	−0.793512	+	0.608555i	$0.791749\pi$
$150$	0	0
$151$	−10.1436	−0.825477	−0.412739	−	0.910850i	$-0.635428\pi$
$151$	−10.1436	−0.825477	−0.412739	+	0.910850i	$0.635428\pi$
$152$	0	0
$153$	8.04570	0.650456
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 7.49744i	− 0.598361i	−0.954197	−	0.299180i	$-0.903287\pi$
$157$	− 7.49744i	− 0.598361i	0.954197	−	0.299180i	$-0.0967132\pi$
$158$	0	0
$159$	−0.843898	−0.0669255
$160$	0	0
$161$	−6.08530	−0.479589
$162$	0	0
$163$	− 8.95189i	− 0.701166i	−0.936532	−	0.350583i	$-0.885983\pi$
$163$	− 8.95189i	− 0.701166i	0.936532	−	0.350583i	$-0.114017\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.49946	0.193414	0.0967070	−	0.995313i	$-0.469169\pi$
$167$	2.49946	0.193414	0.0967070	+	0.995313i	$0.469169\pi$
$168$	0	0
$169$	11.7368	0.902834
$170$	0	0
$171$	8.87221i	0.678475i
$172$	0	0
$173$	− 0.589561i	− 0.0448235i	−0.999749	−	0.0224118i	$-0.992866\pi$
$173$	− 0.589561i	− 0.0448235i	0.999749	−	0.0224118i	$-0.00713448\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.84750	−0.214031
$178$	0	0
$179$	10.3029i	0.770077i	0.922901	+	0.385038i	$0.125812\pi$
$179$	10.3029i	0.770077i	−0.922901	+	0.385038i	$0.874188\pi$
$180$	0	0
$181$	17.8021i	1.32322i	0.749848	+	0.661610i	$0.230127\pi$
$181$	17.8021i	1.32322i	−0.749848	+	0.661610i	$0.769873\pi$
$182$	0	0
$183$	3.41168	0.252199
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.7530i	0.859464i
$188$	0	0
$189$	− 3.32247i	− 0.241674i
$190$	0	0
$191$	18.9069	1.36806	0.684029	−	0.729455i	$-0.260226\pi$
$191$	18.9069	1.36806	0.684029	+	0.729455i	$0.260226\pi$
$192$	0	0
$193$	−10.2705	−0.739288	−0.369644	−	0.929173i	$-0.620520\pi$
$193$	−10.2705	−0.739288	−0.369644	+	0.929173i	$0.620520\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.207142i	0.0147583i	0.999973	+	0.00737913i	$0.00234887\pi$
$197$	0.207142i	0.0147583i	−0.999973	+	0.00737913i	$0.997651\pi$
$198$	0	0
$199$	12.2911	0.871295	0.435647	−	0.900117i	$-0.356519\pi$
$199$	12.2911	0.871295	0.435647	+	0.900117i	$0.356519\pi$
$200$	0	0
$201$	−3.48750	−0.245989
$202$	0	0
$203$	− 8.29809i	− 0.582412i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	14.7528	1.02539
$208$	0	0
$209$	−12.9603	−0.896485
$210$	0	0
$211$	3.91559i	0.269560i	0.990876	+	0.134780i	$0.0430328\pi$
$211$	3.91559i	0.269560i	−0.990876	+	0.134780i	$0.956967\pi$
$212$	0	0
$213$	1.68187i	0.115240i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.12363	0.551468
$218$	0	0
$219$	3.58000i	0.241914i
$220$	0	0
$221$	3.30407i	0.222256i
$222$	0	0
$223$	23.8519	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$223$	23.8519	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.7012i	− 1.63948i	−0.572738	−	0.819738i	$-0.694119\pi$
$227$	− 24.7012i	− 1.63948i	0.572738	−	0.819738i	$-0.305881\pi$
$228$	0	0
$229$	22.0732i	1.45864i	0.684175	+	0.729318i	$0.260163\pi$
$229$	22.0732i	1.45864i	−0.684175	+	0.729318i	$0.739837\pi$
$230$	0	0
$231$	2.31536	0.152340
$232$	0	0
$233$	−19.6085	−1.28460	−0.642299	−	0.766454i	$-0.722019\pi$
$233$	−19.6085	−1.28460	−0.642299	+	0.766454i	$0.722019\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.41374i	− 0.286703i
$238$	0	0
$239$	12.3845	0.801088	0.400544	−	0.916278i	$-0.368821\pi$
$239$	12.3845	0.801088	0.400544	+	0.916278i	$0.368821\pi$
$240$	0	0
$241$	−20.4095	−1.31469	−0.657347	−	0.753588i	$-0.728321\pi$
$241$	−20.4095	−1.31469	−0.657347	+	0.753588i	$0.728321\pi$
$242$	0	0
$243$	12.2669i	0.786925i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.64348	−0.231829
$248$	0	0
$249$	−6.91359	−0.438131
$250$	0	0
$251$	− 22.2016i	− 1.40135i	−0.713479	−	0.700676i	$-0.752882\pi$
$251$	− 22.2016i	− 1.40135i	0.713479	−	0.700676i	$-0.247118\pi$
$252$	0	0
$253$	21.5506i	1.35488i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.0592	0.814614	0.407307	−	0.913291i	$-0.366468\pi$
$257$	13.0592	0.814614	0.407307	+	0.913291i	$0.366468\pi$
$258$	0	0
$259$	11.9860i	0.744776i
$260$	0	0
$261$	20.1174i	1.24523i
$262$	0	0
$263$	−29.2389	−1.80295	−0.901473	−	0.432835i	$-0.857513\pi$
$263$	−29.2389	−1.80295	−0.901473	+	0.432835i	$0.857513\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 5.63867i	− 0.345081i
$268$	0	0
$269$	21.4247i	1.30629i	0.757235	+	0.653143i	$0.226550\pi$
$269$	21.4247i	1.30629i	−0.757235	+	0.653143i	$0.773450\pi$
$270$	0	0
$271$	−1.15797	−0.0703415	−0.0351708	−	0.999381i	$-0.511198\pi$
$271$	−1.15797	−0.0703415	−0.0351708	+	0.999381i	$0.511198\pi$
$272$	0	0
$273$	0.650908	0.0393947
$274$	0	0
$275$	0	0
$276$	0	0
$277$	16.8055i	1.00975i	0.863194	+	0.504873i	$0.168461\pi$
$277$	16.8055i	1.00975i	−0.863194	+	0.504873i	$0.831539\pi$
$278$	0	0
$279$	−19.6944	−1.17907
$280$	0	0
$281$	8.99127	0.536374	0.268187	−	0.963367i	$-0.413575\pi$
$281$	8.99127	0.536374	0.268187	+	0.963367i	$0.413575\pi$
$282$	0	0
$283$	− 15.4397i	− 0.917798i	−0.888489	−	0.458899i	$-0.848244\pi$
$283$	− 15.4397i	− 0.917798i	0.888489	−	0.458899i	$-0.151756\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.35283	−0.138883
$288$	0	0
$289$	−8.35748	−0.491616
$290$	0	0
$291$	9.57773i	0.561457i
$292$	0	0
$293$	− 1.62062i	− 0.0946777i	−0.998879	−	0.0473389i	$-0.984926\pi$
$293$	− 1.62062i	− 0.0946777i	0.998879	−	0.0473389i	$-0.0150741\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−11.7663	−0.682748
$298$	0	0
$299$	6.05843i	0.350368i
$300$	0	0
$301$	7.06691i	0.407330i
$302$	0	0
$303$	0.904838	0.0519816
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 13.3243i	− 0.760458i	−0.924892	−	0.380229i	$-0.875845\pi$
$307$	− 13.3243i	− 0.760458i	0.924892	−	0.380229i	$-0.124155\pi$
$308$	0	0
$309$	− 1.99667i	− 0.113587i
$310$	0	0
$311$	3.20457	0.181714	0.0908572	−	0.995864i	$-0.471039\pi$
$311$	3.20457	0.181714	0.0908572	+	0.995864i	$0.471039\pi$
$312$	0	0
$313$	−24.2715	−1.37190	−0.685952	−	0.727647i	$-0.740614\pi$
$313$	−24.2715	−1.37190	−0.685952	+	0.727647i	$0.740614\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.79177i	0.212967i	0.994314	+	0.106483i	$0.0339591\pi$
$317$	3.79177i	0.212967i	−0.994314	+	0.106483i	$0.966041\pi$
$318$	0	0
$319$	−29.3870	−1.64536
$320$	0	0
$321$	−8.21591	−0.458567
$322$	0	0
$323$	9.53034i	0.530282i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.00806	−0.442847
$328$	0	0
$329$	2.53241	0.139616
$330$	0	0
$331$	2.26196i	0.124328i	0.998066	+	0.0621642i	$0.0198002\pi$
$331$	2.26196i	0.124328i	−0.998066	+	0.0621642i	$0.980200\pi$
$332$	0	0
$333$	− 29.0582i	− 1.59238i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−28.6654	−1.56150	−0.780751	−	0.624843i	$-0.785163\pi$
$337$	−28.6654	−1.56150	−0.780751	+	0.624843i	$0.785163\pi$
$338$	0	0
$339$	1.52396i	0.0827701i
$340$	0	0
$341$	− 28.7692i	− 1.55794i
$342$	0	0
$343$	14.3658	0.775679
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.03845i	0.163112i	0.996669	+	0.0815562i	$0.0259890\pi$
$347$	3.03845i	0.163112i	−0.996669	+	0.0815562i	$0.974011\pi$
$348$	0	0
$349$	− 26.1606i	− 1.40035i	−0.713973	−	0.700173i	$-0.753106\pi$
$349$	− 26.1606i	− 1.40035i	0.713973	−	0.700173i	$-0.246894\pi$
$350$	0	0
$351$	−3.30780	−0.176557
$352$	0	0
$353$	29.4943	1.56983	0.784913	−	0.619606i	$-0.212708\pi$
$353$	29.4943	1.56983	0.784913	+	0.619606i	$0.212708\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.70259i	− 0.0901108i
$358$	0	0
$359$	−8.91327	−0.470424	−0.235212	−	0.971944i	$-0.575578\pi$
$359$	−8.91327	−0.470424	−0.235212	+	0.971944i	$0.575578\pi$
$360$	0	0
$361$	8.49064	0.446876
$362$	0	0
$363$	− 2.55638i	− 0.134175i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.6852	1.18416	0.592079	−	0.805880i	$-0.298307\pi$
$367$	22.6852	1.18416	0.592079	+	0.805880i	$0.298307\pi$
$368$	0	0
$369$	5.70406	0.296942
$370$	0	0
$371$	− 1.85695i	− 0.0964081i
$372$	0	0
$373$	6.46312i	0.334648i	0.985902	+	0.167324i	$0.0535125\pi$
$373$	6.46312i	0.334648i	−0.985902	+	0.167324i	$0.946487\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.26145	−0.425486
$378$	0	0
$379$	− 4.31785i	− 0.221793i	−0.993832	−	0.110897i	$-0.964628\pi$
$379$	− 4.31785i	− 0.221793i	0.993832	−	0.110897i	$-0.0353722\pi$
$380$	0	0
$381$	− 6.29441i	− 0.322472i
$382$	0	0
$383$	−11.1049	−0.567432	−0.283716	−	0.958908i	$-0.591567\pi$
$383$	−11.1049	−0.567432	−0.283716	+	0.958908i	$0.591567\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 17.1326i	− 0.870898i
$388$	0	0
$389$	15.5490i	0.788368i	0.919032	+	0.394184i	$0.128973\pi$
$389$	15.5490i	0.788368i	−0.919032	+	0.394184i	$0.871027\pi$
$390$	0	0
$391$	15.8472	0.801426
$392$	0	0
$393$	6.80472	0.343253
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.9682i	1.35350i	0.736214	+	0.676748i	$0.236611\pi$
$397$	26.9682i	1.35350i	−0.736214	+	0.676748i	$0.763389\pi$
$398$	0	0
$399$	1.87750	0.0939924
$400$	0	0
$401$	36.1092	1.80321	0.901603	−	0.432565i	$-0.142391\pi$
$401$	36.1092	1.80321	0.901603	+	0.432565i	$0.142391\pi$
$402$	0	0
$403$	− 8.08775i	− 0.402880i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	42.4476	2.10405
$408$	0	0
$409$	15.5149	0.767163	0.383581	−	0.923507i	$-0.374691\pi$
$409$	15.5149	0.767163	0.383581	+	0.923507i	$0.374691\pi$
$410$	0	0
$411$	− 6.86755i	− 0.338751i
$412$	0	0
$413$	− 6.26576i	− 0.308318i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.3936	−0.508978
$418$	0	0
$419$	− 26.9362i	− 1.31592i	−0.753054	−	0.657959i	$-0.771420\pi$
$419$	− 26.9362i	− 1.31592i	0.753054	−	0.657959i	$-0.228580\pi$
$420$	0	0
$421$	− 6.14641i	− 0.299557i	−0.988720	−	0.149779i	$-0.952144\pi$
$421$	− 6.14641i	− 0.299557i	0.988720	−	0.149779i	$-0.0478561\pi$
$422$	0	0
$423$	−6.13941	−0.298508
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.50721i	0.363299i
$428$	0	0
$429$	− 2.30514i	− 0.111293i
$430$	0	0
$431$	−36.8901	−1.77693	−0.888467	−	0.458940i	$-0.848229\pi$
$431$	−36.8901	−1.77693	−0.888467	+	0.458940i	$0.848229\pi$
$432$	0	0
$433$	−11.5824	−0.556617	−0.278308	−	0.960492i	$-0.589774\pi$
$433$	−11.5824	−0.556617	−0.278308	+	0.960492i	$0.589774\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.4751i	0.835948i
$438$	0	0
$439$	4.15174	0.198152	0.0990760	−	0.995080i	$-0.468411\pi$
$439$	4.15174	0.198152	0.0990760	+	0.995080i	$0.468411\pi$
$440$	0	0
$441$	−15.6699	−0.746185
$442$	0	0
$443$	15.7327i	0.747482i	0.927533	+	0.373741i	$0.121925\pi$
$443$	15.7327i	0.747482i	−0.927533	+	0.373741i	$0.878075\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−7.62190	−0.360504
$448$	0	0
$449$	−20.6861	−0.976239	−0.488120	−	0.872777i	$-0.662317\pi$
$449$	−20.6861	−0.976239	−0.488120	+	0.872777i	$0.662317\pi$
$450$	0	0
$451$	8.33238i	0.392356i
$452$	0	0
$453$	− 5.20396i	− 0.244503i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.7187	1.34340	0.671702	−	0.740821i	$-0.265564\pi$
$457$	28.7187	1.34340	0.671702	+	0.740821i	$0.265564\pi$
$458$	0	0
$459$	8.65228i	0.403854i
$460$	0	0
$461$	− 27.7063i	− 1.29041i	−0.764008	−	0.645207i	$-0.776771\pi$
$461$	− 27.7063i	− 1.29041i	0.764008	−	0.645207i	$-0.223229\pi$
$462$	0	0
$463$	−26.9572	−1.25281	−0.626404	−	0.779499i	$-0.715474\pi$
$463$	−26.9572	−1.25281	−0.626404	+	0.779499i	$0.715474\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 31.3052i	− 1.44863i	−0.689468	−	0.724316i	$-0.742156\pi$
$467$	− 31.3052i	− 1.44863i	0.689468	−	0.724316i	$-0.257844\pi$
$468$	0	0
$469$	− 7.67404i	− 0.354354i
$470$	0	0
$471$	3.84639	0.177232
$472$	0	0
$473$	25.0269	1.15074
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.50187i	0.206127i
$478$	0	0
$479$	32.6888	1.49359	0.746794	−	0.665056i	$-0.231592\pi$
$479$	32.6888	1.49359	0.746794	+	0.665056i	$0.231592\pi$
$480$	0	0
$481$	11.9331	0.544103
$482$	0	0
$483$	− 3.12193i	− 0.142053i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−17.6043	−0.797726	−0.398863	−	0.917011i	$-0.630595\pi$
$487$	−17.6043	−0.797726	−0.398863	+	0.917011i	$0.630595\pi$
$488$	0	0
$489$	4.59256	0.207683
$490$	0	0
$491$	1.31132i	0.0591790i	0.999562	+	0.0295895i	$0.00942000\pi$
$491$	1.31132i	0.0591790i	−0.999562	+	0.0295895i	$0.990580\pi$
$492$	0	0
$493$	21.6097i	0.973250i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.70087	−0.166007
$498$	0	0
$499$	21.2655i	0.951973i	0.879453	+	0.475987i	$0.157909\pi$
$499$	21.2655i	0.951973i	−0.879453	+	0.475987i	$0.842091\pi$
$500$	0	0
$501$	1.28229i	0.0572885i
$502$	0	0
$503$	34.4786	1.53733	0.768663	−	0.639654i	$-0.220922\pi$
$503$	34.4786	1.53733	0.768663	+	0.639654i	$0.220922\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.02132i	0.267416i
$508$	0	0
$509$	8.61795i	0.381984i	0.981592	+	0.190992i	$0.0611704\pi$
$509$	8.61795i	0.381984i	−0.981592	+	0.190992i	$0.938830\pi$
$510$	0	0
$511$	−7.87759	−0.348484
$512$	0	0
$513$	−9.54110	−0.421250
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8.96833i	− 0.394426i
$518$	0	0
$519$	0.302461	0.0132766
$520$	0	0
$521$	−5.28810	−0.231676	−0.115838	−	0.993268i	$-0.536955\pi$
$521$	−5.28810	−0.231676	−0.115838	+	0.993268i	$0.536955\pi$
$522$	0	0
$523$	18.0301i	0.788400i	0.919025	+	0.394200i	$0.128978\pi$
$523$	18.0301i	0.788400i	−0.919025	+	0.394200i	$0.871022\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21.1553	−0.921540
$528$	0	0
$529$	6.05785	0.263385
$530$	0	0
$531$	15.1903i	0.659204i
$532$	0	0
$533$	2.34244i	0.101463i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.28568	−0.228094
$538$	0	0
$539$	− 22.8902i	− 0.985952i
$540$	0	0
$541$	− 2.53085i	− 0.108810i	−0.998519	−	0.0544050i	$-0.982674\pi$
$541$	− 2.53085i	− 0.108810i	0.998519	−	0.0544050i	$-0.0173262\pi$
$542$	0	0
$543$	−9.13296	−0.391933
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 37.9439i	− 1.62236i	−0.584794	−	0.811182i	$-0.698825\pi$
$547$	− 37.9439i	− 1.62236i	0.584794	−	0.811182i	$-0.301175\pi$
$548$	0	0
$549$	− 18.2000i	− 0.776757i
$550$	0	0
$551$	−23.8295	−1.01517
$552$	0	0
$553$	9.71219	0.413004
$554$	0	0
$555$	0	0
$556$	0	0
$557$	37.0123i	1.56826i	0.620596	+	0.784130i	$0.286891\pi$
$557$	37.0123i	1.56826i	−0.620596	+	0.784130i	$0.713109\pi$
$558$	0	0
$559$	7.03571	0.297579
$560$	0	0
$561$	−6.02960	−0.254570
$562$	0	0
$563$	− 14.2506i	− 0.600590i	−0.953846	−	0.300295i	$-0.902915\pi$
$563$	− 14.2506i	− 0.600590i	0.953846	−	0.300295i	$-0.0970852\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.56411	−0.317663
$568$	0	0
$569$	25.9576	1.08820	0.544101	−	0.839020i	$-0.316871\pi$
$569$	25.9576	1.08820	0.544101	+	0.839020i	$0.316871\pi$
$570$	0	0
$571$	20.3690i	0.852416i	0.904625	+	0.426208i	$0.140151\pi$
$571$	20.3690i	0.852416i	−0.904625	+	0.426208i	$0.859849\pi$
$572$	0	0
$573$	9.69977i	0.405214i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−44.6310	−1.85801	−0.929007	−	0.370063i	$-0.879336\pi$
$577$	−44.6310	−1.85801	−0.929007	+	0.370063i	$0.879336\pi$
$578$	0	0
$579$	− 5.26905i	− 0.218974i
$580$	0	0
$581$	− 15.2130i	− 0.631140i
$582$	0	0
$583$	−6.57625	−0.272360
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.9398i	0.781729i	0.920448	+	0.390865i	$0.127824\pi$
$587$	18.9398i	0.781729i	−0.920448	+	0.390865i	$0.872176\pi$
$588$	0	0
$589$	− 23.3285i	− 0.961236i
$590$	0	0
$591$	−0.106270	−0.00437134
$592$	0	0
$593$	8.22399	0.337719	0.168859	−	0.985640i	$-0.445992\pi$
$593$	8.22399	0.337719	0.168859	+	0.985640i	$0.445992\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.30568i	0.258074i
$598$	0	0
$599$	37.2899	1.52362	0.761812	−	0.647798i	$-0.224310\pi$
$599$	37.2899	1.52362	0.761812	+	0.647798i	$0.224310\pi$
$600$	0	0
$601$	23.7882	0.970341	0.485170	−	0.874420i	$-0.338758\pi$
$601$	23.7882	0.970341	0.485170	+	0.874420i	$0.338758\pi$
$602$	0	0
$603$	18.6045i	0.757632i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.6940	1.32701	0.663505	−	0.748172i	$-0.269068\pi$
$607$	32.6940	1.32701	0.663505	+	0.748172i	$0.269068\pi$
$608$	0	0
$609$	4.25715	0.172508
$610$	0	0
$611$	− 2.52123i	− 0.101998i
$612$	0	0
$613$	− 5.70642i	− 0.230480i	−0.993338	−	0.115240i	$-0.963236\pi$
$613$	− 5.70642i	− 0.230480i	0.993338	−	0.115240i	$-0.0367637\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.9653	0.763514	0.381757	−	0.924263i	$-0.375319\pi$
$617$	18.9653	0.763514	0.381757	+	0.924263i	$0.375319\pi$
$618$	0	0
$619$	− 32.0663i	− 1.28885i	−0.764666	−	0.644427i	$-0.777096\pi$
$619$	− 32.0663i	− 1.28885i	0.764666	−	0.644427i	$-0.222904\pi$
$620$	0	0
$621$	15.8651i	0.636643i
$622$	0	0
$623$	12.4076	0.497099
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 6.64901i	− 0.265536i
$628$	0	0
$629$	− 31.2137i	− 1.24457i
$630$	0	0
$631$	11.4737	0.456762	0.228381	−	0.973572i	$-0.426657\pi$
$631$	11.4737	0.456762	0.228381	+	0.973572i	$0.426657\pi$
$632$	0	0
$633$	−2.00880	−0.0798427
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6.43503i	− 0.254965i
$638$	0	0
$639$	8.97215	0.354933
$640$	0	0
$641$	−0.258289	−0.0102018	−0.00510089	−	0.999987i	$-0.501624\pi$
$641$	−0.258289	−0.0102018	−0.00510089	+	0.999987i	$0.501624\pi$
$642$	0	0
$643$	− 31.8119i	− 1.25454i	−0.778802	−	0.627269i	$-0.784173\pi$
$643$	− 31.8119i	− 1.25454i	0.778802	−	0.627269i	$-0.215827\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.3294	0.956486	0.478243	−	0.878228i	$-0.341274\pi$
$647$	24.3294	0.956486	0.478243	+	0.878228i	$0.341274\pi$
$648$	0	0
$649$	−22.1897	−0.871022
$650$	0	0
$651$	4.16764i	0.163343i
$652$	0	0
$653$	− 39.3785i	− 1.54100i	−0.637440	−	0.770500i	$-0.720007\pi$
$653$	− 39.3785i	− 1.54100i	0.637440	−	0.770500i	$-0.279993\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	19.0979	0.745082
$658$	0	0
$659$	− 33.7063i	− 1.31301i	−0.754321	−	0.656506i	$-0.772034\pi$
$659$	− 33.7063i	− 1.31301i	0.754321	−	0.656506i	$-0.227966\pi$
$660$	0	0
$661$	32.9163i	1.28029i	0.768252	+	0.640147i	$0.221127\pi$
$661$	32.9163i	1.28029i	−0.768252	+	0.640147i	$0.778873\pi$
$662$	0	0
$663$	−1.69508	−0.0658313
$664$	0	0
$665$	0	0
$666$	0	0
$667$	39.6241i	1.53425i
$668$	0	0
$669$	12.2367i	0.473097i
$670$	0	0
$671$	26.5862	1.02635
$672$	0	0
$673$	2.87683	0.110894	0.0554468	−	0.998462i	$-0.482342\pi$
$673$	2.87683	0.110894	0.0554468	+	0.998462i	$0.482342\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 28.6415i	− 1.10078i	−0.834907	−	0.550391i	$-0.814479\pi$
$677$	− 28.6415i	− 1.10078i	0.834907	−	0.550391i	$-0.185521\pi$
$678$	0	0
$679$	−21.0753	−0.808794
$680$	0	0
$681$	12.6724	0.485607
$682$	0	0
$683$	18.3147i	0.700792i	0.936602	+	0.350396i	$0.113953\pi$
$683$	18.3147i	0.700792i	−0.936602	+	0.350396i	$0.886047\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−11.3241	−0.432043
$688$	0	0
$689$	−1.84875	−0.0704318
$690$	0	0
$691$	14.4710i	0.550502i	0.961372	+	0.275251i	$0.0887610\pi$
$691$	14.4710i	0.550502i	−0.961372	+	0.275251i	$0.911239\pi$
$692$	0	0
$693$	− 12.3516i	− 0.469198i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.12718	0.232084
$698$	0	0
$699$	− 10.0597i	− 0.380493i
$700$	0	0
$701$	18.5603i	0.701013i	0.936560	+	0.350506i	$0.113991\pi$
$701$	18.5603i	0.701013i	−0.936560	+	0.350506i	$0.886009\pi$
$702$	0	0
$703$	34.4202	1.29818
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.99104i	0.0748809i
$708$	0	0
$709$	− 43.3934i	− 1.62967i	−0.579690	−	0.814837i	$-0.696826\pi$
$709$	− 43.3934i	− 1.62967i	0.579690	−	0.814837i	$-0.303174\pi$
$710$	0	0
$711$	−23.5456	−0.883029
$712$	0	0
$713$	−38.7910	−1.45273
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.35359i	0.237279i
$718$	0	0
$719$	−14.8496	−0.553797	−0.276899	−	0.960899i	$-0.589307\pi$
$719$	−14.8496	−0.553797	−0.276899	+	0.960899i	$0.589307\pi$
$720$	0	0
$721$	4.39356	0.163625
$722$	0	0
$723$	− 10.4706i	− 0.389408i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−6.22824	−0.230993	−0.115496	−	0.993308i	$-0.536846\pi$
$727$	−6.22824	−0.230993	−0.115496	+	0.993308i	$0.536846\pi$
$728$	0	0
$729$	13.8082	0.511416
$730$	0	0
$731$	− 18.4035i	− 0.680676i
$732$	0	0
$733$	43.2406i	1.59713i	0.601910	+	0.798564i	$0.294407\pi$
$733$	43.2406i	1.59713i	−0.601910	+	0.798564i	$0.705593\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27.1770	−1.00108
$738$	0	0
$739$	− 10.0469i	− 0.369582i	−0.982778	−	0.184791i	$-0.940839\pi$
$739$	− 10.0469i	− 0.369582i	0.982778	−	0.184791i	$-0.0591608\pi$
$740$	0	0
$741$	− 1.86921i	− 0.0686670i
$742$	0	0
$743$	−5.85860	−0.214931	−0.107466	−	0.994209i	$-0.534274\pi$
$743$	−5.85860	−0.214931	−0.107466	+	0.994209i	$0.534274\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	36.8814i	1.34942i
$748$	0	0
$749$	− 18.0786i	− 0.660579i
$750$	0	0
$751$	16.0419	0.585377	0.292688	−	0.956208i	$-0.405450\pi$
$751$	16.0419	0.585377	0.292688	+	0.956208i	$0.405450\pi$
$752$	0	0
$753$	11.3900	0.415075
$754$	0	0
$755$	0	0
$756$	0	0
$757$	15.5665i	0.565773i	0.959153	+	0.282886i	$0.0912919\pi$
$757$	15.5665i	0.565773i	−0.959153	+	0.282886i	$0.908708\pi$
$758$	0	0
$759$	−11.0561	−0.401309
$760$	0	0
$761$	−26.9097	−0.975477	−0.487739	−	0.872990i	$-0.662178\pi$
$761$	−26.9097	−0.975477	−0.487739	+	0.872990i	$0.662178\pi$
$762$	0	0
$763$	− 17.6213i	− 0.637933i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.23810	−0.225245
$768$	0	0
$769$	17.0952	0.616470	0.308235	−	0.951310i	$-0.400262\pi$
$769$	17.0952	0.616470	0.308235	+	0.951310i	$0.400262\pi$
$770$	0	0
$771$	6.69975i	0.241286i
$772$	0	0
$773$	6.32624i	0.227539i	0.993507	+	0.113770i	$0.0362926\pi$
$773$	6.32624i	0.227539i	−0.993507	+	0.113770i	$0.963707\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6.14916	−0.220600
$778$	0	0
$779$	6.75661i	0.242081i
$780$	0	0
$781$	13.1063i	0.468981i
$782$	0	0
$783$	−21.6341	−0.773138
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 53.5766i	− 1.90980i	−0.296929	−	0.954899i	$-0.595963\pi$
$787$	− 53.5766i	− 1.90980i	0.296929	−	0.954899i	$-0.404037\pi$
$788$	0	0
$789$	− 15.0003i	− 0.534026i
$790$	0	0
$791$	−3.35339	−0.119233
$792$	0	0
$793$	7.47406	0.265412
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 41.6834i	− 1.47650i	−0.674527	−	0.738251i	$-0.735652\pi$
$797$	− 41.6834i	− 1.47650i	0.674527	−	0.738251i	$-0.264348\pi$
$798$	0	0
$799$	−6.59483	−0.233308
$800$	0	0
$801$	−30.0802	−1.06283
$802$	0	0
$803$	27.8979i	0.984495i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.9914	−0.386917
$808$	0	0
$809$	−26.9010	−0.945788	−0.472894	−	0.881119i	$-0.656791\pi$
$809$	−26.9010	−0.945788	−0.472894	+	0.881119i	$0.656791\pi$
$810$	0	0
$811$	5.88980i	0.206819i	0.994639	+	0.103409i	$0.0329752\pi$
$811$	5.88980i	0.206819i	−0.994639	+	0.103409i	$0.967025\pi$
$812$	0	0
$813$	− 0.594069i	− 0.0208349i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20.2940	0.709997
$818$	0	0
$819$	− 3.47234i	− 0.121334i
$820$	0	0
$821$	− 48.2445i	− 1.68374i	−0.539677	−	0.841872i	$-0.681454\pi$
$821$	− 48.2445i	− 1.68374i	0.539677	−	0.841872i	$-0.318546\pi$
$822$	0	0
$823$	51.9289	1.81013	0.905063	−	0.425277i	$-0.139823\pi$
$823$	51.9289	1.81013	0.905063	+	0.425277i	$0.139823\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.2849i	0.983562i	0.870719	+	0.491781i	$0.163654\pi$
$827$	28.2849i	0.983562i	−0.870719	+	0.491781i	$0.836346\pi$
$828$	0	0
$829$	− 14.2294i	− 0.494209i	−0.968989	−	0.247104i	$-0.920521\pi$
$829$	− 14.2294i	− 0.494209i	0.968989	−	0.247104i	$-0.0794790\pi$
$830$	0	0
$831$	−8.62168	−0.299083
$832$	0	0
$833$	−16.8323	−0.583203
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 21.1792i	− 0.732060i
$838$	0	0
$839$	−5.04346	−0.174120	−0.0870598	−	0.996203i	$-0.527747\pi$
$839$	−5.04346	−0.174120	−0.0870598	+	0.996203i	$0.527747\pi$
$840$	0	0
$841$	−25.0325	−0.863191
$842$	0	0
$843$	4.61277i	0.158872i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.62517	0.193283
$848$	0	0
$849$	7.92101	0.271848
$850$	0	0
$851$	− 57.2343i	− 1.96197i
$852$	0	0
$853$	46.2026i	1.58195i	0.611850	+	0.790974i	$0.290426\pi$
$853$	46.2026i	1.58195i	−0.611850	+	0.790974i	$0.709574\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.5405	1.41900	0.709499	−	0.704707i	$-0.248921\pi$
$857$	41.5405	1.41900	0.709499	+	0.704707i	$0.248921\pi$
$858$	0	0
$859$	− 37.9644i	− 1.29533i	−0.761925	−	0.647665i	$-0.775746\pi$
$859$	− 37.9644i	− 1.29533i	0.761925	−	0.647665i	$-0.224254\pi$
$860$	0	0
$861$	− 1.20707i	− 0.0411368i
$862$	0	0
$863$	2.89022	0.0983843	0.0491922	−	0.998789i	$-0.484335\pi$
$863$	2.89022	0.0983843	0.0491922	+	0.998789i	$0.484335\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 4.28761i	− 0.145615i
$868$	0	0
$869$	− 34.3949i	− 1.16677i
$870$	0	0
$871$	−7.64015	−0.258877
$872$	0	0
$873$	51.0935	1.72925
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 43.0916i	− 1.45510i	−0.686055	−	0.727550i	$-0.740659\pi$
$877$	− 43.0916i	− 1.45510i	0.686055	−	0.727550i	$-0.259341\pi$
$878$	0	0
$879$	0.831423	0.0280432
$880$	0	0
$881$	−2.06410	−0.0695414	−0.0347707	−	0.999395i	$-0.511070\pi$
$881$	−2.06410	−0.0695414	−0.0347707	+	0.999395i	$0.511070\pi$
$882$	0	0
$883$	− 51.4835i	− 1.73256i	−0.499562	−	0.866278i	$-0.666506\pi$
$883$	− 51.4835i	− 1.73256i	0.499562	−	0.866278i	$-0.333494\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−55.5749	−1.86602	−0.933012	−	0.359845i	$-0.882829\pi$
$887$	−55.5749	−1.86602	−0.933012	+	0.359845i	$0.882829\pi$
$888$	0	0
$889$	13.8505	0.464531
$890$	0	0
$891$	26.7877i	0.897422i
$892$	0	0
$893$	− 7.27229i	− 0.243358i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.10814	−0.103778
$898$	0	0
$899$	− 52.8965i	− 1.76420i
$900$	0	0
$901$	4.83582i	0.161105i
$902$	0	0
$903$	−3.62552	−0.120650
$904$	0	0
$905$	0	0
$906$	0	0
$907$	15.8669i	0.526852i	0.964680	+	0.263426i	$0.0848526\pi$
$907$	15.8669i	0.526852i	−0.964680	+	0.263426i	$0.915147\pi$
$908$	0	0
$909$	− 4.82696i	− 0.160100i
$910$	0	0
$911$	30.4293	1.00817	0.504084	−	0.863655i	$-0.331830\pi$
$911$	30.4293	1.00817	0.504084	+	0.863655i	$0.331830\pi$
$912$	0	0
$913$	−53.8755	−1.78302
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.9734i	0.494465i
$918$	0	0
$919$	15.1819	0.500803	0.250402	−	0.968142i	$-0.419437\pi$
$919$	15.1819	0.500803	0.250402	+	0.968142i	$0.419437\pi$
$920$	0	0
$921$	6.83572	0.225245
$922$	0	0
$923$	3.68452i	0.121278i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10.6515	−0.349840
$928$	0	0
$929$	−8.00028	−0.262481	−0.131240	−	0.991351i	$-0.541896\pi$
$929$	−8.00028	−0.262481	−0.131240	+	0.991351i	$0.541896\pi$
$930$	0	0
$931$	− 18.5614i	− 0.608325i
$932$	0	0
$933$	1.64403i	0.0538231i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.0303	0.327676	0.163838	−	0.986487i	$-0.447613\pi$
$937$	10.0303	0.327676	0.163838	+	0.986487i	$0.447613\pi$
$938$	0	0
$939$	− 12.4519i	− 0.406353i
$940$	0	0
$941$	− 4.23635i	− 0.138101i	−0.997613	−	0.0690506i	$-0.978003\pi$
$941$	− 4.23635i	− 0.138101i	0.997613	−	0.0690506i	$-0.0219970\pi$
$942$	0	0
$943$	11.2350	0.365861
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.33715i	− 0.108443i	−0.998529	−	0.0542213i	$-0.982732\pi$
$947$	− 3.33715i	− 0.108443i	0.998529	−	0.0542213i	$-0.0172677\pi$
$948$	0	0
$949$	7.84281i	0.254588i
$950$	0	0
$951$	−1.94528	−0.0630800
$952$	0	0
$953$	−10.9309	−0.354086	−0.177043	−	0.984203i	$-0.556653\pi$
$953$	−10.9309	−0.354086	−0.177043	+	0.984203i	$0.556653\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 15.0763i	− 0.487349i
$958$	0	0
$959$	15.1117	0.487981
$960$	0	0
$961$	20.7844	0.670463
$962$	0	0
$963$	43.8287i	1.41236i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.7599	1.31075	0.655375	−	0.755304i	$-0.272511\pi$
$967$	40.7599	1.31075	0.655375	+	0.755304i	$0.272511\pi$
$968$	0	0
$969$	−4.88932	−0.157068
$970$	0	0
$971$	− 56.9328i	− 1.82706i	−0.406772	−	0.913530i	$-0.633346\pi$
$971$	− 56.9328i	− 1.82706i	0.406772	−	0.913530i	$-0.366654\pi$
$972$	0	0
$973$	− 22.8706i	− 0.733197i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.1254	0.547892	0.273946	−	0.961745i	$-0.411671\pi$
$977$	17.1254	0.547892	0.273946	+	0.961745i	$0.411671\pi$
$978$	0	0
$979$	− 43.9405i	− 1.40434i
$980$	0	0
$981$	42.7199i	1.36394i
$982$	0	0
$983$	−51.3738	−1.63857	−0.819284	−	0.573388i	$-0.805629\pi$
$983$	−51.3738	−1.63857	−0.819284	+	0.573388i	$0.805629\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.29919i	0.0413538i
$988$	0	0
$989$	− 33.7451i	− 1.07303i
$990$	0	0
$991$	23.9933	0.762171	0.381085	−	0.924540i	$-0.375550\pi$
$991$	23.9933	0.762171	0.381085	+	0.924540i	$0.375550\pi$
$992$	0	0
$993$	−1.16045	−0.0368256
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 7.90228i	− 0.250268i	−0.992140	−	0.125134i	$-0.960064\pi$
$997$	− 7.90228i	− 0.250268i	0.992140	−	0.125134i	$-0.0399361\pi$
$998$	0	0
$999$	31.2489	0.988673

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.d.c.2001.20		40
4.3	odd	2		1000.2.d.c.501.6	yes	40
5.2	odd	4		4000.2.f.c.3249.13		20
5.3	odd	4		4000.2.f.d.3249.8		20
5.4	even	2	inner	4000.2.d.c.2001.21		40
8.3	odd	2		1000.2.d.c.501.5	✓	40
8.5	even	2	inner	4000.2.d.c.2001.19		40
20.3	even	4		1000.2.f.d.749.12		20
20.7	even	4		1000.2.f.c.749.9		20
20.19	odd	2		1000.2.d.c.501.35	yes	40
40.3	even	4		1000.2.f.c.749.10		20
40.13	odd	4		4000.2.f.c.3249.14		20
40.19	odd	2		1000.2.d.c.501.36	yes	40
40.27	even	4		1000.2.f.d.749.11		20
40.29	even	2	inner	4000.2.d.c.2001.22		40
40.37	odd	4		4000.2.f.d.3249.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.d.c.501.5	✓	40	8.3	odd	2
1000.2.d.c.501.6	yes	40	4.3	odd	2
1000.2.d.c.501.35	yes	40	20.19	odd	2
1000.2.d.c.501.36	yes	40	40.19	odd	2
1000.2.f.c.749.9		20	20.7	even	4
1000.2.f.c.749.10		20	40.3	even	4
1000.2.f.d.749.11		20	40.27	even	4
1000.2.f.d.749.12		20	20.3	even	4
4000.2.d.c.2001.19		40	8.5	even	2	inner
4000.2.d.c.2001.20		40	1.1	even	1	trivial
4000.2.d.c.2001.21		40	5.4	even	2	inner
4000.2.d.c.2001.22		40	40.29	even	2	inner
4000.2.f.c.3249.13		20	5.2	odd	4
4000.2.f.c.3249.14		20	40.13	odd	4
4000.2.f.d.3249.7		20	40.37	odd	4
4000.2.f.d.3249.8		20	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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.513027i q^{3} -1.12889 q^{7} +2.73680 q^{9} +O(q^{10})\)	\(q+0.513027i q^{3} -1.12889 q^{7} +2.73680 q^{9} +3.99787i q^{11} +1.12390i q^{13} +2.93982 q^{17} +3.24181i q^{19} -0.579150i q^{21} +5.39053 q^{23} +2.94314i q^{27} +7.35068i q^{29} -7.19613 q^{31} -2.05101 q^{33} -10.6176i q^{37} -0.576592 q^{39} +2.08421 q^{41} -6.26007i q^{43} -2.24328 q^{47} -5.72561 q^{49} +1.50821i q^{51} +1.64494i q^{53} -1.66314 q^{57} +5.55039i q^{59} -6.65010i q^{61} -3.08954 q^{63} +6.79788i q^{67} +2.76549i q^{69} +3.27833 q^{71} +6.97819 q^{73} -4.51314i q^{77} -8.60333 q^{79} +6.70050 q^{81} +13.4761i q^{83} -3.77110 q^{87} -10.9910 q^{89} -1.26876i q^{91} -3.69181i q^{93} +18.6691 q^{97} +10.9414i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 24 q^{9}+O(q^{10}) \) 40 * q - 24 * q^9	\( 40 q - 24 q^{9} - 48 q^{31} - 8 q^{39} + 44 q^{41} + 12 q^{49} - 96 q^{71} - 96 q^{79} - 56 q^{81} - 44 q^{89}+O(q^{100}) \) 40 * q - 24 * q^9 - 48 * q^31 - 8 * q^39 + 44 * q^41 + 12 * q^49 - 96 * q^71 - 96 * q^79 - 56 * q^81 - 44 * q^89

Introduction

L-functions

Modular forms

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Database

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