LMFDB - Embedded newform 3920.2.a.cd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.16448.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 14 $ x^4 - 2x^3 - 7x^2 + 8*x + 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1960)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.18398$ of defining polynomial
Character	$\chi$	$=$	3920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.18398 q^{3} +1.00000 q^{5} +1.76977 q^{9} +O(q^{10})$	$q-2.18398 q^{3} +1.00000 q^{5} +1.76977 q^{9} -3.59819 q^{11} +5.85838 q^{13} -2.18398 q^{15} -6.36121 q^{17} -4.50283 q^{19} -2.62814 q^{23} +1.00000 q^{25} +2.68681 q^{27} +2.05866 q^{29} +6.62814 q^{31} +7.85838 q^{33} -5.91704 q^{37} -12.7946 q^{39} +7.22348 q^{41} +5.34880 q^{43} +1.76977 q^{45} -2.31885 q^{47} +13.8927 q^{51} -4.10777 q^{53} -3.59819 q^{55} +9.83408 q^{57} +5.07107 q^{59} -7.37361 q^{61} +5.85838 q^{65} -3.39667 q^{67} +5.73981 q^{69} -16.7224 q^{71} +14.7184 q^{73} -2.18398 q^{75} +3.25505 q^{79} -11.1772 q^{81} -10.6936 q^{83} -6.36121 q^{85} -4.49607 q^{87} +11.1540 q^{89} -14.4757 q^{93} -4.50283 q^{95} +17.1896 q^{97} -6.36796 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5} + 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 4 * q^5 + 6 * q^9	$ 4 q - 2 q^{3} + 4 q^{5} + 6 q^{9} - 2 q^{11} + 10 q^{13} - 2 q^{15} + 6 q^{17} + 4 q^{23} + 4 q^{25} - 14 q^{27} - 2 q^{29} + 12 q^{31} + 18 q^{33} - 14 q^{39} + 12 q^{41} + 8 q^{43} + 6 q^{45} + 2 q^{47} - 2 q^{51} - 4 q^{53} - 2 q^{55} - 8 q^{57} - 8 q^{59} + 20 q^{61} + 10 q^{65} + 8 q^{67} + 24 q^{69} - 4 q^{71} + 16 q^{73} - 2 q^{75} - 22 q^{79} - 20 q^{81} - 36 q^{83} + 6 q^{85} + 18 q^{87} + 40 q^{89} - 32 q^{93} + 26 q^{97} - 12 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 + 6 * q^9 - 2 * q^11 + 10 * q^13 - 2 * q^15 + 6 * q^17 + 4 * q^23 + 4 * q^25 - 14 * q^27 - 2 * q^29 + 12 * q^31 + 18 * q^33 - 14 * q^39 + 12 * q^41 + 8 * q^43 + 6 * q^45 + 2 * q^47 - 2 * q^51 - 4 * q^53 - 2 * q^55 - 8 * q^57 - 8 * q^59 + 20 * q^61 + 10 * q^65 + 8 * q^67 + 24 * q^69 - 4 * q^71 + 16 * q^73 - 2 * q^75 - 22 * q^79 - 20 * q^81 - 36 * q^83 + 6 * q^85 + 18 * q^87 + 40 * q^89 - 32 * q^93 + 26 * q^97 - 12 * 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
$3$	−2.18398	−1.26092	−0.630461	−	0.776221i	$-0.717134\pi$
$3$	−2.18398	−1.26092	−0.630461	+	0.776221i	$0.717134\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.76977	0.589922
$10$	0	0
$11$	−3.59819	−1.08490	−0.542448	−	0.840089i	$-0.682502\pi$
$11$	−3.59819	−1.08490	−0.542448	+	0.840089i	$0.682502\pi$
$12$	0	0
$13$	5.85838	1.62482	0.812411	−	0.583085i	$-0.198155\pi$
$13$	5.85838	1.62482	0.812411	+	0.583085i	$0.198155\pi$
$14$	0	0
$15$	−2.18398	−0.563901
$16$	0	0
$17$	−6.36121	−1.54282	−0.771410	−	0.636339i	$-0.780448\pi$
$17$	−6.36121	−1.54282	−0.771410	+	0.636339i	$0.780448\pi$
$18$	0	0
$19$	−4.50283	−1.03302	−0.516510	−	0.856281i	$-0.672769\pi$
$19$	−4.50283	−1.03302	−0.516510	+	0.856281i	$0.672769\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.62814	−0.548006	−0.274003	−	0.961729i	$-0.588348\pi$
$23$	−2.62814	−0.548006	−0.274003	+	0.961729i	$0.588348\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.68681	0.517076
$28$	0	0
$29$	2.05866	0.382284	0.191142	−	0.981562i	$-0.438781\pi$
$29$	2.05866	0.382284	0.191142	+	0.981562i	$0.438781\pi$
$30$	0	0
$31$	6.62814	1.19045	0.595225	−	0.803559i	$-0.297063\pi$
$31$	6.62814	1.19045	0.595225	+	0.803559i	$0.297063\pi$
$32$	0	0
$33$	7.85838	1.36797
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.91704	−0.972755	−0.486378	−	0.873749i	$-0.661682\pi$
$37$	−5.91704	−0.972755	−0.486378	+	0.873749i	$0.661682\pi$
$38$	0	0
$39$	−12.7946	−2.04877
$40$	0	0
$41$	7.22348	1.12812	0.564059	−	0.825734i	$-0.309239\pi$
$41$	7.22348	1.12812	0.564059	+	0.825734i	$0.309239\pi$
$42$	0	0
$43$	5.34880	0.815684	0.407842	−	0.913052i	$-0.366281\pi$
$43$	5.34880	0.815684	0.407842	+	0.913052i	$0.366281\pi$
$44$	0	0
$45$	1.76977	0.263821
$46$	0	0
$47$	−2.31885	−0.338239	−0.169119	−	0.985596i	$-0.554092\pi$
$47$	−2.31885	−0.338239	−0.169119	+	0.985596i	$0.554092\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	13.8927	1.94537
$52$	0	0
$53$	−4.10777	−0.564246	−0.282123	−	0.959378i	$-0.591039\pi$
$53$	−4.10777	−0.564246	−0.282123	+	0.959378i	$0.591039\pi$
$54$	0	0
$55$	−3.59819	−0.485180
$56$	0	0
$57$	9.83408	1.30256
$58$	0	0
$59$	5.07107	0.660197	0.330098	−	0.943947i	$-0.392918\pi$
$59$	5.07107	0.660197	0.330098	+	0.943947i	$0.392918\pi$
$60$	0	0
$61$	−7.37361	−0.944094	−0.472047	−	0.881573i	$-0.656485\pi$
$61$	−7.37361	−0.944094	−0.472047	+	0.881573i	$0.656485\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.85838	0.726642
$66$	0	0
$67$	−3.39667	−0.414969	−0.207485	−	0.978238i	$-0.566528\pi$
$67$	−3.39667	−0.414969	−0.207485	+	0.978238i	$0.566528\pi$
$68$	0	0
$69$	5.73981	0.690992
$70$	0	0
$71$	−16.7224	−1.98459	−0.992293	−	0.123917i	$-0.960454\pi$
$71$	−16.7224	−1.98459	−0.992293	+	0.123917i	$0.960454\pi$
$72$	0	0
$73$	14.7184	1.72266	0.861328	−	0.508050i	$-0.169633\pi$
$73$	14.7184	1.72266	0.861328	+	0.508050i	$0.169633\pi$
$74$	0	0
$75$	−2.18398	−0.252184
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.25505	0.366221	0.183111	−	0.983092i	$-0.441383\pi$
$79$	3.25505	0.366221	0.183111	+	0.983092i	$0.441383\pi$
$80$	0	0
$81$	−11.1772	−1.24191
$82$	0	0
$83$	−10.6936	−1.17377	−0.586885	−	0.809670i	$-0.699646\pi$
$83$	−10.6936	−1.17377	−0.586885	+	0.809670i	$0.699646\pi$
$84$	0	0
$85$	−6.36121	−0.689970
$86$	0	0
$87$	−4.49607	−0.482030
$88$	0	0
$89$	11.1540	1.18232	0.591162	−	0.806553i	$-0.298669\pi$
$89$	11.1540	1.18232	0.591162	+	0.806553i	$0.298669\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−14.4757	−1.50106
$94$	0	0
$95$	−4.50283	−0.461980
$96$	0	0
$97$	17.1896	1.74534	0.872671	−	0.488308i	$-0.162386\pi$
$97$	17.1896	1.74534	0.872671	+	0.488308i	$0.162386\pi$
$98$	0	0
$99$	−6.36796	−0.640004
$100$	0	0
$101$	3.44131	0.342423	0.171212	−	0.985234i	$-0.445232\pi$
$101$	3.44131	0.342423	0.171212	+	0.985234i	$0.445232\pi$
$102$	0	0
$103$	11.9757	1.18000	0.590000	−	0.807403i	$-0.299128\pi$
$103$	11.9757	1.18000	0.590000	+	0.807403i	$0.299128\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.6760	1.03209	0.516045	−	0.856562i	$-0.327404\pi$
$107$	10.6760	1.03209	0.516045	+	0.856562i	$0.327404\pi$
$108$	0	0
$109$	−8.33411	−0.798263	−0.399131	−	0.916894i	$-0.630688\pi$
$109$	−8.33411	−0.798263	−0.399131	+	0.916894i	$0.630688\pi$
$110$	0	0
$111$	12.9227	1.22657
$112$	0	0
$113$	−12.1542	−1.14337	−0.571684	−	0.820474i	$-0.693710\pi$
$113$	−12.1542	−1.14337	−0.571684	+	0.820474i	$0.693710\pi$
$114$	0	0
$115$	−2.62814	−0.245076
$116$	0	0
$117$	10.3680	0.958518
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.94699	0.176999
$122$	0	0
$123$	−15.7759	−1.42247
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.82453	0.871786	0.435893	−	0.899998i	$-0.356433\pi$
$127$	9.82453	0.871786	0.435893	+	0.899998i	$0.356433\pi$
$128$	0	0
$129$	−11.6817	−1.02851
$130$	0	0
$131$	−4.95375	−0.432811	−0.216405	−	0.976304i	$-0.569433\pi$
$131$	−4.95375	−0.432811	−0.216405	+	0.976304i	$0.569433\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.68681	0.231243
$136$	0	0
$137$	−11.3137	−0.966595	−0.483298	−	0.875456i	$-0.660561\pi$
$137$	−11.3137	−0.966595	−0.483298	+	0.875456i	$0.660561\pi$
$138$	0	0
$139$	−8.70311	−0.738188	−0.369094	−	0.929392i	$-0.620332\pi$
$139$	−8.70311	−0.738188	−0.369094	+	0.929392i	$0.620332\pi$
$140$	0	0
$141$	5.06431	0.426492
$142$	0	0
$143$	−21.0796	−1.76276
$144$	0	0
$145$	2.05866	0.170963
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.45268	−0.528624	−0.264312	−	0.964437i	$-0.585145\pi$
$149$	−6.45268	−0.528624	−0.264312	+	0.964437i	$0.585145\pi$
$150$	0	0
$151$	4.02606	0.327636	0.163818	−	0.986491i	$-0.447619\pi$
$151$	4.02606	0.327636	0.163818	+	0.986491i	$0.447619\pi$
$152$	0	0
$153$	−11.2578	−0.910143
$154$	0	0
$155$	6.62814	0.532385
$156$	0	0
$157$	15.5260	1.23911	0.619556	−	0.784953i	$-0.287313\pi$
$157$	15.5260	1.23911	0.619556	+	0.784953i	$0.287313\pi$
$158$	0	0
$159$	8.97129	0.711470
$160$	0	0
$161$	0	0
$162$	0	0
$163$	24.9610	1.95510	0.977549	−	0.210710i	$-0.0675774\pi$
$163$	24.9610	1.95510	0.977549	+	0.210710i	$0.0675774\pi$
$164$	0	0
$165$	7.85838	0.611774
$166$	0	0
$167$	16.0739	1.24384	0.621919	−	0.783082i	$-0.286353\pi$
$167$	16.0739	1.24384	0.621919	+	0.783082i	$0.286353\pi$
$168$	0	0
$169$	21.3206	1.64005
$170$	0	0
$171$	−7.96895	−0.609401
$172$	0	0
$173$	−0.0356054	−0.00270703	−0.00135351	−	0.999999i	$-0.500431\pi$
$173$	−0.0356054	−0.00270703	−0.00135351	+	0.999999i	$0.500431\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.0751	−0.832456
$178$	0	0
$179$	4.45268	0.332809	0.166404	−	0.986058i	$-0.446784\pi$
$179$	4.45268	0.332809	0.166404	+	0.986058i	$0.446784\pi$
$180$	0	0
$181$	−22.7377	−1.69008	−0.845039	−	0.534705i	$-0.820423\pi$
$181$	−22.7377	−1.69008	−0.845039	+	0.534705i	$0.820423\pi$
$182$	0	0
$183$	16.1038	1.19043
$184$	0	0
$185$	−5.91704	−0.435029
$186$	0	0
$187$	22.8888	1.67380
$188$	0	0
$189$	0	0
$190$	0	0
$191$	25.5521	1.84888	0.924442	−	0.381323i	$-0.124531\pi$
$191$	25.5521	1.84888	0.924442	+	0.381323i	$0.124531\pi$
$192$	0	0
$193$	21.0904	1.51812	0.759059	−	0.651022i	$-0.225659\pi$
$193$	21.0904	1.51812	0.759059	+	0.651022i	$0.225659\pi$
$194$	0	0
$195$	−12.7946	−0.916239
$196$	0	0
$197$	16.4622	1.17288	0.586442	−	0.809991i	$-0.300528\pi$
$197$	16.4622	1.17288	0.586442	+	0.809991i	$0.300528\pi$
$198$	0	0
$199$	25.9419	1.83897	0.919485	−	0.393126i	$-0.128606\pi$
$199$	25.9419	1.83897	0.919485	+	0.393126i	$0.128606\pi$
$200$	0	0
$201$	7.41825	0.523243
$202$	0	0
$203$	0	0
$204$	0	0
$205$	7.22348	0.504510
$206$	0	0
$207$	−4.65120	−0.323281
$208$	0	0
$209$	16.2020	1.12072
$210$	0	0
$211$	−8.84877	−0.609174	−0.304587	−	0.952484i	$-0.598519\pi$
$211$	−8.84877	−0.609174	−0.304587	+	0.952484i	$0.598519\pi$
$212$	0	0
$213$	36.5214	2.50241
$214$	0	0
$215$	5.34880	0.364785
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−32.1446	−2.17213
$220$	0	0
$221$	−37.2664	−2.50681
$222$	0	0
$223$	−4.06042	−0.271906	−0.135953	−	0.990715i	$-0.543410\pi$
$223$	−4.06042	−0.271906	−0.135953	+	0.990715i	$0.543410\pi$
$224$	0	0
$225$	1.76977	0.117984
$226$	0	0
$227$	11.9606	0.793856	0.396928	−	0.917850i	$-0.370076\pi$
$227$	11.9606	0.793856	0.396928	+	0.917850i	$0.370076\pi$
$228$	0	0
$229$	11.3042	0.747000	0.373500	−	0.927630i	$-0.378158\pi$
$229$	11.3042	0.747000	0.373500	+	0.927630i	$0.378158\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.3059	1.39580	0.697898	−	0.716197i	$-0.254119\pi$
$233$	21.3059	1.39580	0.697898	+	0.716197i	$0.254119\pi$
$234$	0	0
$235$	−2.31885	−0.151265
$236$	0	0
$237$	−7.10896	−0.461776
$238$	0	0
$239$	22.5361	1.45774	0.728871	−	0.684651i	$-0.240045\pi$
$239$	22.5361	1.45774	0.728871	+	0.684651i	$0.240045\pi$
$240$	0	0
$241$	−12.8477	−0.827595	−0.413798	−	0.910369i	$-0.635798\pi$
$241$	−12.8477	−0.827595	−0.413798	+	0.910369i	$0.635798\pi$
$242$	0	0
$243$	16.3504	1.04888
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−26.3793	−1.67847
$248$	0	0
$249$	23.3545	1.48003
$250$	0	0
$251$	−23.7361	−1.49821	−0.749103	−	0.662453i	$-0.769515\pi$
$251$	−23.7361	−1.49821	−0.749103	+	0.662453i	$0.769515\pi$
$252$	0	0
$253$	9.45657	0.594530
$254$	0	0
$255$	13.8927	0.869997
$256$	0	0
$257$	13.2139	0.824262	0.412131	−	0.911125i	$-0.364785\pi$
$257$	13.2139	0.824262	0.412131	+	0.911125i	$0.364785\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.64335	0.225518
$262$	0	0
$263$	28.8532	1.77917	0.889584	−	0.456773i	$-0.150995\pi$
$263$	28.8532	1.77917	0.889584	+	0.456773i	$0.150995\pi$
$264$	0	0
$265$	−4.10777	−0.252338
$266$	0	0
$267$	−24.3602	−1.49082
$268$	0	0
$269$	2.57073	0.156740	0.0783700	−	0.996924i	$-0.475028\pi$
$269$	2.57073	0.156740	0.0783700	+	0.996924i	$0.475028\pi$
$270$	0	0
$271$	−6.74722	−0.409865	−0.204932	−	0.978776i	$-0.565697\pi$
$271$	−6.74722	−0.409865	−0.204932	+	0.978776i	$0.565697\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.59819	−0.216979
$276$	0	0
$277$	−3.25453	−0.195546	−0.0977730	−	0.995209i	$-0.531172\pi$
$277$	−3.25453	−0.195546	−0.0977730	+	0.995209i	$0.531172\pi$
$278$	0	0
$279$	11.7303	0.702273
$280$	0	0
$281$	−2.55422	−0.152372	−0.0761860	−	0.997094i	$-0.524274\pi$
$281$	−2.55422	−0.152372	−0.0761860	+	0.997094i	$0.524274\pi$
$282$	0	0
$283$	12.5136	0.743857	0.371929	−	0.928261i	$-0.378697\pi$
$283$	12.5136	0.743857	0.371929	+	0.928261i	$0.378697\pi$
$284$	0	0
$285$	9.83408	0.582521
$286$	0	0
$287$	0	0
$288$	0	0
$289$	23.4649	1.38029
$290$	0	0
$291$	−37.5418	−2.20074
$292$	0	0
$293$	−25.6574	−1.49892	−0.749460	−	0.662050i	$-0.769687\pi$
$293$	−25.6574	−1.49892	−0.749460	+	0.662050i	$0.769687\pi$
$294$	0	0
$295$	5.07107	0.295249
$296$	0	0
$297$	−9.66765	−0.560974
$298$	0	0
$299$	−15.3967	−0.890412
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.51575	−0.431768
$304$	0	0
$305$	−7.37361	−0.422212
$306$	0	0
$307$	11.9391	0.681398	0.340699	−	0.940172i	$-0.389336\pi$
$307$	11.9391	0.681398	0.340699	+	0.940172i	$0.389336\pi$
$308$	0	0
$309$	−26.1547	−1.48789
$310$	0	0
$311$	−8.49879	−0.481922	−0.240961	−	0.970535i	$-0.577463\pi$
$311$	−8.49879	−0.481922	−0.240961	+	0.970535i	$0.577463\pi$
$312$	0	0
$313$	−1.75612	−0.0992616	−0.0496308	−	0.998768i	$-0.515804\pi$
$313$	−1.75612	−0.0992616	−0.0496308	+	0.998768i	$0.515804\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.96491	0.335023	0.167511	−	0.985870i	$-0.446427\pi$
$317$	5.96491	0.335023	0.167511	+	0.985870i	$0.446427\pi$
$318$	0	0
$319$	−7.40746	−0.414738
$320$	0	0
$321$	−23.3162	−1.30138
$322$	0	0
$323$	28.6434	1.59376
$324$	0	0
$325$	5.85838	0.324964
$326$	0	0
$327$	18.2015	1.00655
$328$	0	0
$329$	0	0
$330$	0	0
$331$	14.1038	0.775216	0.387608	−	0.921824i	$-0.373301\pi$
$331$	14.1038	0.775216	0.387608	+	0.921824i	$0.373301\pi$
$332$	0	0
$333$	−10.4718	−0.573850
$334$	0	0
$335$	−3.39667	−0.185580
$336$	0	0
$337$	−22.8876	−1.24677	−0.623384	−	0.781916i	$-0.714242\pi$
$337$	−22.8876	−1.24677	−0.623384	+	0.781916i	$0.714242\pi$
$338$	0	0
$339$	26.5445	1.44170
$340$	0	0
$341$	−23.8493	−1.29151
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5.73981	0.309021
$346$	0	0
$347$	23.6586	1.27006	0.635030	−	0.772487i	$-0.280988\pi$
$347$	23.6586	1.27006	0.635030	+	0.772487i	$0.280988\pi$
$348$	0	0
$349$	27.7912	1.48763	0.743814	−	0.668386i	$-0.233015\pi$
$349$	27.7912	1.48763	0.743814	+	0.668386i	$0.233015\pi$
$350$	0	0
$351$	15.7403	0.840157
$352$	0	0
$353$	21.1680	1.12666	0.563331	−	0.826231i	$-0.309520\pi$
$353$	21.1680	1.12666	0.563331	+	0.826231i	$0.309520\pi$
$354$	0	0
$355$	−16.7224	−0.887533
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−36.2843	−1.91501	−0.957505	−	0.288416i	$-0.906872\pi$
$359$	−36.2843	−1.91501	−0.957505	+	0.288416i	$0.906872\pi$
$360$	0	0
$361$	1.27545	0.0671289
$362$	0	0
$363$	−4.25219	−0.223182
$364$	0	0
$365$	14.7184	0.770395
$366$	0	0
$367$	31.1586	1.62646	0.813232	−	0.581939i	$-0.197706\pi$
$367$	31.1586	1.62646	0.813232	+	0.581939i	$0.197706\pi$
$368$	0	0
$369$	12.7839	0.665502
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6.67279	−0.345504	−0.172752	−	0.984965i	$-0.555266\pi$
$373$	−6.67279	−0.345504	−0.172752	+	0.984965i	$0.555266\pi$
$374$	0	0
$375$	−2.18398	−0.112780
$376$	0	0
$377$	12.0604	0.621143
$378$	0	0
$379$	−21.3984	−1.09916	−0.549582	−	0.835440i	$-0.685213\pi$
$379$	−21.3984	−1.09916	−0.549582	+	0.835440i	$0.685213\pi$
$380$	0	0
$381$	−21.4566	−1.09925
$382$	0	0
$383$	−3.63453	−0.185716	−0.0928578	−	0.995679i	$-0.529600\pi$
$383$	−3.63453	−0.185716	−0.0928578	+	0.995679i	$0.529600\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.46612	0.481190
$388$	0	0
$389$	21.8952	1.11013	0.555066	−	0.831806i	$-0.312693\pi$
$389$	21.8952	1.11013	0.555066	+	0.831806i	$0.312693\pi$
$390$	0	0
$391$	16.7182	0.845474
$392$	0	0
$393$	10.8189	0.545740
$394$	0	0
$395$	3.25505	0.163779
$396$	0	0
$397$	−17.9757	−0.902175	−0.451087	−	0.892480i	$-0.648964\pi$
$397$	−17.9757	−0.902175	−0.451087	+	0.892480i	$0.648964\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.6343	1.37999	0.689996	−	0.723813i	$-0.257612\pi$
$401$	27.6343	1.37999	0.689996	+	0.723813i	$0.257612\pi$
$402$	0	0
$403$	38.8302	1.93427
$404$	0	0
$405$	−11.1772	−0.555401
$406$	0	0
$407$	21.2907	1.05534
$408$	0	0
$409$	−39.9261	−1.97422	−0.987108	−	0.160053i	$-0.948833\pi$
$409$	−39.9261	−1.97422	−0.987108	+	0.160053i	$0.948833\pi$
$410$	0	0
$411$	24.7089	1.21880
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−10.6936	−0.524926
$416$	0	0
$417$	19.0074	0.930797
$418$	0	0
$419$	−2.89384	−0.141373	−0.0706867	−	0.997499i	$-0.522519\pi$
$419$	−2.89384	−0.141373	−0.0706867	+	0.997499i	$0.522519\pi$
$420$	0	0
$421$	−33.5744	−1.63632	−0.818158	−	0.574993i	$-0.805005\pi$
$421$	−33.5744	−1.63632	−0.818158	+	0.574993i	$0.805005\pi$
$422$	0	0
$423$	−4.10382	−0.199534
$424$	0	0
$425$	−6.36121	−0.308564
$426$	0	0
$427$	0	0
$428$	0	0
$429$	46.0374	2.22270
$430$	0	0
$431$	−0.330882	−0.0159380	−0.00796902	−	0.999968i	$-0.502537\pi$
$431$	−0.330882	−0.0159380	−0.00796902	+	0.999968i	$0.502537\pi$
$432$	0	0
$433$	−0.573752	−0.0275727	−0.0137864	−	0.999905i	$-0.504388\pi$
$433$	−0.573752	−0.0275727	−0.0137864	+	0.999905i	$0.504388\pi$
$434$	0	0
$435$	−4.49607	−0.215570
$436$	0	0
$437$	11.8341	0.566101
$438$	0	0
$439$	27.5165	1.31329	0.656645	−	0.754200i	$-0.271975\pi$
$439$	27.5165	1.31329	0.656645	+	0.754200i	$0.271975\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.7168	0.936771	0.468386	−	0.883524i	$-0.344836\pi$
$443$	19.7168	0.936771	0.468386	+	0.883524i	$0.344836\pi$
$444$	0	0
$445$	11.1540	0.528752
$446$	0	0
$447$	14.0925	0.666553
$448$	0	0
$449$	32.6751	1.54203	0.771016	−	0.636816i	$-0.219749\pi$
$449$	32.6751	1.54203	0.771016	+	0.636816i	$0.219749\pi$
$450$	0	0
$451$	−25.9915	−1.22389
$452$	0	0
$453$	−8.79282	−0.413123
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.75112	0.175470	0.0877350	−	0.996144i	$-0.472037\pi$
$457$	3.75112	0.175470	0.0877350	+	0.996144i	$0.472037\pi$
$458$	0	0
$459$	−17.0913	−0.797755
$460$	0	0
$461$	8.77983	0.408917	0.204459	−	0.978875i	$-0.434457\pi$
$461$	8.77983	0.408917	0.204459	+	0.978875i	$0.434457\pi$
$462$	0	0
$463$	4.78203	0.222240	0.111120	−	0.993807i	$-0.464556\pi$
$463$	4.78203	0.222240	0.111120	+	0.993807i	$0.464556\pi$
$464$	0	0
$465$	−14.4757	−0.671296
$466$	0	0
$467$	−21.9198	−1.01433	−0.507165	−	0.861849i	$-0.669306\pi$
$467$	−21.9198	−1.01433	−0.507165	+	0.861849i	$0.669306\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−33.9085	−1.56242
$472$	0	0
$473$	−19.2460	−0.884933
$474$	0	0
$475$	−4.50283	−0.206604
$476$	0	0
$477$	−7.26980	−0.332861
$478$	0	0
$479$	−5.08861	−0.232505	−0.116252	−	0.993220i	$-0.537088\pi$
$479$	−5.08861	−0.232505	−0.116252	+	0.993220i	$0.537088\pi$
$480$	0	0
$481$	−34.6643	−1.58055
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.1896	0.780541
$486$	0	0
$487$	−8.05035	−0.364796	−0.182398	−	0.983225i	$-0.558386\pi$
$487$	−8.05035	−0.364796	−0.182398	+	0.983225i	$0.558386\pi$
$488$	0	0
$489$	−54.5143	−2.46522
$490$	0	0
$491$	−12.8081	−0.578021	−0.289010	−	0.957326i	$-0.593326\pi$
$491$	−12.8081	−0.578021	−0.289010	+	0.957326i	$0.593326\pi$
$492$	0	0
$493$	−13.0956	−0.589795
$494$	0	0
$495$	−6.36796	−0.286218
$496$	0	0
$497$	0	0
$498$	0	0
$499$	28.5284	1.27710	0.638552	−	0.769578i	$-0.279534\pi$
$499$	28.5284	1.27710	0.638552	+	0.769578i	$0.279534\pi$
$500$	0	0
$501$	−35.1051	−1.56838
$502$	0	0
$503$	6.56948	0.292919	0.146459	−	0.989217i	$-0.453212\pi$
$503$	6.56948	0.292919	0.146459	+	0.989217i	$0.453212\pi$
$504$	0	0
$505$	3.44131	0.153136
$506$	0	0
$507$	−46.5638	−2.06797
$508$	0	0
$509$	24.1574	1.07076	0.535379	−	0.844612i	$-0.320169\pi$
$509$	24.1574	1.07076	0.535379	+	0.844612i	$0.320169\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−12.0982	−0.534150
$514$	0	0
$515$	11.9757	0.527712
$516$	0	0
$517$	8.34366	0.366954
$518$	0	0
$519$	0.0777615	0.00341335
$520$	0	0
$521$	20.8630	0.914024	0.457012	−	0.889460i	$-0.348920\pi$
$521$	20.8630	0.914024	0.457012	+	0.889460i	$0.348920\pi$
$522$	0	0
$523$	12.8708	0.562800	0.281400	−	0.959591i	$-0.409201\pi$
$523$	12.8708	0.562800	0.281400	+	0.959591i	$0.409201\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42.1630	−1.83665
$528$	0	0
$529$	−16.0929	−0.699689
$530$	0	0
$531$	8.97460	0.389465
$532$	0	0
$533$	42.3179	1.83299
$534$	0	0
$535$	10.6760	0.461564
$536$	0	0
$537$	−9.72455	−0.419645
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−24.5361	−1.05489	−0.527446	−	0.849589i	$-0.676850\pi$
$541$	−24.5361	−1.05489	−0.527446	+	0.849589i	$0.676850\pi$
$542$	0	0
$543$	49.6586	2.13105
$544$	0	0
$545$	−8.33411	−0.356994
$546$	0	0
$547$	27.5781	1.17916	0.589578	−	0.807711i	$-0.299294\pi$
$547$	27.5781	1.17916	0.589578	+	0.807711i	$0.299294\pi$
$548$	0	0
$549$	−13.0496	−0.556942
$550$	0	0
$551$	−9.26980	−0.394907
$552$	0	0
$553$	0	0
$554$	0	0
$555$	12.9227	0.548538
$556$	0	0
$557$	−6.60260	−0.279761	−0.139881	−	0.990168i	$-0.544672\pi$
$557$	−6.60260	−0.279761	−0.139881	+	0.990168i	$0.544672\pi$
$558$	0	0
$559$	31.3353	1.32534
$560$	0	0
$561$	−49.9888	−2.11053
$562$	0	0
$563$	1.97466	0.0832221	0.0416110	−	0.999134i	$-0.486751\pi$
$563$	1.97466	0.0832221	0.0416110	+	0.999134i	$0.486751\pi$
$564$	0	0
$565$	−12.1542	−0.511330
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.7281	−0.491666	−0.245833	−	0.969312i	$-0.579061\pi$
$569$	−11.7281	−0.491666	−0.245833	+	0.969312i	$0.579061\pi$
$570$	0	0
$571$	−5.69974	−0.238527	−0.119263	−	0.992863i	$-0.538053\pi$
$571$	−5.69974	−0.238527	−0.119263	+	0.992863i	$0.538053\pi$
$572$	0	0
$573$	−55.8052	−2.33130
$574$	0	0
$575$	−2.62814	−0.109601
$576$	0	0
$577$	23.3343	0.971418	0.485709	−	0.874121i	$-0.338562\pi$
$577$	23.3343	0.971418	0.485709	+	0.874121i	$0.338562\pi$
$578$	0	0
$579$	−46.0609	−1.91423
$580$	0	0
$581$	0	0
$582$	0	0
$583$	14.7806	0.612148
$584$	0	0
$585$	10.3680	0.428662
$586$	0	0
$587$	22.8470	0.942997	0.471498	−	0.881867i	$-0.343713\pi$
$587$	22.8470	0.942997	0.471498	+	0.881867i	$0.343713\pi$
$588$	0	0
$589$	−29.8454	−1.22976
$590$	0	0
$591$	−35.9532	−1.47892
$592$	0	0
$593$	−14.4810	−0.594664	−0.297332	−	0.954774i	$-0.596097\pi$
$593$	−14.4810	−0.594664	−0.297332	+	0.954774i	$0.596097\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−56.6565	−2.31880
$598$	0	0
$599$	3.13524	0.128102	0.0640512	−	0.997947i	$-0.479598\pi$
$599$	3.13524	0.128102	0.0640512	+	0.997947i	$0.479598\pi$
$600$	0	0
$601$	32.2118	1.31395	0.656973	−	0.753914i	$-0.271836\pi$
$601$	32.2118	1.31395	0.656973	+	0.753914i	$0.271836\pi$
$602$	0	0
$603$	−6.01131	−0.244799
$604$	0	0
$605$	1.94699	0.0791565
$606$	0	0
$607$	−11.1586	−0.452913	−0.226456	−	0.974021i	$-0.572714\pi$
$607$	−11.1586	−0.452913	−0.226456	+	0.974021i	$0.572714\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.5847	−0.549578
$612$	0	0
$613$	7.53705	0.304418	0.152209	−	0.988348i	$-0.451361\pi$
$613$	7.53705	0.304418	0.152209	+	0.988348i	$0.451361\pi$
$614$	0	0
$615$	−15.7759	−0.636147
$616$	0	0
$617$	−22.7728	−0.916797	−0.458399	−	0.888747i	$-0.651577\pi$
$617$	−22.7728	−0.916797	−0.458399	+	0.888747i	$0.651577\pi$
$618$	0	0
$619$	0.964766	0.0387772	0.0193886	−	0.999812i	$-0.493828\pi$
$619$	0.964766	0.0387772	0.0193886	+	0.999812i	$0.493828\pi$
$620$	0	0
$621$	−7.06132	−0.283361
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−35.3849	−1.41314
$628$	0	0
$629$	37.6395	1.50079
$630$	0	0
$631$	−6.90059	−0.274708	−0.137354	−	0.990522i	$-0.543860\pi$
$631$	−6.90059	−0.274708	−0.137354	+	0.990522i	$0.543860\pi$
$632$	0	0
$633$	19.3255	0.768121
$634$	0	0
$635$	9.82453	0.389875
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−29.5948	−1.17075
$640$	0	0
$641$	−2.14433	−0.0846961	−0.0423481	−	0.999103i	$-0.513484\pi$
$641$	−2.14433	−0.0846961	−0.0423481	+	0.999103i	$0.513484\pi$
$642$	0	0
$643$	27.4980	1.08441	0.542207	−	0.840245i	$-0.317589\pi$
$643$	27.4980	1.08441	0.542207	+	0.840245i	$0.317589\pi$
$644$	0	0
$645$	−11.6817	−0.459965
$646$	0	0
$647$	−30.9482	−1.21670	−0.608350	−	0.793669i	$-0.708168\pi$
$647$	−30.9482	−1.21670	−0.608350	+	0.793669i	$0.708168\pi$
$648$	0	0
$649$	−18.2467	−0.716245
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−39.4448	−1.54360	−0.771798	−	0.635868i	$-0.780642\pi$
$653$	−39.4448	−1.54360	−0.771798	+	0.635868i	$0.780642\pi$
$654$	0	0
$655$	−4.95375	−0.193559
$656$	0	0
$657$	26.0481	1.01623
$658$	0	0
$659$	−8.80237	−0.342892	−0.171446	−	0.985194i	$-0.554844\pi$
$659$	−8.80237	−0.342892	−0.171446	+	0.985194i	$0.554844\pi$
$660$	0	0
$661$	28.1095	1.09333	0.546667	−	0.837350i	$-0.315896\pi$
$661$	28.1095	1.09333	0.546667	+	0.837350i	$0.315896\pi$
$662$	0	0
$663$	81.3889	3.16088
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.41046	−0.209494
$668$	0	0
$669$	8.86787	0.342852
$670$	0	0
$671$	26.5317	1.02424
$672$	0	0
$673$	−14.7933	−0.570241	−0.285121	−	0.958492i	$-0.592034\pi$
$673$	−14.7933	−0.570241	−0.285121	+	0.958492i	$0.592034\pi$
$674$	0	0
$675$	2.68681	0.103415
$676$	0	0
$677$	−2.71162	−0.104216	−0.0521080	−	0.998641i	$-0.516594\pi$
$677$	−2.71162	−0.104216	−0.0521080	+	0.998641i	$0.516594\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−26.1218	−1.00099
$682$	0	0
$683$	1.62633	0.0622297	0.0311149	−	0.999516i	$-0.490094\pi$
$683$	1.62633	0.0622297	0.0311149	+	0.999516i	$0.490094\pi$
$684$	0	0
$685$	−11.3137	−0.432275
$686$	0	0
$687$	−24.6880	−0.941908
$688$	0	0
$689$	−24.0649	−0.916799
$690$	0	0
$691$	−50.8821	−1.93565	−0.967823	−	0.251632i	$-0.919033\pi$
$691$	−50.8821	−1.93565	−0.967823	+	0.251632i	$0.919033\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.70311	−0.330128
$696$	0	0
$697$	−45.9501	−1.74048
$698$	0	0
$699$	−46.5317	−1.75999
$700$	0	0
$701$	7.93106	0.299552	0.149776	−	0.988720i	$-0.452145\pi$
$701$	7.93106	0.299552	0.149776	+	0.988720i	$0.452145\pi$
$702$	0	0
$703$	26.6434	1.00488
$704$	0	0
$705$	5.06431	0.190733
$706$	0	0
$707$	0	0
$708$	0	0
$709$	17.4922	0.656933	0.328466	−	0.944516i	$-0.393468\pi$
$709$	17.4922	0.656933	0.328466	+	0.944516i	$0.393468\pi$
$710$	0	0
$711$	5.76067	0.216042
$712$	0	0
$713$	−17.4197	−0.652374
$714$	0	0
$715$	−21.0796	−0.788332
$716$	0	0
$717$	−49.2185	−1.83810
$718$	0	0
$719$	−18.1294	−0.676111	−0.338055	−	0.941126i	$-0.609769\pi$
$719$	−18.1294	−0.676111	−0.338055	+	0.941126i	$0.609769\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	28.0592	1.04353
$724$	0	0
$725$	2.05866	0.0764568
$726$	0	0
$727$	−2.42538	−0.0899523	−0.0449761	−	0.998988i	$-0.514321\pi$
$727$	−2.42538	−0.0899523	−0.0449761	+	0.998988i	$0.514321\pi$
$728$	0	0
$729$	−2.17729	−0.0806402
$730$	0	0
$731$	−34.0248	−1.25845
$732$	0	0
$733$	−19.7659	−0.730069	−0.365035	−	0.930994i	$-0.618943\pi$
$733$	−19.7659	−0.730069	−0.365035	+	0.930994i	$0.618943\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.2219	0.450198
$738$	0	0
$739$	−39.9065	−1.46799	−0.733993	−	0.679157i	$-0.762345\pi$
$739$	−39.9065	−1.46799	−0.733993	+	0.679157i	$0.762345\pi$
$740$	0	0
$741$	57.6118	2.11642
$742$	0	0
$743$	−33.0921	−1.21403	−0.607016	−	0.794689i	$-0.707634\pi$
$743$	−33.0921	−1.21403	−0.607016	+	0.794689i	$0.707634\pi$
$744$	0	0
$745$	−6.45268	−0.236408
$746$	0	0
$747$	−18.9251	−0.692433
$748$	0	0
$749$	0	0
$750$	0	0
$751$	6.98531	0.254898	0.127449	−	0.991845i	$-0.459321\pi$
$751$	6.98531	0.254898	0.127449	+	0.991845i	$0.459321\pi$
$752$	0	0
$753$	51.8391	1.88912
$754$	0	0
$755$	4.02606	0.146523
$756$	0	0
$757$	−24.4796	−0.889725	−0.444863	−	0.895599i	$-0.646747\pi$
$757$	−24.4796	−0.889725	−0.444863	+	0.895599i	$0.646747\pi$
$758$	0	0
$759$	−20.6530	−0.749655
$760$	0	0
$761$	9.39511	0.340573	0.170286	−	0.985395i	$-0.445531\pi$
$761$	9.39511	0.340573	0.170286	+	0.985395i	$0.445531\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−11.2578	−0.407028
$766$	0	0
$767$	29.7082	1.07270
$768$	0	0
$769$	29.7183	1.07167	0.535835	−	0.844323i	$-0.319997\pi$
$769$	29.7183	1.07167	0.535835	+	0.844323i	$0.319997\pi$
$770$	0	0
$771$	−28.8590	−1.03933
$772$	0	0
$773$	−38.2108	−1.37435	−0.687173	−	0.726494i	$-0.741148\pi$
$773$	−38.2108	−1.37435	−0.687173	+	0.726494i	$0.741148\pi$
$774$	0	0
$775$	6.62814	0.238090
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.5261	−1.16537
$780$	0	0
$781$	60.1705	2.15307
$782$	0	0
$783$	5.53122	0.197670
$784$	0	0
$785$	15.5260	0.554148
$786$	0	0
$787$	−24.8114	−0.884431	−0.442215	−	0.896909i	$-0.645807\pi$
$787$	−24.8114	−0.884431	−0.442215	+	0.896909i	$0.645807\pi$
$788$	0	0
$789$	−63.0149	−2.24339
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−43.1974	−1.53399
$794$	0	0
$795$	8.97129	0.318179
$796$	0	0
$797$	−22.9296	−0.812210	−0.406105	−	0.913826i	$-0.633113\pi$
$797$	−22.9296	−0.812210	−0.406105	+	0.913826i	$0.633113\pi$
$798$	0	0
$799$	14.7507	0.521841
$800$	0	0
$801$	19.7400	0.697479
$802$	0	0
$803$	−52.9595	−1.86890
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.61441	−0.197637
$808$	0	0
$809$	−34.2720	−1.20494	−0.602470	−	0.798142i	$-0.705817\pi$
$809$	−34.2720	−1.20494	−0.602470	+	0.798142i	$0.705817\pi$
$810$	0	0
$811$	37.5925	1.32005	0.660025	−	0.751244i	$-0.270546\pi$
$811$	37.5925	1.32005	0.660025	+	0.751244i	$0.270546\pi$
$812$	0	0
$813$	14.7358	0.516807
$814$	0	0
$815$	24.9610	0.874346
$816$	0	0
$817$	−24.0847	−0.842618
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.2643	1.09113	0.545565	−	0.838068i	$-0.316315\pi$
$821$	31.2643	1.09113	0.545565	+	0.838068i	$0.316315\pi$
$822$	0	0
$823$	18.4452	0.642959	0.321480	−	0.946916i	$-0.395820\pi$
$823$	18.4452	0.642959	0.321480	+	0.946916i	$0.395820\pi$
$824$	0	0
$825$	7.85838	0.273594
$826$	0	0
$827$	−14.5931	−0.507450	−0.253725	−	0.967276i	$-0.581656\pi$
$827$	−14.5931	−0.507450	−0.253725	+	0.967276i	$0.581656\pi$
$828$	0	0
$829$	12.1057	0.420448	0.210224	−	0.977653i	$-0.432581\pi$
$829$	12.1057	0.420448	0.210224	+	0.977653i	$0.432581\pi$
$830$	0	0
$831$	7.10783	0.246568
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.0739	0.556261
$836$	0	0
$837$	17.8085	0.615553
$838$	0	0
$839$	−13.8245	−0.477276	−0.238638	−	0.971109i	$-0.576701\pi$
$839$	−13.8245	−0.477276	−0.238638	+	0.971109i	$0.576701\pi$
$840$	0	0
$841$	−24.7619	−0.853859
$842$	0	0
$843$	5.57836	0.192129
$844$	0	0
$845$	21.3206	0.733451
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−27.3295	−0.937946
$850$	0	0
$851$	15.5508	0.533076
$852$	0	0
$853$	−15.7912	−0.540680	−0.270340	−	0.962765i	$-0.587136\pi$
$853$	−15.7912	−0.540680	−0.270340	+	0.962765i	$0.587136\pi$
$854$	0	0
$855$	−7.96895	−0.272532
$856$	0	0
$857$	−13.4837	−0.460593	−0.230297	−	0.973120i	$-0.573970\pi$
$857$	−13.4837	−0.460593	−0.230297	+	0.973120i	$0.573970\pi$
$858$	0	0
$859$	−23.8300	−0.813071	−0.406535	−	0.913635i	$-0.633263\pi$
$859$	−23.8300	−0.813071	−0.406535	+	0.913635i	$0.633263\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.35592	−0.0801966	−0.0400983	−	0.999196i	$-0.512767\pi$
$863$	−2.35592	−0.0801966	−0.0400983	+	0.999196i	$0.512767\pi$
$864$	0	0
$865$	−0.0356054	−0.00121062
$866$	0	0
$867$	−51.2469	−1.74044
$868$	0	0
$869$	−11.7123	−0.397312
$870$	0	0
$871$	−19.8990	−0.674251
$872$	0	0
$873$	30.4216	1.02962
$874$	0	0
$875$	0	0
$876$	0	0
$877$	26.2892	0.887725	0.443862	−	0.896095i	$-0.353608\pi$
$877$	26.2892	0.887725	0.443862	+	0.896095i	$0.353608\pi$
$878$	0	0
$879$	56.0352	1.89002
$880$	0	0
$881$	46.2476	1.55812	0.779061	−	0.626948i	$-0.215696\pi$
$881$	46.2476	1.55812	0.779061	+	0.626948i	$0.215696\pi$
$882$	0	0
$883$	−40.1184	−1.35009	−0.675045	−	0.737777i	$-0.735876\pi$
$883$	−40.1184	−1.35009	−0.675045	+	0.737777i	$0.735876\pi$
$884$	0	0
$885$	−11.0751	−0.372286
$886$	0	0
$887$	−32.7685	−1.10026	−0.550130	−	0.835079i	$-0.685422\pi$
$887$	−32.7685	−1.10026	−0.550130	+	0.835079i	$0.685422\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	40.2178	1.34735
$892$	0	0
$893$	10.4414	0.349407
$894$	0	0
$895$	4.45268	0.148837
$896$	0	0
$897$	33.6260	1.12274
$898$	0	0
$899$	13.6451	0.455090
$900$	0	0
$901$	26.1304	0.870529
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−22.7377	−0.755826
$906$	0	0
$907$	−5.48710	−0.182196	−0.0910980	−	0.995842i	$-0.529038\pi$
$907$	−5.48710	−0.182196	−0.0910980	+	0.995842i	$0.529038\pi$
$908$	0	0
$909$	6.09031	0.202003
$910$	0	0
$911$	38.3276	1.26985	0.634924	−	0.772574i	$-0.281031\pi$
$911$	38.3276	1.26985	0.634924	+	0.772574i	$0.281031\pi$
$912$	0	0
$913$	38.4775	1.27342
$914$	0	0
$915$	16.1038	0.532376
$916$	0	0
$917$	0	0
$918$	0	0
$919$	38.6094	1.27361	0.636804	−	0.771026i	$-0.280256\pi$
$919$	38.6094	1.27361	0.636804	+	0.771026i	$0.280256\pi$
$920$	0	0
$921$	−26.0747	−0.859189
$922$	0	0
$923$	−97.9662	−3.22460
$924$	0	0
$925$	−5.91704	−0.194551
$926$	0	0
$927$	21.1942	0.696108
$928$	0	0
$929$	39.3387	1.29066	0.645331	−	0.763903i	$-0.276720\pi$
$929$	39.3387	1.29066	0.645331	+	0.763903i	$0.276720\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	18.5612	0.607666
$934$	0	0
$935$	22.8888	0.748545
$936$	0	0
$937$	−18.4619	−0.603124	−0.301562	−	0.953447i	$-0.597508\pi$
$937$	−18.4619	−0.603124	−0.301562	+	0.953447i	$0.597508\pi$
$938$	0	0
$939$	3.83532	0.125161
$940$	0	0
$941$	−37.6005	−1.22574	−0.612870	−	0.790184i	$-0.709985\pi$
$941$	−37.6005	−1.22574	−0.612870	+	0.790184i	$0.709985\pi$
$942$	0	0
$943$	−18.9844	−0.618216
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.4215	1.28103	0.640513	−	0.767947i	$-0.278722\pi$
$947$	39.4215	1.28103	0.640513	+	0.767947i	$0.278722\pi$
$948$	0	0
$949$	86.2258	2.79901
$950$	0	0
$951$	−13.0272	−0.422437
$952$	0	0
$953$	24.4541	0.792148	0.396074	−	0.918219i	$-0.370372\pi$
$953$	24.4541	0.792148	0.396074	+	0.918219i	$0.370372\pi$
$954$	0	0
$955$	25.5521	0.826846
$956$	0	0
$957$	16.1777	0.522952
$958$	0	0
$959$	0	0
$960$	0	0
$961$	12.9323	0.417171
$962$	0	0
$963$	18.8940	0.608852
$964$	0	0
$965$	21.0904	0.678923
$966$	0	0
$967$	−22.5757	−0.725986	−0.362993	−	0.931792i	$-0.618245\pi$
$967$	−22.5757	−0.725986	−0.362993	+	0.931792i	$0.618245\pi$
$968$	0	0
$969$	−62.5566	−2.00961
$970$	0	0
$971$	−35.1636	−1.12845	−0.564226	−	0.825620i	$-0.690826\pi$
$971$	−35.1636	−1.12845	−0.564226	+	0.825620i	$0.690826\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−12.7946	−0.409754
$976$	0	0
$977$	26.4910	0.847522	0.423761	−	0.905774i	$-0.360710\pi$
$977$	26.4910	0.847522	0.423761	+	0.905774i	$0.360710\pi$
$978$	0	0
$979$	−40.1343	−1.28270
$980$	0	0
$981$	−14.7494	−0.470913
$982$	0	0
$983$	−27.3380	−0.871947	−0.435974	−	0.899960i	$-0.643596\pi$
$983$	−27.3380	−0.871947	−0.435974	+	0.899960i	$0.643596\pi$
$984$	0	0
$985$	16.4622	0.524530
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14.0574	−0.447000
$990$	0	0
$991$	−59.9900	−1.90565	−0.952823	−	0.303527i	$-0.901836\pi$
$991$	−59.9900	−1.90565	−0.952823	+	0.303527i	$0.901836\pi$
$992$	0	0
$993$	−30.8024	−0.977486
$994$	0	0
$995$	25.9419	0.822412
$996$	0	0
$997$	−22.4645	−0.711458	−0.355729	−	0.934589i	$-0.615767\pi$
$997$	−22.4645	−0.711458	−0.355729	+	0.934589i	$0.615767\pi$
$998$	0	0
$999$	−15.8979	−0.502989

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.cd.1.2		4
4.3	odd	2		1960.2.a.y.1.3	yes	4
7.6	odd	2		3920.2.a.ce.1.3		4
20.19	odd	2		9800.2.a.cl.1.2		4
28.3	even	6		1960.2.q.y.961.3		8
28.11	odd	6		1960.2.q.x.961.2		8
28.19	even	6		1960.2.q.y.361.3		8
28.23	odd	6		1960.2.q.x.361.2		8
28.27	even	2		1960.2.a.x.1.2	✓	4
140.139	even	2		9800.2.a.cs.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.a.x.1.2	✓	4	28.27	even	2
1960.2.a.y.1.3	yes	4	4.3	odd	2
1960.2.q.x.361.2		8	28.23	odd	6
1960.2.q.x.961.2		8	28.11	odd	6
1960.2.q.y.361.3		8	28.19	even	6
1960.2.q.y.961.3		8	28.3	even	6
3920.2.a.cd.1.2		4	1.1	even	1	trivial
3920.2.a.ce.1.3		4	7.6	odd	2
9800.2.a.cl.1.2		4	20.19	odd	2
9800.2.a.cs.1.3		4	140.139	even	2

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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q-2.18398 q^{3} +1.00000 q^{5} +1.76977 q^{9} +O(q^{10})\)	\(q-2.18398 q^{3} +1.00000 q^{5} +1.76977 q^{9} -3.59819 q^{11} +5.85838 q^{13} -2.18398 q^{15} -6.36121 q^{17} -4.50283 q^{19} -2.62814 q^{23} +1.00000 q^{25} +2.68681 q^{27} +2.05866 q^{29} +6.62814 q^{31} +7.85838 q^{33} -5.91704 q^{37} -12.7946 q^{39} +7.22348 q^{41} +5.34880 q^{43} +1.76977 q^{45} -2.31885 q^{47} +13.8927 q^{51} -4.10777 q^{53} -3.59819 q^{55} +9.83408 q^{57} +5.07107 q^{59} -7.37361 q^{61} +5.85838 q^{65} -3.39667 q^{67} +5.73981 q^{69} -16.7224 q^{71} +14.7184 q^{73} -2.18398 q^{75} +3.25505 q^{79} -11.1772 q^{81} -10.6936 q^{83} -6.36121 q^{85} -4.49607 q^{87} +11.1540 q^{89} -14.4757 q^{93} -4.50283 q^{95} +17.1896 q^{97} -6.36796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} + 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 4 * q^5 + 6 * q^9	\( 4 q - 2 q^{3} + 4 q^{5} + 6 q^{9} - 2 q^{11} + 10 q^{13} - 2 q^{15} + 6 q^{17} + 4 q^{23} + 4 q^{25} - 14 q^{27} - 2 q^{29} + 12 q^{31} + 18 q^{33} - 14 q^{39} + 12 q^{41} + 8 q^{43} + 6 q^{45} + 2 q^{47} - 2 q^{51} - 4 q^{53} - 2 q^{55} - 8 q^{57} - 8 q^{59} + 20 q^{61} + 10 q^{65} + 8 q^{67} + 24 q^{69} - 4 q^{71} + 16 q^{73} - 2 q^{75} - 22 q^{79} - 20 q^{81} - 36 q^{83} + 6 q^{85} + 18 q^{87} + 40 q^{89} - 32 q^{93} + 26 q^{97} - 12 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 + 6 * q^9 - 2 * q^11 + 10 * q^13 - 2 * q^15 + 6 * q^17 + 4 * q^23 + 4 * q^25 - 14 * q^27 - 2 * q^29 + 12 * q^31 + 18 * q^33 - 14 * q^39 + 12 * q^41 + 8 * q^43 + 6 * q^45 + 2 * q^47 - 2 * q^51 - 4 * q^53 - 2 * q^55 - 8 * q^57 - 8 * q^59 + 20 * q^61 + 10 * q^65 + 8 * q^67 + 24 * q^69 - 4 * q^71 + 16 * q^73 - 2 * q^75 - 22 * q^79 - 20 * q^81 - 36 * q^83 + 6 * q^85 + 18 * q^87 + 40 * q^89 - 32 * q^93 + 26 * q^97 - 12 * q^99

Introduction

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