LMFDB - Embedded newform 3856.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3856,2,Mod(1,3856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3856, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3856 = 2^{4} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3856.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:	$30.7903150194$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.02418$ of defining polynomial
Character	$\chi$	$=$	3856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.93498 q^{3} +1.44091 q^{5} -0.381245 q^{7} +5.61411 q^{9} +O(q^{10})$	$q-2.93498 q^{3} +1.44091 q^{5} -0.381245 q^{7} +5.61411 q^{9} -0.280814 q^{11} -4.00528 q^{13} -4.22904 q^{15} +2.60326 q^{17} +3.86940 q^{19} +1.11895 q^{21} -0.698450 q^{23} -2.92378 q^{25} -7.67238 q^{27} +1.62690 q^{29} -9.73691 q^{31} +0.824185 q^{33} -0.549338 q^{35} +4.41735 q^{37} +11.7554 q^{39} -0.0157665 q^{41} +12.2173 q^{43} +8.08942 q^{45} -7.71890 q^{47} -6.85465 q^{49} -7.64051 q^{51} -4.91852 q^{53} -0.404628 q^{55} -11.3566 q^{57} +14.1596 q^{59} -6.97043 q^{61} -2.14035 q^{63} -5.77125 q^{65} +2.97942 q^{67} +2.04994 q^{69} -7.28180 q^{71} -0.165424 q^{73} +8.58125 q^{75} +0.107059 q^{77} -11.1633 q^{79} +5.67594 q^{81} +14.6259 q^{83} +3.75106 q^{85} -4.77491 q^{87} +14.3374 q^{89} +1.52699 q^{91} +28.5777 q^{93} +5.57544 q^{95} +8.75687 q^{97} -1.57652 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9	$ 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) $ 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9 - 22 * q^11 - 5 * q^13 - 13 * q^15 - 4 * q^17 + 6 * q^19 - 14 * q^21 - 32 * q^23 + 4 * q^25 + 5 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - 15 * q^35 - 8 * q^37 - 31 * q^39 - q^41 + 2 * q^43 - 15 * q^45 - 34 * q^47 - 9 * q^49 + 3 * q^51 + 5 * q^53 + 3 * q^55 - 22 * q^57 - 26 * q^59 - 26 * q^61 + 4 * q^63 - 25 * q^65 - 6 * q^67 - 2 * q^69 - 94 * q^71 - 22 * q^73 - 7 * q^77 - 9 * q^79 + 4 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 3 * q^89 + 20 * q^91 + 12 * q^93 - 33 * q^95 - 29 * q^97 - 36 * 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
$3$	−2.93498	−1.69451	−0.847256	−	0.531185i	$-0.821747\pi$
$3$	−2.93498	−1.69451	−0.847256	+	0.531185i	$0.821747\pi$
$4$	0	0
$5$	1.44091	0.644394	0.322197	−	0.946673i	$-0.395579\pi$
$5$	1.44091	0.644394	0.322197	+	0.946673i	$0.395579\pi$
$6$	0	0
$7$	−0.381245	−0.144097	−0.0720485	−	0.997401i	$-0.522954\pi$
$7$	−0.381245	−0.144097	−0.0720485	+	0.997401i	$0.522954\pi$
$8$	0	0
$9$	5.61411	1.87137
$10$	0	0
$11$	−0.280814	−0.0846688	−0.0423344	−	0.999103i	$-0.513479\pi$
$11$	−0.280814	−0.0846688	−0.0423344	+	0.999103i	$0.513479\pi$
$12$	0	0
$13$	−4.00528	−1.11087	−0.555433	−	0.831561i	$-0.687447\pi$
$13$	−4.00528	−1.11087	−0.555433	+	0.831561i	$0.687447\pi$
$14$	0	0
$15$	−4.22904	−1.09193
$16$	0	0
$17$	2.60326	0.631383	0.315691	−	0.948862i	$-0.397764\pi$
$17$	2.60326	0.631383	0.315691	+	0.948862i	$0.397764\pi$
$18$	0	0
$19$	3.86940	0.887700	0.443850	−	0.896101i	$-0.353612\pi$
$19$	3.86940	0.887700	0.443850	+	0.896101i	$0.353612\pi$
$20$	0	0
$21$	1.11895	0.244174
$22$	0	0
$23$	−0.698450	−0.145637	−0.0728185	−	0.997345i	$-0.523199\pi$
$23$	−0.698450	−0.145637	−0.0728185	+	0.997345i	$0.523199\pi$
$24$	0	0
$25$	−2.92378	−0.584757
$26$	0	0
$27$	−7.67238	−1.47655
$28$	0	0
$29$	1.62690	0.302107	0.151054	−	0.988526i	$-0.451733\pi$
$29$	1.62690	0.302107	0.151054	+	0.988526i	$0.451733\pi$
$30$	0	0
$31$	−9.73691	−1.74880	−0.874401	−	0.485205i	$-0.838745\pi$
$31$	−9.73691	−1.74880	−0.874401	+	0.485205i	$0.838745\pi$
$32$	0	0
$33$	0.824185	0.143472
$34$	0	0
$35$	−0.549338	−0.0928551
$36$	0	0
$37$	4.41735	0.726208	0.363104	−	0.931749i	$-0.381717\pi$
$37$	4.41735	0.726208	0.363104	+	0.931749i	$0.381717\pi$
$38$	0	0
$39$	11.7554	1.88238
$40$	0	0
$41$	−0.0157665	−0.00246231	−0.00123116	−	0.999999i	$-0.500392\pi$
$41$	−0.0157665	−0.00246231	−0.00123116	+	0.999999i	$0.500392\pi$
$42$	0	0
$43$	12.2173	1.86312	0.931562	−	0.363583i	$-0.118447\pi$
$43$	12.2173	1.86312	0.931562	+	0.363583i	$0.118447\pi$
$44$	0	0
$45$	8.08942	1.20590
$46$	0	0
$47$	−7.71890	−1.12592	−0.562958	−	0.826485i	$-0.690337\pi$
$47$	−7.71890	−1.12592	−0.562958	+	0.826485i	$0.690337\pi$
$48$	0	0
$49$	−6.85465	−0.979236
$50$	0	0
$51$	−7.64051	−1.06989
$52$	0	0
$53$	−4.91852	−0.675611	−0.337806	−	0.941216i	$-0.609685\pi$
$53$	−4.91852	−0.675611	−0.337806	+	0.941216i	$0.609685\pi$
$54$	0	0
$55$	−0.404628	−0.0545600
$56$	0	0
$57$	−11.3566	−1.50422
$58$	0	0
$59$	14.1596	1.84342	0.921709	−	0.387882i	$-0.126793\pi$
$59$	14.1596	1.84342	0.921709	+	0.387882i	$0.126793\pi$
$60$	0	0
$61$	−6.97043	−0.892472	−0.446236	−	0.894915i	$-0.647236\pi$
$61$	−6.97043	−0.892472	−0.446236	+	0.894915i	$0.647236\pi$
$62$	0	0
$63$	−2.14035	−0.269659
$64$	0	0
$65$	−5.77125	−0.715835
$66$	0	0
$67$	2.97942	0.363994	0.181997	−	0.983299i	$-0.441744\pi$
$67$	2.97942	0.363994	0.181997	+	0.983299i	$0.441744\pi$
$68$	0	0
$69$	2.04994	0.246784
$70$	0	0
$71$	−7.28180	−0.864191	−0.432095	−	0.901828i	$-0.642226\pi$
$71$	−7.28180	−0.864191	−0.432095	+	0.901828i	$0.642226\pi$
$72$	0	0
$73$	−0.165424	−0.0193614	−0.00968072	−	0.999953i	$-0.503082\pi$
$73$	−0.165424	−0.0193614	−0.00968072	+	0.999953i	$0.503082\pi$
$74$	0	0
$75$	8.58125	0.990878
$76$	0	0
$77$	0.107059	0.0122005
$78$	0	0
$79$	−11.1633	−1.25597	−0.627986	−	0.778225i	$-0.716120\pi$
$79$	−11.1633	−1.25597	−0.627986	+	0.778225i	$0.716120\pi$
$80$	0	0
$81$	5.67594	0.630660
$82$	0	0
$83$	14.6259	1.60540	0.802701	−	0.596382i	$-0.203396\pi$
$83$	14.6259	1.60540	0.802701	+	0.596382i	$0.203396\pi$
$84$	0	0
$85$	3.75106	0.406859
$86$	0	0
$87$	−4.77491	−0.511924
$88$	0	0
$89$	14.3374	1.51976	0.759878	−	0.650066i	$-0.225259\pi$
$89$	14.3374	1.51976	0.759878	+	0.650066i	$0.225259\pi$
$90$	0	0
$91$	1.52699	0.160072
$92$	0	0
$93$	28.5777	2.96337
$94$	0	0
$95$	5.57544	0.572028
$96$	0	0
$97$	8.75687	0.889126	0.444563	−	0.895748i	$-0.353359\pi$
$97$	8.75687	0.889126	0.444563	+	0.895748i	$0.353359\pi$
$98$	0	0
$99$	−1.57652	−0.158447
$100$	0	0
$101$	−5.97227	−0.594263	−0.297132	−	0.954837i	$-0.596030\pi$
$101$	−5.97227	−0.594263	−0.297132	+	0.954837i	$0.596030\pi$
$102$	0	0
$103$	17.0246	1.67749	0.838743	−	0.544527i	$-0.183291\pi$
$103$	17.0246	1.67749	0.838743	+	0.544527i	$0.183291\pi$
$104$	0	0
$105$	1.61230	0.157344
$106$	0	0
$107$	6.06814	0.586629	0.293315	−	0.956016i	$-0.405242\pi$
$107$	6.06814	0.586629	0.293315	+	0.956016i	$0.405242\pi$
$108$	0	0
$109$	−10.0495	−0.962568	−0.481284	−	0.876565i	$-0.659829\pi$
$109$	−10.0495	−0.962568	−0.481284	+	0.876565i	$0.659829\pi$
$110$	0	0
$111$	−12.9648	−1.23057
$112$	0	0
$113$	3.24797	0.305543	0.152772	−	0.988262i	$-0.451180\pi$
$113$	3.24797	0.305543	0.152772	+	0.988262i	$0.451180\pi$
$114$	0	0
$115$	−1.00640	−0.0938476
$116$	0	0
$117$	−22.4861	−2.07884
$118$	0	0
$119$	−0.992478	−0.0909803
$120$	0	0
$121$	−10.9211	−0.992831
$122$	0	0
$123$	0.0462743	0.00417241
$124$	0	0
$125$	−11.4174	−1.02121
$126$	0	0
$127$	−0.309449	−0.0274591	−0.0137296	−	0.999906i	$-0.504370\pi$
$127$	−0.309449	−0.0274591	−0.0137296	+	0.999906i	$0.504370\pi$
$128$	0	0
$129$	−35.8576	−3.15709
$130$	0	0
$131$	−17.9939	−1.57214	−0.786068	−	0.618141i	$-0.787886\pi$
$131$	−17.9939	−1.57214	−0.786068	+	0.618141i	$0.787886\pi$
$132$	0	0
$133$	−1.47519	−0.127915
$134$	0	0
$135$	−11.0552	−0.951479
$136$	0	0
$137$	−8.91168	−0.761377	−0.380688	−	0.924703i	$-0.624313\pi$
$137$	−8.91168	−0.761377	−0.380688	+	0.924703i	$0.624313\pi$
$138$	0	0
$139$	−7.82985	−0.664119	−0.332059	−	0.943258i	$-0.607743\pi$
$139$	−7.82985	−0.664119	−0.332059	+	0.943258i	$0.607743\pi$
$140$	0	0
$141$	22.6548	1.90788
$142$	0	0
$143$	1.12474	0.0940557
$144$	0	0
$145$	2.34421	0.194676
$146$	0	0
$147$	20.1183	1.65933
$148$	0	0
$149$	20.5561	1.68402	0.842010	−	0.539462i	$-0.181372\pi$
$149$	20.5561	1.68402	0.842010	+	0.539462i	$0.181372\pi$
$150$	0	0
$151$	−10.5866	−0.861524	−0.430762	−	0.902466i	$-0.641755\pi$
$151$	−10.5866	−0.861524	−0.430762	+	0.902466i	$0.641755\pi$
$152$	0	0
$153$	14.6150	1.18155
$154$	0	0
$155$	−14.0300	−1.12692
$156$	0	0
$157$	7.79923	0.622447	0.311223	−	0.950337i	$-0.399261\pi$
$157$	7.79923	0.622447	0.311223	+	0.950337i	$0.399261\pi$
$158$	0	0
$159$	14.4358	1.14483
$160$	0	0
$161$	0.266280	0.0209858
$162$	0	0
$163$	−17.1362	−1.34221	−0.671104	−	0.741363i	$-0.734180\pi$
$163$	−17.1362	−1.34221	−0.671104	+	0.741363i	$0.734180\pi$
$164$	0	0
$165$	1.18758	0.0924526
$166$	0	0
$167$	1.88493	0.145860	0.0729302	−	0.997337i	$-0.476765\pi$
$167$	1.88493	0.145860	0.0729302	+	0.997337i	$0.476765\pi$
$168$	0	0
$169$	3.04231	0.234024
$170$	0	0
$171$	21.7232	1.66122
$172$	0	0
$173$	−9.95946	−0.757203	−0.378602	−	0.925560i	$-0.623595\pi$
$173$	−9.95946	−0.757203	−0.378602	+	0.925560i	$0.623595\pi$
$174$	0	0
$175$	1.11468	0.0842616
$176$	0	0
$177$	−41.5581	−3.12369
$178$	0	0
$179$	−25.8971	−1.93564	−0.967820	−	0.251643i	$-0.919029\pi$
$179$	−25.8971	−1.93564	−0.967820	+	0.251643i	$0.919029\pi$
$180$	0	0
$181$	−8.61696	−0.640494	−0.320247	−	0.947334i	$-0.603766\pi$
$181$	−8.61696	−0.640494	−0.320247	+	0.947334i	$0.603766\pi$
$182$	0	0
$183$	20.4581	1.51230
$184$	0	0
$185$	6.36499	0.467964
$186$	0	0
$187$	−0.731033	−0.0534584
$188$	0	0
$189$	2.92505	0.212766
$190$	0	0
$191$	7.40808	0.536030	0.268015	−	0.963415i	$-0.413632\pi$
$191$	7.40808	0.536030	0.268015	+	0.963415i	$0.413632\pi$
$192$	0	0
$193$	−13.0485	−0.939252	−0.469626	−	0.882865i	$-0.655611\pi$
$193$	−13.0485	−0.939252	−0.469626	+	0.882865i	$0.655611\pi$
$194$	0	0
$195$	16.9385	1.21299
$196$	0	0
$197$	−2.32016	−0.165305	−0.0826524	−	0.996578i	$-0.526339\pi$
$197$	−2.32016	−0.165305	−0.0826524	+	0.996578i	$0.526339\pi$
$198$	0	0
$199$	−21.5958	−1.53089	−0.765444	−	0.643502i	$-0.777481\pi$
$199$	−21.5958	−1.53089	−0.765444	+	0.643502i	$0.777481\pi$
$200$	0	0
$201$	−8.74455	−0.616793
$202$	0	0
$203$	−0.620245	−0.0435327
$204$	0	0
$205$	−0.0227180	−0.00158670
$206$	0	0
$207$	−3.92118	−0.272541
$208$	0	0
$209$	−1.08658	−0.0751605
$210$	0	0
$211$	−20.1477	−1.38702	−0.693512	−	0.720445i	$-0.743938\pi$
$211$	−20.1477	−1.38702	−0.693512	+	0.720445i	$0.743938\pi$
$212$	0	0
$213$	21.3719	1.46438
$214$	0	0
$215$	17.6040	1.20059
$216$	0	0
$217$	3.71215	0.251997
$218$	0	0
$219$	0.485517	0.0328082
$220$	0	0
$221$	−10.4268	−0.701382
$222$	0	0
$223$	7.96809	0.533583	0.266792	−	0.963754i	$-0.414036\pi$
$223$	7.96809	0.533583	0.266792	+	0.963754i	$0.414036\pi$
$224$	0	0
$225$	−16.4145	−1.09430
$226$	0	0
$227$	19.2441	1.27728	0.638639	−	0.769506i	$-0.279498\pi$
$227$	19.2441	1.27728	0.638639	+	0.769506i	$0.279498\pi$
$228$	0	0
$229$	12.7475	0.842380	0.421190	−	0.906972i	$-0.361612\pi$
$229$	12.7475	0.842380	0.421190	+	0.906972i	$0.361612\pi$
$230$	0	0
$231$	−0.314216	−0.0206739
$232$	0	0
$233$	−10.2413	−0.670931	−0.335465	−	0.942053i	$-0.608894\pi$
$233$	−10.2413	−0.670931	−0.335465	+	0.942053i	$0.608894\pi$
$234$	0	0
$235$	−11.1222	−0.725534
$236$	0	0
$237$	32.7641	2.12826
$238$	0	0
$239$	−22.4264	−1.45064	−0.725321	−	0.688411i	$-0.758309\pi$
$239$	−22.4264	−1.45064	−0.725321	+	0.688411i	$0.758309\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0	0
$243$	6.35836	0.407889
$244$	0	0
$245$	−9.87692	−0.631014
$246$	0	0
$247$	−15.4980	−0.986116
$248$	0	0
$249$	−42.9267	−2.72037
$250$	0	0
$251$	−14.0729	−0.888274	−0.444137	−	0.895959i	$-0.646490\pi$
$251$	−14.0729	−0.888274	−0.444137	+	0.895959i	$0.646490\pi$
$252$	0	0
$253$	0.196135	0.0123309
$254$	0	0
$255$	−11.0093	−0.689428
$256$	0	0
$257$	1.46464	0.0913619	0.0456809	−	0.998956i	$-0.485454\pi$
$257$	1.46464	0.0913619	0.0456809	+	0.998956i	$0.485454\pi$
$258$	0	0
$259$	−1.68409	−0.104644
$260$	0	0
$261$	9.13358	0.565355
$262$	0	0
$263$	−8.09273	−0.499019	−0.249509	−	0.968372i	$-0.580269\pi$
$263$	−8.09273	−0.499019	−0.249509	+	0.968372i	$0.580269\pi$
$264$	0	0
$265$	−7.08714	−0.435360
$266$	0	0
$267$	−42.0799	−2.57525
$268$	0	0
$269$	−25.2517	−1.53962	−0.769812	−	0.638271i	$-0.779650\pi$
$269$	−25.2517	−1.53962	−0.769812	+	0.638271i	$0.779650\pi$
$270$	0	0
$271$	−6.04109	−0.366970	−0.183485	−	0.983023i	$-0.558738\pi$
$271$	−6.04109	−0.366970	−0.183485	+	0.983023i	$0.558738\pi$
$272$	0	0
$273$	−4.48170	−0.271245
$274$	0	0
$275$	0.821041	0.0495106
$276$	0	0
$277$	−17.9271	−1.07714	−0.538568	−	0.842582i	$-0.681034\pi$
$277$	−17.9271	−1.07714	−0.538568	+	0.842582i	$0.681034\pi$
$278$	0	0
$279$	−54.6641	−3.27266
$280$	0	0
$281$	4.13276	0.246540	0.123270	−	0.992373i	$-0.460662\pi$
$281$	4.13276	0.246540	0.123270	+	0.992373i	$0.460662\pi$
$282$	0	0
$283$	−13.7771	−0.818962	−0.409481	−	0.912319i	$-0.634290\pi$
$283$	−13.7771	−0.818962	−0.409481	+	0.912319i	$0.634290\pi$
$284$	0	0
$285$	−16.3638	−0.969309
$286$	0	0
$287$	0.00601088	0.000354811 0
$288$	0	0
$289$	−10.2230	−0.601356
$290$	0	0
$291$	−25.7013	−1.50663
$292$	0	0
$293$	21.6053	1.26220	0.631098	−	0.775703i	$-0.282605\pi$
$293$	21.6053	1.26220	0.631098	+	0.775703i	$0.282605\pi$
$294$	0	0
$295$	20.4026	1.18789
$296$	0	0
$297$	2.15451	0.125018
$298$	0	0
$299$	2.79749	0.161783
$300$	0	0
$301$	−4.65779	−0.268470
$302$	0	0
$303$	17.5285	1.00699
$304$	0	0
$305$	−10.0437	−0.575103
$306$	0	0
$307$	−20.3178	−1.15960	−0.579800	−	0.814759i	$-0.696869\pi$
$307$	−20.3178	−1.15960	−0.579800	+	0.814759i	$0.696869\pi$
$308$	0	0
$309$	−49.9670	−2.84252
$310$	0	0
$311$	−20.4778	−1.16119	−0.580595	−	0.814193i	$-0.697180\pi$
$311$	−20.4778	−1.16119	−0.580595	+	0.814193i	$0.697180\pi$
$312$	0	0
$313$	1.98489	0.112192	0.0560962	−	0.998425i	$-0.482135\pi$
$313$	1.98489	0.112192	0.0560962	+	0.998425i	$0.482135\pi$
$314$	0	0
$315$	−3.08405	−0.173766
$316$	0	0
$317$	−28.5808	−1.60526	−0.802630	−	0.596477i	$-0.796567\pi$
$317$	−28.5808	−1.60526	−0.802630	+	0.596477i	$0.796567\pi$
$318$	0	0
$319$	−0.456856	−0.0255790
$320$	0	0
$321$	−17.8099	−0.994051
$322$	0	0
$323$	10.0730	0.560479
$324$	0	0
$325$	11.7106	0.649587
$326$	0	0
$327$	29.4951	1.63108
$328$	0	0
$329$	2.94279	0.162241
$330$	0	0
$331$	23.4089	1.28667	0.643334	−	0.765585i	$-0.277551\pi$
$331$	23.4089	1.28667	0.643334	+	0.765585i	$0.277551\pi$
$332$	0	0
$333$	24.7995	1.35900
$334$	0	0
$335$	4.29307	0.234556
$336$	0	0
$337$	2.23389	0.121688	0.0608440	−	0.998147i	$-0.480621\pi$
$337$	2.23389	0.121688	0.0608440	+	0.998147i	$0.480621\pi$
$338$	0	0
$339$	−9.53273	−0.517747
$340$	0	0
$341$	2.73427	0.148069
$342$	0	0
$343$	5.28201	0.285202
$344$	0	0
$345$	2.95377	0.159026
$346$	0	0
$347$	−17.2084	−0.923793	−0.461897	−	0.886934i	$-0.652831\pi$
$347$	−17.2084	−0.923793	−0.461897	+	0.886934i	$0.652831\pi$
$348$	0	0
$349$	21.1996	1.13479	0.567394	−	0.823447i	$-0.307952\pi$
$349$	21.1996	1.13479	0.567394	+	0.823447i	$0.307952\pi$
$350$	0	0
$351$	30.7301	1.64025
$352$	0	0
$353$	−9.42434	−0.501607	−0.250803	−	0.968038i	$-0.580695\pi$
$353$	−9.42434	−0.501607	−0.250803	+	0.968038i	$0.580695\pi$
$354$	0	0
$355$	−10.4924	−0.556879
$356$	0	0
$357$	2.91290	0.154167
$358$	0	0
$359$	−8.50212	−0.448725	−0.224362	−	0.974506i	$-0.572030\pi$
$359$	−8.50212	−0.448725	−0.224362	+	0.974506i	$0.572030\pi$
$360$	0	0
$361$	−4.02777	−0.211988
$362$	0	0
$363$	32.0533	1.68236
$364$	0	0
$365$	−0.238361	−0.0124764
$366$	0	0
$367$	−5.48335	−0.286229	−0.143114	−	0.989706i	$-0.545712\pi$
$367$	−5.48335	−0.286229	−0.143114	+	0.989706i	$0.545712\pi$
$368$	0	0
$369$	−0.0885148	−0.00460790
$370$	0	0
$371$	1.87516	0.0973535
$372$	0	0
$373$	−23.6497	−1.22454	−0.612268	−	0.790650i	$-0.709743\pi$
$373$	−23.6497	−1.22454	−0.612268	+	0.790650i	$0.709743\pi$
$374$	0	0
$375$	33.5100	1.73045
$376$	0	0
$377$	−6.51618	−0.335601
$378$	0	0
$379$	−4.37392	−0.224673	−0.112336	−	0.993670i	$-0.535833\pi$
$379$	−4.37392	−0.224673	−0.112336	+	0.993670i	$0.535833\pi$
$380$	0	0
$381$	0.908227	0.0465299
$382$	0	0
$383$	−22.2150	−1.13514	−0.567568	−	0.823327i	$-0.692115\pi$
$383$	−22.2150	−1.13514	−0.567568	+	0.823327i	$0.692115\pi$
$384$	0	0
$385$	0.154262	0.00786193
$386$	0	0
$387$	68.5894	3.48660
$388$	0	0
$389$	−9.47112	−0.480205	−0.240102	−	0.970748i	$-0.577181\pi$
$389$	−9.47112	−0.480205	−0.240102	+	0.970748i	$0.577181\pi$
$390$	0	0
$391$	−1.81825	−0.0919527
$392$	0	0
$393$	52.8118	2.66400
$394$	0	0
$395$	−16.0853	−0.809340
$396$	0	0
$397$	33.8693	1.69985	0.849925	−	0.526904i	$-0.176647\pi$
$397$	33.8693	1.69985	0.849925	+	0.526904i	$0.176647\pi$
$398$	0	0
$399$	4.32964	0.216753
$400$	0	0
$401$	26.1950	1.30812	0.654059	−	0.756444i	$-0.273065\pi$
$401$	26.1950	1.30812	0.654059	+	0.756444i	$0.273065\pi$
$402$	0	0
$403$	38.9991	1.94268
$404$	0	0
$405$	8.17850	0.406393
$406$	0	0
$407$	−1.24046	−0.0614871
$408$	0	0
$409$	−23.3301	−1.15360	−0.576800	−	0.816885i	$-0.695699\pi$
$409$	−23.3301	−1.15360	−0.576800	+	0.816885i	$0.695699\pi$
$410$	0	0
$411$	26.1556	1.29016
$412$	0	0
$413$	−5.39826	−0.265631
$414$	0	0
$415$	21.0746	1.03451
$416$	0	0
$417$	22.9805	1.12536
$418$	0	0
$419$	17.9680	0.877794	0.438897	−	0.898537i	$-0.355369\pi$
$419$	17.9680	0.877794	0.438897	+	0.898537i	$0.355369\pi$
$420$	0	0
$421$	20.6124	1.00459	0.502294	−	0.864697i	$-0.332489\pi$
$421$	20.6124	1.00459	0.502294	+	0.864697i	$0.332489\pi$
$422$	0	0
$423$	−43.3348	−2.10701
$424$	0	0
$425$	−7.61136	−0.369205
$426$	0	0
$427$	2.65744	0.128602
$428$	0	0
$429$	−3.30110	−0.159378
$430$	0	0
$431$	4.74707	0.228659	0.114329	−	0.993443i	$-0.463528\pi$
$431$	4.74707	0.228659	0.114329	+	0.993443i	$0.463528\pi$
$432$	0	0
$433$	−40.8642	−1.96381	−0.981904	−	0.189378i	$-0.939353\pi$
$433$	−40.8642	−1.96381	−0.981904	+	0.189378i	$0.939353\pi$
$434$	0	0
$435$	−6.88021	−0.329881
$436$	0	0
$437$	−2.70258	−0.129282
$438$	0	0
$439$	23.5388	1.12344	0.561722	−	0.827326i	$-0.310139\pi$
$439$	23.5388	1.12344	0.561722	+	0.827326i	$0.310139\pi$
$440$	0	0
$441$	−38.4828	−1.83251
$442$	0	0
$443$	−27.9145	−1.32626	−0.663129	−	0.748505i	$-0.730772\pi$
$443$	−27.9145	−1.32626	−0.663129	+	0.748505i	$0.730772\pi$
$444$	0	0
$445$	20.6588	0.979321
$446$	0	0
$447$	−60.3317	−2.85359
$448$	0	0
$449$	23.7869	1.12258	0.561288	−	0.827621i	$-0.310306\pi$
$449$	23.7869	1.12258	0.561288	+	0.827621i	$0.310306\pi$
$450$	0	0
$451$	0.00442746	0.000208481 0
$452$	0	0
$453$	31.0714	1.45986
$454$	0	0
$455$	2.20026	0.103150
$456$	0	0
$457$	33.9948	1.59021	0.795106	−	0.606471i	$-0.207415\pi$
$457$	33.9948	1.59021	0.795106	+	0.606471i	$0.207415\pi$
$458$	0	0
$459$	−19.9732	−0.932268
$460$	0	0
$461$	18.0014	0.838410	0.419205	−	0.907892i	$-0.362309\pi$
$461$	18.0014	0.838410	0.419205	+	0.907892i	$0.362309\pi$
$462$	0	0
$463$	−11.9773	−0.556634	−0.278317	−	0.960489i	$-0.589777\pi$
$463$	−11.9773	−0.556634	−0.278317	+	0.960489i	$0.589777\pi$
$464$	0	0
$465$	41.1778	1.90957
$466$	0	0
$467$	−22.0474	−1.02023	−0.510117	−	0.860105i	$-0.670398\pi$
$467$	−22.0474	−1.02023	−0.510117	+	0.860105i	$0.670398\pi$
$468$	0	0
$469$	−1.13589	−0.0524505
$470$	0	0
$471$	−22.8906	−1.05474
$472$	0	0
$473$	−3.43080	−0.157748
$474$	0	0
$475$	−11.3133	−0.519089
$476$	0	0
$477$	−27.6132	−1.26432
$478$	0	0
$479$	−14.1990	−0.648769	−0.324385	−	0.945925i	$-0.605157\pi$
$479$	−14.1990	−0.648769	−0.324385	+	0.945925i	$0.605157\pi$
$480$	0	0
$481$	−17.6927	−0.806720
$482$	0	0
$483$	−0.781528	−0.0355608
$484$	0	0
$485$	12.6178	0.572947
$486$	0	0
$487$	40.8500	1.85109	0.925544	−	0.378639i	$-0.123608\pi$
$487$	40.8500	1.85109	0.925544	+	0.378639i	$0.123608\pi$
$488$	0	0
$489$	50.2944	2.27439
$490$	0	0
$491$	−7.70287	−0.347626	−0.173813	−	0.984779i	$-0.555609\pi$
$491$	−7.70287	−0.347626	−0.173813	+	0.984779i	$0.555609\pi$
$492$	0	0
$493$	4.23523	0.190745
$494$	0	0
$495$	−2.27163	−0.102102
$496$	0	0
$497$	2.77615	0.124527
$498$	0	0
$499$	1.84960	0.0827996	0.0413998	−	0.999143i	$-0.486818\pi$
$499$	1.84960	0.0827996	0.0413998	+	0.999143i	$0.486818\pi$
$500$	0	0
$501$	−5.53224	−0.247162
$502$	0	0
$503$	−16.8921	−0.753181	−0.376590	−	0.926380i	$-0.622904\pi$
$503$	−16.8921	−0.753181	−0.376590	+	0.926380i	$0.622904\pi$
$504$	0	0
$505$	−8.60549	−0.382939
$506$	0	0
$507$	−8.92911	−0.396556
$508$	0	0
$509$	2.10162	0.0931524	0.0465762	−	0.998915i	$-0.485169\pi$
$509$	2.10162	0.0931524	0.0465762	+	0.998915i	$0.485169\pi$
$510$	0	0
$511$	0.0630671	0.00278992
$512$	0	0
$513$	−29.6875	−1.31073
$514$	0	0
$515$	24.5309	1.08096
$516$	0	0
$517$	2.16758	0.0953300
$518$	0	0
$519$	29.2308	1.28309
$520$	0	0
$521$	−1.90175	−0.0833170	−0.0416585	−	0.999132i	$-0.513264\pi$
$521$	−1.90175	−0.0833170	−0.0416585	+	0.999132i	$0.513264\pi$
$522$	0	0
$523$	−2.87743	−0.125821	−0.0629105	−	0.998019i	$-0.520038\pi$
$523$	−2.87743	−0.125821	−0.0629105	+	0.998019i	$0.520038\pi$
$524$	0	0
$525$	−3.27156	−0.142782
$526$	0	0
$527$	−25.3477	−1.10416
$528$	0	0
$529$	−22.5122	−0.978790
$530$	0	0
$531$	79.4934	3.44972
$532$	0	0
$533$	0.0631492	0.00273530
$534$	0	0
$535$	8.74363	0.378020
$536$	0	0
$537$	76.0075	3.27997
$538$	0	0
$539$	1.92489	0.0829107
$540$	0	0
$541$	3.56338	0.153202	0.0766009	−	0.997062i	$-0.475593\pi$
$541$	3.56338	0.153202	0.0766009	+	0.997062i	$0.475593\pi$
$542$	0	0
$543$	25.2906	1.08532
$544$	0	0
$545$	−14.4804	−0.620272
$546$	0	0
$547$	−8.83030	−0.377557	−0.188778	−	0.982020i	$-0.560453\pi$
$547$	−8.83030	−0.377557	−0.188778	+	0.982020i	$0.560453\pi$
$548$	0	0
$549$	−39.1328	−1.67015
$550$	0	0
$551$	6.29511	0.268181
$552$	0	0
$553$	4.25595	0.180982
$554$	0	0
$555$	−18.6811	−0.792970
$556$	0	0
$557$	−8.23306	−0.348846	−0.174423	−	0.984671i	$-0.555806\pi$
$557$	−8.23306	−0.348846	−0.174423	+	0.984671i	$0.555806\pi$
$558$	0	0
$559$	−48.9338	−2.06968
$560$	0	0
$561$	2.14557	0.0905859
$562$	0	0
$563$	40.7923	1.71919	0.859596	−	0.510974i	$-0.170715\pi$
$563$	40.7923	1.71919	0.859596	+	0.510974i	$0.170715\pi$
$564$	0	0
$565$	4.68003	0.196890
$566$	0	0
$567$	−2.16392	−0.0908761
$568$	0	0
$569$	17.6055	0.738063	0.369031	−	0.929417i	$-0.379689\pi$
$569$	17.6055	0.738063	0.369031	+	0.929417i	$0.379689\pi$
$570$	0	0
$571$	−16.2781	−0.681216	−0.340608	−	0.940205i	$-0.610633\pi$
$571$	−16.2781	−0.681216	−0.340608	+	0.940205i	$0.610633\pi$
$572$	0	0
$573$	−21.7426	−0.908309
$574$	0	0
$575$	2.04212	0.0851622
$576$	0	0
$577$	25.7002	1.06991	0.534956	−	0.844880i	$-0.320328\pi$
$577$	25.7002	1.06991	0.534956	+	0.844880i	$0.320328\pi$
$578$	0	0
$579$	38.2971	1.59157
$580$	0	0
$581$	−5.57604	−0.231333
$582$	0	0
$583$	1.38119	0.0572032
$584$	0	0
$585$	−32.4004	−1.33959
$586$	0	0
$587$	−35.7831	−1.47693	−0.738464	−	0.674293i	$-0.764448\pi$
$587$	−35.7831	−1.47693	−0.738464	+	0.674293i	$0.764448\pi$
$588$	0	0
$589$	−37.6760	−1.55241
$590$	0	0
$591$	6.80964	0.280111
$592$	0	0
$593$	−5.87794	−0.241378	−0.120689	−	0.992690i	$-0.538510\pi$
$593$	−5.87794	−0.241378	−0.120689	+	0.992690i	$0.538510\pi$
$594$	0	0
$595$	−1.43007	−0.0586271
$596$	0	0
$597$	63.3834	2.59411
$598$	0	0
$599$	−21.9834	−0.898218	−0.449109	−	0.893477i	$-0.648259\pi$
$599$	−21.9834	−0.898218	−0.449109	+	0.893477i	$0.648259\pi$
$600$	0	0
$601$	21.9020	0.893400	0.446700	−	0.894684i	$-0.352599\pi$
$601$	21.9020	0.893400	0.446700	+	0.894684i	$0.352599\pi$
$602$	0	0
$603$	16.7268	0.681169
$604$	0	0
$605$	−15.7364	−0.639774
$606$	0	0
$607$	−23.4846	−0.953209	−0.476604	−	0.879118i	$-0.658133\pi$
$607$	−23.4846	−0.953209	−0.476604	+	0.879118i	$0.658133\pi$
$608$	0	0
$609$	1.82041	0.0737667
$610$	0	0
$611$	30.9164	1.25074
$612$	0	0
$613$	8.60887	0.347709	0.173854	−	0.984771i	$-0.444378\pi$
$613$	8.60887	0.347709	0.173854	+	0.984771i	$0.444378\pi$
$614$	0	0
$615$	0.0666770	0.00268868
$616$	0	0
$617$	−33.1258	−1.33360	−0.666798	−	0.745238i	$-0.732336\pi$
$617$	−33.1258	−1.33360	−0.666798	+	0.745238i	$0.732336\pi$
$618$	0	0
$619$	15.7965	0.634915	0.317457	−	0.948273i	$-0.397171\pi$
$619$	15.7965	0.634915	0.317457	+	0.948273i	$0.397171\pi$
$620$	0	0
$621$	5.35878	0.215040
$622$	0	0
$623$	−5.46604	−0.218992
$624$	0	0
$625$	−1.83257	−0.0733027
$626$	0	0
$627$	3.18910	0.127360
$628$	0	0
$629$	11.4995	0.458515
$630$	0	0
$631$	41.7674	1.66273	0.831367	−	0.555724i	$-0.187559\pi$
$631$	41.7674	1.66273	0.831367	+	0.555724i	$0.187559\pi$
$632$	0	0
$633$	59.1331	2.35033
$634$	0	0
$635$	−0.445887	−0.0176945
$636$	0	0
$637$	27.4548	1.08780
$638$	0	0
$639$	−40.8809	−1.61722
$640$	0	0
$641$	12.5409	0.495337	0.247669	−	0.968845i	$-0.420336\pi$
$641$	12.5409	0.495337	0.247669	+	0.968845i	$0.420336\pi$
$642$	0	0
$643$	15.0272	0.592616	0.296308	−	0.955092i	$-0.404244\pi$
$643$	15.0272	0.592616	0.296308	+	0.955092i	$0.404244\pi$
$644$	0	0
$645$	−51.6675	−2.03441
$646$	0	0
$647$	−0.131198	−0.00515792	−0.00257896	−	0.999997i	$-0.500821\pi$
$647$	−0.131198	−0.00515792	−0.00257896	+	0.999997i	$0.500821\pi$
$648$	0	0
$649$	−3.97621	−0.156080
$650$	0	0
$651$	−10.8951	−0.427012
$652$	0	0
$653$	29.9259	1.17109	0.585546	−	0.810639i	$-0.300880\pi$
$653$	29.9259	1.17109	0.585546	+	0.810639i	$0.300880\pi$
$654$	0	0
$655$	−25.9276	−1.01307
$656$	0	0
$657$	−0.928710	−0.0362324
$658$	0	0
$659$	38.8457	1.51321	0.756606	−	0.653871i	$-0.226856\pi$
$659$	38.8457	1.51321	0.756606	+	0.653871i	$0.226856\pi$
$660$	0	0
$661$	−41.0165	−1.59536	−0.797678	−	0.603084i	$-0.793938\pi$
$661$	−41.0165	−1.59536	−0.797678	+	0.603084i	$0.793938\pi$
$662$	0	0
$663$	30.6024	1.18850
$664$	0	0
$665$	−2.12561	−0.0824275
$666$	0	0
$667$	−1.13631	−0.0439980
$668$	0	0
$669$	−23.3862	−0.904163
$670$	0	0
$671$	1.95740	0.0755645
$672$	0	0
$673$	8.97729	0.346049	0.173024	−	0.984918i	$-0.444646\pi$
$673$	8.97729	0.346049	0.173024	+	0.984918i	$0.444646\pi$
$674$	0	0
$675$	22.4324	0.863422
$676$	0	0
$677$	4.52843	0.174042	0.0870209	−	0.996206i	$-0.472265\pi$
$677$	4.52843	0.174042	0.0870209	+	0.996206i	$0.472265\pi$
$678$	0	0
$679$	−3.33851	−0.128120
$680$	0	0
$681$	−56.4812	−2.16436
$682$	0	0
$683$	−3.48350	−0.133292	−0.0666462	−	0.997777i	$-0.521230\pi$
$683$	−3.48350	−0.133292	−0.0666462	+	0.997777i	$0.521230\pi$
$684$	0	0
$685$	−12.8409	−0.490626
$686$	0	0
$687$	−37.4138	−1.42742
$688$	0	0
$689$	19.7001	0.750514
$690$	0	0
$691$	44.5762	1.69576	0.847879	−	0.530190i	$-0.177879\pi$
$691$	44.5762	1.69576	0.847879	+	0.530190i	$0.177879\pi$
$692$	0	0
$693$	0.601042	0.0228317
$694$	0	0
$695$	−11.2821	−0.427954
$696$	0	0
$697$	−0.0410442	−0.00155466
$698$	0	0
$699$	30.0581	1.13690
$700$	0	0
$701$	8.90961	0.336511	0.168256	−	0.985743i	$-0.446187\pi$
$701$	8.90961	0.336511	0.168256	+	0.985743i	$0.446187\pi$
$702$	0	0
$703$	17.0925	0.644655
$704$	0	0
$705$	32.6435	1.22943
$706$	0	0
$707$	2.27690	0.0856315
$708$	0	0
$709$	−13.4817	−0.506315	−0.253158	−	0.967425i	$-0.581469\pi$
$709$	−13.4817	−0.506315	−0.253158	+	0.967425i	$0.581469\pi$
$710$	0	0
$711$	−62.6721	−2.35039
$712$	0	0
$713$	6.80075	0.254690
$714$	0	0
$715$	1.62065	0.0606089
$716$	0	0
$717$	65.8210	2.45813
$718$	0	0
$719$	−25.1548	−0.938118	−0.469059	−	0.883167i	$-0.655407\pi$
$719$	−25.1548	−0.938118	−0.469059	+	0.883167i	$0.655407\pi$
$720$	0	0
$721$	−6.49055	−0.241721
$722$	0	0
$723$	−2.93498	−0.109153
$724$	0	0
$725$	−4.75669	−0.176659
$726$	0	0
$727$	1.31218	0.0486659	0.0243330	−	0.999704i	$-0.492254\pi$
$727$	1.31218	0.0486659	0.0243330	+	0.999704i	$0.492254\pi$
$728$	0	0
$729$	−35.6895	−1.32183
$730$	0	0
$731$	31.8048	1.17634
$732$	0	0
$733$	22.8551	0.844173	0.422087	−	0.906556i	$-0.361298\pi$
$733$	22.8551	0.844173	0.422087	+	0.906556i	$0.361298\pi$
$734$	0	0
$735$	28.9886	1.06926
$736$	0	0
$737$	−0.836665	−0.0308190
$738$	0	0
$739$	−10.2085	−0.375525	−0.187762	−	0.982214i	$-0.560123\pi$
$739$	−10.2085	−0.375525	−0.187762	+	0.982214i	$0.560123\pi$
$740$	0	0
$741$	45.4864	1.67099
$742$	0	0
$743$	−26.3806	−0.967810	−0.483905	−	0.875120i	$-0.660782\pi$
$743$	−26.3806	−0.967810	−0.483905	+	0.875120i	$0.660782\pi$
$744$	0	0
$745$	29.6194	1.08517
$746$	0	0
$747$	82.1115	3.00430
$748$	0	0
$749$	−2.31345	−0.0845315
$750$	0	0
$751$	31.8074	1.16067	0.580335	−	0.814378i	$-0.302922\pi$
$751$	31.8074	1.16067	0.580335	+	0.814378i	$0.302922\pi$
$752$	0	0
$753$	41.3037	1.50519
$754$	0	0
$755$	−15.2543	−0.555160
$756$	0	0
$757$	−33.6767	−1.22400	−0.612000	−	0.790858i	$-0.709635\pi$
$757$	−33.6767	−1.22400	−0.612000	+	0.790858i	$0.709635\pi$
$758$	0	0
$759$	−0.575653	−0.0208949
$760$	0	0
$761$	−14.9020	−0.540197	−0.270098	−	0.962833i	$-0.587056\pi$
$761$	−14.9020	−0.540197	−0.270098	+	0.962833i	$0.587056\pi$
$762$	0	0
$763$	3.83132	0.138703
$764$	0	0
$765$	21.0589	0.761385
$766$	0	0
$767$	−56.7131	−2.04779
$768$	0	0
$769$	−23.5992	−0.851010	−0.425505	−	0.904956i	$-0.639904\pi$
$769$	−23.5992	−0.851010	−0.425505	+	0.904956i	$0.639904\pi$
$770$	0	0
$771$	−4.29870	−0.154814
$772$	0	0
$773$	30.4365	1.09472	0.547362	−	0.836896i	$-0.315632\pi$
$773$	30.4365	1.09472	0.547362	+	0.836896i	$0.315632\pi$
$774$	0	0
$775$	28.4686	1.02262
$776$	0	0
$777$	4.94277	0.177321
$778$	0	0
$779$	−0.0610068	−0.00218579
$780$	0	0
$781$	2.04483	0.0731699
$782$	0	0
$783$	−12.4822	−0.446076
$784$	0	0
$785$	11.2380	0.401101
$786$	0	0
$787$	−0.281399	−0.0100308	−0.00501539	−	0.999987i	$-0.501596\pi$
$787$	−0.281399	−0.0100308	−0.00501539	+	0.999987i	$0.501596\pi$
$788$	0	0
$789$	23.7520	0.845593
$790$	0	0
$791$	−1.23827	−0.0440279
$792$	0	0
$793$	27.9185	0.991417
$794$	0	0
$795$	20.8006	0.737722
$796$	0	0
$797$	26.6128	0.942673	0.471336	−	0.881954i	$-0.343772\pi$
$797$	26.6128	0.942673	0.471336	+	0.881954i	$0.343772\pi$
$798$	0	0
$799$	−20.0943	−0.710885
$800$	0	0
$801$	80.4915	2.84403
$802$	0	0
$803$	0.0464535	0.00163931
$804$	0	0
$805$	0.383686	0.0135231
$806$	0	0
$807$	74.1133	2.60891
$808$	0	0
$809$	13.2472	0.465748	0.232874	−	0.972507i	$-0.425187\pi$
$809$	13.2472	0.465748	0.232874	+	0.972507i	$0.425187\pi$
$810$	0	0
$811$	−35.8348	−1.25833	−0.629165	−	0.777272i	$-0.716603\pi$
$811$	−35.8348	−1.25833	−0.629165	+	0.777272i	$0.716603\pi$
$812$	0	0
$813$	17.7305	0.621835
$814$	0	0
$815$	−24.6917	−0.864911
$816$	0	0
$817$	47.2737	1.65390
$818$	0	0
$819$	8.57271	0.299555
$820$	0	0
$821$	−1.76973	−0.0617640	−0.0308820	−	0.999523i	$-0.509832\pi$
$821$	−1.76973	−0.0617640	−0.0308820	+	0.999523i	$0.509832\pi$
$822$	0	0
$823$	29.0119	1.01129	0.505645	−	0.862742i	$-0.331255\pi$
$823$	29.0119	1.01129	0.505645	+	0.862742i	$0.331255\pi$
$824$	0	0
$825$	−2.40974	−0.0838964
$826$	0	0
$827$	15.9224	0.553675	0.276838	−	0.960917i	$-0.410714\pi$
$827$	15.9224	0.553675	0.276838	+	0.960917i	$0.410714\pi$
$828$	0	0
$829$	−31.0294	−1.07770	−0.538849	−	0.842403i	$-0.681141\pi$
$829$	−31.0294	−1.07770	−0.538849	+	0.842403i	$0.681141\pi$
$830$	0	0
$831$	52.6157	1.82522
$832$	0	0
$833$	−17.8444	−0.618273
$834$	0	0
$835$	2.71601	0.0939916
$836$	0	0
$837$	74.7053	2.58219
$838$	0	0
$839$	54.4602	1.88017	0.940087	−	0.340935i	$-0.110744\pi$
$839$	54.4602	1.88017	0.940087	+	0.340935i	$0.110744\pi$
$840$	0	0
$841$	−26.3532	−0.908731
$842$	0	0
$843$	−12.1296	−0.417765
$844$	0	0
$845$	4.38368	0.150803
$846$	0	0
$847$	4.16363	0.143064
$848$	0	0
$849$	40.4355	1.38774
$850$	0	0
$851$	−3.08530	−0.105763
$852$	0	0
$853$	−5.82777	−0.199539	−0.0997694	−	0.995011i	$-0.531811\pi$
$853$	−5.82777	−0.199539	−0.0997694	+	0.995011i	$0.531811\pi$
$854$	0	0
$855$	31.3012	1.07048
$856$	0	0
$857$	−48.9503	−1.67211	−0.836056	−	0.548644i	$-0.815144\pi$
$857$	−48.9503	−1.67211	−0.836056	+	0.548644i	$0.815144\pi$
$858$	0	0
$859$	23.1193	0.788819	0.394409	−	0.918935i	$-0.370949\pi$
$859$	23.1193	0.788819	0.394409	+	0.918935i	$0.370949\pi$
$860$	0	0
$861$	−0.0176418	−0.000601232 0
$862$	0	0
$863$	24.5041	0.834129	0.417065	−	0.908877i	$-0.363059\pi$
$863$	24.5041	0.834129	0.417065	+	0.908877i	$0.363059\pi$
$864$	0	0
$865$	−14.3507	−0.487937
$866$	0	0
$867$	30.0045	1.01900
$868$	0	0
$869$	3.13482	0.106342
$870$	0	0
$871$	−11.9334	−0.404349
$872$	0	0
$873$	49.1621	1.66388
$874$	0	0
$875$	4.35284	0.147153
$876$	0	0
$877$	−11.9231	−0.402614	−0.201307	−	0.979528i	$-0.564519\pi$
$877$	−11.9231	−0.402614	−0.201307	+	0.979528i	$0.564519\pi$
$878$	0	0
$879$	−63.4112	−2.13881
$880$	0	0
$881$	−12.2901	−0.414063	−0.207032	−	0.978334i	$-0.566380\pi$
$881$	−12.2901	−0.414063	−0.207032	+	0.978334i	$0.566380\pi$
$882$	0	0
$883$	48.1400	1.62004	0.810020	−	0.586403i	$-0.199456\pi$
$883$	48.1400	1.62004	0.810020	+	0.586403i	$0.199456\pi$
$884$	0	0
$885$	−59.8813	−2.01289
$886$	0	0
$887$	−20.7341	−0.696184	−0.348092	−	0.937460i	$-0.613170\pi$
$887$	−20.7341	−0.696184	−0.348092	+	0.937460i	$0.613170\pi$
$888$	0	0
$889$	0.117976	0.00395678
$890$	0	0
$891$	−1.59389	−0.0533972
$892$	0	0
$893$	−29.8675	−0.999477
$894$	0	0
$895$	−37.3153	−1.24731
$896$	0	0
$897$	−8.21059	−0.274144
$898$	0	0
$899$	−15.8410	−0.528325
$900$	0	0
$901$	−12.8042	−0.426569
$902$	0	0
$903$	13.6705	0.454926
$904$	0	0
$905$	−12.4162	−0.412730
$906$	0	0
$907$	7.02598	0.233294	0.116647	−	0.993173i	$-0.462785\pi$
$907$	7.02598	0.233294	0.116647	+	0.993173i	$0.462785\pi$
$908$	0	0
$909$	−33.5290	−1.11209
$910$	0	0
$911$	−58.0477	−1.92320	−0.961602	−	0.274446i	$-0.911505\pi$
$911$	−58.0477	−1.92320	−0.961602	+	0.274446i	$0.911505\pi$
$912$	0	0
$913$	−4.10716	−0.135927
$914$	0	0
$915$	29.4782	0.974519
$916$	0	0
$917$	6.86008	0.226540
$918$	0	0
$919$	−17.9524	−0.592194	−0.296097	−	0.955158i	$-0.595685\pi$
$919$	−17.9524	−0.592194	−0.296097	+	0.955158i	$0.595685\pi$
$920$	0	0
$921$	59.6324	1.96496
$922$	0	0
$923$	29.1657	0.960000
$924$	0	0
$925$	−12.9154	−0.424655
$926$	0	0
$927$	95.5782	3.13920
$928$	0	0
$929$	−20.8907	−0.685403	−0.342701	−	0.939444i	$-0.611342\pi$
$929$	−20.8907	−0.685403	−0.342701	+	0.939444i	$0.611342\pi$
$930$	0	0
$931$	−26.5234	−0.869268
$932$	0	0
$933$	60.1019	1.96765
$934$	0	0
$935$	−1.05335	−0.0344483
$936$	0	0
$937$	−39.5712	−1.29273	−0.646367	−	0.763027i	$-0.723712\pi$
$937$	−39.5712	−1.29273	−0.646367	+	0.763027i	$0.723712\pi$
$938$	0	0
$939$	−5.82561	−0.190112
$940$	0	0
$941$	−2.21402	−0.0721750	−0.0360875	−	0.999349i	$-0.511490\pi$
$941$	−2.21402	−0.0721750	−0.0360875	+	0.999349i	$0.511490\pi$
$942$	0	0
$943$	0.0110121	0.000358603 0
$944$	0	0
$945$	4.21473	0.137105
$946$	0	0
$947$	27.2239	0.884659	0.442330	−	0.896853i	$-0.354152\pi$
$947$	27.2239	0.884659	0.442330	+	0.896853i	$0.354152\pi$
$948$	0	0
$949$	0.662571	0.0215080
$950$	0	0
$951$	83.8842	2.72013
$952$	0	0
$953$	−49.1033	−1.59061	−0.795305	−	0.606209i	$-0.792689\pi$
$953$	−49.1033	−1.59061	−0.795305	+	0.606209i	$0.792689\pi$
$954$	0	0
$955$	10.6744	0.345414
$956$	0	0
$957$	1.34086	0.0433440
$958$	0	0
$959$	3.39753	0.109712
$960$	0	0
$961$	63.8075	2.05831
$962$	0	0
$963$	34.0672	1.09780
$964$	0	0
$965$	−18.8017	−0.605248
$966$	0	0
$967$	2.12457	0.0683214	0.0341607	−	0.999416i	$-0.489124\pi$
$967$	2.12457	0.0683214	0.0341607	+	0.999416i	$0.489124\pi$
$968$	0	0
$969$	−29.5642	−0.949738
$970$	0	0
$971$	−32.0749	−1.02933	−0.514666	−	0.857391i	$-0.672084\pi$
$971$	−32.0749	−1.02933	−0.514666	+	0.857391i	$0.672084\pi$
$972$	0	0
$973$	2.98509	0.0956975
$974$	0	0
$975$	−34.3704	−1.10073
$976$	0	0
$977$	−1.85481	−0.0593406	−0.0296703	−	0.999560i	$-0.509446\pi$
$977$	−1.85481	−0.0593406	−0.0296703	+	0.999560i	$0.509446\pi$
$978$	0	0
$979$	−4.02614	−0.128676
$980$	0	0
$981$	−56.4190	−1.80132
$982$	0	0
$983$	48.0786	1.53347	0.766734	−	0.641965i	$-0.221881\pi$
$983$	48.0786	1.53347	0.766734	+	0.641965i	$0.221881\pi$
$984$	0	0
$985$	−3.34314	−0.106521
$986$	0	0
$987$	−8.63703	−0.274920
$988$	0	0
$989$	−8.53319	−0.271340
$990$	0	0
$991$	−45.7585	−1.45357	−0.726783	−	0.686867i	$-0.758985\pi$
$991$	−45.7585	−1.45357	−0.726783	+	0.686867i	$0.758985\pi$
$992$	0	0
$993$	−68.7046	−2.18028
$994$	0	0
$995$	−31.1176	−0.986495
$996$	0	0
$997$	−17.8475	−0.565235	−0.282618	−	0.959233i	$-0.591203\pi$
$997$	−17.8475	−0.565235	−0.282618	+	0.959233i	$0.591203\pi$
$998$	0	0
$999$	−33.8916	−1.07228

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.n.1.1		12
4.3	odd	2		241.2.a.b.1.2	✓	12
12.11	even	2		2169.2.a.h.1.11		12
20.19	odd	2		6025.2.a.h.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.2	✓	12	4.3	odd	2
2169.2.a.h.1.11		12	12.11	even	2
3856.2.a.n.1.1		12	1.1	even	1	trivial
6025.2.a.h.1.11		12	20.19	odd	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:	\( N \)	\(=\)	\( 3856 = 2^{4} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3856.a (trivial)

\(f(q)\)	\(=\)	\(q-2.93498 q^{3} +1.44091 q^{5} -0.381245 q^{7} +5.61411 q^{9} +O(q^{10})\)	\(q-2.93498 q^{3} +1.44091 q^{5} -0.381245 q^{7} +5.61411 q^{9} -0.280814 q^{11} -4.00528 q^{13} -4.22904 q^{15} +2.60326 q^{17} +3.86940 q^{19} +1.11895 q^{21} -0.698450 q^{23} -2.92378 q^{25} -7.67238 q^{27} +1.62690 q^{29} -9.73691 q^{31} +0.824185 q^{33} -0.549338 q^{35} +4.41735 q^{37} +11.7554 q^{39} -0.0157665 q^{41} +12.2173 q^{43} +8.08942 q^{45} -7.71890 q^{47} -6.85465 q^{49} -7.64051 q^{51} -4.91852 q^{53} -0.404628 q^{55} -11.3566 q^{57} +14.1596 q^{59} -6.97043 q^{61} -2.14035 q^{63} -5.77125 q^{65} +2.97942 q^{67} +2.04994 q^{69} -7.28180 q^{71} -0.165424 q^{73} +8.58125 q^{75} +0.107059 q^{77} -11.1633 q^{79} +5.67594 q^{81} +14.6259 q^{83} +3.75106 q^{85} -4.77491 q^{87} +14.3374 q^{89} +1.52699 q^{91} +28.5777 q^{93} +5.57544 q^{95} +8.75687 q^{97} -1.57652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9 - 22 * q^11 - 5 * q^13 - 13 * q^15 - 4 * q^17 + 6 * q^19 - 14 * q^21 - 32 * q^23 + 4 * q^25 + 5 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - 15 * q^35 - 8 * q^37 - 31 * q^39 - q^41 + 2 * q^43 - 15 * q^45 - 34 * q^47 - 9 * q^49 + 3 * q^51 + 5 * q^53 + 3 * q^55 - 22 * q^57 - 26 * q^59 - 26 * q^61 + 4 * q^63 - 25 * q^65 - 6 * q^67 - 2 * q^69 - 94 * q^71 - 22 * q^73 - 7 * q^77 - 9 * q^79 + 4 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 3 * q^89 + 20 * q^91 + 12 * q^93 - 33 * q^95 - 29 * q^97 - 36 * q^99

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