LMFDB - Embedded newform 3016.2.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3016,2,Mod(1,3016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3016, 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("3016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3016 = 2^{3} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3016.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:	$24.0828812496$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 21x^{8} + 40x^{7} + 138x^{6} - 243x^{5} - 318x^{4} + 448x^{3} + 312x^{2} - 240x - 128 $ x^10 - 2x^9 - 21x^8 + 40x^7 + 138x^6 - 243x^5 - 318x^4 + 448x^3 + 312x^2 - 240*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.866798$ of defining polynomial
Character	$\chi$	$=$	3016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.866798 q^{3} +2.19814 q^{5} -4.35947 q^{7} -2.24866 q^{9} +O(q^{10})$	$q-0.866798 q^{3} +2.19814 q^{5} -4.35947 q^{7} -2.24866 q^{9} -3.62729 q^{11} +1.00000 q^{13} -1.90534 q^{15} +0.809117 q^{17} -5.53409 q^{19} +3.77878 q^{21} +3.43856 q^{23} -0.168191 q^{25} +4.54953 q^{27} -1.00000 q^{29} +0.227533 q^{31} +3.14413 q^{33} -9.58271 q^{35} +4.90883 q^{37} -0.866798 q^{39} +6.88191 q^{41} +2.53289 q^{43} -4.94287 q^{45} -2.69537 q^{47} +12.0049 q^{49} -0.701341 q^{51} -3.14083 q^{53} -7.97328 q^{55} +4.79694 q^{57} +12.9502 q^{59} +7.54000 q^{61} +9.80296 q^{63} +2.19814 q^{65} +8.95057 q^{67} -2.98054 q^{69} -7.11287 q^{71} +15.6224 q^{73} +0.145788 q^{75} +15.8130 q^{77} +1.98066 q^{79} +2.80246 q^{81} -1.59895 q^{83} +1.77855 q^{85} +0.866798 q^{87} -8.71529 q^{89} -4.35947 q^{91} -0.197225 q^{93} -12.1647 q^{95} -15.3323 q^{97} +8.15655 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 16 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 + 5 * q^5 - 2 * q^7 + 16 * q^9	$ 10 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 16 q^{9} + 4 q^{11} + 10 q^{13} + 8 q^{15} + 10 q^{17} - q^{19} + q^{21} + 23 q^{23} + 25 q^{25} + 2 q^{27} - 10 q^{29} - 13 q^{31} + 15 q^{33} - 12 q^{35} - 7 q^{37} + 2 q^{39} + 16 q^{41} - 12 q^{43} + 55 q^{45} + 11 q^{47} - 25 q^{51} + 11 q^{53} + 22 q^{55} - 6 q^{57} - 11 q^{59} + 34 q^{61} + 37 q^{63} + 5 q^{65} - 23 q^{67} + 2 q^{69} - 4 q^{71} + 39 q^{73} + 11 q^{75} + 32 q^{77} + 5 q^{79} + 38 q^{81} + 6 q^{83} + 45 q^{85} - 2 q^{87} - 24 q^{89} - 2 q^{91} + 13 q^{93} + 33 q^{95} + 19 q^{97}+O(q^{100}) $ 10 * q + 2 * q^3 + 5 * q^5 - 2 * q^7 + 16 * q^9 + 4 * q^11 + 10 * q^13 + 8 * q^15 + 10 * q^17 - q^19 + q^21 + 23 * q^23 + 25 * q^25 + 2 * q^27 - 10 * q^29 - 13 * q^31 + 15 * q^33 - 12 * q^35 - 7 * q^37 + 2 * q^39 + 16 * q^41 - 12 * q^43 + 55 * q^45 + 11 * q^47 - 25 * q^51 + 11 * q^53 + 22 * q^55 - 6 * q^57 - 11 * q^59 + 34 * q^61 + 37 * q^63 + 5 * q^65 - 23 * q^67 + 2 * q^69 - 4 * q^71 + 39 * q^73 + 11 * q^75 + 32 * q^77 + 5 * q^79 + 38 * q^81 + 6 * q^83 + 45 * q^85 - 2 * q^87 - 24 * q^89 - 2 * q^91 + 13 * q^93 + 33 * q^95 + 19 * q^97

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
$3$	−0.866798	−0.500446	−0.250223	−	0.968188i	$-0.580504\pi$
$3$	−0.866798	−0.500446	−0.250223	+	0.968188i	$0.580504\pi$
$4$	0	0
$5$	2.19814	0.983037	0.491519	−	0.870867i	$-0.336442\pi$
$5$	2.19814	0.983037	0.491519	+	0.870867i	$0.336442\pi$
$6$	0	0
$7$	−4.35947	−1.64772	−0.823862	−	0.566791i	$-0.808185\pi$
$7$	−4.35947	−1.64772	−0.823862	+	0.566791i	$0.808185\pi$
$8$	0	0
$9$	−2.24866	−0.749554
$10$	0	0
$11$	−3.62729	−1.09367	−0.546835	−	0.837241i	$-0.684167\pi$
$11$	−3.62729	−1.09367	−0.546835	+	0.837241i	$0.684167\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.90534	−0.491957
$16$	0	0
$17$	0.809117	0.196240	0.0981198	−	0.995175i	$-0.468717\pi$
$17$	0.809117	0.196240	0.0981198	+	0.995175i	$0.468717\pi$
$18$	0	0
$19$	−5.53409	−1.26961	−0.634803	−	0.772674i	$-0.718919\pi$
$19$	−5.53409	−1.26961	−0.634803	+	0.772674i	$0.718919\pi$
$20$	0	0
$21$	3.77878	0.824597
$22$	0	0
$23$	3.43856	0.716989	0.358495	−	0.933532i	$-0.383290\pi$
$23$	3.43856	0.716989	0.358495	+	0.933532i	$0.383290\pi$
$24$	0	0
$25$	−0.168191	−0.0336382
$26$	0	0
$27$	4.54953	0.875557
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0.227533	0.0408662	0.0204331	−	0.999791i	$-0.493495\pi$
$31$	0.227533	0.0408662	0.0204331	+	0.999791i	$0.493495\pi$
$32$	0	0
$33$	3.14413	0.547322
$34$	0	0
$35$	−9.58271	−1.61977
$36$	0	0
$37$	4.90883	0.807007	0.403504	−	0.914978i	$-0.367792\pi$
$37$	4.90883	0.807007	0.403504	+	0.914978i	$0.367792\pi$
$38$	0	0
$39$	−0.866798	−0.138799
$40$	0	0
$41$	6.88191	1.07477	0.537387	−	0.843336i	$-0.319411\pi$
$41$	6.88191	1.07477	0.537387	+	0.843336i	$0.319411\pi$
$42$	0	0
$43$	2.53289	0.386262	0.193131	−	0.981173i	$-0.438136\pi$
$43$	2.53289	0.386262	0.193131	+	0.981173i	$0.438136\pi$
$44$	0	0
$45$	−4.94287	−0.736839
$46$	0	0
$47$	−2.69537	−0.393160	−0.196580	−	0.980488i	$-0.562984\pi$
$47$	−2.69537	−0.393160	−0.196580	+	0.980488i	$0.562984\pi$
$48$	0	0
$49$	12.0049	1.71499
$50$	0	0
$51$	−0.701341	−0.0982074
$52$	0	0
$53$	−3.14083	−0.431427	−0.215713	−	0.976457i	$-0.569208\pi$
$53$	−3.14083	−0.431427	−0.215713	+	0.976457i	$0.569208\pi$
$54$	0	0
$55$	−7.97328	−1.07512
$56$	0	0
$57$	4.79694	0.635370
$58$	0	0
$59$	12.9502	1.68598	0.842989	−	0.537930i	$-0.180794\pi$
$59$	12.9502	1.68598	0.842989	+	0.537930i	$0.180794\pi$
$60$	0	0
$61$	7.54000	0.965398	0.482699	−	0.875786i	$-0.339656\pi$
$61$	7.54000	0.965398	0.482699	+	0.875786i	$0.339656\pi$
$62$	0	0
$63$	9.80296	1.23506
$64$	0	0
$65$	2.19814	0.272645
$66$	0	0
$67$	8.95057	1.09349	0.546743	−	0.837300i	$-0.315867\pi$
$67$	8.95057	1.09349	0.546743	+	0.837300i	$0.315867\pi$
$68$	0	0
$69$	−2.98054	−0.358815
$70$	0	0
$71$	−7.11287	−0.844143	−0.422071	−	0.906563i	$-0.638697\pi$
$71$	−7.11287	−0.844143	−0.422071	+	0.906563i	$0.638697\pi$
$72$	0	0
$73$	15.6224	1.82847	0.914235	−	0.405185i	$-0.132793\pi$
$73$	15.6224	1.82847	0.914235	+	0.405185i	$0.132793\pi$
$74$	0	0
$75$	0.145788	0.0168341
$76$	0	0
$77$	15.8130	1.80206
$78$	0	0
$79$	1.98066	0.222841	0.111421	−	0.993773i	$-0.464460\pi$
$79$	1.98066	0.222841	0.111421	+	0.993773i	$0.464460\pi$
$80$	0	0
$81$	2.80246	0.311384
$82$	0	0
$83$	−1.59895	−0.175508	−0.0877538	−	0.996142i	$-0.527969\pi$
$83$	−1.59895	−0.175508	−0.0877538	+	0.996142i	$0.527969\pi$
$84$	0	0
$85$	1.77855	0.192911
$86$	0	0
$87$	0.866798	0.0929305
$88$	0	0
$89$	−8.71529	−0.923818	−0.461909	−	0.886927i	$-0.652835\pi$
$89$	−8.71529	−0.923818	−0.461909	+	0.886927i	$0.652835\pi$
$90$	0	0
$91$	−4.35947	−0.456996
$92$	0	0
$93$	−0.197225	−0.0204513
$94$	0	0
$95$	−12.1647	−1.24807
$96$	0	0
$97$	−15.3323	−1.55676	−0.778378	−	0.627796i	$-0.783957\pi$
$97$	−15.3323	−1.55676	−0.778378	+	0.627796i	$0.783957\pi$
$98$	0	0
$99$	8.15655	0.819764
$100$	0	0
$101$	1.74668	0.173801	0.0869005	−	0.996217i	$-0.472304\pi$
$101$	1.74668	0.173801	0.0869005	+	0.996217i	$0.472304\pi$
$102$	0	0
$103$	−3.96219	−0.390406	−0.195203	−	0.980763i	$-0.562537\pi$
$103$	−3.96219	−0.390406	−0.195203	+	0.980763i	$0.562537\pi$
$104$	0	0
$105$	8.30627	0.810609
$106$	0	0
$107$	10.9317	1.05681	0.528406	−	0.848992i	$-0.322790\pi$
$107$	10.9317	1.05681	0.528406	+	0.848992i	$0.322790\pi$
$108$	0	0
$109$	−3.21484	−0.307926	−0.153963	−	0.988077i	$-0.549204\pi$
$109$	−3.21484	−0.307926	−0.153963	+	0.988077i	$0.549204\pi$
$110$	0	0
$111$	−4.25497	−0.403864
$112$	0	0
$113$	13.0885	1.23127	0.615633	−	0.788033i	$-0.288900\pi$
$113$	13.0885	1.23127	0.615633	+	0.788033i	$0.288900\pi$
$114$	0	0
$115$	7.55843	0.704827
$116$	0	0
$117$	−2.24866	−0.207889
$118$	0	0
$119$	−3.52732	−0.323349
$120$	0	0
$121$	2.15723	0.196112
$122$	0	0
$123$	−5.96523	−0.537867
$124$	0	0
$125$	−11.3604	−1.01610
$126$	0	0
$127$	−1.79424	−0.159213	−0.0796066	−	0.996826i	$-0.525366\pi$
$127$	−1.79424	−0.159213	−0.0796066	+	0.996826i	$0.525366\pi$
$128$	0	0
$129$	−2.19551	−0.193304
$130$	0	0
$131$	−2.50063	−0.218481	−0.109241	−	0.994015i	$-0.534842\pi$
$131$	−2.50063	−0.218481	−0.109241	+	0.994015i	$0.534842\pi$
$132$	0	0
$133$	24.1257	2.09196
$134$	0	0
$135$	10.0005	0.860705
$136$	0	0
$137$	−11.2810	−0.963798	−0.481899	−	0.876227i	$-0.660053\pi$
$137$	−11.2810	−0.963798	−0.481899	+	0.876227i	$0.660053\pi$
$138$	0	0
$139$	18.1314	1.53789	0.768945	−	0.639315i	$-0.220782\pi$
$139$	18.1314	1.53789	0.768945	+	0.639315i	$0.220782\pi$
$140$	0	0
$141$	2.33634	0.196756
$142$	0	0
$143$	−3.62729	−0.303329
$144$	0	0
$145$	−2.19814	−0.182545
$146$	0	0
$147$	−10.4059	−0.858261
$148$	0	0
$149$	−6.47830	−0.530723	−0.265361	−	0.964149i	$-0.585491\pi$
$149$	−6.47830	−0.530723	−0.265361	+	0.964149i	$0.585491\pi$
$150$	0	0
$151$	−0.449097	−0.0365470	−0.0182735	−	0.999833i	$-0.505817\pi$
$151$	−0.449097	−0.0365470	−0.0182735	+	0.999833i	$0.505817\pi$
$152$	0	0
$153$	−1.81943	−0.147092
$154$	0	0
$155$	0.500149	0.0401729
$156$	0	0
$157$	2.05118	0.163702	0.0818508	−	0.996645i	$-0.473917\pi$
$157$	2.05118	0.163702	0.0818508	+	0.996645i	$0.473917\pi$
$158$	0	0
$159$	2.72247	0.215906
$160$	0	0
$161$	−14.9903	−1.18140
$162$	0	0
$163$	−13.0998	−1.02606	−0.513029	−	0.858371i	$-0.671477\pi$
$163$	−13.0998	−1.02606	−0.513029	+	0.858371i	$0.671477\pi$
$164$	0	0
$165$	6.91123	0.538038
$166$	0	0
$167$	−3.02735	−0.234264	−0.117132	−	0.993116i	$-0.537370\pi$
$167$	−3.02735	−0.234264	−0.117132	+	0.993116i	$0.537370\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	12.4443	0.951638
$172$	0	0
$173$	2.87327	0.218451	0.109225	−	0.994017i	$-0.465163\pi$
$173$	2.87327	0.218451	0.109225	+	0.994017i	$0.465163\pi$
$174$	0	0
$175$	0.733223	0.0554264
$176$	0	0
$177$	−11.2252	−0.843741
$178$	0	0
$179$	1.33055	0.0994501	0.0497250	−	0.998763i	$-0.484166\pi$
$179$	1.33055	0.0994501	0.0497250	+	0.998763i	$0.484166\pi$
$180$	0	0
$181$	15.5300	1.15433	0.577167	−	0.816626i	$-0.304158\pi$
$181$	15.5300	1.15433	0.577167	+	0.816626i	$0.304158\pi$
$182$	0	0
$183$	−6.53566	−0.483130
$184$	0	0
$185$	10.7903	0.793318
$186$	0	0
$187$	−2.93490	−0.214621
$188$	0	0
$189$	−19.8335	−1.44268
$190$	0	0
$191$	8.87561	0.642216	0.321108	−	0.947043i	$-0.395945\pi$
$191$	8.87561	0.642216	0.321108	+	0.947043i	$0.395945\pi$
$192$	0	0
$193$	14.7284	1.06018	0.530088	−	0.847943i	$-0.322159\pi$
$193$	14.7284	1.06018	0.530088	+	0.847943i	$0.322159\pi$
$194$	0	0
$195$	−1.90534	−0.136444
$196$	0	0
$197$	0.728224	0.0518838	0.0259419	−	0.999663i	$-0.491742\pi$
$197$	0.728224	0.0518838	0.0259419	+	0.999663i	$0.491742\pi$
$198$	0	0
$199$	−9.75714	−0.691665	−0.345833	−	0.938296i	$-0.612403\pi$
$199$	−9.75714	−0.691665	−0.345833	+	0.938296i	$0.612403\pi$
$200$	0	0
$201$	−7.75834	−0.547231
$202$	0	0
$203$	4.35947	0.305975
$204$	0	0
$205$	15.1274	1.05654
$206$	0	0
$207$	−7.73216	−0.537422
$208$	0	0
$209$	20.0737	1.38853
$210$	0	0
$211$	5.49906	0.378571	0.189285	−	0.981922i	$-0.439383\pi$
$211$	5.49906	0.378571	0.189285	+	0.981922i	$0.439383\pi$
$212$	0	0
$213$	6.16543	0.422448
$214$	0	0
$215$	5.56764	0.379710
$216$	0	0
$217$	−0.991923	−0.0673361
$218$	0	0
$219$	−13.5415	−0.915050
$220$	0	0
$221$	0.809117	0.0544271
$222$	0	0
$223$	17.6940	1.18488	0.592440	−	0.805615i	$-0.298165\pi$
$223$	17.6940	1.18488	0.592440	+	0.805615i	$0.298165\pi$
$224$	0	0
$225$	0.378205	0.0252136
$226$	0	0
$227$	−5.78798	−0.384162	−0.192081	−	0.981379i	$-0.561524\pi$
$227$	−5.78798	−0.384162	−0.192081	+	0.981379i	$0.561524\pi$
$228$	0	0
$229$	22.8632	1.51084	0.755422	−	0.655239i	$-0.227432\pi$
$229$	22.8632	1.51084	0.755422	+	0.655239i	$0.227432\pi$
$230$	0	0
$231$	−13.7067	−0.901836
$232$	0	0
$233$	−13.8442	−0.906963	−0.453482	−	0.891266i	$-0.649818\pi$
$233$	−13.8442	−0.906963	−0.453482	+	0.891266i	$0.649818\pi$
$234$	0	0
$235$	−5.92480	−0.386491
$236$	0	0
$237$	−1.71683	−0.111520
$238$	0	0
$239$	8.74900	0.565926	0.282963	−	0.959131i	$-0.408683\pi$
$239$	8.74900	0.565926	0.282963	+	0.959131i	$0.408683\pi$
$240$	0	0
$241$	−18.5363	−1.19403	−0.597014	−	0.802231i	$-0.703646\pi$
$241$	−18.5363	−1.19403	−0.597014	+	0.802231i	$0.703646\pi$
$242$	0	0
$243$	−16.0778	−1.03139
$244$	0	0
$245$	26.3885	1.68590
$246$	0	0
$247$	−5.53409	−0.352126
$248$	0	0
$249$	1.38597	0.0878321
$250$	0	0
$251$	−6.68464	−0.421931	−0.210965	−	0.977494i	$-0.567661\pi$
$251$	−6.68464	−0.421931	−0.210965	+	0.977494i	$0.567661\pi$
$252$	0	0
$253$	−12.4727	−0.784149
$254$	0	0
$255$	−1.54164	−0.0965415
$256$	0	0
$257$	20.9053	1.30404	0.652019	−	0.758203i	$-0.273922\pi$
$257$	20.9053	1.30404	0.652019	+	0.758203i	$0.273922\pi$
$258$	0	0
$259$	−21.3999	−1.32972
$260$	0	0
$261$	2.24866	0.139189
$262$	0	0
$263$	23.7095	1.46199	0.730995	−	0.682383i	$-0.239056\pi$
$263$	23.7095	1.46199	0.730995	+	0.682383i	$0.239056\pi$
$264$	0	0
$265$	−6.90398	−0.424108
$266$	0	0
$267$	7.55439	0.462321
$268$	0	0
$269$	−10.7389	−0.654763	−0.327381	−	0.944892i	$-0.606166\pi$
$269$	−10.7389	−0.654763	−0.327381	+	0.944892i	$0.606166\pi$
$270$	0	0
$271$	27.5160	1.67148	0.835739	−	0.549127i	$-0.185040\pi$
$271$	27.5160	1.67148	0.835739	+	0.549127i	$0.185040\pi$
$272$	0	0
$273$	3.77878	0.228702
$274$	0	0
$275$	0.610077	0.0367891
$276$	0	0
$277$	−18.4171	−1.10658	−0.553289	−	0.832989i	$-0.686628\pi$
$277$	−18.4171	−1.10658	−0.553289	+	0.832989i	$0.686628\pi$
$278$	0	0
$279$	−0.511645	−0.0306314
$280$	0	0
$281$	3.07889	0.183671	0.0918356	−	0.995774i	$-0.470727\pi$
$281$	3.07889	0.183671	0.0918356	+	0.995774i	$0.470727\pi$
$282$	0	0
$283$	−5.30920	−0.315599	−0.157800	−	0.987471i	$-0.550440\pi$
$283$	−5.30920	−0.315599	−0.157800	+	0.987471i	$0.550440\pi$
$284$	0	0
$285$	10.5443	0.624592
$286$	0	0
$287$	−30.0015	−1.77093
$288$	0	0
$289$	−16.3453	−0.961490
$290$	0	0
$291$	13.2900	0.779072
$292$	0	0
$293$	16.8332	0.983405	0.491702	−	0.870763i	$-0.336375\pi$
$293$	16.8332	0.983405	0.491702	+	0.870763i	$0.336375\pi$
$294$	0	0
$295$	28.4664	1.65738
$296$	0	0
$297$	−16.5025	−0.957570
$298$	0	0
$299$	3.43856	0.198857
$300$	0	0
$301$	−11.0421	−0.636454
$302$	0	0
$303$	−1.51402	−0.0869780
$304$	0	0
$305$	16.5740	0.949022
$306$	0	0
$307$	25.9689	1.48213	0.741063	−	0.671436i	$-0.234322\pi$
$307$	25.9689	1.48213	0.741063	+	0.671436i	$0.234322\pi$
$308$	0	0
$309$	3.43442	0.195377
$310$	0	0
$311$	10.3043	0.584304	0.292152	−	0.956372i	$-0.405629\pi$
$311$	10.3043	0.584304	0.292152	+	0.956372i	$0.405629\pi$
$312$	0	0
$313$	30.2526	1.70998	0.854989	−	0.518646i	$-0.173564\pi$
$313$	30.2526	1.70998	0.854989	+	0.518646i	$0.173564\pi$
$314$	0	0
$315$	21.5483	1.21411
$316$	0	0
$317$	−18.4428	−1.03585	−0.517926	−	0.855425i	$-0.673296\pi$
$317$	−18.4428	−1.03585	−0.517926	+	0.855425i	$0.673296\pi$
$318$	0	0
$319$	3.62729	0.203089
$320$	0	0
$321$	−9.47561	−0.528877
$322$	0	0
$323$	−4.47772	−0.249147
$324$	0	0
$325$	−0.168191	−0.00932956
$326$	0	0
$327$	2.78662	0.154100
$328$	0	0
$329$	11.7504	0.647820
$330$	0	0
$331$	−5.65331	−0.310734	−0.155367	−	0.987857i	$-0.549656\pi$
$331$	−5.65331	−0.310734	−0.155367	+	0.987857i	$0.549656\pi$
$332$	0	0
$333$	−11.0383	−0.604895
$334$	0	0
$335$	19.6746	1.07494
$336$	0	0
$337$	−19.5324	−1.06400	−0.532000	−	0.846744i	$-0.678559\pi$
$337$	−19.5324	−1.06400	−0.532000	+	0.846744i	$0.678559\pi$
$338$	0	0
$339$	−11.3451	−0.616183
$340$	0	0
$341$	−0.825329	−0.0446941
$342$	0	0
$343$	−21.8189	−1.17811
$344$	0	0
$345$	−6.55163	−0.352728
$346$	0	0
$347$	−6.63857	−0.356377	−0.178189	−	0.983996i	$-0.557024\pi$
$347$	−6.63857	−0.356377	−0.178189	+	0.983996i	$0.557024\pi$
$348$	0	0
$349$	−4.45417	−0.238426	−0.119213	−	0.992869i	$-0.538037\pi$
$349$	−4.45417	−0.238426	−0.119213	+	0.992869i	$0.538037\pi$
$350$	0	0
$351$	4.54953	0.242836
$352$	0	0
$353$	7.34190	0.390770	0.195385	−	0.980727i	$-0.437404\pi$
$353$	7.34190	0.390770	0.195385	+	0.980727i	$0.437404\pi$
$354$	0	0
$355$	−15.6351	−0.829824
$356$	0	0
$357$	3.05747	0.161819
$358$	0	0
$359$	23.0325	1.21561	0.607804	−	0.794087i	$-0.292051\pi$
$359$	23.0325	1.21561	0.607804	+	0.794087i	$0.292051\pi$
$360$	0	0
$361$	11.6261	0.611901
$362$	0	0
$363$	−1.86988	−0.0981435
$364$	0	0
$365$	34.3403	1.79745
$366$	0	0
$367$	26.9306	1.40576	0.702882	−	0.711306i	$-0.251896\pi$
$367$	26.9306	1.40576	0.702882	+	0.711306i	$0.251896\pi$
$368$	0	0
$369$	−15.4751	−0.805601
$370$	0	0
$371$	13.6924	0.710872
$372$	0	0
$373$	31.9916	1.65646	0.828230	−	0.560388i	$-0.189348\pi$
$373$	31.9916	1.65646	0.828230	+	0.560388i	$0.189348\pi$
$374$	0	0
$375$	9.84717	0.508506
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−11.4552	−0.588413	−0.294207	−	0.955742i	$-0.595055\pi$
$379$	−11.4552	−0.588413	−0.294207	+	0.955742i	$0.595055\pi$
$380$	0	0
$381$	1.55525	0.0796777
$382$	0	0
$383$	0.901149	0.0460466	0.0230233	−	0.999735i	$-0.492671\pi$
$383$	0.901149	0.0460466	0.0230233	+	0.999735i	$0.492671\pi$
$384$	0	0
$385$	34.7593	1.77150
$386$	0	0
$387$	−5.69561	−0.289524
$388$	0	0
$389$	0.198850	0.0100821	0.00504104	−	0.999987i	$-0.498395\pi$
$389$	0.198850	0.0100821	0.00504104	+	0.999987i	$0.498395\pi$
$390$	0	0
$391$	2.78220	0.140702
$392$	0	0
$393$	2.16754	0.109338
$394$	0	0
$395$	4.35376	0.219061
$396$	0	0
$397$	22.6657	1.13756	0.568780	−	0.822490i	$-0.307416\pi$
$397$	22.6657	1.13756	0.568780	+	0.822490i	$0.307416\pi$
$398$	0	0
$399$	−20.9121	−1.04691
$400$	0	0
$401$	−3.32964	−0.166274	−0.0831372	−	0.996538i	$-0.526494\pi$
$401$	−3.32964	−0.166274	−0.0831372	+	0.996538i	$0.526494\pi$
$402$	0	0
$403$	0.227533	0.0113342
$404$	0	0
$405$	6.16019	0.306102
$406$	0	0
$407$	−17.8058	−0.882599
$408$	0	0
$409$	−5.47625	−0.270783	−0.135391	−	0.990792i	$-0.543229\pi$
$409$	−5.47625	−0.270783	−0.135391	+	0.990792i	$0.543229\pi$
$410$	0	0
$411$	9.77833	0.482329
$412$	0	0
$413$	−56.4562	−2.77803
$414$	0	0
$415$	−3.51471	−0.172531
$416$	0	0
$417$	−15.7163	−0.769631
$418$	0	0
$419$	10.9483	0.534860	0.267430	−	0.963577i	$-0.413826\pi$
$419$	10.9483	0.534860	0.267430	+	0.963577i	$0.413826\pi$
$420$	0	0
$421$	0.726095	0.0353877	0.0176939	−	0.999843i	$-0.494368\pi$
$421$	0.726095	0.0353877	0.0176939	+	0.999843i	$0.494368\pi$
$422$	0	0
$423$	6.06098	0.294695
$424$	0	0
$425$	−0.136086	−0.00660115
$426$	0	0
$427$	−32.8704	−1.59071
$428$	0	0
$429$	3.14413	0.151800
$430$	0	0
$431$	−14.7541	−0.710680	−0.355340	−	0.934737i	$-0.615635\pi$
$431$	−14.7541	−0.710680	−0.355340	+	0.934737i	$0.615635\pi$
$432$	0	0
$433$	29.8348	1.43377	0.716884	−	0.697193i	$-0.245568\pi$
$433$	29.8348	1.43377	0.716884	+	0.697193i	$0.245568\pi$
$434$	0	0
$435$	1.90534	0.0913541
$436$	0	0
$437$	−19.0293	−0.910294
$438$	0	0
$439$	−37.5016	−1.78985	−0.894927	−	0.446212i	$-0.852773\pi$
$439$	−37.5016	−1.78985	−0.894927	+	0.446212i	$0.852773\pi$
$440$	0	0
$441$	−26.9951	−1.28548
$442$	0	0
$443$	−34.3849	−1.63368	−0.816839	−	0.576866i	$-0.804275\pi$
$443$	−34.3849	−1.63368	−0.816839	+	0.576866i	$0.804275\pi$
$444$	0	0
$445$	−19.1574	−0.908148
$446$	0	0
$447$	5.61538	0.265598
$448$	0	0
$449$	26.6260	1.25656	0.628280	−	0.777988i	$-0.283759\pi$
$449$	26.6260	1.25656	0.628280	+	0.777988i	$0.283759\pi$
$450$	0	0
$451$	−24.9627	−1.17545
$452$	0	0
$453$	0.389276	0.0182898
$454$	0	0
$455$	−9.58271	−0.449244
$456$	0	0
$457$	1.49872	0.0701071	0.0350535	−	0.999385i	$-0.488840\pi$
$457$	1.49872	0.0701071	0.0350535	+	0.999385i	$0.488840\pi$
$458$	0	0
$459$	3.68110	0.171819
$460$	0	0
$461$	−11.9893	−0.558397	−0.279198	−	0.960233i	$-0.590069\pi$
$461$	−11.9893	−0.558397	−0.279198	+	0.960233i	$0.590069\pi$
$462$	0	0
$463$	−9.20541	−0.427812	−0.213906	−	0.976854i	$-0.568619\pi$
$463$	−9.20541	−0.427812	−0.213906	+	0.976854i	$0.568619\pi$
$464$	0	0
$465$	−0.433528	−0.0201044
$466$	0	0
$467$	19.9910	0.925072	0.462536	−	0.886601i	$-0.346940\pi$
$467$	19.9910	0.925072	0.462536	+	0.886601i	$0.346940\pi$
$468$	0	0
$469$	−39.0197	−1.80176
$470$	0	0
$471$	−1.77796	−0.0819238
$472$	0	0
$473$	−9.18753	−0.422443
$474$	0	0
$475$	0.930784	0.0427073
$476$	0	0
$477$	7.06267	0.323377
$478$	0	0
$479$	−35.9438	−1.64231	−0.821157	−	0.570702i	$-0.806671\pi$
$479$	−35.9438	−1.64231	−0.821157	+	0.570702i	$0.806671\pi$
$480$	0	0
$481$	4.90883	0.223823
$482$	0	0
$483$	12.9936	0.591227
$484$	0	0
$485$	−33.7024	−1.53035
$486$	0	0
$487$	−15.9734	−0.723822	−0.361911	−	0.932213i	$-0.617876\pi$
$487$	−15.9734	−0.723822	−0.361911	+	0.932213i	$0.617876\pi$
$488$	0	0
$489$	11.3549	0.513486
$490$	0	0
$491$	9.28873	0.419194	0.209597	−	0.977788i	$-0.432785\pi$
$491$	9.28873	0.419194	0.209597	+	0.977788i	$0.432785\pi$
$492$	0	0
$493$	−0.809117	−0.0364408
$494$	0	0
$495$	17.9292	0.805858
$496$	0	0
$497$	31.0083	1.39091
$498$	0	0
$499$	14.2188	0.636519	0.318259	−	0.948004i	$-0.396902\pi$
$499$	14.2188	0.636519	0.318259	+	0.948004i	$0.396902\pi$
$500$	0	0
$501$	2.62411	0.117236
$502$	0	0
$503$	22.0456	0.982963	0.491482	−	0.870888i	$-0.336455\pi$
$503$	22.0456	0.982963	0.491482	+	0.870888i	$0.336455\pi$
$504$	0	0
$505$	3.83944	0.170853
$506$	0	0
$507$	−0.866798	−0.0384959
$508$	0	0
$509$	21.1535	0.937614	0.468807	−	0.883301i	$-0.344684\pi$
$509$	21.1535	0.937614	0.468807	+	0.883301i	$0.344684\pi$
$510$	0	0
$511$	−68.1055	−3.01281
$512$	0	0
$513$	−25.1775	−1.11161
$514$	0	0
$515$	−8.70943	−0.383783
$516$	0	0
$517$	9.77690	0.429987
$518$	0	0
$519$	−2.49055	−0.109323
$520$	0	0
$521$	38.5905	1.69068	0.845341	−	0.534227i	$-0.179397\pi$
$521$	38.5905	1.69068	0.845341	+	0.534227i	$0.179397\pi$
$522$	0	0
$523$	−24.8645	−1.08725	−0.543625	−	0.839328i	$-0.682949\pi$
$523$	−24.8645	−1.08725	−0.543625	+	0.839328i	$0.682949\pi$
$524$	0	0
$525$	−0.635556	−0.0277380
$526$	0	0
$527$	0.184101	0.00801956
$528$	0	0
$529$	−11.1763	−0.485926
$530$	0	0
$531$	−29.1207	−1.26373
$532$	0	0
$533$	6.88191	0.298089
$534$	0	0
$535$	24.0295	1.03888
$536$	0	0
$537$	−1.15332	−0.0497694
$538$	0	0
$539$	−43.5454	−1.87563
$540$	0	0
$541$	11.6371	0.500317	0.250158	−	0.968205i	$-0.419517\pi$
$541$	11.6371	0.500317	0.250158	+	0.968205i	$0.419517\pi$
$542$	0	0
$543$	−13.4614	−0.577682
$544$	0	0
$545$	−7.06666	−0.302703
$546$	0	0
$547$	−0.690340	−0.0295168	−0.0147584	−	0.999891i	$-0.504698\pi$
$547$	−0.690340	−0.0295168	−0.0147584	+	0.999891i	$0.504698\pi$
$548$	0	0
$549$	−16.9549	−0.723618
$550$	0	0
$551$	5.53409	0.235760
$552$	0	0
$553$	−8.63461	−0.367181
$554$	0	0
$555$	−9.35300	−0.397013
$556$	0	0
$557$	−40.1097	−1.69950	−0.849751	−	0.527184i	$-0.823248\pi$
$557$	−40.1097	−1.69950	−0.849751	+	0.527184i	$0.823248\pi$
$558$	0	0
$559$	2.53289	0.107130
$560$	0	0
$561$	2.54397	0.107406
$562$	0	0
$563$	−32.1905	−1.35667	−0.678333	−	0.734754i	$-0.737297\pi$
$563$	−32.1905	−1.35667	−0.678333	+	0.734754i	$0.737297\pi$
$564$	0	0
$565$	28.7704	1.21038
$566$	0	0
$567$	−12.2172	−0.513075
$568$	0	0
$569$	20.1691	0.845531	0.422766	−	0.906239i	$-0.361059\pi$
$569$	20.1691	0.845531	0.422766	+	0.906239i	$0.361059\pi$
$570$	0	0
$571$	−6.43260	−0.269196	−0.134598	−	0.990900i	$-0.542974\pi$
$571$	−6.43260	−0.269196	−0.134598	+	0.990900i	$0.542974\pi$
$572$	0	0
$573$	−7.69336	−0.321395
$574$	0	0
$575$	−0.578335	−0.0241182
$576$	0	0
$577$	−27.4692	−1.14356	−0.571779	−	0.820408i	$-0.693747\pi$
$577$	−27.4692	−1.14356	−0.571779	+	0.820408i	$0.693747\pi$
$578$	0	0
$579$	−12.7666	−0.530561
$580$	0	0
$581$	6.97057	0.289188
$582$	0	0
$583$	11.3927	0.471838
$584$	0	0
$585$	−4.94287	−0.204362
$586$	0	0
$587$	−32.2797	−1.33233	−0.666163	−	0.745806i	$-0.732065\pi$
$587$	−32.2797	−1.33233	−0.666163	+	0.745806i	$0.732065\pi$
$588$	0	0
$589$	−1.25919	−0.0518839
$590$	0	0
$591$	−0.631223	−0.0259650
$592$	0	0
$593$	−29.9968	−1.23182	−0.615911	−	0.787815i	$-0.711212\pi$
$593$	−29.9968	−1.23182	−0.615911	+	0.787815i	$0.711212\pi$
$594$	0	0
$595$	−7.75353	−0.317864
$596$	0	0
$597$	8.45747	0.346141
$598$	0	0
$599$	38.5665	1.57579	0.787893	−	0.615812i	$-0.211172\pi$
$599$	38.5665	1.57579	0.787893	+	0.615812i	$0.211172\pi$
$600$	0	0
$601$	12.5601	0.512337	0.256169	−	0.966632i	$-0.417540\pi$
$601$	12.5601	0.512337	0.256169	+	0.966632i	$0.417540\pi$
$602$	0	0
$603$	−20.1268	−0.819627
$604$	0	0
$605$	4.74189	0.192785
$606$	0	0
$607$	−36.0082	−1.46153	−0.730764	−	0.682630i	$-0.760836\pi$
$607$	−36.0082	−1.46153	−0.730764	+	0.682630i	$0.760836\pi$
$608$	0	0
$609$	−3.77878	−0.153124
$610$	0	0
$611$	−2.69537	−0.109043
$612$	0	0
$613$	−34.4848	−1.39283	−0.696414	−	0.717640i	$-0.745222\pi$
$613$	−34.4848	−1.39283	−0.696414	+	0.717640i	$0.745222\pi$
$614$	0	0
$615$	−13.1124	−0.528743
$616$	0	0
$617$	1.81984	0.0732639	0.0366319	−	0.999329i	$-0.488337\pi$
$617$	1.81984	0.0732639	0.0366319	+	0.999329i	$0.488337\pi$
$618$	0	0
$619$	0.718328	0.0288720	0.0144360	−	0.999896i	$-0.495405\pi$
$619$	0.718328	0.0288720	0.0144360	+	0.999896i	$0.495405\pi$
$620$	0	0
$621$	15.6438	0.627765
$622$	0	0
$623$	37.9940	1.52220
$624$	0	0
$625$	−24.1308	−0.965230
$626$	0	0
$627$	−17.3999	−0.694884
$628$	0	0
$629$	3.97182	0.158367
$630$	0	0
$631$	18.1962	0.724378	0.362189	−	0.932105i	$-0.382030\pi$
$631$	18.1962	0.724378	0.362189	+	0.932105i	$0.382030\pi$
$632$	0	0
$633$	−4.76658	−0.189454
$634$	0	0
$635$	−3.94399	−0.156513
$636$	0	0
$637$	12.0049	0.475653
$638$	0	0
$639$	15.9944	0.632730
$640$	0	0
$641$	−0.321087	−0.0126822	−0.00634109	−	0.999980i	$-0.502018\pi$
$641$	−0.321087	−0.0126822	−0.00634109	+	0.999980i	$0.502018\pi$
$642$	0	0
$643$	−19.4353	−0.766455	−0.383227	−	0.923654i	$-0.625187\pi$
$643$	−19.4353	−0.766455	−0.383227	+	0.923654i	$0.625187\pi$
$644$	0	0
$645$	−4.82602	−0.190025
$646$	0	0
$647$	30.5532	1.20117	0.600585	−	0.799561i	$-0.294935\pi$
$647$	30.5532	1.20117	0.600585	+	0.799561i	$0.294935\pi$
$648$	0	0
$649$	−46.9743	−1.84390
$650$	0	0
$651$	0.859797	0.0336981
$652$	0	0
$653$	−26.0930	−1.02110	−0.510549	−	0.859848i	$-0.670558\pi$
$653$	−26.0930	−1.02110	−0.510549	+	0.859848i	$0.670558\pi$
$654$	0	0
$655$	−5.49674	−0.214775
$656$	0	0
$657$	−35.1296	−1.37054
$658$	0	0
$659$	−2.22719	−0.0867591	−0.0433796	−	0.999059i	$-0.513812\pi$
$659$	−2.22719	−0.0867591	−0.0433796	+	0.999059i	$0.513812\pi$
$660$	0	0
$661$	35.6483	1.38656	0.693279	−	0.720669i	$-0.256165\pi$
$661$	35.6483	1.38656	0.693279	+	0.720669i	$0.256165\pi$
$662$	0	0
$663$	−0.701341	−0.0272378
$664$	0	0
$665$	53.0315	2.05647
$666$	0	0
$667$	−3.43856	−0.133142
$668$	0	0
$669$	−15.3371	−0.592968
$670$	0	0
$671$	−27.3498	−1.05583
$672$	0	0
$673$	−16.4224	−0.633039	−0.316519	−	0.948586i	$-0.602514\pi$
$673$	−16.4224	−0.633039	−0.316519	+	0.948586i	$0.602514\pi$
$674$	0	0
$675$	−0.765190	−0.0294522
$676$	0	0
$677$	−22.9119	−0.880576	−0.440288	−	0.897857i	$-0.645124\pi$
$677$	−22.9119	−0.880576	−0.440288	+	0.897857i	$0.645124\pi$
$678$	0	0
$679$	66.8405	2.56510
$680$	0	0
$681$	5.01701	0.192252
$682$	0	0
$683$	41.4800	1.58719	0.793594	−	0.608447i	$-0.208207\pi$
$683$	41.4800	1.58719	0.793594	+	0.608447i	$0.208207\pi$
$684$	0	0
$685$	−24.7971	−0.947450
$686$	0	0
$687$	−19.8178	−0.756096
$688$	0	0
$689$	−3.14083	−0.119656
$690$	0	0
$691$	−25.4177	−0.966933	−0.483467	−	0.875363i	$-0.660623\pi$
$691$	−25.4177	−0.966933	−0.483467	+	0.875363i	$0.660623\pi$
$692$	0	0
$693$	−35.5582	−1.35074
$694$	0	0
$695$	39.8554	1.51180
$696$	0	0
$697$	5.56827	0.210913
$698$	0	0
$699$	12.0001	0.453886
$700$	0	0
$701$	−1.16013	−0.0438175	−0.0219087	−	0.999760i	$-0.506974\pi$
$701$	−1.16013	−0.0438175	−0.0219087	+	0.999760i	$0.506974\pi$
$702$	0	0
$703$	−27.1659	−1.02458
$704$	0	0
$705$	5.13561	0.193418
$706$	0	0
$707$	−7.61459	−0.286376
$708$	0	0
$709$	7.32344	0.275038	0.137519	−	0.990499i	$-0.456087\pi$
$709$	7.32344	0.275038	0.137519	+	0.990499i	$0.456087\pi$
$710$	0	0
$711$	−4.45383	−0.167032
$712$	0	0
$713$	0.782386	0.0293006
$714$	0	0
$715$	−7.97328	−0.298184
$716$	0	0
$717$	−7.58362	−0.283215
$718$	0	0
$719$	−14.4802	−0.540019	−0.270010	−	0.962858i	$-0.587027\pi$
$719$	−14.4802	−0.540019	−0.270010	+	0.962858i	$0.587027\pi$
$720$	0	0
$721$	17.2730	0.643281
$722$	0	0
$723$	16.0672	0.597546
$724$	0	0
$725$	0.168191	0.00624646
$726$	0	0
$727$	43.5365	1.61468	0.807340	−	0.590087i	$-0.200906\pi$
$727$	43.5365	1.61468	0.807340	+	0.590087i	$0.200906\pi$
$728$	0	0
$729$	5.52879	0.204770
$730$	0	0
$731$	2.04940	0.0758000
$732$	0	0
$733$	−3.72585	−0.137617	−0.0688086	−	0.997630i	$-0.521920\pi$
$733$	−3.72585	−0.137617	−0.0688086	+	0.997630i	$0.521920\pi$
$734$	0	0
$735$	−22.8735	−0.843703
$736$	0	0
$737$	−32.4663	−1.19591
$738$	0	0
$739$	8.10272	0.298063	0.149032	−	0.988832i	$-0.452384\pi$
$739$	8.10272	0.298063	0.149032	+	0.988832i	$0.452384\pi$
$740$	0	0
$741$	4.79694	0.176220
$742$	0	0
$743$	0.480988	0.0176457	0.00882287	−	0.999961i	$-0.497192\pi$
$743$	0.480988	0.0176457	0.00882287	+	0.999961i	$0.497192\pi$
$744$	0	0
$745$	−14.2402	−0.521720
$746$	0	0
$747$	3.59550	0.131552
$748$	0	0
$749$	−47.6565	−1.74133
$750$	0	0
$751$	48.5082	1.77009	0.885044	−	0.465507i	$-0.154128\pi$
$751$	48.5082	1.77009	0.885044	+	0.465507i	$0.154128\pi$
$752$	0	0
$753$	5.79423	0.211154
$754$	0	0
$755$	−0.987177	−0.0359270
$756$	0	0
$757$	−42.0009	−1.52655	−0.763274	−	0.646075i	$-0.776410\pi$
$757$	−42.0009	−1.52655	−0.763274	+	0.646075i	$0.776410\pi$
$758$	0	0
$759$	10.8113	0.392424
$760$	0	0
$761$	−24.9513	−0.904484	−0.452242	−	0.891895i	$-0.649376\pi$
$761$	−24.9513	−0.904484	−0.452242	+	0.891895i	$0.649376\pi$
$762$	0	0
$763$	14.0150	0.507377
$764$	0	0
$765$	−3.99936	−0.144597
$766$	0	0
$767$	12.9502	0.467606
$768$	0	0
$769$	42.0942	1.51796	0.758978	−	0.651116i	$-0.225699\pi$
$769$	42.0942	1.51796	0.758978	+	0.651116i	$0.225699\pi$
$770$	0	0
$771$	−18.1207	−0.652601
$772$	0	0
$773$	8.24607	0.296590	0.148295	−	0.988943i	$-0.452621\pi$
$773$	8.24607	0.296590	0.148295	+	0.988943i	$0.452621\pi$
$774$	0	0
$775$	−0.0382690	−0.00137466
$776$	0	0
$777$	18.5494	0.665456
$778$	0	0
$779$	−38.0851	−1.36454
$780$	0	0
$781$	25.8005	0.923213
$782$	0	0
$783$	−4.54953	−0.162587
$784$	0	0
$785$	4.50877	0.160925
$786$	0	0
$787$	−40.8478	−1.45607	−0.728033	−	0.685542i	$-0.759565\pi$
$787$	−40.8478	−1.45607	−0.728033	+	0.685542i	$0.759565\pi$
$788$	0	0
$789$	−20.5514	−0.731647
$790$	0	0
$791$	−57.0591	−2.02879
$792$	0	0
$793$	7.54000	0.267753
$794$	0	0
$795$	5.98436	0.212243
$796$	0	0
$797$	38.3461	1.35829	0.679144	−	0.734005i	$-0.262351\pi$
$797$	38.3461	1.35829	0.679144	+	0.734005i	$0.262351\pi$
$798$	0	0
$799$	−2.18087	−0.0771537
$800$	0	0
$801$	19.5977	0.692452
$802$	0	0
$803$	−56.6671	−1.99974
$804$	0	0
$805$	−32.9507	−1.16136
$806$	0	0
$807$	9.30847	0.327674
$808$	0	0
$809$	−1.31360	−0.0461837	−0.0230919	−	0.999733i	$-0.507351\pi$
$809$	−1.31360	−0.0461837	−0.0230919	+	0.999733i	$0.507351\pi$
$810$	0	0
$811$	55.3838	1.94479	0.972395	−	0.233342i	$-0.0749660\pi$
$811$	55.3838	1.94479	0.972395	+	0.233342i	$0.0749660\pi$
$812$	0	0
$813$	−23.8508	−0.836485
$814$	0	0
$815$	−28.7952	−1.00865
$816$	0	0
$817$	−14.0172	−0.490401
$818$	0	0
$819$	9.80296	0.342543
$820$	0	0
$821$	36.8984	1.28776	0.643882	−	0.765125i	$-0.277323\pi$
$821$	36.8984	1.28776	0.643882	+	0.765125i	$0.277323\pi$
$822$	0	0
$823$	−41.9170	−1.46113	−0.730567	−	0.682841i	$-0.760744\pi$
$823$	−41.9170	−1.46113	−0.730567	+	0.682841i	$0.760744\pi$
$824$	0	0
$825$	−0.528814	−0.0184109
$826$	0	0
$827$	4.12791	0.143542	0.0717708	−	0.997421i	$-0.477135\pi$
$827$	4.12791	0.143542	0.0717708	+	0.997421i	$0.477135\pi$
$828$	0	0
$829$	43.6665	1.51660	0.758300	−	0.651905i	$-0.226030\pi$
$829$	43.6665	1.51660	0.758300	+	0.651905i	$0.226030\pi$
$830$	0	0
$831$	15.9639	0.553783
$832$	0	0
$833$	9.71340	0.336549
$834$	0	0
$835$	−6.65454	−0.230290
$836$	0	0
$837$	1.03517	0.0357807
$838$	0	0
$839$	52.3291	1.80660	0.903300	−	0.429008i	$-0.141137\pi$
$839$	52.3291	1.80660	0.903300	+	0.429008i	$0.141137\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−2.66878	−0.0919175
$844$	0	0
$845$	2.19814	0.0756182
$846$	0	0
$847$	−9.40438	−0.323138
$848$	0	0
$849$	4.60201	0.157941
$850$	0	0
$851$	16.8793	0.578616
$852$	0	0
$853$	−16.7742	−0.574339	−0.287169	−	0.957880i	$-0.592714\pi$
$853$	−16.7742	−0.574339	−0.287169	+	0.957880i	$0.592714\pi$
$854$	0	0
$855$	27.3543	0.935496
$856$	0	0
$857$	13.2516	0.452667	0.226334	−	0.974050i	$-0.427326\pi$
$857$	13.2516	0.452667	0.226334	+	0.974050i	$0.427326\pi$
$858$	0	0
$859$	−20.3091	−0.692936	−0.346468	−	0.938062i	$-0.612619\pi$
$859$	−20.3091	−0.692936	−0.346468	+	0.938062i	$0.612619\pi$
$860$	0	0
$861$	26.0052	0.886256
$862$	0	0
$863$	−12.0234	−0.409280	−0.204640	−	0.978837i	$-0.565602\pi$
$863$	−12.0234	−0.409280	−0.204640	+	0.978837i	$0.565602\pi$
$864$	0	0
$865$	6.31584	0.214745
$866$	0	0
$867$	14.1681	0.481174
$868$	0	0
$869$	−7.18442	−0.243715
$870$	0	0
$871$	8.95057	0.303279
$872$	0	0
$873$	34.4771	1.16687
$874$	0	0
$875$	49.5253	1.67426
$876$	0	0
$877$	−34.5918	−1.16808	−0.584041	−	0.811724i	$-0.698529\pi$
$877$	−34.5918	−1.16808	−0.584041	+	0.811724i	$0.698529\pi$
$878$	0	0
$879$	−14.5910	−0.492141
$880$	0	0
$881$	33.5740	1.13114	0.565569	−	0.824701i	$-0.308657\pi$
$881$	33.5740	1.13114	0.565569	+	0.824701i	$0.308657\pi$
$882$	0	0
$883$	−13.7440	−0.462522	−0.231261	−	0.972892i	$-0.574285\pi$
$883$	−13.7440	−0.462522	−0.231261	+	0.972892i	$0.574285\pi$
$884$	0	0
$885$	−24.6746	−0.829429
$886$	0	0
$887$	13.4578	0.451868	0.225934	−	0.974143i	$-0.427457\pi$
$887$	13.4578	0.451868	0.225934	+	0.974143i	$0.427457\pi$
$888$	0	0
$889$	7.82194	0.262339
$890$	0	0
$891$	−10.1653	−0.340551
$892$	0	0
$893$	14.9164	0.499159
$894$	0	0
$895$	2.92473	0.0977631
$896$	0	0
$897$	−2.98054	−0.0995173
$898$	0	0
$899$	−0.227533	−0.00758865
$900$	0	0
$901$	−2.54130	−0.0846630
$902$	0	0
$903$	9.57123	0.318511
$904$	0	0
$905$	34.1370	1.13475
$906$	0	0
$907$	−21.3789	−0.709875	−0.354937	−	0.934890i	$-0.615498\pi$
$907$	−21.3789	−0.709875	−0.354937	+	0.934890i	$0.615498\pi$
$908$	0	0
$909$	−3.92769	−0.130273
$910$	0	0
$911$	46.4415	1.53868	0.769338	−	0.638841i	$-0.220586\pi$
$911$	46.4415	1.53868	0.769338	+	0.638841i	$0.220586\pi$
$912$	0	0
$913$	5.79986	0.191947
$914$	0	0
$915$	−14.3663	−0.474935
$916$	0	0
$917$	10.9014	0.359997
$918$	0	0
$919$	27.1716	0.896307	0.448154	−	0.893957i	$-0.352082\pi$
$919$	27.1716	0.896307	0.448154	+	0.893957i	$0.352082\pi$
$920$	0	0
$921$	−22.5098	−0.741724
$922$	0	0
$923$	−7.11287	−0.234123
$924$	0	0
$925$	−0.825621	−0.0271463
$926$	0	0
$927$	8.90962	0.292630
$928$	0	0
$929$	28.7343	0.942743	0.471372	−	0.881935i	$-0.343759\pi$
$929$	28.7343	0.942743	0.471372	+	0.881935i	$0.343759\pi$
$930$	0	0
$931$	−66.4364	−2.17737
$932$	0	0
$933$	−8.93176	−0.292413
$934$	0	0
$935$	−6.45132	−0.210981
$936$	0	0
$937$	−42.0740	−1.37450	−0.687249	−	0.726422i	$-0.741182\pi$
$937$	−42.0740	−1.37450	−0.687249	+	0.726422i	$0.741182\pi$
$938$	0	0
$939$	−26.2229	−0.855752
$940$	0	0
$941$	−37.8989	−1.23547	−0.617734	−	0.786387i	$-0.711949\pi$
$941$	−37.8989	−1.23547	−0.617734	+	0.786387i	$0.711949\pi$
$942$	0	0
$943$	23.6639	0.770602
$944$	0	0
$945$	−43.5968	−1.41820
$946$	0	0
$947$	29.7705	0.967411	0.483706	−	0.875231i	$-0.339291\pi$
$947$	29.7705	0.967411	0.483706	+	0.875231i	$0.339291\pi$
$948$	0	0
$949$	15.6224	0.507126
$950$	0	0
$951$	15.9862	0.518388
$952$	0	0
$953$	47.3380	1.53343	0.766714	−	0.641989i	$-0.221890\pi$
$953$	47.3380	1.53343	0.766714	+	0.641989i	$0.221890\pi$
$954$	0	0
$955$	19.5098	0.631322
$956$	0	0
$957$	−3.14413	−0.101635
$958$	0	0
$959$	49.1790	1.58807
$960$	0	0
$961$	−30.9482	−0.998330
$962$	0	0
$963$	−24.5818	−0.792137
$964$	0	0
$965$	32.3751	1.04219
$966$	0	0
$967$	−33.6621	−1.08250	−0.541249	−	0.840862i	$-0.682048\pi$
$967$	−33.6621	−1.08250	−0.541249	+	0.840862i	$0.682048\pi$
$968$	0	0
$969$	3.88128	0.124685
$970$	0	0
$971$	51.1600	1.64180	0.820901	−	0.571070i	$-0.193472\pi$
$971$	51.1600	1.64180	0.820901	+	0.571070i	$0.193472\pi$
$972$	0	0
$973$	−79.0434	−2.53402
$974$	0	0
$975$	0.145788	0.00466894
$976$	0	0
$977$	50.3817	1.61185	0.805926	−	0.592016i	$-0.201668\pi$
$977$	50.3817	1.61185	0.805926	+	0.592016i	$0.201668\pi$
$978$	0	0
$979$	31.6129	1.01035
$980$	0	0
$981$	7.22909	0.230807
$982$	0	0
$983$	16.0901	0.513194	0.256597	−	0.966518i	$-0.417399\pi$
$983$	16.0901	0.513194	0.256597	+	0.966518i	$0.417399\pi$
$984$	0	0
$985$	1.60074	0.0510037
$986$	0	0
$987$	−10.1852	−0.324199
$988$	0	0
$989$	8.70950	0.276946
$990$	0	0
$991$	31.2136	0.991533	0.495766	−	0.868456i	$-0.334887\pi$
$991$	31.2136	0.991533	0.495766	+	0.868456i	$0.334887\pi$
$992$	0	0
$993$	4.90028	0.155506
$994$	0	0
$995$	−21.4475	−0.679932
$996$	0	0
$997$	−61.7232	−1.95479	−0.977397	−	0.211414i	$-0.932193\pi$
$997$	−61.7232	−1.95479	−0.977397	+	0.211414i	$0.932193\pi$
$998$	0	0
$999$	22.3329	0.706581

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3016.2.a.g.1.4	✓	10
4.3	odd	2		6032.2.a.ba.1.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.g.1.4	✓	10	1.1	even	1	trivial
6032.2.a.ba.1.7		10	4.3	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:	\( N \)	\(=\)	\( 3016 = 2^{3} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3016.a (trivial)

\(f(q)\)	\(=\)	\(q-0.866798 q^{3} +2.19814 q^{5} -4.35947 q^{7} -2.24866 q^{9} +O(q^{10})\)	\(q-0.866798 q^{3} +2.19814 q^{5} -4.35947 q^{7} -2.24866 q^{9} -3.62729 q^{11} +1.00000 q^{13} -1.90534 q^{15} +0.809117 q^{17} -5.53409 q^{19} +3.77878 q^{21} +3.43856 q^{23} -0.168191 q^{25} +4.54953 q^{27} -1.00000 q^{29} +0.227533 q^{31} +3.14413 q^{33} -9.58271 q^{35} +4.90883 q^{37} -0.866798 q^{39} +6.88191 q^{41} +2.53289 q^{43} -4.94287 q^{45} -2.69537 q^{47} +12.0049 q^{49} -0.701341 q^{51} -3.14083 q^{53} -7.97328 q^{55} +4.79694 q^{57} +12.9502 q^{59} +7.54000 q^{61} +9.80296 q^{63} +2.19814 q^{65} +8.95057 q^{67} -2.98054 q^{69} -7.11287 q^{71} +15.6224 q^{73} +0.145788 q^{75} +15.8130 q^{77} +1.98066 q^{79} +2.80246 q^{81} -1.59895 q^{83} +1.77855 q^{85} +0.866798 q^{87} -8.71529 q^{89} -4.35947 q^{91} -0.197225 q^{93} -12.1647 q^{95} -15.3323 q^{97} +8.15655 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 16 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 + 5 * q^5 - 2 * q^7 + 16 * q^9	\( 10 q + 2 q^{3} + 5 q^{5} - 2 q^{7} + 16 q^{9} + 4 q^{11} + 10 q^{13} + 8 q^{15} + 10 q^{17} - q^{19} + q^{21} + 23 q^{23} + 25 q^{25} + 2 q^{27} - 10 q^{29} - 13 q^{31} + 15 q^{33} - 12 q^{35} - 7 q^{37} + 2 q^{39} + 16 q^{41} - 12 q^{43} + 55 q^{45} + 11 q^{47} - 25 q^{51} + 11 q^{53} + 22 q^{55} - 6 q^{57} - 11 q^{59} + 34 q^{61} + 37 q^{63} + 5 q^{65} - 23 q^{67} + 2 q^{69} - 4 q^{71} + 39 q^{73} + 11 q^{75} + 32 q^{77} + 5 q^{79} + 38 q^{81} + 6 q^{83} + 45 q^{85} - 2 q^{87} - 24 q^{89} - 2 q^{91} + 13 q^{93} + 33 q^{95} + 19 q^{97}+O(q^{100}) \) 10 * q + 2 * q^3 + 5 * q^5 - 2 * q^7 + 16 * q^9 + 4 * q^11 + 10 * q^13 + 8 * q^15 + 10 * q^17 - q^19 + q^21 + 23 * q^23 + 25 * q^25 + 2 * q^27 - 10 * q^29 - 13 * q^31 + 15 * q^33 - 12 * q^35 - 7 * q^37 + 2 * q^39 + 16 * q^41 - 12 * q^43 + 55 * q^45 + 11 * q^47 - 25 * q^51 + 11 * q^53 + 22 * q^55 - 6 * q^57 - 11 * q^59 + 34 * q^61 + 37 * q^63 + 5 * q^65 - 23 * q^67 + 2 * q^69 - 4 * q^71 + 39 * q^73 + 11 * q^75 + 32 * q^77 + 5 * q^79 + 38 * q^81 + 6 * q^83 + 45 * q^85 - 2 * q^87 - 24 * q^89 - 2 * q^91 + 13 * q^93 + 33 * q^95 + 19 * q^97

Introduction

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