LMFDB - Embedded newform 8044.2.a.a.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8044,2,Mod(1,8044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8044 = 2^{2} \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8044.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:	$64.2316633859$
Analytic rank:	$1$
Dimension:	$80$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56684 q^{3} -2.59165 q^{5} +1.04792 q^{7} +3.58865 q^{9} +O(q^{10})$	$q-2.56684 q^{3} -2.59165 q^{5} +1.04792 q^{7} +3.58865 q^{9} -6.08305 q^{11} +2.47623 q^{13} +6.65234 q^{15} -1.15370 q^{17} +6.51626 q^{19} -2.68984 q^{21} -5.54302 q^{23} +1.71665 q^{25} -1.51098 q^{27} -6.08477 q^{29} -2.21997 q^{31} +15.6142 q^{33} -2.71585 q^{35} -0.188694 q^{37} -6.35609 q^{39} +3.17185 q^{41} +3.91566 q^{43} -9.30053 q^{45} -0.0411246 q^{47} -5.90186 q^{49} +2.96137 q^{51} +8.97669 q^{53} +15.7651 q^{55} -16.7262 q^{57} +4.99880 q^{59} +11.9502 q^{61} +3.76063 q^{63} -6.41753 q^{65} -8.06647 q^{67} +14.2280 q^{69} -3.31307 q^{71} +6.36612 q^{73} -4.40636 q^{75} -6.37455 q^{77} -0.753103 q^{79} -6.88753 q^{81} +2.63626 q^{83} +2.98999 q^{85} +15.6186 q^{87} +2.13984 q^{89} +2.59490 q^{91} +5.69830 q^{93} -16.8879 q^{95} +2.13450 q^{97} -21.8299 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * 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
$3$	−2.56684	−1.48196	−0.740982	−	0.671525i	$-0.765640\pi$
$3$	−2.56684	−1.48196	−0.740982	+	0.671525i	$0.765640\pi$
$4$	0	0
$5$	−2.59165	−1.15902	−0.579511	−	0.814965i	$-0.696756\pi$
$5$	−2.59165	−1.15902	−0.579511	+	0.814965i	$0.696756\pi$
$6$	0	0
$7$	1.04792	0.396077	0.198039	−	0.980194i	$-0.436543\pi$
$7$	1.04792	0.396077	0.198039	+	0.980194i	$0.436543\pi$
$8$	0	0
$9$	3.58865	1.19622
$10$	0	0
$11$	−6.08305	−1.83411	−0.917054	−	0.398764i	$-0.869439\pi$
$11$	−6.08305	−1.83411	−0.917054	+	0.398764i	$0.869439\pi$
$12$	0	0
$13$	2.47623	0.686784	0.343392	−	0.939192i	$-0.388424\pi$
$13$	2.47623	0.686784	0.343392	+	0.939192i	$0.388424\pi$
$14$	0	0
$15$	6.65234	1.71763
$16$	0	0
$17$	−1.15370	−0.279814	−0.139907	−	0.990165i	$-0.544680\pi$
$17$	−1.15370	−0.279814	−0.139907	+	0.990165i	$0.544680\pi$
$18$	0	0
$19$	6.51626	1.49493	0.747467	−	0.664299i	$-0.231270\pi$
$19$	6.51626	1.49493	0.747467	+	0.664299i	$0.231270\pi$
$20$	0	0
$21$	−2.68984	−0.586972
$22$	0	0
$23$	−5.54302	−1.15580	−0.577900	−	0.816107i	$-0.696128\pi$
$23$	−5.54302	−1.15580	−0.577900	+	0.816107i	$0.696128\pi$
$24$	0	0
$25$	1.71665	0.343330
$26$	0	0
$27$	−1.51098	−0.290788
$28$	0	0
$29$	−6.08477	−1.12991	−0.564956	−	0.825121i	$-0.691107\pi$
$29$	−6.08477	−1.12991	−0.564956	+	0.825121i	$0.691107\pi$
$30$	0	0
$31$	−2.21997	−0.398718	−0.199359	−	0.979926i	$-0.563886\pi$
$31$	−2.21997	−0.398718	−0.199359	+	0.979926i	$0.563886\pi$
$32$	0	0
$33$	15.6142	2.71808
$34$	0	0
$35$	−2.71585	−0.459062
$36$	0	0
$37$	−0.188694	−0.0310211	−0.0155105	−	0.999880i	$-0.504937\pi$
$37$	−0.188694	−0.0310211	−0.0155105	+	0.999880i	$0.504937\pi$
$38$	0	0
$39$	−6.35609	−1.01779
$40$	0	0
$41$	3.17185	0.495359	0.247680	−	0.968842i	$-0.420332\pi$
$41$	3.17185	0.495359	0.247680	+	0.968842i	$0.420332\pi$
$42$	0	0
$43$	3.91566	0.597133	0.298566	−	0.954389i	$-0.403492\pi$
$43$	3.91566	0.597133	0.298566	+	0.954389i	$0.403492\pi$
$44$	0	0
$45$	−9.30053	−1.38644
$46$	0	0
$47$	−0.0411246	−0.00599863	−0.00299932	−	0.999996i	$-0.500955\pi$
$47$	−0.0411246	−0.00599863	−0.00299932	+	0.999996i	$0.500955\pi$
$48$	0	0
$49$	−5.90186	−0.843123
$50$	0	0
$51$	2.96137	0.414674
$52$	0	0
$53$	8.97669	1.23304	0.616522	−	0.787338i	$-0.288541\pi$
$53$	8.97669	1.23304	0.616522	+	0.787338i	$0.288541\pi$
$54$	0	0
$55$	15.7651	2.12577
$56$	0	0
$57$	−16.7262	−2.21544
$58$	0	0
$59$	4.99880	0.650788	0.325394	−	0.945579i	$-0.394503\pi$
$59$	4.99880	0.650788	0.325394	+	0.945579i	$0.394503\pi$
$60$	0	0
$61$	11.9502	1.53006	0.765032	−	0.643992i	$-0.222723\pi$
$61$	11.9502	1.53006	0.765032	+	0.643992i	$0.222723\pi$
$62$	0	0
$63$	3.76063	0.473794
$64$	0	0
$65$	−6.41753	−0.795997
$66$	0	0
$67$	−8.06647	−0.985477	−0.492738	−	0.870178i	$-0.664004\pi$
$67$	−8.06647	−0.985477	−0.492738	+	0.870178i	$0.664004\pi$
$68$	0	0
$69$	14.2280	1.71285
$70$	0	0
$71$	−3.31307	−0.393189	−0.196595	−	0.980485i	$-0.562988\pi$
$71$	−3.31307	−0.393189	−0.196595	+	0.980485i	$0.562988\pi$
$72$	0	0
$73$	6.36612	0.745098	0.372549	−	0.928013i	$-0.378484\pi$
$73$	6.36612	0.745098	0.372549	+	0.928013i	$0.378484\pi$
$74$	0	0
$75$	−4.40636	−0.508802
$76$	0	0
$77$	−6.37455	−0.726448
$78$	0	0
$79$	−0.753103	−0.0847307	−0.0423654	−	0.999102i	$-0.513489\pi$
$79$	−0.753103	−0.0847307	−0.0423654	+	0.999102i	$0.513489\pi$
$80$	0	0
$81$	−6.88753	−0.765281
$82$	0	0
$83$	2.63626	0.289367	0.144684	−	0.989478i	$-0.453784\pi$
$83$	2.63626	0.289367	0.144684	+	0.989478i	$0.453784\pi$
$84$	0	0
$85$	2.98999	0.324310
$86$	0	0
$87$	15.6186	1.67449
$88$	0	0
$89$	2.13984	0.226822	0.113411	−	0.993548i	$-0.463822\pi$
$89$	2.13984	0.226822	0.113411	+	0.993548i	$0.463822\pi$
$90$	0	0
$91$	2.59490	0.272019
$92$	0	0
$93$	5.69830	0.590886
$94$	0	0
$95$	−16.8879	−1.73266
$96$	0	0
$97$	2.13450	0.216726	0.108363	−	0.994111i	$-0.465439\pi$
$97$	2.13450	0.216726	0.108363	+	0.994111i	$0.465439\pi$
$98$	0	0
$99$	−21.8299	−2.19399
$100$	0	0
$101$	−1.33043	−0.132383	−0.0661915	−	0.997807i	$-0.521085\pi$
$101$	−1.33043	−0.132383	−0.0661915	+	0.997807i	$0.521085\pi$
$102$	0	0
$103$	12.3262	1.21453	0.607266	−	0.794499i	$-0.292266\pi$
$103$	12.3262	1.21453	0.607266	+	0.794499i	$0.292266\pi$
$104$	0	0
$105$	6.97113	0.680313
$106$	0	0
$107$	−0.371409	−0.0359055	−0.0179527	−	0.999839i	$-0.505715\pi$
$107$	−0.371409	−0.0359055	−0.0179527	+	0.999839i	$0.505715\pi$
$108$	0	0
$109$	6.73877	0.645458	0.322729	−	0.946491i	$-0.395400\pi$
$109$	6.73877	0.645458	0.322729	+	0.946491i	$0.395400\pi$
$110$	0	0
$111$	0.484346	0.0459721
$112$	0	0
$113$	−6.98060	−0.656680	−0.328340	−	0.944560i	$-0.606489\pi$
$113$	−6.98060	−0.656680	−0.328340	+	0.944560i	$0.606489\pi$
$114$	0	0
$115$	14.3656	1.33960
$116$	0	0
$117$	8.88634	0.821543
$118$	0	0
$119$	−1.20899	−0.110828
$120$	0	0
$121$	26.0035	2.36395
$122$	0	0
$123$	−8.14161	−0.734105
$124$	0	0
$125$	8.50930	0.761095
$126$	0	0
$127$	−22.0871	−1.95991	−0.979957	−	0.199207i	$-0.936163\pi$
$127$	−22.0871	−1.95991	−0.979957	+	0.199207i	$0.936163\pi$
$128$	0	0
$129$	−10.0509	−0.884929
$130$	0	0
$131$	18.1115	1.58241	0.791205	−	0.611551i	$-0.209454\pi$
$131$	18.1115	1.58241	0.791205	+	0.611551i	$0.209454\pi$
$132$	0	0
$133$	6.82853	0.592109
$134$	0	0
$135$	3.91592	0.337029
$136$	0	0
$137$	9.17796	0.784126	0.392063	−	0.919938i	$-0.371762\pi$
$137$	9.17796	0.784126	0.392063	+	0.919938i	$0.371762\pi$
$138$	0	0
$139$	4.33970	0.368089	0.184044	−	0.982918i	$-0.441081\pi$
$139$	4.33970	0.368089	0.184044	+	0.982918i	$0.441081\pi$
$140$	0	0
$141$	0.105560	0.00888976
$142$	0	0
$143$	−15.0630	−1.25963
$144$	0	0
$145$	15.7696	1.30959
$146$	0	0
$147$	15.1491	1.24948
$148$	0	0
$149$	10.6339	0.871164	0.435582	−	0.900149i	$-0.356543\pi$
$149$	10.6339	0.871164	0.435582	+	0.900149i	$0.356543\pi$
$150$	0	0
$151$	9.30227	0.757008	0.378504	−	0.925600i	$-0.376439\pi$
$151$	9.30227	0.757008	0.378504	+	0.925600i	$0.376439\pi$
$152$	0	0
$153$	−4.14024	−0.334718
$154$	0	0
$155$	5.75338	0.462123
$156$	0	0
$157$	14.6770	1.17135	0.585677	−	0.810545i	$-0.300829\pi$
$157$	14.6770	1.17135	0.585677	+	0.810545i	$0.300829\pi$
$158$	0	0
$159$	−23.0417	−1.82733
$160$	0	0
$161$	−5.80865	−0.457786
$162$	0	0
$163$	−7.96091	−0.623546	−0.311773	−	0.950157i	$-0.600923\pi$
$163$	−7.96091	−0.623546	−0.311773	+	0.950157i	$0.600923\pi$
$164$	0	0
$165$	−40.4665	−3.15031
$166$	0	0
$167$	−24.8410	−1.92226	−0.961128	−	0.276102i	$-0.910957\pi$
$167$	−24.8410	−1.92226	−0.961128	+	0.276102i	$0.910957\pi$
$168$	0	0
$169$	−6.86827	−0.528328
$170$	0	0
$171$	23.3846	1.78827
$172$	0	0
$173$	8.47046	0.643997	0.321999	−	0.946740i	$-0.395645\pi$
$173$	8.47046	0.643997	0.321999	+	0.946740i	$0.395645\pi$
$174$	0	0
$175$	1.79891	0.135985
$176$	0	0
$177$	−12.8311	−0.964445
$178$	0	0
$179$	21.0257	1.57153	0.785767	−	0.618523i	$-0.212269\pi$
$179$	21.0257	1.57153	0.785767	+	0.618523i	$0.212269\pi$
$180$	0	0
$181$	9.16263	0.681053	0.340526	−	0.940235i	$-0.389395\pi$
$181$	9.16263	0.681053	0.340526	+	0.940235i	$0.389395\pi$
$182$	0	0
$183$	−30.6742	−2.26750
$184$	0	0
$185$	0.489028	0.0359541
$186$	0	0
$187$	7.01802	0.513209
$188$	0	0
$189$	−1.58339	−0.115174
$190$	0	0
$191$	2.69138	0.194742	0.0973708	−	0.995248i	$-0.468957\pi$
$191$	2.69138	0.194742	0.0973708	+	0.995248i	$0.468957\pi$
$192$	0	0
$193$	4.58970	0.330374	0.165187	−	0.986262i	$-0.447177\pi$
$193$	4.58970	0.330374	0.165187	+	0.986262i	$0.447177\pi$
$194$	0	0
$195$	16.4728	1.17964
$196$	0	0
$197$	15.2783	1.08853	0.544267	−	0.838912i	$-0.316808\pi$
$197$	15.2783	1.08853	0.544267	+	0.838912i	$0.316808\pi$
$198$	0	0
$199$	−9.03663	−0.640589	−0.320295	−	0.947318i	$-0.603782\pi$
$199$	−9.03663	−0.640589	−0.320295	+	0.947318i	$0.603782\pi$
$200$	0	0
$201$	20.7053	1.46044
$202$	0	0
$203$	−6.37636	−0.447532
$204$	0	0
$205$	−8.22032	−0.574132
$206$	0	0
$207$	−19.8920	−1.38259
$208$	0	0
$209$	−39.6387	−2.74187
$210$	0	0
$211$	−6.70946	−0.461899	−0.230949	−	0.972966i	$-0.574183\pi$
$211$	−6.70946	−0.461899	−0.230949	+	0.972966i	$0.574183\pi$
$212$	0	0
$213$	8.50412	0.582693
$214$	0	0
$215$	−10.1480	−0.692089
$216$	0	0
$217$	−2.32635	−0.157923
$218$	0	0
$219$	−16.3408	−1.10421
$220$	0	0
$221$	−2.85684	−0.192172
$222$	0	0
$223$	−1.36185	−0.0911961	−0.0455981	−	0.998960i	$-0.514519\pi$
$223$	−1.36185	−0.0911961	−0.0455981	+	0.998960i	$0.514519\pi$
$224$	0	0
$225$	6.16046	0.410697
$226$	0	0
$227$	−17.0286	−1.13023	−0.565114	−	0.825013i	$-0.691168\pi$
$227$	−17.0286	−1.13023	−0.565114	+	0.825013i	$0.691168\pi$
$228$	0	0
$229$	−5.16801	−0.341512	−0.170756	−	0.985313i	$-0.554621\pi$
$229$	−5.16801	−0.341512	−0.170756	+	0.985313i	$0.554621\pi$
$230$	0	0
$231$	16.3624	1.07657
$232$	0	0
$233$	−21.2465	−1.39191	−0.695953	−	0.718087i	$-0.745018\pi$
$233$	−21.2465	−1.39191	−0.695953	+	0.718087i	$0.745018\pi$
$234$	0	0
$235$	0.106580	0.00695254
$236$	0	0
$237$	1.93309	0.125568
$238$	0	0
$239$	16.4982	1.06718	0.533591	−	0.845743i	$-0.320842\pi$
$239$	16.4982	1.06718	0.533591	+	0.845743i	$0.320842\pi$
$240$	0	0
$241$	−17.1027	−1.10168	−0.550842	−	0.834610i	$-0.685693\pi$
$241$	−17.1027	−1.10168	−0.550842	+	0.834610i	$0.685693\pi$
$242$	0	0
$243$	22.2121	1.42491
$244$	0	0
$245$	15.2956	0.977197
$246$	0	0
$247$	16.1358	1.02670
$248$	0	0
$249$	−6.76685	−0.428832
$250$	0	0
$251$	−6.44264	−0.406656	−0.203328	−	0.979111i	$-0.565176\pi$
$251$	−6.44264	−0.406656	−0.203328	+	0.979111i	$0.565176\pi$
$252$	0	0
$253$	33.7185	2.11986
$254$	0	0
$255$	−7.67482	−0.480616
$256$	0	0
$257$	17.5436	1.09434	0.547170	−	0.837021i	$-0.315705\pi$
$257$	17.5436	1.09434	0.547170	+	0.837021i	$0.315705\pi$
$258$	0	0
$259$	−0.197736	−0.0122867
$260$	0	0
$261$	−21.8361	−1.35162
$262$	0	0
$263$	−25.1934	−1.55349	−0.776746	−	0.629814i	$-0.783131\pi$
$263$	−25.1934	−1.55349	−0.776746	+	0.629814i	$0.783131\pi$
$264$	0	0
$265$	−23.2644	−1.42912
$266$	0	0
$267$	−5.49261	−0.336142
$268$	0	0
$269$	−30.8747	−1.88247	−0.941233	−	0.337758i	$-0.890331\pi$
$269$	−30.8747	−1.88247	−0.941233	+	0.337758i	$0.890331\pi$
$270$	0	0
$271$	7.62881	0.463417	0.231709	−	0.972785i	$-0.425568\pi$
$271$	7.62881	0.463417	0.231709	+	0.972785i	$0.425568\pi$
$272$	0	0
$273$	−6.66068	−0.403123
$274$	0	0
$275$	−10.4425	−0.629704
$276$	0	0
$277$	29.9303	1.79834	0.899169	−	0.437601i	$-0.144172\pi$
$277$	29.9303	1.79834	0.899169	+	0.437601i	$0.144172\pi$
$278$	0	0
$279$	−7.96670	−0.476954
$280$	0	0
$281$	−10.4284	−0.622109	−0.311054	−	0.950392i	$-0.600682\pi$
$281$	−10.4284	−0.622109	−0.311054	+	0.950392i	$0.600682\pi$
$282$	0	0
$283$	−3.99418	−0.237429	−0.118715	−	0.992928i	$-0.537877\pi$
$283$	−3.99418	−0.237429	−0.118715	+	0.992928i	$0.537877\pi$
$284$	0	0
$285$	43.3484	2.56774
$286$	0	0
$287$	3.32385	0.196200
$288$	0	0
$289$	−15.6690	−0.921704
$290$	0	0
$291$	−5.47892	−0.321180
$292$	0	0
$293$	−33.1893	−1.93894	−0.969471	−	0.245206i	$-0.921144\pi$
$293$	−33.1893	−1.93894	−0.969471	+	0.245206i	$0.921144\pi$
$294$	0	0
$295$	−12.9551	−0.754277
$296$	0	0
$297$	9.19134	0.533336
$298$	0	0
$299$	−13.7258	−0.793785
$300$	0	0
$301$	4.10330	0.236511
$302$	0	0
$303$	3.41501	0.196187
$304$	0	0
$305$	−30.9707	−1.77338
$306$	0	0
$307$	−4.18409	−0.238798	−0.119399	−	0.992846i	$-0.538097\pi$
$307$	−4.18409	−0.238798	−0.119399	+	0.992846i	$0.538097\pi$
$308$	0	0
$309$	−31.6392	−1.79989
$310$	0	0
$311$	6.44186	0.365284	0.182642	−	0.983179i	$-0.441535\pi$
$311$	6.44186	0.365284	0.182642	+	0.983179i	$0.441535\pi$
$312$	0	0
$313$	12.1645	0.687577	0.343789	−	0.939047i	$-0.388290\pi$
$313$	12.1645	0.687577	0.343789	+	0.939047i	$0.388290\pi$
$314$	0	0
$315$	−9.74623	−0.549138
$316$	0	0
$317$	−1.17666	−0.0660876	−0.0330438	−	0.999454i	$-0.510520\pi$
$317$	−1.17666	−0.0660876	−0.0330438	+	0.999454i	$0.510520\pi$
$318$	0	0
$319$	37.0139	2.07238
$320$	0	0
$321$	0.953347	0.0532106
$322$	0	0
$323$	−7.51783	−0.418303
$324$	0	0
$325$	4.25082	0.235793
$326$	0	0
$327$	−17.2973	−0.956545
$328$	0	0
$329$	−0.0430953	−0.00237592
$330$	0	0
$331$	−24.8755	−1.36728	−0.683641	−	0.729818i	$-0.739605\pi$
$331$	−24.8755	−1.36728	−0.683641	+	0.729818i	$0.739605\pi$
$332$	0	0
$333$	−0.677157	−0.0371080
$334$	0	0
$335$	20.9055	1.14219
$336$	0	0
$337$	−11.9970	−0.653521	−0.326760	−	0.945107i	$-0.605957\pi$
$337$	−11.9970	−0.653521	−0.326760	+	0.945107i	$0.605957\pi$
$338$	0	0
$339$	17.9181	0.973176
$340$	0	0
$341$	13.5042	0.731292
$342$	0	0
$343$	−13.5201	−0.730019
$344$	0	0
$345$	−36.8741	−1.98523
$346$	0	0
$347$	−22.1311	−1.18806	−0.594029	−	0.804444i	$-0.702464\pi$
$347$	−22.1311	−1.18806	−0.594029	+	0.804444i	$0.702464\pi$
$348$	0	0
$349$	−28.0816	−1.50318	−0.751588	−	0.659633i	$-0.770712\pi$
$349$	−28.0816	−1.50318	−0.751588	+	0.659633i	$0.770712\pi$
$350$	0	0
$351$	−3.74153	−0.199708
$352$	0	0
$353$	1.25845	0.0669806	0.0334903	−	0.999439i	$-0.489338\pi$
$353$	1.25845	0.0669806	0.0334903	+	0.999439i	$0.489338\pi$
$354$	0	0
$355$	8.58633	0.455715
$356$	0	0
$357$	3.10328	0.164243
$358$	0	0
$359$	7.70593	0.406703	0.203352	−	0.979106i	$-0.434817\pi$
$359$	7.70593	0.406703	0.203352	+	0.979106i	$0.434817\pi$
$360$	0	0
$361$	23.4617	1.23483
$362$	0	0
$363$	−66.7466	−3.50329
$364$	0	0
$365$	−16.4988	−0.863584
$366$	0	0
$367$	−3.12525	−0.163137	−0.0815684	−	0.996668i	$-0.525993\pi$
$367$	−3.12525	−0.163137	−0.0815684	+	0.996668i	$0.525993\pi$
$368$	0	0
$369$	11.3827	0.592557
$370$	0	0
$371$	9.40687	0.488380
$372$	0	0
$373$	−13.0498	−0.675694	−0.337847	−	0.941201i	$-0.609699\pi$
$373$	−13.0498	−0.675694	−0.337847	+	0.941201i	$0.609699\pi$
$374$	0	0
$375$	−21.8420	−1.12791
$376$	0	0
$377$	−15.0673	−0.776005
$378$	0	0
$379$	21.9645	1.12824	0.564121	−	0.825692i	$-0.309215\pi$
$379$	21.9645	1.12824	0.564121	+	0.825692i	$0.309215\pi$
$380$	0	0
$381$	56.6940	2.90452
$382$	0	0
$383$	−23.5859	−1.20518	−0.602591	−	0.798050i	$-0.705865\pi$
$383$	−23.5859	−1.20518	−0.602591	+	0.798050i	$0.705865\pi$
$384$	0	0
$385$	16.5206	0.841968
$386$	0	0
$387$	14.0519	0.714301
$388$	0	0
$389$	28.0665	1.42303	0.711513	−	0.702673i	$-0.248010\pi$
$389$	28.0665	1.42303	0.711513	+	0.702673i	$0.248010\pi$
$390$	0	0
$391$	6.39500	0.323409
$392$	0	0
$393$	−46.4893	−2.34508
$394$	0	0
$395$	1.95178	0.0982047
$396$	0	0
$397$	20.1378	1.01069	0.505343	−	0.862919i	$-0.331366\pi$
$397$	20.1378	1.01069	0.505343	+	0.862919i	$0.331366\pi$
$398$	0	0
$399$	−17.5277	−0.877484
$400$	0	0
$401$	−6.45200	−0.322197	−0.161099	−	0.986938i	$-0.551504\pi$
$401$	−6.45200	−0.322197	−0.161099	+	0.986938i	$0.551504\pi$
$402$	0	0
$403$	−5.49716	−0.273833
$404$	0	0
$405$	17.8501	0.886977
$406$	0	0
$407$	1.14783	0.0568960
$408$	0	0
$409$	18.0094	0.890506	0.445253	−	0.895405i	$-0.353114\pi$
$409$	18.0094	0.890506	0.445253	+	0.895405i	$0.353114\pi$
$410$	0	0
$411$	−23.5583	−1.16205
$412$	0	0
$413$	5.23835	0.257762
$414$	0	0
$415$	−6.83226	−0.335383
$416$	0	0
$417$	−11.1393	−0.545494
$418$	0	0
$419$	−26.1258	−1.27633	−0.638164	−	0.769901i	$-0.720306\pi$
$419$	−26.1258	−1.27633	−0.638164	+	0.769901i	$0.720306\pi$
$420$	0	0
$421$	−16.8807	−0.822714	−0.411357	−	0.911474i	$-0.634945\pi$
$421$	−16.8807	−0.822714	−0.411357	+	0.911474i	$0.634945\pi$
$422$	0	0
$423$	−0.147582	−0.00717567
$424$	0	0
$425$	−1.98050	−0.0960684
$426$	0	0
$427$	12.5229	0.606023
$428$	0	0
$429$	38.6644	1.86673
$430$	0	0
$431$	−21.4389	−1.03268	−0.516339	−	0.856384i	$-0.672705\pi$
$431$	−21.4389	−1.03268	−0.516339	+	0.856384i	$0.672705\pi$
$432$	0	0
$433$	34.7531	1.67013	0.835064	−	0.550153i	$-0.185431\pi$
$433$	34.7531	1.67013	0.835064	+	0.550153i	$0.185431\pi$
$434$	0	0
$435$	−40.4779	−1.94077
$436$	0	0
$437$	−36.1198	−1.72784
$438$	0	0
$439$	−21.1609	−1.00995	−0.504977	−	0.863133i	$-0.668499\pi$
$439$	−21.1609	−1.00995	−0.504977	+	0.863133i	$0.668499\pi$
$440$	0	0
$441$	−21.1797	−1.00856
$442$	0	0
$443$	8.94182	0.424839	0.212419	−	0.977179i	$-0.431866\pi$
$443$	8.94182	0.424839	0.212419	+	0.977179i	$0.431866\pi$
$444$	0	0
$445$	−5.54570	−0.262892
$446$	0	0
$447$	−27.2955	−1.29103
$448$	0	0
$449$	−23.9301	−1.12933	−0.564666	−	0.825320i	$-0.690995\pi$
$449$	−23.9301	−1.12933	−0.564666	+	0.825320i	$0.690995\pi$
$450$	0	0
$451$	−19.2945	−0.908542
$452$	0	0
$453$	−23.8774	−1.12186
$454$	0	0
$455$	−6.72507	−0.315276
$456$	0	0
$457$	14.2821	0.668088	0.334044	−	0.942557i	$-0.391587\pi$
$457$	14.2821	0.668088	0.334044	+	0.942557i	$0.391587\pi$
$458$	0	0
$459$	1.74322	0.0813664
$460$	0	0
$461$	16.4367	0.765535	0.382768	−	0.923845i	$-0.374971\pi$
$461$	16.4367	0.765535	0.382768	+	0.923845i	$0.374971\pi$
$462$	0	0
$463$	28.0657	1.30432	0.652162	−	0.758080i	$-0.273862\pi$
$463$	28.0657	1.30432	0.652162	+	0.758080i	$0.273862\pi$
$464$	0	0
$465$	−14.7680	−0.684850
$466$	0	0
$467$	−5.05417	−0.233879	−0.116939	−	0.993139i	$-0.537308\pi$
$467$	−5.05417	−0.233879	−0.116939	+	0.993139i	$0.537308\pi$
$468$	0	0
$469$	−8.45303	−0.390325
$470$	0	0
$471$	−37.6735	−1.73590
$472$	0	0
$473$	−23.8191	−1.09521
$474$	0	0
$475$	11.1861	0.513255
$476$	0	0
$477$	32.2142	1.47499
$478$	0	0
$479$	−4.97448	−0.227290	−0.113645	−	0.993521i	$-0.536253\pi$
$479$	−4.97448	−0.227290	−0.113645	+	0.993521i	$0.536253\pi$
$480$	0	0
$481$	−0.467250	−0.0213048
$482$	0	0
$483$	14.9099	0.678422
$484$	0	0
$485$	−5.53189	−0.251190
$486$	0	0
$487$	23.4620	1.06317	0.531583	−	0.847006i	$-0.321597\pi$
$487$	23.4620	1.06317	0.531583	+	0.847006i	$0.321597\pi$
$488$	0	0
$489$	20.4344	0.924073
$490$	0	0
$491$	−6.84329	−0.308834	−0.154417	−	0.988006i	$-0.549350\pi$
$491$	−6.84329	−0.308834	−0.154417	+	0.988006i	$0.549350\pi$
$492$	0	0
$493$	7.02001	0.316165
$494$	0	0
$495$	56.5756	2.54288
$496$	0	0
$497$	−3.47184	−0.155733
$498$	0	0
$499$	−21.0754	−0.943464	−0.471732	−	0.881742i	$-0.656371\pi$
$499$	−21.0754	−0.943464	−0.471732	+	0.881742i	$0.656371\pi$
$500$	0	0
$501$	63.7629	2.84872
$502$	0	0
$503$	30.3823	1.35468	0.677340	−	0.735670i	$-0.263132\pi$
$503$	30.3823	1.35468	0.677340	+	0.735670i	$0.263132\pi$
$504$	0	0
$505$	3.44802	0.153435
$506$	0	0
$507$	17.6297	0.782964
$508$	0	0
$509$	−12.5330	−0.555516	−0.277758	−	0.960651i	$-0.589591\pi$
$509$	−12.5330	−0.555516	−0.277758	+	0.960651i	$0.589591\pi$
$510$	0	0
$511$	6.67119	0.295116
$512$	0	0
$513$	−9.84593	−0.434708
$514$	0	0
$515$	−31.9451	−1.40767
$516$	0	0
$517$	0.250163	0.0110021
$518$	0	0
$519$	−21.7423	−0.954380
$520$	0	0
$521$	29.7312	1.30255	0.651274	−	0.758843i	$-0.274235\pi$
$521$	29.7312	1.30255	0.651274	+	0.758843i	$0.274235\pi$
$522$	0	0
$523$	−25.2785	−1.10535	−0.552675	−	0.833397i	$-0.686393\pi$
$523$	−25.2785	−1.10535	−0.552675	+	0.833397i	$0.686393\pi$
$524$	0	0
$525$	−4.61752	−0.201525
$526$	0	0
$527$	2.56118	0.111567
$528$	0	0
$529$	7.72512	0.335875
$530$	0	0
$531$	17.9390	0.778484
$532$	0	0
$533$	7.85423	0.340205
$534$	0	0
$535$	0.962562	0.0416152
$536$	0	0
$537$	−53.9695	−2.32896
$538$	0	0
$539$	35.9013	1.54638
$540$	0	0
$541$	0.766543	0.0329562	0.0164781	−	0.999864i	$-0.494755\pi$
$541$	0.766543	0.0329562	0.0164781	+	0.999864i	$0.494755\pi$
$542$	0	0
$543$	−23.5190	−1.00930
$544$	0	0
$545$	−17.4645	−0.748099
$546$	0	0
$547$	−12.9480	−0.553617	−0.276809	−	0.960925i	$-0.589277\pi$
$547$	−12.9480	−0.553617	−0.276809	+	0.960925i	$0.589277\pi$
$548$	0	0
$549$	42.8851	1.83029
$550$	0	0
$551$	−39.6499	−1.68914
$552$	0	0
$553$	−0.789193	−0.0335599
$554$	0	0
$555$	−1.25526	−0.0532826
$556$	0	0
$557$	37.0498	1.56985	0.784925	−	0.619591i	$-0.212701\pi$
$557$	37.0498	1.56985	0.784925	+	0.619591i	$0.212701\pi$
$558$	0	0
$559$	9.69609	0.410101
$560$	0	0
$561$	−18.0141	−0.760557
$562$	0	0
$563$	8.07098	0.340151	0.170076	−	0.985431i	$-0.445599\pi$
$563$	8.07098	0.340151	0.170076	+	0.985431i	$0.445599\pi$
$564$	0	0
$565$	18.0913	0.761106
$566$	0	0
$567$	−7.21759	−0.303110
$568$	0	0
$569$	−10.8448	−0.454637	−0.227318	−	0.973821i	$-0.572996\pi$
$569$	−10.8448	−0.454637	−0.227318	+	0.973821i	$0.572996\pi$
$570$	0	0
$571$	−30.9186	−1.29390	−0.646952	−	0.762531i	$-0.723957\pi$
$571$	−30.9186	−1.29390	−0.646952	+	0.762531i	$0.723957\pi$
$572$	0	0
$573$	−6.90834	−0.288600
$574$	0	0
$575$	−9.51543	−0.396821
$576$	0	0
$577$	−12.2403	−0.509569	−0.254785	−	0.966998i	$-0.582005\pi$
$577$	−12.2403	−0.509569	−0.254785	+	0.966998i	$0.582005\pi$
$578$	0	0
$579$	−11.7810	−0.489602
$580$	0	0
$581$	2.76259	0.114612
$582$	0	0
$583$	−54.6056	−2.26153
$584$	0	0
$585$	−23.0303	−0.952185
$586$	0	0
$587$	8.67867	0.358207	0.179103	−	0.983830i	$-0.442680\pi$
$587$	8.67867	0.358207	0.179103	+	0.983830i	$0.442680\pi$
$588$	0	0
$589$	−14.4659	−0.596057
$590$	0	0
$591$	−39.2169	−1.61317
$592$	0	0
$593$	48.1007	1.97526	0.987630	−	0.156801i	$-0.0501181\pi$
$593$	48.1007	1.97526	0.987630	+	0.156801i	$0.0501181\pi$
$594$	0	0
$595$	3.13328	0.128452
$596$	0	0
$597$	23.1955	0.949331
$598$	0	0
$599$	−12.5452	−0.512585	−0.256292	−	0.966599i	$-0.582501\pi$
$599$	−12.5452	−0.512585	−0.256292	+	0.966599i	$0.582501\pi$
$600$	0	0
$601$	−6.93373	−0.282833	−0.141416	−	0.989950i	$-0.545166\pi$
$601$	−6.93373	−0.282833	−0.141416	+	0.989950i	$0.545166\pi$
$602$	0	0
$603$	−28.9478	−1.17884
$604$	0	0
$605$	−67.3918	−2.73987
$606$	0	0
$607$	−22.0774	−0.896093	−0.448047	−	0.894010i	$-0.647880\pi$
$607$	−22.0774	−0.896093	−0.448047	+	0.894010i	$0.647880\pi$
$608$	0	0
$609$	16.3671	0.663227
$610$	0	0
$611$	−0.101834	−0.00411976
$612$	0	0
$613$	18.8583	0.761679	0.380839	−	0.924641i	$-0.375635\pi$
$613$	18.8583	0.761679	0.380839	+	0.924641i	$0.375635\pi$
$614$	0	0
$615$	21.1002	0.850843
$616$	0	0
$617$	−2.31924	−0.0933690	−0.0466845	−	0.998910i	$-0.514866\pi$
$617$	−2.31924	−0.0933690	−0.0466845	+	0.998910i	$0.514866\pi$
$618$	0	0
$619$	−23.8134	−0.957141	−0.478571	−	0.878049i	$-0.658845\pi$
$619$	−23.8134	−0.957141	−0.478571	+	0.878049i	$0.658845\pi$
$620$	0	0
$621$	8.37538	0.336093
$622$	0	0
$623$	2.24238	0.0898390
$624$	0	0
$625$	−30.6364	−1.22545
$626$	0	0
$627$	101.746	4.06335
$628$	0	0
$629$	0.217696	0.00868012
$630$	0	0
$631$	−29.0541	−1.15662	−0.578312	−	0.815816i	$-0.696288\pi$
$631$	−29.0541	−1.15662	−0.578312	+	0.815816i	$0.696288\pi$
$632$	0	0
$633$	17.2221	0.684517
$634$	0	0
$635$	57.2421	2.27158
$636$	0	0
$637$	−14.6144	−0.579043
$638$	0	0
$639$	−11.8895	−0.470340
$640$	0	0
$641$	−19.5556	−0.772398	−0.386199	−	0.922415i	$-0.626212\pi$
$641$	−19.5556	−0.772398	−0.386199	+	0.922415i	$0.626212\pi$
$642$	0	0
$643$	−24.9849	−0.985307	−0.492653	−	0.870226i	$-0.663973\pi$
$643$	−24.9849	−0.985307	−0.492653	+	0.870226i	$0.663973\pi$
$644$	0	0
$645$	26.0483	1.02565
$646$	0	0
$647$	−13.6728	−0.537535	−0.268767	−	0.963205i	$-0.586616\pi$
$647$	−13.6728	−0.537535	−0.268767	+	0.963205i	$0.586616\pi$
$648$	0	0
$649$	−30.4079	−1.19362
$650$	0	0
$651$	5.97137	0.234036
$652$	0	0
$653$	−33.5536	−1.31305	−0.656527	−	0.754303i	$-0.727975\pi$
$653$	−33.5536	−1.31305	−0.656527	+	0.754303i	$0.727975\pi$
$654$	0	0
$655$	−46.9387	−1.83405
$656$	0	0
$657$	22.8458	0.891299
$658$	0	0
$659$	15.8129	0.615984	0.307992	−	0.951389i	$-0.400343\pi$
$659$	15.8129	0.615984	0.307992	+	0.951389i	$0.400343\pi$
$660$	0	0
$661$	21.6062	0.840386	0.420193	−	0.907435i	$-0.361962\pi$
$661$	21.6062	0.840386	0.420193	+	0.907435i	$0.361962\pi$
$662$	0	0
$663$	7.33303	0.284791
$664$	0	0
$665$	−17.6972	−0.686266
$666$	0	0
$667$	33.7280	1.30595
$668$	0	0
$669$	3.49564	0.135149
$670$	0	0
$671$	−72.6935	−2.80630
$672$	0	0
$673$	−17.5584	−0.676827	−0.338414	−	0.940997i	$-0.609890\pi$
$673$	−17.5584	−0.676827	−0.338414	+	0.940997i	$0.609890\pi$
$674$	0	0
$675$	−2.59382	−0.0998361
$676$	0	0
$677$	−16.3480	−0.628304	−0.314152	−	0.949373i	$-0.601720\pi$
$677$	−16.3480	−0.628304	−0.314152	+	0.949373i	$0.601720\pi$
$678$	0	0
$679$	2.23679	0.0858402
$680$	0	0
$681$	43.7097	1.67496
$682$	0	0
$683$	−19.8974	−0.761354	−0.380677	−	0.924708i	$-0.624309\pi$
$683$	−19.8974	−0.761354	−0.380677	+	0.924708i	$0.624309\pi$
$684$	0	0
$685$	−23.7860	−0.908818
$686$	0	0
$687$	13.2654	0.506108
$688$	0	0
$689$	22.2284	0.846834
$690$	0	0
$691$	−44.7109	−1.70088	−0.850441	−	0.526071i	$-0.823665\pi$
$691$	−44.7109	−1.70088	−0.850441	+	0.526071i	$0.823665\pi$
$692$	0	0
$693$	−22.8761	−0.868990
$694$	0	0
$695$	−11.2470	−0.426622
$696$	0	0
$697$	−3.65937	−0.138608
$698$	0	0
$699$	54.5364	2.06276
$700$	0	0
$701$	−45.1474	−1.70519	−0.852596	−	0.522570i	$-0.824973\pi$
$701$	−45.1474	−1.70519	−0.852596	+	0.522570i	$0.824973\pi$
$702$	0	0
$703$	−1.22958	−0.0463744
$704$	0	0
$705$	−0.273575	−0.0103034
$706$	0	0
$707$	−1.39419	−0.0524339
$708$	0	0
$709$	10.6702	0.400727	0.200363	−	0.979722i	$-0.435788\pi$
$709$	10.6702	0.400727	0.200363	+	0.979722i	$0.435788\pi$
$710$	0	0
$711$	−2.70263	−0.101356
$712$	0	0
$713$	12.3053	0.460839
$714$	0	0
$715$	39.0381	1.45994
$716$	0	0
$717$	−42.3483	−1.58153
$718$	0	0
$719$	−23.0372	−0.859141	−0.429571	−	0.903033i	$-0.641335\pi$
$719$	−23.0372	−0.859141	−0.429571	+	0.903033i	$0.641335\pi$
$720$	0	0
$721$	12.9168	0.481048
$722$	0	0
$723$	43.8999	1.63266
$724$	0	0
$725$	−10.4454	−0.387933
$726$	0	0
$727$	−48.4797	−1.79801	−0.899007	−	0.437935i	$-0.855710\pi$
$727$	−48.4797	−1.79801	−0.899007	+	0.437935i	$0.855710\pi$
$728$	0	0
$729$	−36.3522	−1.34638
$730$	0	0
$731$	−4.51751	−0.167086
$732$	0	0
$733$	−38.5239	−1.42291	−0.711457	−	0.702730i	$-0.751964\pi$
$733$	−38.5239	−1.42291	−0.711457	+	0.702730i	$0.751964\pi$
$734$	0	0
$735$	−39.2612	−1.44817
$736$	0	0
$737$	49.0687	1.80747
$738$	0	0
$739$	7.24443	0.266491	0.133245	−	0.991083i	$-0.457460\pi$
$739$	7.24443	0.266491	0.133245	+	0.991083i	$0.457460\pi$
$740$	0	0
$741$	−41.4179	−1.52153
$742$	0	0
$743$	−32.3173	−1.18561	−0.592803	−	0.805348i	$-0.701979\pi$
$743$	−32.3173	−1.18561	−0.592803	+	0.805348i	$0.701979\pi$
$744$	0	0
$745$	−27.5594	−1.00970
$746$	0	0
$747$	9.46062	0.346146
$748$	0	0
$749$	−0.389207	−0.0142213
$750$	0	0
$751$	7.34562	0.268045	0.134023	−	0.990978i	$-0.457210\pi$
$751$	7.34562	0.268045	0.134023	+	0.990978i	$0.457210\pi$
$752$	0	0
$753$	16.5372	0.602649
$754$	0	0
$755$	−24.1082	−0.877389
$756$	0	0
$757$	−28.8438	−1.04834	−0.524172	−	0.851612i	$-0.675625\pi$
$757$	−28.8438	−1.04834	−0.524172	+	0.851612i	$0.675625\pi$
$758$	0	0
$759$	−86.5498	−3.14156
$760$	0	0
$761$	−8.57884	−0.310983	−0.155491	−	0.987837i	$-0.549696\pi$
$761$	−8.57884	−0.310983	−0.155491	+	0.987837i	$0.549696\pi$
$762$	0	0
$763$	7.06171	0.255651
$764$	0	0
$765$	10.7300	0.387946
$766$	0	0
$767$	12.3782	0.446951
$768$	0	0
$769$	−44.7081	−1.61221	−0.806107	−	0.591769i	$-0.798430\pi$
$769$	−44.7081	−1.61221	−0.806107	+	0.591769i	$0.798430\pi$
$770$	0	0
$771$	−45.0316	−1.62177
$772$	0	0
$773$	−25.5720	−0.919762	−0.459881	−	0.887981i	$-0.652108\pi$
$773$	−25.5720	−0.919762	−0.459881	+	0.887981i	$0.652108\pi$
$774$	0	0
$775$	−3.81091	−0.136892
$776$	0	0
$777$	0.507557	0.0182085
$778$	0	0
$779$	20.6686	0.740529
$780$	0	0
$781$	20.1536	0.721152
$782$	0	0
$783$	9.19394	0.328565
$784$	0	0
$785$	−38.0377	−1.35762
$786$	0	0
$787$	−39.5156	−1.40858	−0.704289	−	0.709913i	$-0.748734\pi$
$787$	−39.5156	−1.40858	−0.704289	+	0.709913i	$0.748734\pi$
$788$	0	0
$789$	64.6674	2.30222
$790$	0	0
$791$	−7.31512	−0.260096
$792$	0	0
$793$	29.5914	1.05082
$794$	0	0
$795$	59.7160	2.11791
$796$	0	0
$797$	6.51953	0.230934	0.115467	−	0.993311i	$-0.463164\pi$
$797$	6.51953	0.230934	0.115467	+	0.993311i	$0.463164\pi$
$798$	0	0
$799$	0.0474455	0.00167850
$800$	0	0
$801$	7.67913	0.271329
$802$	0	0
$803$	−38.7254	−1.36659
$804$	0	0
$805$	15.0540	0.530584
$806$	0	0
$807$	79.2504	2.78975
$808$	0	0
$809$	26.8754	0.944888	0.472444	−	0.881361i	$-0.343372\pi$
$809$	26.8754	0.944888	0.472444	+	0.881361i	$0.343372\pi$
$810$	0	0
$811$	51.0145	1.79136	0.895681	−	0.444698i	$-0.146689\pi$
$811$	51.0145	1.79136	0.895681	+	0.444698i	$0.146689\pi$
$812$	0	0
$813$	−19.5819	−0.686768
$814$	0	0
$815$	20.6319	0.722703
$816$	0	0
$817$	25.5155	0.892673
$818$	0	0
$819$	9.31219	0.325394
$820$	0	0
$821$	−10.7847	−0.376389	−0.188195	−	0.982132i	$-0.560264\pi$
$821$	−10.7847	−0.376389	−0.188195	+	0.982132i	$0.560264\pi$
$822$	0	0
$823$	18.3172	0.638496	0.319248	−	0.947671i	$-0.396570\pi$
$823$	18.3172	0.638496	0.319248	+	0.947671i	$0.396570\pi$
$824$	0	0
$825$	26.8041	0.933198
$826$	0	0
$827$	21.5365	0.748897	0.374449	−	0.927248i	$-0.377832\pi$
$827$	21.5365	0.748897	0.374449	+	0.927248i	$0.377832\pi$
$828$	0	0
$829$	−8.10609	−0.281536	−0.140768	−	0.990043i	$-0.544957\pi$
$829$	−8.10609	−0.281536	−0.140768	+	0.990043i	$0.544957\pi$
$830$	0	0
$831$	−76.8263	−2.66507
$832$	0	0
$833$	6.80899	0.235917
$834$	0	0
$835$	64.3793	2.22794
$836$	0	0
$837$	3.35432	0.115942
$838$	0	0
$839$	−17.7823	−0.613912	−0.306956	−	0.951724i	$-0.599310\pi$
$839$	−17.7823	−0.613912	−0.306956	+	0.951724i	$0.599310\pi$
$840$	0	0
$841$	8.02437	0.276702
$842$	0	0
$843$	26.7681	0.921943
$844$	0	0
$845$	17.8001	0.612344
$846$	0	0
$847$	27.2496	0.936306
$848$	0	0
$849$	10.2524	0.351861
$850$	0	0
$851$	1.04593	0.0358542
$852$	0	0
$853$	13.0609	0.447197	0.223599	−	0.974681i	$-0.428219\pi$
$853$	13.0609	0.447197	0.223599	+	0.974681i	$0.428219\pi$
$854$	0	0
$855$	−60.6047	−2.07264
$856$	0	0
$857$	51.8144	1.76995	0.884974	−	0.465641i	$-0.154176\pi$
$857$	51.8144	1.76995	0.884974	+	0.465641i	$0.154176\pi$
$858$	0	0
$859$	−2.11690	−0.0722275	−0.0361138	−	0.999348i	$-0.511498\pi$
$859$	−2.11690	−0.0722275	−0.0361138	+	0.999348i	$0.511498\pi$
$860$	0	0
$861$	−8.53177	−0.290762
$862$	0	0
$863$	29.7543	1.01285	0.506425	−	0.862284i	$-0.330967\pi$
$863$	29.7543	1.01285	0.506425	+	0.862284i	$0.330967\pi$
$864$	0	0
$865$	−21.9525	−0.746406
$866$	0	0
$867$	40.2197	1.36593
$868$	0	0
$869$	4.58116	0.155405
$870$	0	0
$871$	−19.9745	−0.676809
$872$	0	0
$873$	7.65999	0.259252
$874$	0	0
$875$	8.91707	0.301452
$876$	0	0
$877$	11.5746	0.390848	0.195424	−	0.980719i	$-0.437392\pi$
$877$	11.5746	0.390848	0.195424	+	0.980719i	$0.437392\pi$
$878$	0	0
$879$	85.1916	2.87344
$880$	0	0
$881$	−7.68887	−0.259045	−0.129522	−	0.991577i	$-0.541344\pi$
$881$	−7.68887	−0.259045	−0.129522	+	0.991577i	$0.541344\pi$
$882$	0	0
$883$	10.2571	0.345180	0.172590	−	0.984994i	$-0.444787\pi$
$883$	10.2571	0.345180	0.172590	+	0.984994i	$0.444787\pi$
$884$	0	0
$885$	33.2537	1.11781
$886$	0	0
$887$	22.3736	0.751232	0.375616	−	0.926775i	$-0.377431\pi$
$887$	22.3736	0.751232	0.375616	+	0.926775i	$0.377431\pi$
$888$	0	0
$889$	−23.1456	−0.776277
$890$	0	0
$891$	41.8971	1.40361
$892$	0	0
$893$	−0.267978	−0.00896755
$894$	0	0
$895$	−54.4912	−1.82144
$896$	0	0
$897$	35.2320	1.17636
$898$	0	0
$899$	13.5080	0.450517
$900$	0	0
$901$	−10.3564	−0.345023
$902$	0	0
$903$	−10.5325	−0.350500
$904$	0	0
$905$	−23.7463	−0.789354
$906$	0	0
$907$	39.7966	1.32142	0.660712	−	0.750640i	$-0.270254\pi$
$907$	39.7966	1.32142	0.660712	+	0.750640i	$0.270254\pi$
$908$	0	0
$909$	−4.77446	−0.158359
$910$	0	0
$911$	−7.85896	−0.260379	−0.130189	−	0.991489i	$-0.541559\pi$
$911$	−7.85896	−0.260379	−0.130189	+	0.991489i	$0.541559\pi$
$912$	0	0
$913$	−16.0365	−0.530730
$914$	0	0
$915$	79.4967	2.62808
$916$	0	0
$917$	18.9794	0.626756
$918$	0	0
$919$	27.4189	0.904467	0.452234	−	0.891900i	$-0.350627\pi$
$919$	27.4189	0.904467	0.452234	+	0.891900i	$0.350627\pi$
$920$	0	0
$921$	10.7399	0.353891
$922$	0	0
$923$	−8.20394	−0.270036
$924$	0	0
$925$	−0.323921	−0.0106505
$926$	0	0
$927$	44.2343	1.45284
$928$	0	0
$929$	−0.803737	−0.0263697	−0.0131849	−	0.999913i	$-0.504197\pi$
$929$	−0.803737	−0.0263697	−0.0131849	+	0.999913i	$0.504197\pi$
$930$	0	0
$931$	−38.4581	−1.26041
$932$	0	0
$933$	−16.5352	−0.541338
$934$	0	0
$935$	−18.1883	−0.594820
$936$	0	0
$937$	−9.07124	−0.296344	−0.148172	−	0.988962i	$-0.547339\pi$
$937$	−9.07124	−0.296344	−0.148172	+	0.988962i	$0.547339\pi$
$938$	0	0
$939$	−31.2243	−1.01897
$940$	0	0
$941$	−23.1292	−0.753989	−0.376995	−	0.926215i	$-0.623043\pi$
$941$	−23.1292	−0.753989	−0.376995	+	0.926215i	$0.623043\pi$
$942$	0	0
$943$	−17.5816	−0.572536
$944$	0	0
$945$	4.10358	0.133489
$946$	0	0
$947$	42.7573	1.38943	0.694713	−	0.719287i	$-0.255531\pi$
$947$	42.7573	1.38943	0.694713	+	0.719287i	$0.255531\pi$
$948$	0	0
$949$	15.7640	0.511721
$950$	0	0
$951$	3.02029	0.0979395
$952$	0	0
$953$	−40.6948	−1.31823	−0.659117	−	0.752041i	$-0.729070\pi$
$953$	−40.6948	−1.31823	−0.659117	+	0.752041i	$0.729070\pi$
$954$	0	0
$955$	−6.97512	−0.225710
$956$	0	0
$957$	−95.0087	−3.07119
$958$	0	0
$959$	9.61778	0.310574
$960$	0	0
$961$	−26.0717	−0.841024
$962$	0	0
$963$	−1.33286	−0.0429508
$964$	0	0
$965$	−11.8949	−0.382910
$966$	0	0
$967$	23.4969	0.755610	0.377805	−	0.925885i	$-0.376679\pi$
$967$	23.4969	0.755610	0.377805	+	0.925885i	$0.376679\pi$
$968$	0	0
$969$	19.2970	0.619910
$970$	0	0
$971$	−12.0315	−0.386109	−0.193054	−	0.981188i	$-0.561839\pi$
$971$	−12.0315	−0.386109	−0.193054	+	0.981188i	$0.561839\pi$
$972$	0	0
$973$	4.54766	0.145791
$974$	0	0
$975$	−10.9112	−0.349437
$976$	0	0
$977$	53.7593	1.71991	0.859957	−	0.510367i	$-0.170490\pi$
$977$	53.7593	1.71991	0.859957	+	0.510367i	$0.170490\pi$
$978$	0	0
$979$	−13.0167	−0.416016
$980$	0	0
$981$	24.1831	0.772108
$982$	0	0
$983$	27.4837	0.876595	0.438298	−	0.898830i	$-0.355582\pi$
$983$	27.4837	0.876595	0.438298	+	0.898830i	$0.355582\pi$
$984$	0	0
$985$	−39.5960	−1.26163
$986$	0	0
$987$	0.110619	0.00352103
$988$	0	0
$989$	−21.7046	−0.690166
$990$	0	0
$991$	28.5961	0.908384	0.454192	−	0.890904i	$-0.349928\pi$
$991$	28.5961	0.908384	0.454192	+	0.890904i	$0.349928\pi$
$992$	0	0
$993$	63.8514	2.02626
$994$	0	0
$995$	23.4198	0.742457
$996$	0	0
$997$	12.7654	0.404286	0.202143	−	0.979356i	$-0.435209\pi$
$997$	12.7654	0.404286	0.202143	+	0.979356i	$0.435209\pi$
$998$	0	0
$999$	0.285112	0.00902055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.14	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.14	✓	80	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8044 = 2^{2} \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56684 q^{3} -2.59165 q^{5} +1.04792 q^{7} +3.58865 q^{9} +O(q^{10})\)	\(q-2.56684 q^{3} -2.59165 q^{5} +1.04792 q^{7} +3.58865 q^{9} -6.08305 q^{11} +2.47623 q^{13} +6.65234 q^{15} -1.15370 q^{17} +6.51626 q^{19} -2.68984 q^{21} -5.54302 q^{23} +1.71665 q^{25} -1.51098 q^{27} -6.08477 q^{29} -2.21997 q^{31} +15.6142 q^{33} -2.71585 q^{35} -0.188694 q^{37} -6.35609 q^{39} +3.17185 q^{41} +3.91566 q^{43} -9.30053 q^{45} -0.0411246 q^{47} -5.90186 q^{49} +2.96137 q^{51} +8.97669 q^{53} +15.7651 q^{55} -16.7262 q^{57} +4.99880 q^{59} +11.9502 q^{61} +3.76063 q^{63} -6.41753 q^{65} -8.06647 q^{67} +14.2280 q^{69} -3.31307 q^{71} +6.36612 q^{73} -4.40636 q^{75} -6.37455 q^{77} -0.753103 q^{79} -6.88753 q^{81} +2.63626 q^{83} +2.98999 q^{85} +15.6186 q^{87} +2.13984 q^{89} +2.59490 q^{91} +5.69830 q^{93} -16.8879 q^{95} +2.13450 q^{97} -21.8299 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more