LMFDB - Embedded newform 8044.2.a.b.1.10

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:	$0$
Dimension:	$87$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-2.55851 q^{3} +1.25574 q^{5} -4.90748 q^{7} +3.54596 q^{9} +O(q^{10})$	$q-2.55851 q^{3} +1.25574 q^{5} -4.90748 q^{7} +3.54596 q^{9} +6.23904 q^{11} -0.189546 q^{13} -3.21282 q^{15} +4.00924 q^{17} +8.30940 q^{19} +12.5558 q^{21} +7.24427 q^{23} -3.42311 q^{25} -1.39686 q^{27} +3.02171 q^{29} +2.30660 q^{31} -15.9626 q^{33} -6.16253 q^{35} -7.44766 q^{37} +0.484955 q^{39} +4.65752 q^{41} +8.02512 q^{43} +4.45281 q^{45} -0.900752 q^{47} +17.0834 q^{49} -10.2577 q^{51} +5.19469 q^{53} +7.83461 q^{55} -21.2597 q^{57} +6.81475 q^{59} -5.51611 q^{61} -17.4018 q^{63} -0.238021 q^{65} +9.80843 q^{67} -18.5345 q^{69} -6.46456 q^{71} -5.86667 q^{73} +8.75807 q^{75} -30.6180 q^{77} -6.77113 q^{79} -7.06403 q^{81} -11.9104 q^{83} +5.03457 q^{85} -7.73108 q^{87} +15.6543 q^{89} +0.930194 q^{91} -5.90145 q^{93} +10.4345 q^{95} -10.7682 q^{97} +22.1234 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * 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
$3$	−2.55851	−1.47716	−0.738578	−	0.674168i	$-0.764502\pi$
$3$	−2.55851	−1.47716	−0.738578	+	0.674168i	$0.764502\pi$
$4$	0	0
$5$	1.25574	0.561584	0.280792	−	0.959769i	$-0.409403\pi$
$5$	1.25574	0.561584	0.280792	+	0.959769i	$0.409403\pi$
$6$	0	0
$7$	−4.90748	−1.85485	−0.927427	−	0.374003i	$-0.877985\pi$
$7$	−4.90748	−1.85485	−0.927427	+	0.374003i	$0.877985\pi$
$8$	0	0
$9$	3.54596	1.18199
$10$	0	0
$11$	6.23904	1.88114	0.940570	−	0.339600i	$-0.110292\pi$
$11$	6.23904	1.88114	0.940570	+	0.339600i	$0.110292\pi$
$12$	0	0
$13$	−0.189546	−0.0525706	−0.0262853	−	0.999654i	$-0.508368\pi$
$13$	−0.189546	−0.0525706	−0.0262853	+	0.999654i	$0.508368\pi$
$14$	0	0
$15$	−3.21282	−0.829548
$16$	0	0
$17$	4.00924	0.972384	0.486192	−	0.873852i	$-0.338386\pi$
$17$	4.00924	0.972384	0.486192	+	0.873852i	$0.338386\pi$
$18$	0	0
$19$	8.30940	1.90631	0.953154	−	0.302487i	$-0.0978169\pi$
$19$	8.30940	1.90631	0.953154	+	0.302487i	$0.0978169\pi$
$20$	0	0
$21$	12.5558	2.73991
$22$	0	0
$23$	7.24427	1.51054	0.755268	−	0.655417i	$-0.227507\pi$
$23$	7.24427	1.51054	0.755268	+	0.655417i	$0.227507\pi$
$24$	0	0
$25$	−3.42311	−0.684623
$26$	0	0
$27$	−1.39686	−0.268825
$28$	0	0
$29$	3.02171	0.561118	0.280559	−	0.959837i	$-0.409480\pi$
$29$	3.02171	0.561118	0.280559	+	0.959837i	$0.409480\pi$
$30$	0	0
$31$	2.30660	0.414277	0.207138	−	0.978312i	$-0.433585\pi$
$31$	2.30660	0.414277	0.207138	+	0.978312i	$0.433585\pi$
$32$	0	0
$33$	−15.9626	−2.77874
$34$	0	0
$35$	−6.16253	−1.04166
$36$	0	0
$37$	−7.44766	−1.22439	−0.612194	−	0.790708i	$-0.709713\pi$
$37$	−7.44766	−1.22439	−0.612194	+	0.790708i	$0.709713\pi$
$38$	0	0
$39$	0.484955	0.0776549
$40$	0	0
$41$	4.65752	0.727383	0.363692	−	0.931519i	$-0.381516\pi$
$41$	4.65752	0.727383	0.363692	+	0.931519i	$0.381516\pi$
$42$	0	0
$43$	8.02512	1.22382	0.611910	−	0.790927i	$-0.290401\pi$
$43$	8.02512	1.22382	0.611910	+	0.790927i	$0.290401\pi$
$44$	0	0
$45$	4.45281	0.663786
$46$	0	0
$47$	−0.900752	−0.131388	−0.0656941	−	0.997840i	$-0.520926\pi$
$47$	−0.900752	−0.131388	−0.0656941	+	0.997840i	$0.520926\pi$
$48$	0	0
$49$	17.0834	2.44049
$50$	0	0
$51$	−10.2577	−1.43636
$52$	0	0
$53$	5.19469	0.713545	0.356773	−	0.934191i	$-0.383877\pi$
$53$	5.19469	0.713545	0.356773	+	0.934191i	$0.383877\pi$
$54$	0	0
$55$	7.83461	1.05642
$56$	0	0
$57$	−21.2597	−2.81591
$58$	0	0
$59$	6.81475	0.887205	0.443602	−	0.896224i	$-0.353700\pi$
$59$	6.81475	0.887205	0.443602	+	0.896224i	$0.353700\pi$
$60$	0	0
$61$	−5.51611	−0.706265	−0.353133	−	0.935573i	$-0.614884\pi$
$61$	−5.51611	−0.706265	−0.353133	+	0.935573i	$0.614884\pi$
$62$	0	0
$63$	−17.4018	−2.19242
$64$	0	0
$65$	−0.238021	−0.0295228
$66$	0	0
$67$	9.80843	1.19829	0.599145	−	0.800640i	$-0.295507\pi$
$67$	9.80843	1.19829	0.599145	+	0.800640i	$0.295507\pi$
$68$	0	0
$69$	−18.5345	−2.23130
$70$	0	0
$71$	−6.46456	−0.767202	−0.383601	−	0.923499i	$-0.625316\pi$
$71$	−6.46456	−0.767202	−0.383601	+	0.923499i	$0.625316\pi$
$72$	0	0
$73$	−5.86667	−0.686642	−0.343321	−	0.939218i	$-0.611552\pi$
$73$	−5.86667	−0.686642	−0.343321	+	0.939218i	$0.611552\pi$
$74$	0	0
$75$	8.75807	1.01129
$76$	0	0
$77$	−30.6180	−3.48924
$78$	0	0
$79$	−6.77113	−0.761811	−0.380906	−	0.924614i	$-0.624388\pi$
$79$	−6.77113	−0.761811	−0.380906	+	0.924614i	$0.624388\pi$
$80$	0	0
$81$	−7.06403	−0.784892
$82$	0	0
$83$	−11.9104	−1.30733	−0.653666	−	0.756783i	$-0.726770\pi$
$83$	−11.9104	−1.30733	−0.653666	+	0.756783i	$0.726770\pi$
$84$	0	0
$85$	5.03457	0.546075
$86$	0	0
$87$	−7.73108	−0.828859
$88$	0	0
$89$	15.6543	1.65935	0.829676	−	0.558246i	$-0.188525\pi$
$89$	15.6543	1.65935	0.829676	+	0.558246i	$0.188525\pi$
$90$	0	0
$91$	0.930194	0.0975108
$92$	0	0
$93$	−5.90145	−0.611951
$94$	0	0
$95$	10.4345	1.07055
$96$	0	0
$97$	−10.7682	−1.09335	−0.546673	−	0.837346i	$-0.684106\pi$
$97$	−10.7682	−1.09335	−0.546673	+	0.837346i	$0.684106\pi$
$98$	0	0
$99$	22.1234	2.22349
$100$	0	0
$101$	−8.86746	−0.882346	−0.441173	−	0.897422i	$-0.645437\pi$
$101$	−8.86746	−0.882346	−0.441173	+	0.897422i	$0.645437\pi$
$102$	0	0
$103$	−16.7631	−1.65172	−0.825859	−	0.563877i	$-0.809309\pi$
$103$	−16.7631	−1.65172	−0.825859	+	0.563877i	$0.809309\pi$
$104$	0	0
$105$	15.7669	1.53869
$106$	0	0
$107$	15.0563	1.45555	0.727774	−	0.685818i	$-0.240555\pi$
$107$	15.0563	1.45555	0.727774	+	0.685818i	$0.240555\pi$
$108$	0	0
$109$	6.90386	0.661270	0.330635	−	0.943759i	$-0.392737\pi$
$109$	6.90386	0.661270	0.330635	+	0.943759i	$0.392737\pi$
$110$	0	0
$111$	19.0549	1.80861
$112$	0	0
$113$	−14.4844	−1.36258	−0.681291	−	0.732013i	$-0.738581\pi$
$113$	−14.4844	−1.36258	−0.681291	+	0.732013i	$0.738581\pi$
$114$	0	0
$115$	9.09693	0.848293
$116$	0	0
$117$	−0.672123	−0.0621378
$118$	0	0
$119$	−19.6753	−1.80363
$120$	0	0
$121$	27.9256	2.53869
$122$	0	0
$123$	−11.9163	−1.07446
$124$	0	0
$125$	−10.5773	−0.946058
$126$	0	0
$127$	13.4082	1.18979	0.594894	−	0.803804i	$-0.297194\pi$
$127$	13.4082	1.18979	0.594894	+	0.803804i	$0.297194\pi$
$128$	0	0
$129$	−20.5323	−1.80777
$130$	0	0
$131$	13.1055	1.14503	0.572516	−	0.819894i	$-0.305968\pi$
$131$	13.1055	1.14503	0.572516	+	0.819894i	$0.305968\pi$
$132$	0	0
$133$	−40.7783	−3.53592
$134$	0	0
$135$	−1.75409	−0.150968
$136$	0	0
$137$	−14.2077	−1.21385	−0.606924	−	0.794760i	$-0.707597\pi$
$137$	−14.2077	−1.21385	−0.606924	+	0.794760i	$0.707597\pi$
$138$	0	0
$139$	8.95970	0.759952	0.379976	−	0.924996i	$-0.375932\pi$
$139$	8.95970	0.759952	0.379976	+	0.924996i	$0.375932\pi$
$140$	0	0
$141$	2.30458	0.194081
$142$	0	0
$143$	−1.18258	−0.0988926
$144$	0	0
$145$	3.79449	0.315115
$146$	0	0
$147$	−43.7080	−3.60498
$148$	0	0
$149$	14.7323	1.20692	0.603459	−	0.797394i	$-0.293789\pi$
$149$	14.7323	1.20692	0.603459	+	0.797394i	$0.293789\pi$
$150$	0	0
$151$	−13.5467	−1.10241	−0.551206	−	0.834369i	$-0.685832\pi$
$151$	−13.5467	−1.10241	−0.551206	+	0.834369i	$0.685832\pi$
$152$	0	0
$153$	14.2166	1.14935
$154$	0	0
$155$	2.89649	0.232651
$156$	0	0
$157$	20.4314	1.63060	0.815302	−	0.579036i	$-0.196571\pi$
$157$	20.4314	1.63060	0.815302	+	0.579036i	$0.196571\pi$
$158$	0	0
$159$	−13.2907	−1.05402
$160$	0	0
$161$	−35.5512	−2.80182
$162$	0	0
$163$	−1.37337	−0.107571	−0.0537854	−	0.998553i	$-0.517129\pi$
$163$	−1.37337	−0.107571	−0.0537854	+	0.998553i	$0.517129\pi$
$164$	0	0
$165$	−20.0449	−1.56050
$166$	0	0
$167$	−3.02802	−0.234315	−0.117158	−	0.993113i	$-0.537378\pi$
$167$	−3.02802	−0.234315	−0.117158	+	0.993113i	$0.537378\pi$
$168$	0	0
$169$	−12.9641	−0.997236
$170$	0	0
$171$	29.4648	2.25323
$172$	0	0
$173$	3.15913	0.240185	0.120092	−	0.992763i	$-0.461681\pi$
$173$	3.15913	0.240185	0.120092	+	0.992763i	$0.461681\pi$
$174$	0	0
$175$	16.7989	1.26988
$176$	0	0
$177$	−17.4356	−1.31054
$178$	0	0
$179$	−10.0941	−0.754472	−0.377236	−	0.926117i	$-0.623125\pi$
$179$	−10.0941	−0.754472	−0.377236	+	0.926117i	$0.623125\pi$
$180$	0	0
$181$	25.0835	1.86444	0.932220	−	0.361891i	$-0.117869\pi$
$181$	25.0835	1.86444	0.932220	+	0.361891i	$0.117869\pi$
$182$	0	0
$183$	14.1130	1.04326
$184$	0	0
$185$	−9.35233	−0.687597
$186$	0	0
$187$	25.0138	1.82919
$188$	0	0
$189$	6.85505	0.498631
$190$	0	0
$191$	2.96490	0.214533	0.107266	−	0.994230i	$-0.465790\pi$
$191$	2.96490	0.214533	0.107266	+	0.994230i	$0.465790\pi$
$192$	0	0
$193$	16.8794	1.21500	0.607502	−	0.794318i	$-0.292172\pi$
$193$	16.8794	1.21500	0.607502	+	0.794318i	$0.292172\pi$
$194$	0	0
$195$	0.608978	0.0436098
$196$	0	0
$197$	−14.6476	−1.04360	−0.521799	−	0.853068i	$-0.674739\pi$
$197$	−14.6476	−1.04360	−0.521799	+	0.853068i	$0.674739\pi$
$198$	0	0
$199$	−7.70587	−0.546254	−0.273127	−	0.961978i	$-0.588058\pi$
$199$	−7.70587	−0.546254	−0.273127	+	0.961978i	$0.588058\pi$
$200$	0	0
$201$	−25.0950	−1.77006
$202$	0	0
$203$	−14.8290	−1.04079
$204$	0	0
$205$	5.84864	0.408487
$206$	0	0
$207$	25.6879	1.78543
$208$	0	0
$209$	51.8426	3.58603
$210$	0	0
$211$	5.09774	0.350943	0.175471	−	0.984485i	$-0.443855\pi$
$211$	5.09774	0.350943	0.175471	+	0.984485i	$0.443855\pi$
$212$	0	0
$213$	16.5396	1.13328
$214$	0	0
$215$	10.0775	0.687278
$216$	0	0
$217$	−11.3196	−0.768424
$218$	0	0
$219$	15.0099	1.01428
$220$	0	0
$221$	−0.759935	−0.0511188
$222$	0	0
$223$	6.45474	0.432242	0.216121	−	0.976367i	$-0.430660\pi$
$223$	6.45474	0.432242	0.216121	+	0.976367i	$0.430660\pi$
$224$	0	0
$225$	−12.1382	−0.809216
$226$	0	0
$227$	20.8933	1.38674	0.693368	−	0.720584i	$-0.256126\pi$
$227$	20.8933	1.38674	0.693368	+	0.720584i	$0.256126\pi$
$228$	0	0
$229$	8.45752	0.558888	0.279444	−	0.960162i	$-0.409850\pi$
$229$	8.45752	0.558888	0.279444	+	0.960162i	$0.409850\pi$
$230$	0	0
$231$	78.3363	5.15415
$232$	0	0
$233$	3.54565	0.232283	0.116142	−	0.993233i	$-0.462947\pi$
$233$	3.54565	0.232283	0.116142	+	0.993233i	$0.462947\pi$
$234$	0	0
$235$	−1.13111	−0.0737855
$236$	0	0
$237$	17.3240	1.12531
$238$	0	0
$239$	−20.1514	−1.30348	−0.651742	−	0.758441i	$-0.725961\pi$
$239$	−20.1514	−1.30348	−0.651742	+	0.758441i	$0.725961\pi$
$240$	0	0
$241$	−20.4656	−1.31831	−0.659153	−	0.752009i	$-0.729085\pi$
$241$	−20.4656	−1.31831	−0.659153	+	0.752009i	$0.729085\pi$
$242$	0	0
$243$	22.2639	1.42823
$244$	0	0
$245$	21.4523	1.37054
$246$	0	0
$247$	−1.57501	−0.100216
$248$	0	0
$249$	30.4727	1.93113
$250$	0	0
$251$	13.4489	0.848888	0.424444	−	0.905454i	$-0.360470\pi$
$251$	13.4489	0.848888	0.424444	+	0.905454i	$0.360470\pi$
$252$	0	0
$253$	45.1973	2.84153
$254$	0	0
$255$	−12.8810	−0.806638
$256$	0	0
$257$	−9.35772	−0.583718	−0.291859	−	0.956461i	$-0.594274\pi$
$257$	−9.35772	−0.583718	−0.291859	+	0.956461i	$0.594274\pi$
$258$	0	0
$259$	36.5493	2.27106
$260$	0	0
$261$	10.7149	0.663235
$262$	0	0
$263$	2.37480	0.146436	0.0732181	−	0.997316i	$-0.476673\pi$
$263$	2.37480	0.146436	0.0732181	+	0.997316i	$0.476673\pi$
$264$	0	0
$265$	6.52318	0.400716
$266$	0	0
$267$	−40.0516	−2.45112
$268$	0	0
$269$	−0.0621461	−0.00378911	−0.00189456	−	0.999998i	$-0.500603\pi$
$269$	−0.0621461	−0.00378911	−0.00189456	+	0.999998i	$0.500603\pi$
$270$	0	0
$271$	10.8842	0.661166	0.330583	−	0.943777i	$-0.392755\pi$
$271$	10.8842	0.661166	0.330583	+	0.943777i	$0.392755\pi$
$272$	0	0
$273$	−2.37991	−0.144039
$274$	0	0
$275$	−21.3569	−1.28787
$276$	0	0
$277$	−16.2718	−0.977675	−0.488837	−	0.872375i	$-0.662579\pi$
$277$	−16.2718	−0.977675	−0.488837	+	0.872375i	$0.662579\pi$
$278$	0	0
$279$	8.17911	0.489670
$280$	0	0
$281$	−18.0021	−1.07391	−0.536957	−	0.843610i	$-0.680426\pi$
$281$	−18.0021	−1.07391	−0.536957	+	0.843610i	$0.680426\pi$
$282$	0	0
$283$	15.2904	0.908919	0.454460	−	0.890767i	$-0.349832\pi$
$283$	15.2904	0.908919	0.454460	+	0.890767i	$0.349832\pi$
$284$	0	0
$285$	−26.6966	−1.58137
$286$	0	0
$287$	−22.8567	−1.34919
$288$	0	0
$289$	−0.925994	−0.0544702
$290$	0	0
$291$	27.5506	1.61504
$292$	0	0
$293$	−9.79272	−0.572097	−0.286048	−	0.958215i	$-0.592342\pi$
$293$	−9.79272	−0.572097	−0.286048	+	0.958215i	$0.592342\pi$
$294$	0	0
$295$	8.55756	0.498240
$296$	0	0
$297$	−8.71503	−0.505697
$298$	0	0
$299$	−1.37312	−0.0794097
$300$	0	0
$301$	−39.3832	−2.27001
$302$	0	0
$303$	22.6875	1.30336
$304$	0	0
$305$	−6.92680	−0.396628
$306$	0	0
$307$	−32.7815	−1.87094	−0.935469	−	0.353409i	$-0.885022\pi$
$307$	−32.7815	−1.87094	−0.935469	+	0.353409i	$0.885022\pi$
$308$	0	0
$309$	42.8885	2.43984
$310$	0	0
$311$	18.8393	1.06828	0.534140	−	0.845396i	$-0.320635\pi$
$311$	18.8393	1.06828	0.534140	+	0.845396i	$0.320635\pi$
$312$	0	0
$313$	−21.0436	−1.18945	−0.594726	−	0.803928i	$-0.702740\pi$
$313$	−21.0436	−1.18945	−0.594726	+	0.803928i	$0.702740\pi$
$314$	0	0
$315$	−21.8521	−1.23123
$316$	0	0
$317$	−12.5205	−0.703221	−0.351611	−	0.936146i	$-0.614366\pi$
$317$	−12.5205	−0.703221	−0.351611	+	0.936146i	$0.614366\pi$
$318$	0	0
$319$	18.8526	1.05554
$320$	0	0
$321$	−38.5217	−2.15007
$322$	0	0
$323$	33.3144	1.85366
$324$	0	0
$325$	0.648837	0.0359910
$326$	0	0
$327$	−17.6636	−0.976799
$328$	0	0
$329$	4.42042	0.243706
$330$	0	0
$331$	6.83573	0.375725	0.187863	−	0.982195i	$-0.439844\pi$
$331$	6.83573	0.375725	0.187863	+	0.982195i	$0.439844\pi$
$332$	0	0
$333$	−26.4091	−1.44721
$334$	0	0
$335$	12.3169	0.672942
$336$	0	0
$337$	8.99743	0.490121	0.245061	−	0.969508i	$-0.421192\pi$
$337$	8.99743	0.490121	0.245061	+	0.969508i	$0.421192\pi$
$338$	0	0
$339$	37.0586	2.01274
$340$	0	0
$341$	14.3909	0.779313
$342$	0	0
$343$	−49.4842	−2.67189
$344$	0	0
$345$	−23.2746	−1.25306
$346$	0	0
$347$	13.6371	0.732077	0.366038	−	0.930600i	$-0.380714\pi$
$347$	13.6371	0.732077	0.366038	+	0.930600i	$0.380714\pi$
$348$	0	0
$349$	−37.0633	−1.98395	−0.991976	−	0.126426i	$-0.959650\pi$
$349$	−37.0633	−1.98395	−0.991976	+	0.126426i	$0.959650\pi$
$350$	0	0
$351$	0.264768	0.0141323
$352$	0	0
$353$	−32.4727	−1.72835	−0.864175	−	0.503192i	$-0.832159\pi$
$353$	−32.4727	−1.72835	−0.864175	+	0.503192i	$0.832159\pi$
$354$	0	0
$355$	−8.11781	−0.430849
$356$	0	0
$357$	50.3394	2.66424
$358$	0	0
$359$	−21.3379	−1.12617	−0.563085	−	0.826399i	$-0.690386\pi$
$359$	−21.3379	−1.12617	−0.563085	+	0.826399i	$0.690386\pi$
$360$	0	0
$361$	50.0461	2.63401
$362$	0	0
$363$	−71.4478	−3.75004
$364$	0	0
$365$	−7.36702	−0.385607
$366$	0	0
$367$	38.2176	1.99494	0.997470	−	0.0710884i	$-0.0226472\pi$
$367$	38.2176	1.99494	0.997470	+	0.0710884i	$0.0226472\pi$
$368$	0	0
$369$	16.5154	0.859758
$370$	0	0
$371$	−25.4929	−1.32352
$372$	0	0
$373$	36.4158	1.88554	0.942770	−	0.333444i	$-0.108211\pi$
$373$	36.4158	1.88554	0.942770	+	0.333444i	$0.108211\pi$
$374$	0	0
$375$	27.0620	1.39747
$376$	0	0
$377$	−0.572753	−0.0294983
$378$	0	0
$379$	33.7668	1.73448	0.867241	−	0.497889i	$-0.165891\pi$
$379$	33.7668	1.73448	0.867241	+	0.497889i	$0.165891\pi$
$380$	0	0
$381$	−34.3051	−1.75750
$382$	0	0
$383$	−35.2116	−1.79923	−0.899614	−	0.436685i	$-0.856152\pi$
$383$	−35.2116	−1.79923	−0.899614	+	0.436685i	$0.856152\pi$
$384$	0	0
$385$	−38.4482	−1.95950
$386$	0	0
$387$	28.4568	1.44654
$388$	0	0
$389$	−24.8269	−1.25877	−0.629387	−	0.777092i	$-0.716694\pi$
$389$	−24.8269	−1.25877	−0.629387	+	0.777092i	$0.716694\pi$
$390$	0	0
$391$	29.0440	1.46882
$392$	0	0
$393$	−33.5305	−1.69139
$394$	0	0
$395$	−8.50278	−0.427821
$396$	0	0
$397$	33.9942	1.70612	0.853060	−	0.521812i	$-0.174744\pi$
$397$	33.9942	1.70612	0.853060	+	0.521812i	$0.174744\pi$
$398$	0	0
$399$	104.331	5.22311
$400$	0	0
$401$	−23.8192	−1.18947	−0.594737	−	0.803920i	$-0.702744\pi$
$401$	−23.8192	−1.18947	−0.594737	+	0.803920i	$0.702744\pi$
$402$	0	0
$403$	−0.437206	−0.0217788
$404$	0	0
$405$	−8.87059	−0.440783
$406$	0	0
$407$	−46.4662	−2.30324
$408$	0	0
$409$	−39.3288	−1.94468	−0.972342	−	0.233562i	$-0.924962\pi$
$409$	−39.3288	−1.94468	−0.972342	+	0.233562i	$0.924962\pi$
$410$	0	0
$411$	36.3506	1.79304
$412$	0	0
$413$	−33.4433	−1.64564
$414$	0	0
$415$	−14.9563	−0.734177
$416$	0	0
$417$	−22.9235	−1.12257
$418$	0	0
$419$	2.26098	0.110456	0.0552280	−	0.998474i	$-0.482411\pi$
$419$	2.26098	0.110456	0.0552280	+	0.998474i	$0.482411\pi$
$420$	0	0
$421$	−10.0055	−0.487638	−0.243819	−	0.969821i	$-0.578400\pi$
$421$	−10.0055	−0.487638	−0.243819	+	0.969821i	$0.578400\pi$
$422$	0	0
$423$	−3.19403	−0.155299
$424$	0	0
$425$	−13.7241	−0.665716
$426$	0	0
$427$	27.0702	1.31002
$428$	0	0
$429$	3.02565	0.146080
$430$	0	0
$431$	−29.2649	−1.40964	−0.704821	−	0.709385i	$-0.748973\pi$
$431$	−29.2649	−1.40964	−0.704821	+	0.709385i	$0.748973\pi$
$432$	0	0
$433$	17.6696	0.849146	0.424573	−	0.905394i	$-0.360424\pi$
$433$	17.6696	0.849146	0.424573	+	0.905394i	$0.360424\pi$
$434$	0	0
$435$	−9.70823	−0.465474
$436$	0	0
$437$	60.1956	2.87954
$438$	0	0
$439$	28.9392	1.38119	0.690596	−	0.723240i	$-0.257348\pi$
$439$	28.9392	1.38119	0.690596	+	0.723240i	$0.257348\pi$
$440$	0	0
$441$	60.5772	2.88463
$442$	0	0
$443$	9.95875	0.473154	0.236577	−	0.971613i	$-0.423974\pi$
$443$	9.95875	0.473154	0.236577	+	0.971613i	$0.423974\pi$
$444$	0	0
$445$	19.6577	0.931866
$446$	0	0
$447$	−37.6927	−1.78280
$448$	0	0
$449$	35.2084	1.66159	0.830793	−	0.556581i	$-0.187887\pi$
$449$	35.2084	1.66159	0.830793	+	0.556581i	$0.187887\pi$
$450$	0	0
$451$	29.0585	1.36831
$452$	0	0
$453$	34.6593	1.62844
$454$	0	0
$455$	1.16808	0.0547605
$456$	0	0
$457$	3.80107	0.177807	0.0889034	−	0.996040i	$-0.471664\pi$
$457$	3.80107	0.177807	0.0889034	+	0.996040i	$0.471664\pi$
$458$	0	0
$459$	−5.60033	−0.261401
$460$	0	0
$461$	−7.95648	−0.370570	−0.185285	−	0.982685i	$-0.559321\pi$
$461$	−7.95648	−0.370570	−0.185285	+	0.982685i	$0.559321\pi$
$462$	0	0
$463$	−1.54869	−0.0719735	−0.0359868	−	0.999352i	$-0.511457\pi$
$463$	−1.54869	−0.0719735	−0.0359868	+	0.999352i	$0.511457\pi$
$464$	0	0
$465$	−7.41069	−0.343662
$466$	0	0
$467$	8.32360	0.385170	0.192585	−	0.981280i	$-0.438313\pi$
$467$	8.32360	0.385170	0.192585	+	0.981280i	$0.438313\pi$
$468$	0	0
$469$	−48.1347	−2.22266
$470$	0	0
$471$	−52.2739	−2.40865
$472$	0	0
$473$	50.0690	2.30218
$474$	0	0
$475$	−28.4440	−1.30510
$476$	0	0
$477$	18.4202	0.843402
$478$	0	0
$479$	8.85606	0.404644	0.202322	−	0.979319i	$-0.435151\pi$
$479$	8.85606	0.404644	0.202322	+	0.979319i	$0.435151\pi$
$480$	0	0
$481$	1.41167	0.0643668
$482$	0	0
$483$	90.9579	4.13873
$484$	0	0
$485$	−13.5221	−0.614006
$486$	0	0
$487$	17.2868	0.783340	0.391670	−	0.920106i	$-0.371898\pi$
$487$	17.2868	0.783340	0.391670	+	0.920106i	$0.371898\pi$
$488$	0	0
$489$	3.51379	0.158899
$490$	0	0
$491$	−4.66736	−0.210635	−0.105317	−	0.994439i	$-0.533586\pi$
$491$	−4.66736	−0.210635	−0.105317	+	0.994439i	$0.533586\pi$
$492$	0	0
$493$	12.1148	0.545622
$494$	0	0
$495$	27.7813	1.24867
$496$	0	0
$497$	31.7247	1.42305
$498$	0	0
$499$	0.655028	0.0293231	0.0146615	−	0.999893i	$-0.495333\pi$
$499$	0.655028	0.0293231	0.0146615	+	0.999893i	$0.495333\pi$
$500$	0	0
$501$	7.74722	0.346120
$502$	0	0
$503$	−6.79709	−0.303067	−0.151534	−	0.988452i	$-0.548421\pi$
$503$	−6.79709	−0.303067	−0.151534	+	0.988452i	$0.548421\pi$
$504$	0	0
$505$	−11.1352	−0.495512
$506$	0	0
$507$	33.1687	1.47307
$508$	0	0
$509$	11.1171	0.492756	0.246378	−	0.969174i	$-0.420760\pi$
$509$	11.1171	0.492756	0.246378	+	0.969174i	$0.420760\pi$
$510$	0	0
$511$	28.7906	1.27362
$512$	0	0
$513$	−11.6070	−0.512463
$514$	0	0
$515$	−21.0501	−0.927579
$516$	0	0
$517$	−5.61982	−0.247159
$518$	0	0
$519$	−8.08267	−0.354790
$520$	0	0
$521$	20.8718	0.914411	0.457206	−	0.889361i	$-0.348850\pi$
$521$	20.8718	0.914411	0.457206	+	0.889361i	$0.348850\pi$
$522$	0	0
$523$	−11.4975	−0.502752	−0.251376	−	0.967890i	$-0.580883\pi$
$523$	−11.4975	−0.502752	−0.251376	+	0.967890i	$0.580883\pi$
$524$	0	0
$525$	−42.9801	−1.87580
$526$	0	0
$527$	9.24770	0.402836
$528$	0	0
$529$	29.4795	1.28172
$530$	0	0
$531$	24.1649	1.04867
$532$	0	0
$533$	−0.882814	−0.0382389
$534$	0	0
$535$	18.9068	0.817413
$536$	0	0
$537$	25.8260	1.11447
$538$	0	0
$539$	106.584	4.59090
$540$	0	0
$541$	−11.1866	−0.480950	−0.240475	−	0.970655i	$-0.577303\pi$
$541$	−11.1866	−0.480950	−0.240475	+	0.970655i	$0.577303\pi$
$542$	0	0
$543$	−64.1763	−2.75407
$544$	0	0
$545$	8.66946	0.371359
$546$	0	0
$547$	−35.4894	−1.51742	−0.758708	−	0.651431i	$-0.774169\pi$
$547$	−35.4894	−1.51742	−0.758708	+	0.651431i	$0.774169\pi$
$548$	0	0
$549$	−19.5599	−0.834797
$550$	0	0
$551$	25.1086	1.06966
$552$	0	0
$553$	33.2292	1.41305
$554$	0	0
$555$	23.9280	1.01569
$556$	0	0
$557$	16.5063	0.699396	0.349698	−	0.936863i	$-0.386284\pi$
$557$	16.5063	0.699396	0.349698	+	0.936863i	$0.386284\pi$
$558$	0	0
$559$	−1.52113	−0.0643369
$560$	0	0
$561$	−63.9980	−2.70200
$562$	0	0
$563$	26.2870	1.10786	0.553932	−	0.832562i	$-0.313127\pi$
$563$	26.2870	1.10786	0.553932	+	0.832562i	$0.313127\pi$
$564$	0	0
$565$	−18.1887	−0.765204
$566$	0	0
$567$	34.6666	1.45586
$568$	0	0
$569$	−2.31682	−0.0971263	−0.0485631	−	0.998820i	$-0.515464\pi$
$569$	−2.31682	−0.0971263	−0.0485631	+	0.998820i	$0.515464\pi$
$570$	0	0
$571$	−0.0603900	−0.00252724	−0.00126362	−	0.999999i	$-0.500402\pi$
$571$	−0.0603900	−0.00252724	−0.00126362	+	0.999999i	$0.500402\pi$
$572$	0	0
$573$	−7.58573	−0.316898
$574$	0	0
$575$	−24.7980	−1.03415
$576$	0	0
$577$	15.4336	0.642510	0.321255	−	0.946993i	$-0.395895\pi$
$577$	15.4336	0.642510	0.321255	+	0.946993i	$0.395895\pi$
$578$	0	0
$579$	−43.1860	−1.79475
$580$	0	0
$581$	58.4499	2.42491
$582$	0	0
$583$	32.4099	1.34228
$584$	0	0
$585$	−0.844012	−0.0348956
$586$	0	0
$587$	40.4830	1.67091	0.835456	−	0.549557i	$-0.185204\pi$
$587$	40.4830	1.67091	0.835456	+	0.549557i	$0.185204\pi$
$588$	0	0
$589$	19.1664	0.789739
$590$	0	0
$591$	37.4760	1.54156
$592$	0	0
$593$	6.38600	0.262242	0.131121	−	0.991366i	$-0.458142\pi$
$593$	6.38600	0.262242	0.131121	+	0.991366i	$0.458142\pi$
$594$	0	0
$595$	−24.7071	−1.01289
$596$	0	0
$597$	19.7155	0.806903
$598$	0	0
$599$	9.94142	0.406196	0.203098	−	0.979158i	$-0.434899\pi$
$599$	9.94142	0.406196	0.203098	+	0.979158i	$0.434899\pi$
$600$	0	0
$601$	36.1727	1.47551	0.737757	−	0.675067i	$-0.235885\pi$
$601$	36.1727	1.47551	0.737757	+	0.675067i	$0.235885\pi$
$602$	0	0
$603$	34.7804	1.41637
$604$	0	0
$605$	35.0673	1.42569
$606$	0	0
$607$	12.9682	0.526362	0.263181	−	0.964746i	$-0.415228\pi$
$607$	12.9682	0.526362	0.263181	+	0.964746i	$0.415228\pi$
$608$	0	0
$609$	37.9402	1.53741
$610$	0	0
$611$	0.170734	0.00690715
$612$	0	0
$613$	−10.5620	−0.426595	−0.213298	−	0.976987i	$-0.568420\pi$
$613$	−10.5620	−0.426595	−0.213298	+	0.976987i	$0.568420\pi$
$614$	0	0
$615$	−14.9638	−0.603399
$616$	0	0
$617$	−3.28842	−0.132387	−0.0661934	−	0.997807i	$-0.521085\pi$
$617$	−3.28842	−0.132387	−0.0661934	+	0.997807i	$0.521085\pi$
$618$	0	0
$619$	−47.3971	−1.90505	−0.952525	−	0.304461i	$-0.901524\pi$
$619$	−47.3971	−1.90505	−0.952525	+	0.304461i	$0.901524\pi$
$620$	0	0
$621$	−10.1192	−0.406069
$622$	0	0
$623$	−76.8232	−3.07786
$624$	0	0
$625$	3.83328	0.153331
$626$	0	0
$627$	−132.640	−5.29712
$628$	0	0
$629$	−29.8595	−1.19057
$630$	0	0
$631$	9.54505	0.379982	0.189991	−	0.981786i	$-0.439154\pi$
$631$	9.54505	0.379982	0.189991	+	0.981786i	$0.439154\pi$
$632$	0	0
$633$	−13.0426	−0.518397
$634$	0	0
$635$	16.8373	0.668167
$636$	0	0
$637$	−3.23809	−0.128298
$638$	0	0
$639$	−22.9231	−0.906823
$640$	0	0
$641$	8.18506	0.323291	0.161645	−	0.986849i	$-0.448320\pi$
$641$	8.18506	0.323291	0.161645	+	0.986849i	$0.448320\pi$
$642$	0	0
$643$	12.8134	0.505310	0.252655	−	0.967556i	$-0.418696\pi$
$643$	12.8134	0.505310	0.252655	+	0.967556i	$0.418696\pi$
$644$	0	0
$645$	−25.7833	−1.01522
$646$	0	0
$647$	31.3428	1.23221	0.616106	−	0.787663i	$-0.288709\pi$
$647$	31.3428	1.23221	0.616106	+	0.787663i	$0.288709\pi$
$648$	0	0
$649$	42.5175	1.66896
$650$	0	0
$651$	28.9613	1.13508
$652$	0	0
$653$	−15.7922	−0.617996	−0.308998	−	0.951063i	$-0.599994\pi$
$653$	−15.7922	−0.617996	−0.308998	+	0.951063i	$0.599994\pi$
$654$	0	0
$655$	16.4571	0.643032
$656$	0	0
$657$	−20.8030	−0.811602
$658$	0	0
$659$	7.57151	0.294944	0.147472	−	0.989066i	$-0.452886\pi$
$659$	7.57151	0.294944	0.147472	+	0.989066i	$0.452886\pi$
$660$	0	0
$661$	10.4092	0.404870	0.202435	−	0.979296i	$-0.435114\pi$
$661$	10.4092	0.404870	0.202435	+	0.979296i	$0.435114\pi$
$662$	0	0
$663$	1.94430	0.0755104
$664$	0	0
$665$	−51.2069	−1.98572
$666$	0	0
$667$	21.8901	0.847589
$668$	0	0
$669$	−16.5145	−0.638488
$670$	0	0
$671$	−34.4152	−1.32858
$672$	0	0
$673$	−10.3235	−0.397943	−0.198972	−	0.980005i	$-0.563760\pi$
$673$	−10.3235	−0.397943	−0.198972	+	0.980005i	$0.563760\pi$
$674$	0	0
$675$	4.78160	0.184044
$676$	0	0
$677$	1.90539	0.0732300	0.0366150	−	0.999329i	$-0.488342\pi$
$677$	1.90539	0.0732300	0.0366150	+	0.999329i	$0.488342\pi$
$678$	0	0
$679$	52.8448	2.02800
$680$	0	0
$681$	−53.4556	−2.04842
$682$	0	0
$683$	13.6936	0.523969	0.261985	−	0.965072i	$-0.415623\pi$
$683$	13.6936	0.523969	0.261985	+	0.965072i	$0.415623\pi$
$684$	0	0
$685$	−17.8412	−0.681678
$686$	0	0
$687$	−21.6386	−0.825565
$688$	0	0
$689$	−0.984632	−0.0375115
$690$	0	0
$691$	−12.6315	−0.480525	−0.240263	−	0.970708i	$-0.577234\pi$
$691$	−12.6315	−0.480525	−0.240263	+	0.970708i	$0.577234\pi$
$692$	0	0
$693$	−108.570	−4.12424
$694$	0	0
$695$	11.2511	0.426777
$696$	0	0
$697$	18.6731	0.707295
$698$	0	0
$699$	−9.07158	−0.343119
$700$	0	0
$701$	1.49045	0.0562937	0.0281468	−	0.999604i	$-0.491039\pi$
$701$	1.49045	0.0562937	0.0281468	+	0.999604i	$0.491039\pi$
$702$	0	0
$703$	−61.8856	−2.33406
$704$	0	0
$705$	2.89396	0.108993
$706$	0	0
$707$	43.5169	1.63662
$708$	0	0
$709$	−41.9782	−1.57652	−0.788261	−	0.615340i	$-0.789019\pi$
$709$	−41.9782	−1.57652	−0.788261	+	0.615340i	$0.789019\pi$
$710$	0	0
$711$	−24.0102	−0.900452
$712$	0	0
$713$	16.7096	0.625780
$714$	0	0
$715$	−1.48502	−0.0555366
$716$	0	0
$717$	51.5574	1.92545
$718$	0	0
$719$	−36.6663	−1.36742	−0.683712	−	0.729752i	$-0.739635\pi$
$719$	−36.6663	−1.36742	−0.683712	+	0.729752i	$0.739635\pi$
$720$	0	0
$721$	82.2647	3.06370
$722$	0	0
$723$	52.3614	1.94734
$724$	0	0
$725$	−10.3437	−0.384154
$726$	0	0
$727$	7.77126	0.288220	0.144110	−	0.989562i	$-0.453968\pi$
$727$	7.77126	0.288220	0.144110	+	0.989562i	$0.453968\pi$
$728$	0	0
$729$	−35.7704	−1.32483
$730$	0	0
$731$	32.1746	1.19002
$732$	0	0
$733$	−23.8761	−0.881884	−0.440942	−	0.897536i	$-0.645356\pi$
$733$	−23.8761	−0.881884	−0.440942	+	0.897536i	$0.645356\pi$
$734$	0	0
$735$	−54.8860	−2.02450
$736$	0	0
$737$	61.1952	2.25415
$738$	0	0
$739$	50.7598	1.86723	0.933614	−	0.358280i	$-0.116637\pi$
$739$	50.7598	1.86723	0.933614	+	0.358280i	$0.116637\pi$
$740$	0	0
$741$	4.02968	0.148034
$742$	0	0
$743$	−30.9453	−1.13527	−0.567636	−	0.823280i	$-0.692142\pi$
$743$	−30.9453	−1.13527	−0.567636	+	0.823280i	$0.692142\pi$
$744$	0	0
$745$	18.5000	0.677786
$746$	0	0
$747$	−42.2337	−1.54525
$748$	0	0
$749$	−73.8885	−2.69983
$750$	0	0
$751$	12.5697	0.458673	0.229337	−	0.973347i	$-0.426344\pi$
$751$	12.5697	0.458673	0.229337	+	0.973347i	$0.426344\pi$
$752$	0	0
$753$	−34.4092	−1.25394
$754$	0	0
$755$	−17.0111	−0.619098
$756$	0	0
$757$	17.6184	0.640350	0.320175	−	0.947358i	$-0.396258\pi$
$757$	17.6184	0.640350	0.320175	+	0.947358i	$0.396258\pi$
$758$	0	0
$759$	−115.638	−4.19738
$760$	0	0
$761$	35.1037	1.27251	0.636253	−	0.771480i	$-0.280483\pi$
$761$	35.1037	1.27251	0.636253	+	0.771480i	$0.280483\pi$
$762$	0	0
$763$	−33.8806	−1.22656
$764$	0	0
$765$	17.8524	0.645455
$766$	0	0
$767$	−1.29171	−0.0466408
$768$	0	0
$769$	−14.7293	−0.531151	−0.265575	−	0.964090i	$-0.585562\pi$
$769$	−14.7293	−0.531151	−0.265575	+	0.964090i	$0.585562\pi$
$770$	0	0
$771$	23.9418	0.862243
$772$	0	0
$773$	−5.01346	−0.180321	−0.0901607	−	0.995927i	$-0.528738\pi$
$773$	−5.01346	−0.180321	−0.0901607	+	0.995927i	$0.528738\pi$
$774$	0	0
$775$	−7.89574	−0.283623
$776$	0	0
$777$	−93.5116	−3.35471
$778$	0	0
$779$	38.7012	1.38662
$780$	0	0
$781$	−40.3326	−1.44321
$782$	0	0
$783$	−4.22090	−0.150843
$784$	0	0
$785$	25.6566	0.915722
$786$	0	0
$787$	−50.5298	−1.80119	−0.900596	−	0.434658i	$-0.856869\pi$
$787$	−50.5298	−1.80119	−0.900596	+	0.434658i	$0.856869\pi$
$788$	0	0
$789$	−6.07594	−0.216309
$790$	0	0
$791$	71.0822	2.52739
$792$	0	0
$793$	1.04556	0.0371288
$794$	0	0
$795$	−16.6896	−0.591920
$796$	0	0
$797$	−4.78602	−0.169529	−0.0847647	−	0.996401i	$-0.527014\pi$
$797$	−4.78602	−0.169529	−0.0847647	+	0.996401i	$0.527014\pi$
$798$	0	0
$799$	−3.61133	−0.127760
$800$	0	0
$801$	55.5096	1.96133
$802$	0	0
$803$	−36.6024	−1.29167
$804$	0	0
$805$	−44.6430	−1.57346
$806$	0	0
$807$	0.159001	0.00559711
$808$	0	0
$809$	16.6397	0.585021	0.292511	−	0.956262i	$-0.405509\pi$
$809$	16.6397	0.585021	0.292511	+	0.956262i	$0.405509\pi$
$810$	0	0
$811$	23.6075	0.828970	0.414485	−	0.910056i	$-0.363962\pi$
$811$	23.6075	0.828970	0.414485	+	0.910056i	$0.363962\pi$
$812$	0	0
$813$	−27.8472	−0.976645
$814$	0	0
$815$	−1.72460	−0.0604101
$816$	0	0
$817$	66.6840	2.33298
$818$	0	0
$819$	3.29843	0.115257
$820$	0	0
$821$	29.4328	1.02721	0.513607	−	0.858026i	$-0.328309\pi$
$821$	29.4328	1.02721	0.513607	+	0.858026i	$0.328309\pi$
$822$	0	0
$823$	−6.35900	−0.221661	−0.110830	−	0.993839i	$-0.535351\pi$
$823$	−6.35900	−0.221661	−0.110830	+	0.993839i	$0.535351\pi$
$824$	0	0
$825$	54.6419	1.90239
$826$	0	0
$827$	34.8484	1.21180	0.605900	−	0.795541i	$-0.292813\pi$
$827$	34.8484	1.21180	0.605900	+	0.795541i	$0.292813\pi$
$828$	0	0
$829$	29.8931	1.03823	0.519116	−	0.854704i	$-0.326261\pi$
$829$	29.8931	1.03823	0.519116	+	0.854704i	$0.326261\pi$
$830$	0	0
$831$	41.6314	1.44418
$832$	0	0
$833$	68.4915	2.37309
$834$	0	0
$835$	−3.80241	−0.131588
$836$	0	0
$837$	−3.22198	−0.111368
$838$	0	0
$839$	−39.7953	−1.37389	−0.686944	−	0.726711i	$-0.741048\pi$
$839$	−39.7953	−1.37389	−0.686944	+	0.726711i	$0.741048\pi$
$840$	0	0
$841$	−19.8692	−0.685146
$842$	0	0
$843$	46.0584	1.58634
$844$	0	0
$845$	−16.2795	−0.560032
$846$	0	0
$847$	−137.044	−4.70890
$848$	0	0
$849$	−39.1206	−1.34261
$850$	0	0
$851$	−53.9529	−1.84948
$852$	0	0
$853$	29.1314	0.997439	0.498720	−	0.866763i	$-0.333804\pi$
$853$	29.1314	0.997439	0.498720	+	0.866763i	$0.333804\pi$
$854$	0	0
$855$	37.0002	1.26538
$856$	0	0
$857$	−29.4238	−1.00510	−0.502549	−	0.864549i	$-0.667604\pi$
$857$	−29.4238	−1.00510	−0.502549	+	0.864549i	$0.667604\pi$
$858$	0	0
$859$	−30.5376	−1.04193	−0.520965	−	0.853578i	$-0.674428\pi$
$859$	−30.5376	−1.04193	−0.520965	+	0.853578i	$0.674428\pi$
$860$	0	0
$861$	58.4791	1.99296
$862$	0	0
$863$	13.0125	0.442951	0.221475	−	0.975166i	$-0.428913\pi$
$863$	13.0125	0.442951	0.221475	+	0.975166i	$0.428913\pi$
$864$	0	0
$865$	3.96705	0.134884
$866$	0	0
$867$	2.36916	0.0804610
$868$	0	0
$869$	−42.2453	−1.43307
$870$	0	0
$871$	−1.85915	−0.0629948
$872$	0	0
$873$	−38.1837	−1.29232
$874$	0	0
$875$	51.9077	1.75480
$876$	0	0
$877$	−14.8316	−0.500826	−0.250413	−	0.968139i	$-0.580566\pi$
$877$	−14.8316	−0.500826	−0.250413	+	0.968139i	$0.580566\pi$
$878$	0	0
$879$	25.0547	0.845076
$880$	0	0
$881$	37.6312	1.26783	0.633914	−	0.773404i	$-0.281447\pi$
$881$	37.6312	1.26783	0.633914	+	0.773404i	$0.281447\pi$
$882$	0	0
$883$	−35.3845	−1.19078	−0.595391	−	0.803436i	$-0.703003\pi$
$883$	−35.3845	−1.19078	−0.595391	+	0.803436i	$0.703003\pi$
$884$	0	0
$885$	−21.8946	−0.735978
$886$	0	0
$887$	−38.6921	−1.29916	−0.649578	−	0.760295i	$-0.725054\pi$
$887$	−38.6921	−1.29916	−0.649578	+	0.760295i	$0.725054\pi$
$888$	0	0
$889$	−65.8007	−2.20688
$890$	0	0
$891$	−44.0727	−1.47649
$892$	0	0
$893$	−7.48471	−0.250466
$894$	0	0
$895$	−12.6756	−0.423700
$896$	0	0
$897$	3.51314	0.117300
$898$	0	0
$899$	6.96987	0.232458
$900$	0	0
$901$	20.8268	0.693840
$902$	0	0
$903$	100.762	3.35316
$904$	0	0
$905$	31.4984	1.04704
$906$	0	0
$907$	17.0073	0.564717	0.282358	−	0.959309i	$-0.408883\pi$
$907$	17.0073	0.564717	0.282358	+	0.959309i	$0.408883\pi$
$908$	0	0
$909$	−31.4437	−1.04292
$910$	0	0
$911$	−32.2575	−1.06874	−0.534369	−	0.845251i	$-0.679451\pi$
$911$	−32.2575	−1.06874	−0.534369	+	0.845251i	$0.679451\pi$
$912$	0	0
$913$	−74.3091	−2.45927
$914$	0	0
$915$	17.7223	0.585881
$916$	0	0
$917$	−64.3149	−2.12387
$918$	0	0
$919$	27.1142	0.894416	0.447208	−	0.894430i	$-0.352418\pi$
$919$	27.1142	0.894416	0.447208	+	0.894430i	$0.352418\pi$
$920$	0	0
$921$	83.8717	2.76367
$922$	0	0
$923$	1.22533	0.0403322
$924$	0	0
$925$	25.4942	0.838244
$926$	0	0
$927$	−59.4414	−1.95231
$928$	0	0
$929$	22.6313	0.742509	0.371254	−	0.928531i	$-0.378928\pi$
$929$	22.6313	0.742509	0.371254	+	0.928531i	$0.378928\pi$
$930$	0	0
$931$	141.953	4.65232
$932$	0	0
$933$	−48.2006	−1.57802
$934$	0	0
$935$	31.4108	1.02724
$936$	0	0
$937$	25.1883	0.822867	0.411433	−	0.911440i	$-0.365028\pi$
$937$	25.1883	0.822867	0.411433	+	0.911440i	$0.365028\pi$
$938$	0	0
$939$	53.8401	1.75701
$940$	0	0
$941$	6.56880	0.214137	0.107068	−	0.994252i	$-0.465854\pi$
$941$	6.56880	0.214137	0.107068	+	0.994252i	$0.465854\pi$
$942$	0	0
$943$	33.7404	1.09874
$944$	0	0
$945$	8.60816	0.280024
$946$	0	0
$947$	−32.8204	−1.06652	−0.533260	−	0.845952i	$-0.679033\pi$
$947$	−32.8204	−1.06652	−0.533260	+	0.845952i	$0.679033\pi$
$948$	0	0
$949$	1.11200	0.0360971
$950$	0	0
$951$	32.0338	1.03877
$952$	0	0
$953$	−12.0571	−0.390567	−0.195284	−	0.980747i	$-0.562563\pi$
$953$	−12.0571	−0.390567	−0.195284	+	0.980747i	$0.562563\pi$
$954$	0	0
$955$	3.72315	0.120478
$956$	0	0
$957$	−48.2345	−1.55920
$958$	0	0
$959$	69.7242	2.25151
$960$	0	0
$961$	−25.6796	−0.828375
$962$	0	0
$963$	53.3891	1.72044
$964$	0	0
$965$	21.1961	0.682327
$966$	0	0
$967$	−51.4483	−1.65447	−0.827233	−	0.561859i	$-0.810086\pi$
$967$	−51.4483	−1.65447	−0.827233	+	0.561859i	$0.810086\pi$
$968$	0	0
$969$	−85.2351	−2.73815
$970$	0	0
$971$	−25.6136	−0.821979	−0.410989	−	0.911640i	$-0.634817\pi$
$971$	−25.6136	−0.821979	−0.410989	+	0.911640i	$0.634817\pi$
$972$	0	0
$973$	−43.9696	−1.40960
$974$	0	0
$975$	−1.66006	−0.0531643
$976$	0	0
$977$	−26.5542	−0.849543	−0.424772	−	0.905300i	$-0.639646\pi$
$977$	−26.5542	−0.849543	−0.424772	+	0.905300i	$0.639646\pi$
$978$	0	0
$979$	97.6677	3.12147
$980$	0	0
$981$	24.4808	0.781613
$982$	0	0
$983$	32.5679	1.03875	0.519377	−	0.854545i	$-0.326164\pi$
$983$	32.5679	1.03875	0.519377	+	0.854545i	$0.326164\pi$
$984$	0	0
$985$	−18.3936	−0.586069
$986$	0	0
$987$	−11.3097	−0.359992
$988$	0	0
$989$	58.1362	1.84862
$990$	0	0
$991$	16.1270	0.512290	0.256145	−	0.966638i	$-0.417548\pi$
$991$	16.1270	0.512290	0.256145	+	0.966638i	$0.417548\pi$
$992$	0	0
$993$	−17.4893	−0.555005
$994$	0	0
$995$	−9.67657	−0.306768
$996$	0	0
$997$	9.21914	0.291973	0.145987	−	0.989287i	$-0.453364\pi$
$997$	9.21914	0.291973	0.145987	+	0.989287i	$0.453364\pi$
$998$	0	0
$999$	10.4033	0.329146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.b.1.10	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.b.1.10	✓	87	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)

\(f(q)\)	\(=\)	\(q-2.55851 q^{3} +1.25574 q^{5} -4.90748 q^{7} +3.54596 q^{9} +O(q^{10})\)	\(q-2.55851 q^{3} +1.25574 q^{5} -4.90748 q^{7} +3.54596 q^{9} +6.23904 q^{11} -0.189546 q^{13} -3.21282 q^{15} +4.00924 q^{17} +8.30940 q^{19} +12.5558 q^{21} +7.24427 q^{23} -3.42311 q^{25} -1.39686 q^{27} +3.02171 q^{29} +2.30660 q^{31} -15.9626 q^{33} -6.16253 q^{35} -7.44766 q^{37} +0.484955 q^{39} +4.65752 q^{41} +8.02512 q^{43} +4.45281 q^{45} -0.900752 q^{47} +17.0834 q^{49} -10.2577 q^{51} +5.19469 q^{53} +7.83461 q^{55} -21.2597 q^{57} +6.81475 q^{59} -5.51611 q^{61} -17.4018 q^{63} -0.238021 q^{65} +9.80843 q^{67} -18.5345 q^{69} -6.46456 q^{71} -5.86667 q^{73} +8.75807 q^{75} -30.6180 q^{77} -6.77113 q^{79} -7.06403 q^{81} -11.9104 q^{83} +5.03457 q^{85} -7.73108 q^{87} +15.6543 q^{89} +0.930194 q^{91} -5.90145 q^{93} +10.4345 q^{95} -10.7682 q^{97} +22.1234 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * q^99

Introduction

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