LMFDB - Embedded newform 8044.2.a.a.1.19

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			1.19
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)

$f(q)$	$=$	$q-1.93385 q^{3} -1.81561 q^{5} +3.06905 q^{7} +0.739771 q^{9} +O(q^{10})$	$q-1.93385 q^{3} -1.81561 q^{5} +3.06905 q^{7} +0.739771 q^{9} +0.605887 q^{11} -2.49274 q^{13} +3.51111 q^{15} +3.35472 q^{17} +0.713406 q^{19} -5.93507 q^{21} -7.94231 q^{23} -1.70356 q^{25} +4.37094 q^{27} +0.696449 q^{29} +10.9940 q^{31} -1.17169 q^{33} -5.57219 q^{35} +3.13027 q^{37} +4.82059 q^{39} -7.32324 q^{41} +0.307544 q^{43} -1.34313 q^{45} -10.6399 q^{47} +2.41905 q^{49} -6.48752 q^{51} -2.15881 q^{53} -1.10005 q^{55} -1.37962 q^{57} +2.58964 q^{59} -3.85494 q^{61} +2.27039 q^{63} +4.52585 q^{65} +12.5429 q^{67} +15.3592 q^{69} +7.98060 q^{71} +2.68746 q^{73} +3.29443 q^{75} +1.85949 q^{77} -11.1295 q^{79} -10.6721 q^{81} -2.82552 q^{83} -6.09086 q^{85} -1.34683 q^{87} +1.46828 q^{89} -7.65035 q^{91} -21.2607 q^{93} -1.29527 q^{95} +6.88648 q^{97} +0.448217 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.93385	−1.11651	−0.558254	−	0.829670i	$-0.688529\pi$
$3$	−1.93385	−1.11651	−0.558254	+	0.829670i	$0.688529\pi$
$4$	0	0
$5$	−1.81561	−0.811965	−0.405983	−	0.913881i	$-0.633071\pi$
$5$	−1.81561	−0.811965	−0.405983	+	0.913881i	$0.633071\pi$
$6$	0	0
$7$	3.06905	1.15999	0.579995	−	0.814620i	$-0.303054\pi$
$7$	3.06905	1.15999	0.579995	+	0.814620i	$0.303054\pi$
$8$	0	0
$9$	0.739771	0.246590
$10$	0	0
$11$	0.605887	0.182682	0.0913408	−	0.995820i	$-0.470885\pi$
$11$	0.605887	0.182682	0.0913408	+	0.995820i	$0.470885\pi$
$12$	0	0
$13$	−2.49274	−0.691363	−0.345681	−	0.938352i	$-0.612352\pi$
$13$	−2.49274	−0.691363	−0.345681	+	0.938352i	$0.612352\pi$
$14$	0	0
$15$	3.51111	0.906566
$16$	0	0
$17$	3.35472	0.813640	0.406820	−	0.913508i	$-0.366638\pi$
$17$	3.35472	0.813640	0.406820	+	0.913508i	$0.366638\pi$
$18$	0	0
$19$	0.713406	0.163667	0.0818333	−	0.996646i	$-0.473922\pi$
$19$	0.713406	0.163667	0.0818333	+	0.996646i	$0.473922\pi$
$20$	0	0
$21$	−5.93507	−1.29514
$22$	0	0
$23$	−7.94231	−1.65609	−0.828043	−	0.560665i	$-0.810546\pi$
$23$	−7.94231	−1.65609	−0.828043	+	0.560665i	$0.810546\pi$
$24$	0	0
$25$	−1.70356	−0.340713
$26$	0	0
$27$	4.37094	0.841188
$28$	0	0
$29$	0.696449	0.129327	0.0646637	−	0.997907i	$-0.479403\pi$
$29$	0.696449	0.129327	0.0646637	+	0.997907i	$0.479403\pi$
$30$	0	0
$31$	10.9940	1.97458	0.987291	−	0.158923i	$-0.0508020\pi$
$31$	10.9940	1.97458	0.987291	+	0.158923i	$0.0508020\pi$
$32$	0	0
$33$	−1.17169	−0.203966
$34$	0	0
$35$	−5.57219	−0.941872
$36$	0	0
$37$	3.13027	0.514613	0.257307	−	0.966330i	$-0.417165\pi$
$37$	3.13027	0.514613	0.257307	+	0.966330i	$0.417165\pi$
$38$	0	0
$39$	4.82059	0.771912
$40$	0	0
$41$	−7.32324	−1.14370	−0.571849	−	0.820359i	$-0.693774\pi$
$41$	−7.32324	−1.14370	−0.571849	+	0.820359i	$0.693774\pi$
$42$	0	0
$43$	0.307544	0.0468999	0.0234500	−	0.999725i	$-0.492535\pi$
$43$	0.307544	0.0468999	0.0234500	+	0.999725i	$0.492535\pi$
$44$	0	0
$45$	−1.34313	−0.200223
$46$	0	0
$47$	−10.6399	−1.55199	−0.775995	−	0.630739i	$-0.782752\pi$
$47$	−10.6399	−1.55199	−0.775995	+	0.630739i	$0.782752\pi$
$48$	0	0
$49$	2.41905	0.345579
$50$	0	0
$51$	−6.48752	−0.908435
$52$	0	0
$53$	−2.15881	−0.296535	−0.148268	−	0.988947i	$-0.547370\pi$
$53$	−2.15881	−0.296535	−0.148268	+	0.988947i	$0.547370\pi$
$54$	0	0
$55$	−1.10005	−0.148331
$56$	0	0
$57$	−1.37962	−0.182735
$58$	0	0
$59$	2.58964	0.337142	0.168571	−	0.985690i	$-0.446085\pi$
$59$	2.58964	0.337142	0.168571	+	0.985690i	$0.446085\pi$
$60$	0	0
$61$	−3.85494	−0.493575	−0.246787	−	0.969070i	$-0.579375\pi$
$61$	−3.85494	−0.493575	−0.246787	+	0.969070i	$0.579375\pi$
$62$	0	0
$63$	2.27039	0.286042
$64$	0	0
$65$	4.52585	0.561362
$66$	0	0
$67$	12.5429	1.53236	0.766178	−	0.642629i	$-0.222156\pi$
$67$	12.5429	1.53236	0.766178	+	0.642629i	$0.222156\pi$
$68$	0	0
$69$	15.3592	1.84903
$70$	0	0
$71$	7.98060	0.947123	0.473561	−	0.880761i	$-0.342968\pi$
$71$	7.98060	0.947123	0.473561	+	0.880761i	$0.342968\pi$
$72$	0	0
$73$	2.68746	0.314544	0.157272	−	0.987555i	$-0.449730\pi$
$73$	2.68746	0.314544	0.157272	+	0.987555i	$0.449730\pi$
$74$	0	0
$75$	3.29443	0.380408
$76$	0	0
$77$	1.85949	0.211909
$78$	0	0
$79$	−11.1295	−1.25217	−0.626086	−	0.779754i	$-0.715344\pi$
$79$	−11.1295	−1.25217	−0.626086	+	0.779754i	$0.715344\pi$
$80$	0	0
$81$	−10.6721	−1.18578
$82$	0	0
$83$	−2.82552	−0.310141	−0.155071	−	0.987903i	$-0.549561\pi$
$83$	−2.82552	−0.310141	−0.155071	+	0.987903i	$0.549561\pi$
$84$	0	0
$85$	−6.09086	−0.660647
$86$	0	0
$87$	−1.34683	−0.144395
$88$	0	0
$89$	1.46828	0.155637	0.0778187	−	0.996968i	$-0.475204\pi$
$89$	1.46828	0.155637	0.0778187	+	0.996968i	$0.475204\pi$
$90$	0	0
$91$	−7.65035	−0.801974
$92$	0	0
$93$	−21.2607	−2.20464
$94$	0	0
$95$	−1.29527	−0.132892
$96$	0	0
$97$	6.88648	0.699216	0.349608	−	0.936896i	$-0.386315\pi$
$97$	6.88648	0.699216	0.349608	+	0.936896i	$0.386315\pi$
$98$	0	0
$99$	0.448217	0.0450475
$100$	0	0
$101$	−5.12523	−0.509979	−0.254990	−	0.966944i	$-0.582072\pi$
$101$	−5.12523	−0.509979	−0.254990	+	0.966944i	$0.582072\pi$
$102$	0	0
$103$	3.11019	0.306456	0.153228	−	0.988191i	$-0.451033\pi$
$103$	3.11019	0.306456	0.153228	+	0.988191i	$0.451033\pi$
$104$	0	0
$105$	10.7758	1.05161
$106$	0	0
$107$	8.97937	0.868068	0.434034	−	0.900896i	$-0.357090\pi$
$107$	8.97937	0.868068	0.434034	+	0.900896i	$0.357090\pi$
$108$	0	0
$109$	18.0481	1.72870	0.864349	−	0.502892i	$-0.167731\pi$
$109$	18.0481	1.72870	0.864349	+	0.502892i	$0.167731\pi$
$110$	0	0
$111$	−6.05347	−0.574570
$112$	0	0
$113$	−15.0257	−1.41350	−0.706750	−	0.707464i	$-0.749839\pi$
$113$	−15.0257	−1.41350	−0.706750	+	0.707464i	$0.749839\pi$
$114$	0	0
$115$	14.4201	1.34468
$116$	0	0
$117$	−1.84406	−0.170483
$118$	0	0
$119$	10.2958	0.943814
$120$	0	0
$121$	−10.6329	−0.966627
$122$	0	0
$123$	14.1620	1.27695
$124$	0	0
$125$	12.1711	1.08861
$126$	0	0
$127$	−5.88695	−0.522382	−0.261191	−	0.965287i	$-0.584115\pi$
$127$	−5.88695	−0.522382	−0.261191	+	0.965287i	$0.584115\pi$
$128$	0	0
$129$	−0.594743	−0.0523642
$130$	0	0
$131$	7.20923	0.629873	0.314937	−	0.949113i	$-0.398017\pi$
$131$	7.20923	0.629873	0.314937	+	0.949113i	$0.398017\pi$
$132$	0	0
$133$	2.18948	0.189852
$134$	0	0
$135$	−7.93592	−0.683015
$136$	0	0
$137$	−1.28189	−0.109519	−0.0547595	−	0.998500i	$-0.517439\pi$
$137$	−1.28189	−0.109519	−0.0547595	+	0.998500i	$0.517439\pi$
$138$	0	0
$139$	7.47851	0.634319	0.317160	−	0.948372i	$-0.397271\pi$
$139$	7.47851	0.634319	0.317160	+	0.948372i	$0.397271\pi$
$140$	0	0
$141$	20.5760	1.73281
$142$	0	0
$143$	−1.51032	−0.126299
$144$	0	0
$145$	−1.26448	−0.105009
$146$	0	0
$147$	−4.67808	−0.385841
$148$	0	0
$149$	−23.7850	−1.94855	−0.974274	−	0.225369i	$-0.927641\pi$
$149$	−23.7850	−1.94855	−0.974274	+	0.225369i	$0.927641\pi$
$150$	0	0
$151$	10.6795	0.869082	0.434541	−	0.900652i	$-0.356911\pi$
$151$	10.6795	0.869082	0.434541	+	0.900652i	$0.356911\pi$
$152$	0	0
$153$	2.48172	0.200636
$154$	0	0
$155$	−19.9608	−1.60329
$156$	0	0
$157$	19.8815	1.58672	0.793358	−	0.608755i	$-0.208331\pi$
$157$	19.8815	1.58672	0.793358	+	0.608755i	$0.208331\pi$
$158$	0	0
$159$	4.17481	0.331084
$160$	0	0
$161$	−24.3753	−1.92104
$162$	0	0
$163$	−10.2724	−0.804594	−0.402297	−	0.915509i	$-0.631788\pi$
$163$	−10.2724	−0.804594	−0.402297	+	0.915509i	$0.631788\pi$
$164$	0	0
$165$	2.12734	0.165613
$166$	0	0
$167$	23.0719	1.78535	0.892677	−	0.450697i	$-0.148824\pi$
$167$	23.0719	1.78535	0.892677	+	0.450697i	$0.148824\pi$
$168$	0	0
$169$	−6.78623	−0.522017
$170$	0	0
$171$	0.527757	0.0403586
$172$	0	0
$173$	−6.95658	−0.528899	−0.264450	−	0.964400i	$-0.585190\pi$
$173$	−6.95658	−0.528899	−0.264450	+	0.964400i	$0.585190\pi$
$174$	0	0
$175$	−5.22832	−0.395224
$176$	0	0
$177$	−5.00796	−0.376422
$178$	0	0
$179$	−20.4719	−1.53014	−0.765072	−	0.643945i	$-0.777297\pi$
$179$	−20.4719	−1.53014	−0.765072	+	0.643945i	$0.777297\pi$
$180$	0	0
$181$	5.17220	0.384447	0.192223	−	0.981351i	$-0.438430\pi$
$181$	5.17220	0.384447	0.192223	+	0.981351i	$0.438430\pi$
$182$	0	0
$183$	7.45488	0.551080
$184$	0	0
$185$	−5.68335	−0.417848
$186$	0	0
$187$	2.03258	0.148637
$188$	0	0
$189$	13.4146	0.975770
$190$	0	0
$191$	−6.87200	−0.497241	−0.248620	−	0.968601i	$-0.579977\pi$
$191$	−6.87200	−0.497241	−0.248620	+	0.968601i	$0.579977\pi$
$192$	0	0
$193$	−8.77305	−0.631498	−0.315749	−	0.948843i	$-0.602256\pi$
$193$	−8.77305	−0.631498	−0.315749	+	0.948843i	$0.602256\pi$
$194$	0	0
$195$	−8.75231	−0.626766
$196$	0	0
$197$	22.5490	1.60655	0.803273	−	0.595611i	$-0.203090\pi$
$197$	22.5490	1.60655	0.803273	+	0.595611i	$0.203090\pi$
$198$	0	0
$199$	−19.7804	−1.40220	−0.701098	−	0.713065i	$-0.747306\pi$
$199$	−19.7804	−1.40220	−0.701098	+	0.713065i	$0.747306\pi$
$200$	0	0
$201$	−24.2560	−1.71089
$202$	0	0
$203$	2.13744	0.150019
$204$	0	0
$205$	13.2961	0.928643
$206$	0	0
$207$	−5.87548	−0.408374
$208$	0	0
$209$	0.432243	0.0298989
$210$	0	0
$211$	16.6389	1.14547	0.572733	−	0.819742i	$-0.305883\pi$
$211$	16.6389	1.14547	0.572733	+	0.819742i	$0.305883\pi$
$212$	0	0
$213$	−15.4333	−1.05747
$214$	0	0
$215$	−0.558379	−0.0380811
$216$	0	0
$217$	33.7411	2.29050
$218$	0	0
$219$	−5.19715	−0.351191
$220$	0	0
$221$	−8.36246	−0.562520
$222$	0	0
$223$	−0.277749	−0.0185995	−0.00929973	−	0.999957i	$-0.502960\pi$
$223$	−0.277749	−0.0185995	−0.00929973	+	0.999957i	$0.502960\pi$
$224$	0	0
$225$	−1.26025	−0.0840164
$226$	0	0
$227$	−3.34402	−0.221950	−0.110975	−	0.993823i	$-0.535397\pi$
$227$	−3.34402	−0.221950	−0.110975	+	0.993823i	$0.535397\pi$
$228$	0	0
$229$	−22.1703	−1.46505	−0.732526	−	0.680739i	$-0.761659\pi$
$229$	−22.1703	−1.46505	−0.732526	+	0.680739i	$0.761659\pi$
$230$	0	0
$231$	−3.59598	−0.236598
$232$	0	0
$233$	14.2888	0.936092	0.468046	−	0.883704i	$-0.344958\pi$
$233$	14.2888	0.936092	0.468046	+	0.883704i	$0.344958\pi$
$234$	0	0
$235$	19.3179	1.26016
$236$	0	0
$237$	21.5228	1.39806
$238$	0	0
$239$	−1.06528	−0.0689071	−0.0344535	−	0.999406i	$-0.510969\pi$
$239$	−1.06528	−0.0689071	−0.0344535	+	0.999406i	$0.510969\pi$
$240$	0	0
$241$	16.3056	1.05034	0.525168	−	0.850999i	$-0.324003\pi$
$241$	16.3056	1.05034	0.525168	+	0.850999i	$0.324003\pi$
$242$	0	0
$243$	7.52531	0.482749
$244$	0	0
$245$	−4.39205	−0.280598
$246$	0	0
$247$	−1.77834	−0.113153
$248$	0	0
$249$	5.46413	0.346275
$250$	0	0
$251$	14.2129	0.897113	0.448556	−	0.893755i	$-0.351938\pi$
$251$	14.2129	0.897113	0.448556	+	0.893755i	$0.351938\pi$
$252$	0	0
$253$	−4.81214	−0.302536
$254$	0	0
$255$	11.7788	0.737618
$256$	0	0
$257$	−0.0308301	−0.00192313	−0.000961566	−	1.00000i	$-0.500306\pi$
$257$	−0.0308301	−0.00192313	−0.000961566		1.00000i	$0.500306\pi$
$258$	0	0
$259$	9.60695	0.596947
$260$	0	0
$261$	0.515213	0.0318909
$262$	0	0
$263$	1.53505	0.0946554	0.0473277	−	0.998879i	$-0.484930\pi$
$263$	1.53505	0.0946554	0.0473277	+	0.998879i	$0.484930\pi$
$264$	0	0
$265$	3.91955	0.240776
$266$	0	0
$267$	−2.83943	−0.173770
$268$	0	0
$269$	8.25567	0.503357	0.251678	−	0.967811i	$-0.419017\pi$
$269$	8.25567	0.503357	0.251678	+	0.967811i	$0.419017\pi$
$270$	0	0
$271$	−22.2934	−1.35423	−0.677115	−	0.735877i	$-0.736770\pi$
$271$	−22.2934	−1.35423	−0.677115	+	0.735877i	$0.736770\pi$
$272$	0	0
$273$	14.7946	0.895411
$274$	0	0
$275$	−1.03217	−0.0622420
$276$	0	0
$277$	20.3098	1.22030	0.610150	−	0.792286i	$-0.291109\pi$
$277$	20.3098	1.22030	0.610150	+	0.792286i	$0.291109\pi$
$278$	0	0
$279$	8.13304	0.486913
$280$	0	0
$281$	−3.05651	−0.182336	−0.0911681	−	0.995836i	$-0.529060\pi$
$281$	−3.05651	−0.182336	−0.0911681	+	0.995836i	$0.529060\pi$
$282$	0	0
$283$	−26.7086	−1.58766	−0.793829	−	0.608140i	$-0.791916\pi$
$283$	−26.7086	−1.58766	−0.793829	+	0.608140i	$0.791916\pi$
$284$	0	0
$285$	2.50485	0.148375
$286$	0	0
$287$	−22.4754	−1.32668
$288$	0	0
$289$	−5.74584	−0.337991
$290$	0	0
$291$	−13.3174	−0.780680
$292$	0	0
$293$	−6.21918	−0.363329	−0.181664	−	0.983361i	$-0.558148\pi$
$293$	−6.21918	−0.363329	−0.181664	+	0.983361i	$0.558148\pi$
$294$	0	0
$295$	−4.70177	−0.273747
$296$	0	0
$297$	2.64829	0.153670
$298$	0	0
$299$	19.7981	1.14496
$300$	0	0
$301$	0.943866	0.0544035
$302$	0	0
$303$	9.91142	0.569396
$304$	0	0
$305$	6.99907	0.400766
$306$	0	0
$307$	−3.04422	−0.173743	−0.0868713	−	0.996220i	$-0.527687\pi$
$307$	−3.04422	−0.173743	−0.0868713	+	0.996220i	$0.527687\pi$
$308$	0	0
$309$	−6.01463	−0.342160
$310$	0	0
$311$	−16.1687	−0.916841	−0.458421	−	0.888735i	$-0.651585\pi$
$311$	−16.1687	−0.916841	−0.458421	+	0.888735i	$0.651585\pi$
$312$	0	0
$313$	−12.7994	−0.723466	−0.361733	−	0.932282i	$-0.617815\pi$
$313$	−12.7994	−0.723466	−0.361733	+	0.932282i	$0.617815\pi$
$314$	0	0
$315$	−4.12214	−0.232256
$316$	0	0
$317$	−1.54449	−0.0867469	−0.0433735	−	0.999059i	$-0.513811\pi$
$317$	−1.54449	−0.0867469	−0.0433735	+	0.999059i	$0.513811\pi$
$318$	0	0
$319$	0.421969	0.0236257
$320$	0	0
$321$	−17.3647	−0.969205
$322$	0	0
$323$	2.39328	0.133166
$324$	0	0
$325$	4.24655	0.235556
$326$	0	0
$327$	−34.9024	−1.93011
$328$	0	0
$329$	−32.6544	−1.80029
$330$	0	0
$331$	−31.2279	−1.71644	−0.858219	−	0.513283i	$-0.828429\pi$
$331$	−31.2279	−1.71644	−0.858219	+	0.513283i	$0.828429\pi$
$332$	0	0
$333$	2.31568	0.126899
$334$	0	0
$335$	−22.7729	−1.24422
$336$	0	0
$337$	−10.8911	−0.593276	−0.296638	−	0.954990i	$-0.595866\pi$
$337$	−10.8911	−0.593276	−0.296638	+	0.954990i	$0.595866\pi$
$338$	0	0
$339$	29.0574	1.57818
$340$	0	0
$341$	6.66112	0.360720
$342$	0	0
$343$	−14.0591	−0.759123
$344$	0	0
$345$	−27.8863	−1.50135
$346$	0	0
$347$	−9.29342	−0.498897	−0.249448	−	0.968388i	$-0.580249\pi$
$347$	−9.29342	−0.498897	−0.249448	+	0.968388i	$0.580249\pi$
$348$	0	0
$349$	−10.9538	−0.586342	−0.293171	−	0.956060i	$-0.594710\pi$
$349$	−10.9538	−0.586342	−0.293171	+	0.956060i	$0.594710\pi$
$350$	0	0
$351$	−10.8956	−0.581566
$352$	0	0
$353$	−34.0402	−1.81178	−0.905888	−	0.423517i	$-0.860795\pi$
$353$	−34.0402	−1.81178	−0.905888	+	0.423517i	$0.860795\pi$
$354$	0	0
$355$	−14.4896	−0.769031
$356$	0	0
$357$	−19.9105	−1.05378
$358$	0	0
$359$	−14.2564	−0.752426	−0.376213	−	0.926533i	$-0.622774\pi$
$359$	−14.2564	−0.752426	−0.376213	+	0.926533i	$0.622774\pi$
$360$	0	0
$361$	−18.4911	−0.973213
$362$	0	0
$363$	20.5624	1.07925
$364$	0	0
$365$	−4.87938	−0.255399
$366$	0	0
$367$	−24.6179	−1.28505	−0.642523	−	0.766266i	$-0.722112\pi$
$367$	−24.6179	−1.28505	−0.642523	+	0.766266i	$0.722112\pi$
$368$	0	0
$369$	−5.41752	−0.282025
$370$	0	0
$371$	−6.62549	−0.343978
$372$	0	0
$373$	28.6244	1.48212	0.741059	−	0.671440i	$-0.234324\pi$
$373$	28.6244	1.48212	0.741059	+	0.671440i	$0.234324\pi$
$374$	0	0
$375$	−23.5370	−1.21544
$376$	0	0
$377$	−1.73607	−0.0894122
$378$	0	0
$379$	−13.1426	−0.675088	−0.337544	−	0.941310i	$-0.609596\pi$
$379$	−13.1426	−0.675088	−0.337544	+	0.941310i	$0.609596\pi$
$380$	0	0
$381$	11.3845	0.583244
$382$	0	0
$383$	−9.31215	−0.475829	−0.237914	−	0.971286i	$-0.576464\pi$
$383$	−9.31215	−0.475829	−0.237914	+	0.971286i	$0.576464\pi$
$384$	0	0
$385$	−3.37612	−0.172063
$386$	0	0
$387$	0.227512	0.0115651
$388$	0	0
$389$	−16.6417	−0.843769	−0.421884	−	0.906650i	$-0.638631\pi$
$389$	−16.6417	−0.843769	−0.421884	+	0.906650i	$0.638631\pi$
$390$	0	0
$391$	−26.6442	−1.34746
$392$	0	0
$393$	−13.9416	−0.703258
$394$	0	0
$395$	20.2069	1.01672
$396$	0	0
$397$	−6.32093	−0.317238	−0.158619	−	0.987340i	$-0.550704\pi$
$397$	−6.32093	−0.317238	−0.158619	+	0.987340i	$0.550704\pi$
$398$	0	0
$399$	−4.23412	−0.211971
$400$	0	0
$401$	37.1133	1.85335	0.926675	−	0.375865i	$-0.122654\pi$
$401$	37.1133	1.85335	0.926675	+	0.375865i	$0.122654\pi$
$402$	0	0
$403$	−27.4052	−1.36515
$404$	0	0
$405$	19.3763	0.962815
$406$	0	0
$407$	1.89659	0.0940104
$408$	0	0
$409$	5.80676	0.287126	0.143563	−	0.989641i	$-0.454144\pi$
$409$	5.80676	0.287126	0.143563	+	0.989641i	$0.454144\pi$
$410$	0	0
$411$	2.47898	0.122279
$412$	0	0
$413$	7.94771	0.391081
$414$	0	0
$415$	5.13004	0.251824
$416$	0	0
$417$	−14.4623	−0.708223
$418$	0	0
$419$	7.35657	0.359392	0.179696	−	0.983722i	$-0.442489\pi$
$419$	7.35657	0.359392	0.179696	+	0.983722i	$0.442489\pi$
$420$	0	0
$421$	7.49907	0.365482	0.182741	−	0.983161i	$-0.441503\pi$
$421$	7.49907	0.365482	0.182741	+	0.983161i	$0.441503\pi$
$422$	0	0
$423$	−7.87109	−0.382706
$424$	0	0
$425$	−5.71498	−0.277217
$426$	0	0
$427$	−11.8310	−0.572542
$428$	0	0
$429$	2.92073	0.141014
$430$	0	0
$431$	5.12951	0.247080	0.123540	−	0.992340i	$-0.460575\pi$
$431$	5.12951	0.247080	0.123540	+	0.992340i	$0.460575\pi$
$432$	0	0
$433$	−22.0530	−1.05980	−0.529899	−	0.848061i	$-0.677770\pi$
$433$	−22.0530	−1.05980	−0.529899	+	0.848061i	$0.677770\pi$
$434$	0	0
$435$	2.44531	0.117244
$436$	0	0
$437$	−5.66609	−0.271046
$438$	0	0
$439$	−8.73650	−0.416971	−0.208485	−	0.978025i	$-0.566853\pi$
$439$	−8.73650	−0.416971	−0.208485	+	0.978025i	$0.566853\pi$
$440$	0	0
$441$	1.78954	0.0852163
$442$	0	0
$443$	20.6225	0.979804	0.489902	−	0.871778i	$-0.337033\pi$
$443$	20.6225	0.979804	0.489902	+	0.871778i	$0.337033\pi$
$444$	0	0
$445$	−2.66582	−0.126372
$446$	0	0
$447$	45.9967	2.17557
$448$	0	0
$449$	9.88271	0.466394	0.233197	−	0.972430i	$-0.425081\pi$
$449$	9.88271	0.466394	0.233197	+	0.972430i	$0.425081\pi$
$450$	0	0
$451$	−4.43706	−0.208933
$452$	0	0
$453$	−20.6524	−0.970337
$454$	0	0
$455$	13.8900	0.651175
$456$	0	0
$457$	−2.54949	−0.119260	−0.0596302	−	0.998221i	$-0.518992\pi$
$457$	−2.54949	−0.119260	−0.0596302	+	0.998221i	$0.518992\pi$
$458$	0	0
$459$	14.6633	0.684424
$460$	0	0
$461$	−16.5012	−0.768536	−0.384268	−	0.923222i	$-0.625546\pi$
$461$	−16.5012	−0.768536	−0.384268	+	0.923222i	$0.625546\pi$
$462$	0	0
$463$	8.82899	0.410318	0.205159	−	0.978729i	$-0.434229\pi$
$463$	8.82899	0.410318	0.205159	+	0.978729i	$0.434229\pi$
$464$	0	0
$465$	38.6012	1.79009
$466$	0	0
$467$	−25.3956	−1.17517	−0.587585	−	0.809162i	$-0.699921\pi$
$467$	−25.3956	−1.17517	−0.587585	+	0.809162i	$0.699921\pi$
$468$	0	0
$469$	38.4947	1.77752
$470$	0	0
$471$	−38.4478	−1.77158
$472$	0	0
$473$	0.186336	0.00856776
$474$	0	0
$475$	−1.21533	−0.0557633
$476$	0	0
$477$	−1.59702	−0.0731227
$478$	0	0
$479$	−6.07633	−0.277635	−0.138817	−	0.990318i	$-0.544330\pi$
$479$	−6.07633	−0.277635	−0.138817	+	0.990318i	$0.544330\pi$
$480$	0	0
$481$	−7.80296	−0.355785
$482$	0	0
$483$	47.1382	2.14486
$484$	0	0
$485$	−12.5032	−0.567739
$486$	0	0
$487$	−15.2106	−0.689259	−0.344630	−	0.938739i	$-0.611996\pi$
$487$	−15.2106	−0.689259	−0.344630	+	0.938739i	$0.611996\pi$
$488$	0	0
$489$	19.8652	0.898336
$490$	0	0
$491$	−16.3062	−0.735888	−0.367944	−	0.929848i	$-0.619938\pi$
$491$	−16.3062	−0.735888	−0.367944	+	0.929848i	$0.619938\pi$
$492$	0	0
$493$	2.33639	0.105226
$494$	0	0
$495$	−0.813787	−0.0365770
$496$	0	0
$497$	24.4928	1.09865
$498$	0	0
$499$	−2.89538	−0.129615	−0.0648075	−	0.997898i	$-0.520643\pi$
$499$	−2.89538	−0.129615	−0.0648075	+	0.997898i	$0.520643\pi$
$500$	0	0
$501$	−44.6175	−1.99336
$502$	0	0
$503$	−18.2973	−0.815835	−0.407918	−	0.913019i	$-0.633745\pi$
$503$	−18.2973	−0.815835	−0.407918	+	0.913019i	$0.633745\pi$
$504$	0	0
$505$	9.30541	0.414085
$506$	0	0
$507$	13.1235	0.582837
$508$	0	0
$509$	8.06258	0.357368	0.178684	−	0.983907i	$-0.442816\pi$
$509$	8.06258	0.357368	0.178684	+	0.983907i	$0.442816\pi$
$510$	0	0
$511$	8.24795	0.364868
$512$	0	0
$513$	3.11826	0.137674
$514$	0	0
$515$	−5.64688	−0.248831
$516$	0	0
$517$	−6.44658	−0.283520
$518$	0	0
$519$	13.4530	0.590520
$520$	0	0
$521$	2.40565	0.105394	0.0526968	−	0.998611i	$-0.483218\pi$
$521$	2.40565	0.105394	0.0526968	+	0.998611i	$0.483218\pi$
$522$	0	0
$523$	−35.5662	−1.55520	−0.777601	−	0.628758i	$-0.783564\pi$
$523$	−35.5662	−1.55520	−0.777601	+	0.628758i	$0.783564\pi$
$524$	0	0
$525$	10.1108	0.441270
$526$	0	0
$527$	36.8818	1.60660
$528$	0	0
$529$	40.0802	1.74262
$530$	0	0
$531$	1.91574	0.0831359
$532$	0	0
$533$	18.2550	0.790711
$534$	0	0
$535$	−16.3030	−0.704841
$536$	0	0
$537$	39.5896	1.70842
$538$	0	0
$539$	1.46567	0.0631309
$540$	0	0
$541$	−45.1008	−1.93904	−0.969518	−	0.245019i	$-0.921206\pi$
$541$	−45.1008	−1.93904	−0.969518	+	0.245019i	$0.921206\pi$
$542$	0	0
$543$	−10.0023	−0.429238
$544$	0	0
$545$	−32.7684	−1.40364
$546$	0	0
$547$	−19.2655	−0.823731	−0.411866	−	0.911245i	$-0.635123\pi$
$547$	−19.2655	−0.823731	−0.411866	+	0.911245i	$0.635123\pi$
$548$	0	0
$549$	−2.85177	−0.121711
$550$	0	0
$551$	0.496851	0.0211666
$552$	0	0
$553$	−34.1571	−1.45251
$554$	0	0
$555$	10.9907	0.466531
$556$	0	0
$557$	−27.0472	−1.14603	−0.573013	−	0.819546i	$-0.694226\pi$
$557$	−27.0472	−1.14603	−0.573013	+	0.819546i	$0.694226\pi$
$558$	0	0
$559$	−0.766627	−0.0324249
$560$	0	0
$561$	−3.93070	−0.165954
$562$	0	0
$563$	−38.6632	−1.62946	−0.814730	−	0.579840i	$-0.803115\pi$
$563$	−38.6632	−1.62946	−0.814730	+	0.579840i	$0.803115\pi$
$564$	0	0
$565$	27.2808	1.14771
$566$	0	0
$567$	−32.7530	−1.37550
$568$	0	0
$569$	−34.7039	−1.45486	−0.727431	−	0.686181i	$-0.759286\pi$
$569$	−34.7039	−1.45486	−0.727431	+	0.686181i	$0.759286\pi$
$570$	0	0
$571$	−19.5680	−0.818896	−0.409448	−	0.912334i	$-0.634279\pi$
$571$	−19.5680	−0.818896	−0.409448	+	0.912334i	$0.634279\pi$
$572$	0	0
$573$	13.2894	0.555173
$574$	0	0
$575$	13.5302	0.564249
$576$	0	0
$577$	18.2763	0.760851	0.380426	−	0.924812i	$-0.375778\pi$
$577$	18.2763	0.760851	0.380426	+	0.924812i	$0.375778\pi$
$578$	0	0
$579$	16.9657	0.705072
$580$	0	0
$581$	−8.67166	−0.359761
$582$	0	0
$583$	−1.30799	−0.0541716
$584$	0	0
$585$	3.34809	0.138426
$586$	0	0
$587$	−39.1023	−1.61393	−0.806963	−	0.590602i	$-0.798890\pi$
$587$	−39.1023	−1.61393	−0.806963	+	0.590602i	$0.798890\pi$
$588$	0	0
$589$	7.84320	0.323173
$590$	0	0
$591$	−43.6063	−1.79372
$592$	0	0
$593$	−36.4256	−1.49582	−0.747909	−	0.663801i	$-0.768942\pi$
$593$	−36.4256	−1.49582	−0.747909	+	0.663801i	$0.768942\pi$
$594$	0	0
$595$	−18.6931	−0.766344
$596$	0	0
$597$	38.2523	1.56556
$598$	0	0
$599$	28.4672	1.16314	0.581569	−	0.813497i	$-0.302439\pi$
$599$	28.4672	1.16314	0.581569	+	0.813497i	$0.302439\pi$
$600$	0	0
$601$	1.70899	0.0697112	0.0348556	−	0.999392i	$-0.488903\pi$
$601$	1.70899	0.0697112	0.0348556	+	0.999392i	$0.488903\pi$
$602$	0	0
$603$	9.27884	0.377864
$604$	0	0
$605$	19.3052	0.784868
$606$	0	0
$607$	−30.1955	−1.22560	−0.612799	−	0.790239i	$-0.709956\pi$
$607$	−30.1955	−1.22560	−0.612799	+	0.790239i	$0.709956\pi$
$608$	0	0
$609$	−4.13348	−0.167497
$610$	0	0
$611$	26.5226	1.07299
$612$	0	0
$613$	3.37824	0.136446	0.0682228	−	0.997670i	$-0.478267\pi$
$613$	3.37824	0.136446	0.0682228	+	0.997670i	$0.478267\pi$
$614$	0	0
$615$	−25.7127	−1.03684
$616$	0	0
$617$	−40.8561	−1.64481	−0.822403	−	0.568906i	$-0.807367\pi$
$617$	−40.8561	−1.64481	−0.822403	+	0.568906i	$0.807367\pi$
$618$	0	0
$619$	19.6657	0.790432	0.395216	−	0.918588i	$-0.370670\pi$
$619$	19.6657	0.790432	0.395216	+	0.918588i	$0.370670\pi$
$620$	0	0
$621$	−34.7154	−1.39308
$622$	0	0
$623$	4.50622	0.180538
$624$	0	0
$625$	−13.5801	−0.543202
$626$	0	0
$627$	−0.835893	−0.0333824
$628$	0	0
$629$	10.5012	0.418710
$630$	0	0
$631$	−7.55587	−0.300794	−0.150397	−	0.988626i	$-0.548055\pi$
$631$	−7.55587	−0.300794	−0.150397	+	0.988626i	$0.548055\pi$
$632$	0	0
$633$	−32.1770	−1.27892
$634$	0	0
$635$	10.6884	0.424156
$636$	0	0
$637$	−6.03008	−0.238920
$638$	0	0
$639$	5.90381	0.233551
$640$	0	0
$641$	30.0952	1.18869	0.594344	−	0.804211i	$-0.297412\pi$
$641$	30.0952	1.18869	0.594344	+	0.804211i	$0.297412\pi$
$642$	0	0
$643$	49.4931	1.95182	0.975908	−	0.218181i	$-0.0700125\pi$
$643$	49.4931	1.95182	0.975908	+	0.218181i	$0.0700125\pi$
$644$	0	0
$645$	1.07982	0.0425179
$646$	0	0
$647$	−21.4854	−0.844679	−0.422340	−	0.906438i	$-0.638791\pi$
$647$	−21.4854	−0.844679	−0.422340	+	0.906438i	$0.638791\pi$
$648$	0	0
$649$	1.56903	0.0615896
$650$	0	0
$651$	−65.2502	−2.55736
$652$	0	0
$653$	30.0367	1.17543	0.587713	−	0.809069i	$-0.300028\pi$
$653$	30.0367	1.17543	0.587713	+	0.809069i	$0.300028\pi$
$654$	0	0
$655$	−13.0891	−0.511435
$656$	0	0
$657$	1.98811	0.0775634
$658$	0	0
$659$	−27.0971	−1.05555	−0.527777	−	0.849383i	$-0.676974\pi$
$659$	−27.0971	−1.05555	−0.527777	+	0.849383i	$0.676974\pi$
$660$	0	0
$661$	33.5122	1.30347	0.651737	−	0.758445i	$-0.274041\pi$
$661$	33.5122	1.30347	0.651737	+	0.758445i	$0.274041\pi$
$662$	0	0
$663$	16.1717	0.628058
$664$	0	0
$665$	−3.97524	−0.154153
$666$	0	0
$667$	−5.53141	−0.214177
$668$	0	0
$669$	0.537125	0.0207665
$670$	0	0
$671$	−2.33566	−0.0901671
$672$	0	0
$673$	36.0199	1.38847	0.694233	−	0.719750i	$-0.255744\pi$
$673$	36.0199	1.38847	0.694233	+	0.719750i	$0.255744\pi$
$674$	0	0
$675$	−7.44618	−0.286603
$676$	0	0
$677$	3.62680	0.139389	0.0696946	−	0.997568i	$-0.477798\pi$
$677$	3.62680	0.139389	0.0696946	+	0.997568i	$0.477798\pi$
$678$	0	0
$679$	21.1349	0.811084
$680$	0	0
$681$	6.46682	0.247809
$682$	0	0
$683$	−6.00867	−0.229915	−0.114958	−	0.993370i	$-0.536673\pi$
$683$	−6.00867	−0.229915	−0.114958	+	0.993370i	$0.536673\pi$
$684$	0	0
$685$	2.32741	0.0889256
$686$	0	0
$687$	42.8739	1.63574
$688$	0	0
$689$	5.38136	0.205013
$690$	0	0
$691$	−5.35871	−0.203855	−0.101927	−	0.994792i	$-0.532501\pi$
$691$	−5.35871	−0.203855	−0.101927	+	0.994792i	$0.532501\pi$
$692$	0	0
$693$	1.37560	0.0522547
$694$	0	0
$695$	−13.5781	−0.515045
$696$	0	0
$697$	−24.5674	−0.930558
$698$	0	0
$699$	−27.6324	−1.04515
$700$	0	0
$701$	21.3590	0.806719	0.403359	−	0.915042i	$-0.367842\pi$
$701$	21.3590	0.806719	0.403359	+	0.915042i	$0.367842\pi$
$702$	0	0
$703$	2.23316	0.0842251
$704$	0	0
$705$	−37.3579	−1.40698
$706$	0	0
$707$	−15.7296	−0.591571
$708$	0	0
$709$	8.39590	0.315315	0.157657	−	0.987494i	$-0.449606\pi$
$709$	8.39590	0.315315	0.157657	+	0.987494i	$0.449606\pi$
$710$	0	0
$711$	−8.23331	−0.308773
$712$	0	0
$713$	−87.3178	−3.27008
$714$	0	0
$715$	2.74215	0.102551
$716$	0	0
$717$	2.06009	0.0769353
$718$	0	0
$719$	4.05532	0.151238	0.0756190	−	0.997137i	$-0.475907\pi$
$719$	4.05532	0.151238	0.0756190	+	0.997137i	$0.475907\pi$
$720$	0	0
$721$	9.54531	0.355486
$722$	0	0
$723$	−31.5325	−1.17271
$724$	0	0
$725$	−1.18645	−0.0440635
$726$	0	0
$727$	−16.8767	−0.625924	−0.312962	−	0.949766i	$-0.601321\pi$
$727$	−16.8767	−0.625924	−0.312962	+	0.949766i	$0.601321\pi$
$728$	0	0
$729$	17.4633	0.646791
$730$	0	0
$731$	1.03172	0.0381597
$732$	0	0
$733$	−3.91939	−0.144766	−0.0723829	−	0.997377i	$-0.523060\pi$
$733$	−3.91939	−0.144766	−0.0723829	+	0.997377i	$0.523060\pi$
$734$	0	0
$735$	8.49356	0.313290
$736$	0	0
$737$	7.59956	0.279933
$738$	0	0
$739$	26.4049	0.971318	0.485659	−	0.874148i	$-0.338580\pi$
$739$	26.4049	0.971318	0.485659	+	0.874148i	$0.338580\pi$
$740$	0	0
$741$	3.43904	0.126336
$742$	0	0
$743$	−4.97836	−0.182638	−0.0913192	−	0.995822i	$-0.529108\pi$
$743$	−4.97836	−0.182638	−0.0913192	+	0.995822i	$0.529108\pi$
$744$	0	0
$745$	43.1844	1.58215
$746$	0	0
$747$	−2.09024	−0.0764778
$748$	0	0
$749$	27.5581	1.00695
$750$	0	0
$751$	25.4939	0.930286	0.465143	−	0.885235i	$-0.346003\pi$
$751$	25.4939	0.930286	0.465143	+	0.885235i	$0.346003\pi$
$752$	0	0
$753$	−27.4857	−1.00163
$754$	0	0
$755$	−19.3897	−0.705664
$756$	0	0
$757$	44.4752	1.61648	0.808239	−	0.588854i	$-0.200421\pi$
$757$	44.4752	1.61648	0.808239	+	0.588854i	$0.200421\pi$
$758$	0	0
$759$	9.30594	0.337784
$760$	0	0
$761$	−35.5381	−1.28826	−0.644128	−	0.764918i	$-0.722780\pi$
$761$	−35.5381	−1.28826	−0.644128	+	0.764918i	$0.722780\pi$
$762$	0	0
$763$	55.3906	2.00527
$764$	0	0
$765$	−4.50584	−0.162909
$766$	0	0
$767$	−6.45530	−0.233087
$768$	0	0
$769$	−36.5519	−1.31809	−0.659047	−	0.752102i	$-0.729040\pi$
$769$	−36.5519	−1.31809	−0.659047	+	0.752102i	$0.729040\pi$
$770$	0	0
$771$	0.0596208	0.00214719
$772$	0	0
$773$	−7.61294	−0.273818	−0.136909	−	0.990584i	$-0.543717\pi$
$773$	−7.61294	−0.273818	−0.136909	+	0.990584i	$0.543717\pi$
$774$	0	0
$775$	−18.7290	−0.672765
$776$	0	0
$777$	−18.5784	−0.666496
$778$	0	0
$779$	−5.22445	−0.187185
$780$	0	0
$781$	4.83534	0.173022
$782$	0	0
$783$	3.04414	0.108789
$784$	0	0
$785$	−36.0970	−1.28836
$786$	0	0
$787$	−27.7777	−0.990169	−0.495085	−	0.868845i	$-0.664863\pi$
$787$	−27.7777	−0.990169	−0.495085	+	0.868845i	$0.664863\pi$
$788$	0	0
$789$	−2.96856	−0.105683
$790$	0	0
$791$	−46.1146	−1.63965
$792$	0	0
$793$	9.60938	0.341239
$794$	0	0
$795$	−7.57982	−0.268829
$796$	0	0
$797$	12.7822	0.452769	0.226384	−	0.974038i	$-0.427309\pi$
$797$	12.7822	0.452769	0.226384	+	0.974038i	$0.427309\pi$
$798$	0	0
$799$	−35.6939	−1.26276
$800$	0	0
$801$	1.08619	0.0383786
$802$	0	0
$803$	1.62830	0.0574614
$804$	0	0
$805$	44.2560	1.55982
$806$	0	0
$807$	−15.9652	−0.562002
$808$	0	0
$809$	−39.9525	−1.40465	−0.702327	−	0.711854i	$-0.747856\pi$
$809$	−39.9525	−1.40465	−0.702327	+	0.711854i	$0.747856\pi$
$810$	0	0
$811$	−14.3604	−0.504261	−0.252131	−	0.967693i	$-0.581131\pi$
$811$	−14.3604	−0.504261	−0.252131	+	0.967693i	$0.581131\pi$
$812$	0	0
$813$	43.1121	1.51201
$814$	0	0
$815$	18.6506	0.653302
$816$	0	0
$817$	0.219403	0.00767596
$818$	0	0
$819$	−5.65950	−0.197759
$820$	0	0
$821$	−45.0204	−1.57122	−0.785611	−	0.618721i	$-0.787651\pi$
$821$	−45.0204	−1.57122	−0.785611	+	0.618721i	$0.787651\pi$
$822$	0	0
$823$	−3.38750	−0.118081	−0.0590404	−	0.998256i	$-0.518804\pi$
$823$	−3.38750	−0.118081	−0.0590404	+	0.998256i	$0.518804\pi$
$824$	0	0
$825$	1.99605	0.0694937
$826$	0	0
$827$	9.03366	0.314131	0.157066	−	0.987588i	$-0.449797\pi$
$827$	9.03366	0.314131	0.157066	+	0.987588i	$0.449797\pi$
$828$	0	0
$829$	6.51735	0.226357	0.113178	−	0.993575i	$-0.463897\pi$
$829$	6.51735	0.226357	0.113178	+	0.993575i	$0.463897\pi$
$830$	0	0
$831$	−39.2761	−1.36247
$832$	0	0
$833$	8.11524	0.281177
$834$	0	0
$835$	−41.8895	−1.44965
$836$	0	0
$837$	48.0542	1.66099
$838$	0	0
$839$	−28.5341	−0.985107	−0.492553	−	0.870282i	$-0.663936\pi$
$839$	−28.5341	−0.985107	−0.492553	+	0.870282i	$0.663936\pi$
$840$	0	0
$841$	−28.5150	−0.983274
$842$	0	0
$843$	5.91083	0.203580
$844$	0	0
$845$	12.3211	0.423860
$846$	0	0
$847$	−32.6329	−1.12128
$848$	0	0
$849$	51.6503	1.77263
$850$	0	0
$851$	−24.8616	−0.852244
$852$	0	0
$853$	17.3921	0.595495	0.297748	−	0.954645i	$-0.403765\pi$
$853$	17.3921	0.595495	0.297748	+	0.954645i	$0.403765\pi$
$854$	0	0
$855$	−0.958201	−0.0327698
$856$	0	0
$857$	−17.9813	−0.614230	−0.307115	−	0.951672i	$-0.599364\pi$
$857$	−17.9813	−0.614230	−0.307115	+	0.951672i	$0.599364\pi$
$858$	0	0
$859$	−8.05434	−0.274810	−0.137405	−	0.990515i	$-0.543876\pi$
$859$	−8.05434	−0.274810	−0.137405	+	0.990515i	$0.543876\pi$
$860$	0	0
$861$	43.4640	1.48125
$862$	0	0
$863$	15.2078	0.517680	0.258840	−	0.965920i	$-0.416660\pi$
$863$	15.2078	0.517680	0.258840	+	0.965920i	$0.416660\pi$
$864$	0	0
$865$	12.6304	0.429448
$866$	0	0
$867$	11.1116	0.377369
$868$	0	0
$869$	−6.74324	−0.228749
$870$	0	0
$871$	−31.2662	−1.05941
$872$	0	0
$873$	5.09441	0.172420
$874$	0	0
$875$	37.3535	1.26278
$876$	0	0
$877$	19.5740	0.660967	0.330484	−	0.943812i	$-0.392788\pi$
$877$	19.5740	0.660967	0.330484	+	0.943812i	$0.392788\pi$
$878$	0	0
$879$	12.0270	0.405659
$880$	0	0
$881$	39.4726	1.32986	0.664932	−	0.746904i	$-0.268460\pi$
$881$	39.4726	1.32986	0.664932	+	0.746904i	$0.268460\pi$
$882$	0	0
$883$	−35.3214	−1.18866	−0.594329	−	0.804222i	$-0.702582\pi$
$883$	−35.3214	−1.18866	−0.594329	+	0.804222i	$0.702582\pi$
$884$	0	0
$885$	9.09250	0.305641
$886$	0	0
$887$	−35.2980	−1.18519	−0.592596	−	0.805500i	$-0.701897\pi$
$887$	−35.2980	−1.18519	−0.592596	+	0.805500i	$0.701897\pi$
$888$	0	0
$889$	−18.0673	−0.605958
$890$	0	0
$891$	−6.46605	−0.216621
$892$	0	0
$893$	−7.59058	−0.254009
$894$	0	0
$895$	37.1690	1.24242
$896$	0	0
$897$	−38.2866	−1.27835
$898$	0	0
$899$	7.65677	0.255368
$900$	0	0
$901$	−7.24220	−0.241273
$902$	0	0
$903$	−1.82529	−0.0607420
$904$	0	0
$905$	−9.39069	−0.312157
$906$	0	0
$907$	54.3717	1.80538	0.902692	−	0.430287i	$-0.141588\pi$
$907$	54.3717	1.80538	0.902692	+	0.430287i	$0.141588\pi$
$908$	0	0
$909$	−3.79149	−0.125756
$910$	0	0
$911$	−35.8668	−1.18832	−0.594160	−	0.804347i	$-0.702515\pi$
$911$	−35.8668	−1.18832	−0.594160	+	0.804347i	$0.702515\pi$
$912$	0	0
$913$	−1.71195	−0.0566571
$914$	0	0
$915$	−13.5351	−0.447458
$916$	0	0
$917$	22.1255	0.730647
$918$	0	0
$919$	28.1104	0.927276	0.463638	−	0.886025i	$-0.346544\pi$
$919$	28.1104	0.927276	0.463638	+	0.886025i	$0.346544\pi$
$920$	0	0
$921$	5.88706	0.193985
$922$	0	0
$923$	−19.8936	−0.654805
$924$	0	0
$925$	−5.33262	−0.175335
$926$	0	0
$927$	2.30082	0.0755690
$928$	0	0
$929$	−8.44598	−0.277104	−0.138552	−	0.990355i	$-0.544245\pi$
$929$	−8.44598	−0.277104	−0.138552	+	0.990355i	$0.544245\pi$
$930$	0	0
$931$	1.72577	0.0565597
$932$	0	0
$933$	31.2678	1.02366
$934$	0	0
$935$	−3.69037	−0.120688
$936$	0	0
$937$	32.5637	1.06381	0.531904	−	0.846805i	$-0.321477\pi$
$937$	32.5637	1.06381	0.531904	+	0.846805i	$0.321477\pi$
$938$	0	0
$939$	24.7521	0.807755
$940$	0	0
$941$	−4.40571	−0.143622	−0.0718110	−	0.997418i	$-0.522878\pi$
$941$	−4.40571	−0.143622	−0.0718110	+	0.997418i	$0.522878\pi$
$942$	0	0
$943$	58.1634	1.89406
$944$	0	0
$945$	−24.3557	−0.792292
$946$	0	0
$947$	40.0851	1.30259	0.651295	−	0.758825i	$-0.274226\pi$
$947$	40.0851	1.30259	0.651295	+	0.758825i	$0.274226\pi$
$948$	0	0
$949$	−6.69916	−0.217464
$950$	0	0
$951$	2.98680	0.0968536
$952$	0	0
$953$	39.4177	1.27687	0.638433	−	0.769678i	$-0.279583\pi$
$953$	39.4177	1.27687	0.638433	+	0.769678i	$0.279583\pi$
$954$	0	0
$955$	12.4769	0.403742
$956$	0	0
$957$	−0.816025	−0.0263783
$958$	0	0
$959$	−3.93417	−0.127041
$960$	0	0
$961$	89.8682	2.89897
$962$	0	0
$963$	6.64267	0.214057
$964$	0	0
$965$	15.9284	0.512754
$966$	0	0
$967$	25.7828	0.829119	0.414559	−	0.910022i	$-0.363936\pi$
$967$	25.7828	0.829119	0.414559	+	0.910022i	$0.363936\pi$
$968$	0	0
$969$	−4.62824	−0.148681
$970$	0	0
$971$	−22.6625	−0.727273	−0.363637	−	0.931541i	$-0.618465\pi$
$971$	−22.6625	−0.727273	−0.363637	+	0.931541i	$0.618465\pi$
$972$	0	0
$973$	22.9519	0.735805
$974$	0	0
$975$	−8.21218	−0.263000
$976$	0	0
$977$	60.2449	1.92741	0.963703	−	0.266977i	$-0.0860248\pi$
$977$	60.2449	1.92741	0.963703	+	0.266977i	$0.0860248\pi$
$978$	0	0
$979$	0.889611	0.0284321
$980$	0	0
$981$	13.3515	0.426280
$982$	0	0
$983$	−37.1415	−1.18463	−0.592314	−	0.805707i	$-0.701786\pi$
$983$	−37.1415	−1.18463	−0.592314	+	0.805707i	$0.701786\pi$
$984$	0	0
$985$	−40.9401	−1.30446
$986$	0	0
$987$	63.1486	2.01004
$988$	0	0
$989$	−2.44260	−0.0776703
$990$	0	0
$991$	41.0445	1.30382	0.651911	−	0.758296i	$-0.273968\pi$
$991$	41.0445	1.30382	0.651911	+	0.758296i	$0.273968\pi$
$992$	0	0
$993$	60.3900	1.91642
$994$	0	0
$995$	35.9135	1.13853
$996$	0	0
$997$	−1.95803	−0.0620113	−0.0310056	−	0.999519i	$-0.509871\pi$
$997$	−1.95803	−0.0620113	−0.0310056	+	0.999519i	$0.509871\pi$
$998$	0	0
$999$	13.6822	0.432887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.19	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.19	✓	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)

\(f(q)\)	\(=\)	\(q-1.93385 q^{3} -1.81561 q^{5} +3.06905 q^{7} +0.739771 q^{9} +O(q^{10})\)	\(q-1.93385 q^{3} -1.81561 q^{5} +3.06905 q^{7} +0.739771 q^{9} +0.605887 q^{11} -2.49274 q^{13} +3.51111 q^{15} +3.35472 q^{17} +0.713406 q^{19} -5.93507 q^{21} -7.94231 q^{23} -1.70356 q^{25} +4.37094 q^{27} +0.696449 q^{29} +10.9940 q^{31} -1.17169 q^{33} -5.57219 q^{35} +3.13027 q^{37} +4.82059 q^{39} -7.32324 q^{41} +0.307544 q^{43} -1.34313 q^{45} -10.6399 q^{47} +2.41905 q^{49} -6.48752 q^{51} -2.15881 q^{53} -1.10005 q^{55} -1.37962 q^{57} +2.58964 q^{59} -3.85494 q^{61} +2.27039 q^{63} +4.52585 q^{65} +12.5429 q^{67} +15.3592 q^{69} +7.98060 q^{71} +2.68746 q^{73} +3.29443 q^{75} +1.85949 q^{77} -11.1295 q^{79} -10.6721 q^{81} -2.82552 q^{83} -6.09086 q^{85} -1.34683 q^{87} +1.46828 q^{89} -7.65035 q^{91} -21.2607 q^{93} -1.29527 q^{95} +6.88648 q^{97} +0.448217 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 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

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Database

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