LMFDB - Embedded newform 8044.2.a.a.1.16

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			1.16
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)

$f(q)$	$=$	$q-2.27117 q^{3} -3.46385 q^{5} -0.730522 q^{7} +2.15820 q^{9} +O(q^{10})$	$q-2.27117 q^{3} -3.46385 q^{5} -0.730522 q^{7} +2.15820 q^{9} +4.31652 q^{11} -4.69306 q^{13} +7.86698 q^{15} -7.59580 q^{17} -1.72935 q^{19} +1.65914 q^{21} +5.06230 q^{23} +6.99825 q^{25} +1.91187 q^{27} +4.64463 q^{29} -3.37495 q^{31} -9.80354 q^{33} +2.53042 q^{35} -11.8233 q^{37} +10.6587 q^{39} -8.18233 q^{41} +11.8466 q^{43} -7.47568 q^{45} +8.98645 q^{47} -6.46634 q^{49} +17.2513 q^{51} +7.60726 q^{53} -14.9518 q^{55} +3.92765 q^{57} -9.91577 q^{59} -7.53932 q^{61} -1.57661 q^{63} +16.2560 q^{65} +13.4595 q^{67} -11.4973 q^{69} +1.09133 q^{71} +15.7181 q^{73} -15.8942 q^{75} -3.15331 q^{77} -0.255289 q^{79} -10.8168 q^{81} +10.0207 q^{83} +26.3107 q^{85} -10.5487 q^{87} +18.0601 q^{89} +3.42838 q^{91} +7.66508 q^{93} +5.99022 q^{95} -11.2424 q^{97} +9.31591 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.27117	−1.31126	−0.655629	−	0.755083i	$-0.727597\pi$
$3$	−2.27117	−1.31126	−0.655629	+	0.755083i	$0.727597\pi$
$4$	0	0
$5$	−3.46385	−1.54908	−0.774540	−	0.632525i	$-0.782019\pi$
$5$	−3.46385	−1.54908	−0.774540	+	0.632525i	$0.782019\pi$
$6$	0	0
$7$	−0.730522	−0.276111	−0.138056	−	0.990424i	$-0.544085\pi$
$7$	−0.730522	−0.276111	−0.138056	+	0.990424i	$0.544085\pi$
$8$	0	0
$9$	2.15820	0.719400
$10$	0	0
$11$	4.31652	1.30148	0.650740	−	0.759301i	$-0.274459\pi$
$11$	4.31652	1.30148	0.650740	+	0.759301i	$0.274459\pi$
$12$	0	0
$13$	−4.69306	−1.30162	−0.650810	−	0.759241i	$-0.725570\pi$
$13$	−4.69306	−1.30162	−0.650810	+	0.759241i	$0.725570\pi$
$14$	0	0
$15$	7.86698	2.03125
$16$	0	0
$17$	−7.59580	−1.84225	−0.921126	−	0.389264i	$-0.872729\pi$
$17$	−7.59580	−1.84225	−0.921126	+	0.389264i	$0.872729\pi$
$18$	0	0
$19$	−1.72935	−0.396741	−0.198371	−	0.980127i	$-0.563565\pi$
$19$	−1.72935	−0.396741	−0.198371	+	0.980127i	$0.563565\pi$
$20$	0	0
$21$	1.65914	0.362053
$22$	0	0
$23$	5.06230	1.05556	0.527781	−	0.849380i	$-0.323024\pi$
$23$	5.06230	1.05556	0.527781	+	0.849380i	$0.323024\pi$
$24$	0	0
$25$	6.99825	1.39965
$26$	0	0
$27$	1.91187	0.367940
$28$	0	0
$29$	4.64463	0.862486	0.431243	−	0.902236i	$-0.358075\pi$
$29$	4.64463	0.862486	0.431243	+	0.902236i	$0.358075\pi$
$30$	0	0
$31$	−3.37495	−0.606159	−0.303080	−	0.952965i	$-0.598015\pi$
$31$	−3.37495	−0.606159	−0.303080	+	0.952965i	$0.598015\pi$
$32$	0	0
$33$	−9.80354	−1.70658
$34$	0	0
$35$	2.53042	0.427719
$36$	0	0
$37$	−11.8233	−1.94374	−0.971872	−	0.235508i	$-0.924325\pi$
$37$	−11.8233	−1.94374	−0.971872	+	0.235508i	$0.924325\pi$
$38$	0	0
$39$	10.6587	1.70676
$40$	0	0
$41$	−8.18233	−1.27786	−0.638932	−	0.769263i	$-0.720624\pi$
$41$	−8.18233	−1.27786	−0.638932	+	0.769263i	$0.720624\pi$
$42$	0	0
$43$	11.8466	1.80659	0.903295	−	0.429020i	$-0.141141\pi$
$43$	11.8466	1.80659	0.903295	+	0.429020i	$0.141141\pi$
$44$	0	0
$45$	−7.47568	−1.11441
$46$	0	0
$47$	8.98645	1.31081	0.655404	−	0.755278i	$-0.272498\pi$
$47$	8.98645	1.31081	0.655404	+	0.755278i	$0.272498\pi$
$48$	0	0
$49$	−6.46634	−0.923763
$50$	0	0
$51$	17.2513	2.41567
$52$	0	0
$53$	7.60726	1.04494	0.522469	−	0.852659i	$-0.325011\pi$
$53$	7.60726	1.04494	0.522469	+	0.852659i	$0.325011\pi$
$54$	0	0
$55$	−14.9518	−2.01610
$56$	0	0
$57$	3.92765	0.520230
$58$	0	0
$59$	−9.91577	−1.29092	−0.645462	−	0.763793i	$-0.723335\pi$
$59$	−9.91577	−1.29092	−0.645462	+	0.763793i	$0.723335\pi$
$60$	0	0
$61$	−7.53932	−0.965311	−0.482656	−	0.875810i	$-0.660328\pi$
$61$	−7.53932	−0.965311	−0.482656	+	0.875810i	$0.660328\pi$
$62$	0	0
$63$	−1.57661	−0.198634
$64$	0	0
$65$	16.2560	2.01631
$66$	0	0
$67$	13.4595	1.64434	0.822171	−	0.569241i	$-0.192763\pi$
$67$	13.4595	1.64434	0.822171	+	0.569241i	$0.192763\pi$
$68$	0	0
$69$	−11.4973	−1.38412
$70$	0	0
$71$	1.09133	0.129517	0.0647586	−	0.997901i	$-0.479372\pi$
$71$	1.09133	0.129517	0.0647586	+	0.997901i	$0.479372\pi$
$72$	0	0
$73$	15.7181	1.83967	0.919833	−	0.392310i	$-0.128324\pi$
$73$	15.7181	1.83967	0.919833	+	0.392310i	$0.128324\pi$
$74$	0	0
$75$	−15.8942	−1.83530
$76$	0	0
$77$	−3.15331	−0.359353
$78$	0	0
$79$	−0.255289	−0.0287222	−0.0143611	−	0.999897i	$-0.504571\pi$
$79$	−0.255289	−0.0287222	−0.0143611	+	0.999897i	$0.504571\pi$
$80$	0	0
$81$	−10.8168	−1.20186
$82$	0	0
$83$	10.0207	1.09991	0.549957	−	0.835193i	$-0.314644\pi$
$83$	10.0207	1.09991	0.549957	+	0.835193i	$0.314644\pi$
$84$	0	0
$85$	26.3107	2.85380
$86$	0	0
$87$	−10.5487	−1.13094
$88$	0	0
$89$	18.0601	1.91437	0.957183	−	0.289483i	$-0.0934834\pi$
$89$	18.0601	1.91437	0.957183	+	0.289483i	$0.0934834\pi$
$90$	0	0
$91$	3.42838	0.359392
$92$	0	0
$93$	7.66508	0.794832
$94$	0	0
$95$	5.99022	0.614584
$96$	0	0
$97$	−11.2424	−1.14149	−0.570744	−	0.821128i	$-0.693345\pi$
$97$	−11.2424	−1.14149	−0.570744	+	0.821128i	$0.693345\pi$
$98$	0	0
$99$	9.31591	0.936284
$100$	0	0
$101$	12.4528	1.23910	0.619552	−	0.784955i	$-0.287314\pi$
$101$	12.4528	1.23910	0.619552	+	0.784955i	$0.287314\pi$
$102$	0	0
$103$	16.8467	1.65995	0.829975	−	0.557801i	$-0.188355\pi$
$103$	16.8467	1.65995	0.829975	+	0.557801i	$0.188355\pi$
$104$	0	0
$105$	−5.74700	−0.560850
$106$	0	0
$107$	−12.2373	−1.18302	−0.591511	−	0.806297i	$-0.701469\pi$
$107$	−12.2373	−1.18302	−0.591511	+	0.806297i	$0.701469\pi$
$108$	0	0
$109$	−2.52600	−0.241947	−0.120973	−	0.992656i	$-0.538602\pi$
$109$	−2.52600	−0.241947	−0.120973	+	0.992656i	$0.538602\pi$
$110$	0	0
$111$	26.8528	2.54875
$112$	0	0
$113$	−10.5100	−0.988701	−0.494351	−	0.869263i	$-0.664594\pi$
$113$	−10.5100	−0.988701	−0.494351	+	0.869263i	$0.664594\pi$
$114$	0	0
$115$	−17.5350	−1.63515
$116$	0	0
$117$	−10.1285	−0.936385
$118$	0	0
$119$	5.54890	0.508667
$120$	0	0
$121$	7.63235	0.693850
$122$	0	0
$123$	18.5834	1.67561
$124$	0	0
$125$	−6.92163	−0.619090
$126$	0	0
$127$	−2.47421	−0.219551	−0.109776	−	0.993956i	$-0.535013\pi$
$127$	−2.47421	−0.219551	−0.109776	+	0.993956i	$0.535013\pi$
$128$	0	0
$129$	−26.9056	−2.36891
$130$	0	0
$131$	7.86440	0.687116	0.343558	−	0.939131i	$-0.388368\pi$
$131$	7.86440	0.687116	0.343558	+	0.939131i	$0.388368\pi$
$132$	0	0
$133$	1.26333	0.109545
$134$	0	0
$135$	−6.62243	−0.569968
$136$	0	0
$137$	−5.65342	−0.483004	−0.241502	−	0.970400i	$-0.577640\pi$
$137$	−5.65342	−0.483004	−0.241502	+	0.970400i	$0.577640\pi$
$138$	0	0
$139$	−4.06332	−0.344647	−0.172323	−	0.985040i	$-0.555127\pi$
$139$	−4.06332	−0.344647	−0.172323	+	0.985040i	$0.555127\pi$
$140$	0	0
$141$	−20.4097	−1.71881
$142$	0	0
$143$	−20.2577	−1.69403
$144$	0	0
$145$	−16.0883	−1.33606
$146$	0	0
$147$	14.6861	1.21129
$148$	0	0
$149$	1.63879	0.134255	0.0671275	−	0.997744i	$-0.478617\pi$
$149$	1.63879	0.134255	0.0671275	+	0.997744i	$0.478617\pi$
$150$	0	0
$151$	13.4150	1.09170	0.545849	−	0.837884i	$-0.316207\pi$
$151$	13.4150	1.09170	0.545849	+	0.837884i	$0.316207\pi$
$152$	0	0
$153$	−16.3933	−1.32532
$154$	0	0
$155$	11.6903	0.938989
$156$	0	0
$157$	20.3862	1.62699	0.813497	−	0.581570i	$-0.197561\pi$
$157$	20.3862	1.62699	0.813497	+	0.581570i	$0.197561\pi$
$158$	0	0
$159$	−17.2774	−1.37018
$160$	0	0
$161$	−3.69812	−0.291453
$162$	0	0
$163$	0.441090	0.0345488	0.0172744	−	0.999851i	$-0.494501\pi$
$163$	0.441090	0.0345488	0.0172744	+	0.999851i	$0.494501\pi$
$164$	0	0
$165$	33.9580	2.64363
$166$	0	0
$167$	17.5006	1.35424	0.677118	−	0.735874i	$-0.263229\pi$
$167$	17.5006	1.35424	0.677118	+	0.735874i	$0.263229\pi$
$168$	0	0
$169$	9.02477	0.694213
$170$	0	0
$171$	−3.73229	−0.285415
$172$	0	0
$173$	−8.00230	−0.608404	−0.304202	−	0.952608i	$-0.598390\pi$
$173$	−8.00230	−0.608404	−0.304202	+	0.952608i	$0.598390\pi$
$174$	0	0
$175$	−5.11237	−0.386459
$176$	0	0
$177$	22.5204	1.69273
$178$	0	0
$179$	8.10421	0.605737	0.302869	−	0.953032i	$-0.402056\pi$
$179$	8.10421	0.605737	0.302869	+	0.953032i	$0.402056\pi$
$180$	0	0
$181$	4.76300	0.354031	0.177016	−	0.984208i	$-0.443356\pi$
$181$	4.76300	0.354031	0.177016	+	0.984208i	$0.443356\pi$
$182$	0	0
$183$	17.1231	1.26577
$184$	0	0
$185$	40.9543	3.01102
$186$	0	0
$187$	−32.7874	−2.39765
$188$	0	0
$189$	−1.39666	−0.101592
$190$	0	0
$191$	−19.0973	−1.38183	−0.690917	−	0.722934i	$-0.742793\pi$
$191$	−19.0973	−1.38183	−0.690917	+	0.722934i	$0.742793\pi$
$192$	0	0
$193$	−8.21566	−0.591376	−0.295688	−	0.955285i	$-0.595549\pi$
$193$	−8.21566	−0.591376	−0.295688	+	0.955285i	$0.595549\pi$
$194$	0	0
$195$	−36.9202	−2.64391
$196$	0	0
$197$	20.4997	1.46054	0.730272	−	0.683156i	$-0.239393\pi$
$197$	20.4997	1.46054	0.730272	+	0.683156i	$0.239393\pi$
$198$	0	0
$199$	5.16920	0.366435	0.183217	−	0.983072i	$-0.441349\pi$
$199$	5.16920	0.366435	0.183217	+	0.983072i	$0.441349\pi$
$200$	0	0
$201$	−30.5688	−2.15616
$202$	0	0
$203$	−3.39300	−0.238142
$204$	0	0
$205$	28.3423	1.97952
$206$	0	0
$207$	10.9255	0.759372
$208$	0	0
$209$	−7.46479	−0.516351
$210$	0	0
$211$	−6.16412	−0.424356	−0.212178	−	0.977231i	$-0.568056\pi$
$211$	−6.16412	−0.424356	−0.212178	+	0.977231i	$0.568056\pi$
$212$	0	0
$213$	−2.47860	−0.169831
$214$	0	0
$215$	−41.0348	−2.79855
$216$	0	0
$217$	2.46548	0.167367
$218$	0	0
$219$	−35.6985	−2.41228
$220$	0	0
$221$	35.6475	2.39791
$222$	0	0
$223$	7.34292	0.491718	0.245859	−	0.969306i	$-0.420930\pi$
$223$	7.34292	0.491718	0.245859	+	0.969306i	$0.420930\pi$
$224$	0	0
$225$	15.1036	1.00691
$226$	0	0
$227$	−28.3187	−1.87958	−0.939790	−	0.341754i	$-0.888979\pi$
$227$	−28.3187	−1.87958	−0.939790	+	0.341754i	$0.888979\pi$
$228$	0	0
$229$	−27.5148	−1.81823	−0.909114	−	0.416548i	$-0.863240\pi$
$229$	−27.5148	−1.81823	−0.909114	+	0.416548i	$0.863240\pi$
$230$	0	0
$231$	7.16170	0.471205
$232$	0	0
$233$	−2.74339	−0.179726	−0.0898629	−	0.995954i	$-0.528643\pi$
$233$	−2.74339	−0.179726	−0.0898629	+	0.995954i	$0.528643\pi$
$234$	0	0
$235$	−31.1277	−2.03055
$236$	0	0
$237$	0.579803	0.0376623
$238$	0	0
$239$	−2.11386	−0.136735	−0.0683673	−	0.997660i	$-0.521779\pi$
$239$	−2.11386	−0.136735	−0.0683673	+	0.997660i	$0.521779\pi$
$240$	0	0
$241$	−17.1536	−1.10496	−0.552479	−	0.833527i	$-0.686318\pi$
$241$	−17.1536	−1.10496	−0.552479	+	0.833527i	$0.686318\pi$
$242$	0	0
$243$	18.8311	1.20801
$244$	0	0
$245$	22.3984	1.43098
$246$	0	0
$247$	8.11596	0.516406
$248$	0	0
$249$	−22.7587	−1.44227
$250$	0	0
$251$	3.48953	0.220257	0.110129	−	0.993917i	$-0.464874\pi$
$251$	3.48953	0.220257	0.110129	+	0.993917i	$0.464874\pi$
$252$	0	0
$253$	21.8515	1.37379
$254$	0	0
$255$	−59.7560	−3.74207
$256$	0	0
$257$	19.9229	1.24276	0.621378	−	0.783511i	$-0.286573\pi$
$257$	19.9229	1.24276	0.621378	+	0.783511i	$0.286573\pi$
$258$	0	0
$259$	8.63721	0.536690
$260$	0	0
$261$	10.0240	0.620472
$262$	0	0
$263$	−13.4335	−0.828346	−0.414173	−	0.910198i	$-0.635929\pi$
$263$	−13.4335	−0.828346	−0.414173	+	0.910198i	$0.635929\pi$
$264$	0	0
$265$	−26.3504	−1.61869
$266$	0	0
$267$	−41.0175	−2.51023
$268$	0	0
$269$	−19.4841	−1.18797	−0.593983	−	0.804478i	$-0.702445\pi$
$269$	−19.4841	−1.18797	−0.593983	+	0.804478i	$0.702445\pi$
$270$	0	0
$271$	−9.53315	−0.579097	−0.289549	−	0.957163i	$-0.593505\pi$
$271$	−9.53315	−0.579097	−0.289549	+	0.957163i	$0.593505\pi$
$272$	0	0
$273$	−7.78642	−0.471256
$274$	0	0
$275$	30.2081	1.82162
$276$	0	0
$277$	19.4889	1.17097	0.585487	−	0.810682i	$-0.300903\pi$
$277$	19.4889	1.17097	0.585487	+	0.810682i	$0.300903\pi$
$278$	0	0
$279$	−7.28382	−0.436071
$280$	0	0
$281$	−28.1192	−1.67745	−0.838725	−	0.544556i	$-0.816698\pi$
$281$	−28.1192	−1.67745	−0.838725	+	0.544556i	$0.816698\pi$
$282$	0	0
$283$	28.1425	1.67290	0.836450	−	0.548044i	$-0.184627\pi$
$283$	28.1425	1.67290	0.836450	+	0.548044i	$0.184627\pi$
$284$	0	0
$285$	−13.6048	−0.805878
$286$	0	0
$287$	5.97737	0.352833
$288$	0	0
$289$	40.6962	2.39389
$290$	0	0
$291$	25.5333	1.49679
$292$	0	0
$293$	2.08499	0.121806	0.0609032	−	0.998144i	$-0.480602\pi$
$293$	2.08499	0.121806	0.0609032	+	0.998144i	$0.480602\pi$
$294$	0	0
$295$	34.3467	1.99974
$296$	0	0
$297$	8.25263	0.478866
$298$	0	0
$299$	−23.7577	−1.37394
$300$	0	0
$301$	−8.65420	−0.498820
$302$	0	0
$303$	−28.2825	−1.62479
$304$	0	0
$305$	26.1151	1.49534
$306$	0	0
$307$	18.2994	1.04440	0.522202	−	0.852822i	$-0.325111\pi$
$307$	18.2994	1.04440	0.522202	+	0.852822i	$0.325111\pi$
$308$	0	0
$309$	−38.2616	−2.17662
$310$	0	0
$311$	−27.7855	−1.57557	−0.787786	−	0.615949i	$-0.788773\pi$
$311$	−27.7855	−1.57557	−0.787786	+	0.615949i	$0.788773\pi$
$312$	0	0
$313$	11.3659	0.642441	0.321220	−	0.947004i	$-0.395907\pi$
$313$	11.3659	0.642441	0.321220	+	0.947004i	$0.395907\pi$
$314$	0	0
$315$	5.46115	0.307701
$316$	0	0
$317$	−11.2440	−0.631526	−0.315763	−	0.948838i	$-0.602260\pi$
$317$	−11.2440	−0.631526	−0.315763	+	0.948838i	$0.602260\pi$
$318$	0	0
$319$	20.0486	1.12251
$320$	0	0
$321$	27.7929	1.55125
$322$	0	0
$323$	13.1358	0.730897
$324$	0	0
$325$	−32.8432	−1.82181
$326$	0	0
$327$	5.73697	0.317255
$328$	0	0
$329$	−6.56480	−0.361929
$330$	0	0
$331$	13.1467	0.722610	0.361305	−	0.932448i	$-0.382331\pi$
$331$	13.1467	0.722610	0.361305	+	0.932448i	$0.382331\pi$
$332$	0	0
$333$	−25.5171	−1.39833
$334$	0	0
$335$	−46.6217	−2.54722
$336$	0	0
$337$	−21.3258	−1.16169	−0.580844	−	0.814015i	$-0.697277\pi$
$337$	−21.3258	−1.16169	−0.580844	+	0.814015i	$0.697277\pi$
$338$	0	0
$339$	23.8700	1.29644
$340$	0	0
$341$	−14.5680	−0.788904
$342$	0	0
$343$	9.83746	0.531173
$344$	0	0
$345$	39.8250	2.14411
$346$	0	0
$347$	23.3803	1.25512	0.627561	−	0.778568i	$-0.284053\pi$
$347$	23.3803	1.25512	0.627561	+	0.778568i	$0.284053\pi$
$348$	0	0
$349$	0.665717	0.0356350	0.0178175	−	0.999841i	$-0.494328\pi$
$349$	0.665717	0.0356350	0.0178175	+	0.999841i	$0.494328\pi$
$350$	0	0
$351$	−8.97251	−0.478917
$352$	0	0
$353$	−22.5991	−1.20283	−0.601415	−	0.798937i	$-0.705396\pi$
$353$	−22.5991	−1.20283	−0.601415	+	0.798937i	$0.705396\pi$
$354$	0	0
$355$	−3.78021	−0.200633
$356$	0	0
$357$	−12.6025	−0.666994
$358$	0	0
$359$	15.9481	0.841709	0.420854	−	0.907128i	$-0.361730\pi$
$359$	15.9481	0.841709	0.420854	+	0.907128i	$0.361730\pi$
$360$	0	0
$361$	−16.0093	−0.842597
$362$	0	0
$363$	−17.3343	−0.909817
$364$	0	0
$365$	−54.4452	−2.84979
$366$	0	0
$367$	−7.03084	−0.367007	−0.183503	−	0.983019i	$-0.558744\pi$
$367$	−7.03084	−0.367007	−0.183503	+	0.983019i	$0.558744\pi$
$368$	0	0
$369$	−17.6591	−0.919296
$370$	0	0
$371$	−5.55727	−0.288519
$372$	0	0
$373$	−12.2347	−0.633487	−0.316744	−	0.948511i	$-0.602590\pi$
$373$	−12.2347	−0.633487	−0.316744	+	0.948511i	$0.602590\pi$
$374$	0	0
$375$	15.7202	0.811787
$376$	0	0
$377$	−21.7975	−1.12263
$378$	0	0
$379$	−14.4661	−0.743075	−0.371537	−	0.928418i	$-0.621169\pi$
$379$	−14.4661	−0.743075	−0.371537	+	0.928418i	$0.621169\pi$
$380$	0	0
$381$	5.61935	0.287888
$382$	0	0
$383$	−4.47825	−0.228828	−0.114414	−	0.993433i	$-0.536499\pi$
$383$	−4.47825	−0.228828	−0.114414	+	0.993433i	$0.536499\pi$
$384$	0	0
$385$	10.9226	0.556667
$386$	0	0
$387$	25.5673	1.29966
$388$	0	0
$389$	1.31787	0.0668188	0.0334094	−	0.999442i	$-0.489363\pi$
$389$	1.31787	0.0668188	0.0334094	+	0.999442i	$0.489363\pi$
$390$	0	0
$391$	−38.4522	−1.94461
$392$	0	0
$393$	−17.8614	−0.900987
$394$	0	0
$395$	0.884282	0.0444930
$396$	0	0
$397$	13.9016	0.697701	0.348850	−	0.937178i	$-0.386572\pi$
$397$	13.9016	0.697701	0.348850	+	0.937178i	$0.386572\pi$
$398$	0	0
$399$	−2.86924	−0.143641
$400$	0	0
$401$	−1.40952	−0.0703882	−0.0351941	−	0.999380i	$-0.511205\pi$
$401$	−1.40952	−0.0703882	−0.0351941	+	0.999380i	$0.511205\pi$
$402$	0	0
$403$	15.8388	0.788989
$404$	0	0
$405$	37.4677	1.86178
$406$	0	0
$407$	−51.0357	−2.52975
$408$	0	0
$409$	−8.63919	−0.427181	−0.213590	−	0.976923i	$-0.568516\pi$
$409$	−8.63919	−0.427181	−0.213590	+	0.976923i	$0.568516\pi$
$410$	0	0
$411$	12.8399	0.633343
$412$	0	0
$413$	7.24369	0.356439
$414$	0	0
$415$	−34.7102	−1.70385
$416$	0	0
$417$	9.22849	0.451921
$418$	0	0
$419$	23.7035	1.15799	0.578996	−	0.815330i	$-0.303445\pi$
$419$	23.7035	1.15799	0.578996	+	0.815330i	$0.303445\pi$
$420$	0	0
$421$	−6.74219	−0.328594	−0.164297	−	0.986411i	$-0.552536\pi$
$421$	−6.74219	−0.328594	−0.164297	+	0.986411i	$0.552536\pi$
$422$	0	0
$423$	19.3945	0.942995
$424$	0	0
$425$	−53.1573	−2.57851
$426$	0	0
$427$	5.50764	0.266533
$428$	0	0
$429$	46.0086	2.22131
$430$	0	0
$431$	0.575261	0.0277093	0.0138547	−	0.999904i	$-0.495590\pi$
$431$	0.575261	0.0277093	0.0138547	+	0.999904i	$0.495590\pi$
$432$	0	0
$433$	−14.1019	−0.677694	−0.338847	−	0.940841i	$-0.610037\pi$
$433$	−14.1019	−0.677694	−0.338847	+	0.940841i	$0.610037\pi$
$434$	0	0
$435$	36.5392	1.75192
$436$	0	0
$437$	−8.75451	−0.418785
$438$	0	0
$439$	−8.82923	−0.421396	−0.210698	−	0.977551i	$-0.567574\pi$
$439$	−8.82923	−0.421396	−0.210698	+	0.977551i	$0.567574\pi$
$440$	0	0
$441$	−13.9556	−0.664555
$442$	0	0
$443$	−28.5866	−1.35819	−0.679096	−	0.734050i	$-0.737628\pi$
$443$	−28.5866	−1.35819	−0.679096	+	0.734050i	$0.737628\pi$
$444$	0	0
$445$	−62.5574	−2.96551
$446$	0	0
$447$	−3.72197	−0.176043
$448$	0	0
$449$	−30.8750	−1.45708	−0.728540	−	0.685003i	$-0.759801\pi$
$449$	−30.8750	−1.45708	−0.728540	+	0.685003i	$0.759801\pi$
$450$	0	0
$451$	−35.3192	−1.66312
$452$	0	0
$453$	−30.4677	−1.43150
$454$	0	0
$455$	−11.8754	−0.556727
$456$	0	0
$457$	16.8764	0.789447	0.394723	−	0.918800i	$-0.370840\pi$
$457$	16.8764	0.789447	0.394723	+	0.918800i	$0.370840\pi$
$458$	0	0
$459$	−14.5222	−0.677838
$460$	0	0
$461$	−20.3136	−0.946097	−0.473048	−	0.881037i	$-0.656846\pi$
$461$	−20.3136	−0.946097	−0.473048	+	0.881037i	$0.656846\pi$
$462$	0	0
$463$	−5.89664	−0.274040	−0.137020	−	0.990568i	$-0.543752\pi$
$463$	−5.89664	−0.274040	−0.137020	+	0.990568i	$0.543752\pi$
$464$	0	0
$465$	−26.5507	−1.23126
$466$	0	0
$467$	−20.1077	−0.930474	−0.465237	−	0.885186i	$-0.654031\pi$
$467$	−20.1077	−0.930474	−0.465237	+	0.885186i	$0.654031\pi$
$468$	0	0
$469$	−9.83247	−0.454021
$470$	0	0
$471$	−46.3004	−2.13341
$472$	0	0
$473$	51.1361	2.35124
$474$	0	0
$475$	−12.1025	−0.555299
$476$	0	0
$477$	16.4180	0.751727
$478$	0	0
$479$	10.7362	0.490549	0.245275	−	0.969454i	$-0.421122\pi$
$479$	10.7362	0.490549	0.245275	+	0.969454i	$0.421122\pi$
$480$	0	0
$481$	55.4876	2.53002
$482$	0	0
$483$	8.39905	0.382170
$484$	0	0
$485$	38.9418	1.76826
$486$	0	0
$487$	5.16574	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$487$	5.16574	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$488$	0	0
$489$	−1.00179	−0.0453025
$490$	0	0
$491$	−32.7860	−1.47961	−0.739805	−	0.672821i	$-0.765082\pi$
$491$	−32.7860	−1.47961	−0.739805	+	0.672821i	$0.765082\pi$
$492$	0	0
$493$	−35.2797	−1.58892
$494$	0	0
$495$	−32.2689	−1.45038
$496$	0	0
$497$	−0.797242	−0.0357612
$498$	0	0
$499$	−26.3480	−1.17950	−0.589748	−	0.807587i	$-0.700773\pi$
$499$	−26.3480	−1.17950	−0.589748	+	0.807587i	$0.700773\pi$
$500$	0	0
$501$	−39.7468	−1.77575
$502$	0	0
$503$	−39.7107	−1.77061	−0.885307	−	0.465008i	$-0.846052\pi$
$503$	−39.7107	−1.77061	−0.885307	+	0.465008i	$0.846052\pi$
$504$	0	0
$505$	−43.1348	−1.91947
$506$	0	0
$507$	−20.4968	−0.910293
$508$	0	0
$509$	17.0890	0.757458	0.378729	−	0.925508i	$-0.376361\pi$
$509$	17.0890	0.757458	0.378729	+	0.925508i	$0.376361\pi$
$510$	0	0
$511$	−11.4824	−0.507953
$512$	0	0
$513$	−3.30630	−0.145977
$514$	0	0
$515$	−58.3543	−2.57140
$516$	0	0
$517$	38.7902	1.70599
$518$	0	0
$519$	18.1746	0.797775
$520$	0	0
$521$	−30.0826	−1.31794	−0.658971	−	0.752168i	$-0.729008\pi$
$521$	−30.0826	−1.31794	−0.658971	+	0.752168i	$0.729008\pi$
$522$	0	0
$523$	5.34593	0.233761	0.116881	−	0.993146i	$-0.462711\pi$
$523$	5.34593	0.233761	0.116881	+	0.993146i	$0.462711\pi$
$524$	0	0
$525$	11.6111	0.506748
$526$	0	0
$527$	25.6355	1.11670
$528$	0	0
$529$	2.62690	0.114213
$530$	0	0
$531$	−21.4002	−0.928690
$532$	0	0
$533$	38.4001	1.66329
$534$	0	0
$535$	42.3881	1.83260
$536$	0	0
$537$	−18.4060	−0.794278
$538$	0	0
$539$	−27.9121	−1.20226
$540$	0	0
$541$	7.73360	0.332494	0.166247	−	0.986084i	$-0.446835\pi$
$541$	7.73360	0.332494	0.166247	+	0.986084i	$0.446835\pi$
$542$	0	0
$543$	−10.8176	−0.464227
$544$	0	0
$545$	8.74968	0.374795
$546$	0	0
$547$	−7.65137	−0.327149	−0.163574	−	0.986531i	$-0.552302\pi$
$547$	−7.65137	−0.327149	−0.163574	+	0.986531i	$0.552302\pi$
$548$	0	0
$549$	−16.2714	−0.694445
$550$	0	0
$551$	−8.03221	−0.342184
$552$	0	0
$553$	0.186494	0.00793053
$554$	0	0
$555$	−93.0140	−3.94822
$556$	0	0
$557$	−4.49100	−0.190290	−0.0951449	−	0.995463i	$-0.530331\pi$
$557$	−4.49100	−0.190290	−0.0951449	+	0.995463i	$0.530331\pi$
$558$	0	0
$559$	−55.5968	−2.35149
$560$	0	0
$561$	74.4657	3.14395
$562$	0	0
$563$	37.6174	1.58538	0.792692	−	0.609623i	$-0.208679\pi$
$563$	37.6174	1.58538	0.792692	+	0.609623i	$0.208679\pi$
$564$	0	0
$565$	36.4052	1.53158
$566$	0	0
$567$	7.90189	0.331848
$568$	0	0
$569$	1.09267	0.0458069	0.0229035	−	0.999738i	$-0.492709\pi$
$569$	1.09267	0.0458069	0.0229035	+	0.999738i	$0.492709\pi$
$570$	0	0
$571$	34.3735	1.43849	0.719243	−	0.694759i	$-0.244489\pi$
$571$	34.3735	1.43849	0.719243	+	0.694759i	$0.244489\pi$
$572$	0	0
$573$	43.3732	1.81194
$574$	0	0
$575$	35.4273	1.47742
$576$	0	0
$577$	−41.9497	−1.74639	−0.873195	−	0.487370i	$-0.837956\pi$
$577$	−41.9497	−1.74639	−0.873195	+	0.487370i	$0.837956\pi$
$578$	0	0
$579$	18.6591	0.775447
$580$	0	0
$581$	−7.32034	−0.303699
$582$	0	0
$583$	32.8369	1.35996
$584$	0	0
$585$	35.0838	1.45054
$586$	0	0
$587$	−9.11663	−0.376284	−0.188142	−	0.982142i	$-0.560246\pi$
$587$	−9.11663	−0.376284	−0.188142	+	0.982142i	$0.560246\pi$
$588$	0	0
$589$	5.83649	0.240488
$590$	0	0
$591$	−46.5583	−1.91515
$592$	0	0
$593$	−2.91615	−0.119752	−0.0598760	−	0.998206i	$-0.519071\pi$
$593$	−2.91615	−0.119752	−0.0598760	+	0.998206i	$0.519071\pi$
$594$	0	0
$595$	−19.2206	−0.787966
$596$	0	0
$597$	−11.7401	−0.480491
$598$	0	0
$599$	28.4010	1.16043	0.580217	−	0.814462i	$-0.302968\pi$
$599$	28.4010	1.16043	0.580217	+	0.814462i	$0.302968\pi$
$600$	0	0
$601$	−24.2008	−0.987173	−0.493587	−	0.869697i	$-0.664314\pi$
$601$	−24.2008	−0.987173	−0.493587	+	0.869697i	$0.664314\pi$
$602$	0	0
$603$	29.0483	1.18294
$604$	0	0
$605$	−26.4373	−1.07483
$606$	0	0
$607$	−9.00233	−0.365393	−0.182697	−	0.983169i	$-0.558483\pi$
$607$	−9.00233	−0.365393	−0.182697	+	0.983169i	$0.558483\pi$
$608$	0	0
$609$	7.70608	0.312266
$610$	0	0
$611$	−42.1739	−1.70617
$612$	0	0
$613$	25.9915	1.04979	0.524893	−	0.851168i	$-0.324105\pi$
$613$	25.9915	1.04979	0.524893	+	0.851168i	$0.324105\pi$
$614$	0	0
$615$	−64.3702	−2.59566
$616$	0	0
$617$	24.7857	0.997835	0.498918	−	0.866649i	$-0.333731\pi$
$617$	24.7857	0.997835	0.498918	+	0.866649i	$0.333731\pi$
$618$	0	0
$619$	2.83398	0.113907	0.0569537	−	0.998377i	$-0.481861\pi$
$619$	2.83398	0.113907	0.0569537	+	0.998377i	$0.481861\pi$
$620$	0	0
$621$	9.67846	0.388383
$622$	0	0
$623$	−13.1933	−0.528578
$624$	0	0
$625$	−11.0158	−0.440630
$626$	0	0
$627$	16.9538	0.677069
$628$	0	0
$629$	89.8077	3.58087
$630$	0	0
$631$	−4.71230	−0.187594	−0.0937969	−	0.995591i	$-0.529900\pi$
$631$	−4.71230	−0.187594	−0.0937969	+	0.995591i	$0.529900\pi$
$632$	0	0
$633$	13.9998	0.556440
$634$	0	0
$635$	8.57031	0.340102
$636$	0	0
$637$	30.3469	1.20239
$638$	0	0
$639$	2.35531	0.0931747
$640$	0	0
$641$	−33.5466	−1.32501	−0.662505	−	0.749058i	$-0.730507\pi$
$641$	−33.5466	−1.32501	−0.662505	+	0.749058i	$0.730507\pi$
$642$	0	0
$643$	−22.7984	−0.899083	−0.449541	−	0.893260i	$-0.648413\pi$
$643$	−22.7984	−0.899083	−0.449541	+	0.893260i	$0.648413\pi$
$644$	0	0
$645$	93.1970	3.66963
$646$	0	0
$647$	4.41683	0.173644	0.0868218	−	0.996224i	$-0.472329\pi$
$647$	4.41683	0.173644	0.0868218	+	0.996224i	$0.472329\pi$
$648$	0	0
$649$	−42.8016	−1.68011
$650$	0	0
$651$	−5.59951	−0.219462
$652$	0	0
$653$	−15.6035	−0.610613	−0.305307	−	0.952254i	$-0.598759\pi$
$653$	−15.6035	−0.610613	−0.305307	+	0.952254i	$0.598759\pi$
$654$	0	0
$655$	−27.2411	−1.06440
$656$	0	0
$657$	33.9228	1.32346
$658$	0	0
$659$	−0.112403	−0.00437859	−0.00218929	−	0.999998i	$-0.500697\pi$
$659$	−0.112403	−0.00437859	−0.00218929	+	0.999998i	$0.500697\pi$
$660$	0	0
$661$	6.29203	0.244731	0.122366	−	0.992485i	$-0.460952\pi$
$661$	6.29203	0.244731	0.122366	+	0.992485i	$0.460952\pi$
$662$	0	0
$663$	−80.9615	−3.14428
$664$	0	0
$665$	−4.37599	−0.169694
$666$	0	0
$667$	23.5125	0.910408
$668$	0	0
$669$	−16.6770	−0.644770
$670$	0	0
$671$	−32.5436	−1.25633
$672$	0	0
$673$	−19.6329	−0.756792	−0.378396	−	0.925644i	$-0.623524\pi$
$673$	−19.6329	−0.756792	−0.378396	+	0.925644i	$0.623524\pi$
$674$	0	0
$675$	13.3797	0.514987
$676$	0	0
$677$	−8.44925	−0.324731	−0.162365	−	0.986731i	$-0.551912\pi$
$677$	−8.44925	−0.324731	−0.162365	+	0.986731i	$0.551912\pi$
$678$	0	0
$679$	8.21279	0.315178
$680$	0	0
$681$	64.3165	2.46461
$682$	0	0
$683$	−23.4926	−0.898917	−0.449459	−	0.893301i	$-0.648383\pi$
$683$	−23.4926	−0.898917	−0.449459	+	0.893301i	$0.648383\pi$
$684$	0	0
$685$	19.5826	0.748212
$686$	0	0
$687$	62.4906	2.38417
$688$	0	0
$689$	−35.7013	−1.36011
$690$	0	0
$691$	50.9203	1.93710	0.968549	−	0.248822i	$-0.0800434\pi$
$691$	50.9203	1.93710	0.968549	+	0.248822i	$0.0800434\pi$
$692$	0	0
$693$	−6.80548	−0.258519
$694$	0	0
$695$	14.0747	0.533885
$696$	0	0
$697$	62.1513	2.35415
$698$	0	0
$699$	6.23071	0.235667
$700$	0	0
$701$	30.2794	1.14364	0.571819	−	0.820380i	$-0.306238\pi$
$701$	30.2794	1.14364	0.571819	+	0.820380i	$0.306238\pi$
$702$	0	0
$703$	20.4467	0.771163
$704$	0	0
$705$	70.6962	2.66257
$706$	0	0
$707$	−9.09708	−0.342131
$708$	0	0
$709$	1.19894	0.0450271	0.0225136	−	0.999747i	$-0.492833\pi$
$709$	1.19894	0.0450271	0.0225136	+	0.999747i	$0.492833\pi$
$710$	0	0
$711$	−0.550964	−0.0206628
$712$	0	0
$713$	−17.0850	−0.639839
$714$	0	0
$715$	70.1695	2.62419
$716$	0	0
$717$	4.80094	0.179294
$718$	0	0
$719$	−6.14391	−0.229129	−0.114565	−	0.993416i	$-0.536547\pi$
$719$	−6.14391	−0.229129	−0.114565	+	0.993416i	$0.536547\pi$
$720$	0	0
$721$	−12.3068	−0.458331
$722$	0	0
$723$	38.9586	1.44889
$724$	0	0
$725$	32.5043	1.20718
$726$	0	0
$727$	1.20130	0.0445536	0.0222768	−	0.999752i	$-0.492908\pi$
$727$	1.20130	0.0445536	0.0222768	+	0.999752i	$0.492908\pi$
$728$	0	0
$729$	−10.3182	−0.382156
$730$	0	0
$731$	−89.9845	−3.32820
$732$	0	0
$733$	20.4761	0.756301	0.378151	−	0.925744i	$-0.376560\pi$
$733$	20.4761	0.756301	0.378151	+	0.925744i	$0.376560\pi$
$734$	0	0
$735$	−50.8705	−1.87639
$736$	0	0
$737$	58.0983	2.14008
$738$	0	0
$739$	26.1071	0.960365	0.480182	−	0.877169i	$-0.340571\pi$
$739$	26.1071	0.960365	0.480182	+	0.877169i	$0.340571\pi$
$740$	0	0
$741$	−18.4327	−0.677142
$742$	0	0
$743$	46.7875	1.71647	0.858233	−	0.513261i	$-0.171563\pi$
$743$	46.7875	1.71647	0.858233	+	0.513261i	$0.171563\pi$
$744$	0	0
$745$	−5.67652	−0.207972
$746$	0	0
$747$	21.6266	0.791278
$748$	0	0
$749$	8.93960	0.326646
$750$	0	0
$751$	−15.6221	−0.570060	−0.285030	−	0.958519i	$-0.592004\pi$
$751$	−15.6221	−0.570060	−0.285030	+	0.958519i	$0.592004\pi$
$752$	0	0
$753$	−7.92530	−0.288814
$754$	0	0
$755$	−46.4675	−1.69113
$756$	0	0
$757$	−22.4879	−0.817337	−0.408668	−	0.912683i	$-0.634007\pi$
$757$	−22.4879	−0.817337	−0.408668	+	0.912683i	$0.634007\pi$
$758$	0	0
$759$	−49.6285	−1.80140
$760$	0	0
$761$	−7.04350	−0.255327	−0.127663	−	0.991818i	$-0.540748\pi$
$761$	−7.04350	−0.255327	−0.127663	+	0.991818i	$0.540748\pi$
$762$	0	0
$763$	1.84530	0.0668043
$764$	0	0
$765$	56.7838	2.05302
$766$	0	0
$767$	46.5353	1.68029
$768$	0	0
$769$	12.4599	0.449317	0.224658	−	0.974438i	$-0.427873\pi$
$769$	12.4599	0.449317	0.224658	+	0.974438i	$0.427873\pi$
$770$	0	0
$771$	−45.2482	−1.62957
$772$	0	0
$773$	3.21235	0.115540	0.0577701	−	0.998330i	$-0.481601\pi$
$773$	3.21235	0.115540	0.0577701	+	0.998330i	$0.481601\pi$
$774$	0	0
$775$	−23.6188	−0.848411
$776$	0	0
$777$	−19.6165	−0.703740
$778$	0	0
$779$	14.1501	0.506982
$780$	0	0
$781$	4.71076	0.168564
$782$	0	0
$783$	8.87993	0.317343
$784$	0	0
$785$	−70.6146	−2.52034
$786$	0	0
$787$	20.7904	0.741097	0.370549	−	0.928813i	$-0.379170\pi$
$787$	20.7904	0.741097	0.370549	+	0.928813i	$0.379170\pi$
$788$	0	0
$789$	30.5097	1.08618
$790$	0	0
$791$	7.67781	0.272992
$792$	0	0
$793$	35.3825	1.25647
$794$	0	0
$795$	59.8461	2.12252
$796$	0	0
$797$	−20.3798	−0.721888	−0.360944	−	0.932587i	$-0.617545\pi$
$797$	−20.3798	−0.721888	−0.360944	+	0.932587i	$0.617545\pi$
$798$	0	0
$799$	−68.2593	−2.41484
$800$	0	0
$801$	38.9773	1.37719
$802$	0	0
$803$	67.8476	2.39429
$804$	0	0
$805$	12.8097	0.451484
$806$	0	0
$807$	44.2516	1.55773
$808$	0	0
$809$	39.8972	1.40271	0.701355	−	0.712812i	$-0.252579\pi$
$809$	39.8972	1.40271	0.701355	+	0.712812i	$0.252579\pi$
$810$	0	0
$811$	5.88623	0.206693	0.103347	−	0.994645i	$-0.467045\pi$
$811$	5.88623	0.206693	0.103347	+	0.994645i	$0.467045\pi$
$812$	0	0
$813$	21.6514	0.759347
$814$	0	0
$815$	−1.52787	−0.0535189
$816$	0	0
$817$	−20.4870	−0.716748
$818$	0	0
$819$	7.39913	0.258546
$820$	0	0
$821$	34.6693	1.20997	0.604984	−	0.796238i	$-0.293180\pi$
$821$	34.6693	1.20997	0.604984	+	0.796238i	$0.293180\pi$
$822$	0	0
$823$	28.6598	0.999017	0.499508	−	0.866309i	$-0.333514\pi$
$823$	28.6598	0.999017	0.499508	+	0.866309i	$0.333514\pi$
$824$	0	0
$825$	−68.6076	−2.38861
$826$	0	0
$827$	21.1206	0.734435	0.367217	−	0.930135i	$-0.380311\pi$
$827$	21.1206	0.734435	0.367217	+	0.930135i	$0.380311\pi$
$828$	0	0
$829$	47.1098	1.63619	0.818095	−	0.575083i	$-0.195030\pi$
$829$	47.1098	1.63619	0.818095	+	0.575083i	$0.195030\pi$
$830$	0	0
$831$	−44.2625	−1.53545
$832$	0	0
$833$	49.1170	1.70180
$834$	0	0
$835$	−60.6194	−2.09782
$836$	0	0
$837$	−6.45247	−0.223030
$838$	0	0
$839$	−14.5734	−0.503129	−0.251564	−	0.967841i	$-0.580945\pi$
$839$	−14.5734	−0.503129	−0.251564	+	0.967841i	$0.580945\pi$
$840$	0	0
$841$	−7.42743	−0.256118
$842$	0	0
$843$	63.8633	2.19957
$844$	0	0
$845$	−31.2605	−1.07539
$846$	0	0
$847$	−5.57560	−0.191580
$848$	0	0
$849$	−63.9164	−2.19360
$850$	0	0
$851$	−59.8533	−2.05175
$852$	0	0
$853$	12.8459	0.439836	0.219918	−	0.975518i	$-0.429421\pi$
$853$	12.8459	0.439836	0.219918	+	0.975518i	$0.429421\pi$
$854$	0	0
$855$	12.9281	0.442131
$856$	0	0
$857$	−23.8349	−0.814184	−0.407092	−	0.913387i	$-0.633457\pi$
$857$	−23.8349	−0.814184	−0.407092	+	0.913387i	$0.633457\pi$
$858$	0	0
$859$	4.79890	0.163736	0.0818682	−	0.996643i	$-0.473911\pi$
$859$	4.79890	0.163736	0.0818682	+	0.996643i	$0.473911\pi$
$860$	0	0
$861$	−13.5756	−0.462655
$862$	0	0
$863$	−41.4072	−1.40952	−0.704758	−	0.709448i	$-0.748944\pi$
$863$	−41.4072	−1.40952	−0.704758	+	0.709448i	$0.748944\pi$
$864$	0	0
$865$	27.7188	0.942466
$866$	0	0
$867$	−92.4279	−3.13902
$868$	0	0
$869$	−1.10196	−0.0373814
$870$	0	0
$871$	−63.1663	−2.14031
$872$	0	0
$873$	−24.2632	−0.821186
$874$	0	0
$875$	5.05640	0.170938
$876$	0	0
$877$	37.0568	1.25132	0.625660	−	0.780096i	$-0.284830\pi$
$877$	37.0568	1.25132	0.625660	+	0.780096i	$0.284830\pi$
$878$	0	0
$879$	−4.73536	−0.159720
$880$	0	0
$881$	40.0403	1.34899	0.674496	−	0.738279i	$-0.264361\pi$
$881$	40.0403	1.34899	0.674496	+	0.738279i	$0.264361\pi$
$882$	0	0
$883$	29.8353	1.00404	0.502019	−	0.864857i	$-0.332591\pi$
$883$	29.8353	1.00404	0.502019	+	0.864857i	$0.332591\pi$
$884$	0	0
$885$	−78.0072	−2.62218
$886$	0	0
$887$	−14.6793	−0.492884	−0.246442	−	0.969158i	$-0.579261\pi$
$887$	−14.6793	−0.492884	−0.246442	+	0.969158i	$0.579261\pi$
$888$	0	0
$889$	1.80747	0.0606205
$890$	0	0
$891$	−46.6908	−1.56420
$892$	0	0
$893$	−15.5407	−0.520051
$894$	0	0
$895$	−28.0718	−0.938335
$896$	0	0
$897$	53.9576	1.80159
$898$	0	0
$899$	−15.6754	−0.522804
$900$	0	0
$901$	−57.7832	−1.92504
$902$	0	0
$903$	19.6551	0.654082
$904$	0	0
$905$	−16.4983	−0.548423
$906$	0	0
$907$	38.9659	1.29384	0.646920	−	0.762558i	$-0.276057\pi$
$907$	38.9659	1.29384	0.646920	+	0.762558i	$0.276057\pi$
$908$	0	0
$909$	26.8757	0.891411
$910$	0	0
$911$	16.0472	0.531669	0.265834	−	0.964019i	$-0.414353\pi$
$911$	16.0472	0.531669	0.265834	+	0.964019i	$0.414353\pi$
$912$	0	0
$913$	43.2545	1.43152
$914$	0	0
$915$	−59.3117	−1.96078
$916$	0	0
$917$	−5.74512	−0.189721
$918$	0	0
$919$	−18.5907	−0.613251	−0.306626	−	0.951830i	$-0.599200\pi$
$919$	−18.5907	−0.613251	−0.306626	+	0.951830i	$0.599200\pi$
$920$	0	0
$921$	−41.5611	−1.36948
$922$	0	0
$923$	−5.12168	−0.168582
$924$	0	0
$925$	−82.7427	−2.72056
$926$	0	0
$927$	36.3584	1.19417
$928$	0	0
$929$	−58.5523	−1.92104	−0.960520	−	0.278212i	$-0.910258\pi$
$929$	−58.5523	−1.92104	−0.960520	+	0.278212i	$0.910258\pi$
$930$	0	0
$931$	11.1826	0.366495
$932$	0	0
$933$	63.1055	2.06598
$934$	0	0
$935$	113.571	3.71416
$936$	0	0
$937$	37.1582	1.21390	0.606952	−	0.794738i	$-0.292392\pi$
$937$	37.1582	1.21390	0.606952	+	0.794738i	$0.292392\pi$
$938$	0	0
$939$	−25.8139	−0.842406
$940$	0	0
$941$	54.1858	1.76641	0.883203	−	0.468990i	$-0.155382\pi$
$941$	54.1858	1.76641	0.883203	+	0.468990i	$0.155382\pi$
$942$	0	0
$943$	−41.4214	−1.34887
$944$	0	0
$945$	4.83783	0.157375
$946$	0	0
$947$	−51.2571	−1.66563	−0.832816	−	0.553550i	$-0.813273\pi$
$947$	−51.2571	−1.66563	−0.832816	+	0.553550i	$0.813273\pi$
$948$	0	0
$949$	−73.7660	−2.39455
$950$	0	0
$951$	25.5370	0.828094
$952$	0	0
$953$	−34.4210	−1.11501	−0.557503	−	0.830175i	$-0.688240\pi$
$953$	−34.4210	−1.11501	−0.557503	+	0.830175i	$0.688240\pi$
$954$	0	0
$955$	66.1502	2.14057
$956$	0	0
$957$	−45.5338	−1.47190
$958$	0	0
$959$	4.12995	0.133363
$960$	0	0
$961$	−19.6097	−0.632571
$962$	0	0
$963$	−26.4105	−0.851066
$964$	0	0
$965$	28.4578	0.916089
$966$	0	0
$967$	6.08882	0.195803	0.0979017	−	0.995196i	$-0.468787\pi$
$967$	6.08882	0.195803	0.0979017	+	0.995196i	$0.468787\pi$
$968$	0	0
$969$	−29.8337	−0.958395
$970$	0	0
$971$	−11.2801	−0.361995	−0.180997	−	0.983484i	$-0.557933\pi$
$971$	−11.2801	−0.361995	−0.180997	+	0.983484i	$0.557933\pi$
$972$	0	0
$973$	2.96835	0.0951608
$974$	0	0
$975$	74.5923	2.38887
$976$	0	0
$977$	54.4460	1.74188	0.870941	−	0.491387i	$-0.163510\pi$
$977$	54.4460	1.74188	0.870941	+	0.491387i	$0.163510\pi$
$978$	0	0
$979$	77.9568	2.49151
$980$	0	0
$981$	−5.45161	−0.174057
$982$	0	0
$983$	−51.5966	−1.64567	−0.822837	−	0.568277i	$-0.807610\pi$
$983$	−51.5966	−1.64567	−0.822837	+	0.568277i	$0.807610\pi$
$984$	0	0
$985$	−71.0079	−2.26250
$986$	0	0
$987$	14.9097	0.474583
$988$	0	0
$989$	59.9711	1.90697
$990$	0	0
$991$	28.1772	0.895080	0.447540	−	0.894264i	$-0.352300\pi$
$991$	28.1772	0.895080	0.447540	+	0.894264i	$0.352300\pi$
$992$	0	0
$993$	−29.8585	−0.947529
$994$	0	0
$995$	−17.9053	−0.567637
$996$	0	0
$997$	−0.413572	−0.0130980	−0.00654899	−	0.999979i	$-0.502085\pi$
$997$	−0.413572	−0.0130980	−0.00654899	+	0.999979i	$0.502085\pi$
$998$	0	0
$999$	−22.6047	−0.715181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.16	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.16	✓	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)

\(f(q)\)	\(=\)	\(q-2.27117 q^{3} -3.46385 q^{5} -0.730522 q^{7} +2.15820 q^{9} +O(q^{10})\)	\(q-2.27117 q^{3} -3.46385 q^{5} -0.730522 q^{7} +2.15820 q^{9} +4.31652 q^{11} -4.69306 q^{13} +7.86698 q^{15} -7.59580 q^{17} -1.72935 q^{19} +1.65914 q^{21} +5.06230 q^{23} +6.99825 q^{25} +1.91187 q^{27} +4.64463 q^{29} -3.37495 q^{31} -9.80354 q^{33} +2.53042 q^{35} -11.8233 q^{37} +10.6587 q^{39} -8.18233 q^{41} +11.8466 q^{43} -7.47568 q^{45} +8.98645 q^{47} -6.46634 q^{49} +17.2513 q^{51} +7.60726 q^{53} -14.9518 q^{55} +3.92765 q^{57} -9.91577 q^{59} -7.53932 q^{61} -1.57661 q^{63} +16.2560 q^{65} +13.4595 q^{67} -11.4973 q^{69} +1.09133 q^{71} +15.7181 q^{73} -15.8942 q^{75} -3.15331 q^{77} -0.255289 q^{79} -10.8168 q^{81} +10.0207 q^{83} +26.3107 q^{85} -10.5487 q^{87} +18.0601 q^{89} +3.42838 q^{91} +7.66508 q^{93} +5.99022 q^{95} -11.2424 q^{97} +9.31591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 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

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Database

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