LMFDB - Embedded newform 8048.2.a.r.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 22 x^{10} + 67 x^{9} + 180 x^{8} - 383 x^{7} - 674 x^{6} + 875 x^{5} + 1077 x^{4} + \cdots + 2 $ x^12 - 4x^11 - 22x^10 + 67x^9 + 180x^8 - 383x^7 - 674x^6 + 875x^5 + 1077x^4 - 704x^3 - 467x^2 + 210*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.33502$ of defining polynomial
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.656123 q^{3} +4.07967 q^{5} +0.827612 q^{7} -2.56950 q^{9} +O(q^{10})$	$q-0.656123 q^{3} +4.07967 q^{5} +0.827612 q^{7} -2.56950 q^{9} +3.65273 q^{11} +1.01942 q^{13} -2.67677 q^{15} +3.12716 q^{17} -4.20968 q^{19} -0.543015 q^{21} -6.99846 q^{23} +11.6437 q^{25} +3.65428 q^{27} +5.92375 q^{29} +1.93174 q^{31} -2.39664 q^{33} +3.37638 q^{35} +7.83866 q^{37} -0.668864 q^{39} +0.429262 q^{41} -1.87562 q^{43} -10.4827 q^{45} -9.98732 q^{47} -6.31506 q^{49} -2.05180 q^{51} -8.00130 q^{53} +14.9020 q^{55} +2.76207 q^{57} +13.2850 q^{59} +8.78970 q^{61} -2.12655 q^{63} +4.15889 q^{65} +6.21701 q^{67} +4.59185 q^{69} -10.9469 q^{71} +13.3974 q^{73} -7.63970 q^{75} +3.02305 q^{77} +6.97796 q^{79} +5.31085 q^{81} +16.1238 q^{83} +12.7578 q^{85} -3.88671 q^{87} -1.61645 q^{89} +0.843682 q^{91} -1.26746 q^{93} -17.1741 q^{95} -14.0367 q^{97} -9.38571 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9	$ 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9} - 18 q^{11} - 4 q^{13} + 2 q^{15} + 12 q^{17} + 7 q^{21} + 9 q^{23} + 25 q^{25} + 18 q^{27} + 34 q^{29} + 11 q^{31} + 4 q^{33} - 21 q^{35} - 22 q^{37} - 13 q^{39} + 32 q^{41} + 8 q^{43} + 13 q^{45} - 24 q^{47} + 36 q^{49} - 16 q^{51} - 2 q^{53} + 12 q^{55} + 26 q^{57} - 26 q^{59} + 12 q^{61} - 5 q^{63} + 66 q^{65} + 21 q^{67} + 20 q^{69} - 50 q^{71} + 17 q^{73} + 14 q^{75} + 25 q^{77} + 9 q^{79} + 48 q^{81} - 25 q^{83} + 24 q^{85} + 10 q^{87} + 21 q^{89} + 9 q^{91} + 31 q^{93} - 22 q^{95} + 18 q^{97} - 102 q^{99}+O(q^{100}) $ 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9 - 18 * q^11 - 4 * q^13 + 2 * q^15 + 12 * q^17 + 7 * q^21 + 9 * q^23 + 25 * q^25 + 18 * q^27 + 34 * q^29 + 11 * q^31 + 4 * q^33 - 21 * q^35 - 22 * q^37 - 13 * q^39 + 32 * q^41 + 8 * q^43 + 13 * q^45 - 24 * q^47 + 36 * q^49 - 16 * q^51 - 2 * q^53 + 12 * q^55 + 26 * q^57 - 26 * q^59 + 12 * q^61 - 5 * q^63 + 66 * q^65 + 21 * q^67 + 20 * q^69 - 50 * q^71 + 17 * q^73 + 14 * q^75 + 25 * q^77 + 9 * q^79 + 48 * q^81 - 25 * q^83 + 24 * q^85 + 10 * q^87 + 21 * q^89 + 9 * q^91 + 31 * q^93 - 22 * q^95 + 18 * q^97 - 102 * 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
$3$	−0.656123	−0.378813	−0.189406	−	0.981899i	$-0.560656\pi$
$3$	−0.656123	−0.378813	−0.189406	+	0.981899i	$0.560656\pi$
$4$	0	0
$5$	4.07967	1.82448	0.912242	−	0.409652i	$-0.134350\pi$
$5$	4.07967	1.82448	0.912242	+	0.409652i	$0.134350\pi$
$6$	0	0
$7$	0.827612	0.312808	0.156404	−	0.987693i	$-0.450010\pi$
$7$	0.827612	0.312808	0.156404	+	0.987693i	$0.450010\pi$
$8$	0	0
$9$	−2.56950	−0.856501
$10$	0	0
$11$	3.65273	1.10134	0.550670	−	0.834723i	$-0.314372\pi$
$11$	3.65273	1.10134	0.550670	+	0.834723i	$0.314372\pi$
$12$	0	0
$13$	1.01942	0.282736	0.141368	−	0.989957i	$-0.454850\pi$
$13$	1.01942	0.282736	0.141368	+	0.989957i	$0.454850\pi$
$14$	0	0
$15$	−2.67677	−0.691138
$16$	0	0
$17$	3.12716	0.758447	0.379223	−	0.925305i	$-0.376191\pi$
$17$	3.12716	0.758447	0.379223	+	0.925305i	$0.376191\pi$
$18$	0	0
$19$	−4.20968	−0.965766	−0.482883	−	0.875685i	$-0.660410\pi$
$19$	−4.20968	−0.965766	−0.482883	+	0.875685i	$0.660410\pi$
$20$	0	0
$21$	−0.543015	−0.118496
$22$	0	0
$23$	−6.99846	−1.45928	−0.729640	−	0.683831i	$-0.760312\pi$
$23$	−6.99846	−1.45928	−0.729640	+	0.683831i	$0.760312\pi$
$24$	0	0
$25$	11.6437	2.32874
$26$	0	0
$27$	3.65428	0.703267
$28$	0	0
$29$	5.92375	1.10001	0.550006	−	0.835161i	$-0.314625\pi$
$29$	5.92375	1.10001	0.550006	+	0.835161i	$0.314625\pi$
$30$	0	0
$31$	1.93174	0.346951	0.173476	−	0.984838i	$-0.444500\pi$
$31$	1.93174	0.346951	0.173476	+	0.984838i	$0.444500\pi$
$32$	0	0
$33$	−2.39664	−0.417202
$34$	0	0
$35$	3.37638	0.570713
$36$	0	0
$37$	7.83866	1.28867	0.644334	−	0.764744i	$-0.277135\pi$
$37$	7.83866	1.28867	0.644334	+	0.764744i	$0.277135\pi$
$38$	0	0
$39$	−0.668864	−0.107104
$40$	0	0
$41$	0.429262	0.0670394	0.0335197	−	0.999438i	$-0.489328\pi$
$41$	0.429262	0.0670394	0.0335197	+	0.999438i	$0.489328\pi$
$42$	0	0
$43$	−1.87562	−0.286029	−0.143015	−	0.989721i	$-0.545680\pi$
$43$	−1.87562	−0.286029	−0.143015	+	0.989721i	$0.545680\pi$
$44$	0	0
$45$	−10.4827	−1.56267
$46$	0	0
$47$	−9.98732	−1.45680	−0.728400	−	0.685152i	$-0.759736\pi$
$47$	−9.98732	−1.45680	−0.728400	+	0.685152i	$0.759736\pi$
$48$	0	0
$49$	−6.31506	−0.902151
$50$	0	0
$51$	−2.05180	−0.287309
$52$	0	0
$53$	−8.00130	−1.09906	−0.549531	−	0.835473i	$-0.685194\pi$
$53$	−8.00130	−1.09906	−0.549531	+	0.835473i	$0.685194\pi$
$54$	0	0
$55$	14.9020	2.00938
$56$	0	0
$57$	2.76207	0.365845
$58$	0	0
$59$	13.2850	1.72956	0.864782	−	0.502148i	$-0.167457\pi$
$59$	13.2850	1.72956	0.864782	+	0.502148i	$0.167457\pi$
$60$	0	0
$61$	8.78970	1.12541	0.562703	−	0.826659i	$-0.309761\pi$
$61$	8.78970	1.12541	0.562703	+	0.826659i	$0.309761\pi$
$62$	0	0
$63$	−2.12655	−0.267920
$64$	0	0
$65$	4.15889	0.515846
$66$	0	0
$67$	6.21701	0.759528	0.379764	−	0.925083i	$-0.376005\pi$
$67$	6.21701	0.759528	0.379764	+	0.925083i	$0.376005\pi$
$68$	0	0
$69$	4.59185	0.552794
$70$	0	0
$71$	−10.9469	−1.29916	−0.649580	−	0.760294i	$-0.725055\pi$
$71$	−10.9469	−1.29916	−0.649580	+	0.760294i	$0.725055\pi$
$72$	0	0
$73$	13.3974	1.56805	0.784024	−	0.620731i	$-0.213164\pi$
$73$	13.3974	1.56805	0.784024	+	0.620731i	$0.213164\pi$
$74$	0	0
$75$	−7.63970	−0.882157
$76$	0	0
$77$	3.02305	0.344508
$78$	0	0
$79$	6.97796	0.785082	0.392541	−	0.919734i	$-0.371596\pi$
$79$	6.97796	0.785082	0.392541	+	0.919734i	$0.371596\pi$
$80$	0	0
$81$	5.31085	0.590094
$82$	0	0
$83$	16.1238	1.76982	0.884910	−	0.465762i	$-0.154220\pi$
$83$	16.1238	1.76982	0.884910	+	0.465762i	$0.154220\pi$
$84$	0	0
$85$	12.7578	1.38377
$86$	0	0
$87$	−3.88671	−0.416699
$88$	0	0
$89$	−1.61645	−0.171343	−0.0856717	−	0.996323i	$-0.527304\pi$
$89$	−1.61645	−0.171343	−0.0856717	+	0.996323i	$0.527304\pi$
$90$	0	0
$91$	0.843682	0.0884419
$92$	0	0
$93$	−1.26746	−0.131430
$94$	0	0
$95$	−17.1741	−1.76202
$96$	0	0
$97$	−14.0367	−1.42521	−0.712607	−	0.701563i	$-0.752486\pi$
$97$	−14.0367	−1.42521	−0.712607	+	0.701563i	$0.752486\pi$
$98$	0	0
$99$	−9.38571	−0.943299
$100$	0	0
$101$	0.0559113	0.00556339	0.00278169	−	0.999996i	$-0.499115\pi$
$101$	0.0559113	0.00556339	0.00278169	+	0.999996i	$0.499115\pi$
$102$	0	0
$103$	11.8641	1.16900	0.584502	−	0.811392i	$-0.301290\pi$
$103$	11.8641	1.16900	0.584502	+	0.811392i	$0.301290\pi$
$104$	0	0
$105$	−2.21532	−0.216193
$106$	0	0
$107$	−9.11517	−0.881197	−0.440598	−	0.897704i	$-0.645234\pi$
$107$	−9.11517	−0.881197	−0.440598	+	0.897704i	$0.645234\pi$
$108$	0	0
$109$	−3.20164	−0.306661	−0.153331	−	0.988175i	$-0.549000\pi$
$109$	−3.20164	−0.306661	−0.153331	+	0.988175i	$0.549000\pi$
$110$	0	0
$111$	−5.14313	−0.488164
$112$	0	0
$113$	−6.06795	−0.570825	−0.285412	−	0.958405i	$-0.592131\pi$
$113$	−6.06795	−0.570825	−0.285412	+	0.958405i	$0.592131\pi$
$114$	0	0
$115$	−28.5514	−2.66243
$116$	0	0
$117$	−2.61940	−0.242163
$118$	0	0
$119$	2.58807	0.237248
$120$	0	0
$121$	2.34247	0.212952
$122$	0	0
$123$	−0.281649	−0.0253954
$124$	0	0
$125$	27.1041	2.42427
$126$	0	0
$127$	12.5463	1.11330	0.556651	−	0.830747i	$-0.312086\pi$
$127$	12.5463	1.11330	0.556651	+	0.830747i	$0.312086\pi$
$128$	0	0
$129$	1.23064	0.108352
$130$	0	0
$131$	10.4573	0.913660	0.456830	−	0.889554i	$-0.348985\pi$
$131$	10.4573	0.913660	0.456830	+	0.889554i	$0.348985\pi$
$132$	0	0
$133$	−3.48398	−0.302099
$134$	0	0
$135$	14.9083	1.28310
$136$	0	0
$137$	−1.94361	−0.166054	−0.0830270	−	0.996547i	$-0.526459\pi$
$137$	−1.94361	−0.166054	−0.0830270	+	0.996547i	$0.526459\pi$
$138$	0	0
$139$	1.12875	0.0957393	0.0478697	−	0.998854i	$-0.484757\pi$
$139$	1.12875	0.0957393	0.0478697	+	0.998854i	$0.484757\pi$
$140$	0	0
$141$	6.55292	0.551855
$142$	0	0
$143$	3.72366	0.311388
$144$	0	0
$145$	24.1669	2.00695
$146$	0	0
$147$	4.14346	0.341747
$148$	0	0
$149$	19.0650	1.56187	0.780934	−	0.624613i	$-0.214743\pi$
$149$	19.0650	1.56187	0.780934	+	0.624613i	$0.214743\pi$
$150$	0	0
$151$	2.06796	0.168288	0.0841441	−	0.996454i	$-0.473184\pi$
$151$	2.06796	0.168288	0.0841441	+	0.996454i	$0.473184\pi$
$152$	0	0
$153$	−8.03523	−0.649610
$154$	0	0
$155$	7.88087	0.633007
$156$	0	0
$157$	−2.32431	−0.185500	−0.0927501	−	0.995689i	$-0.529566\pi$
$157$	−2.32431	−0.185500	−0.0927501	+	0.995689i	$0.529566\pi$
$158$	0	0
$159$	5.24984	0.416339
$160$	0	0
$161$	−5.79201	−0.456474
$162$	0	0
$163$	12.7044	0.995087	0.497544	−	0.867439i	$-0.334235\pi$
$163$	12.7044	0.995087	0.497544	+	0.867439i	$0.334235\pi$
$164$	0	0
$165$	−9.77752	−0.761179
$166$	0	0
$167$	11.1013	0.859041	0.429521	−	0.903057i	$-0.358683\pi$
$167$	11.1013	0.859041	0.429521	+	0.903057i	$0.358683\pi$
$168$	0	0
$169$	−11.9608	−0.920061
$170$	0	0
$171$	10.8168	0.827180
$172$	0	0
$173$	16.4587	1.25133	0.625664	−	0.780092i	$-0.284828\pi$
$173$	16.4587	1.25133	0.625664	+	0.780092i	$0.284828\pi$
$174$	0	0
$175$	9.63646	0.728448
$176$	0	0
$177$	−8.71662	−0.655181
$178$	0	0
$179$	−3.22776	−0.241254	−0.120627	−	0.992698i	$-0.538491\pi$
$179$	−3.22776	−0.241254	−0.120627	+	0.992698i	$0.538491\pi$
$180$	0	0
$181$	−24.5474	−1.82459	−0.912296	−	0.409532i	$-0.865692\pi$
$181$	−24.5474	−1.82459	−0.912296	+	0.409532i	$0.865692\pi$
$182$	0	0
$183$	−5.76713	−0.426319
$184$	0	0
$185$	31.9791	2.35115
$186$	0	0
$187$	11.4227	0.835308
$188$	0	0
$189$	3.02432	0.219987
$190$	0	0
$191$	−11.8200	−0.855269	−0.427634	−	0.903952i	$-0.640653\pi$
$191$	−11.8200	−0.855269	−0.427634	+	0.903952i	$0.640653\pi$
$192$	0	0
$193$	−4.42089	−0.318223	−0.159111	−	0.987261i	$-0.550863\pi$
$193$	−4.42089	−0.318223	−0.159111	+	0.987261i	$0.550863\pi$
$194$	0	0
$195$	−2.72874	−0.195409
$196$	0	0
$197$	20.1220	1.43363	0.716817	−	0.697262i	$-0.245598\pi$
$197$	20.1220	1.43363	0.716817	+	0.697262i	$0.245598\pi$
$198$	0	0
$199$	7.55742	0.535731	0.267866	−	0.963456i	$-0.413682\pi$
$199$	7.55742	0.535731	0.267866	+	0.963456i	$0.413682\pi$
$200$	0	0
$201$	−4.07912	−0.287719
$202$	0	0
$203$	4.90256	0.344092
$204$	0	0
$205$	1.75125	0.122312
$206$	0	0
$207$	17.9826	1.24987
$208$	0	0
$209$	−15.3768	−1.06364
$210$	0	0
$211$	−0.519024	−0.0357311	−0.0178655	−	0.999840i	$-0.505687\pi$
$211$	−0.519024	−0.0357311	−0.0178655	+	0.999840i	$0.505687\pi$
$212$	0	0
$213$	7.18252	0.492138
$214$	0	0
$215$	−7.65190	−0.521856
$216$	0	0
$217$	1.59873	0.108529
$218$	0	0
$219$	−8.79035	−0.593997
$220$	0	0
$221$	3.18788	0.214440
$222$	0	0
$223$	−20.0815	−1.34476	−0.672379	−	0.740207i	$-0.734727\pi$
$223$	−20.0815	−1.34476	−0.672379	+	0.740207i	$0.734727\pi$
$224$	0	0
$225$	−29.9185	−1.99457
$226$	0	0
$227$	18.7337	1.24340	0.621698	−	0.783257i	$-0.286443\pi$
$227$	18.7337	1.24340	0.621698	+	0.783257i	$0.286443\pi$
$228$	0	0
$229$	−7.71597	−0.509886	−0.254943	−	0.966956i	$-0.582057\pi$
$229$	−7.71597	−0.509886	−0.254943	+	0.966956i	$0.582057\pi$
$230$	0	0
$231$	−1.98349	−0.130504
$232$	0	0
$233$	−13.7327	−0.899659	−0.449830	−	0.893114i	$-0.648515\pi$
$233$	−13.7327	−0.899659	−0.449830	+	0.893114i	$0.648515\pi$
$234$	0	0
$235$	−40.7450	−2.65791
$236$	0	0
$237$	−4.57841	−0.297399
$238$	0	0
$239$	−22.5330	−1.45754	−0.728769	−	0.684759i	$-0.759907\pi$
$239$	−22.5330	−1.45754	−0.728769	+	0.684759i	$0.759907\pi$
$240$	0	0
$241$	−9.67520	−0.623234	−0.311617	−	0.950208i	$-0.600871\pi$
$241$	−9.67520	−0.623234	−0.311617	+	0.950208i	$0.600871\pi$
$242$	0	0
$243$	−14.4474	−0.926802
$244$	0	0
$245$	−25.7634	−1.64596
$246$	0	0
$247$	−4.29142	−0.273057
$248$	0	0
$249$	−10.5792	−0.670431
$250$	0	0
$251$	−6.94583	−0.438417	−0.219208	−	0.975678i	$-0.570347\pi$
$251$	−6.94583	−0.438417	−0.219208	+	0.975678i	$0.570347\pi$
$252$	0	0
$253$	−25.5635	−1.60716
$254$	0	0
$255$	−8.37066	−0.524191
$256$	0	0
$257$	7.43215	0.463605	0.231802	−	0.972763i	$-0.425538\pi$
$257$	7.43215	0.463605	0.231802	+	0.972763i	$0.425538\pi$
$258$	0	0
$259$	6.48736	0.403105
$260$	0	0
$261$	−15.2211	−0.942161
$262$	0	0
$263$	10.3744	0.639710	0.319855	−	0.947466i	$-0.396366\pi$
$263$	10.3744	0.639710	0.319855	+	0.947466i	$0.396366\pi$
$264$	0	0
$265$	−32.6427	−2.00522
$266$	0	0
$267$	1.06059	0.0649071
$268$	0	0
$269$	14.3855	0.877098	0.438549	−	0.898707i	$-0.355493\pi$
$269$	14.3855	0.877098	0.438549	+	0.898707i	$0.355493\pi$
$270$	0	0
$271$	−3.96558	−0.240892	−0.120446	−	0.992720i	$-0.538432\pi$
$271$	−3.96558	−0.240892	−0.120446	+	0.992720i	$0.538432\pi$
$272$	0	0
$273$	−0.553559	−0.0335029
$274$	0	0
$275$	42.5314	2.56474
$276$	0	0
$277$	8.75012	0.525744	0.262872	−	0.964831i	$-0.415330\pi$
$277$	8.75012	0.525744	0.262872	+	0.964831i	$0.415330\pi$
$278$	0	0
$279$	−4.96362	−0.297164
$280$	0	0
$281$	19.2887	1.15067	0.575333	−	0.817919i	$-0.304873\pi$
$281$	19.2887	1.15067	0.575333	+	0.817919i	$0.304873\pi$
$282$	0	0
$283$	7.13668	0.424232	0.212116	−	0.977245i	$-0.431965\pi$
$283$	7.13668	0.424232	0.212116	+	0.977245i	$0.431965\pi$
$284$	0	0
$285$	11.2683	0.667478
$286$	0	0
$287$	0.355262	0.0209704
$288$	0	0
$289$	−7.22090	−0.424759
$290$	0	0
$291$	9.20983	0.539890
$292$	0	0
$293$	15.3448	0.896452	0.448226	−	0.893920i	$-0.352056\pi$
$293$	15.3448	0.896452	0.448226	+	0.893920i	$0.352056\pi$
$294$	0	0
$295$	54.1985	3.15556
$296$	0	0
$297$	13.3481	0.774536
$298$	0	0
$299$	−7.13436	−0.412590
$300$	0	0
$301$	−1.55228	−0.0894721
$302$	0	0
$303$	−0.0366847	−0.00210748
$304$	0	0
$305$	35.8591	2.05329
$306$	0	0
$307$	1.37575	0.0785180	0.0392590	−	0.999229i	$-0.487500\pi$
$307$	1.37575	0.0785180	0.0392590	+	0.999229i	$0.487500\pi$
$308$	0	0
$309$	−7.78431	−0.442834
$310$	0	0
$311$	−24.1027	−1.36674	−0.683370	−	0.730072i	$-0.739487\pi$
$311$	−24.1027	−1.36674	−0.683370	+	0.730072i	$0.739487\pi$
$312$	0	0
$313$	−18.0561	−1.02059	−0.510296	−	0.859999i	$-0.670464\pi$
$313$	−18.0561	−1.02059	−0.510296	+	0.859999i	$0.670464\pi$
$314$	0	0
$315$	−8.67562	−0.488816
$316$	0	0
$317$	−33.0954	−1.85882	−0.929412	−	0.369045i	$-0.879685\pi$
$317$	−33.0954	−1.85882	−0.929412	+	0.369045i	$0.879685\pi$
$318$	0	0
$319$	21.6379	1.21149
$320$	0	0
$321$	5.98068	0.333809
$322$	0	0
$323$	−13.1643	−0.732482
$324$	0	0
$325$	11.8698	0.658418
$326$	0	0
$327$	2.10067	0.116167
$328$	0	0
$329$	−8.26562	−0.455699
$330$	0	0
$331$	10.4728	0.575636	0.287818	−	0.957685i	$-0.407070\pi$
$331$	10.4728	0.575636	0.287818	+	0.957685i	$0.407070\pi$
$332$	0	0
$333$	−20.1414	−1.10374
$334$	0	0
$335$	25.3633	1.38575
$336$	0	0
$337$	−27.6057	−1.50378	−0.751890	−	0.659289i	$-0.770857\pi$
$337$	−27.6057	−1.50378	−0.751890	+	0.659289i	$0.770857\pi$
$338$	0	0
$339$	3.98132	0.216236
$340$	0	0
$341$	7.05615	0.382112
$342$	0	0
$343$	−11.0197	−0.595008
$344$	0	0
$345$	18.7332	1.00856
$346$	0	0
$347$	7.10820	0.381588	0.190794	−	0.981630i	$-0.438894\pi$
$347$	7.10820	0.381588	0.190794	+	0.981630i	$0.438894\pi$
$348$	0	0
$349$	23.9211	1.28047	0.640235	−	0.768179i	$-0.278837\pi$
$349$	23.9211	1.28047	0.640235	+	0.768179i	$0.278837\pi$
$350$	0	0
$351$	3.72524	0.198839
$352$	0	0
$353$	−26.3395	−1.40191	−0.700956	−	0.713205i	$-0.747243\pi$
$353$	−26.3395	−1.40191	−0.700956	+	0.713205i	$0.747243\pi$
$354$	0	0
$355$	−44.6598	−2.37029
$356$	0	0
$357$	−1.69809	−0.0898726
$358$	0	0
$359$	13.5796	0.716705	0.358353	−	0.933586i	$-0.383339\pi$
$359$	13.5796	0.716705	0.358353	+	0.933586i	$0.383339\pi$
$360$	0	0
$361$	−1.27861	−0.0672952
$362$	0	0
$363$	−1.53695	−0.0806690
$364$	0	0
$365$	54.6570	2.86088
$366$	0	0
$367$	35.9960	1.87898	0.939488	−	0.342583i	$-0.111302\pi$
$367$	35.9960	1.87898	0.939488	+	0.342583i	$0.111302\pi$
$368$	0	0
$369$	−1.10299	−0.0574193
$370$	0	0
$371$	−6.62197	−0.343795
$372$	0	0
$373$	3.10144	0.160587	0.0802933	−	0.996771i	$-0.474414\pi$
$373$	3.10144	0.160587	0.0802933	+	0.996771i	$0.474414\pi$
$374$	0	0
$375$	−17.7836	−0.918343
$376$	0	0
$377$	6.03877	0.311013
$378$	0	0
$379$	0.107046	0.00549859	0.00274929	−	0.999996i	$-0.499125\pi$
$379$	0.107046	0.00549859	0.00274929	+	0.999996i	$0.499125\pi$
$380$	0	0
$381$	−8.23190	−0.421733
$382$	0	0
$383$	−20.2718	−1.03584	−0.517922	−	0.855428i	$-0.673294\pi$
$383$	−20.2718	−1.03584	−0.517922	+	0.855428i	$0.673294\pi$
$384$	0	0
$385$	12.3330	0.628549
$386$	0	0
$387$	4.81941	0.244984
$388$	0	0
$389$	−36.3434	−1.84268	−0.921341	−	0.388755i	$-0.872905\pi$
$389$	−36.3434	−1.84268	−0.921341	+	0.388755i	$0.872905\pi$
$390$	0	0
$391$	−21.8853	−1.10679
$392$	0	0
$393$	−6.86129	−0.346106
$394$	0	0
$395$	28.4678	1.43237
$396$	0	0
$397$	29.3155	1.47130	0.735651	−	0.677361i	$-0.236876\pi$
$397$	29.3155	1.47130	0.735651	+	0.677361i	$0.236876\pi$
$398$	0	0
$399$	2.28592	0.114439
$400$	0	0
$401$	24.9935	1.24812	0.624058	−	0.781378i	$-0.285483\pi$
$401$	24.9935	1.24812	0.624058	+	0.781378i	$0.285483\pi$
$402$	0	0
$403$	1.96925	0.0980955
$404$	0	0
$405$	21.6665	1.07662
$406$	0	0
$407$	28.6325	1.41926
$408$	0	0
$409$	12.0369	0.595185	0.297592	−	0.954693i	$-0.403816\pi$
$409$	12.0369	0.595185	0.297592	+	0.954693i	$0.403816\pi$
$410$	0	0
$411$	1.27525	0.0629034
$412$	0	0
$413$	10.9948	0.541021
$414$	0	0
$415$	65.7799	3.22901
$416$	0	0
$417$	−0.740599	−0.0362673
$418$	0	0
$419$	−6.69023	−0.326839	−0.163419	−	0.986557i	$-0.552252\pi$
$419$	−6.69023	−0.326839	−0.163419	+	0.986557i	$0.552252\pi$
$420$	0	0
$421$	1.06238	0.0517771	0.0258885	−	0.999665i	$-0.491759\pi$
$421$	1.06238	0.0517771	0.0258885	+	0.999665i	$0.491759\pi$
$422$	0	0
$423$	25.6624	1.24775
$424$	0	0
$425$	36.4117	1.76623
$426$	0	0
$427$	7.27446	0.352036
$428$	0	0
$429$	−2.44318	−0.117958
$430$	0	0
$431$	−1.42868	−0.0688171	−0.0344085	−	0.999408i	$-0.510955\pi$
$431$	−1.42868	−0.0688171	−0.0344085	+	0.999408i	$0.510955\pi$
$432$	0	0
$433$	−29.8991	−1.43686	−0.718430	−	0.695599i	$-0.755139\pi$
$433$	−29.8991	−1.43686	−0.718430	+	0.695599i	$0.755139\pi$
$434$	0	0
$435$	−15.8565	−0.760260
$436$	0	0
$437$	29.4613	1.40932
$438$	0	0
$439$	−21.6986	−1.03562	−0.517808	−	0.855497i	$-0.673252\pi$
$439$	−21.6986	−1.03562	−0.517808	+	0.855497i	$0.673252\pi$
$440$	0	0
$441$	16.2266	0.772693
$442$	0	0
$443$	−27.2994	−1.29703	−0.648516	−	0.761201i	$-0.724610\pi$
$443$	−27.2994	−1.29703	−0.648516	+	0.761201i	$0.724610\pi$
$444$	0	0
$445$	−6.59458	−0.312613
$446$	0	0
$447$	−12.5090	−0.591656
$448$	0	0
$449$	−14.7271	−0.695016	−0.347508	−	0.937677i	$-0.612972\pi$
$449$	−14.7271	−0.695016	−0.347508	+	0.937677i	$0.612972\pi$
$450$	0	0
$451$	1.56798	0.0738332
$452$	0	0
$453$	−1.35684	−0.0637498
$454$	0	0
$455$	3.44194	0.161361
$456$	0	0
$457$	−40.2029	−1.88061	−0.940305	−	0.340332i	$-0.889460\pi$
$457$	−40.2029	−1.88061	−0.940305	+	0.340332i	$0.889460\pi$
$458$	0	0
$459$	11.4275	0.533390
$460$	0	0
$461$	17.0667	0.794874	0.397437	−	0.917629i	$-0.369900\pi$
$461$	17.0667	0.794874	0.397437	+	0.917629i	$0.369900\pi$
$462$	0	0
$463$	−4.18450	−0.194470	−0.0972350	−	0.995261i	$-0.531000\pi$
$463$	−4.18450	−0.194470	−0.0972350	+	0.995261i	$0.531000\pi$
$464$	0	0
$465$	−5.17083	−0.239791
$466$	0	0
$467$	15.3038	0.708176	0.354088	−	0.935212i	$-0.384791\pi$
$467$	15.3038	0.708176	0.354088	+	0.935212i	$0.384791\pi$
$468$	0	0
$469$	5.14527	0.237586
$470$	0	0
$471$	1.52503	0.0702699
$472$	0	0
$473$	−6.85114	−0.315016
$474$	0	0
$475$	−49.0162	−2.24902
$476$	0	0
$477$	20.5594	0.941348
$478$	0	0
$479$	30.0685	1.37386	0.686932	−	0.726722i	$-0.258957\pi$
$479$	30.0685	1.37386	0.686932	+	0.726722i	$0.258957\pi$
$480$	0	0
$481$	7.99087	0.364352
$482$	0	0
$483$	3.80027	0.172918
$484$	0	0
$485$	−57.2652	−2.60028
$486$	0	0
$487$	24.7944	1.12354	0.561772	−	0.827292i	$-0.310120\pi$
$487$	24.7944	1.12354	0.561772	+	0.827292i	$0.310120\pi$
$488$	0	0
$489$	−8.33567	−0.376952
$490$	0	0
$491$	−19.8418	−0.895449	−0.447725	−	0.894172i	$-0.647765\pi$
$491$	−19.8418	−0.895449	−0.447725	+	0.894172i	$0.647765\pi$
$492$	0	0
$493$	18.5245	0.834301
$494$	0	0
$495$	−38.2906	−1.72103
$496$	0	0
$497$	−9.05979	−0.406387
$498$	0	0
$499$	37.9026	1.69675	0.848376	−	0.529394i	$-0.177581\pi$
$499$	37.9026	1.69675	0.848376	+	0.529394i	$0.177581\pi$
$500$	0	0
$501$	−7.28380	−0.325416
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	0.228100	0.0101503
$506$	0	0
$507$	7.84775	0.348531
$508$	0	0
$509$	15.4778	0.686043	0.343021	−	0.939328i	$-0.388550\pi$
$509$	15.4778	0.686043	0.343021	+	0.939328i	$0.388550\pi$
$510$	0	0
$511$	11.0878	0.490497
$512$	0	0
$513$	−15.3833	−0.679191
$514$	0	0
$515$	48.4016	2.13283
$516$	0	0
$517$	−36.4810	−1.60443
$518$	0	0
$519$	−10.7989	−0.474020
$520$	0	0
$521$	41.7097	1.82734	0.913668	−	0.406462i	$-0.133238\pi$
$521$	41.7097	1.82734	0.913668	+	0.406462i	$0.133238\pi$
$522$	0	0
$523$	−42.3591	−1.85223	−0.926117	−	0.377235i	$-0.876875\pi$
$523$	−42.3591	−1.85223	−0.926117	+	0.377235i	$0.876875\pi$
$524$	0	0
$525$	−6.32271	−0.275946
$526$	0	0
$527$	6.04086	0.263144
$528$	0	0
$529$	25.9785	1.12950
$530$	0	0
$531$	−34.1359	−1.48137
$532$	0	0
$533$	0.437597	0.0189544
$534$	0	0
$535$	−37.1869	−1.60773
$536$	0	0
$537$	2.11781	0.0913903
$538$	0	0
$539$	−23.0672	−0.993576
$540$	0	0
$541$	27.1811	1.16861	0.584303	−	0.811535i	$-0.301368\pi$
$541$	27.1811	1.16861	0.584303	+	0.811535i	$0.301368\pi$
$542$	0	0
$543$	16.1061	0.691179
$544$	0	0
$545$	−13.0616	−0.559498
$546$	0	0
$547$	32.7735	1.40129	0.700647	−	0.713508i	$-0.252895\pi$
$547$	32.7735	1.40129	0.700647	+	0.713508i	$0.252895\pi$
$548$	0	0
$549$	−22.5852	−0.963911
$550$	0	0
$551$	−24.9371	−1.06235
$552$	0	0
$553$	5.77504	0.245580
$554$	0	0
$555$	−20.9823	−0.890647
$556$	0	0
$557$	−11.6853	−0.495122	−0.247561	−	0.968872i	$-0.579629\pi$
$557$	−11.6853	−0.495122	−0.247561	+	0.968872i	$0.579629\pi$
$558$	0	0
$559$	−1.91204	−0.0808706
$560$	0	0
$561$	−7.49468	−0.316426
$562$	0	0
$563$	−34.8146	−1.46726	−0.733630	−	0.679549i	$-0.762176\pi$
$563$	−34.8146	−1.46726	−0.733630	+	0.679549i	$0.762176\pi$
$564$	0	0
$565$	−24.7552	−1.04146
$566$	0	0
$567$	4.39532	0.184586
$568$	0	0
$569$	16.9988	0.712628	0.356314	−	0.934366i	$-0.384033\pi$
$569$	16.9988	0.712628	0.356314	+	0.934366i	$0.384033\pi$
$570$	0	0
$571$	12.0790	0.505492	0.252746	−	0.967533i	$-0.418666\pi$
$571$	12.0790	0.505492	0.252746	+	0.967533i	$0.418666\pi$
$572$	0	0
$573$	7.75541	0.323987
$574$	0	0
$575$	−81.4880	−3.39828
$576$	0	0
$577$	28.9182	1.20388	0.601940	−	0.798541i	$-0.294395\pi$
$577$	28.9182	1.20388	0.601940	+	0.798541i	$0.294395\pi$
$578$	0	0
$579$	2.90065	0.120547
$580$	0	0
$581$	13.3443	0.553613
$582$	0	0
$583$	−29.2266	−1.21044
$584$	0	0
$585$	−10.6863	−0.441823
$586$	0	0
$587$	24.0071	0.990879	0.495439	−	0.868643i	$-0.335007\pi$
$587$	24.0071	0.990879	0.495439	+	0.868643i	$0.335007\pi$
$588$	0	0
$589$	−8.13202	−0.335074
$590$	0	0
$591$	−13.2025	−0.543079
$592$	0	0
$593$	−7.86849	−0.323120	−0.161560	−	0.986863i	$-0.551653\pi$
$593$	−7.86849	−0.323120	−0.161560	+	0.986863i	$0.551653\pi$
$594$	0	0
$595$	10.5585	0.432855
$596$	0	0
$597$	−4.95860	−0.202942
$598$	0	0
$599$	−6.81401	−0.278413	−0.139206	−	0.990263i	$-0.544455\pi$
$599$	−6.81401	−0.278413	−0.139206	+	0.990263i	$0.544455\pi$
$600$	0	0
$601$	22.7195	0.926747	0.463373	−	0.886163i	$-0.346639\pi$
$601$	22.7195	0.926747	0.463373	+	0.886163i	$0.346639\pi$
$602$	0	0
$603$	−15.9746	−0.650536
$604$	0	0
$605$	9.55651	0.388527
$606$	0	0
$607$	−10.7015	−0.434361	−0.217181	−	0.976131i	$-0.569686\pi$
$607$	−10.7015	−0.434361	−0.217181	+	0.976131i	$0.569686\pi$
$608$	0	0
$609$	−3.21669	−0.130347
$610$	0	0
$611$	−10.1813	−0.411889
$612$	0	0
$613$	26.7949	1.08223	0.541117	−	0.840947i	$-0.318002\pi$
$613$	26.7949	1.08223	0.541117	+	0.840947i	$0.318002\pi$
$614$	0	0
$615$	−1.14903	−0.0463335
$616$	0	0
$617$	28.0353	1.12866	0.564329	−	0.825550i	$-0.309135\pi$
$617$	28.0353	1.12866	0.564329	+	0.825550i	$0.309135\pi$
$618$	0	0
$619$	−19.8287	−0.796983	−0.398492	−	0.917172i	$-0.630466\pi$
$619$	−19.8287	−0.796983	−0.398492	+	0.917172i	$0.630466\pi$
$620$	0	0
$621$	−25.5743	−1.02626
$622$	0	0
$623$	−1.33779	−0.0535975
$624$	0	0
$625$	52.3573	2.09429
$626$	0	0
$627$	10.0891	0.402920
$628$	0	0
$629$	24.5127	0.977385
$630$	0	0
$631$	−15.1993	−0.605076	−0.302538	−	0.953137i	$-0.597834\pi$
$631$	−15.1993	−0.605076	−0.302538	+	0.953137i	$0.597834\pi$
$632$	0	0
$633$	0.340544	0.0135354
$634$	0	0
$635$	51.1846	2.03120
$636$	0	0
$637$	−6.43768	−0.255070
$638$	0	0
$639$	28.1281	1.11273
$640$	0	0
$641$	0.390249	0.0154139	0.00770695	−	0.999970i	$-0.497547\pi$
$641$	0.390249	0.0154139	0.00770695	+	0.999970i	$0.497547\pi$
$642$	0	0
$643$	4.54344	0.179176	0.0895879	−	0.995979i	$-0.471445\pi$
$643$	4.54344	0.179176	0.0895879	+	0.995979i	$0.471445\pi$
$644$	0	0
$645$	5.02059	0.197686
$646$	0	0
$647$	44.3067	1.74187	0.870937	−	0.491394i	$-0.163512\pi$
$647$	44.3067	1.74187	0.870937	+	0.491394i	$0.163512\pi$
$648$	0	0
$649$	48.5267	1.90484
$650$	0	0
$651$	−1.04897	−0.0411122
$652$	0	0
$653$	−19.7415	−0.772546	−0.386273	−	0.922385i	$-0.626238\pi$
$653$	−19.7415	−0.772546	−0.386273	+	0.922385i	$0.626238\pi$
$654$	0	0
$655$	42.6624	1.66696
$656$	0	0
$657$	−34.4246	−1.34303
$658$	0	0
$659$	35.6313	1.38800	0.693999	−	0.719976i	$-0.255847\pi$
$659$	35.6313	1.38800	0.693999	+	0.719976i	$0.255847\pi$
$660$	0	0
$661$	4.60081	0.178951	0.0894755	−	0.995989i	$-0.471481\pi$
$661$	4.60081	0.178951	0.0894755	+	0.995989i	$0.471481\pi$
$662$	0	0
$663$	−2.09164	−0.0812326
$664$	0	0
$665$	−14.2135	−0.551175
$666$	0	0
$667$	−41.4571	−1.60523
$668$	0	0
$669$	13.1759	0.509411
$670$	0	0
$671$	32.1065	1.23946
$672$	0	0
$673$	36.9042	1.42255	0.711277	−	0.702912i	$-0.248117\pi$
$673$	36.9042	1.42255	0.711277	+	0.702912i	$0.248117\pi$
$674$	0	0
$675$	42.5494	1.63773
$676$	0	0
$677$	−25.5262	−0.981051	−0.490526	−	0.871427i	$-0.663195\pi$
$677$	−25.5262	−0.981051	−0.490526	+	0.871427i	$0.663195\pi$
$678$	0	0
$679$	−11.6170	−0.445818
$680$	0	0
$681$	−12.2916	−0.471015
$682$	0	0
$683$	−16.3747	−0.626562	−0.313281	−	0.949660i	$-0.601428\pi$
$683$	−16.3747	−0.626562	−0.313281	+	0.949660i	$0.601428\pi$
$684$	0	0
$685$	−7.92929	−0.302963
$686$	0	0
$687$	5.06263	0.193151
$688$	0	0
$689$	−8.15667	−0.310744
$690$	0	0
$691$	26.4422	1.00591	0.502955	−	0.864312i	$-0.332246\pi$
$691$	26.4422	1.00591	0.502955	+	0.864312i	$0.332246\pi$
$692$	0	0
$693$	−7.76772	−0.295071
$694$	0	0
$695$	4.60493	0.174675
$696$	0	0
$697$	1.34237	0.0508458
$698$	0	0
$699$	9.01035	0.340803
$700$	0	0
$701$	−19.6590	−0.742512	−0.371256	−	0.928531i	$-0.621073\pi$
$701$	−19.6590	−0.742512	−0.371256	+	0.928531i	$0.621073\pi$
$702$	0	0
$703$	−32.9982	−1.24455
$704$	0	0
$705$	26.7337	1.00685
$706$	0	0
$707$	0.0462729	0.00174027
$708$	0	0
$709$	−18.3448	−0.688955	−0.344477	−	0.938795i	$-0.611944\pi$
$709$	−18.3448	−0.688955	−0.344477	+	0.938795i	$0.611944\pi$
$710$	0	0
$711$	−17.9299	−0.672424
$712$	0	0
$713$	−13.5192	−0.506299
$714$	0	0
$715$	15.1913	0.568123
$716$	0	0
$717$	14.7844	0.552135
$718$	0	0
$719$	10.5400	0.393074	0.196537	−	0.980496i	$-0.437030\pi$
$719$	10.5400	0.393074	0.196537	+	0.980496i	$0.437030\pi$
$720$	0	0
$721$	9.81887	0.365674
$722$	0	0
$723$	6.34812	0.236089
$724$	0	0
$725$	68.9744	2.56164
$726$	0	0
$727$	29.0789	1.07848	0.539238	−	0.842153i	$-0.318712\pi$
$727$	29.0789	1.07848	0.539238	+	0.842153i	$0.318712\pi$
$728$	0	0
$729$	−6.45326	−0.239010
$730$	0	0
$731$	−5.86535	−0.216938
$732$	0	0
$733$	−48.5065	−1.79163	−0.895814	−	0.444429i	$-0.853406\pi$
$733$	−48.5065	−1.79163	−0.895814	+	0.444429i	$0.853406\pi$
$734$	0	0
$735$	16.9039	0.623511
$736$	0	0
$737$	22.7091	0.836499
$738$	0	0
$739$	2.28173	0.0839349	0.0419674	−	0.999119i	$-0.486637\pi$
$739$	2.28173	0.0839349	0.0419674	+	0.999119i	$0.486637\pi$
$740$	0	0
$741$	2.81570	0.103437
$742$	0	0
$743$	40.9558	1.50252	0.751261	−	0.660005i	$-0.229446\pi$
$743$	40.9558	1.50252	0.751261	+	0.660005i	$0.229446\pi$
$744$	0	0
$745$	77.7791	2.84960
$746$	0	0
$747$	−41.4302	−1.51585
$748$	0	0
$749$	−7.54382	−0.275645
$750$	0	0
$751$	31.5760	1.15222	0.576112	−	0.817371i	$-0.304569\pi$
$751$	31.5760	1.15222	0.576112	+	0.817371i	$0.304569\pi$
$752$	0	0
$753$	4.55732	0.166078
$754$	0	0
$755$	8.43660	0.307039
$756$	0	0
$757$	14.9705	0.544113	0.272056	−	0.962281i	$-0.412296\pi$
$757$	14.9705	0.544113	0.272056	+	0.962281i	$0.412296\pi$
$758$	0	0
$759$	16.7728	0.608815
$760$	0	0
$761$	−38.0399	−1.37895	−0.689473	−	0.724311i	$-0.742158\pi$
$761$	−38.0399	−1.37895	−0.689473	+	0.724311i	$0.742158\pi$
$762$	0	0
$763$	−2.64971	−0.0959260
$764$	0	0
$765$	−32.7811	−1.18520
$766$	0	0
$767$	13.5430	0.489009
$768$	0	0
$769$	−8.90082	−0.320972	−0.160486	−	0.987038i	$-0.551306\pi$
$769$	−8.90082	−0.320972	−0.160486	+	0.987038i	$0.551306\pi$
$770$	0	0
$771$	−4.87641	−0.175620
$772$	0	0
$773$	20.0877	0.722503	0.361251	−	0.932468i	$-0.382350\pi$
$773$	20.0877	0.722503	0.361251	+	0.932468i	$0.382350\pi$
$774$	0	0
$775$	22.4926	0.807960
$776$	0	0
$777$	−4.25651	−0.152701
$778$	0	0
$779$	−1.80705	−0.0647444
$780$	0	0
$781$	−39.9862	−1.43082
$782$	0	0
$783$	21.6470	0.773602
$784$	0	0
$785$	−9.48242	−0.338442
$786$	0	0
$787$	−19.0644	−0.679572	−0.339786	−	0.940503i	$-0.610355\pi$
$787$	−19.0644	−0.679572	−0.339786	+	0.940503i	$0.610355\pi$
$788$	0	0
$789$	−6.80686	−0.242330
$790$	0	0
$791$	−5.02190	−0.178558
$792$	0	0
$793$	8.96038	0.318192
$794$	0	0
$795$	21.4176	0.759604
$796$	0	0
$797$	−48.6270	−1.72246	−0.861228	−	0.508218i	$-0.830304\pi$
$797$	−48.6270	−1.72246	−0.861228	+	0.508218i	$0.830304\pi$
$798$	0	0
$799$	−31.2319	−1.10491
$800$	0	0
$801$	4.15347	0.146756
$802$	0	0
$803$	48.9371	1.72695
$804$	0	0
$805$	−23.6295	−0.832830
$806$	0	0
$807$	−9.43865	−0.332256
$808$	0	0
$809$	35.9972	1.26560	0.632798	−	0.774317i	$-0.281906\pi$
$809$	35.9972	1.26560	0.632798	+	0.774317i	$0.281906\pi$
$810$	0	0
$811$	−28.3337	−0.994932	−0.497466	−	0.867484i	$-0.665736\pi$
$811$	−28.3337	−0.994932	−0.497466	+	0.867484i	$0.665736\pi$
$812$	0	0
$813$	2.60191	0.0912529
$814$	0	0
$815$	51.8298	1.81552
$816$	0	0
$817$	7.89575	0.276237
$818$	0	0
$819$	−2.16784	−0.0757505
$820$	0	0
$821$	−35.3222	−1.23275	−0.616377	−	0.787451i	$-0.711400\pi$
$821$	−35.3222	−1.23275	−0.616377	+	0.787451i	$0.711400\pi$
$822$	0	0
$823$	−46.8943	−1.63463	−0.817316	−	0.576190i	$-0.804539\pi$
$823$	−46.8943	−1.63463	−0.817316	+	0.576190i	$0.804539\pi$
$824$	0	0
$825$	−27.9058	−0.971556
$826$	0	0
$827$	24.1986	0.841466	0.420733	−	0.907184i	$-0.361773\pi$
$827$	24.1986	0.841466	0.420733	+	0.907184i	$0.361773\pi$
$828$	0	0
$829$	−29.2621	−1.01631	−0.508157	−	0.861264i	$-0.669673\pi$
$829$	−29.2621	−1.01631	−0.508157	+	0.861264i	$0.669673\pi$
$830$	0	0
$831$	−5.74116	−0.199159
$832$	0	0
$833$	−19.7482	−0.684234
$834$	0	0
$835$	45.2895	1.56731
$836$	0	0
$837$	7.05913	0.243999
$838$	0	0
$839$	24.8493	0.857895	0.428947	−	0.903330i	$-0.358885\pi$
$839$	24.8493	0.857895	0.428947	+	0.903330i	$0.358885\pi$
$840$	0	0
$841$	6.09078	0.210027
$842$	0	0
$843$	−12.6557	−0.435887
$844$	0	0
$845$	−48.7961	−1.67864
$846$	0	0
$847$	1.93866	0.0666130
$848$	0	0
$849$	−4.68254	−0.160704
$850$	0	0
$851$	−54.8585	−1.88053
$852$	0	0
$853$	−41.1391	−1.40858	−0.704289	−	0.709914i	$-0.748734\pi$
$853$	−41.1391	−1.40858	−0.704289	+	0.709914i	$0.748734\pi$
$854$	0	0
$855$	44.1289	1.50918
$856$	0	0
$857$	47.5380	1.62387	0.811933	−	0.583750i	$-0.198415\pi$
$857$	47.5380	1.62387	0.811933	+	0.583750i	$0.198415\pi$
$858$	0	0
$859$	−3.76888	−0.128593	−0.0642963	−	0.997931i	$-0.520480\pi$
$859$	−3.76888	−0.128593	−0.0642963	+	0.997931i	$0.520480\pi$
$860$	0	0
$861$	−0.233096	−0.00794388
$862$	0	0
$863$	−30.4615	−1.03692	−0.518461	−	0.855101i	$-0.673495\pi$
$863$	−30.4615	−1.03692	−0.518461	+	0.855101i	$0.673495\pi$
$864$	0	0
$865$	67.1459	2.28303
$866$	0	0
$867$	4.73780	0.160904
$868$	0	0
$869$	25.4887	0.864643
$870$	0	0
$871$	6.33773	0.214746
$872$	0	0
$873$	36.0674	1.22070
$874$	0	0
$875$	22.4317	0.758329
$876$	0	0
$877$	−32.8507	−1.10929	−0.554645	−	0.832087i	$-0.687146\pi$
$877$	−32.8507	−1.10929	−0.554645	+	0.832087i	$0.687146\pi$
$878$	0	0
$879$	−10.0681	−0.339588
$880$	0	0
$881$	13.2854	0.447596	0.223798	−	0.974636i	$-0.428154\pi$
$881$	13.2854	0.447596	0.223798	+	0.974636i	$0.428154\pi$
$882$	0	0
$883$	6.44108	0.216760	0.108380	−	0.994110i	$-0.465434\pi$
$883$	6.44108	0.216760	0.108380	+	0.994110i	$0.465434\pi$
$884$	0	0
$885$	−35.5609	−1.19537
$886$	0	0
$887$	20.9626	0.703857	0.351928	−	0.936027i	$-0.385526\pi$
$887$	20.9626	0.703857	0.351928	+	0.936027i	$0.385526\pi$
$888$	0	0
$889$	10.3834	0.348249
$890$	0	0
$891$	19.3991	0.649895
$892$	0	0
$893$	42.0434	1.40693
$894$	0	0
$895$	−13.1682	−0.440164
$896$	0	0
$897$	4.68102	0.156295
$898$	0	0
$899$	11.4432	0.381651
$900$	0	0
$901$	−25.0213	−0.833581
$902$	0	0
$903$	1.01849	0.0338932
$904$	0	0
$905$	−100.145	−3.32894
$906$	0	0
$907$	−22.9319	−0.761441	−0.380721	−	0.924690i	$-0.624324\pi$
$907$	−22.9319	−0.761441	−0.380721	+	0.924690i	$0.624324\pi$
$908$	0	0
$909$	−0.143664	−0.00476504
$910$	0	0
$911$	16.9964	0.563117	0.281558	−	0.959544i	$-0.409149\pi$
$911$	16.9964	0.563117	0.281558	+	0.959544i	$0.409149\pi$
$912$	0	0
$913$	58.8961	1.94917
$914$	0	0
$915$	−23.5280	−0.777811
$916$	0	0
$917$	8.65459	0.285800
$918$	0	0
$919$	−40.1274	−1.32368	−0.661840	−	0.749645i	$-0.730224\pi$
$919$	−40.1274	−1.32368	−0.661840	+	0.749645i	$0.730224\pi$
$920$	0	0
$921$	−0.902660	−0.0297436
$922$	0	0
$923$	−11.1595	−0.367319
$924$	0	0
$925$	91.2710	3.00097
$926$	0	0
$927$	−30.4848	−1.00125
$928$	0	0
$929$	25.5140	0.837087	0.418544	−	0.908197i	$-0.362541\pi$
$929$	25.5140	0.837087	0.418544	+	0.908197i	$0.362541\pi$
$930$	0	0
$931$	26.5844	0.871267
$932$	0	0
$933$	15.8144	0.517739
$934$	0	0
$935$	46.6007	1.52401
$936$	0	0
$937$	33.8573	1.10607	0.553035	−	0.833158i	$-0.313470\pi$
$937$	33.8573	1.10607	0.553035	+	0.833158i	$0.313470\pi$
$938$	0	0
$939$	11.8470	0.386613
$940$	0	0
$941$	−42.5547	−1.38724	−0.693622	−	0.720339i	$-0.743986\pi$
$941$	−42.5547	−1.38724	−0.693622	+	0.720339i	$0.743986\pi$
$942$	0	0
$943$	−3.00417	−0.0978293
$944$	0	0
$945$	12.3382	0.401363
$946$	0	0
$947$	−13.4986	−0.438645	−0.219322	−	0.975652i	$-0.570385\pi$
$947$	−13.4986	−0.438645	−0.219322	+	0.975652i	$0.570385\pi$
$948$	0	0
$949$	13.6575	0.443343
$950$	0	0
$951$	21.7147	0.704146
$952$	0	0
$953$	−7.30869	−0.236752	−0.118376	−	0.992969i	$-0.537769\pi$
$953$	−7.30869	−0.236752	−0.118376	+	0.992969i	$0.537769\pi$
$954$	0	0
$955$	−48.2219	−1.56042
$956$	0	0
$957$	−14.1971	−0.458928
$958$	0	0
$959$	−1.60856	−0.0519430
$960$	0	0
$961$	−27.2684	−0.879625
$962$	0	0
$963$	23.4214	0.754746
$964$	0	0
$965$	−18.0358	−0.580592
$966$	0	0
$967$	−16.1507	−0.519371	−0.259686	−	0.965693i	$-0.583619\pi$
$967$	−16.1507	−0.519371	−0.259686	+	0.965693i	$0.583619\pi$
$968$	0	0
$969$	8.63742	0.277474
$970$	0	0
$971$	−38.9620	−1.25035	−0.625175	−	0.780485i	$-0.714972\pi$
$971$	−38.9620	−1.25035	−0.625175	+	0.780485i	$0.714972\pi$
$972$	0	0
$973$	0.934166	0.0299480
$974$	0	0
$975$	−7.78805	−0.249417
$976$	0	0
$977$	9.95826	0.318593	0.159296	−	0.987231i	$-0.449077\pi$
$977$	9.95826	0.318593	0.159296	+	0.987231i	$0.449077\pi$
$978$	0	0
$979$	−5.90446	−0.188708
$980$	0	0
$981$	8.22661	0.262655
$982$	0	0
$983$	29.1463	0.929623	0.464811	−	0.885410i	$-0.346122\pi$
$983$	29.1463	0.929623	0.464811	+	0.885410i	$0.346122\pi$
$984$	0	0
$985$	82.0911	2.61564
$986$	0	0
$987$	5.42327	0.172625
$988$	0	0
$989$	13.1264	0.417397
$990$	0	0
$991$	−7.83004	−0.248730	−0.124365	−	0.992237i	$-0.539689\pi$
$991$	−7.83004	−0.248730	−0.124365	+	0.992237i	$0.539689\pi$
$992$	0	0
$993$	−6.87144	−0.218058
$994$	0	0
$995$	30.8318	0.977433
$996$	0	0
$997$	−2.70406	−0.0856384	−0.0428192	−	0.999083i	$-0.513634\pi$
$997$	−2.70406	−0.0856384	−0.0428192	+	0.999083i	$0.513634\pi$
$998$	0	0
$999$	28.6447	0.906277

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.r.1.5		12
4.3	odd	2		1006.2.a.i.1.8	✓	12
12.11	even	2		9054.2.a.bj.1.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.i.1.8	✓	12	4.3	odd	2
8048.2.a.r.1.5		12	1.1	even	1	trivial
9054.2.a.bj.1.2		12	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-0.656123 q^{3} +4.07967 q^{5} +0.827612 q^{7} -2.56950 q^{9} +O(q^{10})\)	\(q-0.656123 q^{3} +4.07967 q^{5} +0.827612 q^{7} -2.56950 q^{9} +3.65273 q^{11} +1.01942 q^{13} -2.67677 q^{15} +3.12716 q^{17} -4.20968 q^{19} -0.543015 q^{21} -6.99846 q^{23} +11.6437 q^{25} +3.65428 q^{27} +5.92375 q^{29} +1.93174 q^{31} -2.39664 q^{33} +3.37638 q^{35} +7.83866 q^{37} -0.668864 q^{39} +0.429262 q^{41} -1.87562 q^{43} -10.4827 q^{45} -9.98732 q^{47} -6.31506 q^{49} -2.05180 q^{51} -8.00130 q^{53} +14.9020 q^{55} +2.76207 q^{57} +13.2850 q^{59} +8.78970 q^{61} -2.12655 q^{63} +4.15889 q^{65} +6.21701 q^{67} +4.59185 q^{69} -10.9469 q^{71} +13.3974 q^{73} -7.63970 q^{75} +3.02305 q^{77} +6.97796 q^{79} +5.31085 q^{81} +16.1238 q^{83} +12.7578 q^{85} -3.88671 q^{87} -1.61645 q^{89} +0.843682 q^{91} -1.26746 q^{93} -17.1741 q^{95} -14.0367 q^{97} -9.38571 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9	\( 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9} - 18 q^{11} - 4 q^{13} + 2 q^{15} + 12 q^{17} + 7 q^{21} + 9 q^{23} + 25 q^{25} + 18 q^{27} + 34 q^{29} + 11 q^{31} + 4 q^{33} - 21 q^{35} - 22 q^{37} - 13 q^{39} + 32 q^{41} + 8 q^{43} + 13 q^{45} - 24 q^{47} + 36 q^{49} - 16 q^{51} - 2 q^{53} + 12 q^{55} + 26 q^{57} - 26 q^{59} + 12 q^{61} - 5 q^{63} + 66 q^{65} + 21 q^{67} + 20 q^{69} - 50 q^{71} + 17 q^{73} + 14 q^{75} + 25 q^{77} + 9 q^{79} + 48 q^{81} - 25 q^{83} + 24 q^{85} + 10 q^{87} + 21 q^{89} + 9 q^{91} + 31 q^{93} - 22 q^{95} + 18 q^{97} - 102 q^{99}+O(q^{100}) \) 12 * q + 3 * q^3 + 7 * q^5 + 2 * q^7 + 27 * q^9 - 18 * q^11 - 4 * q^13 + 2 * q^15 + 12 * q^17 + 7 * q^21 + 9 * q^23 + 25 * q^25 + 18 * q^27 + 34 * q^29 + 11 * q^31 + 4 * q^33 - 21 * q^35 - 22 * q^37 - 13 * q^39 + 32 * q^41 + 8 * q^43 + 13 * q^45 - 24 * q^47 + 36 * q^49 - 16 * q^51 - 2 * q^53 + 12 * q^55 + 26 * q^57 - 26 * q^59 + 12 * q^61 - 5 * q^63 + 66 * q^65 + 21 * q^67 + 20 * q^69 - 50 * q^71 + 17 * q^73 + 14 * q^75 + 25 * q^77 + 9 * q^79 + 48 * q^81 - 25 * q^83 + 24 * q^85 + 10 * q^87 + 21 * q^89 + 9 * q^91 + 31 * q^93 - 22 * q^95 + 18 * q^97 - 102 * q^99

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