LMFDB - Embedded newform 8048.2.a.r.1.1

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			1.1
Root			$2.60861$ 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)

$f(q)$	$=$	$q-2.92999 q^{3} -1.53308 q^{5} +4.18455 q^{7} +5.58485 q^{9} +O(q^{10})$	$q-2.92999 q^{3} -1.53308 q^{5} +4.18455 q^{7} +5.58485 q^{9} -3.61828 q^{11} +1.36077 q^{13} +4.49191 q^{15} +2.36421 q^{17} -8.18166 q^{19} -12.2607 q^{21} +3.08634 q^{23} -2.64967 q^{25} -7.57357 q^{27} +5.68095 q^{29} -8.16339 q^{31} +10.6015 q^{33} -6.41525 q^{35} +1.96839 q^{37} -3.98705 q^{39} +9.60349 q^{41} +9.19968 q^{43} -8.56201 q^{45} -6.84043 q^{47} +10.5105 q^{49} -6.92711 q^{51} -5.91778 q^{53} +5.54711 q^{55} +23.9722 q^{57} +5.95834 q^{59} +5.60973 q^{61} +23.3701 q^{63} -2.08617 q^{65} +0.102632 q^{67} -9.04295 q^{69} -0.534601 q^{71} -8.60811 q^{73} +7.76351 q^{75} -15.1409 q^{77} +16.6121 q^{79} +5.43596 q^{81} -12.4370 q^{83} -3.62452 q^{85} -16.6451 q^{87} +2.87710 q^{89} +5.69421 q^{91} +23.9187 q^{93} +12.5431 q^{95} -5.21359 q^{97} -20.2076 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.92999	−1.69163	−0.845815	−	0.533476i	$-0.820886\pi$
$3$	−2.92999	−1.69163	−0.845815	+	0.533476i	$0.820886\pi$
$4$	0	0
$5$	−1.53308	−0.685614	−0.342807	−	0.939406i	$-0.611378\pi$
$5$	−1.53308	−0.685614	−0.342807	+	0.939406i	$0.611378\pi$
$6$	0	0
$7$	4.18455	1.58161	0.790806	−	0.612067i	$-0.209662\pi$
$7$	4.18455	1.58161	0.790806	+	0.612067i	$0.209662\pi$
$8$	0	0
$9$	5.58485	1.86162
$10$	0	0
$11$	−3.61828	−1.09095	−0.545477	−	0.838126i	$-0.683651\pi$
$11$	−3.61828	−1.09095	−0.545477	+	0.838126i	$0.683651\pi$
$12$	0	0
$13$	1.36077	0.377410	0.188705	−	0.982034i	$-0.439571\pi$
$13$	1.36077	0.377410	0.188705	+	0.982034i	$0.439571\pi$
$14$	0	0
$15$	4.49191	1.15981
$16$	0	0
$17$	2.36421	0.573405	0.286702	−	0.958020i	$-0.407441\pi$
$17$	2.36421	0.573405	0.286702	+	0.958020i	$0.407441\pi$
$18$	0	0
$19$	−8.18166	−1.87700	−0.938500	−	0.345278i	$-0.887784\pi$
$19$	−8.18166	−1.87700	−0.938500	+	0.345278i	$0.887784\pi$
$20$	0	0
$21$	−12.2607	−2.67550
$22$	0	0
$23$	3.08634	0.643547	0.321773	−	0.946817i	$-0.395721\pi$
$23$	3.08634	0.643547	0.321773	+	0.946817i	$0.395721\pi$
$24$	0	0
$25$	−2.64967	−0.529934
$26$	0	0
$27$	−7.57357	−1.45754
$28$	0	0
$29$	5.68095	1.05493	0.527463	−	0.849578i	$-0.323143\pi$
$29$	5.68095	1.05493	0.527463	+	0.849578i	$0.323143\pi$
$30$	0	0
$31$	−8.16339	−1.46619	−0.733094	−	0.680127i	$-0.761925\pi$
$31$	−8.16339	−1.46619	−0.733094	+	0.680127i	$0.761925\pi$
$32$	0	0
$33$	10.6015	1.84549
$34$	0	0
$35$	−6.41525	−1.08437
$36$	0	0
$37$	1.96839	0.323602	0.161801	−	0.986823i	$-0.448270\pi$
$37$	1.96839	0.323602	0.161801	+	0.986823i	$0.448270\pi$
$38$	0	0
$39$	−3.98705	−0.638438
$40$	0	0
$41$	9.60349	1.49981	0.749907	−	0.661543i	$-0.230098\pi$
$41$	9.60349	1.49981	0.749907	+	0.661543i	$0.230098\pi$
$42$	0	0
$43$	9.19968	1.40294	0.701469	−	0.712700i	$-0.252528\pi$
$43$	9.19968	1.40294	0.701469	+	0.712700i	$0.252528\pi$
$44$	0	0
$45$	−8.56201	−1.27635
$46$	0	0
$47$	−6.84043	−0.997779	−0.498890	−	0.866666i	$-0.666259\pi$
$47$	−6.84043	−0.997779	−0.498890	+	0.866666i	$0.666259\pi$
$48$	0	0
$49$	10.5105	1.50150
$50$	0	0
$51$	−6.92711	−0.969990
$52$	0	0
$53$	−5.91778	−0.812870	−0.406435	−	0.913680i	$-0.633228\pi$
$53$	−5.91778	−0.812870	−0.406435	+	0.913680i	$0.633228\pi$
$54$	0	0
$55$	5.54711	0.747972
$56$	0	0
$57$	23.9722	3.17519
$58$	0	0
$59$	5.95834	0.775710	0.387855	−	0.921720i	$-0.373216\pi$
$59$	5.95834	0.775710	0.387855	+	0.921720i	$0.373216\pi$
$60$	0	0
$61$	5.60973	0.718252	0.359126	−	0.933289i	$-0.383075\pi$
$61$	5.60973	0.718252	0.359126	+	0.933289i	$0.383075\pi$
$62$	0	0
$63$	23.3701	2.94435
$64$	0	0
$65$	−2.08617	−0.258757
$66$	0	0
$67$	0.102632	0.0125385	0.00626927	−	0.999980i	$-0.498004\pi$
$67$	0.102632	0.0125385	0.00626927	+	0.999980i	$0.498004\pi$
$68$	0	0
$69$	−9.04295	−1.08864
$70$	0	0
$71$	−0.534601	−0.0634454	−0.0317227	−	0.999497i	$-0.510099\pi$
$71$	−0.534601	−0.0634454	−0.0317227	+	0.999497i	$0.510099\pi$
$72$	0	0
$73$	−8.60811	−1.00750	−0.503751	−	0.863849i	$-0.668047\pi$
$73$	−8.60811	−1.00750	−0.503751	+	0.863849i	$0.668047\pi$
$74$	0	0
$75$	7.76351	0.896453
$76$	0	0
$77$	−15.1409	−1.72546
$78$	0	0
$79$	16.6121	1.86901	0.934503	−	0.355955i	$-0.115845\pi$
$79$	16.6121	1.86901	0.934503	+	0.355955i	$0.115845\pi$
$80$	0	0
$81$	5.43596	0.603996
$82$	0	0
$83$	−12.4370	−1.36514	−0.682569	−	0.730821i	$-0.739138\pi$
$83$	−12.4370	−1.36514	−0.682569	+	0.730821i	$0.739138\pi$
$84$	0	0
$85$	−3.62452	−0.393134
$86$	0	0
$87$	−16.6451	−1.78455
$88$	0	0
$89$	2.87710	0.304972	0.152486	−	0.988306i	$-0.451272\pi$
$89$	2.87710	0.304972	0.152486	+	0.988306i	$0.451272\pi$
$90$	0	0
$91$	5.69421	0.596916
$92$	0	0
$93$	23.9187	2.48025
$94$	0	0
$95$	12.5431	1.28690
$96$	0	0
$97$	−5.21359	−0.529360	−0.264680	−	0.964336i	$-0.585266\pi$
$97$	−5.21359	−0.529360	−0.264680	+	0.964336i	$0.585266\pi$
$98$	0	0
$99$	−20.2076	−2.03094
$100$	0	0
$101$	5.09242	0.506715	0.253357	−	0.967373i	$-0.418465\pi$
$101$	5.09242	0.506715	0.253357	+	0.967373i	$0.418465\pi$
$102$	0	0
$103$	6.75710	0.665797	0.332898	−	0.942963i	$-0.391973\pi$
$103$	6.75710	0.665797	0.332898	+	0.942963i	$0.391973\pi$
$104$	0	0
$105$	18.7966	1.83436
$106$	0	0
$107$	−3.75160	−0.362681	−0.181340	−	0.983420i	$-0.558044\pi$
$107$	−3.75160	−0.362681	−0.181340	+	0.983420i	$0.558044\pi$
$108$	0	0
$109$	−7.49487	−0.717878	−0.358939	−	0.933361i	$-0.616861\pi$
$109$	−7.49487	−0.717878	−0.358939	+	0.933361i	$0.616861\pi$
$110$	0	0
$111$	−5.76737	−0.547415
$112$	0	0
$113$	−18.4007	−1.73099	−0.865496	−	0.500916i	$-0.832997\pi$
$113$	−18.4007	−1.73099	−0.865496	+	0.500916i	$0.832997\pi$
$114$	0	0
$115$	−4.73161	−0.441224
$116$	0	0
$117$	7.59969	0.702592
$118$	0	0
$119$	9.89315	0.906904
$120$	0	0
$121$	2.09197	0.190179
$122$	0	0
$123$	−28.1381	−2.53713
$124$	0	0
$125$	11.7275	1.04894
$126$	0	0
$127$	12.0533	1.06956	0.534779	−	0.844992i	$-0.320395\pi$
$127$	12.0533	1.06956	0.534779	+	0.844992i	$0.320395\pi$
$128$	0	0
$129$	−26.9550	−2.37325
$130$	0	0
$131$	−8.32064	−0.726978	−0.363489	−	0.931599i	$-0.618415\pi$
$131$	−8.32064	−0.726978	−0.363489	+	0.931599i	$0.618415\pi$
$132$	0	0
$133$	−34.2366	−2.96869
$134$	0	0
$135$	11.6109	0.999306
$136$	0	0
$137$	−12.5057	−1.06844	−0.534218	−	0.845347i	$-0.679394\pi$
$137$	−12.5057	−1.06844	−0.534218	+	0.845347i	$0.679394\pi$
$138$	0	0
$139$	−16.4852	−1.39826	−0.699129	−	0.714996i	$-0.746429\pi$
$139$	−16.4852	−1.39826	−0.699129	+	0.714996i	$0.746429\pi$
$140$	0	0
$141$	20.0424	1.68787
$142$	0	0
$143$	−4.92365	−0.411737
$144$	0	0
$145$	−8.70934	−0.723272
$146$	0	0
$147$	−30.7956	−2.53998
$148$	0	0
$149$	10.1113	0.828349	0.414175	−	0.910197i	$-0.364070\pi$
$149$	10.1113	0.828349	0.414175	+	0.910197i	$0.364070\pi$
$150$	0	0
$151$	3.31097	0.269443	0.134721	−	0.990884i	$-0.456986\pi$
$151$	3.31097	0.269443	0.134721	+	0.990884i	$0.456986\pi$
$152$	0	0
$153$	13.2037	1.06746
$154$	0	0
$155$	12.5151	1.00524
$156$	0	0
$157$	15.2703	1.21870	0.609351	−	0.792901i	$-0.291430\pi$
$157$	15.2703	1.21870	0.609351	+	0.792901i	$0.291430\pi$
$158$	0	0
$159$	17.3390	1.37508
$160$	0	0
$161$	12.9150	1.01784
$162$	0	0
$163$	7.56428	0.592480	0.296240	−	0.955113i	$-0.404267\pi$
$163$	7.56428	0.592480	0.296240	+	0.955113i	$0.404267\pi$
$164$	0	0
$165$	−16.2530	−1.26529
$166$	0	0
$167$	−16.3380	−1.26427	−0.632135	−	0.774858i	$-0.717821\pi$
$167$	−16.3380	−1.26427	−0.632135	+	0.774858i	$0.717821\pi$
$168$	0	0
$169$	−11.1483	−0.857562
$170$	0	0
$171$	−45.6933	−3.49425
$172$	0	0
$173$	−11.8905	−0.904017	−0.452008	−	0.892014i	$-0.649292\pi$
$173$	−11.8905	−0.904017	−0.452008	+	0.892014i	$0.649292\pi$
$174$	0	0
$175$	−11.0877	−0.838150
$176$	0	0
$177$	−17.4579	−1.31222
$178$	0	0
$179$	−5.48216	−0.409756	−0.204878	−	0.978788i	$-0.565680\pi$
$179$	−5.48216	−0.409756	−0.204878	+	0.978788i	$0.565680\pi$
$180$	0	0
$181$	9.79714	0.728216	0.364108	−	0.931357i	$-0.381374\pi$
$181$	9.79714	0.728216	0.364108	+	0.931357i	$0.381374\pi$
$182$	0	0
$183$	−16.4365	−1.21502
$184$	0	0
$185$	−3.01770	−0.221866
$186$	0	0
$187$	−8.55438	−0.625558
$188$	0	0
$189$	−31.6920	−2.30525
$190$	0	0
$191$	14.6451	1.05968	0.529840	−	0.848098i	$-0.322252\pi$
$191$	14.6451	1.05968	0.529840	+	0.848098i	$0.322252\pi$
$192$	0	0
$193$	9.16792	0.659921	0.329961	−	0.943995i	$-0.392965\pi$
$193$	9.16792	0.659921	0.329961	+	0.943995i	$0.392965\pi$
$194$	0	0
$195$	6.11245	0.437722
$196$	0	0
$197$	−6.43336	−0.458358	−0.229179	−	0.973384i	$-0.573604\pi$
$197$	−6.43336	−0.458358	−0.229179	+	0.973384i	$0.573604\pi$
$198$	0	0
$199$	26.4251	1.87322	0.936612	−	0.350369i	$-0.113944\pi$
$199$	26.4251	1.87322	0.936612	+	0.350369i	$0.113944\pi$
$200$	0	0
$201$	−0.300712	−0.0212106
$202$	0	0
$203$	23.7722	1.66848
$204$	0	0
$205$	−14.7229	−1.02829
$206$	0	0
$207$	17.2367	1.19804
$208$	0	0
$209$	29.6035	2.04772
$210$	0	0
$211$	16.7526	1.15330	0.576648	−	0.816993i	$-0.304360\pi$
$211$	16.7526	1.15330	0.576648	+	0.816993i	$0.304360\pi$
$212$	0	0
$213$	1.56638	0.107326
$214$	0	0
$215$	−14.1038	−0.961873
$216$	0	0
$217$	−34.1601	−2.31894
$218$	0	0
$219$	25.2217	1.70432
$220$	0	0
$221$	3.21715	0.216409
$222$	0	0
$223$	5.50930	0.368930	0.184465	−	0.982839i	$-0.440945\pi$
$223$	5.50930	0.368930	0.184465	+	0.982839i	$0.440945\pi$
$224$	0	0
$225$	−14.7980	−0.986533
$226$	0	0
$227$	−19.4252	−1.28930	−0.644648	−	0.764480i	$-0.722996\pi$
$227$	−19.4252	−1.28930	−0.644648	+	0.764480i	$0.722996\pi$
$228$	0	0
$229$	−22.4932	−1.48639	−0.743196	−	0.669074i	$-0.766691\pi$
$229$	−22.4932	−1.48639	−0.743196	+	0.669074i	$0.766691\pi$
$230$	0	0
$231$	44.3627	2.91885
$232$	0	0
$233$	9.17472	0.601056	0.300528	−	0.953773i	$-0.402837\pi$
$233$	9.17472	0.601056	0.300528	+	0.953773i	$0.402837\pi$
$234$	0	0
$235$	10.4869	0.684091
$236$	0	0
$237$	−48.6733	−3.16167
$238$	0	0
$239$	−29.3644	−1.89943	−0.949713	−	0.313121i	$-0.898625\pi$
$239$	−29.3644	−1.89943	−0.949713	+	0.313121i	$0.898625\pi$
$240$	0	0
$241$	−13.8561	−0.892547	−0.446273	−	0.894897i	$-0.647249\pi$
$241$	−13.8561	−0.892547	−0.446273	+	0.894897i	$0.647249\pi$
$242$	0	0
$243$	6.79340	0.435797
$244$	0	0
$245$	−16.1134	−1.02945
$246$	0	0
$247$	−11.1334	−0.708399
$248$	0	0
$249$	36.4403	2.30931
$250$	0	0
$251$	−6.54983	−0.413422	−0.206711	−	0.978402i	$-0.566276\pi$
$251$	−6.54983	−0.413422	−0.206711	+	0.978402i	$0.566276\pi$
$252$	0	0
$253$	−11.1673	−0.702080
$254$	0	0
$255$	10.6198	0.665038
$256$	0	0
$257$	25.4281	1.58616	0.793082	−	0.609115i	$-0.208475\pi$
$257$	25.4281	1.58616	0.793082	+	0.609115i	$0.208475\pi$
$258$	0	0
$259$	8.23684	0.511812
$260$	0	0
$261$	31.7272	1.96387
$262$	0	0
$263$	23.0364	1.42048	0.710242	−	0.703957i	$-0.248585\pi$
$263$	23.0364	1.42048	0.710242	+	0.703957i	$0.248585\pi$
$264$	0	0
$265$	9.07243	0.557315
$266$	0	0
$267$	−8.42989	−0.515901
$268$	0	0
$269$	−13.1416	−0.801259	−0.400629	−	0.916240i	$-0.631209\pi$
$269$	−13.1416	−0.801259	−0.400629	+	0.916240i	$0.631209\pi$
$270$	0	0
$271$	27.2158	1.65324	0.826622	−	0.562758i	$-0.190260\pi$
$271$	27.2158	1.65324	0.826622	+	0.562758i	$0.190260\pi$
$272$	0	0
$273$	−16.6840	−1.00976
$274$	0	0
$275$	9.58726	0.578133
$276$	0	0
$277$	19.7973	1.18950	0.594752	−	0.803909i	$-0.297250\pi$
$277$	19.7973	1.18950	0.594752	+	0.803909i	$0.297250\pi$
$278$	0	0
$279$	−45.5913	−2.72948
$280$	0	0
$281$	5.15740	0.307665	0.153832	−	0.988097i	$-0.450838\pi$
$281$	5.15740	0.307665	0.153832	+	0.988097i	$0.450838\pi$
$282$	0	0
$283$	4.63508	0.275527	0.137763	−	0.990465i	$-0.456009\pi$
$283$	4.63508	0.275527	0.137763	+	0.990465i	$0.456009\pi$
$284$	0	0
$285$	−36.7512	−2.17695
$286$	0	0
$287$	40.1863	2.37212
$288$	0	0
$289$	−11.4105	−0.671207
$290$	0	0
$291$	15.2758	0.895481
$292$	0	0
$293$	9.75579	0.569939	0.284970	−	0.958537i	$-0.408016\pi$
$293$	9.75579	0.569939	0.284970	+	0.958537i	$0.408016\pi$
$294$	0	0
$295$	−9.13461	−0.531837
$296$	0	0
$297$	27.4033	1.59010
$298$	0	0
$299$	4.19980	0.242881
$300$	0	0
$301$	38.4965	2.21890
$302$	0	0
$303$	−14.9207	−0.857174
$304$	0	0
$305$	−8.60016	−0.492444
$306$	0	0
$307$	−23.4531	−1.33854	−0.669270	−	0.743020i	$-0.733393\pi$
$307$	−23.4531	−1.33854	−0.669270	+	0.743020i	$0.733393\pi$
$308$	0	0
$309$	−19.7982	−1.12628
$310$	0	0
$311$	−2.14629	−0.121705	−0.0608524	−	0.998147i	$-0.519382\pi$
$311$	−2.14629	−0.121705	−0.0608524	+	0.998147i	$0.519382\pi$
$312$	0	0
$313$	24.5672	1.38862	0.694309	−	0.719677i	$-0.255710\pi$
$313$	24.5672	1.38862	0.694309	+	0.719677i	$0.255710\pi$
$314$	0	0
$315$	−35.8282	−2.01869
$316$	0	0
$317$	4.40170	0.247224	0.123612	−	0.992331i	$-0.460552\pi$
$317$	4.40170	0.247224	0.123612	+	0.992331i	$0.460552\pi$
$318$	0	0
$319$	−20.5553	−1.15087
$320$	0	0
$321$	10.9921	0.613522
$322$	0	0
$323$	−19.3431	−1.07628
$324$	0	0
$325$	−3.60559	−0.200002
$326$	0	0
$327$	21.9599	1.21439
$328$	0	0
$329$	−28.6241	−1.57810
$330$	0	0
$331$	23.5150	1.29250	0.646251	−	0.763125i	$-0.276336\pi$
$331$	23.5150	1.29250	0.646251	+	0.763125i	$0.276336\pi$
$332$	0	0
$333$	10.9932	0.602422
$334$	0	0
$335$	−0.157343	−0.00859659
$336$	0	0
$337$	−21.2855	−1.15950	−0.579749	−	0.814795i	$-0.696849\pi$
$337$	−21.2855	−1.15950	−0.579749	+	0.814795i	$0.696849\pi$
$338$	0	0
$339$	53.9139	2.92820
$340$	0	0
$341$	29.5375	1.59954
$342$	0	0
$343$	14.6897	0.793171
$344$	0	0
$345$	13.8636	0.746389
$346$	0	0
$347$	−1.01644	−0.0545653	−0.0272827	−	0.999628i	$-0.508685\pi$
$347$	−1.01644	−0.0545653	−0.0272827	+	0.999628i	$0.508685\pi$
$348$	0	0
$349$	−1.45343	−0.0778004	−0.0389002	−	0.999243i	$-0.512385\pi$
$349$	−1.45343	−0.0778004	−0.0389002	+	0.999243i	$0.512385\pi$
$350$	0	0
$351$	−10.3059	−0.550088
$352$	0	0
$353$	−14.1438	−0.752799	−0.376400	−	0.926457i	$-0.622838\pi$
$353$	−14.1438	−0.752799	−0.376400	+	0.926457i	$0.622838\pi$
$354$	0	0
$355$	0.819585	0.0434991
$356$	0	0
$357$	−28.9869	−1.53415
$358$	0	0
$359$	16.2294	0.856554	0.428277	−	0.903648i	$-0.359121\pi$
$359$	16.2294	0.856554	0.428277	+	0.903648i	$0.359121\pi$
$360$	0	0
$361$	47.9395	2.52313
$362$	0	0
$363$	−6.12945	−0.321713
$364$	0	0
$365$	13.1969	0.690758
$366$	0	0
$367$	−0.949246	−0.0495503	−0.0247751	−	0.999693i	$-0.507887\pi$
$367$	−0.949246	−0.0495503	−0.0247751	+	0.999693i	$0.507887\pi$
$368$	0	0
$369$	53.6340	2.79208
$370$	0	0
$371$	−24.7633	−1.28564
$372$	0	0
$373$	34.5665	1.78978	0.894892	−	0.446282i	$-0.147252\pi$
$373$	34.5665	1.78978	0.894892	+	0.446282i	$0.147252\pi$
$374$	0	0
$375$	−34.3616	−1.77443
$376$	0	0
$377$	7.73047	0.398140
$378$	0	0
$379$	22.1305	1.13677	0.568383	−	0.822764i	$-0.307569\pi$
$379$	22.1305	1.13677	0.568383	+	0.822764i	$0.307569\pi$
$380$	0	0
$381$	−35.3161	−1.80930
$382$	0	0
$383$	15.2092	0.777152	0.388576	−	0.921417i	$-0.372967\pi$
$383$	15.2092	0.777152	0.388576	+	0.921417i	$0.372967\pi$
$384$	0	0
$385$	23.2122	1.18300
$386$	0	0
$387$	51.3788	2.61173
$388$	0	0
$389$	33.4426	1.69561	0.847805	−	0.530308i	$-0.177924\pi$
$389$	33.4426	1.69561	0.847805	+	0.530308i	$0.177924\pi$
$390$	0	0
$391$	7.29676	0.369013
$392$	0	0
$393$	24.3794	1.22978
$394$	0	0
$395$	−25.4676	−1.28142
$396$	0	0
$397$	13.7728	0.691235	0.345618	−	0.938375i	$-0.387669\pi$
$397$	13.7728	0.691235	0.345618	+	0.938375i	$0.387669\pi$
$398$	0	0
$399$	100.313	5.02192
$400$	0	0
$401$	−14.6911	−0.733640	−0.366820	−	0.930292i	$-0.619553\pi$
$401$	−14.6911	−0.733640	−0.366820	+	0.930292i	$0.619553\pi$
$402$	0	0
$403$	−11.1085	−0.553354
$404$	0	0
$405$	−8.33376	−0.414108
$406$	0	0
$407$	−7.12220	−0.353034
$408$	0	0
$409$	−19.5228	−0.965343	−0.482671	−	0.875802i	$-0.660333\pi$
$409$	−19.5228	−0.965343	−0.482671	+	0.875802i	$0.660333\pi$
$410$	0	0
$411$	36.6417	1.80740
$412$	0	0
$413$	24.9330	1.22687
$414$	0	0
$415$	19.0669	0.935958
$416$	0	0
$417$	48.3015	2.36534
$418$	0	0
$419$	11.0617	0.540400	0.270200	−	0.962804i	$-0.412910\pi$
$419$	11.0617	0.540400	0.270200	+	0.962804i	$0.412910\pi$
$420$	0	0
$421$	38.1012	1.85694	0.928470	−	0.371407i	$-0.121125\pi$
$421$	38.1012	1.85694	0.928470	+	0.371407i	$0.121125\pi$
$422$	0	0
$423$	−38.2027	−1.85748
$424$	0	0
$425$	−6.26437	−0.303867
$426$	0	0
$427$	23.4742	1.13600
$428$	0	0
$429$	14.4263	0.696506
$430$	0	0
$431$	20.1345	0.969846	0.484923	−	0.874557i	$-0.338848\pi$
$431$	20.1345	0.969846	0.484923	+	0.874557i	$0.338848\pi$
$432$	0	0
$433$	21.5098	1.03370	0.516848	−	0.856077i	$-0.327105\pi$
$433$	21.5098	1.03370	0.516848	+	0.856077i	$0.327105\pi$
$434$	0	0
$435$	25.5183	1.22351
$436$	0	0
$437$	−25.2514	−1.20794
$438$	0	0
$439$	27.8707	1.33020	0.665098	−	0.746756i	$-0.268390\pi$
$439$	27.8707	1.33020	0.665098	+	0.746756i	$0.268390\pi$
$440$	0	0
$441$	58.6993	2.79521
$442$	0	0
$443$	26.0833	1.23925	0.619627	−	0.784896i	$-0.287284\pi$
$443$	26.0833	1.23925	0.619627	+	0.784896i	$0.287284\pi$
$444$	0	0
$445$	−4.41083	−0.209093
$446$	0	0
$447$	−29.6260	−1.40126
$448$	0	0
$449$	−33.0299	−1.55878	−0.779388	−	0.626542i	$-0.784470\pi$
$449$	−33.0299	−1.55878	−0.779388	+	0.626542i	$0.784470\pi$
$450$	0	0
$451$	−34.7482	−1.63623
$452$	0	0
$453$	−9.70110	−0.455798
$454$	0	0
$455$	−8.72968	−0.409254
$456$	0	0
$457$	−23.4939	−1.09900	−0.549500	−	0.835494i	$-0.685182\pi$
$457$	−23.4939	−1.09900	−0.549500	+	0.835494i	$0.685182\pi$
$458$	0	0
$459$	−17.9055	−0.835758
$460$	0	0
$461$	−36.3545	−1.69320	−0.846599	−	0.532231i	$-0.821354\pi$
$461$	−36.3545	−1.69320	−0.846599	+	0.532231i	$0.821354\pi$
$462$	0	0
$463$	−33.1722	−1.54164	−0.770822	−	0.637050i	$-0.780154\pi$
$463$	−33.1722	−1.54164	−0.770822	+	0.637050i	$0.780154\pi$
$464$	0	0
$465$	−36.6692	−1.70049
$466$	0	0
$467$	−2.08487	−0.0964763	−0.0482381	−	0.998836i	$-0.515361\pi$
$467$	−2.08487	−0.0964763	−0.0482381	+	0.998836i	$0.515361\pi$
$468$	0	0
$469$	0.429470	0.0198311
$470$	0	0
$471$	−44.7418	−2.06159
$472$	0	0
$473$	−33.2870	−1.53054
$474$	0	0
$475$	21.6787	0.994686
$476$	0	0
$477$	−33.0499	−1.51325
$478$	0	0
$479$	−3.25389	−0.148674	−0.0743370	−	0.997233i	$-0.523684\pi$
$479$	−3.25389	−0.148674	−0.0743370	+	0.997233i	$0.523684\pi$
$480$	0	0
$481$	2.67853	0.122131
$482$	0	0
$483$	−37.8407	−1.72181
$484$	0	0
$485$	7.99284	0.362936
$486$	0	0
$487$	−8.99983	−0.407821	−0.203911	−	0.978990i	$-0.565365\pi$
$487$	−8.99983	−0.407821	−0.203911	+	0.978990i	$0.565365\pi$
$488$	0	0
$489$	−22.1633	−1.00226
$490$	0	0
$491$	16.7967	0.758024	0.379012	−	0.925392i	$-0.376264\pi$
$491$	16.7967	0.758024	0.379012	+	0.925392i	$0.376264\pi$
$492$	0	0
$493$	13.4310	0.604900
$494$	0	0
$495$	30.9798	1.39244
$496$	0	0
$497$	−2.23706	−0.100346
$498$	0	0
$499$	−1.54356	−0.0690992	−0.0345496	−	0.999403i	$-0.511000\pi$
$499$	−1.54356	−0.0690992	−0.0345496	+	0.999403i	$0.511000\pi$
$500$	0	0
$501$	47.8701	2.13868
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−7.80708	−0.347410
$506$	0	0
$507$	32.6644	1.45068
$508$	0	0
$509$	−25.2647	−1.11984	−0.559918	−	0.828548i	$-0.689168\pi$
$509$	−25.2647	−1.11984	−0.559918	+	0.828548i	$0.689168\pi$
$510$	0	0
$511$	−36.0211	−1.59348
$512$	0	0
$513$	61.9644	2.73579
$514$	0	0
$515$	−10.3592	−0.456479
$516$	0	0
$517$	24.7506	1.08853
$518$	0	0
$519$	34.8390	1.52926
$520$	0	0
$521$	−26.0037	−1.13924	−0.569622	−	0.821907i	$-0.692910\pi$
$521$	−26.0037	−1.13924	−0.569622	+	0.821907i	$0.692910\pi$
$522$	0	0
$523$	24.5777	1.07471	0.537355	−	0.843356i	$-0.319424\pi$
$523$	24.5777	1.07471	0.537355	+	0.843356i	$0.319424\pi$
$524$	0	0
$525$	32.4868	1.41784
$526$	0	0
$527$	−19.3000	−0.840720
$528$	0	0
$529$	−13.4745	−0.585847
$530$	0	0
$531$	33.2764	1.44407
$532$	0	0
$533$	13.0682	0.566045
$534$	0	0
$535$	5.75150	0.248659
$536$	0	0
$537$	16.0627	0.693156
$538$	0	0
$539$	−38.0298	−1.63806
$540$	0	0
$541$	36.2404	1.55810	0.779048	−	0.626964i	$-0.215703\pi$
$541$	36.2404	1.55810	0.779048	+	0.626964i	$0.215703\pi$
$542$	0	0
$543$	−28.7055	−1.23187
$544$	0	0
$545$	11.4902	0.492187
$546$	0	0
$547$	−43.7278	−1.86967	−0.934834	−	0.355085i	$-0.884452\pi$
$547$	−43.7278	−1.86967	−0.934834	+	0.355085i	$0.884452\pi$
$548$	0	0
$549$	31.3295	1.33711
$550$	0	0
$551$	−46.4796	−1.98010
$552$	0	0
$553$	69.5141	2.95604
$554$	0	0
$555$	8.84183	0.375315
$556$	0	0
$557$	13.2150	0.559938	0.279969	−	0.960009i	$-0.409676\pi$
$557$	13.2150	0.559938	0.279969	+	0.960009i	$0.409676\pi$
$558$	0	0
$559$	12.5187	0.529483
$560$	0	0
$561$	25.0642	1.05821
$562$	0	0
$563$	−3.41042	−0.143732	−0.0718660	−	0.997414i	$-0.522895\pi$
$563$	−3.41042	−0.143732	−0.0718660	+	0.997414i	$0.522895\pi$
$564$	0	0
$565$	28.2097	1.18679
$566$	0	0
$567$	22.7471	0.955287
$568$	0	0
$569$	−13.4895	−0.565509	−0.282754	−	0.959192i	$-0.591248\pi$
$569$	−13.4895	−0.565509	−0.282754	+	0.959192i	$0.591248\pi$
$570$	0	0
$571$	24.0472	1.00634	0.503172	−	0.864186i	$-0.332166\pi$
$571$	24.0472	1.00634	0.503172	+	0.864186i	$0.332166\pi$
$572$	0	0
$573$	−42.9099	−1.79259
$574$	0	0
$575$	−8.17779	−0.341037
$576$	0	0
$577$	10.1329	0.421838	0.210919	−	0.977503i	$-0.432354\pi$
$577$	10.1329	0.421838	0.210919	+	0.977503i	$0.432354\pi$
$578$	0	0
$579$	−26.8619	−1.11634
$580$	0	0
$581$	−52.0433	−2.15912
$582$	0	0
$583$	21.4122	0.886803
$584$	0	0
$585$	−11.6509	−0.481707
$586$	0	0
$587$	5.65124	0.233252	0.116626	−	0.993176i	$-0.462792\pi$
$587$	5.65124	0.233252	0.116626	+	0.993176i	$0.462792\pi$
$588$	0	0
$589$	66.7901	2.75204
$590$	0	0
$591$	18.8497	0.775372
$592$	0	0
$593$	32.4864	1.33406	0.667029	−	0.745032i	$-0.267566\pi$
$593$	32.4864	1.33406	0.667029	+	0.745032i	$0.267566\pi$
$594$	0	0
$595$	−15.1670	−0.621786
$596$	0	0
$597$	−77.4252	−3.16880
$598$	0	0
$599$	−18.5837	−0.759308	−0.379654	−	0.925129i	$-0.623957\pi$
$599$	−18.5837	−0.759308	−0.379654	+	0.925129i	$0.623957\pi$
$600$	0	0
$601$	−7.04626	−0.287423	−0.143711	−	0.989620i	$-0.545904\pi$
$601$	−7.04626	−0.287423	−0.143711	+	0.989620i	$0.545904\pi$
$602$	0	0
$603$	0.573186	0.0233419
$604$	0	0
$605$	−3.20715	−0.130389
$606$	0	0
$607$	23.2975	0.945615	0.472808	−	0.881166i	$-0.343241\pi$
$607$	23.2975	0.945615	0.472808	+	0.881166i	$0.343241\pi$
$608$	0	0
$609$	−69.6524	−2.82246
$610$	0	0
$611$	−9.30826	−0.376572
$612$	0	0
$613$	33.3494	1.34697	0.673486	−	0.739200i	$-0.264796\pi$
$613$	33.3494	1.34697	0.673486	+	0.739200i	$0.264796\pi$
$614$	0	0
$615$	43.1380	1.73949
$616$	0	0
$617$	42.9323	1.72839	0.864195	−	0.503158i	$-0.167829\pi$
$617$	42.9323	1.72839	0.864195	+	0.503158i	$0.167829\pi$
$618$	0	0
$619$	38.8135	1.56005	0.780023	−	0.625751i	$-0.215207\pi$
$619$	38.8135	1.56005	0.780023	+	0.625751i	$0.215207\pi$
$620$	0	0
$621$	−23.3746	−0.937992
$622$	0	0
$623$	12.0394	0.482348
$624$	0	0
$625$	−4.73090	−0.189236
$626$	0	0
$627$	−86.7381	−3.46399
$628$	0	0
$629$	4.65369	0.185555
$630$	0	0
$631$	36.1962	1.44095	0.720474	−	0.693482i	$-0.243924\pi$
$631$	36.1962	1.44095	0.720474	+	0.693482i	$0.243924\pi$
$632$	0	0
$633$	−49.0850	−1.95095
$634$	0	0
$635$	−18.4787	−0.733303
$636$	0	0
$637$	14.3023	0.566679
$638$	0	0
$639$	−2.98566	−0.118111
$640$	0	0
$641$	19.5121	0.770683	0.385342	−	0.922774i	$-0.374084\pi$
$641$	19.5121	0.770683	0.385342	+	0.922774i	$0.374084\pi$
$642$	0	0
$643$	34.2094	1.34909	0.674543	−	0.738236i	$-0.264341\pi$
$643$	34.2094	1.34909	0.674543	+	0.738236i	$0.264341\pi$
$644$	0	0
$645$	41.3241	1.62713
$646$	0	0
$647$	11.0615	0.434874	0.217437	−	0.976074i	$-0.430230\pi$
$647$	11.0615	0.434874	0.217437	+	0.976074i	$0.430230\pi$
$648$	0	0
$649$	−21.5590	−0.846263
$650$	0	0
$651$	100.089	3.92279
$652$	0	0
$653$	−29.8137	−1.16670	−0.583350	−	0.812221i	$-0.698258\pi$
$653$	−29.8137	−1.16670	−0.583350	+	0.812221i	$0.698258\pi$
$654$	0	0
$655$	12.7562	0.498426
$656$	0	0
$657$	−48.0750	−1.87558
$658$	0	0
$659$	−32.7668	−1.27641	−0.638206	−	0.769865i	$-0.720323\pi$
$659$	−32.7668	−1.27641	−0.638206	+	0.769865i	$0.720323\pi$
$660$	0	0
$661$	8.00778	0.311467	0.155733	−	0.987799i	$-0.450226\pi$
$661$	8.00778	0.311467	0.155733	+	0.987799i	$0.450226\pi$
$662$	0	0
$663$	−9.42621	−0.366084
$664$	0	0
$665$	52.4873	2.03537
$666$	0	0
$667$	17.5334	0.678894
$668$	0	0
$669$	−16.1422	−0.624093
$670$	0	0
$671$	−20.2976	−0.783580
$672$	0	0
$673$	−5.45241	−0.210175	−0.105087	−	0.994463i	$-0.533512\pi$
$673$	−5.45241	−0.210175	−0.105087	+	0.994463i	$0.533512\pi$
$674$	0	0
$675$	20.0675	0.772397
$676$	0	0
$677$	23.1904	0.891278	0.445639	−	0.895213i	$-0.352977\pi$
$677$	23.1904	0.891278	0.445639	+	0.895213i	$0.352977\pi$
$678$	0	0
$679$	−21.8165	−0.837242
$680$	0	0
$681$	56.9157	2.18101
$682$	0	0
$683$	29.6846	1.13585	0.567925	−	0.823081i	$-0.307747\pi$
$683$	29.6846	1.13585	0.567925	+	0.823081i	$0.307747\pi$
$684$	0	0
$685$	19.1723	0.732534
$686$	0	0
$687$	65.9049	2.51443
$688$	0	0
$689$	−8.05275	−0.306785
$690$	0	0
$691$	35.4953	1.35031	0.675153	−	0.737678i	$-0.264078\pi$
$691$	35.4953	1.35031	0.675153	+	0.737678i	$0.264078\pi$
$692$	0	0
$693$	−84.5595	−3.21215
$694$	0	0
$695$	25.2731	0.958664
$696$	0	0
$697$	22.7047	0.860001
$698$	0	0
$699$	−26.8818	−1.01676
$700$	0	0
$701$	2.61646	0.0988222	0.0494111	−	0.998779i	$-0.484266\pi$
$701$	2.61646	0.0988222	0.0494111	+	0.998779i	$0.484266\pi$
$702$	0	0
$703$	−16.1047	−0.607401
$704$	0	0
$705$	−30.7266	−1.15723
$706$	0	0
$707$	21.3095	0.801426
$708$	0	0
$709$	−16.3928	−0.615643	−0.307822	−	0.951444i	$-0.599600\pi$
$709$	−16.3928	−0.615643	−0.307822	+	0.951444i	$0.599600\pi$
$710$	0	0
$711$	92.7759	3.47937
$712$	0	0
$713$	−25.1950	−0.943561
$714$	0	0
$715$	7.54835	0.282292
$716$	0	0
$717$	86.0375	3.21313
$718$	0	0
$719$	−17.4123	−0.649369	−0.324685	−	0.945822i	$-0.605258\pi$
$719$	−17.4123	−0.649369	−0.324685	+	0.945822i	$0.605258\pi$
$720$	0	0
$721$	28.2754	1.05303
$722$	0	0
$723$	40.5981	1.50986
$724$	0	0
$725$	−15.0526	−0.559041
$726$	0	0
$727$	18.4569	0.684529	0.342264	−	0.939604i	$-0.388806\pi$
$727$	18.4569	0.684529	0.342264	+	0.939604i	$0.388806\pi$
$728$	0	0
$729$	−36.2125	−1.34120
$730$	0	0
$731$	21.7500	0.804451
$732$	0	0
$733$	22.4283	0.828408	0.414204	−	0.910184i	$-0.364060\pi$
$733$	22.4283	0.828408	0.414204	+	0.910184i	$0.364060\pi$
$734$	0	0
$735$	47.2120	1.74144
$736$	0	0
$737$	−0.371353	−0.0136790
$738$	0	0
$739$	9.97234	0.366838	0.183419	−	0.983035i	$-0.441283\pi$
$739$	9.97234	0.366838	0.183419	+	0.983035i	$0.441283\pi$
$740$	0	0
$741$	32.6206	1.19835
$742$	0	0
$743$	3.20111	0.117437	0.0587187	−	0.998275i	$-0.481298\pi$
$743$	3.20111	0.117437	0.0587187	+	0.998275i	$0.481298\pi$
$744$	0	0
$745$	−15.5014	−0.567927
$746$	0	0
$747$	−69.4588	−2.54136
$748$	0	0
$749$	−15.6988	−0.573620
$750$	0	0
$751$	14.0297	0.511950	0.255975	−	0.966683i	$-0.417604\pi$
$751$	14.0297	0.511950	0.255975	+	0.966683i	$0.417604\pi$
$752$	0	0
$753$	19.1909	0.699357
$754$	0	0
$755$	−5.07597	−0.184734
$756$	0	0
$757$	11.2123	0.407518	0.203759	−	0.979021i	$-0.434684\pi$
$757$	11.2123	0.407518	0.203759	+	0.979021i	$0.434684\pi$
$758$	0	0
$759$	32.7200	1.18766
$760$	0	0
$761$	4.14946	0.150418	0.0752088	−	0.997168i	$-0.476038\pi$
$761$	4.14946	0.150418	0.0752088	+	0.997168i	$0.476038\pi$
$762$	0	0
$763$	−31.3627	−1.13540
$764$	0	0
$765$	−20.2424	−0.731865
$766$	0	0
$767$	8.10794	0.292761
$768$	0	0
$769$	25.4860	0.919048	0.459524	−	0.888165i	$-0.348020\pi$
$769$	25.4860	0.919048	0.459524	+	0.888165i	$0.348020\pi$
$770$	0	0
$771$	−74.5042	−2.68320
$772$	0	0
$773$	−43.7624	−1.57403	−0.787013	−	0.616936i	$-0.788374\pi$
$773$	−43.7624	−1.57403	−0.787013	+	0.616936i	$0.788374\pi$
$774$	0	0
$775$	21.6303	0.776983
$776$	0	0
$777$	−24.1339	−0.865798
$778$	0	0
$779$	−78.5725	−2.81515
$780$	0	0
$781$	1.93434	0.0692160
$782$	0	0
$783$	−43.0251	−1.53759
$784$	0	0
$785$	−23.4106	−0.835559
$786$	0	0
$787$	−19.2346	−0.685639	−0.342820	−	0.939401i	$-0.611382\pi$
$787$	−19.2346	−0.685639	−0.342820	+	0.939401i	$0.611382\pi$
$788$	0	0
$789$	−67.4964	−2.40294
$790$	0	0
$791$	−76.9987	−2.73776
$792$	0	0
$793$	7.63356	0.271076
$794$	0	0
$795$	−26.5821	−0.942771
$796$	0	0
$797$	40.7497	1.44343	0.721714	−	0.692191i	$-0.243355\pi$
$797$	40.7497	1.44343	0.721714	+	0.692191i	$0.243355\pi$
$798$	0	0
$799$	−16.1722	−0.572131
$800$	0	0
$801$	16.0682	0.567741
$802$	0	0
$803$	31.1466	1.09914
$804$	0	0
$805$	−19.7996	−0.697846
$806$	0	0
$807$	38.5048	1.35543
$808$	0	0
$809$	50.7630	1.78473	0.892366	−	0.451312i	$-0.149044\pi$
$809$	50.7630	1.78473	0.892366	+	0.451312i	$0.149044\pi$
$810$	0	0
$811$	−28.4968	−1.00066	−0.500330	−	0.865835i	$-0.666788\pi$
$811$	−28.4968	−1.00066	−0.500330	+	0.865835i	$0.666788\pi$
$812$	0	0
$813$	−79.7421	−2.79668
$814$	0	0
$815$	−11.5966	−0.406213
$816$	0	0
$817$	−75.2686	−2.63331
$818$	0	0
$819$	31.8013	1.11123
$820$	0	0
$821$	10.9996	0.383890	0.191945	−	0.981406i	$-0.438520\pi$
$821$	10.9996	0.383890	0.191945	+	0.981406i	$0.438520\pi$
$822$	0	0
$823$	4.15311	0.144768	0.0723841	−	0.997377i	$-0.476939\pi$
$823$	4.15311	0.144768	0.0723841	+	0.997377i	$0.476939\pi$
$824$	0	0
$825$	−28.0906	−0.977988
$826$	0	0
$827$	21.8177	0.758674	0.379337	−	0.925259i	$-0.376152\pi$
$827$	21.8177	0.758674	0.379337	+	0.925259i	$0.376152\pi$
$828$	0	0
$829$	35.3419	1.22748	0.613738	−	0.789510i	$-0.289665\pi$
$829$	35.3419	1.22748	0.613738	+	0.789510i	$0.289665\pi$
$830$	0	0
$831$	−58.0059	−2.01220
$832$	0	0
$833$	24.8489	0.860965
$834$	0	0
$835$	25.0474	0.866801
$836$	0	0
$837$	61.8261	2.13702
$838$	0	0
$839$	19.9086	0.687321	0.343661	−	0.939094i	$-0.388333\pi$
$839$	19.9086	0.687321	0.343661	+	0.939094i	$0.388333\pi$
$840$	0	0
$841$	3.27319	0.112869
$842$	0	0
$843$	−15.1111	−0.520456
$844$	0	0
$845$	17.0912	0.587956
$846$	0	0
$847$	8.75396	0.300789
$848$	0	0
$849$	−13.5807	−0.466089
$850$	0	0
$851$	6.07513	0.208253
$852$	0	0
$853$	−28.4986	−0.975772	−0.487886	−	0.872907i	$-0.662232\pi$
$853$	−28.4986	−0.975772	−0.487886	+	0.872907i	$0.662232\pi$
$854$	0	0
$855$	70.0514	2.39571
$856$	0	0
$857$	−3.74958	−0.128083	−0.0640416	−	0.997947i	$-0.520399\pi$
$857$	−3.74958	−0.128083	−0.0640416	+	0.997947i	$0.520399\pi$
$858$	0	0
$859$	−23.4382	−0.799702	−0.399851	−	0.916580i	$-0.630938\pi$
$859$	−23.4382	−0.799702	−0.399851	+	0.916580i	$0.630938\pi$
$860$	0	0
$861$	−117.746	−4.01276
$862$	0	0
$863$	−2.18386	−0.0743395	−0.0371697	−	0.999309i	$-0.511834\pi$
$863$	−2.18386	−0.0743395	−0.0371697	+	0.999309i	$0.511834\pi$
$864$	0	0
$865$	18.2290	0.619806
$866$	0	0
$867$	33.4327	1.13543
$868$	0	0
$869$	−60.1072	−2.03900
$870$	0	0
$871$	0.139659	0.00473217
$872$	0	0
$873$	−29.1171	−0.985464
$874$	0	0
$875$	49.0745	1.65902
$876$	0	0
$877$	41.6346	1.40590	0.702951	−	0.711238i	$-0.251865\pi$
$877$	41.6346	1.40590	0.702951	+	0.711238i	$0.251865\pi$
$878$	0	0
$879$	−28.5844	−0.964127
$880$	0	0
$881$	14.4346	0.486314	0.243157	−	0.969987i	$-0.421817\pi$
$881$	14.4346	0.486314	0.243157	+	0.969987i	$0.421817\pi$
$882$	0	0
$883$	6.65089	0.223820	0.111910	−	0.993718i	$-0.464303\pi$
$883$	6.65089	0.223820	0.111910	+	0.993718i	$0.464303\pi$
$884$	0	0
$885$	26.7643	0.899673
$886$	0	0
$887$	−28.7906	−0.966692	−0.483346	−	0.875429i	$-0.660579\pi$
$887$	−28.7906	−0.966692	−0.483346	+	0.875429i	$0.660579\pi$
$888$	0	0
$889$	50.4376	1.69162
$890$	0	0
$891$	−19.6689	−0.658932
$892$	0	0
$893$	55.9660	1.87283
$894$	0	0
$895$	8.40459	0.280934
$896$	0	0
$897$	−12.3054	−0.410865
$898$	0	0
$899$	−46.3758	−1.54672
$900$	0	0
$901$	−13.9909	−0.466104
$902$	0	0
$903$	−112.794	−3.75356
$904$	0	0
$905$	−15.0198	−0.499275
$906$	0	0
$907$	20.3304	0.675061	0.337530	−	0.941315i	$-0.390408\pi$
$907$	20.3304	0.675061	0.337530	+	0.941315i	$0.390408\pi$
$908$	0	0
$909$	28.4404	0.943308
$910$	0	0
$911$	49.6946	1.64645	0.823227	−	0.567712i	$-0.192171\pi$
$911$	49.6946	1.64645	0.823227	+	0.567712i	$0.192171\pi$
$912$	0	0
$913$	45.0006	1.48930
$914$	0	0
$915$	25.1984	0.833033
$916$	0	0
$917$	−34.8182	−1.14980
$918$	0	0
$919$	46.7416	1.54186	0.770931	−	0.636918i	$-0.219791\pi$
$919$	46.7416	1.54186	0.770931	+	0.636918i	$0.219791\pi$
$920$	0	0
$921$	68.7174	2.26431
$922$	0	0
$923$	−0.727469	−0.0239449
$924$	0	0
$925$	−5.21559	−0.171488
$926$	0	0
$927$	37.7374	1.23946
$928$	0	0
$929$	−2.98110	−0.0978066	−0.0489033	−	0.998804i	$-0.515573\pi$
$929$	−2.98110	−0.0978066	−0.0489033	+	0.998804i	$0.515573\pi$
$930$	0	0
$931$	−85.9930	−2.81831
$932$	0	0
$933$	6.28860	0.205880
$934$	0	0
$935$	13.1145	0.428891
$936$	0	0
$937$	17.2491	0.563502	0.281751	−	0.959488i	$-0.409085\pi$
$937$	17.2491	0.563502	0.281751	+	0.959488i	$0.409085\pi$
$938$	0	0
$939$	−71.9815	−2.34903
$940$	0	0
$941$	−16.5888	−0.540780	−0.270390	−	0.962751i	$-0.587153\pi$
$941$	−16.5888	−0.540780	−0.270390	+	0.962751i	$0.587153\pi$
$942$	0	0
$943$	29.6397	0.965201
$944$	0	0
$945$	48.5863	1.58051
$946$	0	0
$947$	41.0333	1.33340	0.666702	−	0.745325i	$-0.267706\pi$
$947$	41.0333	1.33340	0.666702	+	0.745325i	$0.267706\pi$
$948$	0	0
$949$	−11.7137	−0.380242
$950$	0	0
$951$	−12.8969	−0.418212
$952$	0	0
$953$	−56.0176	−1.81459	−0.907293	−	0.420498i	$-0.861855\pi$
$953$	−56.0176	−1.81459	−0.907293	+	0.420498i	$0.861855\pi$
$954$	0	0
$955$	−22.4520	−0.726530
$956$	0	0
$957$	60.2268	1.94686
$958$	0	0
$959$	−52.3308	−1.68985
$960$	0	0
$961$	35.6410	1.14971
$962$	0	0
$963$	−20.9521	−0.675172
$964$	0	0
$965$	−14.0551	−0.452451
$966$	0	0
$967$	30.8571	0.992298	0.496149	−	0.868237i	$-0.334747\pi$
$967$	30.8571	0.992298	0.496149	+	0.868237i	$0.334747\pi$
$968$	0	0
$969$	56.6752	1.82067
$970$	0	0
$971$	20.1094	0.645342	0.322671	−	0.946511i	$-0.395419\pi$
$971$	20.1094	0.645342	0.322671	+	0.946511i	$0.395419\pi$
$972$	0	0
$973$	−68.9832	−2.21150
$974$	0	0
$975$	10.5644	0.338330
$976$	0	0
$977$	−32.6229	−1.04370	−0.521849	−	0.853038i	$-0.674758\pi$
$977$	−32.6229	−1.04370	−0.521849	+	0.853038i	$0.674758\pi$
$978$	0	0
$979$	−10.4102	−0.332711
$980$	0	0
$981$	−41.8577	−1.33641
$982$	0	0
$983$	−37.8245	−1.20641	−0.603207	−	0.797585i	$-0.706111\pi$
$983$	−37.8245	−1.20641	−0.603207	+	0.797585i	$0.706111\pi$
$984$	0	0
$985$	9.86285	0.314256
$986$	0	0
$987$	83.8684	2.66956
$988$	0	0
$989$	28.3934	0.902856
$990$	0	0
$991$	−39.4906	−1.25446	−0.627230	−	0.778834i	$-0.715811\pi$
$991$	−39.4906	−1.25446	−0.627230	+	0.778834i	$0.715811\pi$
$992$	0	0
$993$	−68.8988	−2.18644
$994$	0	0
$995$	−40.5117	−1.28431
$996$	0	0
$997$	−56.6265	−1.79338	−0.896689	−	0.442661i	$-0.854035\pi$
$997$	−56.6265	−1.79338	−0.896689	+	0.442661i	$0.854035\pi$
$998$	0	0
$999$	−14.9078	−0.471661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.r.1.1		12
4.3	odd	2		1006.2.a.i.1.12	✓	12
12.11	even	2		9054.2.a.bj.1.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.i.1.12	✓	12	4.3	odd	2
8048.2.a.r.1.1		12	1.1	even	1	trivial
9054.2.a.bj.1.10		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)

\(f(q)\)	\(=\)	\(q-2.92999 q^{3} -1.53308 q^{5} +4.18455 q^{7} +5.58485 q^{9} +O(q^{10})\)	\(q-2.92999 q^{3} -1.53308 q^{5} +4.18455 q^{7} +5.58485 q^{9} -3.61828 q^{11} +1.36077 q^{13} +4.49191 q^{15} +2.36421 q^{17} -8.18166 q^{19} -12.2607 q^{21} +3.08634 q^{23} -2.64967 q^{25} -7.57357 q^{27} +5.68095 q^{29} -8.16339 q^{31} +10.6015 q^{33} -6.41525 q^{35} +1.96839 q^{37} -3.98705 q^{39} +9.60349 q^{41} +9.19968 q^{43} -8.56201 q^{45} -6.84043 q^{47} +10.5105 q^{49} -6.92711 q^{51} -5.91778 q^{53} +5.54711 q^{55} +23.9722 q^{57} +5.95834 q^{59} +5.60973 q^{61} +23.3701 q^{63} -2.08617 q^{65} +0.102632 q^{67} -9.04295 q^{69} -0.534601 q^{71} -8.60811 q^{73} +7.76351 q^{75} -15.1409 q^{77} +16.6121 q^{79} +5.43596 q^{81} -12.4370 q^{83} -3.62452 q^{85} -16.6451 q^{87} +2.87710 q^{89} +5.69421 q^{91} +23.9187 q^{93} +12.5431 q^{95} -5.21359 q^{97} -20.2076 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} + 7 q^{5} + 2 q^{7} + 27 q^{9}+O(q^{10}) \) 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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