LMFDB - Embedded newform 2004.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2004,2,Mod(1,2004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2004, 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("2004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2004 = 2^{2} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2004.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:	$16.0020205651$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 29x^{7} - 7x^{6} + 266x^{5} + 69x^{4} - 901x^{3} - 199x^{2} + 875x + 391 $ x^9 - 29x^7 - 7x^6 + 266x^5 + 69x^4 - 901x^3 - 199x^2 + 875*x + 391
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.16840$ of defining polynomial
Character	$\chi$	$=$	2004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.16840 q^{5} -0.230890 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.16840 q^{5} -0.230890 q^{7} +1.00000 q^{9} +0.534912 q^{11} +2.40467 q^{13} -3.16840 q^{15} +3.83643 q^{17} -7.74980 q^{19} -0.230890 q^{21} +5.14190 q^{23} +5.03875 q^{25} +1.00000 q^{27} +0.602944 q^{29} -4.43840 q^{31} +0.534912 q^{33} +0.731551 q^{35} +8.52496 q^{37} +2.40467 q^{39} +1.40231 q^{41} -3.66794 q^{43} -3.16840 q^{45} +12.4809 q^{47} -6.94669 q^{49} +3.83643 q^{51} +4.85598 q^{53} -1.69482 q^{55} -7.74980 q^{57} +9.66413 q^{59} -1.49131 q^{61} -0.230890 q^{63} -7.61896 q^{65} +12.0615 q^{67} +5.14190 q^{69} +11.4038 q^{71} +4.16407 q^{73} +5.03875 q^{75} -0.123506 q^{77} -7.52893 q^{79} +1.00000 q^{81} +7.44885 q^{83} -12.1553 q^{85} +0.602944 q^{87} +9.40223 q^{89} -0.555214 q^{91} -4.43840 q^{93} +24.5544 q^{95} +12.2203 q^{97} +0.534912 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9} + 7 q^{11} + 6 q^{13} + 9 q^{15} + 7 q^{17} + 2 q^{19} + 2 q^{21} + 19 q^{23} + 22 q^{25} + 9 q^{27} + 13 q^{29} + 12 q^{31} + 7 q^{33} + 4 q^{35} + 15 q^{37} + 6 q^{39} + 18 q^{41} - 6 q^{43} + 9 q^{45} + 25 q^{47} + 19 q^{49} + 7 q^{51} + 17 q^{53} - 3 q^{55} + 2 q^{57} + 3 q^{59} + 14 q^{61} + 2 q^{63} + 14 q^{65} - 4 q^{67} + 19 q^{69} + 17 q^{71} - 20 q^{73} + 22 q^{75} + 14 q^{77} - 8 q^{79} + 9 q^{81} - q^{83} + 5 q^{85} + 13 q^{87} + 36 q^{89} - 41 q^{91} + 12 q^{93} + 5 q^{95} + 31 q^{97} + 7 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9 + 7 * q^11 + 6 * q^13 + 9 * q^15 + 7 * q^17 + 2 * q^19 + 2 * q^21 + 19 * q^23 + 22 * q^25 + 9 * q^27 + 13 * q^29 + 12 * q^31 + 7 * q^33 + 4 * q^35 + 15 * q^37 + 6 * q^39 + 18 * q^41 - 6 * q^43 + 9 * q^45 + 25 * q^47 + 19 * q^49 + 7 * q^51 + 17 * q^53 - 3 * q^55 + 2 * q^57 + 3 * q^59 + 14 * q^61 + 2 * q^63 + 14 * q^65 - 4 * q^67 + 19 * q^69 + 17 * q^71 - 20 * q^73 + 22 * q^75 + 14 * q^77 - 8 * q^79 + 9 * q^81 - q^83 + 5 * q^85 + 13 * q^87 + 36 * q^89 - 41 * q^91 + 12 * q^93 + 5 * q^95 + 31 * q^97 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.16840	−1.41695	−0.708476	−	0.705735i	$-0.750617\pi$
$5$	−3.16840	−1.41695	−0.708476	+	0.705735i	$0.750617\pi$
$6$	0	0
$7$	−0.230890	−0.0872681	−0.0436341	−	0.999048i	$-0.513894\pi$
$7$	−0.230890	−0.0872681	−0.0436341	+	0.999048i	$0.513894\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.534912	0.161282	0.0806411	−	0.996743i	$-0.474303\pi$
$11$	0.534912	0.161282	0.0806411	+	0.996743i	$0.474303\pi$
$12$	0	0
$13$	2.40467	0.666936	0.333468	−	0.942761i	$-0.391781\pi$
$13$	2.40467	0.666936	0.333468	+	0.942761i	$0.391781\pi$
$14$	0	0
$15$	−3.16840	−0.818077
$16$	0	0
$17$	3.83643	0.930471	0.465235	−	0.885187i	$-0.345970\pi$
$17$	3.83643	0.930471	0.465235	+	0.885187i	$0.345970\pi$
$18$	0	0
$19$	−7.74980	−1.77793	−0.888963	−	0.457980i	$-0.848573\pi$
$19$	−7.74980	−1.77793	−0.888963	+	0.457980i	$0.848573\pi$
$20$	0	0
$21$	−0.230890	−0.0503843
$22$	0	0
$23$	5.14190	1.07216	0.536080	−	0.844167i	$-0.319905\pi$
$23$	5.14190	1.07216	0.536080	+	0.844167i	$0.319905\pi$
$24$	0	0
$25$	5.03875	1.00775
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.602944	0.111964	0.0559820	−	0.998432i	$-0.482171\pi$
$29$	0.602944	0.111964	0.0559820	+	0.998432i	$0.482171\pi$
$30$	0	0
$31$	−4.43840	−0.797161	−0.398580	−	0.917133i	$-0.630497\pi$
$31$	−4.43840	−0.797161	−0.398580	+	0.917133i	$0.630497\pi$
$32$	0	0
$33$	0.534912	0.0931163
$34$	0	0
$35$	0.731551	0.123655
$36$	0	0
$37$	8.52496	1.40149	0.700747	−	0.713410i	$-0.252850\pi$
$37$	8.52496	1.40149	0.700747	+	0.713410i	$0.252850\pi$
$38$	0	0
$39$	2.40467	0.385056
$40$	0	0
$41$	1.40231	0.219004	0.109502	−	0.993987i	$-0.465074\pi$
$41$	1.40231	0.219004	0.109502	+	0.993987i	$0.465074\pi$
$42$	0	0
$43$	−3.66794	−0.559356	−0.279678	−	0.960094i	$-0.590228\pi$
$43$	−3.66794	−0.559356	−0.279678	+	0.960094i	$0.590228\pi$
$44$	0	0
$45$	−3.16840	−0.472317
$46$	0	0
$47$	12.4809	1.82052	0.910262	−	0.414032i	$-0.135880\pi$
$47$	12.4809	1.82052	0.910262	+	0.414032i	$0.135880\pi$
$48$	0	0
$49$	−6.94669	−0.992384
$50$	0	0
$51$	3.83643	0.537208
$52$	0	0
$53$	4.85598	0.667020	0.333510	−	0.942747i	$-0.391767\pi$
$53$	4.85598	0.667020	0.333510	+	0.942747i	$0.391767\pi$
$54$	0	0
$55$	−1.69482	−0.228529
$56$	0	0
$57$	−7.74980	−1.02649
$58$	0	0
$59$	9.66413	1.25816	0.629081	−	0.777339i	$-0.283431\pi$
$59$	9.66413	1.25816	0.629081	+	0.777339i	$0.283431\pi$
$60$	0	0
$61$	−1.49131	−0.190942	−0.0954712	−	0.995432i	$-0.530436\pi$
$61$	−1.49131	−0.190942	−0.0954712	+	0.995432i	$0.530436\pi$
$62$	0	0
$63$	−0.230890	−0.0290894
$64$	0	0
$65$	−7.61896	−0.945015
$66$	0	0
$67$	12.0615	1.47355	0.736773	−	0.676140i	$-0.236348\pi$
$67$	12.0615	1.47355	0.736773	+	0.676140i	$0.236348\pi$
$68$	0	0
$69$	5.14190	0.619012
$70$	0	0
$71$	11.4038	1.35338	0.676691	−	0.736267i	$-0.263413\pi$
$71$	11.4038	1.35338	0.676691	+	0.736267i	$0.263413\pi$
$72$	0	0
$73$	4.16407	0.487368	0.243684	−	0.969855i	$-0.421644\pi$
$73$	4.16407	0.487368	0.243684	+	0.969855i	$0.421644\pi$
$74$	0	0
$75$	5.03875	0.581825
$76$	0	0
$77$	−0.123506	−0.0140748
$78$	0	0
$79$	−7.52893	−0.847071	−0.423536	−	0.905879i	$-0.639211\pi$
$79$	−7.52893	−0.847071	−0.423536	+	0.905879i	$0.639211\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.44885	0.817617	0.408809	−	0.912620i	$-0.365944\pi$
$83$	7.44885	0.817617	0.408809	+	0.912620i	$0.365944\pi$
$84$	0	0
$85$	−12.1553	−1.31843
$86$	0	0
$87$	0.602944	0.0646424
$88$	0	0
$89$	9.40223	0.996635	0.498317	−	0.866995i	$-0.333951\pi$
$89$	9.40223	0.996635	0.498317	+	0.866995i	$0.333951\pi$
$90$	0	0
$91$	−0.555214	−0.0582022
$92$	0	0
$93$	−4.43840	−0.460241
$94$	0	0
$95$	24.5544	2.51923
$96$	0	0
$97$	12.2203	1.24078	0.620391	−	0.784293i	$-0.286974\pi$
$97$	12.2203	1.24078	0.620391	+	0.784293i	$0.286974\pi$
$98$	0	0
$99$	0.534912	0.0537607
$100$	0	0
$101$	3.01977	0.300478	0.150239	−	0.988650i	$-0.451996\pi$
$101$	3.01977	0.300478	0.150239	+	0.988650i	$0.451996\pi$
$102$	0	0
$103$	11.2753	1.11099	0.555496	−	0.831519i	$-0.312528\pi$
$103$	11.2753	1.11099	0.555496	+	0.831519i	$0.312528\pi$
$104$	0	0
$105$	0.731551	0.0713920
$106$	0	0
$107$	−20.1981	−1.95263	−0.976314	−	0.216360i	$-0.930582\pi$
$107$	−20.1981	−1.95263	−0.976314	+	0.216360i	$0.930582\pi$
$108$	0	0
$109$	0.443113	0.0424425	0.0212212	−	0.999775i	$-0.493245\pi$
$109$	0.443113	0.0424425	0.0212212	+	0.999775i	$0.493245\pi$
$110$	0	0
$111$	8.52496	0.809153
$112$	0	0
$113$	12.4535	1.17152	0.585762	−	0.810483i	$-0.300795\pi$
$113$	12.4535	1.17152	0.585762	+	0.810483i	$0.300795\pi$
$114$	0	0
$115$	−16.2916	−1.51920
$116$	0	0
$117$	2.40467	0.222312
$118$	0	0
$119$	−0.885792	−0.0812004
$120$	0	0
$121$	−10.7139	−0.973988
$122$	0	0
$123$	1.40231	0.126442
$124$	0	0
$125$	−0.122782	−0.0109819
$126$	0	0
$127$	20.7466	1.84096	0.920480	−	0.390790i	$-0.127798\pi$
$127$	20.7466	1.84096	0.920480	+	0.390790i	$0.127798\pi$
$128$	0	0
$129$	−3.66794	−0.322944
$130$	0	0
$131$	−2.73611	−0.239055	−0.119527	−	0.992831i	$-0.538138\pi$
$131$	−2.73611	−0.239055	−0.119527	+	0.992831i	$0.538138\pi$
$132$	0	0
$133$	1.78935	0.155156
$134$	0	0
$135$	−3.16840	−0.272692
$136$	0	0
$137$	5.01418	0.428391	0.214195	−	0.976791i	$-0.431287\pi$
$137$	5.01418	0.428391	0.214195	+	0.976791i	$0.431287\pi$
$138$	0	0
$139$	−0.0795501	−0.00674735	−0.00337367	−	0.999994i	$-0.501074\pi$
$139$	−0.0795501	−0.00674735	−0.00337367	+	0.999994i	$0.501074\pi$
$140$	0	0
$141$	12.4809	1.05108
$142$	0	0
$143$	1.28629	0.107565
$144$	0	0
$145$	−1.91037	−0.158647
$146$	0	0
$147$	−6.94669	−0.572953
$148$	0	0
$149$	−14.9835	−1.22749	−0.613746	−	0.789503i	$-0.710338\pi$
$149$	−14.9835	−1.22749	−0.613746	+	0.789503i	$0.710338\pi$
$150$	0	0
$151$	−16.8126	−1.36819	−0.684097	−	0.729391i	$-0.739803\pi$
$151$	−16.8126	−1.36819	−0.684097	+	0.729391i	$0.739803\pi$
$152$	0	0
$153$	3.83643	0.310157
$154$	0	0
$155$	14.0626	1.12954
$156$	0	0
$157$	8.50629	0.678876	0.339438	−	0.940628i	$-0.389763\pi$
$157$	8.50629	0.678876	0.339438	+	0.940628i	$0.389763\pi$
$158$	0	0
$159$	4.85598	0.385104
$160$	0	0
$161$	−1.18721	−0.0935653
$162$	0	0
$163$	−15.6607	−1.22664	−0.613320	−	0.789835i	$-0.710166\pi$
$163$	−15.6607	−1.22664	−0.613320	+	0.789835i	$0.710166\pi$
$164$	0	0
$165$	−1.69482	−0.131941
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−7.21756	−0.555197
$170$	0	0
$171$	−7.74980	−0.592642
$172$	0	0
$173$	−17.1435	−1.30340	−0.651700	−	0.758477i	$-0.725944\pi$
$173$	−17.1435	−1.30340	−0.651700	+	0.758477i	$0.725944\pi$
$174$	0	0
$175$	−1.16340	−0.0879445
$176$	0	0
$177$	9.66413	0.726401
$178$	0	0
$179$	−26.4251	−1.97510	−0.987551	−	0.157297i	$-0.949722\pi$
$179$	−26.4251	−1.97510	−0.987551	+	0.157297i	$0.949722\pi$
$180$	0	0
$181$	3.49842	0.260036	0.130018	−	0.991512i	$-0.458497\pi$
$181$	3.49842	0.260036	0.130018	+	0.991512i	$0.458497\pi$
$182$	0	0
$183$	−1.49131	−0.110241
$184$	0	0
$185$	−27.0105	−1.98585
$186$	0	0
$187$	2.05215	0.150068
$188$	0	0
$189$	−0.230890	−0.0167948
$190$	0	0
$191$	−21.6136	−1.56390	−0.781952	−	0.623339i	$-0.785776\pi$
$191$	−21.6136	−1.56390	−0.781952	+	0.623339i	$0.785776\pi$
$192$	0	0
$193$	−8.20183	−0.590381	−0.295190	−	0.955438i	$-0.595383\pi$
$193$	−8.20183	−0.590381	−0.295190	+	0.955438i	$0.595383\pi$
$194$	0	0
$195$	−7.61896	−0.545605
$196$	0	0
$197$	2.73231	0.194669	0.0973345	−	0.995252i	$-0.468968\pi$
$197$	2.73231	0.194669	0.0973345	+	0.995252i	$0.468968\pi$
$198$	0	0
$199$	24.5661	1.74144	0.870721	−	0.491778i	$-0.163653\pi$
$199$	24.5661	1.74144	0.870721	+	0.491778i	$0.163653\pi$
$200$	0	0
$201$	12.0615	0.850753
$202$	0	0
$203$	−0.139214	−0.00977088
$204$	0	0
$205$	−4.44308	−0.310318
$206$	0	0
$207$	5.14190	0.357387
$208$	0	0
$209$	−4.14546	−0.286748
$210$	0	0
$211$	−13.0986	−0.901748	−0.450874	−	0.892588i	$-0.648888\pi$
$211$	−13.0986	−0.901748	−0.450874	+	0.892588i	$0.648888\pi$
$212$	0	0
$213$	11.4038	0.781376
$214$	0	0
$215$	11.6215	0.792580
$216$	0	0
$217$	1.02478	0.0695667
$218$	0	0
$219$	4.16407	0.281382
$220$	0	0
$221$	9.22535	0.620564
$222$	0	0
$223$	−3.90957	−0.261804	−0.130902	−	0.991395i	$-0.541787\pi$
$223$	−3.90957	−0.261804	−0.130902	+	0.991395i	$0.541787\pi$
$224$	0	0
$225$	5.03875	0.335917
$226$	0	0
$227$	6.24862	0.414736	0.207368	−	0.978263i	$-0.433510\pi$
$227$	6.24862	0.414736	0.207368	+	0.978263i	$0.433510\pi$
$228$	0	0
$229$	25.5918	1.69116	0.845578	−	0.533851i	$-0.179256\pi$
$229$	25.5918	1.69116	0.845578	+	0.533851i	$0.179256\pi$
$230$	0	0
$231$	−0.123506	−0.00812608
$232$	0	0
$233$	−13.0269	−0.853421	−0.426711	−	0.904388i	$-0.640328\pi$
$233$	−13.0269	−0.853421	−0.426711	+	0.904388i	$0.640328\pi$
$234$	0	0
$235$	−39.5444	−2.57959
$236$	0	0
$237$	−7.52893	−0.489057
$238$	0	0
$239$	8.54827	0.552942	0.276471	−	0.961022i	$-0.410835\pi$
$239$	8.54827	0.552942	0.276471	+	0.961022i	$0.410835\pi$
$240$	0	0
$241$	−16.8836	−1.08757	−0.543783	−	0.839226i	$-0.683008\pi$
$241$	−16.8836	−1.08757	−0.543783	+	0.839226i	$0.683008\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	22.0099	1.40616
$246$	0	0
$247$	−18.6357	−1.18576
$248$	0	0
$249$	7.44885	0.472051
$250$	0	0
$251$	−1.92874	−0.121741	−0.0608704	−	0.998146i	$-0.519388\pi$
$251$	−1.92874	−0.121741	−0.0608704	+	0.998146i	$0.519388\pi$
$252$	0	0
$253$	2.75046	0.172920
$254$	0	0
$255$	−12.1553	−0.761197
$256$	0	0
$257$	18.9215	1.18029	0.590145	−	0.807297i	$-0.299071\pi$
$257$	18.9215	1.18029	0.590145	+	0.807297i	$0.299071\pi$
$258$	0	0
$259$	−1.96833	−0.122306
$260$	0	0
$261$	0.602944	0.0373213
$262$	0	0
$263$	−8.99207	−0.554475	−0.277237	−	0.960801i	$-0.589419\pi$
$263$	−8.99207	−0.554475	−0.277237	+	0.960801i	$0.589419\pi$
$264$	0	0
$265$	−15.3857	−0.945135
$266$	0	0
$267$	9.40223	0.575407
$268$	0	0
$269$	11.8434	0.722102	0.361051	−	0.932546i	$-0.382418\pi$
$269$	11.8434	0.722102	0.361051	+	0.932546i	$0.382418\pi$
$270$	0	0
$271$	3.28934	0.199813	0.0999065	−	0.994997i	$-0.468146\pi$
$271$	3.28934	0.199813	0.0999065	+	0.994997i	$0.468146\pi$
$272$	0	0
$273$	−0.555214	−0.0336031
$274$	0	0
$275$	2.69529	0.162532
$276$	0	0
$277$	8.25296	0.495872	0.247936	−	0.968776i	$-0.420248\pi$
$277$	8.25296	0.495872	0.247936	+	0.968776i	$0.420248\pi$
$278$	0	0
$279$	−4.43840	−0.265720
$280$	0	0
$281$	15.3602	0.916313	0.458156	−	0.888872i	$-0.348510\pi$
$281$	15.3602	0.916313	0.458156	+	0.888872i	$0.348510\pi$
$282$	0	0
$283$	11.8821	0.706320	0.353160	−	0.935563i	$-0.385107\pi$
$283$	11.8821	0.706320	0.353160	+	0.935563i	$0.385107\pi$
$284$	0	0
$285$	24.5544	1.45448
$286$	0	0
$287$	−0.323779	−0.0191121
$288$	0	0
$289$	−2.28181	−0.134224
$290$	0	0
$291$	12.2203	0.716366
$292$	0	0
$293$	−22.7646	−1.32992	−0.664960	−	0.746879i	$-0.731551\pi$
$293$	−22.7646	−1.32992	−0.664960	+	0.746879i	$0.731551\pi$
$294$	0	0
$295$	−30.6198	−1.78276
$296$	0	0
$297$	0.534912	0.0310388
$298$	0	0
$299$	12.3646	0.715062
$300$	0	0
$301$	0.846890	0.0488140
$302$	0	0
$303$	3.01977	0.173481
$304$	0	0
$305$	4.72506	0.270556
$306$	0	0
$307$	−33.2061	−1.89517	−0.947585	−	0.319503i	$-0.896484\pi$
$307$	−33.2061	−1.89517	−0.947585	+	0.319503i	$0.896484\pi$
$308$	0	0
$309$	11.2753	0.641432
$310$	0	0
$311$	−11.8693	−0.673047	−0.336523	−	0.941675i	$-0.609251\pi$
$311$	−11.8693	−0.673047	−0.336523	+	0.941675i	$0.609251\pi$
$312$	0	0
$313$	−8.72134	−0.492959	−0.246480	−	0.969148i	$-0.579274\pi$
$313$	−8.72134	−0.492959	−0.246480	+	0.969148i	$0.579274\pi$
$314$	0	0
$315$	0.731551	0.0412182
$316$	0	0
$317$	21.0893	1.18449	0.592246	−	0.805757i	$-0.298241\pi$
$317$	21.0893	1.18449	0.592246	+	0.805757i	$0.298241\pi$
$318$	0	0
$319$	0.322522	0.0180578
$320$	0	0
$321$	−20.1981	−1.12735
$322$	0	0
$323$	−29.7315	−1.65431
$324$	0	0
$325$	12.1165	0.672105
$326$	0	0
$327$	0.443113	0.0245042
$328$	0	0
$329$	−2.88171	−0.158874
$330$	0	0
$331$	−10.9529	−0.602026	−0.301013	−	0.953620i	$-0.597325\pi$
$331$	−10.9529	−0.602026	−0.301013	+	0.953620i	$0.597325\pi$
$332$	0	0
$333$	8.52496	0.467165
$334$	0	0
$335$	−38.2156	−2.08794
$336$	0	0
$337$	−16.1884	−0.881839	−0.440920	−	0.897547i	$-0.645348\pi$
$337$	−16.1884	−0.881839	−0.440920	+	0.897547i	$0.645348\pi$
$338$	0	0
$339$	12.4535	0.676379
$340$	0	0
$341$	−2.37416	−0.128568
$342$	0	0
$343$	3.22015	0.173872
$344$	0	0
$345$	−16.2916	−0.877109
$346$	0	0
$347$	5.29375	0.284183	0.142092	−	0.989854i	$-0.454617\pi$
$347$	5.29375	0.284183	0.142092	+	0.989854i	$0.454617\pi$
$348$	0	0
$349$	33.7932	1.80891	0.904453	−	0.426573i	$-0.140279\pi$
$349$	33.7932	1.80891	0.904453	+	0.426573i	$0.140279\pi$
$350$	0	0
$351$	2.40467	0.128352
$352$	0	0
$353$	14.1450	0.752864	0.376432	−	0.926444i	$-0.377151\pi$
$353$	14.1450	0.752864	0.376432	+	0.926444i	$0.377151\pi$
$354$	0	0
$355$	−36.1318	−1.91768
$356$	0	0
$357$	−0.885792	−0.0468811
$358$	0	0
$359$	−9.66904	−0.510312	−0.255156	−	0.966900i	$-0.582127\pi$
$359$	−9.66904	−0.510312	−0.255156	+	0.966900i	$0.582127\pi$
$360$	0	0
$361$	41.0593	2.16102
$362$	0	0
$363$	−10.7139	−0.562332
$364$	0	0
$365$	−13.1934	−0.690577
$366$	0	0
$367$	13.1304	0.685399	0.342700	−	0.939445i	$-0.388659\pi$
$367$	13.1304	0.685399	0.342700	+	0.939445i	$0.388659\pi$
$368$	0	0
$369$	1.40231	0.0730014
$370$	0	0
$371$	−1.12120	−0.0582096
$372$	0	0
$373$	−3.67027	−0.190039	−0.0950197	−	0.995475i	$-0.530291\pi$
$373$	−3.67027	−0.190039	−0.0950197	+	0.995475i	$0.530291\pi$
$374$	0	0
$375$	−0.122782	−0.00634042
$376$	0	0
$377$	1.44988	0.0746727
$378$	0	0
$379$	21.9612	1.12807	0.564036	−	0.825750i	$-0.309248\pi$
$379$	21.9612	1.12807	0.564036	+	0.825750i	$0.309248\pi$
$380$	0	0
$381$	20.7466	1.06288
$382$	0	0
$383$	13.5332	0.691515	0.345757	−	0.938324i	$-0.387622\pi$
$383$	13.5332	0.691515	0.345757	+	0.938324i	$0.387622\pi$
$384$	0	0
$385$	0.391315	0.0199433
$386$	0	0
$387$	−3.66794	−0.186452
$388$	0	0
$389$	−5.09521	−0.258337	−0.129169	−	0.991623i	$-0.541231\pi$
$389$	−5.09521	−0.258337	−0.129169	+	0.991623i	$0.541231\pi$
$390$	0	0
$391$	19.7265	0.997613
$392$	0	0
$393$	−2.73611	−0.138018
$394$	0	0
$395$	23.8547	1.20026
$396$	0	0
$397$	10.9338	0.548751	0.274376	−	0.961623i	$-0.411529\pi$
$397$	10.9338	0.548751	0.274376	+	0.961623i	$0.411529\pi$
$398$	0	0
$399$	1.78935	0.0895795
$400$	0	0
$401$	11.6204	0.580296	0.290148	−	0.956982i	$-0.406295\pi$
$401$	11.6204	0.580296	0.290148	+	0.956982i	$0.406295\pi$
$402$	0	0
$403$	−10.6729	−0.531655
$404$	0	0
$405$	−3.16840	−0.157439
$406$	0	0
$407$	4.56011	0.226036
$408$	0	0
$409$	24.7043	1.22155	0.610774	−	0.791805i	$-0.290858\pi$
$409$	24.7043	1.22155	0.610774	+	0.791805i	$0.290858\pi$
$410$	0	0
$411$	5.01418	0.247331
$412$	0	0
$413$	−2.23135	−0.109797
$414$	0	0
$415$	−23.6009	−1.15852
$416$	0	0
$417$	−0.0795501	−0.00389558
$418$	0	0
$419$	−17.9832	−0.878535	−0.439267	−	0.898356i	$-0.644762\pi$
$419$	−17.9832	−0.878535	−0.439267	+	0.898356i	$0.644762\pi$
$420$	0	0
$421$	−20.8040	−1.01393	−0.506963	−	0.861968i	$-0.669232\pi$
$421$	−20.8040	−1.01393	−0.506963	+	0.861968i	$0.669232\pi$
$422$	0	0
$423$	12.4809	0.606841
$424$	0	0
$425$	19.3308	0.937682
$426$	0	0
$427$	0.344328	0.0166632
$428$	0	0
$429$	1.28629	0.0621026
$430$	0	0
$431$	33.8156	1.62884	0.814419	−	0.580277i	$-0.197056\pi$
$431$	33.8156	1.62884	0.814419	+	0.580277i	$0.197056\pi$
$432$	0	0
$433$	19.3923	0.931936	0.465968	−	0.884802i	$-0.345706\pi$
$433$	19.3923	0.931936	0.465968	+	0.884802i	$0.345706\pi$
$434$	0	0
$435$	−1.91037	−0.0915951
$436$	0	0
$437$	−39.8486	−1.90622
$438$	0	0
$439$	−24.3827	−1.16372	−0.581861	−	0.813288i	$-0.697675\pi$
$439$	−24.3827	−1.16372	−0.581861	+	0.813288i	$0.697675\pi$
$440$	0	0
$441$	−6.94669	−0.330795
$442$	0	0
$443$	3.26405	0.155080	0.0775399	−	0.996989i	$-0.475293\pi$
$443$	3.26405	0.155080	0.0775399	+	0.996989i	$0.475293\pi$
$444$	0	0
$445$	−29.7900	−1.41218
$446$	0	0
$447$	−14.9835	−0.708693
$448$	0	0
$449$	7.62351	0.359775	0.179888	−	0.983687i	$-0.442427\pi$
$449$	7.62351	0.359775	0.179888	+	0.983687i	$0.442427\pi$
$450$	0	0
$451$	0.750113	0.0353215
$452$	0	0
$453$	−16.8126	−0.789927
$454$	0	0
$455$	1.75914	0.0824697
$456$	0	0
$457$	−20.3938	−0.953980	−0.476990	−	0.878909i	$-0.658272\pi$
$457$	−20.3938	−0.953980	−0.476990	+	0.878909i	$0.658272\pi$
$458$	0	0
$459$	3.83643	0.179069
$460$	0	0
$461$	−4.34942	−0.202573	−0.101286	−	0.994857i	$-0.532296\pi$
$461$	−4.34942	−0.202573	−0.101286	+	0.994857i	$0.532296\pi$
$462$	0	0
$463$	−34.5171	−1.60415	−0.802074	−	0.597225i	$-0.796270\pi$
$463$	−34.5171	−1.60415	−0.802074	+	0.597225i	$0.796270\pi$
$464$	0	0
$465$	14.0626	0.652139
$466$	0	0
$467$	23.4037	1.08300	0.541498	−	0.840702i	$-0.317858\pi$
$467$	23.4037	1.08300	0.541498	+	0.840702i	$0.317858\pi$
$468$	0	0
$469$	−2.78488	−0.128594
$470$	0	0
$471$	8.50629	0.391949
$472$	0	0
$473$	−1.96203	−0.0902141
$474$	0	0
$475$	−39.0493	−1.79170
$476$	0	0
$477$	4.85598	0.222340
$478$	0	0
$479$	4.86695	0.222377	0.111188	−	0.993799i	$-0.464534\pi$
$479$	4.86695	0.222377	0.111188	+	0.993799i	$0.464534\pi$
$480$	0	0
$481$	20.4997	0.934707
$482$	0	0
$483$	−1.18721	−0.0540200
$484$	0	0
$485$	−38.7187	−1.75813
$486$	0	0
$487$	21.2862	0.964570	0.482285	−	0.876014i	$-0.339807\pi$
$487$	21.2862	0.964570	0.482285	+	0.876014i	$0.339807\pi$
$488$	0	0
$489$	−15.6607	−0.708201
$490$	0	0
$491$	−27.1615	−1.22578	−0.612891	−	0.790168i	$-0.709993\pi$
$491$	−27.1615	−1.22578	−0.612891	+	0.790168i	$0.709993\pi$
$492$	0	0
$493$	2.31315	0.104179
$494$	0	0
$495$	−1.69482	−0.0761763
$496$	0	0
$497$	−2.63302	−0.118107
$498$	0	0
$499$	−29.7335	−1.33105	−0.665526	−	0.746375i	$-0.731793\pi$
$499$	−29.7335	−1.33105	−0.665526	+	0.746375i	$0.731793\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	25.1206	1.12007	0.560037	−	0.828467i	$-0.310787\pi$
$503$	25.1206	1.12007	0.560037	+	0.828467i	$0.310787\pi$
$504$	0	0
$505$	−9.56783	−0.425763
$506$	0	0
$507$	−7.21756	−0.320543
$508$	0	0
$509$	−9.98009	−0.442360	−0.221180	−	0.975233i	$-0.570991\pi$
$509$	−9.98009	−0.442360	−0.221180	+	0.975233i	$0.570991\pi$
$510$	0	0
$511$	−0.961442	−0.0425317
$512$	0	0
$513$	−7.74980	−0.342162
$514$	0	0
$515$	−35.7248	−1.57422
$516$	0	0
$517$	6.67618	0.293618
$518$	0	0
$519$	−17.1435	−0.752518
$520$	0	0
$521$	−36.5986	−1.60341	−0.801707	−	0.597717i	$-0.796075\pi$
$521$	−36.5986	−1.60341	−0.801707	+	0.597717i	$0.796075\pi$
$522$	0	0
$523$	−19.5958	−0.856865	−0.428433	−	0.903574i	$-0.640934\pi$
$523$	−19.5958	−0.856865	−0.428433	+	0.903574i	$0.640934\pi$
$524$	0	0
$525$	−1.16340	−0.0507748
$526$	0	0
$527$	−17.0276	−0.741735
$528$	0	0
$529$	3.43910	0.149526
$530$	0	0
$531$	9.66413	0.419388
$532$	0	0
$533$	3.37210	0.146062
$534$	0	0
$535$	63.9957	2.76678
$536$	0	0
$537$	−26.4251	−1.14033
$538$	0	0
$539$	−3.71587	−0.160054
$540$	0	0
$541$	40.4214	1.73785	0.868927	−	0.494941i	$-0.164810\pi$
$541$	40.4214	1.73785	0.868927	+	0.494941i	$0.164810\pi$
$542$	0	0
$543$	3.49842	0.150132
$544$	0	0
$545$	−1.40396	−0.0601389
$546$	0	0
$547$	−29.7069	−1.27018	−0.635088	−	0.772440i	$-0.719036\pi$
$547$	−29.7069	−1.27018	−0.635088	+	0.772440i	$0.719036\pi$
$548$	0	0
$549$	−1.49131	−0.0636475
$550$	0	0
$551$	−4.67269	−0.199063
$552$	0	0
$553$	1.73835	0.0739223
$554$	0	0
$555$	−27.0105	−1.14653
$556$	0	0
$557$	41.9127	1.77590	0.887950	−	0.459940i	$-0.152129\pi$
$557$	41.9127	1.77590	0.887950	+	0.459940i	$0.152129\pi$
$558$	0	0
$559$	−8.82020	−0.373055
$560$	0	0
$561$	2.05215	0.0866420
$562$	0	0
$563$	−25.6677	−1.08176	−0.540882	−	0.841099i	$-0.681909\pi$
$563$	−25.6677	−1.08176	−0.540882	+	0.841099i	$0.681909\pi$
$564$	0	0
$565$	−39.4575	−1.65999
$566$	0	0
$567$	−0.230890	−0.00969646
$568$	0	0
$569$	20.1215	0.843538	0.421769	−	0.906703i	$-0.361409\pi$
$569$	20.1215	0.843538	0.421769	+	0.906703i	$0.361409\pi$
$570$	0	0
$571$	−2.50467	−0.104817	−0.0524085	−	0.998626i	$-0.516690\pi$
$571$	−2.50467	−0.104817	−0.0524085	+	0.998626i	$0.516690\pi$
$572$	0	0
$573$	−21.6136	−0.902920
$574$	0	0
$575$	25.9087	1.08047
$576$	0	0
$577$	8.39885	0.349649	0.174824	−	0.984600i	$-0.444064\pi$
$577$	8.39885	0.349649	0.174824	+	0.984600i	$0.444064\pi$
$578$	0	0
$579$	−8.20183	−0.340857
$580$	0	0
$581$	−1.71986	−0.0713519
$582$	0	0
$583$	2.59752	0.107578
$584$	0	0
$585$	−7.61896	−0.315005
$586$	0	0
$587$	−5.75584	−0.237569	−0.118784	−	0.992920i	$-0.537900\pi$
$587$	−5.75584	−0.237569	−0.118784	+	0.992920i	$0.537900\pi$
$588$	0	0
$589$	34.3967	1.41729
$590$	0	0
$591$	2.73231	0.112392
$592$	0	0
$593$	−1.97459	−0.0810867	−0.0405434	−	0.999178i	$-0.512909\pi$
$593$	−1.97459	−0.0810867	−0.0405434	+	0.999178i	$0.512909\pi$
$594$	0	0
$595$	2.80654	0.115057
$596$	0	0
$597$	24.5661	1.00542
$598$	0	0
$599$	15.4063	0.629486	0.314743	−	0.949177i	$-0.398082\pi$
$599$	15.4063	0.629486	0.314743	+	0.949177i	$0.398082\pi$
$600$	0	0
$601$	−36.5939	−1.49269	−0.746347	−	0.665557i	$-0.768194\pi$
$601$	−36.5939	−1.49269	−0.746347	+	0.665557i	$0.768194\pi$
$602$	0	0
$603$	12.0615	0.491182
$604$	0	0
$605$	33.9458	1.38009
$606$	0	0
$607$	−17.4777	−0.709398	−0.354699	−	0.934981i	$-0.615417\pi$
$607$	−17.4777	−0.709398	−0.354699	+	0.934981i	$0.615417\pi$
$608$	0	0
$609$	−0.139214	−0.00564122
$610$	0	0
$611$	30.0124	1.21417
$612$	0	0
$613$	−27.2259	−1.09964	−0.549821	−	0.835282i	$-0.685304\pi$
$613$	−27.2259	−1.09964	−0.549821	+	0.835282i	$0.685304\pi$
$614$	0	0
$615$	−4.44308	−0.179162
$616$	0	0
$617$	−22.0487	−0.887648	−0.443824	−	0.896114i	$-0.646378\pi$
$617$	−22.0487	−0.887648	−0.443824	+	0.896114i	$0.646378\pi$
$618$	0	0
$619$	−20.9615	−0.842513	−0.421257	−	0.906942i	$-0.638411\pi$
$619$	−20.9615	−0.842513	−0.421257	+	0.906942i	$0.638411\pi$
$620$	0	0
$621$	5.14190	0.206337
$622$	0	0
$623$	−2.17088	−0.0869744
$624$	0	0
$625$	−24.8047	−0.992190
$626$	0	0
$627$	−4.14546	−0.165554
$628$	0	0
$629$	32.7054	1.30405
$630$	0	0
$631$	−23.8805	−0.950668	−0.475334	−	0.879805i	$-0.657673\pi$
$631$	−23.8805	−0.950668	−0.475334	+	0.879805i	$0.657673\pi$
$632$	0	0
$633$	−13.0986	−0.520624
$634$	0	0
$635$	−65.7334	−2.60855
$636$	0	0
$637$	−16.7045	−0.661857
$638$	0	0
$639$	11.4038	0.451128
$640$	0	0
$641$	5.17791	0.204515	0.102258	−	0.994758i	$-0.467393\pi$
$641$	5.17791	0.204515	0.102258	+	0.994758i	$0.467393\pi$
$642$	0	0
$643$	22.0105	0.868010	0.434005	−	0.900911i	$-0.357100\pi$
$643$	22.0105	0.868010	0.434005	+	0.900911i	$0.357100\pi$
$644$	0	0
$645$	11.6215	0.457596
$646$	0	0
$647$	−7.73591	−0.304130	−0.152065	−	0.988370i	$-0.548592\pi$
$647$	−7.73591	−0.304130	−0.152065	+	0.988370i	$0.548592\pi$
$648$	0	0
$649$	5.16946	0.202919
$650$	0	0
$651$	1.02478	0.0401644
$652$	0	0
$653$	14.5886	0.570896	0.285448	−	0.958394i	$-0.407858\pi$
$653$	14.5886	0.570896	0.285448	+	0.958394i	$0.407858\pi$
$654$	0	0
$655$	8.66908	0.338729
$656$	0	0
$657$	4.16407	0.162456
$658$	0	0
$659$	−20.8462	−0.812053	−0.406027	−	0.913861i	$-0.633086\pi$
$659$	−20.8462	−0.812053	−0.406027	+	0.913861i	$0.633086\pi$
$660$	0	0
$661$	−21.8177	−0.848609	−0.424305	−	0.905519i	$-0.639481\pi$
$661$	−21.8177	−0.848609	−0.424305	+	0.905519i	$0.639481\pi$
$662$	0	0
$663$	9.22535	0.358283
$664$	0	0
$665$	−5.66937	−0.219849
$666$	0	0
$667$	3.10028	0.120043
$668$	0	0
$669$	−3.90957	−0.151153
$670$	0	0
$671$	−0.797719	−0.0307956
$672$	0	0
$673$	21.9795	0.847247	0.423623	−	0.905838i	$-0.360758\pi$
$673$	21.9795	0.847247	0.423623	+	0.905838i	$0.360758\pi$
$674$	0	0
$675$	5.03875	0.193942
$676$	0	0
$677$	−31.8610	−1.22452	−0.612260	−	0.790657i	$-0.709739\pi$
$677$	−31.8610	−1.22452	−0.612260	+	0.790657i	$0.709739\pi$
$678$	0	0
$679$	−2.82154	−0.108281
$680$	0	0
$681$	6.24862	0.239448
$682$	0	0
$683$	29.1729	1.11627	0.558134	−	0.829751i	$-0.311517\pi$
$683$	29.1729	1.11627	0.558134	+	0.829751i	$0.311517\pi$
$684$	0	0
$685$	−15.8869	−0.607009
$686$	0	0
$687$	25.5918	0.976390
$688$	0	0
$689$	11.6770	0.444859
$690$	0	0
$691$	−3.35450	−0.127611	−0.0638056	−	0.997962i	$-0.520324\pi$
$691$	−3.35450	−0.127611	−0.0638056	+	0.997962i	$0.520324\pi$
$692$	0	0
$693$	−0.123506	−0.00469160
$694$	0	0
$695$	0.252046	0.00956066
$696$	0	0
$697$	5.37987	0.203777
$698$	0	0
$699$	−13.0269	−0.492723
$700$	0	0
$701$	36.7230	1.38701	0.693504	−	0.720453i	$-0.256066\pi$
$701$	36.7230	1.38701	0.693504	+	0.720453i	$0.256066\pi$
$702$	0	0
$703$	−66.0667	−2.49175
$704$	0	0
$705$	−39.5444	−1.48933
$706$	0	0
$707$	−0.697234	−0.0262222
$708$	0	0
$709$	−10.3615	−0.389136	−0.194568	−	0.980889i	$-0.562330\pi$
$709$	−10.3615	−0.389136	−0.194568	+	0.980889i	$0.562330\pi$
$710$	0	0
$711$	−7.52893	−0.282357
$712$	0	0
$713$	−22.8218	−0.854684
$714$	0	0
$715$	−4.07547	−0.152414
$716$	0	0
$717$	8.54827	0.319241
$718$	0	0
$719$	19.4718	0.726175	0.363088	−	0.931755i	$-0.381723\pi$
$719$	19.4718	0.726175	0.363088	+	0.931755i	$0.381723\pi$
$720$	0	0
$721$	−2.60336	−0.0969542
$722$	0	0
$723$	−16.8836	−0.627906
$724$	0	0
$725$	3.03809	0.112832
$726$	0	0
$727$	−39.9158	−1.48040	−0.740198	−	0.672389i	$-0.765268\pi$
$727$	−39.9158	−1.48040	−0.740198	+	0.672389i	$0.765268\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.0718	−0.520464
$732$	0	0
$733$	51.3551	1.89684	0.948422	−	0.317009i	$-0.102679\pi$
$733$	51.3551	1.89684	0.948422	+	0.317009i	$0.102679\pi$
$734$	0	0
$735$	22.0099	0.811847
$736$	0	0
$737$	6.45184	0.237657
$738$	0	0
$739$	7.93272	0.291810	0.145905	−	0.989299i	$-0.453391\pi$
$739$	7.93272	0.291810	0.145905	+	0.989299i	$0.453391\pi$
$740$	0	0
$741$	−18.6357	−0.684600
$742$	0	0
$743$	−31.2211	−1.14539	−0.572695	−	0.819769i	$-0.694102\pi$
$743$	−31.2211	−1.14539	−0.572695	+	0.819769i	$0.694102\pi$
$744$	0	0
$745$	47.4736	1.73930
$746$	0	0
$747$	7.44885	0.272539
$748$	0	0
$749$	4.66354	0.170402
$750$	0	0
$751$	−15.5552	−0.567619	−0.283810	−	0.958881i	$-0.591598\pi$
$751$	−15.5552	−0.567619	−0.283810	+	0.958881i	$0.591598\pi$
$752$	0	0
$753$	−1.92874	−0.0702871
$754$	0	0
$755$	53.2692	1.93866
$756$	0	0
$757$	12.1429	0.441340	0.220670	−	0.975348i	$-0.429176\pi$
$757$	12.1429	0.441340	0.220670	+	0.975348i	$0.429176\pi$
$758$	0	0
$759$	2.75046	0.0998355
$760$	0	0
$761$	50.6261	1.83519	0.917597	−	0.397513i	$-0.130127\pi$
$761$	50.6261	1.83519	0.917597	+	0.397513i	$0.130127\pi$
$762$	0	0
$763$	−0.102310	−0.00370388
$764$	0	0
$765$	−12.1553	−0.439477
$766$	0	0
$767$	23.2391	0.839114
$768$	0	0
$769$	2.38129	0.0858716	0.0429358	−	0.999078i	$-0.486329\pi$
$769$	2.38129	0.0858716	0.0429358	+	0.999078i	$0.486329\pi$
$770$	0	0
$771$	18.9215	0.681441
$772$	0	0
$773$	43.0022	1.54668	0.773340	−	0.633991i	$-0.218584\pi$
$773$	43.0022	1.54668	0.773340	+	0.633991i	$0.218584\pi$
$774$	0	0
$775$	−22.3640	−0.803339
$776$	0	0
$777$	−1.96833	−0.0706133
$778$	0	0
$779$	−10.8676	−0.389373
$780$	0	0
$781$	6.10003	0.218276
$782$	0	0
$783$	0.602944	0.0215475
$784$	0	0
$785$	−26.9513	−0.961934
$786$	0	0
$787$	4.51599	0.160978	0.0804889	−	0.996756i	$-0.474352\pi$
$787$	4.51599	0.160978	0.0804889	+	0.996756i	$0.474352\pi$
$788$	0	0
$789$	−8.99207	−0.320126
$790$	0	0
$791$	−2.87538	−0.102237
$792$	0	0
$793$	−3.58611	−0.127346
$794$	0	0
$795$	−15.3857	−0.545674
$796$	0	0
$797$	−37.2925	−1.32097	−0.660484	−	0.750840i	$-0.729649\pi$
$797$	−37.2925	−1.32097	−0.660484	+	0.750840i	$0.729649\pi$
$798$	0	0
$799$	47.8820	1.69394
$800$	0	0
$801$	9.40223	0.332212
$802$	0	0
$803$	2.22741	0.0786038
$804$	0	0
$805$	3.76156	0.132578
$806$	0	0
$807$	11.8434	0.416906
$808$	0	0
$809$	11.4322	0.401935	0.200967	−	0.979598i	$-0.435592\pi$
$809$	11.4322	0.401935	0.200967	+	0.979598i	$0.435592\pi$
$810$	0	0
$811$	−29.1204	−1.02256	−0.511278	−	0.859415i	$-0.670828\pi$
$811$	−29.1204	−1.02256	−0.511278	+	0.859415i	$0.670828\pi$
$812$	0	0
$813$	3.28934	0.115362
$814$	0	0
$815$	49.6193	1.73809
$816$	0	0
$817$	28.4258	0.994493
$818$	0	0
$819$	−0.555214	−0.0194007
$820$	0	0
$821$	−24.3833	−0.850982	−0.425491	−	0.904963i	$-0.639899\pi$
$821$	−24.3833	−0.850982	−0.425491	+	0.904963i	$0.639899\pi$
$822$	0	0
$823$	33.2495	1.15901	0.579503	−	0.814970i	$-0.303247\pi$
$823$	33.2495	1.15901	0.579503	+	0.814970i	$0.303247\pi$
$824$	0	0
$825$	2.69529	0.0938380
$826$	0	0
$827$	24.7980	0.862312	0.431156	−	0.902277i	$-0.358106\pi$
$827$	24.7980	0.862312	0.431156	+	0.902277i	$0.358106\pi$
$828$	0	0
$829$	0.0759189	0.00263677	0.00131839	−	0.999999i	$-0.499580\pi$
$829$	0.0759189	0.00263677	0.00131839	+	0.999999i	$0.499580\pi$
$830$	0	0
$831$	8.25296	0.286292
$832$	0	0
$833$	−26.6505	−0.923384
$834$	0	0
$835$	−3.16840	−0.109647
$836$	0	0
$837$	−4.43840	−0.153414
$838$	0	0
$839$	−36.3774	−1.25589	−0.627944	−	0.778259i	$-0.716103\pi$
$839$	−36.3774	−1.25589	−0.627944	+	0.778259i	$0.716103\pi$
$840$	0	0
$841$	−28.6365	−0.987464
$842$	0	0
$843$	15.3602	0.529033
$844$	0	0
$845$	22.8681	0.786687
$846$	0	0
$847$	2.47372	0.0849981
$848$	0	0
$849$	11.8821	0.407794
$850$	0	0
$851$	43.8345	1.50263
$852$	0	0
$853$	−29.0407	−0.994334	−0.497167	−	0.867655i	$-0.665626\pi$
$853$	−29.0407	−0.994334	−0.497167	+	0.867655i	$0.665626\pi$
$854$	0	0
$855$	24.5544	0.839744
$856$	0	0
$857$	41.1252	1.40481	0.702406	−	0.711777i	$-0.252109\pi$
$857$	41.1252	1.40481	0.702406	+	0.711777i	$0.252109\pi$
$858$	0	0
$859$	8.21249	0.280206	0.140103	−	0.990137i	$-0.455257\pi$
$859$	8.21249	0.280206	0.140103	+	0.990137i	$0.455257\pi$
$860$	0	0
$861$	−0.323779	−0.0110344
$862$	0	0
$863$	9.51360	0.323847	0.161923	−	0.986803i	$-0.448230\pi$
$863$	9.51360	0.323847	0.161923	+	0.986803i	$0.448230\pi$
$864$	0	0
$865$	54.3176	1.84685
$866$	0	0
$867$	−2.28181	−0.0774944
$868$	0	0
$869$	−4.02732	−0.136617
$870$	0	0
$871$	29.0039	0.982761
$872$	0	0
$873$	12.2203	0.413594
$874$	0	0
$875$	0.0283490	0.000958372 0
$876$	0	0
$877$	−15.4443	−0.521517	−0.260758	−	0.965404i	$-0.583973\pi$
$877$	−15.4443	−0.521517	−0.260758	+	0.965404i	$0.583973\pi$
$878$	0	0
$879$	−22.7646	−0.767830
$880$	0	0
$881$	8.94074	0.301221	0.150611	−	0.988593i	$-0.451876\pi$
$881$	8.94074	0.301221	0.150611	+	0.988593i	$0.451876\pi$
$882$	0	0
$883$	32.6275	1.09800	0.549001	−	0.835822i	$-0.315008\pi$
$883$	32.6275	1.09800	0.549001	+	0.835822i	$0.315008\pi$
$884$	0	0
$885$	−30.6198	−1.02927
$886$	0	0
$887$	−45.4740	−1.52687	−0.763434	−	0.645886i	$-0.776488\pi$
$887$	−45.4740	−1.52687	−0.763434	+	0.645886i	$0.776488\pi$
$888$	0	0
$889$	−4.79017	−0.160657
$890$	0	0
$891$	0.534912	0.0179202
$892$	0	0
$893$	−96.7243	−3.23676
$894$	0	0
$895$	83.7252	2.79862
$896$	0	0
$897$	12.3646	0.412841
$898$	0	0
$899$	−2.67611	−0.0892533
$900$	0	0
$901$	18.6296	0.620642
$902$	0	0
$903$	0.846890	0.0281827
$904$	0	0
$905$	−11.0844	−0.368458
$906$	0	0
$907$	21.4979	0.713826	0.356913	−	0.934138i	$-0.383829\pi$
$907$	21.4979	0.713826	0.356913	+	0.934138i	$0.383829\pi$
$908$	0	0
$909$	3.01977	0.100159
$910$	0	0
$911$	44.8385	1.48556	0.742782	−	0.669533i	$-0.233506\pi$
$911$	44.8385	1.48556	0.742782	+	0.669533i	$0.233506\pi$
$912$	0	0
$913$	3.98448	0.131867
$914$	0	0
$915$	4.72506	0.156206
$916$	0	0
$917$	0.631739	0.0208619
$918$	0	0
$919$	46.0171	1.51796	0.758982	−	0.651112i	$-0.225697\pi$
$919$	46.0171	1.51796	0.758982	+	0.651112i	$0.225697\pi$
$920$	0	0
$921$	−33.2061	−1.09418
$922$	0	0
$923$	27.4224	0.902619
$924$	0	0
$925$	42.9552	1.41236
$926$	0	0
$927$	11.2753	0.370331
$928$	0	0
$929$	11.8004	0.387159	0.193580	−	0.981085i	$-0.437990\pi$
$929$	11.8004	0.387159	0.193580	+	0.981085i	$0.437990\pi$
$930$	0	0
$931$	53.8354	1.76438
$932$	0	0
$933$	−11.8693	−0.388584
$934$	0	0
$935$	−6.50204	−0.212639
$936$	0	0
$937$	42.2945	1.38170	0.690850	−	0.722998i	$-0.257237\pi$
$937$	42.2945	1.38170	0.690850	+	0.722998i	$0.257237\pi$
$938$	0	0
$939$	−8.72134	−0.284610
$940$	0	0
$941$	6.50734	0.212133	0.106067	−	0.994359i	$-0.466174\pi$
$941$	6.50734	0.212133	0.106067	+	0.994359i	$0.466174\pi$
$942$	0	0
$943$	7.21054	0.234807
$944$	0	0
$945$	0.731551	0.0237973
$946$	0	0
$947$	−2.52120	−0.0819278	−0.0409639	−	0.999161i	$-0.513043\pi$
$947$	−2.52120	−0.0819278	−0.0409639	+	0.999161i	$0.513043\pi$
$948$	0	0
$949$	10.0132	0.325043
$950$	0	0
$951$	21.0893	0.683867
$952$	0	0
$953$	23.7358	0.768879	0.384439	−	0.923150i	$-0.374395\pi$
$953$	23.7358	0.768879	0.384439	+	0.923150i	$0.374395\pi$
$954$	0	0
$955$	68.4804	2.21597
$956$	0	0
$957$	0.322522	0.0104257
$958$	0	0
$959$	−1.15772	−0.0373848
$960$	0	0
$961$	−11.3006	−0.364534
$962$	0	0
$963$	−20.1981	−0.650876
$964$	0	0
$965$	25.9867	0.836541
$966$	0	0
$967$	12.9742	0.417223	0.208612	−	0.977999i	$-0.433106\pi$
$967$	12.9742	0.417223	0.208612	+	0.977999i	$0.433106\pi$
$968$	0	0
$969$	−29.7315	−0.955115
$970$	0	0
$971$	−38.0031	−1.21958	−0.609789	−	0.792564i	$-0.708746\pi$
$971$	−38.0031	−1.21958	−0.609789	+	0.792564i	$0.708746\pi$
$972$	0	0
$973$	0.0183673	0.000588828 0
$974$	0	0
$975$	12.1165	0.388040
$976$	0	0
$977$	32.3943	1.03639	0.518193	−	0.855264i	$-0.326605\pi$
$977$	32.3943	1.03639	0.518193	+	0.855264i	$0.326605\pi$
$978$	0	0
$979$	5.02937	0.160739
$980$	0	0
$981$	0.443113	0.0141475
$982$	0	0
$983$	−13.9912	−0.446249	−0.223124	−	0.974790i	$-0.571626\pi$
$983$	−13.9912	−0.446249	−0.223124	+	0.974790i	$0.571626\pi$
$984$	0	0
$985$	−8.65705	−0.275836
$986$	0	0
$987$	−2.88171	−0.0917258
$988$	0	0
$989$	−18.8602	−0.599719
$990$	0	0
$991$	−6.34209	−0.201463	−0.100732	−	0.994914i	$-0.532118\pi$
$991$	−6.34209	−0.201463	−0.100732	+	0.994914i	$0.532118\pi$
$992$	0	0
$993$	−10.9529	−0.347580
$994$	0	0
$995$	−77.8351	−2.46754
$996$	0	0
$997$	39.1554	1.24006	0.620032	−	0.784576i	$-0.287119\pi$
$997$	39.1554	1.24006	0.620032	+	0.784576i	$0.287119\pi$
$998$	0	0
$999$	8.52496	0.269718

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2004.2.a.d.1.1	✓	9
3.2	odd	2		6012.2.a.h.1.9		9
4.3	odd	2		8016.2.a.bb.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.d.1.1	✓	9	1.1	even	1	trivial
6012.2.a.h.1.9		9	3.2	odd	2
8016.2.a.bb.1.1		9	4.3	odd	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:	\( N \)	\(=\)	\( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.16840 q^{5} -0.230890 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.16840 q^{5} -0.230890 q^{7} +1.00000 q^{9} +0.534912 q^{11} +2.40467 q^{13} -3.16840 q^{15} +3.83643 q^{17} -7.74980 q^{19} -0.230890 q^{21} +5.14190 q^{23} +5.03875 q^{25} +1.00000 q^{27} +0.602944 q^{29} -4.43840 q^{31} +0.534912 q^{33} +0.731551 q^{35} +8.52496 q^{37} +2.40467 q^{39} +1.40231 q^{41} -3.66794 q^{43} -3.16840 q^{45} +12.4809 q^{47} -6.94669 q^{49} +3.83643 q^{51} +4.85598 q^{53} -1.69482 q^{55} -7.74980 q^{57} +9.66413 q^{59} -1.49131 q^{61} -0.230890 q^{63} -7.61896 q^{65} +12.0615 q^{67} +5.14190 q^{69} +11.4038 q^{71} +4.16407 q^{73} +5.03875 q^{75} -0.123506 q^{77} -7.52893 q^{79} +1.00000 q^{81} +7.44885 q^{83} -12.1553 q^{85} +0.602944 q^{87} +9.40223 q^{89} -0.555214 q^{91} -4.43840 q^{93} +24.5544 q^{95} +12.2203 q^{97} +0.534912 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9} + 7 q^{11} + 6 q^{13} + 9 q^{15} + 7 q^{17} + 2 q^{19} + 2 q^{21} + 19 q^{23} + 22 q^{25} + 9 q^{27} + 13 q^{29} + 12 q^{31} + 7 q^{33} + 4 q^{35} + 15 q^{37} + 6 q^{39} + 18 q^{41} - 6 q^{43} + 9 q^{45} + 25 q^{47} + 19 q^{49} + 7 q^{51} + 17 q^{53} - 3 q^{55} + 2 q^{57} + 3 q^{59} + 14 q^{61} + 2 q^{63} + 14 q^{65} - 4 q^{67} + 19 q^{69} + 17 q^{71} - 20 q^{73} + 22 q^{75} + 14 q^{77} - 8 q^{79} + 9 q^{81} - q^{83} + 5 q^{85} + 13 q^{87} + 36 q^{89} - 41 q^{91} + 12 q^{93} + 5 q^{95} + 31 q^{97} + 7 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9 + 7 * q^11 + 6 * q^13 + 9 * q^15 + 7 * q^17 + 2 * q^19 + 2 * q^21 + 19 * q^23 + 22 * q^25 + 9 * q^27 + 13 * q^29 + 12 * q^31 + 7 * q^33 + 4 * q^35 + 15 * q^37 + 6 * q^39 + 18 * q^41 - 6 * q^43 + 9 * q^45 + 25 * q^47 + 19 * q^49 + 7 * q^51 + 17 * q^53 - 3 * q^55 + 2 * q^57 + 3 * q^59 + 14 * q^61 + 2 * q^63 + 14 * q^65 - 4 * q^67 + 19 * q^69 + 17 * q^71 - 20 * q^73 + 22 * q^75 + 14 * q^77 - 8 * q^79 + 9 * q^81 - q^83 + 5 * q^85 + 13 * q^87 + 36 * q^89 - 41 * q^91 + 12 * q^93 + 5 * q^95 + 31 * q^97 + 7 * q^99

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