LMFDB - Embedded newform 2004.2.a.d.1.8

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			1.8
Root			$-3.19525$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +4.19525 q^{5} +1.43344 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.19525 q^{5} +1.43344 q^{7} +1.00000 q^{9} +5.55413 q^{11} -2.46575 q^{13} +4.19525 q^{15} -0.151828 q^{17} +0.370208 q^{19} +1.43344 q^{21} -5.22489 q^{23} +12.6001 q^{25} +1.00000 q^{27} -9.43548 q^{29} +6.03892 q^{31} +5.55413 q^{33} +6.01363 q^{35} +9.91270 q^{37} -2.46575 q^{39} -8.03882 q^{41} -12.3613 q^{43} +4.19525 q^{45} -5.19319 q^{47} -4.94525 q^{49} -0.151828 q^{51} +6.02724 q^{53} +23.3010 q^{55} +0.370208 q^{57} -6.13698 q^{59} -4.70024 q^{61} +1.43344 q^{63} -10.3445 q^{65} -6.05931 q^{67} -5.22489 q^{69} +3.38205 q^{71} -13.2561 q^{73} +12.6001 q^{75} +7.96151 q^{77} -5.34461 q^{79} +1.00000 q^{81} +1.83389 q^{83} -0.636958 q^{85} -9.43548 q^{87} +7.69442 q^{89} -3.53451 q^{91} +6.03892 q^{93} +1.55312 q^{95} +18.7449 q^{97} +5.55413 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.19525	1.87617	0.938087	−	0.346400i	$-0.112596\pi$
$5$	4.19525	1.87617	0.938087	+	0.346400i	$0.112596\pi$
$6$	0	0
$7$	1.43344	0.541789	0.270894	−	0.962609i	$-0.412681\pi$
$7$	1.43344	0.541789	0.270894	+	0.962609i	$0.412681\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.55413	1.67463	0.837317	−	0.546718i	$-0.184123\pi$
$11$	5.55413	1.67463	0.837317	+	0.546718i	$0.184123\pi$
$12$	0	0
$13$	−2.46575	−0.683877	−0.341938	−	0.939722i	$-0.611083\pi$
$13$	−2.46575	−0.683877	−0.341938	+	0.939722i	$0.611083\pi$
$14$	0	0
$15$	4.19525	1.08321
$16$	0	0
$17$	−0.151828	−0.0368237	−0.0184119	−	0.999830i	$-0.505861\pi$
$17$	−0.151828	−0.0368237	−0.0184119	+	0.999830i	$0.505861\pi$
$18$	0	0
$19$	0.370208	0.0849316	0.0424658	−	0.999098i	$-0.486479\pi$
$19$	0.370208	0.0849316	0.0424658	+	0.999098i	$0.486479\pi$
$20$	0	0
$21$	1.43344	0.312802
$22$	0	0
$23$	−5.22489	−1.08946	−0.544732	−	0.838610i	$-0.683369\pi$
$23$	−5.22489	−1.08946	−0.544732	+	0.838610i	$0.683369\pi$
$24$	0	0
$25$	12.6001	2.52003
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.43548	−1.75212	−0.876062	−	0.482198i	$-0.839839\pi$
$29$	−9.43548	−1.75212	−0.876062	+	0.482198i	$0.839839\pi$
$30$	0	0
$31$	6.03892	1.08462	0.542311	−	0.840178i	$-0.317549\pi$
$31$	6.03892	1.08462	0.542311	+	0.840178i	$0.317549\pi$
$32$	0	0
$33$	5.55413	0.966850
$34$	0	0
$35$	6.01363	1.01649
$36$	0	0
$37$	9.91270	1.62964	0.814819	−	0.579715i	$-0.196836\pi$
$37$	9.91270	1.62964	0.814819	+	0.579715i	$0.196836\pi$
$38$	0	0
$39$	−2.46575	−0.394837
$40$	0	0
$41$	−8.03882	−1.25545	−0.627727	−	0.778434i	$-0.716014\pi$
$41$	−8.03882	−1.25545	−0.627727	+	0.778434i	$0.716014\pi$
$42$	0	0
$43$	−12.3613	−1.88508	−0.942539	−	0.334097i	$-0.891569\pi$
$43$	−12.3613	−1.88508	−0.942539	+	0.334097i	$0.891569\pi$
$44$	0	0
$45$	4.19525	0.625391
$46$	0	0
$47$	−5.19319	−0.757505	−0.378752	−	0.925498i	$-0.623647\pi$
$47$	−5.19319	−0.757505	−0.378752	+	0.925498i	$0.623647\pi$
$48$	0	0
$49$	−4.94525	−0.706465
$50$	0	0
$51$	−0.151828	−0.0212602
$52$	0	0
$53$	6.02724	0.827906	0.413953	−	0.910298i	$-0.364148\pi$
$53$	6.02724	0.827906	0.413953	+	0.910298i	$0.364148\pi$
$54$	0	0
$55$	23.3010	3.14190
$56$	0	0
$57$	0.370208	0.0490353
$58$	0	0
$59$	−6.13698	−0.798966	−0.399483	−	0.916741i	$-0.630810\pi$
$59$	−6.13698	−0.798966	−0.399483	+	0.916741i	$0.630810\pi$
$60$	0	0
$61$	−4.70024	−0.601805	−0.300902	−	0.953655i	$-0.597288\pi$
$61$	−4.70024	−0.601805	−0.300902	+	0.953655i	$0.597288\pi$
$62$	0	0
$63$	1.43344	0.180596
$64$	0	0
$65$	−10.3445	−1.28307
$66$	0	0
$67$	−6.05931	−0.740263	−0.370132	−	0.928979i	$-0.620687\pi$
$67$	−6.05931	−0.740263	−0.370132	+	0.928979i	$0.620687\pi$
$68$	0	0
$69$	−5.22489	−0.629003
$70$	0	0
$71$	3.38205	0.401376	0.200688	−	0.979655i	$-0.435682\pi$
$71$	3.38205	0.401376	0.200688	+	0.979655i	$0.435682\pi$
$72$	0	0
$73$	−13.2561	−1.55151	−0.775756	−	0.631033i	$-0.782631\pi$
$73$	−13.2561	−1.55151	−0.775756	+	0.631033i	$0.782631\pi$
$74$	0	0
$75$	12.6001	1.45494
$76$	0	0
$77$	7.96151	0.907298
$78$	0	0
$79$	−5.34461	−0.601315	−0.300658	−	0.953732i	$-0.597206\pi$
$79$	−5.34461	−0.601315	−0.300658	+	0.953732i	$0.597206\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.83389	0.201296	0.100648	−	0.994922i	$-0.467908\pi$
$83$	1.83389	0.201296	0.100648	+	0.994922i	$0.467908\pi$
$84$	0	0
$85$	−0.636958	−0.0690877
$86$	0	0
$87$	−9.43548	−1.01159
$88$	0	0
$89$	7.69442	0.815607	0.407804	−	0.913070i	$-0.366295\pi$
$89$	7.69442	0.815607	0.407804	+	0.913070i	$0.366295\pi$
$90$	0	0
$91$	−3.53451	−0.370517
$92$	0	0
$93$	6.03892	0.626207
$94$	0	0
$95$	1.55312	0.159346
$96$	0	0
$97$	18.7449	1.90325	0.951626	−	0.307260i	$-0.0994121\pi$
$97$	18.7449	1.90325	0.951626	+	0.307260i	$0.0994121\pi$
$98$	0	0
$99$	5.55413	0.558211
$100$	0	0
$101$	8.67739	0.863433	0.431717	−	0.902009i	$-0.357908\pi$
$101$	8.67739	0.863433	0.431717	+	0.902009i	$0.357908\pi$
$102$	0	0
$103$	−17.6281	−1.73695	−0.868476	−	0.495731i	$-0.834900\pi$
$103$	−17.6281	−1.73695	−0.868476	+	0.495731i	$0.834900\pi$
$104$	0	0
$105$	6.01363	0.586871
$106$	0	0
$107$	−5.19885	−0.502591	−0.251296	−	0.967910i	$-0.580857\pi$
$107$	−5.19885	−0.502591	−0.251296	+	0.967910i	$0.580857\pi$
$108$	0	0
$109$	−6.27648	−0.601178	−0.300589	−	0.953754i	$-0.597183\pi$
$109$	−6.27648	−0.601178	−0.300589	+	0.953754i	$0.597183\pi$
$110$	0	0
$111$	9.91270	0.940872
$112$	0	0
$113$	17.8990	1.68379	0.841897	−	0.539638i	$-0.181439\pi$
$113$	17.8990	1.68379	0.841897	+	0.539638i	$0.181439\pi$
$114$	0	0
$115$	−21.9197	−2.04403
$116$	0	0
$117$	−2.46575	−0.227959
$118$	0	0
$119$	−0.217636	−0.0199507
$120$	0	0
$121$	19.8484	1.80440
$122$	0	0
$123$	−8.03882	−0.724836
$124$	0	0
$125$	31.8845	2.85183
$126$	0	0
$127$	−14.1924	−1.25937	−0.629686	−	0.776850i	$-0.716816\pi$
$127$	−14.1924	−1.25937	−0.629686	+	0.776850i	$0.716816\pi$
$128$	0	0
$129$	−12.3613	−1.08835
$130$	0	0
$131$	−6.35032	−0.554830	−0.277415	−	0.960750i	$-0.589478\pi$
$131$	−6.35032	−0.554830	−0.277415	+	0.960750i	$0.589478\pi$
$132$	0	0
$133$	0.530671	0.0460150
$134$	0	0
$135$	4.19525	0.361070
$136$	0	0
$137$	−5.75224	−0.491447	−0.245724	−	0.969340i	$-0.579026\pi$
$137$	−5.75224	−0.491447	−0.245724	+	0.969340i	$0.579026\pi$
$138$	0	0
$139$	−7.26012	−0.615795	−0.307898	−	0.951420i	$-0.599625\pi$
$139$	−7.26012	−0.615795	−0.307898	+	0.951420i	$0.599625\pi$
$140$	0	0
$141$	−5.19319	−0.437346
$142$	0	0
$143$	−13.6951	−1.14524
$144$	0	0
$145$	−39.5842	−3.28729
$146$	0	0
$147$	−4.94525	−0.407878
$148$	0	0
$149$	9.28898	0.760983	0.380491	−	0.924784i	$-0.375755\pi$
$149$	9.28898	0.760983	0.380491	+	0.924784i	$0.375755\pi$
$150$	0	0
$151$	13.5816	1.10526	0.552629	−	0.833428i	$-0.313625\pi$
$151$	13.5816	1.10526	0.552629	+	0.833428i	$0.313625\pi$
$152$	0	0
$153$	−0.151828	−0.0122746
$154$	0	0
$155$	25.3348	2.03494
$156$	0	0
$157$	6.50310	0.519004	0.259502	−	0.965743i	$-0.416442\pi$
$157$	6.50310	0.519004	0.259502	+	0.965743i	$0.416442\pi$
$158$	0	0
$159$	6.02724	0.477992
$160$	0	0
$161$	−7.48956	−0.590260
$162$	0	0
$163$	2.73367	0.214118	0.107059	−	0.994253i	$-0.465857\pi$
$163$	2.73367	0.214118	0.107059	+	0.994253i	$0.465857\pi$
$164$	0	0
$165$	23.3010	1.81398
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−6.92006	−0.532312
$170$	0	0
$171$	0.370208	0.0283105
$172$	0	0
$173$	−2.63491	−0.200329	−0.100164	−	0.994971i	$-0.531937\pi$
$173$	−2.63491	−0.200329	−0.100164	+	0.994971i	$0.531937\pi$
$174$	0	0
$175$	18.0615	1.36532
$176$	0	0
$177$	−6.13698	−0.461283
$178$	0	0
$179$	4.99385	0.373258	0.186629	−	0.982430i	$-0.440244\pi$
$179$	4.99385	0.373258	0.186629	+	0.982430i	$0.440244\pi$
$180$	0	0
$181$	10.5484	0.784055	0.392027	−	0.919954i	$-0.371774\pi$
$181$	10.5484	0.784055	0.392027	+	0.919954i	$0.371774\pi$
$182$	0	0
$183$	−4.70024	−0.347452
$184$	0	0
$185$	41.5863	3.05748
$186$	0	0
$187$	−0.843274	−0.0616663
$188$	0	0
$189$	1.43344	0.104267
$190$	0	0
$191$	−11.9556	−0.865074	−0.432537	−	0.901616i	$-0.642381\pi$
$191$	−11.9556	−0.865074	−0.432537	+	0.901616i	$0.642381\pi$
$192$	0	0
$193$	−5.12509	−0.368912	−0.184456	−	0.982841i	$-0.559052\pi$
$193$	−5.12509	−0.368912	−0.184456	+	0.982841i	$0.559052\pi$
$194$	0	0
$195$	−10.3445	−0.740782
$196$	0	0
$197$	26.0322	1.85471	0.927357	−	0.374177i	$-0.122075\pi$
$197$	26.0322	1.85471	0.927357	+	0.374177i	$0.122075\pi$
$198$	0	0
$199$	21.5874	1.53029	0.765143	−	0.643860i	$-0.222668\pi$
$199$	21.5874	1.53029	0.765143	+	0.643860i	$0.222668\pi$
$200$	0	0
$201$	−6.05931	−0.427391
$202$	0	0
$203$	−13.5252	−0.949282
$204$	0	0
$205$	−33.7249	−2.35545
$206$	0	0
$207$	−5.22489	−0.363155
$208$	0	0
$209$	2.05619	0.142229
$210$	0	0
$211$	17.3410	1.19381	0.596903	−	0.802313i	$-0.296398\pi$
$211$	17.3410	1.19381	0.596903	+	0.802313i	$0.296398\pi$
$212$	0	0
$213$	3.38205	0.231734
$214$	0	0
$215$	−51.8587	−3.53673
$216$	0	0
$217$	8.65642	0.587636
$218$	0	0
$219$	−13.2561	−0.895766
$220$	0	0
$221$	0.374371	0.0251829
$222$	0	0
$223$	11.0841	0.742247	0.371124	−	0.928584i	$-0.378973\pi$
$223$	11.0841	0.742247	0.371124	+	0.928584i	$0.378973\pi$
$224$	0	0
$225$	12.6001	0.840009
$226$	0	0
$227$	5.62010	0.373019	0.186510	−	0.982453i	$-0.440282\pi$
$227$	5.62010	0.373019	0.186510	+	0.982453i	$0.440282\pi$
$228$	0	0
$229$	6.45735	0.426714	0.213357	−	0.976974i	$-0.431560\pi$
$229$	6.45735	0.426714	0.213357	+	0.976974i	$0.431560\pi$
$230$	0	0
$231$	7.96151	0.523829
$232$	0	0
$233$	−19.4269	−1.27270	−0.636349	−	0.771401i	$-0.719556\pi$
$233$	−19.4269	−1.27270	−0.636349	+	0.771401i	$0.719556\pi$
$234$	0	0
$235$	−21.7867	−1.42121
$236$	0	0
$237$	−5.34461	−0.347169
$238$	0	0
$239$	16.2016	1.04799	0.523997	−	0.851720i	$-0.324440\pi$
$239$	16.2016	1.04799	0.523997	+	0.851720i	$0.324440\pi$
$240$	0	0
$241$	22.1313	1.42560	0.712800	−	0.701367i	$-0.247427\pi$
$241$	22.1313	1.42560	0.712800	+	0.701367i	$0.247427\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−20.7466	−1.32545
$246$	0	0
$247$	−0.912842	−0.0580828
$248$	0	0
$249$	1.83389	0.116218
$250$	0	0
$251$	8.70622	0.549532	0.274766	−	0.961511i	$-0.411400\pi$
$251$	8.70622	0.549532	0.274766	+	0.961511i	$0.411400\pi$
$252$	0	0
$253$	−29.0197	−1.82446
$254$	0	0
$255$	−0.636958	−0.0398878
$256$	0	0
$257$	−8.73325	−0.544765	−0.272383	−	0.962189i	$-0.587812\pi$
$257$	−8.73325	−0.544765	−0.272383	+	0.962189i	$0.587812\pi$
$258$	0	0
$259$	14.2093	0.882920
$260$	0	0
$261$	−9.43548	−0.584042
$262$	0	0
$263$	−17.1621	−1.05826	−0.529131	−	0.848540i	$-0.677482\pi$
$263$	−17.1621	−1.05826	−0.529131	+	0.848540i	$0.677482\pi$
$264$	0	0
$265$	25.2858	1.55329
$266$	0	0
$267$	7.69442	0.470891
$268$	0	0
$269$	23.1618	1.41220	0.706101	−	0.708112i	$-0.250453\pi$
$269$	23.1618	1.41220	0.706101	+	0.708112i	$0.250453\pi$
$270$	0	0
$271$	−27.3302	−1.66019	−0.830097	−	0.557619i	$-0.811715\pi$
$271$	−27.3302	−1.66019	−0.830097	+	0.557619i	$0.811715\pi$
$272$	0	0
$273$	−3.53451	−0.213918
$274$	0	0
$275$	69.9828	4.22012
$276$	0	0
$277$	26.8048	1.61055	0.805274	−	0.592904i	$-0.202018\pi$
$277$	26.8048	1.61055	0.805274	+	0.592904i	$0.202018\pi$
$278$	0	0
$279$	6.03892	0.361541
$280$	0	0
$281$	19.5511	1.16632	0.583161	−	0.812357i	$-0.301816\pi$
$281$	19.5511	1.16632	0.583161	+	0.812357i	$0.301816\pi$
$282$	0	0
$283$	9.70821	0.577093	0.288546	−	0.957466i	$-0.406828\pi$
$283$	9.70821	0.577093	0.288546	+	0.957466i	$0.406828\pi$
$284$	0	0
$285$	1.55312	0.0919987
$286$	0	0
$287$	−11.5232	−0.680191
$288$	0	0
$289$	−16.9769	−0.998644
$290$	0	0
$291$	18.7449	1.09884
$292$	0	0
$293$	19.0093	1.11053	0.555267	−	0.831672i	$-0.312616\pi$
$293$	19.0093	1.11053	0.555267	+	0.831672i	$0.312616\pi$
$294$	0	0
$295$	−25.7462	−1.49900
$296$	0	0
$297$	5.55413	0.322283
$298$	0	0
$299$	12.8833	0.745060
$300$	0	0
$301$	−17.7191	−1.02131
$302$	0	0
$303$	8.67739	0.498503
$304$	0	0
$305$	−19.7187	−1.12909
$306$	0	0
$307$	−15.8337	−0.903677	−0.451838	−	0.892100i	$-0.649232\pi$
$307$	−15.8337	−0.903677	−0.451838	+	0.892100i	$0.649232\pi$
$308$	0	0
$309$	−17.6281	−1.00283
$310$	0	0
$311$	−7.16509	−0.406295	−0.203148	−	0.979148i	$-0.565117\pi$
$311$	−7.16509	−0.406295	−0.203148	+	0.979148i	$0.565117\pi$
$312$	0	0
$313$	−11.2093	−0.633586	−0.316793	−	0.948495i	$-0.602606\pi$
$313$	−11.2093	−0.633586	−0.316793	+	0.948495i	$0.602606\pi$
$314$	0	0
$315$	6.01363	0.338830
$316$	0	0
$317$	−33.7170	−1.89374	−0.946868	−	0.321621i	$-0.895772\pi$
$317$	−33.7170	−1.89374	−0.946868	+	0.321621i	$0.895772\pi$
$318$	0	0
$319$	−52.4059	−2.93417
$320$	0	0
$321$	−5.19885	−0.290171
$322$	0	0
$323$	−0.0562080	−0.00312750
$324$	0	0
$325$	−31.0688	−1.72339
$326$	0	0
$327$	−6.27648	−0.347090
$328$	0	0
$329$	−7.44412	−0.410408
$330$	0	0
$331$	0.859249	0.0472286	0.0236143	−	0.999721i	$-0.492483\pi$
$331$	0.859249	0.0472286	0.0236143	+	0.999721i	$0.492483\pi$
$332$	0	0
$333$	9.91270	0.543213
$334$	0	0
$335$	−25.4203	−1.38886
$336$	0	0
$337$	−23.0099	−1.25343	−0.626714	−	0.779250i	$-0.715600\pi$
$337$	−23.0099	−1.25343	−0.626714	+	0.779250i	$0.715600\pi$
$338$	0	0
$339$	17.8990	0.972139
$340$	0	0
$341$	33.5410	1.81635
$342$	0	0
$343$	−17.1228	−0.924544
$344$	0	0
$345$	−21.9197	−1.18012
$346$	0	0
$347$	−1.36127	−0.0730768	−0.0365384	−	0.999332i	$-0.511633\pi$
$347$	−1.36127	−0.0730768	−0.0365384	+	0.999332i	$0.511633\pi$
$348$	0	0
$349$	−12.4163	−0.664629	−0.332315	−	0.943169i	$-0.607830\pi$
$349$	−12.4163	−0.664629	−0.332315	+	0.943169i	$0.607830\pi$
$350$	0	0
$351$	−2.46575	−0.131612
$352$	0	0
$353$	9.14770	0.486883	0.243442	−	0.969916i	$-0.421724\pi$
$353$	9.14770	0.486883	0.243442	+	0.969916i	$0.421724\pi$
$354$	0	0
$355$	14.1886	0.753050
$356$	0	0
$357$	−0.217636	−0.0115185
$358$	0	0
$359$	35.7239	1.88544	0.942718	−	0.333592i	$-0.108261\pi$
$359$	35.7239	1.88544	0.942718	+	0.333592i	$0.108261\pi$
$360$	0	0
$361$	−18.8629	−0.992787
$362$	0	0
$363$	19.8484	1.04177
$364$	0	0
$365$	−55.6128	−2.91091
$366$	0	0
$367$	−15.4781	−0.807949	−0.403975	−	0.914770i	$-0.632372\pi$
$367$	−15.4781	−0.807949	−0.403975	+	0.914770i	$0.632372\pi$
$368$	0	0
$369$	−8.03882	−0.418485
$370$	0	0
$371$	8.63968	0.448550
$372$	0	0
$373$	5.05019	0.261489	0.130744	−	0.991416i	$-0.458263\pi$
$373$	5.05019	0.261489	0.130744	+	0.991416i	$0.458263\pi$
$374$	0	0
$375$	31.8845	1.64651
$376$	0	0
$377$	23.2656	1.19824
$378$	0	0
$379$	4.06656	0.208885	0.104443	−	0.994531i	$-0.466694\pi$
$379$	4.06656	0.208885	0.104443	+	0.994531i	$0.466694\pi$
$380$	0	0
$381$	−14.1924	−0.727099
$382$	0	0
$383$	−3.04908	−0.155801	−0.0779005	−	0.996961i	$-0.524822\pi$
$383$	−3.04908	−0.155801	−0.0779005	+	0.996961i	$0.524822\pi$
$384$	0	0
$385$	33.4005	1.70225
$386$	0	0
$387$	−12.3613	−0.628359
$388$	0	0
$389$	−15.1210	−0.766665	−0.383333	−	0.923610i	$-0.625224\pi$
$389$	−15.1210	−0.766665	−0.383333	+	0.923610i	$0.625224\pi$
$390$	0	0
$391$	0.793286	0.0401182
$392$	0	0
$393$	−6.35032	−0.320331
$394$	0	0
$395$	−22.4220	−1.12817
$396$	0	0
$397$	−28.9642	−1.45367	−0.726836	−	0.686811i	$-0.759010\pi$
$397$	−28.9642	−1.45367	−0.726836	+	0.686811i	$0.759010\pi$
$398$	0	0
$399$	0.530671	0.0265668
$400$	0	0
$401$	−22.2008	−1.10865	−0.554327	−	0.832299i	$-0.687024\pi$
$401$	−22.2008	−1.10865	−0.554327	+	0.832299i	$0.687024\pi$
$402$	0	0
$403$	−14.8905	−0.741748
$404$	0	0
$405$	4.19525	0.208464
$406$	0	0
$407$	55.0565	2.72905
$408$	0	0
$409$	35.7127	1.76588	0.882939	−	0.469489i	$-0.155562\pi$
$409$	35.7127	1.76588	0.882939	+	0.469489i	$0.155562\pi$
$410$	0	0
$411$	−5.75224	−0.283737
$412$	0	0
$413$	−8.79698	−0.432871
$414$	0	0
$415$	7.69364	0.377666
$416$	0	0
$417$	−7.26012	−0.355529
$418$	0	0
$419$	−9.16518	−0.447748	−0.223874	−	0.974618i	$-0.571870\pi$
$419$	−9.16518	−0.447748	−0.223874	+	0.974618i	$0.571870\pi$
$420$	0	0
$421$	−10.2729	−0.500671	−0.250336	−	0.968159i	$-0.580541\pi$
$421$	−10.2729	−0.500671	−0.250336	+	0.968159i	$0.580541\pi$
$422$	0	0
$423$	−5.19319	−0.252502
$424$	0	0
$425$	−1.91306	−0.0927968
$426$	0	0
$427$	−6.73751	−0.326051
$428$	0	0
$429$	−13.6951	−0.661207
$430$	0	0
$431$	−3.83372	−0.184664	−0.0923319	−	0.995728i	$-0.529432\pi$
$431$	−3.83372	−0.184664	−0.0923319	+	0.995728i	$0.529432\pi$
$432$	0	0
$433$	5.96106	0.286470	0.143235	−	0.989689i	$-0.454249\pi$
$433$	5.96106	0.286470	0.143235	+	0.989689i	$0.454249\pi$
$434$	0	0
$435$	−39.5842	−1.89792
$436$	0	0
$437$	−1.93430	−0.0925300
$438$	0	0
$439$	2.00545	0.0957149	0.0478574	−	0.998854i	$-0.484761\pi$
$439$	2.00545	0.0957149	0.0478574	+	0.998854i	$0.484761\pi$
$440$	0	0
$441$	−4.94525	−0.235488
$442$	0	0
$443$	4.02748	0.191351	0.0956757	−	0.995413i	$-0.469499\pi$
$443$	4.02748	0.191351	0.0956757	+	0.995413i	$0.469499\pi$
$444$	0	0
$445$	32.2800	1.53022
$446$	0	0
$447$	9.28898	0.439354
$448$	0	0
$449$	−7.75720	−0.366085	−0.183042	−	0.983105i	$-0.558595\pi$
$449$	−7.75720	−0.366085	−0.183042	+	0.983105i	$0.558595\pi$
$450$	0	0
$451$	−44.6487	−2.10243
$452$	0	0
$453$	13.5816	0.638121
$454$	0	0
$455$	−14.8281	−0.695154
$456$	0	0
$457$	−0.952815	−0.0445708	−0.0222854	−	0.999752i	$-0.507094\pi$
$457$	−0.952815	−0.0445708	−0.0222854	+	0.999752i	$0.507094\pi$
$458$	0	0
$459$	−0.151828	−0.00708673
$460$	0	0
$461$	0.288626	0.0134427	0.00672133	−	0.999977i	$-0.497861\pi$
$461$	0.288626	0.0134427	0.00672133	+	0.999977i	$0.497861\pi$
$462$	0	0
$463$	−37.0632	−1.72248	−0.861238	−	0.508202i	$-0.830310\pi$
$463$	−37.0632	−1.72248	−0.861238	+	0.508202i	$0.830310\pi$
$464$	0	0
$465$	25.3348	1.17487
$466$	0	0
$467$	18.5601	0.858860	0.429430	−	0.903100i	$-0.358714\pi$
$467$	18.5601	0.858860	0.429430	+	0.903100i	$0.358714\pi$
$468$	0	0
$469$	−8.68565	−0.401066
$470$	0	0
$471$	6.50310	0.299647
$472$	0	0
$473$	−68.6562	−3.15682
$474$	0	0
$475$	4.66467	0.214030
$476$	0	0
$477$	6.02724	0.275969
$478$	0	0
$479$	−20.1854	−0.922295	−0.461147	−	0.887324i	$-0.652562\pi$
$479$	−20.1854	−0.922295	−0.461147	+	0.887324i	$0.652562\pi$
$480$	0	0
$481$	−24.4423	−1.11447
$482$	0	0
$483$	−7.48956	−0.340787
$484$	0	0
$485$	78.6394	3.57083
$486$	0	0
$487$	−35.2838	−1.59886	−0.799430	−	0.600760i	$-0.794865\pi$
$487$	−35.2838	−1.59886	−0.799430	+	0.600760i	$0.794865\pi$
$488$	0	0
$489$	2.73367	0.123621
$490$	0	0
$491$	−25.3811	−1.14543	−0.572715	−	0.819754i	$-0.694110\pi$
$491$	−25.3811	−1.14543	−0.572715	+	0.819754i	$0.694110\pi$
$492$	0	0
$493$	1.43257	0.0645198
$494$	0	0
$495$	23.3010	1.04730
$496$	0	0
$497$	4.84796	0.217461
$498$	0	0
$499$	−26.0391	−1.16567	−0.582836	−	0.812590i	$-0.698057\pi$
$499$	−26.0391	−1.16567	−0.582836	+	0.812590i	$0.698057\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	36.5383	1.62916	0.814580	−	0.580051i	$-0.196967\pi$
$503$	36.5383	1.62916	0.814580	+	0.580051i	$0.196967\pi$
$504$	0	0
$505$	36.4039	1.61995
$506$	0	0
$507$	−6.92006	−0.307331
$508$	0	0
$509$	13.7212	0.608181	0.304091	−	0.952643i	$-0.401647\pi$
$509$	13.7212	0.608181	0.304091	+	0.952643i	$0.401647\pi$
$510$	0	0
$511$	−19.0018	−0.840592
$512$	0	0
$513$	0.370208	0.0163451
$514$	0	0
$515$	−73.9545	−3.25882
$516$	0	0
$517$	−28.8437	−1.26854
$518$	0	0
$519$	−2.63491	−0.115660
$520$	0	0
$521$	−4.28133	−0.187568	−0.0937842	−	0.995593i	$-0.529896\pi$
$521$	−4.28133	−0.187568	−0.0937842	+	0.995593i	$0.529896\pi$
$522$	0	0
$523$	13.7039	0.599228	0.299614	−	0.954061i	$-0.403142\pi$
$523$	13.7039	0.599228	0.299614	+	0.954061i	$0.403142\pi$
$524$	0	0
$525$	18.0615	0.788269
$526$	0	0
$527$	−0.916879	−0.0399399
$528$	0	0
$529$	4.29948	0.186934
$530$	0	0
$531$	−6.13698	−0.266322
$532$	0	0
$533$	19.8218	0.858576
$534$	0	0
$535$	−21.8105	−0.942949
$536$	0	0
$537$	4.99385	0.215501
$538$	0	0
$539$	−27.4666	−1.18307
$540$	0	0
$541$	−10.9801	−0.472073	−0.236037	−	0.971744i	$-0.575849\pi$
$541$	−10.9801	−0.472073	−0.236037	+	0.971744i	$0.575849\pi$
$542$	0	0
$543$	10.5484	0.452674
$544$	0	0
$545$	−26.3314	−1.12791
$546$	0	0
$547$	16.5207	0.706375	0.353188	−	0.935553i	$-0.385098\pi$
$547$	16.5207	0.706375	0.353188	+	0.935553i	$0.385098\pi$
$548$	0	0
$549$	−4.70024	−0.200602
$550$	0	0
$551$	−3.49309	−0.148811
$552$	0	0
$553$	−7.66116	−0.325786
$554$	0	0
$555$	41.5863	1.76524
$556$	0	0
$557$	42.4409	1.79828	0.899140	−	0.437661i	$-0.144193\pi$
$557$	42.4409	1.79828	0.899140	+	0.437661i	$0.144193\pi$
$558$	0	0
$559$	30.4799	1.28916
$560$	0	0
$561$	−0.843274	−0.0356031
$562$	0	0
$563$	−9.07135	−0.382312	−0.191156	−	0.981560i	$-0.561224\pi$
$563$	−9.07135	−0.382312	−0.191156	+	0.981560i	$0.561224\pi$
$564$	0	0
$565$	75.0907	3.15909
$566$	0	0
$567$	1.43344	0.0601988
$568$	0	0
$569$	22.1508	0.928611	0.464305	−	0.885675i	$-0.346304\pi$
$569$	22.1508	0.928611	0.464305	+	0.885675i	$0.346304\pi$
$570$	0	0
$571$	20.7569	0.868649	0.434325	−	0.900756i	$-0.356987\pi$
$571$	20.7569	0.868649	0.434325	+	0.900756i	$0.356987\pi$
$572$	0	0
$573$	−11.9556	−0.499451
$574$	0	0
$575$	−65.8343	−2.74548
$576$	0	0
$577$	19.8491	0.826328	0.413164	−	0.910657i	$-0.364424\pi$
$577$	19.8491	0.826328	0.413164	+	0.910657i	$0.364424\pi$
$578$	0	0
$579$	−5.12509	−0.212991
$580$	0	0
$581$	2.62877	0.109060
$582$	0	0
$583$	33.4761	1.38644
$584$	0	0
$585$	−10.3445	−0.427691
$586$	0	0
$587$	−15.1902	−0.626968	−0.313484	−	0.949593i	$-0.601496\pi$
$587$	−15.1902	−0.626968	−0.313484	+	0.949593i	$0.601496\pi$
$588$	0	0
$589$	2.23566	0.0921187
$590$	0	0
$591$	26.0322	1.07082
$592$	0	0
$593$	−29.5658	−1.21412	−0.607061	−	0.794655i	$-0.707652\pi$
$593$	−29.5658	−1.21412	−0.607061	+	0.794655i	$0.707652\pi$
$594$	0	0
$595$	−0.913039	−0.0374310
$596$	0	0
$597$	21.5874	0.883512
$598$	0	0
$599$	46.2469	1.88960	0.944798	−	0.327653i	$-0.106258\pi$
$599$	46.2469	1.88960	0.944798	+	0.327653i	$0.106258\pi$
$600$	0	0
$601$	8.66483	0.353446	0.176723	−	0.984261i	$-0.443450\pi$
$601$	8.66483	0.353446	0.176723	+	0.984261i	$0.443450\pi$
$602$	0	0
$603$	−6.05931	−0.246754
$604$	0	0
$605$	83.2690	3.38537
$606$	0	0
$607$	4.70252	0.190869	0.0954347	−	0.995436i	$-0.469576\pi$
$607$	4.70252	0.190869	0.0954347	+	0.995436i	$0.469576\pi$
$608$	0	0
$609$	−13.5252	−0.548068
$610$	0	0
$611$	12.8051	0.518040
$612$	0	0
$613$	−25.5121	−1.03042	−0.515212	−	0.857063i	$-0.672287\pi$
$613$	−25.5121	−1.03042	−0.515212	+	0.857063i	$0.672287\pi$
$614$	0	0
$615$	−33.7249	−1.35992
$616$	0	0
$617$	−10.1033	−0.406745	−0.203372	−	0.979101i	$-0.565190\pi$
$617$	−10.1033	−0.406745	−0.203372	+	0.979101i	$0.565190\pi$
$618$	0	0
$619$	−32.4406	−1.30390	−0.651948	−	0.758264i	$-0.726048\pi$
$619$	−32.4406	−1.30390	−0.651948	+	0.758264i	$0.726048\pi$
$620$	0	0
$621$	−5.22489	−0.209668
$622$	0	0
$623$	11.0295	0.441887
$624$	0	0
$625$	70.7627	2.83051
$626$	0	0
$627$	2.05619	0.0821161
$628$	0	0
$629$	−1.50503	−0.0600094
$630$	0	0
$631$	8.97751	0.357389	0.178695	−	0.983905i	$-0.442813\pi$
$631$	8.97751	0.357389	0.178695	+	0.983905i	$0.442813\pi$
$632$	0	0
$633$	17.3410	0.689245
$634$	0	0
$635$	−59.5407	−2.36280
$636$	0	0
$637$	12.1938	0.483135
$638$	0	0
$639$	3.38205	0.133792
$640$	0	0
$641$	34.1975	1.35072	0.675361	−	0.737488i	$-0.263988\pi$
$641$	34.1975	1.35072	0.675361	+	0.737488i	$0.263988\pi$
$642$	0	0
$643$	−29.3516	−1.15751	−0.578757	−	0.815500i	$-0.696462\pi$
$643$	−29.3516	−1.15751	−0.578757	+	0.815500i	$0.696462\pi$
$644$	0	0
$645$	−51.8587	−2.04193
$646$	0	0
$647$	−19.4963	−0.766480	−0.383240	−	0.923649i	$-0.625192\pi$
$647$	−19.4963	−0.766480	−0.383240	+	0.923649i	$0.625192\pi$
$648$	0	0
$649$	−34.0856	−1.33798
$650$	0	0
$651$	8.65642	0.339272
$652$	0	0
$653$	23.8805	0.934516	0.467258	−	0.884121i	$-0.345242\pi$
$653$	23.8805	0.934516	0.467258	+	0.884121i	$0.345242\pi$
$654$	0	0
$655$	−26.6412	−1.04096
$656$	0	0
$657$	−13.2561	−0.517171
$658$	0	0
$659$	4.22675	0.164651	0.0823255	−	0.996605i	$-0.473765\pi$
$659$	4.22675	0.164651	0.0823255	+	0.996605i	$0.473765\pi$
$660$	0	0
$661$	28.7013	1.11635	0.558176	−	0.829723i	$-0.311501\pi$
$661$	28.7013	1.11635	0.558176	+	0.829723i	$0.311501\pi$
$662$	0	0
$663$	0.374371	0.0145394
$664$	0	0
$665$	2.22630	0.0863321
$666$	0	0
$667$	49.2994	1.90888
$668$	0	0
$669$	11.0841	0.428537
$670$	0	0
$671$	−26.1058	−1.00780
$672$	0	0
$673$	38.2119	1.47296	0.736479	−	0.676460i	$-0.236487\pi$
$673$	38.2119	1.47296	0.736479	+	0.676460i	$0.236487\pi$
$674$	0	0
$675$	12.6001	0.484979
$676$	0	0
$677$	−27.3849	−1.05249	−0.526243	−	0.850334i	$-0.676400\pi$
$677$	−27.3849	−1.05249	−0.526243	+	0.850334i	$0.676400\pi$
$678$	0	0
$679$	26.8696	1.03116
$680$	0	0
$681$	5.62010	0.215363
$682$	0	0
$683$	−13.5496	−0.518459	−0.259230	−	0.965816i	$-0.583469\pi$
$683$	−13.5496	−0.518459	−0.259230	+	0.965816i	$0.583469\pi$
$684$	0	0
$685$	−24.1321	−0.922040
$686$	0	0
$687$	6.45735	0.246363
$688$	0	0
$689$	−14.8617	−0.566186
$690$	0	0
$691$	−45.6661	−1.73722	−0.868610	−	0.495496i	$-0.834986\pi$
$691$	−45.6661	−1.73722	−0.868610	+	0.495496i	$0.834986\pi$
$692$	0	0
$693$	7.96151	0.302433
$694$	0	0
$695$	−30.4580	−1.15534
$696$	0	0
$697$	1.22052	0.0462305
$698$	0	0
$699$	−19.4269	−0.734793
$700$	0	0
$701$	−2.52975	−0.0955472	−0.0477736	−	0.998858i	$-0.515213\pi$
$701$	−2.52975	−0.0955472	−0.0477736	+	0.998858i	$0.515213\pi$
$702$	0	0
$703$	3.66976	0.138408
$704$	0	0
$705$	−21.7867	−0.820536
$706$	0	0
$707$	12.4385	0.467798
$708$	0	0
$709$	3.67220	0.137912	0.0689562	−	0.997620i	$-0.478033\pi$
$709$	3.67220	0.137912	0.0689562	+	0.997620i	$0.478033\pi$
$710$	0	0
$711$	−5.34461	−0.200438
$712$	0	0
$713$	−31.5527	−1.18166
$714$	0	0
$715$	−57.4545	−2.14868
$716$	0	0
$717$	16.2016	0.605060
$718$	0	0
$719$	39.6398	1.47831	0.739157	−	0.673533i	$-0.235224\pi$
$719$	39.6398	1.47831	0.739157	+	0.673533i	$0.235224\pi$
$720$	0	0
$721$	−25.2688	−0.941061
$722$	0	0
$723$	22.1313	0.823071
$724$	0	0
$725$	−118.888	−4.41540
$726$	0	0
$727$	40.4405	1.49986	0.749928	−	0.661520i	$-0.230088\pi$
$727$	40.4405	1.49986	0.749928	+	0.661520i	$0.230088\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.87679	0.0694156
$732$	0	0
$733$	−22.7634	−0.840784	−0.420392	−	0.907343i	$-0.638107\pi$
$733$	−22.7634	−0.840784	−0.420392	+	0.907343i	$0.638107\pi$
$734$	0	0
$735$	−20.7466	−0.765249
$736$	0	0
$737$	−33.6542	−1.23967
$738$	0	0
$739$	−2.74636	−0.101026	−0.0505132	−	0.998723i	$-0.516086\pi$
$739$	−2.74636	−0.101026	−0.0505132	+	0.998723i	$0.516086\pi$
$740$	0	0
$741$	−0.912842	−0.0335341
$742$	0	0
$743$	13.5644	0.497631	0.248815	−	0.968551i	$-0.419959\pi$
$743$	13.5644	0.497631	0.248815	+	0.968551i	$0.419959\pi$
$744$	0	0
$745$	38.9696	1.42774
$746$	0	0
$747$	1.83389	0.0670986
$748$	0	0
$749$	−7.45223	−0.272298
$750$	0	0
$751$	−48.4821	−1.76914	−0.884568	−	0.466411i	$-0.845547\pi$
$751$	−48.4821	−1.76914	−0.884568	+	0.466411i	$0.845547\pi$
$752$	0	0
$753$	8.70622	0.317272
$754$	0	0
$755$	56.9784	2.07365
$756$	0	0
$757$	25.7718	0.936692	0.468346	−	0.883545i	$-0.344850\pi$
$757$	25.7718	0.936692	0.468346	+	0.883545i	$0.344850\pi$
$758$	0	0
$759$	−29.0197	−1.05335
$760$	0	0
$761$	−15.9928	−0.579739	−0.289869	−	0.957066i	$-0.593612\pi$
$761$	−15.9928	−0.579739	−0.289869	+	0.957066i	$0.593612\pi$
$762$	0	0
$763$	−8.99695	−0.325711
$764$	0	0
$765$	−0.636958	−0.0230292
$766$	0	0
$767$	15.1323	0.546395
$768$	0	0
$769$	−35.3541	−1.27490	−0.637451	−	0.770491i	$-0.720011\pi$
$769$	−35.3541	−1.27490	−0.637451	+	0.770491i	$0.720011\pi$
$770$	0	0
$771$	−8.73325	−0.314520
$772$	0	0
$773$	22.7866	0.819578	0.409789	−	0.912180i	$-0.365602\pi$
$773$	22.7866	0.819578	0.409789	+	0.912180i	$0.365602\pi$
$774$	0	0
$775$	76.0912	2.73328
$776$	0	0
$777$	14.2093	0.509754
$778$	0	0
$779$	−2.97604	−0.106628
$780$	0	0
$781$	18.7844	0.672157
$782$	0	0
$783$	−9.43548	−0.337197
$784$	0	0
$785$	27.2821	0.973741
$786$	0	0
$787$	12.5300	0.446645	0.223322	−	0.974745i	$-0.428310\pi$
$787$	12.5300	0.446645	0.223322	+	0.974745i	$0.428310\pi$
$788$	0	0
$789$	−17.1621	−0.610988
$790$	0	0
$791$	25.6571	0.912261
$792$	0	0
$793$	11.5896	0.411560
$794$	0	0
$795$	25.2858	0.896795
$796$	0	0
$797$	−2.33137	−0.0825815	−0.0412907	−	0.999147i	$-0.513147\pi$
$797$	−2.33137	−0.0825815	−0.0412907	+	0.999147i	$0.513147\pi$
$798$	0	0
$799$	0.788473	0.0278942
$800$	0	0
$801$	7.69442	0.271869
$802$	0	0
$803$	−73.6263	−2.59821
$804$	0	0
$805$	−31.4206	−1.10743
$806$	0	0
$807$	23.1618	0.815335
$808$	0	0
$809$	19.6327	0.690250	0.345125	−	0.938557i	$-0.387836\pi$
$809$	19.6327	0.690250	0.345125	+	0.938557i	$0.387836\pi$
$810$	0	0
$811$	1.25282	0.0439926	0.0219963	−	0.999758i	$-0.492998\pi$
$811$	1.25282	0.0439926	0.0219963	+	0.999758i	$0.492998\pi$
$812$	0	0
$813$	−27.3302	−0.958514
$814$	0	0
$815$	11.4684	0.401722
$816$	0	0
$817$	−4.57625	−0.160103
$818$	0	0
$819$	−3.53451	−0.123506
$820$	0	0
$821$	−4.53393	−0.158235	−0.0791176	−	0.996865i	$-0.525210\pi$
$821$	−4.53393	−0.158235	−0.0791176	+	0.996865i	$0.525210\pi$
$822$	0	0
$823$	2.81623	0.0981677	0.0490838	−	0.998795i	$-0.484370\pi$
$823$	2.81623	0.0981677	0.0490838	+	0.998795i	$0.484370\pi$
$824$	0	0
$825$	69.9828	2.43649
$826$	0	0
$827$	−10.2247	−0.355546	−0.177773	−	0.984071i	$-0.556889\pi$
$827$	−10.2247	−0.355546	−0.177773	+	0.984071i	$0.556889\pi$
$828$	0	0
$829$	3.66108	0.127155	0.0635773	−	0.997977i	$-0.479749\pi$
$829$	3.66108	0.127155	0.0635773	+	0.997977i	$0.479749\pi$
$830$	0	0
$831$	26.8048	0.929850
$832$	0	0
$833$	0.750829	0.0260147
$834$	0	0
$835$	4.19525	0.145183
$836$	0	0
$837$	6.03892	0.208736
$838$	0	0
$839$	19.7190	0.680774	0.340387	−	0.940285i	$-0.389442\pi$
$839$	19.7190	0.680774	0.340387	+	0.940285i	$0.389442\pi$
$840$	0	0
$841$	60.0283	2.06994
$842$	0	0
$843$	19.5511	0.673376
$844$	0	0
$845$	−29.0314	−0.998710
$846$	0	0
$847$	28.4514	0.977603
$848$	0	0
$849$	9.70821	0.333185
$850$	0	0
$851$	−51.7928	−1.77543
$852$	0	0
$853$	57.5393	1.97011	0.985055	−	0.172241i	$-0.0551008\pi$
$853$	57.5393	1.97011	0.985055	+	0.172241i	$0.0551008\pi$
$854$	0	0
$855$	1.55312	0.0531155
$856$	0	0
$857$	−43.1197	−1.47294	−0.736471	−	0.676469i	$-0.763509\pi$
$857$	−43.1197	−1.47294	−0.736471	+	0.676469i	$0.763509\pi$
$858$	0	0
$859$	−27.5816	−0.941071	−0.470535	−	0.882381i	$-0.655939\pi$
$859$	−27.5816	−0.941071	−0.470535	+	0.882381i	$0.655939\pi$
$860$	0	0
$861$	−11.5232	−0.392708
$862$	0	0
$863$	46.9199	1.59717	0.798586	−	0.601881i	$-0.205582\pi$
$863$	46.9199	1.59717	0.798586	+	0.601881i	$0.205582\pi$
$864$	0	0
$865$	−11.0541	−0.375852
$866$	0	0
$867$	−16.9769	−0.576567
$868$	0	0
$869$	−29.6846	−1.00698
$870$	0	0
$871$	14.9408	0.506249
$872$	0	0
$873$	18.7449	0.634417
$874$	0	0
$875$	45.7044	1.54509
$876$	0	0
$877$	−16.2003	−0.547046	−0.273523	−	0.961866i	$-0.588189\pi$
$877$	−16.2003	−0.547046	−0.273523	+	0.961866i	$0.588189\pi$
$878$	0	0
$879$	19.0093	0.641167
$880$	0	0
$881$	−30.8652	−1.03988	−0.519938	−	0.854204i	$-0.674045\pi$
$881$	−30.8652	−1.03988	−0.519938	+	0.854204i	$0.674045\pi$
$882$	0	0
$883$	−22.8934	−0.770423	−0.385211	−	0.922828i	$-0.625871\pi$
$883$	−22.8934	−0.770423	−0.385211	+	0.922828i	$0.625871\pi$
$884$	0	0
$885$	−25.7462	−0.865448
$886$	0	0
$887$	−12.1260	−0.407152	−0.203576	−	0.979059i	$-0.565256\pi$
$887$	−12.1260	−0.407152	−0.203576	+	0.979059i	$0.565256\pi$
$888$	0	0
$889$	−20.3439	−0.682313
$890$	0	0
$891$	5.55413	0.186070
$892$	0	0
$893$	−1.92256	−0.0643361
$894$	0	0
$895$	20.9505	0.700297
$896$	0	0
$897$	12.8833	0.430161
$898$	0	0
$899$	−56.9801	−1.90039
$900$	0	0
$901$	−0.915106	−0.0304866
$902$	0	0
$903$	−17.7191	−0.589656
$904$	0	0
$905$	44.2531	1.47102
$906$	0	0
$907$	−27.7756	−0.922274	−0.461137	−	0.887329i	$-0.652558\pi$
$907$	−27.7756	−0.922274	−0.461137	+	0.887329i	$0.652558\pi$
$908$	0	0
$909$	8.67739	0.287811
$910$	0	0
$911$	−15.6800	−0.519502	−0.259751	−	0.965676i	$-0.583641\pi$
$911$	−15.6800	−0.519502	−0.259751	+	0.965676i	$0.583641\pi$
$912$	0	0
$913$	10.1857	0.337097
$914$	0	0
$915$	−19.7187	−0.651880
$916$	0	0
$917$	−9.10280	−0.300601
$918$	0	0
$919$	−40.1610	−1.32479	−0.662395	−	0.749154i	$-0.730460\pi$
$919$	−40.1610	−1.32479	−0.662395	+	0.749154i	$0.730460\pi$
$920$	0	0
$921$	−15.8337	−0.521738
$922$	0	0
$923$	−8.33930	−0.274491
$924$	0	0
$925$	124.901	4.10673
$926$	0	0
$927$	−17.6281	−0.578984
$928$	0	0
$929$	7.74113	0.253978	0.126989	−	0.991904i	$-0.459469\pi$
$929$	7.74113	0.253978	0.126989	+	0.991904i	$0.459469\pi$
$930$	0	0
$931$	−1.83077	−0.0600012
$932$	0	0
$933$	−7.16509	−0.234575
$934$	0	0
$935$	−3.53775	−0.115697
$936$	0	0
$937$	9.97774	0.325959	0.162979	−	0.986629i	$-0.447890\pi$
$937$	9.97774	0.325959	0.162979	+	0.986629i	$0.447890\pi$
$938$	0	0
$939$	−11.2093	−0.365801
$940$	0	0
$941$	42.7867	1.39481	0.697404	−	0.716679i	$-0.254339\pi$
$941$	42.7867	1.39481	0.697404	+	0.716679i	$0.254339\pi$
$942$	0	0
$943$	42.0020	1.36777
$944$	0	0
$945$	6.01363	0.195624
$946$	0	0
$947$	53.9412	1.75285	0.876427	−	0.481534i	$-0.159920\pi$
$947$	53.9412	1.75285	0.876427	+	0.481534i	$0.159920\pi$
$948$	0	0
$949$	32.6863	1.06104
$950$	0	0
$951$	−33.7170	−1.09335
$952$	0	0
$953$	−28.9693	−0.938408	−0.469204	−	0.883090i	$-0.655459\pi$
$953$	−28.9693	−0.938408	−0.469204	+	0.883090i	$0.655459\pi$
$954$	0	0
$955$	−50.1566	−1.62303
$956$	0	0
$957$	−52.4059	−1.69404
$958$	0	0
$959$	−8.24548	−0.266261
$960$	0	0
$961$	5.46858	0.176406
$962$	0	0
$963$	−5.19885	−0.167530
$964$	0	0
$965$	−21.5010	−0.692143
$966$	0	0
$967$	−9.27228	−0.298176	−0.149088	−	0.988824i	$-0.547634\pi$
$967$	−9.27228	−0.298176	−0.149088	+	0.988824i	$0.547634\pi$
$968$	0	0
$969$	−0.0562080	−0.00180566
$970$	0	0
$971$	32.9233	1.05656	0.528279	−	0.849071i	$-0.322838\pi$
$971$	32.9233	1.05656	0.528279	+	0.849071i	$0.322838\pi$
$972$	0	0
$973$	−10.4069	−0.333631
$974$	0	0
$975$	−31.0688	−0.994999
$976$	0	0
$977$	39.9873	1.27931	0.639654	−	0.768663i	$-0.279078\pi$
$977$	39.9873	1.27931	0.639654	+	0.768663i	$0.279078\pi$
$978$	0	0
$979$	42.7358	1.36584
$980$	0	0
$981$	−6.27648	−0.200393
$982$	0	0
$983$	−6.71964	−0.214323	−0.107162	−	0.994242i	$-0.534176\pi$
$983$	−6.71964	−0.214323	−0.107162	+	0.994242i	$0.534176\pi$
$984$	0	0
$985$	109.211	3.47977
$986$	0	0
$987$	−7.44412	−0.236949
$988$	0	0
$989$	64.5863	2.05373
$990$	0	0
$991$	39.0547	1.24061	0.620306	−	0.784360i	$-0.287008\pi$
$991$	39.0547	1.24061	0.620306	+	0.784360i	$0.287008\pi$
$992$	0	0
$993$	0.859249	0.0272674
$994$	0	0
$995$	90.5644	2.87108
$996$	0	0
$997$	23.1084	0.731849	0.365924	−	0.930645i	$-0.380753\pi$
$997$	23.1084	0.731849	0.365924	+	0.930645i	$0.380753\pi$
$998$	0	0
$999$	9.91270	0.313624

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2004.2.a.d.1.8	✓	9
3.2	odd	2		6012.2.a.h.1.2		9
4.3	odd	2		8016.2.a.bb.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.d.1.8	✓	9	1.1	even	1	trivial
6012.2.a.h.1.2		9	3.2	odd	2
8016.2.a.bb.1.8		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.19525 q^{5} +1.43344 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.19525 q^{5} +1.43344 q^{7} +1.00000 q^{9} +5.55413 q^{11} -2.46575 q^{13} +4.19525 q^{15} -0.151828 q^{17} +0.370208 q^{19} +1.43344 q^{21} -5.22489 q^{23} +12.6001 q^{25} +1.00000 q^{27} -9.43548 q^{29} +6.03892 q^{31} +5.55413 q^{33} +6.01363 q^{35} +9.91270 q^{37} -2.46575 q^{39} -8.03882 q^{41} -12.3613 q^{43} +4.19525 q^{45} -5.19319 q^{47} -4.94525 q^{49} -0.151828 q^{51} +6.02724 q^{53} +23.3010 q^{55} +0.370208 q^{57} -6.13698 q^{59} -4.70024 q^{61} +1.43344 q^{63} -10.3445 q^{65} -6.05931 q^{67} -5.22489 q^{69} +3.38205 q^{71} -13.2561 q^{73} +12.6001 q^{75} +7.96151 q^{77} -5.34461 q^{79} +1.00000 q^{81} +1.83389 q^{83} -0.636958 q^{85} -9.43548 q^{87} +7.69442 q^{89} -3.53451 q^{91} +6.03892 q^{93} +1.55312 q^{95} +18.7449 q^{97} +5.55413 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) 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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