LMFDB - Embedded newform 2004.2.a.d.1.9

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			1.9
Root			$-3.21153$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +4.21153 q^{5} -5.13720 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.21153 q^{5} -5.13720 q^{7} +1.00000 q^{9} -1.59181 q^{11} +5.05424 q^{13} +4.21153 q^{15} +0.754874 q^{17} -1.66875 q^{19} -5.13720 q^{21} +6.04218 q^{23} +12.7370 q^{25} +1.00000 q^{27} -4.24703 q^{29} -1.07105 q^{31} -1.59181 q^{33} -21.6355 q^{35} +4.94651 q^{37} +5.05424 q^{39} +3.04535 q^{41} +12.5625 q^{43} +4.21153 q^{45} +9.35819 q^{47} +19.3908 q^{49} +0.754874 q^{51} -8.52323 q^{53} -6.70395 q^{55} -1.66875 q^{57} -3.98059 q^{59} +8.18677 q^{61} -5.13720 q^{63} +21.2861 q^{65} -4.99504 q^{67} +6.04218 q^{69} +6.73050 q^{71} -6.06414 q^{73} +12.7370 q^{75} +8.17744 q^{77} -13.5695 q^{79} +1.00000 q^{81} -5.95533 q^{83} +3.17917 q^{85} -4.24703 q^{87} +7.83467 q^{89} -25.9646 q^{91} -1.07105 q^{93} -7.02797 q^{95} +3.42397 q^{97} -1.59181 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.21153	1.88345	0.941726	−	0.336380i	$-0.109203\pi$
$5$	4.21153	1.88345	0.941726	+	0.336380i	$0.109203\pi$
$6$	0	0
$7$	−5.13720	−1.94168	−0.970840	−	0.239730i	$-0.922941\pi$
$7$	−5.13720	−1.94168	−0.970840	+	0.239730i	$0.922941\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.59181	−0.479949	−0.239974	−	0.970779i	$-0.577139\pi$
$11$	−1.59181	−0.479949	−0.239974	+	0.970779i	$0.577139\pi$
$12$	0	0
$13$	5.05424	1.40179	0.700897	−	0.713263i	$-0.252783\pi$
$13$	5.05424	1.40179	0.700897	+	0.713263i	$0.252783\pi$
$14$	0	0
$15$	4.21153	1.08741
$16$	0	0
$17$	0.754874	0.183084	0.0915419	−	0.995801i	$-0.470820\pi$
$17$	0.754874	0.183084	0.0915419	+	0.995801i	$0.470820\pi$
$18$	0	0
$19$	−1.66875	−0.382836	−0.191418	−	0.981509i	$-0.561309\pi$
$19$	−1.66875	−0.382836	−0.191418	+	0.981509i	$0.561309\pi$
$20$	0	0
$21$	−5.13720	−1.12103
$22$	0	0
$23$	6.04218	1.25988	0.629941	−	0.776643i	$-0.283079\pi$
$23$	6.04218	1.25988	0.629941	+	0.776643i	$0.283079\pi$
$24$	0	0
$25$	12.7370	2.54739
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.24703	−0.788654	−0.394327	−	0.918970i	$-0.629022\pi$
$29$	−4.24703	−0.788654	−0.394327	+	0.918970i	$0.629022\pi$
$30$	0	0
$31$	−1.07105	−0.192367	−0.0961834	−	0.995364i	$-0.530664\pi$
$31$	−1.07105	−0.192367	−0.0961834	+	0.995364i	$0.530664\pi$
$32$	0	0
$33$	−1.59181	−0.277098
$34$	0	0
$35$	−21.6355	−3.65706
$36$	0	0
$37$	4.94651	0.813201	0.406600	−	0.913606i	$-0.366714\pi$
$37$	4.94651	0.813201	0.406600	+	0.913606i	$0.366714\pi$
$38$	0	0
$39$	5.05424	0.809326
$40$	0	0
$41$	3.04535	0.475603	0.237802	−	0.971314i	$-0.423573\pi$
$41$	3.04535	0.475603	0.237802	+	0.971314i	$0.423573\pi$
$42$	0	0
$43$	12.5625	1.91577	0.957884	−	0.287154i	$-0.0927091\pi$
$43$	12.5625	1.91577	0.957884	+	0.287154i	$0.0927091\pi$
$44$	0	0
$45$	4.21153	0.627818
$46$	0	0
$47$	9.35819	1.36503	0.682516	−	0.730871i	$-0.260886\pi$
$47$	9.35819	1.36503	0.682516	+	0.730871i	$0.260886\pi$
$48$	0	0
$49$	19.3908	2.77012
$50$	0	0
$51$	0.754874	0.105704
$52$	0	0
$53$	−8.52323	−1.17076	−0.585378	−	0.810761i	$-0.699054\pi$
$53$	−8.52323	−1.17076	−0.585378	+	0.810761i	$0.699054\pi$
$54$	0	0
$55$	−6.70395	−0.903961
$56$	0	0
$57$	−1.66875	−0.221031
$58$	0	0
$59$	−3.98059	−0.518229	−0.259115	−	0.965847i	$-0.583431\pi$
$59$	−3.98059	−0.518229	−0.259115	+	0.965847i	$0.583431\pi$
$60$	0	0
$61$	8.18677	1.04821	0.524104	−	0.851654i	$-0.324400\pi$
$61$	8.18677	1.04821	0.524104	+	0.851654i	$0.324400\pi$
$62$	0	0
$63$	−5.13720	−0.647226
$64$	0	0
$65$	21.2861	2.64021
$66$	0	0
$67$	−4.99504	−0.610242	−0.305121	−	0.952314i	$-0.598697\pi$
$67$	−4.99504	−0.610242	−0.305121	+	0.952314i	$0.598697\pi$
$68$	0	0
$69$	6.04218	0.727393
$70$	0	0
$71$	6.73050	0.798763	0.399382	−	0.916785i	$-0.369225\pi$
$71$	6.73050	0.798763	0.399382	+	0.916785i	$0.369225\pi$
$72$	0	0
$73$	−6.06414	−0.709754	−0.354877	−	0.934913i	$-0.615477\pi$
$73$	−6.06414	−0.709754	−0.354877	+	0.934913i	$0.615477\pi$
$74$	0	0
$75$	12.7370	1.47074
$76$	0	0
$77$	8.17744	0.931906
$78$	0	0
$79$	−13.5695	−1.52669	−0.763345	−	0.645991i	$-0.776444\pi$
$79$	−13.5695	−1.52669	−0.763345	+	0.645991i	$0.776444\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.95533	−0.653683	−0.326841	−	0.945079i	$-0.605984\pi$
$83$	−5.95533	−0.653683	−0.326841	+	0.945079i	$0.605984\pi$
$84$	0	0
$85$	3.17917	0.344830
$86$	0	0
$87$	−4.24703	−0.455330
$88$	0	0
$89$	7.83467	0.830473	0.415237	−	0.909713i	$-0.363699\pi$
$89$	7.83467	0.830473	0.415237	+	0.909713i	$0.363699\pi$
$90$	0	0
$91$	−25.9646	−2.72183
$92$	0	0
$93$	−1.07105	−0.111063
$94$	0	0
$95$	−7.02797	−0.721054
$96$	0	0
$97$	3.42397	0.347652	0.173826	−	0.984776i	$-0.444387\pi$
$97$	3.42397	0.347652	0.173826	+	0.984776i	$0.444387\pi$
$98$	0	0
$99$	−1.59181	−0.159983
$100$	0	0
$101$	−17.3384	−1.72523	−0.862617	−	0.505858i	$-0.831176\pi$
$101$	−17.3384	−1.72523	−0.862617	+	0.505858i	$0.831176\pi$
$102$	0	0
$103$	10.2605	1.01100	0.505498	−	0.862828i	$-0.331309\pi$
$103$	10.2605	1.01100	0.505498	+	0.862828i	$0.331309\pi$
$104$	0	0
$105$	−21.6355	−2.11140
$106$	0	0
$107$	13.1324	1.26955	0.634777	−	0.772696i	$-0.281092\pi$
$107$	13.1324	1.26955	0.634777	+	0.772696i	$0.281092\pi$
$108$	0	0
$109$	−4.53965	−0.434820	−0.217410	−	0.976080i	$-0.569761\pi$
$109$	−4.53965	−0.434820	−0.217410	+	0.976080i	$0.569761\pi$
$110$	0	0
$111$	4.94651	0.469502
$112$	0	0
$113$	−20.4713	−1.92578	−0.962889	−	0.269897i	$-0.913010\pi$
$113$	−20.4713	−1.92578	−0.962889	+	0.269897i	$0.913010\pi$
$114$	0	0
$115$	25.4468	2.37293
$116$	0	0
$117$	5.05424	0.467264
$118$	0	0
$119$	−3.87794	−0.355490
$120$	0	0
$121$	−8.46614	−0.769649
$122$	0	0
$123$	3.04535	0.274590
$124$	0	0
$125$	32.5845	2.91444
$126$	0	0
$127$	−17.7560	−1.57559	−0.787795	−	0.615937i	$-0.788778\pi$
$127$	−17.7560	−1.57559	−0.787795	+	0.615937i	$0.788778\pi$
$128$	0	0
$129$	12.5625	1.10607
$130$	0	0
$131$	−19.4096	−1.69582	−0.847912	−	0.530137i	$-0.822141\pi$
$131$	−19.4096	−1.69582	−0.847912	+	0.530137i	$0.822141\pi$
$132$	0	0
$133$	8.57268	0.743345
$134$	0	0
$135$	4.21153	0.362471
$136$	0	0
$137$	9.29492	0.794118	0.397059	−	0.917793i	$-0.370031\pi$
$137$	9.29492	0.794118	0.397059	+	0.917793i	$0.370031\pi$
$138$	0	0
$139$	13.0428	1.10628	0.553140	−	0.833089i	$-0.313430\pi$
$139$	13.0428	1.10628	0.553140	+	0.833089i	$0.313430\pi$
$140$	0	0
$141$	9.35819	0.788101
$142$	0	0
$143$	−8.04538	−0.672789
$144$	0	0
$145$	−17.8865	−1.48539
$146$	0	0
$147$	19.3908	1.59933
$148$	0	0
$149$	5.33359	0.436945	0.218473	−	0.975843i	$-0.429893\pi$
$149$	5.33359	0.436945	0.218473	+	0.975843i	$0.429893\pi$
$150$	0	0
$151$	−18.2184	−1.48259	−0.741294	−	0.671180i	$-0.765788\pi$
$151$	−18.2184	−1.48259	−0.741294	+	0.671180i	$0.765788\pi$
$152$	0	0
$153$	0.754874	0.0610280
$154$	0	0
$155$	−4.51077	−0.362314
$156$	0	0
$157$	−9.86754	−0.787516	−0.393758	−	0.919214i	$-0.628825\pi$
$157$	−9.86754	−0.787516	−0.393758	+	0.919214i	$0.628825\pi$
$158$	0	0
$159$	−8.52323	−0.675936
$160$	0	0
$161$	−31.0399	−2.44629
$162$	0	0
$163$	−7.64582	−0.598867	−0.299433	−	0.954117i	$-0.596798\pi$
$163$	−7.64582	−0.598867	−0.299433	+	0.954117i	$0.596798\pi$
$164$	0	0
$165$	−6.70395	−0.521902
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	12.5453	0.965024
$170$	0	0
$171$	−1.66875	−0.127612
$172$	0	0
$173$	2.13111	0.162025	0.0810127	−	0.996713i	$-0.474185\pi$
$173$	2.13111	0.162025	0.0810127	+	0.996713i	$0.474185\pi$
$174$	0	0
$175$	−65.4324	−4.94622
$176$	0	0
$177$	−3.98059	−0.299200
$178$	0	0
$179$	12.2841	0.918155	0.459078	−	0.888396i	$-0.348180\pi$
$179$	12.2841	0.918155	0.459078	+	0.888396i	$0.348180\pi$
$180$	0	0
$181$	−12.6588	−0.940924	−0.470462	−	0.882420i	$-0.655913\pi$
$181$	−12.6588	−0.940924	−0.470462	+	0.882420i	$0.655913\pi$
$182$	0	0
$183$	8.18677	0.605183
$184$	0	0
$185$	20.8324	1.53163
$186$	0	0
$187$	−1.20162	−0.0878709
$188$	0	0
$189$	−5.13720	−0.373676
$190$	0	0
$191$	14.9840	1.08420	0.542101	−	0.840313i	$-0.317629\pi$
$191$	14.9840	1.08420	0.542101	+	0.840313i	$0.317629\pi$
$192$	0	0
$193$	−19.2434	−1.38517	−0.692586	−	0.721335i	$-0.743529\pi$
$193$	−19.2434	−1.38517	−0.692586	+	0.721335i	$0.743529\pi$
$194$	0	0
$195$	21.2861	1.52433
$196$	0	0
$197$	−3.73605	−0.266182	−0.133091	−	0.991104i	$-0.542490\pi$
$197$	−3.73605	−0.266182	−0.133091	+	0.991104i	$0.542490\pi$
$198$	0	0
$199$	17.3082	1.22694	0.613472	−	0.789717i	$-0.289772\pi$
$199$	17.3082	1.22694	0.613472	+	0.789717i	$0.289772\pi$
$200$	0	0
$201$	−4.99504	−0.352323
$202$	0	0
$203$	21.8179	1.53131
$204$	0	0
$205$	12.8256	0.895776
$206$	0	0
$207$	6.04218	0.419961
$208$	0	0
$209$	2.65632	0.183742
$210$	0	0
$211$	10.3219	0.710587	0.355293	−	0.934755i	$-0.384381\pi$
$211$	10.3219	0.710587	0.355293	+	0.934755i	$0.384381\pi$
$212$	0	0
$213$	6.73050	0.461166
$214$	0	0
$215$	52.9075	3.60826
$216$	0	0
$217$	5.50221	0.373514
$218$	0	0
$219$	−6.06414	−0.409777
$220$	0	0
$221$	3.81531	0.256646
$222$	0	0
$223$	17.3854	1.16421	0.582107	−	0.813113i	$-0.302229\pi$
$223$	17.3854	1.16421	0.582107	+	0.813113i	$0.302229\pi$
$224$	0	0
$225$	12.7370	0.849131
$226$	0	0
$227$	22.0068	1.46065	0.730323	−	0.683102i	$-0.239370\pi$
$227$	22.0068	1.46065	0.730323	+	0.683102i	$0.239370\pi$
$228$	0	0
$229$	−10.2875	−0.679815	−0.339907	−	0.940459i	$-0.610396\pi$
$229$	−10.2875	−0.679815	−0.339907	+	0.940459i	$0.610396\pi$
$230$	0	0
$231$	8.17744	0.538036
$232$	0	0
$233$	−5.06675	−0.331934	−0.165967	−	0.986131i	$-0.553074\pi$
$233$	−5.06675	−0.331934	−0.165967	+	0.986131i	$0.553074\pi$
$234$	0	0
$235$	39.4123	2.57097
$236$	0	0
$237$	−13.5695	−0.881434
$238$	0	0
$239$	2.30330	0.148988	0.0744939	−	0.997221i	$-0.476266\pi$
$239$	2.30330	0.148988	0.0744939	+	0.997221i	$0.476266\pi$
$240$	0	0
$241$	−0.625570	−0.0402965	−0.0201483	−	0.999797i	$-0.506414\pi$
$241$	−0.625570	−0.0402965	−0.0201483	+	0.999797i	$0.506414\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	81.6650	5.21739
$246$	0	0
$247$	−8.43423	−0.536657
$248$	0	0
$249$	−5.95533	−0.377404
$250$	0	0
$251$	−22.4644	−1.41794	−0.708970	−	0.705238i	$-0.750840\pi$
$251$	−22.4644	−1.41794	−0.708970	+	0.705238i	$0.750840\pi$
$252$	0	0
$253$	−9.61800	−0.604678
$254$	0	0
$255$	3.17917	0.199088
$256$	0	0
$257$	−13.6816	−0.853436	−0.426718	−	0.904385i	$-0.640330\pi$
$257$	−13.6816	−0.853436	−0.426718	+	0.904385i	$0.640330\pi$
$258$	0	0
$259$	−25.4112	−1.57897
$260$	0	0
$261$	−4.24703	−0.262885
$262$	0	0
$263$	4.63785	0.285982	0.142991	−	0.989724i	$-0.454328\pi$
$263$	4.63785	0.285982	0.142991	+	0.989724i	$0.454328\pi$
$264$	0	0
$265$	−35.8958	−2.20506
$266$	0	0
$267$	7.83467	0.479474
$268$	0	0
$269$	24.6489	1.50287	0.751434	−	0.659808i	$-0.229363\pi$
$269$	24.6489	1.50287	0.751434	+	0.659808i	$0.229363\pi$
$270$	0	0
$271$	11.6032	0.704844	0.352422	−	0.935841i	$-0.385358\pi$
$271$	11.6032	0.704844	0.352422	+	0.935841i	$0.385358\pi$
$272$	0	0
$273$	−25.9646	−1.57145
$274$	0	0
$275$	−20.2748	−1.22262
$276$	0	0
$277$	−0.516066	−0.0310074	−0.0155037	−	0.999880i	$-0.504935\pi$
$277$	−0.516066	−0.0310074	−0.0155037	+	0.999880i	$0.504935\pi$
$278$	0	0
$279$	−1.07105	−0.0641222
$280$	0	0
$281$	−7.67914	−0.458099	−0.229050	−	0.973415i	$-0.573562\pi$
$281$	−7.67914	−0.458099	−0.229050	+	0.973415i	$0.573562\pi$
$282$	0	0
$283$	−15.5410	−0.923819	−0.461909	−	0.886927i	$-0.652835\pi$
$283$	−15.5410	−0.923819	−0.461909	+	0.886927i	$0.652835\pi$
$284$	0	0
$285$	−7.02797	−0.416301
$286$	0	0
$287$	−15.6445	−0.923469
$288$	0	0
$289$	−16.4302	−0.966480
$290$	0	0
$291$	3.42397	0.200717
$292$	0	0
$293$	7.27759	0.425161	0.212581	−	0.977144i	$-0.431813\pi$
$293$	7.27759	0.425161	0.212581	+	0.977144i	$0.431813\pi$
$294$	0	0
$295$	−16.7644	−0.976060
$296$	0	0
$297$	−1.59181	−0.0923662
$298$	0	0
$299$	30.5386	1.76609
$300$	0	0
$301$	−64.5363	−3.71981
$302$	0	0
$303$	−17.3384	−0.996064
$304$	0	0
$305$	34.4788	1.97425
$306$	0	0
$307$	−0.121465	−0.00693237	−0.00346619	−	0.999994i	$-0.501103\pi$
$307$	−0.121465	−0.00693237	−0.00346619	+	0.999994i	$0.501103\pi$
$308$	0	0
$309$	10.2605	0.583699
$310$	0	0
$311$	−16.6650	−0.944984	−0.472492	−	0.881335i	$-0.656645\pi$
$311$	−16.6650	−0.944984	−0.472492	+	0.881335i	$0.656645\pi$
$312$	0	0
$313$	−33.1709	−1.87493	−0.937464	−	0.348083i	$-0.886833\pi$
$313$	−33.1709	−1.87493	−0.937464	+	0.348083i	$0.886833\pi$
$314$	0	0
$315$	−21.6355	−1.21902
$316$	0	0
$317$	23.8173	1.33771	0.668856	−	0.743392i	$-0.266784\pi$
$317$	23.8173	1.33771	0.668856	+	0.743392i	$0.266784\pi$
$318$	0	0
$319$	6.76047	0.378513
$320$	0	0
$321$	13.1324	0.732977
$322$	0	0
$323$	−1.25969	−0.0700912
$324$	0	0
$325$	64.3757	3.57092
$326$	0	0
$327$	−4.53965	−0.251043
$328$	0	0
$329$	−48.0749	−2.65045
$330$	0	0
$331$	1.84876	0.101617	0.0508085	−	0.998708i	$-0.483820\pi$
$331$	1.84876	0.101617	0.0508085	+	0.998708i	$0.483820\pi$
$332$	0	0
$333$	4.94651	0.271067
$334$	0	0
$335$	−21.0368	−1.14936
$336$	0	0
$337$	12.0656	0.657256	0.328628	−	0.944460i	$-0.393414\pi$
$337$	12.0656	0.657256	0.328628	+	0.944460i	$0.393414\pi$
$338$	0	0
$339$	−20.4713	−1.11185
$340$	0	0
$341$	1.70491	0.0923261
$342$	0	0
$343$	−63.6541	−3.43700
$344$	0	0
$345$	25.4468	1.37001
$346$	0	0
$347$	−5.33156	−0.286213	−0.143106	−	0.989707i	$-0.545709\pi$
$347$	−5.33156	−0.286213	−0.143106	+	0.989707i	$0.545709\pi$
$348$	0	0
$349$	17.1869	0.919994	0.459997	−	0.887920i	$-0.347850\pi$
$349$	17.1869	0.919994	0.459997	+	0.887920i	$0.347850\pi$
$350$	0	0
$351$	5.05424	0.269775
$352$	0	0
$353$	−36.7612	−1.95660	−0.978301	−	0.207189i	$-0.933568\pi$
$353$	−36.7612	−1.95660	−0.978301	+	0.207189i	$0.933568\pi$
$354$	0	0
$355$	28.3457	1.50443
$356$	0	0
$357$	−3.87794	−0.205242
$358$	0	0
$359$	16.2669	0.858536	0.429268	−	0.903177i	$-0.358772\pi$
$359$	16.2669	0.858536	0.429268	+	0.903177i	$0.358772\pi$
$360$	0	0
$361$	−16.2153	−0.853436
$362$	0	0
$363$	−8.46614	−0.444357
$364$	0	0
$365$	−25.5393	−1.33679
$366$	0	0
$367$	−24.3923	−1.27327	−0.636633	−	0.771167i	$-0.719673\pi$
$367$	−24.3923	−1.27327	−0.636633	+	0.771167i	$0.719673\pi$
$368$	0	0
$369$	3.04535	0.158534
$370$	0	0
$371$	43.7855	2.27323
$372$	0	0
$373$	−19.3889	−1.00392	−0.501959	−	0.864892i	$-0.667387\pi$
$373$	−19.3889	−1.00392	−0.501959	+	0.864892i	$0.667387\pi$
$374$	0	0
$375$	32.5845	1.68265
$376$	0	0
$377$	−21.4655	−1.10553
$378$	0	0
$379$	12.6031	0.647375	0.323688	−	0.946164i	$-0.395077\pi$
$379$	12.6031	0.647375	0.323688	+	0.946164i	$0.395077\pi$
$380$	0	0
$381$	−17.7560	−0.909668
$382$	0	0
$383$	−11.7144	−0.598576	−0.299288	−	0.954163i	$-0.596749\pi$
$383$	−11.7144	−0.598576	−0.299288	+	0.954163i	$0.596749\pi$
$384$	0	0
$385$	34.4395	1.75520
$386$	0	0
$387$	12.5625	0.638590
$388$	0	0
$389$	−5.63603	−0.285758	−0.142879	−	0.989740i	$-0.545636\pi$
$389$	−5.63603	−0.285758	−0.142879	+	0.989740i	$0.545636\pi$
$390$	0	0
$391$	4.56109	0.230664
$392$	0	0
$393$	−19.4096	−0.979085
$394$	0	0
$395$	−57.1484	−2.87545
$396$	0	0
$397$	−29.3054	−1.47080	−0.735399	−	0.677634i	$-0.763005\pi$
$397$	−29.3054	−1.47080	−0.735399	+	0.677634i	$0.763005\pi$
$398$	0	0
$399$	8.57268	0.429171
$400$	0	0
$401$	21.4310	1.07021	0.535106	−	0.844785i	$-0.320271\pi$
$401$	21.4310	1.07021	0.535106	+	0.844785i	$0.320271\pi$
$402$	0	0
$403$	−5.41335	−0.269658
$404$	0	0
$405$	4.21153	0.209273
$406$	0	0
$407$	−7.87390	−0.390295
$408$	0	0
$409$	14.9959	0.741498	0.370749	−	0.928733i	$-0.379101\pi$
$409$	14.9959	0.741498	0.370749	+	0.928733i	$0.379101\pi$
$410$	0	0
$411$	9.29492	0.458484
$412$	0	0
$413$	20.4491	1.00623
$414$	0	0
$415$	−25.0811	−1.23118
$416$	0	0
$417$	13.0428	0.638710
$418$	0	0
$419$	−27.7911	−1.35769	−0.678843	−	0.734283i	$-0.737518\pi$
$419$	−27.7911	−1.35769	−0.678843	+	0.734283i	$0.737518\pi$
$420$	0	0
$421$	18.1443	0.884300	0.442150	−	0.896941i	$-0.354216\pi$
$421$	18.1443	0.884300	0.442150	+	0.896941i	$0.354216\pi$
$422$	0	0
$423$	9.35819	0.455011
$424$	0	0
$425$	9.61481	0.466387
$426$	0	0
$427$	−42.0571	−2.03528
$428$	0	0
$429$	−8.04538	−0.388435
$430$	0	0
$431$	−11.8731	−0.571905	−0.285952	−	0.958244i	$-0.592310\pi$
$431$	−11.8731	−0.571905	−0.285952	+	0.958244i	$0.592310\pi$
$432$	0	0
$433$	7.57113	0.363845	0.181923	−	0.983313i	$-0.441768\pi$
$433$	7.57113	0.363845	0.181923	+	0.983313i	$0.441768\pi$
$434$	0	0
$435$	−17.8865	−0.857592
$436$	0	0
$437$	−10.0829	−0.482328
$438$	0	0
$439$	26.3797	1.25904	0.629518	−	0.776986i	$-0.283252\pi$
$439$	26.3797	1.25904	0.629518	+	0.776986i	$0.283252\pi$
$440$	0	0
$441$	19.3908	0.923372
$442$	0	0
$443$	−22.2140	−1.05542	−0.527709	−	0.849425i	$-0.676949\pi$
$443$	−22.2140	−1.05542	−0.527709	+	0.849425i	$0.676949\pi$
$444$	0	0
$445$	32.9959	1.56416
$446$	0	0
$447$	5.33359	0.252270
$448$	0	0
$449$	11.2554	0.531176	0.265588	−	0.964087i	$-0.414434\pi$
$449$	11.2554	0.531176	0.265588	+	0.964087i	$0.414434\pi$
$450$	0	0
$451$	−4.84761	−0.228265
$452$	0	0
$453$	−18.2184	−0.855973
$454$	0	0
$455$	−109.351	−5.12644
$456$	0	0
$457$	9.97107	0.466427	0.233213	−	0.972426i	$-0.425076\pi$
$457$	9.97107	0.466427	0.233213	+	0.972426i	$0.425076\pi$
$458$	0	0
$459$	0.754874	0.0352345
$460$	0	0
$461$	−19.6426	−0.914847	−0.457423	−	0.889249i	$-0.651228\pi$
$461$	−19.6426	−0.914847	−0.457423	+	0.889249i	$0.651228\pi$
$462$	0	0
$463$	24.5210	1.13959	0.569793	−	0.821788i	$-0.307023\pi$
$463$	24.5210	1.13959	0.569793	+	0.821788i	$0.307023\pi$
$464$	0	0
$465$	−4.51077	−0.209182
$466$	0	0
$467$	14.0943	0.652205	0.326102	−	0.945334i	$-0.394265\pi$
$467$	14.0943	0.652205	0.326102	+	0.945334i	$0.394265\pi$
$468$	0	0
$469$	25.6605	1.18489
$470$	0	0
$471$	−9.86754	−0.454672
$472$	0	0
$473$	−19.9972	−0.919471
$474$	0	0
$475$	−21.2548	−0.975235
$476$	0	0
$477$	−8.52323	−0.390252
$478$	0	0
$479$	5.44493	0.248785	0.124393	−	0.992233i	$-0.460302\pi$
$479$	5.44493	0.248785	0.124393	+	0.992233i	$0.460302\pi$
$480$	0	0
$481$	25.0008	1.13994
$482$	0	0
$483$	−31.0399	−1.41236
$484$	0	0
$485$	14.4201	0.654785
$486$	0	0
$487$	−30.1077	−1.36431	−0.682155	−	0.731208i	$-0.738957\pi$
$487$	−30.1077	−1.36431	−0.682155	+	0.731208i	$0.738957\pi$
$488$	0	0
$489$	−7.64582	−0.345756
$490$	0	0
$491$	2.44218	0.110214	0.0551070	−	0.998480i	$-0.482450\pi$
$491$	2.44218	0.110214	0.0551070	+	0.998480i	$0.482450\pi$
$492$	0	0
$493$	−3.20597	−0.144390
$494$	0	0
$495$	−6.70395	−0.301320
$496$	0	0
$497$	−34.5759	−1.55094
$498$	0	0
$499$	−0.782658	−0.0350366	−0.0175183	−	0.999847i	$-0.505577\pi$
$499$	−0.782658	−0.0350366	−0.0175183	+	0.999847i	$0.505577\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	9.28226	0.413876	0.206938	−	0.978354i	$-0.433650\pi$
$503$	9.28226	0.413876	0.206938	+	0.978354i	$0.433650\pi$
$504$	0	0
$505$	−73.0211	−3.24940
$506$	0	0
$507$	12.5453	0.557157
$508$	0	0
$509$	−0.573049	−0.0254000	−0.0127000	−	0.999919i	$-0.504043\pi$
$509$	−0.573049	−0.0254000	−0.0127000	+	0.999919i	$0.504043\pi$
$510$	0	0
$511$	31.1527	1.37812
$512$	0	0
$513$	−1.66875	−0.0736769
$514$	0	0
$515$	43.2123	1.90416
$516$	0	0
$517$	−14.8964	−0.655145
$518$	0	0
$519$	2.13111	0.0935454
$520$	0	0
$521$	−33.0780	−1.44918	−0.724588	−	0.689183i	$-0.757970\pi$
$521$	−33.0780	−1.44918	−0.724588	+	0.689183i	$0.757970\pi$
$522$	0	0
$523$	−29.7761	−1.30202	−0.651010	−	0.759069i	$-0.725654\pi$
$523$	−29.7761	−1.30202	−0.651010	+	0.759069i	$0.725654\pi$
$524$	0	0
$525$	−65.4324	−2.85570
$526$	0	0
$527$	−0.808510	−0.0352192
$528$	0	0
$529$	13.5079	0.587301
$530$	0	0
$531$	−3.98059	−0.172743
$532$	0	0
$533$	15.3919	0.666697
$534$	0	0
$535$	55.3073	2.39114
$536$	0	0
$537$	12.2841	0.530097
$538$	0	0
$539$	−30.8665	−1.32951
$540$	0	0
$541$	35.6923	1.53453	0.767266	−	0.641329i	$-0.221616\pi$
$541$	35.6923	1.53453	0.767266	+	0.641329i	$0.221616\pi$
$542$	0	0
$543$	−12.6588	−0.543243
$544$	0	0
$545$	−19.1189	−0.818962
$546$	0	0
$547$	0.440380	0.0188293	0.00941464	−	0.999956i	$-0.497003\pi$
$547$	0.440380	0.0188293	0.00941464	+	0.999956i	$0.497003\pi$
$548$	0	0
$549$	8.18677	0.349403
$550$	0	0
$551$	7.08721	0.301925
$552$	0	0
$553$	69.7093	2.96434
$554$	0	0
$555$	20.8324	0.884284
$556$	0	0
$557$	−5.81205	−0.246264	−0.123132	−	0.992390i	$-0.539294\pi$
$557$	−5.81205	−0.246264	−0.123132	+	0.992390i	$0.539294\pi$
$558$	0	0
$559$	63.4940	2.68551
$560$	0	0
$561$	−1.20162	−0.0507323
$562$	0	0
$563$	4.04845	0.170622	0.0853109	−	0.996354i	$-0.472812\pi$
$563$	4.04845	0.170622	0.0853109	+	0.996354i	$0.472812\pi$
$564$	0	0
$565$	−86.2155	−3.62711
$566$	0	0
$567$	−5.13720	−0.215742
$568$	0	0
$569$	23.2702	0.975538	0.487769	−	0.872973i	$-0.337811\pi$
$569$	23.2702	0.975538	0.487769	+	0.872973i	$0.337811\pi$
$570$	0	0
$571$	−31.8512	−1.33293	−0.666466	−	0.745536i	$-0.732194\pi$
$571$	−31.8512	−1.33293	−0.666466	+	0.745536i	$0.732194\pi$
$572$	0	0
$573$	14.9840	0.625964
$574$	0	0
$575$	76.9591	3.20941
$576$	0	0
$577$	−12.0290	−0.500773	−0.250387	−	0.968146i	$-0.580558\pi$
$577$	−12.0290	−0.500773	−0.250387	+	0.968146i	$0.580558\pi$
$578$	0	0
$579$	−19.2434	−0.799730
$580$	0	0
$581$	30.5937	1.26924
$582$	0	0
$583$	13.5674	0.561902
$584$	0	0
$585$	21.2861	0.880070
$586$	0	0
$587$	4.88626	0.201678	0.100839	−	0.994903i	$-0.467847\pi$
$587$	4.88626	0.201678	0.100839	+	0.994903i	$0.467847\pi$
$588$	0	0
$589$	1.78731	0.0736450
$590$	0	0
$591$	−3.73605	−0.153680
$592$	0	0
$593$	−19.0611	−0.782746	−0.391373	−	0.920232i	$-0.628000\pi$
$593$	−19.0611	−0.782746	−0.391373	+	0.920232i	$0.628000\pi$
$594$	0	0
$595$	−16.3321	−0.669549
$596$	0	0
$597$	17.3082	0.708376
$598$	0	0
$599$	−12.2482	−0.500447	−0.250224	−	0.968188i	$-0.580504\pi$
$599$	−12.2482	−0.500447	−0.250224	+	0.968188i	$0.580504\pi$
$600$	0	0
$601$	3.87869	0.158215	0.0791076	−	0.996866i	$-0.474793\pi$
$601$	3.87869	0.158215	0.0791076	+	0.996866i	$0.474793\pi$
$602$	0	0
$603$	−4.99504	−0.203414
$604$	0	0
$605$	−35.6554	−1.44960
$606$	0	0
$607$	−12.6407	−0.513071	−0.256535	−	0.966535i	$-0.582581\pi$
$607$	−12.6407	−0.513071	−0.256535	+	0.966535i	$0.582581\pi$
$608$	0	0
$609$	21.8179	0.884104
$610$	0	0
$611$	47.2985	1.91349
$612$	0	0
$613$	−4.03966	−0.163160	−0.0815802	−	0.996667i	$-0.525997\pi$
$613$	−4.03966	−0.163160	−0.0815802	+	0.996667i	$0.525997\pi$
$614$	0	0
$615$	12.8256	0.517176
$616$	0	0
$617$	42.8979	1.72700	0.863502	−	0.504345i	$-0.168266\pi$
$617$	42.8979	1.72700	0.863502	+	0.504345i	$0.168266\pi$
$618$	0	0
$619$	−9.22024	−0.370593	−0.185296	−	0.982683i	$-0.559324\pi$
$619$	−9.22024	−0.370593	−0.185296	+	0.982683i	$0.559324\pi$
$620$	0	0
$621$	6.04218	0.242464
$622$	0	0
$623$	−40.2483	−1.61251
$624$	0	0
$625$	73.5456	2.94182
$626$	0	0
$627$	2.65632	0.106083
$628$	0	0
$629$	3.73399	0.148884
$630$	0	0
$631$	−3.58214	−0.142603	−0.0713014	−	0.997455i	$-0.522715\pi$
$631$	−3.58214	−0.142603	−0.0713014	+	0.997455i	$0.522715\pi$
$632$	0	0
$633$	10.3219	0.410258
$634$	0	0
$635$	−74.7799	−2.96755
$636$	0	0
$637$	98.0058	3.88313
$638$	0	0
$639$	6.73050	0.266254
$640$	0	0
$641$	−3.55994	−0.140609	−0.0703046	−	0.997526i	$-0.522397\pi$
$641$	−3.55994	−0.140609	−0.0703046	+	0.997526i	$0.522397\pi$
$642$	0	0
$643$	−15.3558	−0.605573	−0.302787	−	0.953058i	$-0.597917\pi$
$643$	−15.3558	−0.605573	−0.302787	+	0.953058i	$0.597917\pi$
$644$	0	0
$645$	52.9075	2.08323
$646$	0	0
$647$	−25.8947	−1.01803	−0.509013	−	0.860759i	$-0.669989\pi$
$647$	−25.8947	−1.01803	−0.509013	+	0.860759i	$0.669989\pi$
$648$	0	0
$649$	6.33635	0.248723
$650$	0	0
$651$	5.50221	0.215649
$652$	0	0
$653$	9.96529	0.389972	0.194986	−	0.980806i	$-0.437534\pi$
$653$	9.96529	0.389972	0.194986	+	0.980806i	$0.437534\pi$
$654$	0	0
$655$	−81.7441	−3.19400
$656$	0	0
$657$	−6.06414	−0.236585
$658$	0	0
$659$	23.3366	0.909065	0.454533	−	0.890730i	$-0.349806\pi$
$659$	23.3366	0.909065	0.454533	+	0.890730i	$0.349806\pi$
$660$	0	0
$661$	21.7943	0.847702	0.423851	−	0.905732i	$-0.360678\pi$
$661$	21.7943	0.847702	0.423851	+	0.905732i	$0.360678\pi$
$662$	0	0
$663$	3.81531	0.148174
$664$	0	0
$665$	36.1041	1.40006
$666$	0	0
$667$	−25.6613	−0.993611
$668$	0	0
$669$	17.3854	0.672159
$670$	0	0
$671$	−13.0318	−0.503086
$672$	0	0
$673$	−24.8535	−0.958034	−0.479017	−	0.877806i	$-0.659007\pi$
$673$	−24.8535	−0.958034	−0.479017	+	0.877806i	$0.659007\pi$
$674$	0	0
$675$	12.7370	0.490246
$676$	0	0
$677$	−11.6686	−0.448459	−0.224229	−	0.974536i	$-0.571987\pi$
$677$	−11.6686	−0.448459	−0.224229	+	0.974536i	$0.571987\pi$
$678$	0	0
$679$	−17.5896	−0.675028
$680$	0	0
$681$	22.0068	0.843304
$682$	0	0
$683$	−40.3092	−1.54239	−0.771195	−	0.636599i	$-0.780341\pi$
$683$	−40.3092	−1.54239	−0.771195	+	0.636599i	$0.780341\pi$
$684$	0	0
$685$	39.1458	1.49568
$686$	0	0
$687$	−10.2875	−0.392491
$688$	0	0
$689$	−43.0784	−1.64116
$690$	0	0
$691$	−0.648820	−0.0246823	−0.0123411	−	0.999924i	$-0.503928\pi$
$691$	−0.648820	−0.0246823	−0.0123411	+	0.999924i	$0.503928\pi$
$692$	0	0
$693$	8.17744	0.310635
$694$	0	0
$695$	54.9303	2.08362
$696$	0	0
$697$	2.29885	0.0870753
$698$	0	0
$699$	−5.06675	−0.191642
$700$	0	0
$701$	−14.2448	−0.538017	−0.269009	−	0.963138i	$-0.586696\pi$
$701$	−14.2448	−0.538017	−0.269009	+	0.963138i	$0.586696\pi$
$702$	0	0
$703$	−8.25446	−0.311323
$704$	0	0
$705$	39.4123	1.48435
$706$	0	0
$707$	89.0707	3.34985
$708$	0	0
$709$	33.6480	1.26368	0.631839	−	0.775100i	$-0.282300\pi$
$709$	33.6480	1.26368	0.631839	+	0.775100i	$0.282300\pi$
$710$	0	0
$711$	−13.5695	−0.508896
$712$	0	0
$713$	−6.47149	−0.242359
$714$	0	0
$715$	−33.8834	−1.26717
$716$	0	0
$717$	2.30330	0.0860182
$718$	0	0
$719$	17.0334	0.635239	0.317619	−	0.948218i	$-0.397117\pi$
$719$	17.0334	0.635239	0.317619	+	0.948218i	$0.397117\pi$
$720$	0	0
$721$	−52.7102	−1.96303
$722$	0	0
$723$	−0.625570	−0.0232652
$724$	0	0
$725$	−54.0943	−2.00901
$726$	0	0
$727$	7.31178	0.271179	0.135589	−	0.990765i	$-0.456707\pi$
$727$	7.31178	0.271179	0.135589	+	0.990765i	$0.456707\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.48313	0.350746
$732$	0	0
$733$	17.4158	0.643267	0.321634	−	0.946864i	$-0.395768\pi$
$733$	17.4158	0.643267	0.321634	+	0.946864i	$0.395768\pi$
$734$	0	0
$735$	81.6650	3.01226
$736$	0	0
$737$	7.95116	0.292885
$738$	0	0
$739$	−27.3162	−1.00484	−0.502421	−	0.864623i	$-0.667557\pi$
$739$	−27.3162	−1.00484	−0.502421	+	0.864623i	$0.667557\pi$
$740$	0	0
$741$	−8.43423	−0.309839
$742$	0	0
$743$	33.4574	1.22743	0.613716	−	0.789526i	$-0.289674\pi$
$743$	33.4574	1.22743	0.613716	+	0.789526i	$0.289674\pi$
$744$	0	0
$745$	22.4626	0.822965
$746$	0	0
$747$	−5.95533	−0.217894
$748$	0	0
$749$	−67.4636	−2.46506
$750$	0	0
$751$	−9.04026	−0.329884	−0.164942	−	0.986303i	$-0.552744\pi$
$751$	−9.04026	−0.329884	−0.164942	+	0.986303i	$0.552744\pi$
$752$	0	0
$753$	−22.4644	−0.818649
$754$	0	0
$755$	−76.7271	−2.79239
$756$	0	0
$757$	−29.0912	−1.05734	−0.528669	−	0.848828i	$-0.677309\pi$
$757$	−29.0912	−1.05734	−0.528669	+	0.848828i	$0.677309\pi$
$758$	0	0
$759$	−9.61800	−0.349111
$760$	0	0
$761$	25.4421	0.922274	0.461137	−	0.887329i	$-0.347442\pi$
$761$	25.4421	0.922274	0.461137	+	0.887329i	$0.347442\pi$
$762$	0	0
$763$	23.3211	0.844280
$764$	0	0
$765$	3.17917	0.114943
$766$	0	0
$767$	−20.1189	−0.726450
$768$	0	0
$769$	−16.4143	−0.591915	−0.295958	−	0.955201i	$-0.595639\pi$
$769$	−16.4143	−0.591915	−0.295958	+	0.955201i	$0.595639\pi$
$770$	0	0
$771$	−13.6816	−0.492731
$772$	0	0
$773$	46.6409	1.67756	0.838778	−	0.544474i	$-0.183271\pi$
$773$	46.6409	1.67756	0.838778	+	0.544474i	$0.183271\pi$
$774$	0	0
$775$	−13.6420	−0.490034
$776$	0	0
$777$	−25.4112	−0.911621
$778$	0	0
$779$	−5.08191	−0.182078
$780$	0	0
$781$	−10.7137	−0.383365
$782$	0	0
$783$	−4.24703	−0.151777
$784$	0	0
$785$	−41.5574	−1.48325
$786$	0	0
$787$	4.52468	0.161287	0.0806437	−	0.996743i	$-0.474302\pi$
$787$	4.52468	0.161287	0.0806437	+	0.996743i	$0.474302\pi$
$788$	0	0
$789$	4.63785	0.165112
$790$	0	0
$791$	105.165	3.73924
$792$	0	0
$793$	41.3779	1.46937
$794$	0	0
$795$	−35.8958	−1.27309
$796$	0	0
$797$	−5.76977	−0.204376	−0.102188	−	0.994765i	$-0.532584\pi$
$797$	−5.76977	−0.204376	−0.102188	+	0.994765i	$0.532584\pi$
$798$	0	0
$799$	7.06425	0.249915
$800$	0	0
$801$	7.83467	0.276824
$802$	0	0
$803$	9.65296	0.340646
$804$	0	0
$805$	−130.725	−4.60746
$806$	0	0
$807$	24.6489	0.867681
$808$	0	0
$809$	−2.89016	−0.101613	−0.0508063	−	0.998709i	$-0.516179\pi$
$809$	−2.89016	−0.101613	−0.0508063	+	0.998709i	$0.516179\pi$
$810$	0	0
$811$	−23.7963	−0.835600	−0.417800	−	0.908539i	$-0.637199\pi$
$811$	−23.7963	−0.835600	−0.417800	+	0.908539i	$0.637199\pi$
$812$	0	0
$813$	11.6032	0.406942
$814$	0	0
$815$	−32.2006	−1.12794
$816$	0	0
$817$	−20.9637	−0.733426
$818$	0	0
$819$	−25.9646	−0.907277
$820$	0	0
$821$	15.4083	0.537752	0.268876	−	0.963175i	$-0.413348\pi$
$821$	15.4083	0.537752	0.268876	+	0.963175i	$0.413348\pi$
$822$	0	0
$823$	−53.0392	−1.84883	−0.924416	−	0.381386i	$-0.875447\pi$
$823$	−53.0392	−1.84883	−0.924416	+	0.381386i	$0.875447\pi$
$824$	0	0
$825$	−20.2748	−0.705879
$826$	0	0
$827$	−37.1164	−1.29066	−0.645332	−	0.763902i	$-0.723281\pi$
$827$	−37.1164	−1.29066	−0.645332	+	0.763902i	$0.723281\pi$
$828$	0	0
$829$	−33.9601	−1.17948	−0.589742	−	0.807592i	$-0.700771\pi$
$829$	−33.9601	−1.17948	−0.589742	+	0.807592i	$0.700771\pi$
$830$	0	0
$831$	−0.516066	−0.0179021
$832$	0	0
$833$	14.6376	0.507164
$834$	0	0
$835$	4.21153	0.145746
$836$	0	0
$837$	−1.07105	−0.0370210
$838$	0	0
$839$	22.1451	0.764533	0.382267	−	0.924052i	$-0.375144\pi$
$839$	22.1451	0.764533	0.382267	+	0.924052i	$0.375144\pi$
$840$	0	0
$841$	−10.9627	−0.378025
$842$	0	0
$843$	−7.67914	−0.264484
$844$	0	0
$845$	52.8349	1.81758
$846$	0	0
$847$	43.4923	1.49441
$848$	0	0
$849$	−15.5410	−0.533367
$850$	0	0
$851$	29.8877	1.02454
$852$	0	0
$853$	22.8091	0.780970	0.390485	−	0.920609i	$-0.372307\pi$
$853$	22.8091	0.780970	0.390485	+	0.920609i	$0.372307\pi$
$854$	0	0
$855$	−7.02797	−0.240351
$856$	0	0
$857$	53.9748	1.84374	0.921872	−	0.387493i	$-0.126659\pi$
$857$	53.9748	1.84374	0.921872	+	0.387493i	$0.126659\pi$
$858$	0	0
$859$	36.9731	1.26151	0.630754	−	0.775983i	$-0.282746\pi$
$859$	36.9731	1.26151	0.630754	+	0.775983i	$0.282746\pi$
$860$	0	0
$861$	−15.6445	−0.533165
$862$	0	0
$863$	20.2798	0.690331	0.345165	−	0.938542i	$-0.387823\pi$
$863$	20.2798	0.690331	0.345165	+	0.938542i	$0.387823\pi$
$864$	0	0
$865$	8.97524	0.305167
$866$	0	0
$867$	−16.4302	−0.557998
$868$	0	0
$869$	21.6001	0.732732
$870$	0	0
$871$	−25.2461	−0.855432
$872$	0	0
$873$	3.42397	0.115884
$874$	0	0
$875$	−167.393	−5.65891
$876$	0	0
$877$	38.8159	1.31072	0.655360	−	0.755317i	$-0.272517\pi$
$877$	38.8159	1.31072	0.655360	+	0.755317i	$0.272517\pi$
$878$	0	0
$879$	7.27759	0.245467
$880$	0	0
$881$	−25.0400	−0.843618	−0.421809	−	0.906685i	$-0.638605\pi$
$881$	−25.0400	−0.843618	−0.421809	+	0.906685i	$0.638605\pi$
$882$	0	0
$883$	−9.66634	−0.325298	−0.162649	−	0.986684i	$-0.552004\pi$
$883$	−9.66634	−0.325298	−0.162649	+	0.986684i	$0.552004\pi$
$884$	0	0
$885$	−16.7644	−0.563529
$886$	0	0
$887$	−22.8139	−0.766015	−0.383008	−	0.923745i	$-0.625112\pi$
$887$	−22.8139	−0.766015	−0.383008	+	0.923745i	$0.625112\pi$
$888$	0	0
$889$	91.2162	3.05929
$890$	0	0
$891$	−1.59181	−0.0533276
$892$	0	0
$893$	−15.6164	−0.522584
$894$	0	0
$895$	51.7347	1.72930
$896$	0	0
$897$	30.5386	1.01965
$898$	0	0
$899$	4.54879	0.151711
$900$	0	0
$901$	−6.43396	−0.214346
$902$	0	0
$903$	−64.5363	−2.14763
$904$	0	0
$905$	−53.3130	−1.77219
$906$	0	0
$907$	43.6100	1.44805	0.724023	−	0.689775i	$-0.242291\pi$
$907$	43.6100	1.44805	0.724023	+	0.689775i	$0.242291\pi$
$908$	0	0
$909$	−17.3384	−0.575078
$910$	0	0
$911$	47.3364	1.56832	0.784162	−	0.620556i	$-0.213093\pi$
$911$	47.3364	1.56832	0.784162	+	0.620556i	$0.213093\pi$
$912$	0	0
$913$	9.47976	0.313734
$914$	0	0
$915$	34.4788	1.13983
$916$	0	0
$917$	99.7110	3.29275
$918$	0	0
$919$	3.69738	0.121965	0.0609826	−	0.998139i	$-0.480577\pi$
$919$	3.69738	0.121965	0.0609826	+	0.998139i	$0.480577\pi$
$920$	0	0
$921$	−0.121465	−0.00400241
$922$	0	0
$923$	34.0175	1.11970
$924$	0	0
$925$	63.0035	2.07154
$926$	0	0
$927$	10.2605	0.336999
$928$	0	0
$929$	−19.5929	−0.642821	−0.321410	−	0.946940i	$-0.604157\pi$
$929$	−19.5929	−0.642821	−0.321410	+	0.946940i	$0.604157\pi$
$930$	0	0
$931$	−32.3583	−1.06050
$932$	0	0
$933$	−16.6650	−0.545587
$934$	0	0
$935$	−5.06064	−0.165501
$936$	0	0
$937$	−46.2138	−1.50974	−0.754869	−	0.655875i	$-0.772300\pi$
$937$	−46.2138	−1.50974	−0.754869	+	0.655875i	$0.772300\pi$
$938$	0	0
$939$	−33.1709	−1.08249
$940$	0	0
$941$	−16.1732	−0.527231	−0.263616	−	0.964628i	$-0.584915\pi$
$941$	−16.1732	−0.527231	−0.263616	+	0.964628i	$0.584915\pi$
$942$	0	0
$943$	18.4005	0.599204
$944$	0	0
$945$	−21.6355	−0.703802
$946$	0	0
$947$	−19.7305	−0.641154	−0.320577	−	0.947223i	$-0.603877\pi$
$947$	−19.7305	−0.641154	−0.320577	+	0.947223i	$0.603877\pi$
$948$	0	0
$949$	−30.6496	−0.994929
$950$	0	0
$951$	23.8173	0.772329
$952$	0	0
$953$	44.2858	1.43456	0.717279	−	0.696786i	$-0.245387\pi$
$953$	44.2858	1.43456	0.717279	+	0.696786i	$0.245387\pi$
$954$	0	0
$955$	63.1054	2.04204
$956$	0	0
$957$	6.76047	0.218535
$958$	0	0
$959$	−47.7498	−1.54192
$960$	0	0
$961$	−29.8528	−0.962995
$962$	0	0
$963$	13.1324	0.423184
$964$	0	0
$965$	−81.0443	−2.60891
$966$	0	0
$967$	−19.9140	−0.640391	−0.320195	−	0.947352i	$-0.603749\pi$
$967$	−19.9140	−0.640391	−0.320195	+	0.947352i	$0.603749\pi$
$968$	0	0
$969$	−1.25969	−0.0404672
$970$	0	0
$971$	−30.1118	−0.966334	−0.483167	−	0.875528i	$-0.660514\pi$
$971$	−30.1118	−0.966334	−0.483167	+	0.875528i	$0.660514\pi$
$972$	0	0
$973$	−67.0037	−2.14804
$974$	0	0
$975$	64.3757	2.06167
$976$	0	0
$977$	10.6796	0.341672	0.170836	−	0.985299i	$-0.445353\pi$
$977$	10.6796	0.341672	0.170836	+	0.985299i	$0.445353\pi$
$978$	0	0
$979$	−12.4713	−0.398585
$980$	0	0
$981$	−4.53965	−0.144940
$982$	0	0
$983$	−43.7358	−1.39496	−0.697478	−	0.716606i	$-0.745695\pi$
$983$	−43.7358	−1.39496	−0.697478	+	0.716606i	$0.745695\pi$
$984$	0	0
$985$	−15.7345	−0.501342
$986$	0	0
$987$	−48.0749	−1.53024
$988$	0	0
$989$	75.9051	2.41364
$990$	0	0
$991$	−19.5304	−0.620402	−0.310201	−	0.950671i	$-0.600396\pi$
$991$	−19.5304	−0.620402	−0.310201	+	0.950671i	$0.600396\pi$
$992$	0	0
$993$	1.84876	0.0586685
$994$	0	0
$995$	72.8938	2.31089
$996$	0	0
$997$	−5.72889	−0.181436	−0.0907179	−	0.995877i	$-0.528916\pi$
$997$	−5.72889	−0.181436	−0.0907179	+	0.995877i	$0.528916\pi$
$998$	0	0
$999$	4.94651	0.156501

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.21153 q^{5} -5.13720 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.21153 q^{5} -5.13720 q^{7} +1.00000 q^{9} -1.59181 q^{11} +5.05424 q^{13} +4.21153 q^{15} +0.754874 q^{17} -1.66875 q^{19} -5.13720 q^{21} +6.04218 q^{23} +12.7370 q^{25} +1.00000 q^{27} -4.24703 q^{29} -1.07105 q^{31} -1.59181 q^{33} -21.6355 q^{35} +4.94651 q^{37} +5.05424 q^{39} +3.04535 q^{41} +12.5625 q^{43} +4.21153 q^{45} +9.35819 q^{47} +19.3908 q^{49} +0.754874 q^{51} -8.52323 q^{53} -6.70395 q^{55} -1.66875 q^{57} -3.98059 q^{59} +8.18677 q^{61} -5.13720 q^{63} +21.2861 q^{65} -4.99504 q^{67} +6.04218 q^{69} +6.73050 q^{71} -6.06414 q^{73} +12.7370 q^{75} +8.17744 q^{77} -13.5695 q^{79} +1.00000 q^{81} -5.95533 q^{83} +3.17917 q^{85} -4.24703 q^{87} +7.83467 q^{89} -25.9646 q^{91} -1.07105 q^{93} -7.02797 q^{95} +3.42397 q^{97} -1.59181 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) 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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