LMFDB - Embedded newform 2004.2.a.a.1.2

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:	$1$
Dimension:	$5$
Coefficient field:	5.5.149169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - 3x^{2} + 4x + 1 $ x^5 - 6x^3 - 3x^2 + 4*x + 1
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			1.2
Root			$-0.228573$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.71918 q^{5} -2.29630 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.71918 q^{5} -2.29630 q^{7} +1.00000 q^{9} +2.71918 q^{11} -0.130922 q^{13} +2.71918 q^{15} +5.29280 q^{17} +2.49495 q^{19} +2.29630 q^{21} +2.05324 q^{23} +2.39395 q^{25} -1.00000 q^{27} -5.85977 q^{29} +0.277324 q^{31} -2.71918 q^{33} +6.24405 q^{35} +9.10161 q^{37} +0.130922 q^{39} -8.78322 q^{41} -7.14952 q^{43} -2.71918 q^{45} -1.16618 q^{47} -1.72701 q^{49} -5.29280 q^{51} -1.51392 q^{53} -7.39395 q^{55} -2.49495 q^{57} +6.58556 q^{59} -3.25671 q^{61} -2.29630 q^{63} +0.356000 q^{65} -11.6688 q^{67} -2.05324 q^{69} -10.5813 q^{71} -13.9938 q^{73} -2.39395 q^{75} -6.24405 q^{77} -1.32793 q^{79} +1.00000 q^{81} +15.3931 q^{83} -14.3921 q^{85} +5.85977 q^{87} -0.650613 q^{89} +0.300636 q^{91} -0.277324 q^{93} -6.78421 q^{95} -16.5435 q^{97} +2.71918 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9} + 3 q^{11} - 4 q^{13} + 3 q^{15} - 7 q^{17} - 2 q^{19} + 2 q^{21} + 13 q^{23} - 2 q^{25} - 5 q^{27} - 3 q^{29} - 12 q^{31} - 3 q^{33} + 10 q^{35} - 7 q^{37} + 4 q^{39} - 16 q^{41} - 3 q^{45} + q^{47} - 17 q^{49} + 7 q^{51} + 3 q^{53} - 23 q^{55} + 2 q^{57} + q^{59} - 22 q^{61} - 2 q^{63} - 20 q^{65} + 2 q^{67} - 13 q^{69} + 9 q^{71} - 28 q^{73} + 2 q^{75} - 10 q^{77} - 28 q^{79} + 5 q^{81} + 7 q^{83} - 11 q^{85} + 3 q^{87} - 30 q^{89} - 13 q^{91} + 12 q^{93} + 3 q^{95} - 33 q^{97} + 3 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9 + 3 * q^11 - 4 * q^13 + 3 * q^15 - 7 * q^17 - 2 * q^19 + 2 * q^21 + 13 * q^23 - 2 * q^25 - 5 * q^27 - 3 * q^29 - 12 * q^31 - 3 * q^33 + 10 * q^35 - 7 * q^37 + 4 * q^39 - 16 * q^41 - 3 * q^45 + q^47 - 17 * q^49 + 7 * q^51 + 3 * q^53 - 23 * q^55 + 2 * q^57 + q^59 - 22 * q^61 - 2 * q^63 - 20 * q^65 + 2 * q^67 - 13 * q^69 + 9 * q^71 - 28 * q^73 + 2 * q^75 - 10 * q^77 - 28 * q^79 + 5 * q^81 + 7 * q^83 - 11 * q^85 + 3 * q^87 - 30 * q^89 - 13 * q^91 + 12 * q^93 + 3 * q^95 - 33 * q^97 + 3 * 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
$5$	−2.71918	−1.21606	−0.608028	−	0.793916i	$-0.708039\pi$
$5$	−2.71918	−1.21606	−0.608028	+	0.793916i	$0.708039\pi$
$6$	0	0
$7$	−2.29630	−0.867919	−0.433960	−	0.900932i	$-0.642884\pi$
$7$	−2.29630	−0.867919	−0.433960	+	0.900932i	$0.642884\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.71918	0.819864	0.409932	−	0.912116i	$-0.365552\pi$
$11$	2.71918	0.819864	0.409932	+	0.912116i	$0.365552\pi$
$12$	0	0
$13$	−0.130922	−0.0363112	−0.0181556	−	0.999835i	$-0.505779\pi$
$13$	−0.130922	−0.0363112	−0.0181556	+	0.999835i	$0.505779\pi$
$14$	0	0
$15$	2.71918	0.702090
$16$	0	0
$17$	5.29280	1.28369	0.641847	−	0.766833i	$-0.278169\pi$
$17$	5.29280	1.28369	0.641847	+	0.766833i	$0.278169\pi$
$18$	0	0
$19$	2.49495	0.572380	0.286190	−	0.958173i	$-0.407611\pi$
$19$	2.49495	0.572380	0.286190	+	0.958173i	$0.407611\pi$
$20$	0	0
$21$	2.29630	0.501093
$22$	0	0
$23$	2.05324	0.428129	0.214065	−	0.976819i	$-0.431330\pi$
$23$	2.05324	0.428129	0.214065	+	0.976819i	$0.431330\pi$
$24$	0	0
$25$	2.39395	0.478790
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.85977	−1.08813	−0.544066	−	0.839042i	$-0.683116\pi$
$29$	−5.85977	−1.08813	−0.544066	+	0.839042i	$0.683116\pi$
$30$	0	0
$31$	0.277324	0.0498088	0.0249044	−	0.999690i	$-0.492072\pi$
$31$	0.277324	0.0498088	0.0249044	+	0.999690i	$0.492072\pi$
$32$	0	0
$33$	−2.71918	−0.473349
$34$	0	0
$35$	6.24405	1.05544
$36$	0	0
$37$	9.10161	1.49630	0.748148	−	0.663532i	$-0.230943\pi$
$37$	9.10161	1.49630	0.748148	+	0.663532i	$0.230943\pi$
$38$	0	0
$39$	0.130922	0.0209643
$40$	0	0
$41$	−8.78322	−1.37171	−0.685854	−	0.727739i	$-0.740571\pi$
$41$	−8.78322	−1.37171	−0.685854	+	0.727739i	$0.740571\pi$
$42$	0	0
$43$	−7.14952	−1.09029	−0.545146	−	0.838341i	$-0.683526\pi$
$43$	−7.14952	−1.09029	−0.545146	+	0.838341i	$0.683526\pi$
$44$	0	0
$45$	−2.71918	−0.405352
$46$	0	0
$47$	−1.16618	−0.170104	−0.0850521	−	0.996377i	$-0.527106\pi$
$47$	−1.16618	−0.170104	−0.0850521	+	0.996377i	$0.527106\pi$
$48$	0	0
$49$	−1.72701	−0.246716
$50$	0	0
$51$	−5.29280	−0.741141
$52$	0	0
$53$	−1.51392	−0.207953	−0.103976	−	0.994580i	$-0.533157\pi$
$53$	−1.51392	−0.207953	−0.103976	+	0.994580i	$0.533157\pi$
$54$	0	0
$55$	−7.39395	−0.997000
$56$	0	0
$57$	−2.49495	−0.330464
$58$	0	0
$59$	6.58556	0.857367	0.428684	−	0.903455i	$-0.358977\pi$
$59$	6.58556	0.857367	0.428684	+	0.903455i	$0.358977\pi$
$60$	0	0
$61$	−3.25671	−0.416979	−0.208489	−	0.978025i	$-0.566855\pi$
$61$	−3.25671	−0.416979	−0.208489	+	0.978025i	$0.566855\pi$
$62$	0	0
$63$	−2.29630	−0.289306
$64$	0	0
$65$	0.356000	0.0441564
$66$	0	0
$67$	−11.6688	−1.42557	−0.712784	−	0.701384i	$-0.752566\pi$
$67$	−11.6688	−1.42557	−0.712784	+	0.701384i	$0.752566\pi$
$68$	0	0
$69$	−2.05324	−0.247181
$70$	0	0
$71$	−10.5813	−1.25577	−0.627883	−	0.778308i	$-0.716078\pi$
$71$	−10.5813	−1.25577	−0.627883	+	0.778308i	$0.716078\pi$
$72$	0	0
$73$	−13.9938	−1.63785	−0.818926	−	0.573899i	$-0.805430\pi$
$73$	−13.9938	−1.63785	−0.818926	+	0.573899i	$0.805430\pi$
$74$	0	0
$75$	−2.39395	−0.276429
$76$	0	0
$77$	−6.24405	−0.711576
$78$	0	0
$79$	−1.32793	−0.149404	−0.0747018	−	0.997206i	$-0.523800\pi$
$79$	−1.32793	−0.149404	−0.0747018	+	0.997206i	$0.523800\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.3931	1.68961	0.844805	−	0.535074i	$-0.179716\pi$
$83$	15.3931	1.68961	0.844805	+	0.535074i	$0.179716\pi$
$84$	0	0
$85$	−14.3921	−1.56104
$86$	0	0
$87$	5.85977	0.628233
$88$	0	0
$89$	−0.650613	−0.0689649	−0.0344824	−	0.999405i	$-0.510978\pi$
$89$	−0.650613	−0.0689649	−0.0344824	+	0.999405i	$0.510978\pi$
$90$	0	0
$91$	0.300636	0.0315152
$92$	0	0
$93$	−0.277324	−0.0287571
$94$	0	0
$95$	−6.78421	−0.696045
$96$	0	0
$97$	−16.5435	−1.67973	−0.839867	−	0.542792i	$-0.817367\pi$
$97$	−16.5435	−1.67973	−0.839867	+	0.542792i	$0.817367\pi$
$98$	0	0
$99$	2.71918	0.273288
$100$	0	0
$101$	−3.96140	−0.394174	−0.197087	−	0.980386i	$-0.563148\pi$
$101$	−3.96140	−0.394174	−0.197087	+	0.980386i	$0.563148\pi$
$102$	0	0
$103$	−3.85793	−0.380134	−0.190067	−	0.981771i	$-0.560870\pi$
$103$	−3.85793	−0.380134	−0.190067	+	0.981771i	$0.560870\pi$
$104$	0	0
$105$	−6.24405	−0.609357
$106$	0	0
$107$	13.1188	1.26824	0.634121	−	0.773234i	$-0.281362\pi$
$107$	13.1188	1.26824	0.634121	+	0.773234i	$0.281362\pi$
$108$	0	0
$109$	2.33140	0.223308	0.111654	−	0.993747i	$-0.464385\pi$
$109$	2.33140	0.223308	0.111654	+	0.993747i	$0.464385\pi$
$110$	0	0
$111$	−9.10161	−0.863887
$112$	0	0
$113$	13.1549	1.23751	0.618754	−	0.785585i	$-0.287638\pi$
$113$	13.1549	1.23751	0.618754	+	0.785585i	$0.287638\pi$
$114$	0	0
$115$	−5.58312	−0.520629
$116$	0	0
$117$	−0.130922	−0.0121037
$118$	0	0
$119$	−12.1539	−1.11414
$120$	0	0
$121$	−3.60605	−0.327823
$122$	0	0
$123$	8.78322	0.791956
$124$	0	0
$125$	7.08633	0.633820
$126$	0	0
$127$	−3.91471	−0.347374	−0.173687	−	0.984801i	$-0.555568\pi$
$127$	−3.91471	−0.347374	−0.173687	+	0.984801i	$0.555568\pi$
$128$	0	0
$129$	7.14952	0.629480
$130$	0	0
$131$	−8.22025	−0.718207	−0.359104	−	0.933298i	$-0.616917\pi$
$131$	−8.22025	−0.718207	−0.359104	+	0.933298i	$0.616917\pi$
$132$	0	0
$133$	−5.72914	−0.496780
$134$	0	0
$135$	2.71918	0.234030
$136$	0	0
$137$	9.47880	0.809828	0.404914	−	0.914355i	$-0.367301\pi$
$137$	9.47880	0.809828	0.404914	+	0.914355i	$0.367301\pi$
$138$	0	0
$139$	5.15967	0.437638	0.218819	−	0.975765i	$-0.429780\pi$
$139$	5.15967	0.437638	0.218819	+	0.975765i	$0.429780\pi$
$140$	0	0
$141$	1.16618	0.0982097
$142$	0	0
$143$	−0.356000	−0.0297702
$144$	0	0
$145$	15.9338	1.32323
$146$	0	0
$147$	1.72701	0.142442
$148$	0	0
$149$	−13.5693	−1.11164	−0.555821	−	0.831302i	$-0.687596\pi$
$149$	−13.5693	−1.11164	−0.555821	+	0.831302i	$0.687596\pi$
$150$	0	0
$151$	−21.5430	−1.75315	−0.876573	−	0.481268i	$-0.840176\pi$
$151$	−21.5430	−1.75315	−0.876573	+	0.481268i	$0.840176\pi$
$152$	0	0
$153$	5.29280	0.427898
$154$	0	0
$155$	−0.754094	−0.0605703
$156$	0	0
$157$	−18.1301	−1.44694	−0.723470	−	0.690356i	$-0.757454\pi$
$157$	−18.1301	−1.44694	−0.723470	+	0.690356i	$0.757454\pi$
$158$	0	0
$159$	1.51392	0.120062
$160$	0	0
$161$	−4.71484	−0.371582
$162$	0	0
$163$	9.67728	0.757983	0.378992	−	0.925400i	$-0.376271\pi$
$163$	9.67728	0.757983	0.378992	+	0.925400i	$0.376271\pi$
$164$	0	0
$165$	7.39395	0.575618
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.9829	−0.998681
$170$	0	0
$171$	2.49495	0.190793
$172$	0	0
$173$	−14.3167	−1.08848	−0.544240	−	0.838929i	$-0.683182\pi$
$173$	−14.3167	−1.08848	−0.544240	+	0.838929i	$0.683182\pi$
$174$	0	0
$175$	−5.49722	−0.415551
$176$	0	0
$177$	−6.58556	−0.495001
$178$	0	0
$179$	15.6388	1.16890	0.584449	−	0.811430i	$-0.301311\pi$
$179$	15.6388	1.16890	0.584449	+	0.811430i	$0.301311\pi$
$180$	0	0
$181$	−2.20115	−0.163610	−0.0818051	−	0.996648i	$-0.526069\pi$
$181$	−2.20115	−0.163610	−0.0818051	+	0.996648i	$0.526069\pi$
$182$	0	0
$183$	3.25671	0.240743
$184$	0	0
$185$	−24.7489	−1.81958
$186$	0	0
$187$	14.3921	1.05245
$188$	0	0
$189$	2.29630	0.167031
$190$	0	0
$191$	16.9130	1.22378	0.611891	−	0.790942i	$-0.290409\pi$
$191$	16.9130	1.22378	0.611891	+	0.790942i	$0.290409\pi$
$192$	0	0
$193$	−0.243254	−0.0175098	−0.00875491	−	0.999962i	$-0.502787\pi$
$193$	−0.243254	−0.0175098	−0.00875491	+	0.999962i	$0.502787\pi$
$194$	0	0
$195$	−0.356000	−0.0254937
$196$	0	0
$197$	−8.16812	−0.581954	−0.290977	−	0.956730i	$-0.593980\pi$
$197$	−8.16812	−0.581954	−0.290977	+	0.956730i	$0.593980\pi$
$198$	0	0
$199$	2.34023	0.165894	0.0829471	−	0.996554i	$-0.473567\pi$
$199$	2.34023	0.165894	0.0829471	+	0.996554i	$0.473567\pi$
$200$	0	0
$201$	11.6688	0.823052
$202$	0	0
$203$	13.4558	0.944411
$204$	0	0
$205$	23.8832	1.66807
$206$	0	0
$207$	2.05324	0.142710
$208$	0	0
$209$	6.78421	0.469274
$210$	0	0
$211$	8.03191	0.552939	0.276470	−	0.961023i	$-0.410835\pi$
$211$	8.03191	0.552939	0.276470	+	0.961023i	$0.410835\pi$
$212$	0	0
$213$	10.5813	0.725017
$214$	0	0
$215$	19.4408	1.32585
$216$	0	0
$217$	−0.636818	−0.0432301
$218$	0	0
$219$	13.9938	0.945614
$220$	0	0
$221$	−0.692944	−0.0466124
$222$	0	0
$223$	−21.4275	−1.43489	−0.717447	−	0.696613i	$-0.754690\pi$
$223$	−21.4275	−1.43489	−0.717447	+	0.696613i	$0.754690\pi$
$224$	0	0
$225$	2.39395	0.159597
$226$	0	0
$227$	−1.04560	−0.0693988	−0.0346994	−	0.999398i	$-0.511047\pi$
$227$	−1.04560	−0.0693988	−0.0346994	+	0.999398i	$0.511047\pi$
$228$	0	0
$229$	−14.0173	−0.926287	−0.463144	−	0.886283i	$-0.653279\pi$
$229$	−14.0173	−0.926287	−0.463144	+	0.886283i	$0.653279\pi$
$230$	0	0
$231$	6.24405	0.410829
$232$	0	0
$233$	9.82759	0.643827	0.321913	−	0.946769i	$-0.395674\pi$
$233$	9.82759	0.643827	0.321913	+	0.946769i	$0.395674\pi$
$234$	0	0
$235$	3.17104	0.206856
$236$	0	0
$237$	1.32793	0.0862582
$238$	0	0
$239$	2.39155	0.154696	0.0773482	−	0.997004i	$-0.475355\pi$
$239$	2.39155	0.154696	0.0773482	+	0.997004i	$0.475355\pi$
$240$	0	0
$241$	9.59329	0.617958	0.308979	−	0.951069i	$-0.400013\pi$
$241$	9.59329	0.617958	0.308979	+	0.951069i	$0.400013\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.69606	0.300020
$246$	0	0
$247$	−0.326643	−0.0207838
$248$	0	0
$249$	−15.3931	−0.975497
$250$	0	0
$251$	−16.4626	−1.03911	−0.519555	−	0.854437i	$-0.673902\pi$
$251$	−16.4626	−1.03911	−0.519555	+	0.854437i	$0.673902\pi$
$252$	0	0
$253$	5.58312	0.351008
$254$	0	0
$255$	14.3921	0.901268
$256$	0	0
$257$	−24.4004	−1.52205	−0.761027	−	0.648720i	$-0.775305\pi$
$257$	−24.4004	−1.52205	−0.761027	+	0.648720i	$0.775305\pi$
$258$	0	0
$259$	−20.9000	−1.29866
$260$	0	0
$261$	−5.85977	−0.362711
$262$	0	0
$263$	9.24379	0.569997	0.284998	−	0.958528i	$-0.408007\pi$
$263$	9.24379	0.569997	0.284998	+	0.958528i	$0.408007\pi$
$264$	0	0
$265$	4.11663	0.252882
$266$	0	0
$267$	0.650613	0.0398169
$268$	0	0
$269$	−26.6107	−1.62248	−0.811242	−	0.584710i	$-0.801208\pi$
$269$	−26.6107	−1.62248	−0.811242	+	0.584710i	$0.801208\pi$
$270$	0	0
$271$	−25.3880	−1.54221	−0.771104	−	0.636709i	$-0.780295\pi$
$271$	−25.3880	−1.54221	−0.771104	+	0.636709i	$0.780295\pi$
$272$	0	0
$273$	−0.300636	−0.0181953
$274$	0	0
$275$	6.50958	0.392543
$276$	0	0
$277$	−2.72565	−0.163768	−0.0818842	−	0.996642i	$-0.526094\pi$
$277$	−2.72565	−0.163768	−0.0818842	+	0.996642i	$0.526094\pi$
$278$	0	0
$279$	0.277324	0.0166029
$280$	0	0
$281$	7.66109	0.457022	0.228511	−	0.973541i	$-0.426614\pi$
$281$	7.66109	0.457022	0.228511	+	0.973541i	$0.426614\pi$
$282$	0	0
$283$	27.5196	1.63587	0.817934	−	0.575312i	$-0.195119\pi$
$283$	27.5196	1.63587	0.817934	+	0.575312i	$0.195119\pi$
$284$	0	0
$285$	6.78421	0.401862
$286$	0	0
$287$	20.1689	1.19053
$288$	0	0
$289$	11.0138	0.647869
$290$	0	0
$291$	16.5435	0.969795
$292$	0	0
$293$	−3.58746	−0.209582	−0.104791	−	0.994494i	$-0.533417\pi$
$293$	−3.58746	−0.209582	−0.104791	+	0.994494i	$0.533417\pi$
$294$	0	0
$295$	−17.9073	−1.04261
$296$	0	0
$297$	−2.71918	−0.157783
$298$	0	0
$299$	−0.268814	−0.0155459
$300$	0	0
$301$	16.4174	0.946285
$302$	0	0
$303$	3.96140	0.227577
$304$	0	0
$305$	8.85558	0.507069
$306$	0	0
$307$	3.27758	0.187062	0.0935308	−	0.995616i	$-0.470185\pi$
$307$	3.27758	0.187062	0.0935308	+	0.995616i	$0.470185\pi$
$308$	0	0
$309$	3.85793	0.219470
$310$	0	0
$311$	−24.2430	−1.37469	−0.687347	−	0.726329i	$-0.741225\pi$
$311$	−24.2430	−1.37469	−0.687347	+	0.726329i	$0.741225\pi$
$312$	0	0
$313$	−5.06956	−0.286548	−0.143274	−	0.989683i	$-0.545763\pi$
$313$	−5.06956	−0.286548	−0.143274	+	0.989683i	$0.545763\pi$
$314$	0	0
$315$	6.24405	0.351813
$316$	0	0
$317$	−19.5721	−1.09928	−0.549639	−	0.835403i	$-0.685235\pi$
$317$	−19.5721	−1.09928	−0.549639	+	0.835403i	$0.685235\pi$
$318$	0	0
$319$	−15.9338	−0.892120
$320$	0	0
$321$	−13.1188	−0.732220
$322$	0	0
$323$	13.2053	0.734760
$324$	0	0
$325$	−0.313420	−0.0173854
$326$	0	0
$327$	−2.33140	−0.128927
$328$	0	0
$329$	2.67789	0.147637
$330$	0	0
$331$	−21.8103	−1.19880	−0.599401	−	0.800449i	$-0.704595\pi$
$331$	−21.8103	−1.19880	−0.599401	+	0.800449i	$0.704595\pi$
$332$	0	0
$333$	9.10161	0.498765
$334$	0	0
$335$	31.7295	1.73357
$336$	0	0
$337$	12.4451	0.677930	0.338965	−	0.940799i	$-0.389923\pi$
$337$	12.4451	0.677930	0.338965	+	0.940799i	$0.389923\pi$
$338$	0	0
$339$	−13.1549	−0.714476
$340$	0	0
$341$	0.754094	0.0408365
$342$	0	0
$343$	20.0398	1.08205
$344$	0	0
$345$	5.58312	0.300585
$346$	0	0
$347$	24.4597	1.31306	0.656532	−	0.754298i	$-0.272023\pi$
$347$	24.4597	1.31306	0.656532	+	0.754298i	$0.272023\pi$
$348$	0	0
$349$	−10.6692	−0.571107	−0.285553	−	0.958363i	$-0.592177\pi$
$349$	−10.6692	−0.571107	−0.285553	+	0.958363i	$0.592177\pi$
$350$	0	0
$351$	0.130922	0.00698809
$352$	0	0
$353$	−23.5466	−1.25326	−0.626629	−	0.779318i	$-0.715566\pi$
$353$	−23.5466	−1.25326	−0.626629	+	0.779318i	$0.715566\pi$
$354$	0	0
$355$	28.7724	1.52708
$356$	0	0
$357$	12.1539	0.643250
$358$	0	0
$359$	−16.1523	−0.852484	−0.426242	−	0.904609i	$-0.640163\pi$
$359$	−16.1523	−0.852484	−0.426242	+	0.904609i	$0.640163\pi$
$360$	0	0
$361$	−12.7752	−0.672381
$362$	0	0
$363$	3.60605	0.189269
$364$	0	0
$365$	38.0517	1.99172
$366$	0	0
$367$	3.80007	0.198362	0.0991810	−	0.995069i	$-0.468378\pi$
$367$	3.80007	0.198362	0.0991810	+	0.995069i	$0.468378\pi$
$368$	0	0
$369$	−8.78322	−0.457236
$370$	0	0
$371$	3.47641	0.180486
$372$	0	0
$373$	19.5124	1.01031	0.505156	−	0.863028i	$-0.331435\pi$
$373$	19.5124	1.01031	0.505156	+	0.863028i	$0.331435\pi$
$374$	0	0
$375$	−7.08633	−0.365936
$376$	0	0
$377$	0.767172	0.0395113
$378$	0	0
$379$	−0.918888	−0.0472001	−0.0236000	−	0.999721i	$-0.507513\pi$
$379$	−0.918888	−0.0472001	−0.0236000	+	0.999721i	$0.507513\pi$
$380$	0	0
$381$	3.91471	0.200557
$382$	0	0
$383$	17.2962	0.883792	0.441896	−	0.897066i	$-0.354306\pi$
$383$	17.2962	0.883792	0.441896	+	0.897066i	$0.354306\pi$
$384$	0	0
$385$	16.9787	0.865315
$386$	0	0
$387$	−7.14952	−0.363431
$388$	0	0
$389$	−31.7218	−1.60836	−0.804181	−	0.594385i	$-0.797396\pi$
$389$	−31.7218	−1.60836	−0.804181	+	0.594385i	$0.797396\pi$
$390$	0	0
$391$	10.8674	0.549587
$392$	0	0
$393$	8.22025	0.414657
$394$	0	0
$395$	3.61088	0.181683
$396$	0	0
$397$	−4.53970	−0.227841	−0.113920	−	0.993490i	$-0.536341\pi$
$397$	−4.53970	−0.227841	−0.113920	+	0.993490i	$0.536341\pi$
$398$	0	0
$399$	5.72914	0.286816
$400$	0	0
$401$	−5.11757	−0.255559	−0.127780	−	0.991803i	$-0.540785\pi$
$401$	−5.11757	−0.255559	−0.127780	+	0.991803i	$0.540785\pi$
$402$	0	0
$403$	−0.0363078	−0.00180862
$404$	0	0
$405$	−2.71918	−0.135117
$406$	0	0
$407$	24.7489	1.22676
$408$	0	0
$409$	−9.84326	−0.486718	−0.243359	−	0.969936i	$-0.578249\pi$
$409$	−9.84326	−0.486718	−0.243359	+	0.969936i	$0.578249\pi$
$410$	0	0
$411$	−9.47880	−0.467555
$412$	0	0
$413$	−15.1224	−0.744126
$414$	0	0
$415$	−41.8566	−2.05466
$416$	0	0
$417$	−5.15967	−0.252670
$418$	0	0
$419$	22.5715	1.10269	0.551345	−	0.834277i	$-0.314115\pi$
$419$	22.5715	1.10269	0.551345	+	0.834277i	$0.314115\pi$
$420$	0	0
$421$	6.15702	0.300075	0.150037	−	0.988680i	$-0.452061\pi$
$421$	6.15702	0.300075	0.150037	+	0.988680i	$0.452061\pi$
$422$	0	0
$423$	−1.16618	−0.0567014
$424$	0	0
$425$	12.6707	0.614619
$426$	0	0
$427$	7.47837	0.361904
$428$	0	0
$429$	0.356000	0.0171879
$430$	0	0
$431$	−0.521526	−0.0251210	−0.0125605	−	0.999921i	$-0.503998\pi$
$431$	−0.521526	−0.0251210	−0.0125605	+	0.999921i	$0.503998\pi$
$432$	0	0
$433$	6.11364	0.293803	0.146901	−	0.989151i	$-0.453070\pi$
$433$	6.11364	0.293803	0.146901	+	0.989151i	$0.453070\pi$
$434$	0	0
$435$	−15.9338	−0.763966
$436$	0	0
$437$	5.12272	0.245053
$438$	0	0
$439$	−13.0116	−0.621012	−0.310506	−	0.950571i	$-0.600498\pi$
$439$	−13.0116	−0.621012	−0.310506	+	0.950571i	$0.600498\pi$
$440$	0	0
$441$	−1.72701	−0.0822387
$442$	0	0
$443$	4.28260	0.203473	0.101736	−	0.994811i	$-0.467560\pi$
$443$	4.28260	0.203473	0.101736	+	0.994811i	$0.467560\pi$
$444$	0	0
$445$	1.76914	0.0838651
$446$	0	0
$447$	13.5693	0.641807
$448$	0	0
$449$	24.3609	1.14966	0.574831	−	0.818272i	$-0.305068\pi$
$449$	24.3609	1.14966	0.574831	+	0.818272i	$0.305068\pi$
$450$	0	0
$451$	−23.8832	−1.12461
$452$	0	0
$453$	21.5430	1.01218
$454$	0	0
$455$	−0.817483	−0.0383242
$456$	0	0
$457$	−6.69421	−0.313142	−0.156571	−	0.987667i	$-0.550044\pi$
$457$	−6.69421	−0.313142	−0.156571	+	0.987667i	$0.550044\pi$
$458$	0	0
$459$	−5.29280	−0.247047
$460$	0	0
$461$	−2.94810	−0.137307	−0.0686533	−	0.997641i	$-0.521870\pi$
$461$	−2.94810	−0.137307	−0.0686533	+	0.997641i	$0.521870\pi$
$462$	0	0
$463$	42.4252	1.97167	0.985833	−	0.167730i	$-0.0536436\pi$
$463$	42.4252	1.97167	0.985833	+	0.167730i	$0.0536436\pi$
$464$	0	0
$465$	0.754094	0.0349703
$466$	0	0
$467$	11.7744	0.544853	0.272427	−	0.962177i	$-0.412174\pi$
$467$	11.7744	0.544853	0.272427	+	0.962177i	$0.412174\pi$
$468$	0	0
$469$	26.7950	1.23728
$470$	0	0
$471$	18.1301	0.835391
$472$	0	0
$473$	−19.4408	−0.893891
$474$	0	0
$475$	5.97277	0.274050
$476$	0	0
$477$	−1.51392	−0.0693177
$478$	0	0
$479$	7.48528	0.342011	0.171006	−	0.985270i	$-0.445298\pi$
$479$	7.48528	0.342011	0.171006	+	0.985270i	$0.445298\pi$
$480$	0	0
$481$	−1.19160	−0.0543323
$482$	0	0
$483$	4.71484	0.214533
$484$	0	0
$485$	44.9847	2.04265
$486$	0	0
$487$	32.7190	1.48264	0.741319	−	0.671153i	$-0.234200\pi$
$487$	32.7190	1.48264	0.741319	+	0.671153i	$0.234200\pi$
$488$	0	0
$489$	−9.67728	−0.437622
$490$	0	0
$491$	−15.0119	−0.677477	−0.338738	−	0.940881i	$-0.610000\pi$
$491$	−15.0119	−0.677477	−0.338738	+	0.940881i	$0.610000\pi$
$492$	0	0
$493$	−31.0146	−1.39683
$494$	0	0
$495$	−7.39395	−0.332333
$496$	0	0
$497$	24.2978	1.08990
$498$	0	0
$499$	25.5908	1.14560	0.572801	−	0.819694i	$-0.305857\pi$
$499$	25.5908	1.14560	0.572801	+	0.819694i	$0.305857\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−4.56804	−0.203679	−0.101840	−	0.994801i	$-0.532473\pi$
$503$	−4.56804	−0.203679	−0.101840	+	0.994801i	$0.532473\pi$
$504$	0	0
$505$	10.7718	0.479337
$506$	0	0
$507$	12.9829	0.576589
$508$	0	0
$509$	−19.6850	−0.872524	−0.436262	−	0.899820i	$-0.643698\pi$
$509$	−19.6850	−0.872524	−0.436262	+	0.899820i	$0.643698\pi$
$510$	0	0
$511$	32.1340	1.42152
$512$	0	0
$513$	−2.49495	−0.110155
$514$	0	0
$515$	10.4904	0.462263
$516$	0	0
$517$	−3.17104	−0.139462
$518$	0	0
$519$	14.3167	0.628435
$520$	0	0
$521$	25.8709	1.13343	0.566713	−	0.823915i	$-0.308215\pi$
$521$	25.8709	1.13343	0.566713	+	0.823915i	$0.308215\pi$
$522$	0	0
$523$	9.33225	0.408071	0.204035	−	0.978964i	$-0.434594\pi$
$523$	9.33225	0.408071	0.204035	+	0.978964i	$0.434594\pi$
$524$	0	0
$525$	5.49722	0.239918
$526$	0	0
$527$	1.46782	0.0639393
$528$	0	0
$529$	−18.7842	−0.816705
$530$	0	0
$531$	6.58556	0.285789
$532$	0	0
$533$	1.14992	0.0498084
$534$	0	0
$535$	−35.6724	−1.54225
$536$	0	0
$537$	−15.6388	−0.674864
$538$	0	0
$539$	−4.69606	−0.202274
$540$	0	0
$541$	8.57617	0.368718	0.184359	−	0.982859i	$-0.440979\pi$
$541$	8.57617	0.368718	0.184359	+	0.982859i	$0.440979\pi$
$542$	0	0
$543$	2.20115	0.0944604
$544$	0	0
$545$	−6.33951	−0.271555
$546$	0	0
$547$	−15.3088	−0.654556	−0.327278	−	0.944928i	$-0.606131\pi$
$547$	−15.3088	−0.654556	−0.327278	+	0.944928i	$0.606131\pi$
$548$	0	0
$549$	−3.25671	−0.138993
$550$	0	0
$551$	−14.6198	−0.622825
$552$	0	0
$553$	3.04932	0.129670
$554$	0	0
$555$	24.7489	1.05053
$556$	0	0
$557$	−8.22455	−0.348486	−0.174243	−	0.984703i	$-0.555748\pi$
$557$	−8.22455	−0.348486	−0.174243	+	0.984703i	$0.555748\pi$
$558$	0	0
$559$	0.936028	0.0395898
$560$	0	0
$561$	−14.3921	−0.607635
$562$	0	0
$563$	−21.3271	−0.898830	−0.449415	−	0.893323i	$-0.648368\pi$
$563$	−21.3271	−0.898830	−0.449415	+	0.893323i	$0.648368\pi$
$564$	0	0
$565$	−35.7705	−1.50488
$566$	0	0
$567$	−2.29630	−0.0964355
$568$	0	0
$569$	10.5531	0.442407	0.221204	−	0.975228i	$-0.429001\pi$
$569$	10.5531	0.442407	0.221204	+	0.975228i	$0.429001\pi$
$570$	0	0
$571$	6.43677	0.269371	0.134685	−	0.990888i	$-0.456998\pi$
$571$	6.43677	0.269371	0.134685	+	0.990888i	$0.456998\pi$
$572$	0	0
$573$	−16.9130	−0.706551
$574$	0	0
$575$	4.91535	0.204984
$576$	0	0
$577$	−37.1930	−1.54836	−0.774182	−	0.632963i	$-0.781839\pi$
$577$	−37.1930	−1.54836	−0.774182	+	0.632963i	$0.781839\pi$
$578$	0	0
$579$	0.243254	0.0101093
$580$	0	0
$581$	−35.3471	−1.46645
$582$	0	0
$583$	−4.11663	−0.170493
$584$	0	0
$585$	0.356000	0.0147188
$586$	0	0
$587$	43.7880	1.80733	0.903663	−	0.428245i	$-0.140868\pi$
$587$	43.7880	1.80733	0.903663	+	0.428245i	$0.140868\pi$
$588$	0	0
$589$	0.691908	0.0285096
$590$	0	0
$591$	8.16812	0.335992
$592$	0	0
$593$	−29.7756	−1.22274	−0.611369	−	0.791346i	$-0.709381\pi$
$593$	−29.7756	−1.22274	−0.611369	+	0.791346i	$0.709381\pi$
$594$	0	0
$595$	33.0485	1.35486
$596$	0	0
$597$	−2.34023	−0.0957791
$598$	0	0
$599$	−18.7131	−0.764595	−0.382298	−	0.924039i	$-0.624867\pi$
$599$	−18.7131	−0.764595	−0.382298	+	0.924039i	$0.624867\pi$
$600$	0	0
$601$	15.6102	0.636753	0.318376	−	0.947964i	$-0.396862\pi$
$601$	15.6102	0.636753	0.318376	+	0.947964i	$0.396862\pi$
$602$	0	0
$603$	−11.6688	−0.475189
$604$	0	0
$605$	9.80551	0.398651
$606$	0	0
$607$	−27.5079	−1.11651	−0.558256	−	0.829669i	$-0.688529\pi$
$607$	−27.5079	−1.11651	−0.558256	+	0.829669i	$0.688529\pi$
$608$	0	0
$609$	−13.4558	−0.545256
$610$	0	0
$611$	0.152678	0.00617668
$612$	0	0
$613$	29.9599	1.21007	0.605034	−	0.796199i	$-0.293159\pi$
$613$	29.9599	1.21007	0.605034	+	0.796199i	$0.293159\pi$
$614$	0	0
$615$	−23.8832	−0.963062
$616$	0	0
$617$	25.1911	1.01416	0.507078	−	0.861900i	$-0.330726\pi$
$617$	25.1911	1.01416	0.507078	+	0.861900i	$0.330726\pi$
$618$	0	0
$619$	−12.6419	−0.508120	−0.254060	−	0.967188i	$-0.581766\pi$
$619$	−12.6419	−0.508120	−0.254060	+	0.967188i	$0.581766\pi$
$620$	0	0
$621$	−2.05324	−0.0823936
$622$	0	0
$623$	1.49400	0.0598559
$624$	0	0
$625$	−31.2388	−1.24955
$626$	0	0
$627$	−6.78421	−0.270935
$628$	0	0
$629$	48.1731	1.92079
$630$	0	0
$631$	−46.9107	−1.86749	−0.933744	−	0.357942i	$-0.883478\pi$
$631$	−46.9107	−1.86749	−0.933744	+	0.357942i	$0.883478\pi$
$632$	0	0
$633$	−8.03191	−0.319240
$634$	0	0
$635$	10.6448	0.422426
$636$	0	0
$637$	0.226104	0.00895856
$638$	0	0
$639$	−10.5813	−0.418589
$640$	0	0
$641$	−7.50494	−0.296427	−0.148214	−	0.988955i	$-0.547352\pi$
$641$	−7.50494	−0.296427	−0.148214	+	0.988955i	$0.547352\pi$
$642$	0	0
$643$	28.5667	1.12656	0.563281	−	0.826266i	$-0.309539\pi$
$643$	28.5667	1.12656	0.563281	+	0.826266i	$0.309539\pi$
$644$	0	0
$645$	−19.4408	−0.765483
$646$	0	0
$647$	−12.0671	−0.474405	−0.237202	−	0.971460i	$-0.576230\pi$
$647$	−12.0671	−0.474405	−0.237202	+	0.971460i	$0.576230\pi$
$648$	0	0
$649$	17.9073	0.702925
$650$	0	0
$651$	0.636818	0.0249589
$652$	0	0
$653$	10.8332	0.423934	0.211967	−	0.977277i	$-0.432013\pi$
$653$	10.8332	0.423934	0.211967	+	0.977277i	$0.432013\pi$
$654$	0	0
$655$	22.3524	0.873379
$656$	0	0
$657$	−13.9938	−0.545951
$658$	0	0
$659$	17.3077	0.674211	0.337106	−	0.941467i	$-0.390552\pi$
$659$	17.3077	0.674211	0.337106	+	0.941467i	$0.390552\pi$
$660$	0	0
$661$	9.65536	0.375550	0.187775	−	0.982212i	$-0.439872\pi$
$661$	9.65536	0.375550	0.187775	+	0.982212i	$0.439872\pi$
$662$	0	0
$663$	0.692944	0.0269117
$664$	0	0
$665$	15.5786	0.604111
$666$	0	0
$667$	−12.0315	−0.465861
$668$	0	0
$669$	21.4275	0.828437
$670$	0	0
$671$	−8.85558	−0.341866
$672$	0	0
$673$	−2.27136	−0.0875546	−0.0437773	−	0.999041i	$-0.513939\pi$
$673$	−2.27136	−0.0875546	−0.0437773	+	0.999041i	$0.513939\pi$
$674$	0	0
$675$	−2.39395	−0.0921432
$676$	0	0
$677$	42.4522	1.63157	0.815785	−	0.578356i	$-0.196305\pi$
$677$	42.4522	1.63157	0.815785	+	0.578356i	$0.196305\pi$
$678$	0	0
$679$	37.9887	1.45787
$680$	0	0
$681$	1.04560	0.0400674
$682$	0	0
$683$	26.2727	1.00530	0.502649	−	0.864491i	$-0.332359\pi$
$683$	26.2727	1.00530	0.502649	+	0.864491i	$0.332359\pi$
$684$	0	0
$685$	−25.7746	−0.984796
$686$	0	0
$687$	14.0173	0.534792
$688$	0	0
$689$	0.198205	0.00755102
$690$	0	0
$691$	4.24914	0.161645	0.0808225	−	0.996729i	$-0.474245\pi$
$691$	4.24914	0.161645	0.0808225	+	0.996729i	$0.474245\pi$
$692$	0	0
$693$	−6.24405	−0.237192
$694$	0	0
$695$	−14.0301	−0.532192
$696$	0	0
$697$	−46.4879	−1.76085
$698$	0	0
$699$	−9.82759	−0.371714
$700$	0	0
$701$	15.8974	0.600437	0.300219	−	0.953870i	$-0.402940\pi$
$701$	15.8974	0.600437	0.300219	+	0.953870i	$0.402940\pi$
$702$	0	0
$703$	22.7080	0.856450
$704$	0	0
$705$	−3.17104	−0.119428
$706$	0	0
$707$	9.09656	0.342111
$708$	0	0
$709$	38.6638	1.45205	0.726025	−	0.687668i	$-0.241366\pi$
$709$	38.6638	1.45205	0.726025	+	0.687668i	$0.241366\pi$
$710$	0	0
$711$	−1.32793	−0.0498012
$712$	0	0
$713$	0.569412	0.0213246
$714$	0	0
$715$	0.968029	0.0362022
$716$	0	0
$717$	−2.39155	−0.0893140
$718$	0	0
$719$	−22.7418	−0.848126	−0.424063	−	0.905633i	$-0.639397\pi$
$719$	−22.7418	−0.848126	−0.424063	+	0.905633i	$0.639397\pi$
$720$	0	0
$721$	8.85897	0.329925
$722$	0	0
$723$	−9.59329	−0.356778
$724$	0	0
$725$	−14.0280	−0.520986
$726$	0	0
$727$	−37.0957	−1.37580	−0.687902	−	0.725803i	$-0.741468\pi$
$727$	−37.0957	−1.37580	−0.687902	+	0.725803i	$0.741468\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−37.8410	−1.39960
$732$	0	0
$733$	−28.6271	−1.05737	−0.528683	−	0.848819i	$-0.677314\pi$
$733$	−28.6271	−1.05737	−0.528683	+	0.848819i	$0.677314\pi$
$734$	0	0
$735$	−4.69606	−0.173217
$736$	0	0
$737$	−31.7295	−1.16877
$738$	0	0
$739$	1.48706	0.0547024	0.0273512	−	0.999626i	$-0.491293\pi$
$739$	1.48706	0.0547024	0.0273512	+	0.999626i	$0.491293\pi$
$740$	0	0
$741$	0.326643	0.0119995
$742$	0	0
$743$	−29.0313	−1.06506	−0.532528	−	0.846413i	$-0.678758\pi$
$743$	−29.0313	−1.06506	−0.532528	+	0.846413i	$0.678758\pi$
$744$	0	0
$745$	36.8975	1.35182
$746$	0	0
$747$	15.3931	0.563204
$748$	0	0
$749$	−30.1247	−1.10073
$750$	0	0
$751$	−44.7534	−1.63307	−0.816537	−	0.577293i	$-0.804109\pi$
$751$	−44.7534	−1.63307	−0.816537	+	0.577293i	$0.804109\pi$
$752$	0	0
$753$	16.4626	0.599931
$754$	0	0
$755$	58.5794	2.13192
$756$	0	0
$757$	24.5663	0.892878	0.446439	−	0.894814i	$-0.352692\pi$
$757$	24.5663	0.892878	0.446439	+	0.894814i	$0.352692\pi$
$758$	0	0
$759$	−5.58312	−0.202655
$760$	0	0
$761$	−16.5622	−0.600379	−0.300189	−	0.953880i	$-0.597050\pi$
$761$	−16.5622	−0.600379	−0.300189	+	0.953880i	$0.597050\pi$
$762$	0	0
$763$	−5.35360	−0.193813
$764$	0	0
$765$	−14.3921	−0.520347
$766$	0	0
$767$	−0.862194	−0.0311320
$768$	0	0
$769$	−13.4445	−0.484820	−0.242410	−	0.970174i	$-0.577938\pi$
$769$	−13.4445	−0.484820	−0.242410	+	0.970174i	$0.577938\pi$
$770$	0	0
$771$	24.4004	0.878758
$772$	0	0
$773$	52.2257	1.87843	0.939214	−	0.343333i	$-0.111556\pi$
$773$	52.2257	1.87843	0.939214	+	0.343333i	$0.111556\pi$
$774$	0	0
$775$	0.663899	0.0238480
$776$	0	0
$777$	20.9000	0.749784
$778$	0	0
$779$	−21.9137	−0.785138
$780$	0	0
$781$	−28.7724	−1.02956
$782$	0	0
$783$	5.85977	0.209411
$784$	0	0
$785$	49.2990	1.75956
$786$	0	0
$787$	−12.8705	−0.458782	−0.229391	−	0.973334i	$-0.573674\pi$
$787$	−12.8705	−0.458782	−0.229391	+	0.973334i	$0.573674\pi$
$788$	0	0
$789$	−9.24379	−0.329088
$790$	0	0
$791$	−30.2076	−1.07406
$792$	0	0
$793$	0.426374	0.0151410
$794$	0	0
$795$	−4.11663	−0.146002
$796$	0	0
$797$	−6.46623	−0.229046	−0.114523	−	0.993421i	$-0.536534\pi$
$797$	−6.46623	−0.229046	−0.114523	+	0.993421i	$0.536534\pi$
$798$	0	0
$799$	−6.17234	−0.218362
$800$	0	0
$801$	−0.650613	−0.0229883
$802$	0	0
$803$	−38.0517	−1.34282
$804$	0	0
$805$	12.8205	0.451864
$806$	0	0
$807$	26.6107	0.936742
$808$	0	0
$809$	−15.0052	−0.527554	−0.263777	−	0.964584i	$-0.584968\pi$
$809$	−15.0052	−0.527554	−0.263777	+	0.964584i	$0.584968\pi$
$810$	0	0
$811$	20.6709	0.725852	0.362926	−	0.931818i	$-0.381778\pi$
$811$	20.6709	0.725852	0.362926	+	0.931818i	$0.381778\pi$
$812$	0	0
$813$	25.3880	0.890394
$814$	0	0
$815$	−26.3143	−0.921749
$816$	0	0
$817$	−17.8377	−0.624061
$818$	0	0
$819$	0.300636	0.0105051
$820$	0	0
$821$	−27.2144	−0.949787	−0.474894	−	0.880043i	$-0.657514\pi$
$821$	−27.2144	−0.949787	−0.474894	+	0.880043i	$0.657514\pi$
$822$	0	0
$823$	−38.5039	−1.34216	−0.671081	−	0.741384i	$-0.734170\pi$
$823$	−38.5039	−1.34216	−0.671081	+	0.741384i	$0.734170\pi$
$824$	0	0
$825$	−6.50958	−0.226635
$826$	0	0
$827$	−38.3481	−1.33349	−0.666747	−	0.745284i	$-0.732314\pi$
$827$	−38.3481	−1.33349	−0.666747	+	0.745284i	$0.732314\pi$
$828$	0	0
$829$	0.845959	0.0293814	0.0146907	−	0.999892i	$-0.495324\pi$
$829$	0.845959	0.0293814	0.0146907	+	0.999892i	$0.495324\pi$
$830$	0	0
$831$	2.72565	0.0945518
$832$	0	0
$833$	−9.14074	−0.316708
$834$	0	0
$835$	−2.71918	−0.0941012
$836$	0	0
$837$	−0.277324	−0.00958572
$838$	0	0
$839$	21.2959	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$839$	21.2959	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$840$	0	0
$841$	5.33689	0.184031
$842$	0	0
$843$	−7.66109	−0.263862
$844$	0	0
$845$	35.3028	1.21445
$846$	0	0
$847$	8.28057	0.284524
$848$	0	0
$849$	−27.5196	−0.944469
$850$	0	0
$851$	18.6878	0.640608
$852$	0	0
$853$	−5.56671	−0.190601	−0.0953003	−	0.995449i	$-0.530381\pi$
$853$	−5.56671	−0.190601	−0.0953003	+	0.995449i	$0.530381\pi$
$854$	0	0
$855$	−6.78421	−0.232015
$856$	0	0
$857$	−8.55275	−0.292156	−0.146078	−	0.989273i	$-0.546665\pi$
$857$	−8.55275	−0.292156	−0.146078	+	0.989273i	$0.546665\pi$
$858$	0	0
$859$	16.4240	0.560378	0.280189	−	0.959945i	$-0.409603\pi$
$859$	16.4240	0.560378	0.280189	+	0.959945i	$0.409603\pi$
$860$	0	0
$861$	−20.1689	−0.687354
$862$	0	0
$863$	33.2268	1.13105	0.565526	−	0.824730i	$-0.308673\pi$
$863$	33.2268	1.13105	0.565526	+	0.824730i	$0.308673\pi$
$864$	0	0
$865$	38.9298	1.32365
$866$	0	0
$867$	−11.0138	−0.374047
$868$	0	0
$869$	−3.61088	−0.122491
$870$	0	0
$871$	1.52770	0.0517640
$872$	0	0
$873$	−16.5435	−0.559912
$874$	0	0
$875$	−16.2723	−0.550105
$876$	0	0
$877$	−4.82148	−0.162810	−0.0814049	−	0.996681i	$-0.525941\pi$
$877$	−4.82148	−0.162810	−0.0814049	+	0.996681i	$0.525941\pi$
$878$	0	0
$879$	3.58746	0.121002
$880$	0	0
$881$	−15.4398	−0.520180	−0.260090	−	0.965584i	$-0.583752\pi$
$881$	−15.4398	−0.520180	−0.260090	+	0.965584i	$0.583752\pi$
$882$	0	0
$883$	7.82614	0.263371	0.131685	−	0.991292i	$-0.457961\pi$
$883$	7.82614	0.263371	0.131685	+	0.991292i	$0.457961\pi$
$884$	0	0
$885$	17.9073	0.601949
$886$	0	0
$887$	−22.8275	−0.766474	−0.383237	−	0.923650i	$-0.625191\pi$
$887$	−22.8275	−0.766474	−0.383237	+	0.923650i	$0.625191\pi$
$888$	0	0
$889$	8.98934	0.301493
$890$	0	0
$891$	2.71918	0.0910960
$892$	0	0
$893$	−2.90954	−0.0973642
$894$	0	0
$895$	−42.5247	−1.42145
$896$	0	0
$897$	0.268814	0.00897542
$898$	0	0
$899$	−1.62505	−0.0541986
$900$	0	0
$901$	−8.01289	−0.266948
$902$	0	0
$903$	−16.4174	−0.546338
$904$	0	0
$905$	5.98533	0.198959
$906$	0	0
$907$	−35.9162	−1.19258	−0.596289	−	0.802770i	$-0.703359\pi$
$907$	−35.9162	−1.19258	−0.596289	+	0.802770i	$0.703359\pi$
$908$	0	0
$909$	−3.96140	−0.131391
$910$	0	0
$911$	16.2051	0.536899	0.268449	−	0.963294i	$-0.413489\pi$
$911$	16.2051	0.536899	0.268449	+	0.963294i	$0.413489\pi$
$912$	0	0
$913$	41.8566	1.38525
$914$	0	0
$915$	−8.85558	−0.292756
$916$	0	0
$917$	18.8762	0.623346
$918$	0	0
$919$	44.6831	1.47396	0.736979	−	0.675915i	$-0.236251\pi$
$919$	44.6831	1.47396	0.736979	+	0.675915i	$0.236251\pi$
$920$	0	0
$921$	−3.27758	−0.108000
$922$	0	0
$923$	1.38532	0.0455983
$924$	0	0
$925$	21.7888	0.716411
$926$	0	0
$927$	−3.85793	−0.126711
$928$	0	0
$929$	−45.4092	−1.48983	−0.744914	−	0.667161i	$-0.767509\pi$
$929$	−45.4092	−1.48983	−0.744914	+	0.667161i	$0.767509\pi$
$930$	0	0
$931$	−4.30880	−0.141215
$932$	0	0
$933$	24.2430	0.793680
$934$	0	0
$935$	−39.1347	−1.27984
$936$	0	0
$937$	−47.6798	−1.55763	−0.778815	−	0.627253i	$-0.784179\pi$
$937$	−47.6798	−1.55763	−0.778815	+	0.627253i	$0.784179\pi$
$938$	0	0
$939$	5.06956	0.165439
$940$	0	0
$941$	−35.3734	−1.15314	−0.576570	−	0.817048i	$-0.695609\pi$
$941$	−35.3734	−1.15314	−0.576570	+	0.817048i	$0.695609\pi$
$942$	0	0
$943$	−18.0340	−0.587269
$944$	0	0
$945$	−6.24405	−0.203119
$946$	0	0
$947$	−50.4841	−1.64051	−0.820257	−	0.571995i	$-0.806170\pi$
$947$	−50.4841	−1.64051	−0.820257	+	0.571995i	$0.806170\pi$
$948$	0	0
$949$	1.83210	0.0594723
$950$	0	0
$951$	19.5721	0.634668
$952$	0	0
$953$	25.9394	0.840259	0.420130	−	0.907464i	$-0.361985\pi$
$953$	25.9394	0.840259	0.420130	+	0.907464i	$0.361985\pi$
$954$	0	0
$955$	−45.9896	−1.48819
$956$	0	0
$957$	15.9338	0.515066
$958$	0	0
$959$	−21.7661	−0.702866
$960$	0	0
$961$	−30.9231	−0.997519
$962$	0	0
$963$	13.1188	0.422747
$964$	0	0
$965$	0.661453	0.0212929
$966$	0	0
$967$	9.50298	0.305595	0.152798	−	0.988257i	$-0.451172\pi$
$967$	9.50298	0.305595	0.152798	+	0.988257i	$0.451172\pi$
$968$	0	0
$969$	−13.2053	−0.424214
$970$	0	0
$971$	−22.0186	−0.706609	−0.353305	−	0.935508i	$-0.614942\pi$
$971$	−22.0186	−0.706609	−0.353305	+	0.935508i	$0.614942\pi$
$972$	0	0
$973$	−11.8481	−0.379834
$974$	0	0
$975$	0.313420	0.0100375
$976$	0	0
$977$	−49.7763	−1.59248	−0.796242	−	0.604978i	$-0.793182\pi$
$977$	−49.7763	−1.59248	−0.796242	+	0.604978i	$0.793182\pi$
$978$	0	0
$979$	−1.76914	−0.0565418
$980$	0	0
$981$	2.33140	0.0744360
$982$	0	0
$983$	−31.4675	−1.00366	−0.501828	−	0.864967i	$-0.667339\pi$
$983$	−31.4675	−1.00366	−0.501828	+	0.864967i	$0.667339\pi$
$984$	0	0
$985$	22.2106	0.707689
$986$	0	0
$987$	−2.67789	−0.0852381
$988$	0	0
$989$	−14.6797	−0.466786
$990$	0	0
$991$	28.2915	0.898708	0.449354	−	0.893354i	$-0.351654\pi$
$991$	28.2915	0.898708	0.449354	+	0.893354i	$0.351654\pi$
$992$	0	0
$993$	21.8103	0.692129
$994$	0	0
$995$	−6.36350	−0.201737
$996$	0	0
$997$	48.7224	1.54305	0.771527	−	0.636197i	$-0.219494\pi$
$997$	48.7224	1.54305	0.771527	+	0.636197i	$0.219494\pi$
$998$	0	0
$999$	−9.10161	−0.287962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2004.2.a.a.1.2	✓	5
3.2	odd	2		6012.2.a.e.1.4		5
4.3	odd	2		8016.2.a.t.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.a.1.2	✓	5	1.1	even	1	trivial
6012.2.a.e.1.4		5	3.2	odd	2
8016.2.a.t.1.2		5	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.71918 q^{5} -2.29630 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.71918 q^{5} -2.29630 q^{7} +1.00000 q^{9} +2.71918 q^{11} -0.130922 q^{13} +2.71918 q^{15} +5.29280 q^{17} +2.49495 q^{19} +2.29630 q^{21} +2.05324 q^{23} +2.39395 q^{25} -1.00000 q^{27} -5.85977 q^{29} +0.277324 q^{31} -2.71918 q^{33} +6.24405 q^{35} +9.10161 q^{37} +0.130922 q^{39} -8.78322 q^{41} -7.14952 q^{43} -2.71918 q^{45} -1.16618 q^{47} -1.72701 q^{49} -5.29280 q^{51} -1.51392 q^{53} -7.39395 q^{55} -2.49495 q^{57} +6.58556 q^{59} -3.25671 q^{61} -2.29630 q^{63} +0.356000 q^{65} -11.6688 q^{67} -2.05324 q^{69} -10.5813 q^{71} -13.9938 q^{73} -2.39395 q^{75} -6.24405 q^{77} -1.32793 q^{79} +1.00000 q^{81} +15.3931 q^{83} -14.3921 q^{85} +5.85977 q^{87} -0.650613 q^{89} +0.300636 q^{91} -0.277324 q^{93} -6.78421 q^{95} -16.5435 q^{97} +2.71918 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - 3 q^{5} - 2 q^{7} + 5 q^{9} + 3 q^{11} - 4 q^{13} + 3 q^{15} - 7 q^{17} - 2 q^{19} + 2 q^{21} + 13 q^{23} - 2 q^{25} - 5 q^{27} - 3 q^{29} - 12 q^{31} - 3 q^{33} + 10 q^{35} - 7 q^{37} + 4 q^{39} - 16 q^{41} - 3 q^{45} + q^{47} - 17 q^{49} + 7 q^{51} + 3 q^{53} - 23 q^{55} + 2 q^{57} + q^{59} - 22 q^{61} - 2 q^{63} - 20 q^{65} + 2 q^{67} - 13 q^{69} + 9 q^{71} - 28 q^{73} + 2 q^{75} - 10 q^{77} - 28 q^{79} + 5 q^{81} + 7 q^{83} - 11 q^{85} + 3 q^{87} - 30 q^{89} - 13 q^{91} + 12 q^{93} + 3 q^{95} - 33 q^{97} + 3 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - 3 * q^5 - 2 * q^7 + 5 * q^9 + 3 * q^11 - 4 * q^13 + 3 * q^15 - 7 * q^17 - 2 * q^19 + 2 * q^21 + 13 * q^23 - 2 * q^25 - 5 * q^27 - 3 * q^29 - 12 * q^31 - 3 * q^33 + 10 * q^35 - 7 * q^37 + 4 * q^39 - 16 * q^41 - 3 * q^45 + q^47 - 17 * q^49 + 7 * q^51 + 3 * q^53 - 23 * q^55 + 2 * q^57 + q^59 - 22 * q^61 - 2 * q^63 - 20 * q^65 + 2 * q^67 - 13 * q^69 + 9 * q^71 - 28 * q^73 + 2 * q^75 - 10 * q^77 - 28 * q^79 + 5 * q^81 + 7 * q^83 - 11 * q^85 + 3 * q^87 - 30 * q^89 - 13 * q^91 + 12 * q^93 + 3 * q^95 - 33 * q^97 + 3 * q^99

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