LMFDB - Embedded newform 2004.2.a.c.1.6

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} - x^{8} - 31x^{7} + 24x^{6} + 293x^{5} - 101x^{4} - 864x^{3} - 278x^{2} + 24x + 2 $ x^9 - x^8 - 31x^7 + 24x^6 + 293x^5 - 101x^4 - 864x^3 - 278x^2 + 24*x + 2
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			1.6
Root			$0.111665$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.111665 q^{5} -3.78430 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.111665 q^{5} -3.78430 q^{7} +1.00000 q^{9} -5.64268 q^{11} +6.56111 q^{13} -0.111665 q^{15} -2.37904 q^{17} +4.46447 q^{19} +3.78430 q^{21} -2.90056 q^{23} -4.98753 q^{25} -1.00000 q^{27} -0.383613 q^{29} -5.50063 q^{31} +5.64268 q^{33} -0.422573 q^{35} +6.40869 q^{37} -6.56111 q^{39} +4.20918 q^{41} +3.54888 q^{43} +0.111665 q^{45} -0.255709 q^{47} +7.32096 q^{49} +2.37904 q^{51} +12.1901 q^{53} -0.630088 q^{55} -4.46447 q^{57} +9.65173 q^{59} -2.24962 q^{61} -3.78430 q^{63} +0.732645 q^{65} +10.6436 q^{67} +2.90056 q^{69} -6.99033 q^{71} +1.95501 q^{73} +4.98753 q^{75} +21.3536 q^{77} +14.0719 q^{79} +1.00000 q^{81} +2.74400 q^{83} -0.265655 q^{85} +0.383613 q^{87} +9.44387 q^{89} -24.8292 q^{91} +5.50063 q^{93} +0.498524 q^{95} -6.83182 q^{97} -5.64268 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} + q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 + q^5 + 2 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} + q^{5} + 2 q^{7} + 9 q^{9} - 9 q^{11} + 10 q^{13} - q^{15} + 7 q^{17} - 2 q^{19} - 2 q^{21} - 3 q^{23} + 18 q^{25} - 9 q^{27} + 5 q^{29} + 12 q^{31} + 9 q^{33} - 6 q^{35} + 15 q^{37} - 10 q^{39} + 14 q^{41} + 6 q^{43} + q^{45} - 3 q^{47} + 27 q^{49} - 7 q^{51} + 9 q^{53} + 19 q^{55} + 2 q^{57} - 9 q^{59} + 30 q^{61} + 2 q^{63} + 28 q^{65} + 16 q^{67} + 3 q^{69} - 3 q^{71} + 32 q^{73} - 18 q^{75} + 18 q^{77} + 24 q^{79} + 9 q^{81} - 3 q^{83} + 37 q^{85} - 5 q^{87} + 46 q^{89} + 33 q^{91} - 12 q^{93} + 11 q^{95} + 43 q^{97} - 9 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 + q^5 + 2 * q^7 + 9 * q^9 - 9 * q^11 + 10 * q^13 - q^15 + 7 * q^17 - 2 * q^19 - 2 * q^21 - 3 * q^23 + 18 * q^25 - 9 * q^27 + 5 * q^29 + 12 * q^31 + 9 * q^33 - 6 * q^35 + 15 * q^37 - 10 * q^39 + 14 * q^41 + 6 * q^43 + q^45 - 3 * q^47 + 27 * q^49 - 7 * q^51 + 9 * q^53 + 19 * q^55 + 2 * q^57 - 9 * q^59 + 30 * q^61 + 2 * q^63 + 28 * q^65 + 16 * q^67 + 3 * q^69 - 3 * q^71 + 32 * q^73 - 18 * q^75 + 18 * q^77 + 24 * q^79 + 9 * q^81 - 3 * q^83 + 37 * q^85 - 5 * q^87 + 46 * q^89 + 33 * q^91 - 12 * q^93 + 11 * q^95 + 43 * q^97 - 9 * 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
$5$	0.111665	0.0499380	0.0249690	−	0.999688i	$-0.492051\pi$
$5$	0.111665	0.0499380	0.0249690	+	0.999688i	$0.492051\pi$
$6$	0	0
$7$	−3.78430	−1.43033	−0.715166	−	0.698954i	$-0.753649\pi$
$7$	−3.78430	−1.43033	−0.715166	+	0.698954i	$0.753649\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.64268	−1.70133	−0.850666	−	0.525707i	$-0.823801\pi$
$11$	−5.64268	−1.70133	−0.850666	+	0.525707i	$0.823801\pi$
$12$	0	0
$13$	6.56111	1.81972	0.909862	−	0.414911i	$-0.136187\pi$
$13$	6.56111	1.81972	0.909862	+	0.414911i	$0.136187\pi$
$14$	0	0
$15$	−0.111665	−0.0288317
$16$	0	0
$17$	−2.37904	−0.577003	−0.288501	−	0.957479i	$-0.593157\pi$
$17$	−2.37904	−0.577003	−0.288501	+	0.957479i	$0.593157\pi$
$18$	0	0
$19$	4.46447	1.02422	0.512109	−	0.858920i	$-0.328864\pi$
$19$	4.46447	1.02422	0.512109	+	0.858920i	$0.328864\pi$
$20$	0	0
$21$	3.78430	0.825803
$22$	0	0
$23$	−2.90056	−0.604809	−0.302404	−	0.953180i	$-0.597789\pi$
$23$	−2.90056	−0.604809	−0.302404	+	0.953180i	$0.597789\pi$
$24$	0	0
$25$	−4.98753	−0.997506
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.383613	−0.0712351	−0.0356175	−	0.999365i	$-0.511340\pi$
$29$	−0.383613	−0.0712351	−0.0356175	+	0.999365i	$0.511340\pi$
$30$	0	0
$31$	−5.50063	−0.987943	−0.493971	−	0.869478i	$-0.664455\pi$
$31$	−5.50063	−0.987943	−0.493971	+	0.869478i	$0.664455\pi$
$32$	0	0
$33$	5.64268	0.982264
$34$	0	0
$35$	−0.422573	−0.0714280
$36$	0	0
$37$	6.40869	1.05358	0.526791	−	0.849995i	$-0.323395\pi$
$37$	6.40869	1.05358	0.526791	+	0.849995i	$0.323395\pi$
$38$	0	0
$39$	−6.56111	−1.05062
$40$	0	0
$41$	4.20918	0.657364	0.328682	−	0.944441i	$-0.393396\pi$
$41$	4.20918	0.657364	0.328682	+	0.944441i	$0.393396\pi$
$42$	0	0
$43$	3.54888	0.541199	0.270600	−	0.962692i	$-0.412778\pi$
$43$	3.54888	0.541199	0.270600	+	0.962692i	$0.412778\pi$
$44$	0	0
$45$	0.111665	0.0166460
$46$	0	0
$47$	−0.255709	−0.0372990	−0.0186495	−	0.999826i	$-0.505937\pi$
$47$	−0.255709	−0.0372990	−0.0186495	+	0.999826i	$0.505937\pi$
$48$	0	0
$49$	7.32096	1.04585
$50$	0	0
$51$	2.37904	0.333133
$52$	0	0
$53$	12.1901	1.67443	0.837217	−	0.546870i	$-0.184181\pi$
$53$	12.1901	1.67443	0.837217	+	0.546870i	$0.184181\pi$
$54$	0	0
$55$	−0.630088	−0.0849611
$56$	0	0
$57$	−4.46447	−0.591333
$58$	0	0
$59$	9.65173	1.25655	0.628274	−	0.777992i	$-0.283762\pi$
$59$	9.65173	1.25655	0.628274	+	0.777992i	$0.283762\pi$
$60$	0	0
$61$	−2.24962	−0.288034	−0.144017	−	0.989575i	$-0.546002\pi$
$61$	−2.24962	−0.288034	−0.144017	+	0.989575i	$0.546002\pi$
$62$	0	0
$63$	−3.78430	−0.476778
$64$	0	0
$65$	0.732645	0.0908734
$66$	0	0
$67$	10.6436	1.30032	0.650161	−	0.759796i	$-0.274701\pi$
$67$	10.6436	1.30032	0.650161	+	0.759796i	$0.274701\pi$
$68$	0	0
$69$	2.90056	0.349187
$70$	0	0
$71$	−6.99033	−0.829600	−0.414800	−	0.909913i	$-0.636148\pi$
$71$	−6.99033	−0.829600	−0.414800	+	0.909913i	$0.636148\pi$
$72$	0	0
$73$	1.95501	0.228817	0.114408	−	0.993434i	$-0.463503\pi$
$73$	1.95501	0.228817	0.114408	+	0.993434i	$0.463503\pi$
$74$	0	0
$75$	4.98753	0.575910
$76$	0	0
$77$	21.3536	2.43347
$78$	0	0
$79$	14.0719	1.58321	0.791606	−	0.611031i	$-0.209245\pi$
$79$	14.0719	1.58321	0.791606	+	0.611031i	$0.209245\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.74400	0.301193	0.150596	−	0.988595i	$-0.451881\pi$
$83$	2.74400	0.301193	0.150596	+	0.988595i	$0.451881\pi$
$84$	0	0
$85$	−0.265655	−0.0288144
$86$	0	0
$87$	0.383613	0.0411276
$88$	0	0
$89$	9.44387	1.00105	0.500524	−	0.865723i	$-0.333141\pi$
$89$	9.44387	1.00105	0.500524	+	0.865723i	$0.333141\pi$
$90$	0	0
$91$	−24.8292	−2.60281
$92$	0	0
$93$	5.50063	0.570389
$94$	0	0
$95$	0.498524	0.0511474
$96$	0	0
$97$	−6.83182	−0.693666	−0.346833	−	0.937927i	$-0.612743\pi$
$97$	−6.83182	−0.693666	−0.346833	+	0.937927i	$0.612743\pi$
$98$	0	0
$99$	−5.64268	−0.567110
$100$	0	0
$101$	−17.7721	−1.76839	−0.884197	−	0.467113i	$-0.845294\pi$
$101$	−17.7721	−1.76839	−0.884197	+	0.467113i	$0.845294\pi$
$102$	0	0
$103$	−4.33419	−0.427061	−0.213530	−	0.976936i	$-0.568496\pi$
$103$	−4.33419	−0.427061	−0.213530	+	0.976936i	$0.568496\pi$
$104$	0	0
$105$	0.422573	0.0412389
$106$	0	0
$107$	−6.79832	−0.657219	−0.328609	−	0.944466i	$-0.606580\pi$
$107$	−6.79832	−0.657219	−0.328609	+	0.944466i	$0.606580\pi$
$108$	0	0
$109$	1.55808	0.149237	0.0746187	−	0.997212i	$-0.476226\pi$
$109$	1.55808	0.149237	0.0746187	+	0.997212i	$0.476226\pi$
$110$	0	0
$111$	−6.40869	−0.608286
$112$	0	0
$113$	9.86093	0.927638	0.463819	−	0.885930i	$-0.346479\pi$
$113$	9.86093	0.927638	0.463819	+	0.885930i	$0.346479\pi$
$114$	0	0
$115$	−0.323891	−0.0302029
$116$	0	0
$117$	6.56111	0.606575
$118$	0	0
$119$	9.00302	0.825306
$120$	0	0
$121$	20.8398	1.89453
$122$	0	0
$123$	−4.20918	−0.379529
$124$	0	0
$125$	−1.11526	−0.0997515
$126$	0	0
$127$	18.1734	1.61263	0.806313	−	0.591489i	$-0.201459\pi$
$127$	18.1734	1.61263	0.806313	+	0.591489i	$0.201459\pi$
$128$	0	0
$129$	−3.54888	−0.312462
$130$	0	0
$131$	20.0741	1.75389	0.876943	−	0.480595i	$-0.159579\pi$
$131$	20.0741	1.75389	0.876943	+	0.480595i	$0.159579\pi$
$132$	0	0
$133$	−16.8949	−1.46497
$134$	0	0
$135$	−0.111665	−0.00961057
$136$	0	0
$137$	−4.79003	−0.409240	−0.204620	−	0.978841i	$-0.565596\pi$
$137$	−4.79003	−0.409240	−0.204620	+	0.978841i	$0.565596\pi$
$138$	0	0
$139$	−0.465853	−0.0395132	−0.0197566	−	0.999805i	$-0.506289\pi$
$139$	−0.465853	−0.0395132	−0.0197566	+	0.999805i	$0.506289\pi$
$140$	0	0
$141$	0.255709	0.0215346
$142$	0	0
$143$	−37.0222	−3.09595
$144$	0	0
$145$	−0.0428360	−0.00355734
$146$	0	0
$147$	−7.32096	−0.603823
$148$	0	0
$149$	3.48816	0.285761	0.142880	−	0.989740i	$-0.454364\pi$
$149$	3.48816	0.285761	0.142880	+	0.989740i	$0.454364\pi$
$150$	0	0
$151$	22.1406	1.80177	0.900887	−	0.434054i	$-0.142917\pi$
$151$	22.1406	1.80177	0.900887	+	0.434054i	$0.142917\pi$
$152$	0	0
$153$	−2.37904	−0.192334
$154$	0	0
$155$	−0.614227	−0.0493359
$156$	0	0
$157$	−9.55137	−0.762282	−0.381141	−	0.924517i	$-0.624469\pi$
$157$	−9.55137	−0.762282	−0.381141	+	0.924517i	$0.624469\pi$
$158$	0	0
$159$	−12.1901	−0.966735
$160$	0	0
$161$	10.9766	0.865078
$162$	0	0
$163$	6.60472	0.517322	0.258661	−	0.965968i	$-0.416719\pi$
$163$	6.60472	0.517322	0.258661	+	0.965968i	$0.416719\pi$
$164$	0	0
$165$	0.630088	0.0490523
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	30.0481	2.31140
$170$	0	0
$171$	4.46447	0.341406
$172$	0	0
$173$	−8.21939	−0.624909	−0.312454	−	0.949933i	$-0.601151\pi$
$173$	−8.21939	−0.624909	−0.312454	+	0.949933i	$0.601151\pi$
$174$	0	0
$175$	18.8743	1.42677
$176$	0	0
$177$	−9.65173	−0.725468
$178$	0	0
$179$	15.9969	1.19567	0.597834	−	0.801620i	$-0.296028\pi$
$179$	15.9969	1.19567	0.597834	+	0.801620i	$0.296028\pi$
$180$	0	0
$181$	−5.74031	−0.426674	−0.213337	−	0.976979i	$-0.568433\pi$
$181$	−5.74031	−0.426674	−0.213337	+	0.976979i	$0.568433\pi$
$182$	0	0
$183$	2.24962	0.166297
$184$	0	0
$185$	0.715625	0.0526138
$186$	0	0
$187$	13.4242	0.981673
$188$	0	0
$189$	3.78430	0.275268
$190$	0	0
$191$	5.80254	0.419857	0.209929	−	0.977717i	$-0.432677\pi$
$191$	5.80254	0.419857	0.209929	+	0.977717i	$0.432677\pi$
$192$	0	0
$193$	−7.18082	−0.516887	−0.258443	−	0.966026i	$-0.583210\pi$
$193$	−7.18082	−0.516887	−0.258443	+	0.966026i	$0.583210\pi$
$194$	0	0
$195$	−0.732645	−0.0524658
$196$	0	0
$197$	−25.2091	−1.79607	−0.898036	−	0.439922i	$-0.855006\pi$
$197$	−25.2091	−1.79607	−0.898036	+	0.439922i	$0.855006\pi$
$198$	0	0
$199$	22.7651	1.61377	0.806887	−	0.590705i	$-0.201150\pi$
$199$	22.7651	1.61377	0.806887	+	0.590705i	$0.201150\pi$
$200$	0	0
$201$	−10.6436	−0.750742
$202$	0	0
$203$	1.45171	0.101890
$204$	0	0
$205$	0.470017	0.0328274
$206$	0	0
$207$	−2.90056	−0.201603
$208$	0	0
$209$	−25.1915	−1.74254
$210$	0	0
$211$	9.86999	0.679478	0.339739	−	0.940520i	$-0.389661\pi$
$211$	9.86999	0.679478	0.339739	+	0.940520i	$0.389661\pi$
$212$	0	0
$213$	6.99033	0.478970
$214$	0	0
$215$	0.396285	0.0270264
$216$	0	0
$217$	20.8161	1.41309
$218$	0	0
$219$	−1.95501	−0.132107
$220$	0	0
$221$	−15.6092	−1.04999
$222$	0	0
$223$	−12.5826	−0.842595	−0.421298	−	0.906922i	$-0.638425\pi$
$223$	−12.5826	−0.842595	−0.421298	+	0.906922i	$0.638425\pi$
$224$	0	0
$225$	−4.98753	−0.332502
$226$	0	0
$227$	−10.4014	−0.690366	−0.345183	−	0.938535i	$-0.612183\pi$
$227$	−10.4014	−0.690366	−0.345183	+	0.938535i	$0.612183\pi$
$228$	0	0
$229$	15.9538	1.05426	0.527129	−	0.849785i	$-0.323268\pi$
$229$	15.9538	1.05426	0.527129	+	0.849785i	$0.323268\pi$
$230$	0	0
$231$	−21.3536	−1.40496
$232$	0	0
$233$	11.2954	0.739989	0.369994	−	0.929034i	$-0.379360\pi$
$233$	11.2954	0.739989	0.369994	+	0.929034i	$0.379360\pi$
$234$	0	0
$235$	−0.0285537	−0.00186264
$236$	0	0
$237$	−14.0719	−0.914068
$238$	0	0
$239$	−1.08621	−0.0702609	−0.0351305	−	0.999383i	$-0.511185\pi$
$239$	−1.08621	−0.0702609	−0.0351305	+	0.999383i	$0.511185\pi$
$240$	0	0
$241$	9.60646	0.618806	0.309403	−	0.950931i	$-0.399871\pi$
$241$	9.60646	0.618806	0.309403	+	0.950931i	$0.399871\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.817493	0.0522277
$246$	0	0
$247$	29.2919	1.86380
$248$	0	0
$249$	−2.74400	−0.173894
$250$	0	0
$251$	−27.2234	−1.71833	−0.859163	−	0.511702i	$-0.829015\pi$
$251$	−27.2234	−1.71833	−0.859163	+	0.511702i	$0.829015\pi$
$252$	0	0
$253$	16.3669	1.02898
$254$	0	0
$255$	0.265655	0.0166360
$256$	0	0
$257$	−3.40522	−0.212412	−0.106206	−	0.994344i	$-0.533870\pi$
$257$	−3.40522	−0.212412	−0.106206	+	0.994344i	$0.533870\pi$
$258$	0	0
$259$	−24.2524	−1.50697
$260$	0	0
$261$	−0.383613	−0.0237450
$262$	0	0
$263$	−30.5618	−1.88452	−0.942260	−	0.334883i	$-0.891303\pi$
$263$	−30.5618	−1.88452	−0.942260	+	0.334883i	$0.891303\pi$
$264$	0	0
$265$	1.36120	0.0836179
$266$	0	0
$267$	−9.44387	−0.577955
$268$	0	0
$269$	26.0375	1.58754	0.793768	−	0.608221i	$-0.208117\pi$
$269$	26.0375	1.58754	0.793768	+	0.608221i	$0.208117\pi$
$270$	0	0
$271$	22.1407	1.34495	0.672476	−	0.740119i	$-0.265231\pi$
$271$	22.1407	1.34495	0.672476	+	0.740119i	$0.265231\pi$
$272$	0	0
$273$	24.8292	1.50273
$274$	0	0
$275$	28.1430	1.69709
$276$	0	0
$277$	0.894169	0.0537254	0.0268627	−	0.999639i	$-0.491448\pi$
$277$	0.894169	0.0537254	0.0268627	+	0.999639i	$0.491448\pi$
$278$	0	0
$279$	−5.50063	−0.329314
$280$	0	0
$281$	−31.2159	−1.86219	−0.931093	−	0.364781i	$-0.881144\pi$
$281$	−31.2159	−1.86219	−0.931093	+	0.364781i	$0.881144\pi$
$282$	0	0
$283$	7.65557	0.455076	0.227538	−	0.973769i	$-0.426932\pi$
$283$	7.65557	0.455076	0.227538	+	0.973769i	$0.426932\pi$
$284$	0	0
$285$	−0.498524	−0.0295300
$286$	0	0
$287$	−15.9288	−0.940249
$288$	0	0
$289$	−11.3402	−0.667068
$290$	0	0
$291$	6.83182	0.400488
$292$	0	0
$293$	19.1688	1.11985	0.559926	−	0.828543i	$-0.310830\pi$
$293$	19.1688	1.11985	0.559926	+	0.828543i	$0.310830\pi$
$294$	0	0
$295$	1.07776	0.0627495
$296$	0	0
$297$	5.64268	0.327421
$298$	0	0
$299$	−19.0309	−1.10059
$300$	0	0
$301$	−13.4301	−0.774095
$302$	0	0
$303$	17.7721	1.02098
$304$	0	0
$305$	−0.251203	−0.0143839
$306$	0	0
$307$	−10.8020	−0.616504	−0.308252	−	0.951305i	$-0.599744\pi$
$307$	−10.8020	−0.616504	−0.308252	+	0.951305i	$0.599744\pi$
$308$	0	0
$309$	4.33419	0.246564
$310$	0	0
$311$	−20.9080	−1.18558	−0.592791	−	0.805356i	$-0.701974\pi$
$311$	−20.9080	−1.18558	−0.592791	+	0.805356i	$0.701974\pi$
$312$	0	0
$313$	−5.96365	−0.337085	−0.168543	−	0.985694i	$-0.553906\pi$
$313$	−5.96365	−0.337085	−0.168543	+	0.985694i	$0.553906\pi$
$314$	0	0
$315$	−0.422573	−0.0238093
$316$	0	0
$317$	16.6322	0.934155	0.467078	−	0.884216i	$-0.345307\pi$
$317$	16.6322	0.934155	0.467078	+	0.884216i	$0.345307\pi$
$318$	0	0
$319$	2.16460	0.121194
$320$	0	0
$321$	6.79832	0.379445
$322$	0	0
$323$	−10.6212	−0.590977
$324$	0	0
$325$	−32.7237	−1.81519
$326$	0	0
$327$	−1.55808	−0.0861623
$328$	0	0
$329$	0.967680	0.0533499
$330$	0	0
$331$	−33.3573	−1.83348	−0.916741	−	0.399483i	$-0.869190\pi$
$331$	−33.3573	−1.83348	−0.916741	+	0.399483i	$0.869190\pi$
$332$	0	0
$333$	6.40869	0.351194
$334$	0	0
$335$	1.18852	0.0649355
$336$	0	0
$337$	−33.8458	−1.84370	−0.921848	−	0.387552i	$-0.873321\pi$
$337$	−33.8458	−1.84370	−0.921848	+	0.387552i	$0.873321\pi$
$338$	0	0
$339$	−9.86093	−0.535572
$340$	0	0
$341$	31.0383	1.68082
$342$	0	0
$343$	−1.21461	−0.0655829
$344$	0	0
$345$	0.323891	0.0174377
$346$	0	0
$347$	−24.7213	−1.32711	−0.663553	−	0.748129i	$-0.730952\pi$
$347$	−24.7213	−1.32711	−0.663553	+	0.748129i	$0.730952\pi$
$348$	0	0
$349$	6.75243	0.361449	0.180725	−	0.983534i	$-0.442156\pi$
$349$	6.75243	0.361449	0.180725	+	0.983534i	$0.442156\pi$
$350$	0	0
$351$	−6.56111	−0.350206
$352$	0	0
$353$	13.6342	0.725673	0.362837	−	0.931853i	$-0.381808\pi$
$353$	13.6342	0.725673	0.362837	+	0.931853i	$0.381808\pi$
$354$	0	0
$355$	−0.780574	−0.0414285
$356$	0	0
$357$	−9.00302	−0.476491
$358$	0	0
$359$	9.36561	0.494298	0.247149	−	0.968978i	$-0.420506\pi$
$359$	9.36561	0.494298	0.247149	+	0.968978i	$0.420506\pi$
$360$	0	0
$361$	0.931464	0.0490244
$362$	0	0
$363$	−20.8398	−1.09381
$364$	0	0
$365$	0.218306	0.0114267
$366$	0	0
$367$	−19.1131	−0.997696	−0.498848	−	0.866690i	$-0.666243\pi$
$367$	−19.1131	−0.997696	−0.498848	+	0.866690i	$0.666243\pi$
$368$	0	0
$369$	4.20918	0.219121
$370$	0	0
$371$	−46.1309	−2.39500
$372$	0	0
$373$	25.2383	1.30679	0.653394	−	0.757018i	$-0.273344\pi$
$373$	25.2383	1.30679	0.653394	+	0.757018i	$0.273344\pi$
$374$	0	0
$375$	1.11526	0.0575915
$376$	0	0
$377$	−2.51692	−0.129628
$378$	0	0
$379$	−8.57586	−0.440513	−0.220256	−	0.975442i	$-0.570689\pi$
$379$	−8.57586	−0.440513	−0.220256	+	0.975442i	$0.570689\pi$
$380$	0	0
$381$	−18.1734	−0.931051
$382$	0	0
$383$	11.9850	0.612403	0.306202	−	0.951967i	$-0.400942\pi$
$383$	11.9850	0.612403	0.306202	+	0.951967i	$0.400942\pi$
$384$	0	0
$385$	2.38445	0.121523
$386$	0	0
$387$	3.54888	0.180400
$388$	0	0
$389$	3.72072	0.188648	0.0943240	−	0.995542i	$-0.469931\pi$
$389$	3.72072	0.188648	0.0943240	+	0.995542i	$0.469931\pi$
$390$	0	0
$391$	6.90056	0.348976
$392$	0	0
$393$	−20.0741	−1.01261
$394$	0	0
$395$	1.57134	0.0790625
$396$	0	0
$397$	−10.8404	−0.544065	−0.272032	−	0.962288i	$-0.587696\pi$
$397$	−10.8404	−0.544065	−0.272032	+	0.962288i	$0.587696\pi$
$398$	0	0
$399$	16.8949	0.845803
$400$	0	0
$401$	8.14250	0.406617	0.203308	−	0.979115i	$-0.434831\pi$
$401$	8.14250	0.406617	0.203308	+	0.979115i	$0.434831\pi$
$402$	0	0
$403$	−36.0902	−1.79778
$404$	0	0
$405$	0.111665	0.00554867
$406$	0	0
$407$	−36.1622	−1.79249
$408$	0	0
$409$	−1.26565	−0.0625821	−0.0312911	−	0.999510i	$-0.509962\pi$
$409$	−1.26565	−0.0625821	−0.0312911	+	0.999510i	$0.509962\pi$
$410$	0	0
$411$	4.79003	0.236275
$412$	0	0
$413$	−36.5251	−1.79728
$414$	0	0
$415$	0.306408	0.0150410
$416$	0	0
$417$	0.465853	0.0228129
$418$	0	0
$419$	−21.4317	−1.04701	−0.523503	−	0.852024i	$-0.675375\pi$
$419$	−21.4317	−1.04701	−0.523503	+	0.852024i	$0.675375\pi$
$420$	0	0
$421$	1.89122	0.0921722	0.0460861	−	0.998937i	$-0.485325\pi$
$421$	1.89122	0.0921722	0.0460861	+	0.998937i	$0.485325\pi$
$422$	0	0
$423$	−0.255709	−0.0124330
$424$	0	0
$425$	11.8656	0.575564
$426$	0	0
$427$	8.51325	0.411985
$428$	0	0
$429$	37.0222	1.78745
$430$	0	0
$431$	24.7831	1.19376	0.596880	−	0.802330i	$-0.296407\pi$
$431$	24.7831	1.19376	0.596880	+	0.802330i	$0.296407\pi$
$432$	0	0
$433$	17.1885	0.826027	0.413013	−	0.910725i	$-0.364476\pi$
$433$	17.1885	0.826027	0.413013	+	0.910725i	$0.364476\pi$
$434$	0	0
$435$	0.0428360	0.00205383
$436$	0	0
$437$	−12.9495	−0.619457
$438$	0	0
$439$	28.0523	1.33886	0.669432	−	0.742873i	$-0.266537\pi$
$439$	28.0523	1.33886	0.669432	+	0.742873i	$0.266537\pi$
$440$	0	0
$441$	7.32096	0.348617
$442$	0	0
$443$	3.02151	0.143556	0.0717782	−	0.997421i	$-0.477133\pi$
$443$	3.02151	0.143556	0.0717782	+	0.997421i	$0.477133\pi$
$444$	0	0
$445$	1.05455	0.0499903
$446$	0	0
$447$	−3.48816	−0.164984
$448$	0	0
$449$	−8.17729	−0.385910	−0.192955	−	0.981208i	$-0.561807\pi$
$449$	−8.17729	−0.385910	−0.192955	+	0.981208i	$0.561807\pi$
$450$	0	0
$451$	−23.7511	−1.11839
$452$	0	0
$453$	−22.1406	−1.04025
$454$	0	0
$455$	−2.77255	−0.129979
$456$	0	0
$457$	8.58642	0.401656	0.200828	−	0.979627i	$-0.435637\pi$
$457$	8.58642	0.401656	0.200828	+	0.979627i	$0.435637\pi$
$458$	0	0
$459$	2.37904	0.111044
$460$	0	0
$461$	0.243997	0.0113641	0.00568204	−	0.999984i	$-0.498191\pi$
$461$	0.243997	0.0113641	0.00568204	+	0.999984i	$0.498191\pi$
$462$	0	0
$463$	−34.5035	−1.60351	−0.801757	−	0.597650i	$-0.796101\pi$
$463$	−34.5035	−1.60351	−0.801757	+	0.597650i	$0.796101\pi$
$464$	0	0
$465$	0.614227	0.0284841
$466$	0	0
$467$	24.5335	1.13528	0.567638	−	0.823278i	$-0.307857\pi$
$467$	24.5335	1.13528	0.567638	+	0.823278i	$0.307857\pi$
$468$	0	0
$469$	−40.2786	−1.85989
$470$	0	0
$471$	9.55137	0.440104
$472$	0	0
$473$	−20.0252	−0.920760
$474$	0	0
$475$	−22.2667	−1.02166
$476$	0	0
$477$	12.1901	0.558145
$478$	0	0
$479$	28.8205	1.31684	0.658421	−	0.752649i	$-0.271224\pi$
$479$	28.8205	1.31684	0.658421	+	0.752649i	$0.271224\pi$
$480$	0	0
$481$	42.0481	1.91723
$482$	0	0
$483$	−10.9766	−0.499453
$484$	0	0
$485$	−0.762874	−0.0346403
$486$	0	0
$487$	13.3903	0.606772	0.303386	−	0.952868i	$-0.401883\pi$
$487$	13.3903	0.606772	0.303386	+	0.952868i	$0.401883\pi$
$488$	0	0
$489$	−6.60472	−0.298676
$490$	0	0
$491$	29.2208	1.31872	0.659358	−	0.751829i	$-0.270828\pi$
$491$	29.2208	1.31872	0.659358	+	0.751829i	$0.270828\pi$
$492$	0	0
$493$	0.912631	0.0411028
$494$	0	0
$495$	−0.630088	−0.0283204
$496$	0	0
$497$	26.4535	1.18660
$498$	0	0
$499$	−43.4891	−1.94684	−0.973419	−	0.229030i	$-0.926445\pi$
$499$	−43.4891	−1.94684	−0.973419	+	0.229030i	$0.926445\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−21.9720	−0.979683	−0.489841	−	0.871812i	$-0.662945\pi$
$503$	−21.9720	−0.979683	−0.489841	+	0.871812i	$0.662945\pi$
$504$	0	0
$505$	−1.98452	−0.0883101
$506$	0	0
$507$	−30.0481	−1.33449
$508$	0	0
$509$	−4.71792	−0.209118	−0.104559	−	0.994519i	$-0.533343\pi$
$509$	−4.71792	−0.209118	−0.104559	+	0.994519i	$0.533343\pi$
$510$	0	0
$511$	−7.39836	−0.327284
$512$	0	0
$513$	−4.46447	−0.197111
$514$	0	0
$515$	−0.483977	−0.0213266
$516$	0	0
$517$	1.44288	0.0634579
$518$	0	0
$519$	8.21939	0.360791
$520$	0	0
$521$	12.2758	0.537814	0.268907	−	0.963166i	$-0.413338\pi$
$521$	12.2758	0.537814	0.268907	+	0.963166i	$0.413338\pi$
$522$	0	0
$523$	−27.7110	−1.21172	−0.605859	−	0.795572i	$-0.707170\pi$
$523$	−27.7110	−1.21172	−0.605859	+	0.795572i	$0.707170\pi$
$524$	0	0
$525$	−18.8743	−0.823744
$526$	0	0
$527$	13.0862	0.570046
$528$	0	0
$529$	−14.5867	−0.634206
$530$	0	0
$531$	9.65173	0.418849
$532$	0	0
$533$	27.6169	1.19622
$534$	0	0
$535$	−0.759133	−0.0328202
$536$	0	0
$537$	−15.9969	−0.690319
$538$	0	0
$539$	−41.3098	−1.77934
$540$	0	0
$541$	37.8180	1.62592	0.812962	−	0.582317i	$-0.197854\pi$
$541$	37.8180	1.62592	0.812962	+	0.582317i	$0.197854\pi$
$542$	0	0
$543$	5.74031	0.246340
$544$	0	0
$545$	0.173983	0.00745262
$546$	0	0
$547$	35.9100	1.53540	0.767701	−	0.640808i	$-0.221401\pi$
$547$	35.9100	1.53540	0.767701	+	0.640808i	$0.221401\pi$
$548$	0	0
$549$	−2.24962	−0.0960115
$550$	0	0
$551$	−1.71263	−0.0729603
$552$	0	0
$553$	−53.2524	−2.26452
$554$	0	0
$555$	−0.715625	−0.0303766
$556$	0	0
$557$	−11.6341	−0.492951	−0.246475	−	0.969149i	$-0.579272\pi$
$557$	−11.6341	−0.492951	−0.246475	+	0.969149i	$0.579272\pi$
$558$	0	0
$559$	23.2846	0.984834
$560$	0	0
$561$	−13.4242	−0.566769
$562$	0	0
$563$	17.3452	0.731013	0.365506	−	0.930809i	$-0.380896\pi$
$563$	17.3452	0.731013	0.365506	+	0.930809i	$0.380896\pi$
$564$	0	0
$565$	1.10112	0.0463244
$566$	0	0
$567$	−3.78430	−0.158926
$568$	0	0
$569$	7.42505	0.311274	0.155637	−	0.987814i	$-0.450257\pi$
$569$	7.42505	0.311274	0.155637	+	0.987814i	$0.450257\pi$
$570$	0	0
$571$	19.8276	0.829761	0.414881	−	0.909876i	$-0.363823\pi$
$571$	19.8276	0.829761	0.414881	+	0.909876i	$0.363823\pi$
$572$	0	0
$573$	−5.80254	−0.242405
$574$	0	0
$575$	14.4666	0.603301
$576$	0	0
$577$	20.3484	0.847113	0.423557	−	0.905870i	$-0.360781\pi$
$577$	20.3484	0.847113	0.423557	+	0.905870i	$0.360781\pi$
$578$	0	0
$579$	7.18082	0.298425
$580$	0	0
$581$	−10.3841	−0.430806
$582$	0	0
$583$	−68.7846	−2.84877
$584$	0	0
$585$	0.732645	0.0302911
$586$	0	0
$587$	−26.3887	−1.08918	−0.544590	−	0.838702i	$-0.683315\pi$
$587$	−26.3887	−1.08918	−0.544590	+	0.838702i	$0.683315\pi$
$588$	0	0
$589$	−24.5574	−1.01187
$590$	0	0
$591$	25.2091	1.03696
$592$	0	0
$593$	−1.98631	−0.0815679	−0.0407839	−	0.999168i	$-0.512986\pi$
$593$	−1.98631	−0.0815679	−0.0407839	+	0.999168i	$0.512986\pi$
$594$	0	0
$595$	1.00532	0.0412141
$596$	0	0
$597$	−22.7651	−0.931713
$598$	0	0
$599$	19.4186	0.793422	0.396711	−	0.917943i	$-0.370151\pi$
$599$	19.4186	0.793422	0.396711	+	0.917943i	$0.370151\pi$
$600$	0	0
$601$	6.95870	0.283851	0.141926	−	0.989877i	$-0.454671\pi$
$601$	6.95870	0.283851	0.141926	+	0.989877i	$0.454671\pi$
$602$	0	0
$603$	10.6436	0.433441
$604$	0	0
$605$	2.32707	0.0946090
$606$	0	0
$607$	19.0403	0.772822	0.386411	−	0.922327i	$-0.373715\pi$
$607$	19.0403	0.772822	0.386411	+	0.922327i	$0.373715\pi$
$608$	0	0
$609$	−1.45171	−0.0588261
$610$	0	0
$611$	−1.67773	−0.0678738
$612$	0	0
$613$	−18.3275	−0.740242	−0.370121	−	0.928984i	$-0.620684\pi$
$613$	−18.3275	−0.740242	−0.370121	+	0.928984i	$0.620684\pi$
$614$	0	0
$615$	−0.470017	−0.0189529
$616$	0	0
$617$	2.68917	0.108262	0.0541310	−	0.998534i	$-0.482761\pi$
$617$	2.68917	0.108262	0.0541310	+	0.998534i	$0.482761\pi$
$618$	0	0
$619$	−30.7700	−1.23675	−0.618376	−	0.785882i	$-0.712209\pi$
$619$	−30.7700	−1.23675	−0.618376	+	0.785882i	$0.712209\pi$
$620$	0	0
$621$	2.90056	0.116396
$622$	0	0
$623$	−35.7385	−1.43183
$624$	0	0
$625$	24.8131	0.992525
$626$	0	0
$627$	25.1915	1.00605
$628$	0	0
$629$	−15.2465	−0.607920
$630$	0	0
$631$	36.0018	1.43321	0.716604	−	0.697480i	$-0.245695\pi$
$631$	36.0018	1.43321	0.716604	+	0.697480i	$0.245695\pi$
$632$	0	0
$633$	−9.86999	−0.392297
$634$	0	0
$635$	2.02933	0.0805314
$636$	0	0
$637$	48.0336	1.90316
$638$	0	0
$639$	−6.99033	−0.276533
$640$	0	0
$641$	−37.1562	−1.46758	−0.733790	−	0.679376i	$-0.762250\pi$
$641$	−37.1562	−1.46758	−0.733790	+	0.679376i	$0.762250\pi$
$642$	0	0
$643$	45.1189	1.77931	0.889657	−	0.456630i	$-0.150944\pi$
$643$	45.1189	1.77931	0.889657	+	0.456630i	$0.150944\pi$
$644$	0	0
$645$	−0.396285	−0.0156037
$646$	0	0
$647$	−17.7005	−0.695879	−0.347939	−	0.937517i	$-0.613118\pi$
$647$	−17.7005	−0.695879	−0.347939	+	0.937517i	$0.613118\pi$
$648$	0	0
$649$	−54.4616	−2.13780
$650$	0	0
$651$	−20.8161	−0.815846
$652$	0	0
$653$	4.14301	0.162129	0.0810643	−	0.996709i	$-0.474168\pi$
$653$	4.14301	0.162129	0.0810643	+	0.996709i	$0.474168\pi$
$654$	0	0
$655$	2.24157	0.0875855
$656$	0	0
$657$	1.95501	0.0762723
$658$	0	0
$659$	23.7952	0.926930	0.463465	−	0.886115i	$-0.346606\pi$
$659$	23.7952	0.926930	0.463465	+	0.886115i	$0.346606\pi$
$660$	0	0
$661$	−9.35583	−0.363900	−0.181950	−	0.983308i	$-0.558241\pi$
$661$	−9.35583	−0.363900	−0.181950	+	0.983308i	$0.558241\pi$
$662$	0	0
$663$	15.6092	0.606210
$664$	0	0
$665$	−1.88657	−0.0731579
$666$	0	0
$667$	1.11269	0.0430836
$668$	0	0
$669$	12.5826	0.486473
$670$	0	0
$671$	12.6939	0.490042
$672$	0	0
$673$	11.8403	0.456410	0.228205	−	0.973613i	$-0.426714\pi$
$673$	11.8403	0.456410	0.228205	+	0.973613i	$0.426714\pi$
$674$	0	0
$675$	4.98753	0.191970
$676$	0	0
$677$	5.85878	0.225171	0.112586	−	0.993642i	$-0.464087\pi$
$677$	5.85878	0.225171	0.112586	+	0.993642i	$0.464087\pi$
$678$	0	0
$679$	25.8537	0.992174
$680$	0	0
$681$	10.4014	0.398583
$682$	0	0
$683$	−23.1803	−0.886968	−0.443484	−	0.896282i	$-0.646258\pi$
$683$	−23.1803	−0.886968	−0.443484	+	0.896282i	$0.646258\pi$
$684$	0	0
$685$	−0.534878	−0.0204366
$686$	0	0
$687$	−15.9538	−0.608677
$688$	0	0
$689$	79.9804	3.04701
$690$	0	0
$691$	28.7086	1.09213	0.546063	−	0.837744i	$-0.316126\pi$
$691$	28.7086	1.09213	0.546063	+	0.837744i	$0.316126\pi$
$692$	0	0
$693$	21.3536	0.811157
$694$	0	0
$695$	−0.0520194	−0.00197321
$696$	0	0
$697$	−10.0138	−0.379301
$698$	0	0
$699$	−11.2954	−0.427233
$700$	0	0
$701$	49.3971	1.86570	0.932851	−	0.360261i	$-0.117312\pi$
$701$	49.3971	1.86570	0.932851	+	0.360261i	$0.117312\pi$
$702$	0	0
$703$	28.6114	1.07910
$704$	0	0
$705$	0.0285537	0.00107539
$706$	0	0
$707$	67.2552	2.52939
$708$	0	0
$709$	9.82705	0.369063	0.184531	−	0.982827i	$-0.440923\pi$
$709$	9.82705	0.369063	0.184531	+	0.982827i	$0.440923\pi$
$710$	0	0
$711$	14.0719	0.527738
$712$	0	0
$713$	15.9549	0.597516
$714$	0	0
$715$	−4.13408	−0.154606
$716$	0	0
$717$	1.08621	0.0405652
$718$	0	0
$719$	−45.3562	−1.69150	−0.845751	−	0.533578i	$-0.820847\pi$
$719$	−45.3562	−1.69150	−0.845751	+	0.533578i	$0.820847\pi$
$720$	0	0
$721$	16.4019	0.610839
$722$	0	0
$723$	−9.60646	−0.357268
$724$	0	0
$725$	1.91328	0.0710574
$726$	0	0
$727$	41.9767	1.55683	0.778414	−	0.627751i	$-0.216024\pi$
$727$	41.9767	1.55683	0.778414	+	0.627751i	$0.216024\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.44294	−0.312274
$732$	0	0
$733$	24.1292	0.891234	0.445617	−	0.895224i	$-0.352984\pi$
$733$	24.1292	0.891234	0.445617	+	0.895224i	$0.352984\pi$
$734$	0	0
$735$	−0.817493	−0.0301537
$736$	0	0
$737$	−60.0584	−2.21228
$738$	0	0
$739$	−15.4710	−0.569108	−0.284554	−	0.958660i	$-0.591846\pi$
$739$	−15.4710	−0.569108	−0.284554	+	0.958660i	$0.591846\pi$
$740$	0	0
$741$	−29.2919	−1.07606
$742$	0	0
$743$	−2.61820	−0.0960525	−0.0480263	−	0.998846i	$-0.515293\pi$
$743$	−2.61820	−0.0960525	−0.0480263	+	0.998846i	$0.515293\pi$
$744$	0	0
$745$	0.389504	0.0142703
$746$	0	0
$747$	2.74400	0.100398
$748$	0	0
$749$	25.7269	0.940041
$750$	0	0
$751$	−19.3233	−0.705116	−0.352558	−	0.935790i	$-0.614688\pi$
$751$	−19.3233	−0.705116	−0.352558	+	0.935790i	$0.614688\pi$
$752$	0	0
$753$	27.2234	0.992076
$754$	0	0
$755$	2.47232	0.0899770
$756$	0	0
$757$	22.3807	0.813441	0.406720	−	0.913553i	$-0.366672\pi$
$757$	22.3807	0.813441	0.406720	+	0.913553i	$0.366672\pi$
$758$	0	0
$759$	−16.3669	−0.594082
$760$	0	0
$761$	−32.3260	−1.17182	−0.585909	−	0.810377i	$-0.699262\pi$
$761$	−32.3260	−1.17182	−0.585909	+	0.810377i	$0.699262\pi$
$762$	0	0
$763$	−5.89627	−0.213459
$764$	0	0
$765$	−0.265655	−0.00960479
$766$	0	0
$767$	63.3260	2.28657
$768$	0	0
$769$	−29.1841	−1.05241	−0.526203	−	0.850359i	$-0.676385\pi$
$769$	−29.1841	−1.05241	−0.526203	+	0.850359i	$0.676385\pi$
$770$	0	0
$771$	3.40522	0.122636
$772$	0	0
$773$	−22.4024	−0.805757	−0.402878	−	0.915253i	$-0.631990\pi$
$773$	−22.4024	−0.805757	−0.402878	+	0.915253i	$0.631990\pi$
$774$	0	0
$775$	27.4346	0.985479
$776$	0	0
$777$	24.2524	0.870051
$778$	0	0
$779$	18.7917	0.673284
$780$	0	0
$781$	39.4442	1.41142
$782$	0	0
$783$	0.383613	0.0137092
$784$	0	0
$785$	−1.06655	−0.0380669
$786$	0	0
$787$	12.6867	0.452230	0.226115	−	0.974101i	$-0.427397\pi$
$787$	12.6867	0.452230	0.226115	+	0.974101i	$0.427397\pi$
$788$	0	0
$789$	30.5618	1.08803
$790$	0	0
$791$	−37.3168	−1.32683
$792$	0	0
$793$	−14.7600	−0.524143
$794$	0	0
$795$	−1.36120	−0.0482768
$796$	0	0
$797$	−39.8473	−1.41146	−0.705732	−	0.708479i	$-0.749382\pi$
$797$	−39.8473	−1.41146	−0.705732	+	0.708479i	$0.749382\pi$
$798$	0	0
$799$	0.608342	0.0215216
$800$	0	0
$801$	9.44387	0.333683
$802$	0	0
$803$	−11.0315	−0.389293
$804$	0	0
$805$	1.22570	0.0432003
$806$	0	0
$807$	−26.0375	−0.916564
$808$	0	0
$809$	35.9994	1.26567	0.632835	−	0.774286i	$-0.281891\pi$
$809$	35.9994	1.26567	0.632835	+	0.774286i	$0.281891\pi$
$810$	0	0
$811$	30.4720	1.07002	0.535009	−	0.844846i	$-0.320308\pi$
$811$	30.4720	1.07002	0.535009	+	0.844846i	$0.320308\pi$
$812$	0	0
$813$	−22.1407	−0.776509
$814$	0	0
$815$	0.737515	0.0258340
$816$	0	0
$817$	15.8439	0.554307
$818$	0	0
$819$	−24.8292	−0.867604
$820$	0	0
$821$	14.6784	0.512278	0.256139	−	0.966640i	$-0.417549\pi$
$821$	14.6784	0.512278	0.256139	+	0.966640i	$0.417549\pi$
$822$	0	0
$823$	29.7719	1.03778	0.518891	−	0.854840i	$-0.326345\pi$
$823$	29.7719	1.03778	0.518891	+	0.854840i	$0.326345\pi$
$824$	0	0
$825$	−28.1430	−0.979815
$826$	0	0
$827$	0.393491	0.0136830	0.00684152	−	0.999977i	$-0.497822\pi$
$827$	0.393491	0.0136830	0.00684152	+	0.999977i	$0.497822\pi$
$828$	0	0
$829$	−20.3936	−0.708299	−0.354149	−	0.935189i	$-0.615230\pi$
$829$	−20.3936	−0.708299	−0.354149	+	0.935189i	$0.615230\pi$
$830$	0	0
$831$	−0.894169	−0.0310184
$832$	0	0
$833$	−17.4169	−0.603459
$834$	0	0
$835$	−0.111665	−0.00386432
$836$	0	0
$837$	5.50063	0.190130
$838$	0	0
$839$	30.2050	1.04279	0.521396	−	0.853315i	$-0.325411\pi$
$839$	30.2050	1.04279	0.521396	+	0.853315i	$0.325411\pi$
$840$	0	0
$841$	−28.8528	−0.994926
$842$	0	0
$843$	31.2159	1.07513
$844$	0	0
$845$	3.35532	0.115426
$846$	0	0
$847$	−78.8642	−2.70981
$848$	0	0
$849$	−7.65557	−0.262738
$850$	0	0
$851$	−18.5888	−0.637216
$852$	0	0
$853$	33.9989	1.16410	0.582051	−	0.813153i	$-0.302251\pi$
$853$	33.9989	1.16410	0.582051	+	0.813153i	$0.302251\pi$
$854$	0	0
$855$	0.498524	0.0170491
$856$	0	0
$857$	0.110035	0.00375871	0.00187935	−	0.999998i	$-0.499402\pi$
$857$	0.110035	0.00375871	0.00187935	+	0.999998i	$0.499402\pi$
$858$	0	0
$859$	−16.0084	−0.546199	−0.273100	−	0.961986i	$-0.588049\pi$
$859$	−16.0084	−0.546199	−0.273100	+	0.961986i	$0.588049\pi$
$860$	0	0
$861$	15.9288	0.542853
$862$	0	0
$863$	−44.6719	−1.52065	−0.760325	−	0.649542i	$-0.774960\pi$
$863$	−44.6719	−1.52065	−0.760325	+	0.649542i	$0.774960\pi$
$864$	0	0
$865$	−0.917816	−0.0312067
$866$	0	0
$867$	11.3402	0.385132
$868$	0	0
$869$	−79.4032	−2.69357
$870$	0	0
$871$	69.8338	2.36623
$872$	0	0
$873$	−6.83182	−0.231222
$874$	0	0
$875$	4.22047	0.142678
$876$	0	0
$877$	−18.4516	−0.623067	−0.311533	−	0.950235i	$-0.600843\pi$
$877$	−18.4516	−0.623067	−0.311533	+	0.950235i	$0.600843\pi$
$878$	0	0
$879$	−19.1688	−0.646547
$880$	0	0
$881$	33.5042	1.12879	0.564393	−	0.825506i	$-0.309110\pi$
$881$	33.5042	1.12879	0.564393	+	0.825506i	$0.309110\pi$
$882$	0	0
$883$	−31.7850	−1.06965	−0.534826	−	0.844962i	$-0.679623\pi$
$883$	−31.7850	−1.06965	−0.534826	+	0.844962i	$0.679623\pi$
$884$	0	0
$885$	−1.07776	−0.0362284
$886$	0	0
$887$	−32.0665	−1.07669	−0.538344	−	0.842725i	$-0.680950\pi$
$887$	−32.0665	−1.07669	−0.538344	+	0.842725i	$0.680950\pi$
$888$	0	0
$889$	−68.7736	−2.30659
$890$	0	0
$891$	−5.64268	−0.189037
$892$	0	0
$893$	−1.14160	−0.0382023
$894$	0	0
$895$	1.78629	0.0597092
$896$	0	0
$897$	19.0309	0.635423
$898$	0	0
$899$	2.11011	0.0703762
$900$	0	0
$901$	−29.0007	−0.966153
$902$	0	0
$903$	13.4301	0.446924
$904$	0	0
$905$	−0.640990	−0.0213072
$906$	0	0
$907$	−15.9240	−0.528747	−0.264373	−	0.964420i	$-0.585165\pi$
$907$	−15.9240	−0.528747	−0.264373	+	0.964420i	$0.585165\pi$
$908$	0	0
$909$	−17.7721	−0.589465
$910$	0	0
$911$	6.50065	0.215376	0.107688	−	0.994185i	$-0.465655\pi$
$911$	6.50065	0.215376	0.107688	+	0.994185i	$0.465655\pi$
$912$	0	0
$913$	−15.4835	−0.512429
$914$	0	0
$915$	0.251203	0.00830453
$916$	0	0
$917$	−75.9667	−2.50864
$918$	0	0
$919$	4.81381	0.158793	0.0793965	−	0.996843i	$-0.474701\pi$
$919$	4.81381	0.158793	0.0793965	+	0.996843i	$0.474701\pi$
$920$	0	0
$921$	10.8020	0.355939
$922$	0	0
$923$	−45.8643	−1.50964
$924$	0	0
$925$	−31.9635	−1.05095
$926$	0	0
$927$	−4.33419	−0.142354
$928$	0	0
$929$	53.8321	1.76617	0.883087	−	0.469210i	$-0.155461\pi$
$929$	53.8321	1.76617	0.883087	+	0.469210i	$0.155461\pi$
$930$	0	0
$931$	32.6842	1.07118
$932$	0	0
$933$	20.9080	0.684496
$934$	0	0
$935$	1.49901	0.0490228
$936$	0	0
$937$	23.0027	0.751465	0.375732	−	0.926728i	$-0.377391\pi$
$937$	23.0027	0.751465	0.375732	+	0.926728i	$0.377391\pi$
$938$	0	0
$939$	5.96365	0.194616
$940$	0	0
$941$	−5.83773	−0.190304	−0.0951522	−	0.995463i	$-0.530334\pi$
$941$	−5.83773	−0.190304	−0.0951522	+	0.995463i	$0.530334\pi$
$942$	0	0
$943$	−12.2090	−0.397579
$944$	0	0
$945$	0.422573	0.0137463
$946$	0	0
$947$	−50.9102	−1.65436	−0.827180	−	0.561937i	$-0.810056\pi$
$947$	−50.9102	−1.65436	−0.827180	+	0.561937i	$0.810056\pi$
$948$	0	0
$949$	12.8270	0.416383
$950$	0	0
$951$	−16.6322	−0.539335
$952$	0	0
$953$	−48.7272	−1.57843	−0.789215	−	0.614117i	$-0.789512\pi$
$953$	−48.7272	−1.57843	−0.789215	+	0.614117i	$0.789512\pi$
$954$	0	0
$955$	0.647939	0.0209668
$956$	0	0
$957$	−2.16460	−0.0699717
$958$	0	0
$959$	18.1269	0.585350
$960$	0	0
$961$	−0.743054	−0.0239695
$962$	0	0
$963$	−6.79832	−0.219073
$964$	0	0
$965$	−0.801844	−0.0258123
$966$	0	0
$967$	−40.3000	−1.29596	−0.647981	−	0.761656i	$-0.724386\pi$
$967$	−40.3000	−1.29596	−0.647981	+	0.761656i	$0.724386\pi$
$968$	0	0
$969$	10.6212	0.341201
$970$	0	0
$971$	−15.3107	−0.491345	−0.245672	−	0.969353i	$-0.579009\pi$
$971$	−15.3107	−0.491345	−0.245672	+	0.969353i	$0.579009\pi$
$972$	0	0
$973$	1.76293	0.0565170
$974$	0	0
$975$	32.7237	1.04800
$976$	0	0
$977$	10.4387	0.333963	0.166982	−	0.985960i	$-0.446598\pi$
$977$	10.4387	0.333963	0.166982	+	0.985960i	$0.446598\pi$
$978$	0	0
$979$	−53.2887	−1.70311
$980$	0	0
$981$	1.55808	0.0497458
$982$	0	0
$983$	52.6192	1.67829	0.839146	−	0.543906i	$-0.183055\pi$
$983$	52.6192	1.67829	0.839146	+	0.543906i	$0.183055\pi$
$984$	0	0
$985$	−2.81497	−0.0896923
$986$	0	0
$987$	−0.967680	−0.0308016
$988$	0	0
$989$	−10.2938	−0.327322
$990$	0	0
$991$	1.26644	0.0402298	0.0201149	−	0.999798i	$-0.493597\pi$
$991$	1.26644	0.0402298	0.0201149	+	0.999798i	$0.493597\pi$
$992$	0	0
$993$	33.3573	1.05856
$994$	0	0
$995$	2.54206	0.0805887
$996$	0	0
$997$	55.4566	1.75633	0.878164	−	0.478359i	$-0.158768\pi$
$997$	55.4566	1.75633	0.878164	+	0.478359i	$0.158768\pi$
$998$	0	0
$999$	−6.40869	−0.202762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2004.2.a.c.1.6	✓	9
3.2	odd	2		6012.2.a.i.1.4		9
4.3	odd	2		8016.2.a.bc.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.c.1.6	✓	9	1.1	even	1	trivial
6012.2.a.i.1.4		9	3.2	odd	2
8016.2.a.bc.1.6		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.111665 q^{5} -3.78430 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.111665 q^{5} -3.78430 q^{7} +1.00000 q^{9} -5.64268 q^{11} +6.56111 q^{13} -0.111665 q^{15} -2.37904 q^{17} +4.46447 q^{19} +3.78430 q^{21} -2.90056 q^{23} -4.98753 q^{25} -1.00000 q^{27} -0.383613 q^{29} -5.50063 q^{31} +5.64268 q^{33} -0.422573 q^{35} +6.40869 q^{37} -6.56111 q^{39} +4.20918 q^{41} +3.54888 q^{43} +0.111665 q^{45} -0.255709 q^{47} +7.32096 q^{49} +2.37904 q^{51} +12.1901 q^{53} -0.630088 q^{55} -4.46447 q^{57} +9.65173 q^{59} -2.24962 q^{61} -3.78430 q^{63} +0.732645 q^{65} +10.6436 q^{67} +2.90056 q^{69} -6.99033 q^{71} +1.95501 q^{73} +4.98753 q^{75} +21.3536 q^{77} +14.0719 q^{79} +1.00000 q^{81} +2.74400 q^{83} -0.265655 q^{85} +0.383613 q^{87} +9.44387 q^{89} -24.8292 q^{91} +5.50063 q^{93} +0.498524 q^{95} -6.83182 q^{97} -5.64268 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} + q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 + q^5 + 2 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} + q^{5} + 2 q^{7} + 9 q^{9} - 9 q^{11} + 10 q^{13} - q^{15} + 7 q^{17} - 2 q^{19} - 2 q^{21} - 3 q^{23} + 18 q^{25} - 9 q^{27} + 5 q^{29} + 12 q^{31} + 9 q^{33} - 6 q^{35} + 15 q^{37} - 10 q^{39} + 14 q^{41} + 6 q^{43} + q^{45} - 3 q^{47} + 27 q^{49} - 7 q^{51} + 9 q^{53} + 19 q^{55} + 2 q^{57} - 9 q^{59} + 30 q^{61} + 2 q^{63} + 28 q^{65} + 16 q^{67} + 3 q^{69} - 3 q^{71} + 32 q^{73} - 18 q^{75} + 18 q^{77} + 24 q^{79} + 9 q^{81} - 3 q^{83} + 37 q^{85} - 5 q^{87} + 46 q^{89} + 33 q^{91} - 12 q^{93} + 11 q^{95} + 43 q^{97} - 9 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 + q^5 + 2 * q^7 + 9 * q^9 - 9 * q^11 + 10 * q^13 - q^15 + 7 * q^17 - 2 * q^19 - 2 * q^21 - 3 * q^23 + 18 * q^25 - 9 * q^27 + 5 * q^29 + 12 * q^31 + 9 * q^33 - 6 * q^35 + 15 * q^37 - 10 * q^39 + 14 * q^41 + 6 * q^43 + q^45 - 3 * q^47 + 27 * q^49 - 7 * q^51 + 9 * q^53 + 19 * q^55 + 2 * q^57 - 9 * q^59 + 30 * q^61 + 2 * q^63 + 28 * q^65 + 16 * q^67 + 3 * q^69 - 3 * q^71 + 32 * q^73 - 18 * q^75 + 18 * q^77 + 24 * q^79 + 9 * q^81 - 3 * q^83 + 37 * q^85 - 5 * q^87 + 46 * q^89 + 33 * q^91 - 12 * q^93 + 11 * q^95 + 43 * q^97 - 9 * q^99

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