LMFDB - Embedded newform 2124.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2124,2,Mod(1,2124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2124, 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("2124.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2124 = 2^{2} \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2124.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:	$16.9602253893$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.22896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 4x + 3 $ x^4 - 9x^2 - 4x + 3
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			$-2.65924$ of defining polynomial
Character	$\chi$	$=$	2124.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.89142 q^{5} +1.71126 q^{7} +O(q^{10})$	$q-1.89142 q^{5} +1.71126 q^{7} +0.839403 q^{11} -1.94798 q^{13} -4.19035 q^{17} +1.07158 q^{19} -0.340756 q^{23} -1.42253 q^{25} -4.65924 q^{29} +9.22947 q^{31} -3.23672 q^{35} -3.13833 q^{37} -5.59704 q^{41} +1.01019 q^{43} -10.4986 q^{47} -4.07158 q^{49} +9.10587 q^{53} -1.58766 q^{55} +1.00000 q^{59} -0.592494 q^{61} +3.68445 q^{65} -1.37051 q^{67} -11.2099 q^{71} +8.50884 q^{73} +1.43644 q^{77} -11.4198 q^{79} -3.28420 q^{83} +7.92571 q^{85} -2.60268 q^{89} -3.33351 q^{91} -2.02681 q^{95} -11.2965 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^7	$ 4 q + 2 q^{7} - 10 q^{11} - 2 q^{13} - 4 q^{17} - 6 q^{19} - 12 q^{23} + 4 q^{25} - 8 q^{29} - 8 q^{31} - 12 q^{35} + 6 q^{37} - 16 q^{41} - 6 q^{43} - 18 q^{47} - 6 q^{49} - 4 q^{53} - 6 q^{55} + 4 q^{59} - 18 q^{65} + 10 q^{67} - 16 q^{71} - 14 q^{77} + 12 q^{79} - 22 q^{83} - 6 q^{85} + 2 q^{89} + 20 q^{91} - 36 q^{95} + 6 q^{97}+O(q^{100}) $ 4 * q + 2 * q^7 - 10 * q^11 - 2 * q^13 - 4 * q^17 - 6 * q^19 - 12 * q^23 + 4 * q^25 - 8 * q^29 - 8 * q^31 - 12 * q^35 + 6 * q^37 - 16 * q^41 - 6 * q^43 - 18 * q^47 - 6 * q^49 - 4 * q^53 - 6 * q^55 + 4 * q^59 - 18 * q^65 + 10 * q^67 - 16 * q^71 - 14 * q^77 + 12 * q^79 - 22 * q^83 - 6 * q^85 + 2 * q^89 + 20 * q^91 - 36 * q^95 + 6 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.89142	−0.845869	−0.422935	−	0.906160i	$-0.639000\pi$
$5$	−1.89142	−0.845869	−0.422935	+	0.906160i	$0.639000\pi$
$6$	0	0
$7$	1.71126	0.646797	0.323398	−	0.946263i	$-0.395175\pi$
$7$	1.71126	0.646797	0.323398	+	0.946263i	$0.395175\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.839403	0.253089	0.126545	−	0.991961i	$-0.459611\pi$
$11$	0.839403	0.253089	0.126545	+	0.991961i	$0.459611\pi$
$12$	0	0
$13$	−1.94798	−0.540273	−0.270136	−	0.962822i	$-0.587069\pi$
$13$	−1.94798	−0.540273	−0.270136	+	0.962822i	$0.587069\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.19035	−1.01631	−0.508154	−	0.861266i	$-0.669672\pi$
$17$	−4.19035	−1.01631	−0.508154	+	0.861266i	$0.669672\pi$
$18$	0	0
$19$	1.07158	0.245837	0.122919	−	0.992417i	$-0.460775\pi$
$19$	1.07158	0.245837	0.122919	+	0.992417i	$0.460775\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.340756	−0.0710525	−0.0355262	−	0.999369i	$-0.511311\pi$
$23$	−0.340756	−0.0710525	−0.0355262	+	0.999369i	$0.511311\pi$
$24$	0	0
$25$	−1.42253	−0.284505
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.65924	−0.865200	−0.432600	−	0.901586i	$-0.642404\pi$
$29$	−4.65924	−0.865200	−0.432600	+	0.901586i	$0.642404\pi$
$30$	0	0
$31$	9.22947	1.65766	0.828831	−	0.559499i	$-0.189006\pi$
$31$	9.22947	1.65766	0.828831	+	0.559499i	$0.189006\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.23672	−0.547105
$36$	0	0
$37$	−3.13833	−0.515938	−0.257969	−	0.966153i	$-0.583053\pi$
$37$	−3.13833	−0.515938	−0.257969	+	0.966153i	$0.583053\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.59704	−0.874110	−0.437055	−	0.899435i	$-0.643979\pi$
$41$	−5.59704	−0.874110	−0.437055	+	0.899435i	$0.643979\pi$
$42$	0	0
$43$	1.01019	0.154053	0.0770263	−	0.997029i	$-0.475457\pi$
$43$	1.01019	0.154053	0.0770263	+	0.997029i	$0.475457\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.4986	−1.53139	−0.765693	−	0.643207i	$-0.777604\pi$
$47$	−10.4986	−1.53139	−0.765693	+	0.643207i	$0.777604\pi$
$48$	0	0
$49$	−4.07158	−0.581654
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.10587	1.25079	0.625394	−	0.780309i	$-0.284938\pi$
$53$	9.10587	1.25079	0.625394	+	0.780309i	$0.284938\pi$
$54$	0	0
$55$	−1.58766	−0.214081
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−0.592494	−0.0758611	−0.0379306	−	0.999280i	$-0.512077\pi$
$61$	−0.592494	−0.0758611	−0.0379306	+	0.999280i	$0.512077\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.68445	0.457000
$66$	0	0
$67$	−1.37051	−0.167434	−0.0837170	−	0.996490i	$-0.526679\pi$
$67$	−1.37051	−0.167434	−0.0837170	+	0.996490i	$0.526679\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.2099	−1.33037	−0.665186	−	0.746678i	$-0.731648\pi$
$71$	−11.2099	−1.33037	−0.665186	+	0.746678i	$0.731648\pi$
$72$	0	0
$73$	8.50884	0.995884	0.497942	−	0.867210i	$-0.334089\pi$
$73$	8.50884	0.995884	0.497942	+	0.867210i	$0.334089\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.43644	0.163697
$78$	0	0
$79$	−11.4198	−1.28483	−0.642415	−	0.766357i	$-0.722067\pi$
$79$	−11.4198	−1.28483	−0.642415	+	0.766357i	$0.722067\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.28420	−0.360487	−0.180244	−	0.983622i	$-0.557689\pi$
$83$	−3.28420	−0.360487	−0.180244	+	0.983622i	$0.557689\pi$
$84$	0	0
$85$	7.92571	0.859664
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.60268	−0.275884	−0.137942	−	0.990440i	$-0.544049\pi$
$89$	−2.60268	−0.275884	−0.137942	+	0.990440i	$0.544049\pi$
$90$	0	0
$91$	−3.33351	−0.349447
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.02681	−0.207946
$96$	0	0
$97$	−11.2965	−1.14699	−0.573493	−	0.819210i	$-0.694412\pi$
$97$	−11.2965	−1.14699	−0.573493	+	0.819210i	$0.694412\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.7801	−1.47068	−0.735339	−	0.677699i	$-0.762977\pi$
$101$	−14.7801	−1.47068	−0.735339	+	0.677699i	$0.762977\pi$
$102$	0	0
$103$	1.67881	0.165418	0.0827088	−	0.996574i	$-0.473643\pi$
$103$	1.67881	0.165418	0.0827088	+	0.996574i	$0.473643\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.93136	0.476733	0.238366	−	0.971175i	$-0.423388\pi$
$107$	4.93136	0.476733	0.238366	+	0.971175i	$0.423388\pi$
$108$	0	0
$109$	−11.8793	−1.13783	−0.568917	−	0.822395i	$-0.692638\pi$
$109$	−11.8793	−1.13783	−0.568917	+	0.822395i	$0.692638\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.3332	−1.25428	−0.627142	−	0.778905i	$-0.715775\pi$
$113$	−13.3332	−1.25428	−0.627142	+	0.778905i	$0.715775\pi$
$114$	0	0
$115$	0.644513	0.0601011
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.17079	−0.657345
$120$	0	0
$121$	−10.2954	−0.935946
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1477	1.08652
$126$	0	0
$127$	−3.83375	−0.340191	−0.170095	−	0.985428i	$-0.554408\pi$
$127$	−3.83375	−0.340191	−0.170095	+	0.985428i	$0.554408\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.6166	−1.10232	−0.551159	−	0.834400i	$-0.685814\pi$
$131$	−12.6166	−1.10232	−0.551159	+	0.834400i	$0.685814\pi$
$132$	0	0
$133$	1.83375	0.159007
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.63514	−0.139700	−0.0698498	−	0.997558i	$-0.522252\pi$
$137$	−1.63514	−0.139700	−0.0698498	+	0.997558i	$0.522252\pi$
$138$	0	0
$139$	10.6788	0.905764	0.452882	−	0.891570i	$-0.350396\pi$
$139$	10.6788	0.905764	0.452882	+	0.891570i	$0.350396\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.63514	−0.136737
$144$	0	0
$145$	8.81259	0.731846
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.825490	0.0676267	0.0338134	−	0.999428i	$-0.489235\pi$
$149$	0.825490	0.0676267	0.0338134	+	0.999428i	$0.489235\pi$
$150$	0	0
$151$	0.922772	0.0750941	0.0375471	−	0.999295i	$-0.488046\pi$
$151$	0.922772	0.0750941	0.0375471	+	0.999295i	$0.488046\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−17.4568	−1.40217
$156$	0	0
$157$	6.45146	0.514882	0.257441	−	0.966294i	$-0.417121\pi$
$157$	6.45146	0.514882	0.257441	+	0.966294i	$0.417121\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.583123	−0.0459565
$162$	0	0
$163$	5.39007	0.422183	0.211091	−	0.977466i	$-0.432298\pi$
$163$	5.39007	0.422183	0.211091	+	0.977466i	$0.432298\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.40296	−0.340712	−0.170356	−	0.985383i	$-0.554492\pi$
$167$	−4.40296	−0.340712	−0.170356	+	0.985383i	$0.554492\pi$
$168$	0	0
$169$	−9.20537	−0.708105
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.61447	0.274803	0.137402	−	0.990515i	$-0.456125\pi$
$173$	3.61447	0.274803	0.137402	+	0.990515i	$0.456125\pi$
$174$	0	0
$175$	−2.43431	−0.184017
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.13751	−0.309252	−0.154626	−	0.987973i	$-0.549417\pi$
$179$	−4.13751	−0.309252	−0.154626	+	0.987973i	$0.549417\pi$
$180$	0	0
$181$	8.59544	0.638894	0.319447	−	0.947604i	$-0.396503\pi$
$181$	8.59544	0.638894	0.319447	+	0.947604i	$0.396503\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.93590	0.436416
$186$	0	0
$187$	−3.51739	−0.257217
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−22.0652	−1.59658	−0.798289	−	0.602274i	$-0.794261\pi$
$191$	−22.0652	−1.59658	−0.798289	+	0.602274i	$0.794261\pi$
$192$	0	0
$193$	2.74101	0.197302	0.0986512	−	0.995122i	$-0.468547\pi$
$193$	2.74101	0.197302	0.0986512	+	0.995122i	$0.468547\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.63968	0.544305	0.272152	−	0.962254i	$-0.412265\pi$
$197$	7.63968	0.544305	0.272152	+	0.962254i	$0.412265\pi$
$198$	0	0
$199$	16.6045	1.17706	0.588532	−	0.808474i	$-0.299706\pi$
$199$	16.6045	1.17706	0.588532	+	0.808474i	$0.299706\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.97319	−0.559608
$204$	0	0
$205$	10.5864	0.739383
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.899487	0.0622188
$210$	0	0
$211$	−18.0316	−1.24135	−0.620673	−	0.784070i	$-0.713140\pi$
$211$	−18.0316	−1.24135	−0.620673	+	0.784070i	$0.713140\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.91069	−0.130308
$216$	0	0
$217$	15.7940	1.07217
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.16272	0.549084
$222$	0	0
$223$	6.85413	0.458987	0.229493	−	0.973310i	$-0.426293\pi$
$223$	6.85413	0.458987	0.229493	+	0.973310i	$0.426293\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.48121	0.496545	0.248273	−	0.968690i	$-0.420137\pi$
$227$	7.48121	0.496545	0.248273	+	0.968690i	$0.420137\pi$
$228$	0	0
$229$	9.55067	0.631126	0.315563	−	0.948905i	$-0.397807\pi$
$229$	9.55067	0.631126	0.315563	+	0.948905i	$0.397807\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.0595024	−0.00389813	−0.00194907	−	0.999998i	$-0.500620\pi$
$233$	−0.0595024	−0.00389813	−0.00194907	+	0.999998i	$0.500620\pi$
$234$	0	0
$235$	19.8574	1.29535
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.7311	1.08224	0.541121	−	0.840945i	$-0.318000\pi$
$239$	16.7311	1.08224	0.541121	+	0.840945i	$0.318000\pi$
$240$	0	0
$241$	−16.0595	−1.03448	−0.517242	−	0.855839i	$-0.673041\pi$
$241$	−16.0595	−1.03448	−0.517242	+	0.855839i	$0.673041\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	7.70107	0.492004
$246$	0	0
$247$	−2.08742	−0.132819
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−0.346116	−0.0218466	−0.0109233	−	0.999940i	$-0.503477\pi$
$251$	−0.346116	−0.0218466	−0.0109233	+	0.999940i	$0.503477\pi$
$252$	0	0
$253$	−0.286031	−0.0179826
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.26758	0.266204	0.133102	−	0.991102i	$-0.457506\pi$
$257$	4.26758	0.266204	0.133102	+	0.991102i	$0.457506\pi$
$258$	0	0
$259$	−5.37051	−0.333707
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.73054	0.353360	0.176680	−	0.984268i	$-0.443464\pi$
$263$	5.73054	0.353360	0.176680	+	0.984268i	$0.443464\pi$
$264$	0	0
$265$	−17.2230	−1.05800
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.7557	1.50938	0.754692	−	0.656079i	$-0.227786\pi$
$269$	24.7557	1.50938	0.754692	+	0.656079i	$0.227786\pi$
$270$	0	0
$271$	1.94909	0.118399	0.0591993	−	0.998246i	$-0.481145\pi$
$271$	1.94909	0.118399	0.0591993	+	0.998246i	$0.481145\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.19407	−0.0720052
$276$	0	0
$277$	−2.85684	−0.171651	−0.0858254	−	0.996310i	$-0.527353\pi$
$277$	−2.85684	−0.171651	−0.0858254	+	0.996310i	$0.527353\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	30.0450	1.79233	0.896167	−	0.443716i	$-0.146340\pi$
$281$	30.0450	1.79233	0.896167	+	0.443716i	$0.146340\pi$
$282$	0	0
$283$	−21.1978	−1.26008	−0.630040	−	0.776563i	$-0.716962\pi$
$283$	−21.1978	−1.26008	−0.630040	+	0.776563i	$0.716962\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.57800	−0.565371
$288$	0	0
$289$	0.559020	0.0328835
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.7587	0.628532	0.314266	−	0.949335i	$-0.398242\pi$
$293$	10.7587	0.628532	0.314266	+	0.949335i	$0.398242\pi$
$294$	0	0
$295$	−1.89142	−0.110123
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.663786	0.0383877
$300$	0	0
$301$	1.72870	0.0996406
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.12066	0.0641686
$306$	0	0
$307$	−11.3877	−0.649928	−0.324964	−	0.945726i	$-0.605352\pi$
$307$	−11.3877	−0.649928	−0.324964	+	0.945726i	$0.605352\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.4375	−0.705268	−0.352634	−	0.935761i	$-0.614714\pi$
$311$	−12.4375	−0.705268	−0.352634	+	0.935761i	$0.614714\pi$
$312$	0	0
$313$	13.0734	0.738953	0.369477	−	0.929240i	$-0.379537\pi$
$313$	13.0734	0.738953	0.369477	+	0.929240i	$0.379537\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.3413	1.47947	0.739737	−	0.672896i	$-0.234950\pi$
$317$	26.3413	1.47947	0.739737	+	0.672896i	$0.234950\pi$
$318$	0	0
$319$	−3.91098	−0.218973
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.49029	−0.249847
$324$	0	0
$325$	2.77105	0.153710
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−17.9659	−0.990495
$330$	0	0
$331$	1.64877	0.0906244	0.0453122	−	0.998973i	$-0.485572\pi$
$331$	1.64877	0.0906244	0.0453122	+	0.998973i	$0.485572\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.59221	0.141627
$336$	0	0
$337$	20.2096	1.10089	0.550444	−	0.834872i	$-0.314458\pi$
$337$	20.2096	1.10089	0.550444	+	0.834872i	$0.314458\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.74724	0.419537
$342$	0	0
$343$	−18.9464	−1.02301
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.8257	−1.33271	−0.666357	−	0.745632i	$-0.732147\pi$
$347$	−24.8257	−1.33271	−0.666357	+	0.745632i	$0.732147\pi$
$348$	0	0
$349$	0.793032	0.0424501	0.0212250	−	0.999775i	$-0.493243\pi$
$349$	0.793032	0.0424501	0.0212250	+	0.999775i	$0.493243\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.70378	0.0906830	0.0453415	−	0.998972i	$-0.485562\pi$
$353$	1.70378	0.0906830	0.0453415	+	0.998972i	$0.485562\pi$
$354$	0	0
$355$	21.2027	1.12532
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.0504	0.952665	0.476332	−	0.879265i	$-0.341966\pi$
$359$	18.0504	0.952665	0.476332	+	0.879265i	$0.341966\pi$
$360$	0	0
$361$	−17.8517	−0.939564
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−16.0938	−0.842388
$366$	0	0
$367$	12.8525	0.670897	0.335448	−	0.942059i	$-0.391112\pi$
$367$	12.8525	0.670897	0.335448	+	0.942059i	$0.391112\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15.5825	0.809005
$372$	0	0
$373$	1.07158	0.0554843	0.0277422	−	0.999615i	$-0.491168\pi$
$373$	1.07158	0.0554843	0.0277422	+	0.999615i	$0.491168\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.07612	0.467444
$378$	0	0
$379$	0.801009	0.0411451	0.0205725	−	0.999788i	$-0.493451\pi$
$379$	0.801009	0.0411451	0.0205725	+	0.999788i	$0.493451\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.54240	−0.181008	−0.0905041	−	0.995896i	$-0.528848\pi$
$383$	−3.54240	−0.181008	−0.0905041	+	0.995896i	$0.528848\pi$
$384$	0	0
$385$	−2.71691	−0.138467
$386$	0	0
$387$	0	0
$388$	0	0
$389$	28.9882	1.46976	0.734880	−	0.678197i	$-0.237238\pi$
$389$	28.9882	1.46976	0.734880	+	0.678197i	$0.237238\pi$
$390$	0	0
$391$	1.42789	0.0722113
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21.5997	1.08680
$396$	0	0
$397$	−13.9962	−0.702449	−0.351224	−	0.936291i	$-0.614235\pi$
$397$	−13.9962	−0.702449	−0.351224	+	0.936291i	$0.614235\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.18581	−0.159092	−0.0795458	−	0.996831i	$-0.525347\pi$
$401$	−3.18581	−0.159092	−0.0795458	+	0.996831i	$0.525347\pi$
$402$	0	0
$403$	−17.9788	−0.895590
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.63432	−0.130579
$408$	0	0
$409$	−10.5775	−0.523022	−0.261511	−	0.965200i	$-0.584221\pi$
$409$	−10.5775	−0.523022	−0.261511	+	0.965200i	$0.584221\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.71126	0.0842057
$414$	0	0
$415$	6.21180	0.304925
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15.3877	0.751736	0.375868	−	0.926673i	$-0.377345\pi$
$419$	15.3877	0.751736	0.375868	+	0.926673i	$0.377345\pi$
$420$	0	0
$421$	13.5040	0.658145	0.329073	−	0.944305i	$-0.393264\pi$
$421$	13.5040	0.658145	0.329073	+	0.944305i	$0.393264\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.96088	0.289145
$426$	0	0
$427$	−1.01391	−0.0490667
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.31153	0.304016	0.152008	−	0.988379i	$-0.451426\pi$
$431$	6.31153	0.304016	0.152008	+	0.988379i	$0.451426\pi$
$432$	0	0
$433$	−3.43190	−0.164926	−0.0824632	−	0.996594i	$-0.526279\pi$
$433$	−3.43190	−0.164926	−0.0824632	+	0.996594i	$0.526279\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.365147	−0.0174674
$438$	0	0
$439$	−20.6780	−0.986908	−0.493454	−	0.869772i	$-0.664266\pi$
$439$	−20.6780	−0.986908	−0.493454	+	0.869772i	$0.664266\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.523038	−0.0248503	−0.0124251	−	0.999923i	$-0.503955\pi$
$443$	−0.523038	−0.0248503	−0.0124251	+	0.999923i	$0.503955\pi$
$444$	0	0
$445$	4.92277	0.233362
$446$	0	0
$447$	0	0
$448$	0	0
$449$	28.3504	1.33794	0.668968	−	0.743291i	$-0.266736\pi$
$449$	28.3504	1.33794	0.668968	+	0.743291i	$0.266736\pi$
$450$	0	0
$451$	−4.69817	−0.221228
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.30507	0.295586
$456$	0	0
$457$	−8.73729	−0.408713	−0.204357	−	0.978897i	$-0.565510\pi$
$457$	−8.73729	−0.408713	−0.204357	+	0.978897i	$0.565510\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.24126	0.430408	0.215204	−	0.976569i	$-0.430958\pi$
$461$	9.24126	0.430408	0.215204	+	0.976569i	$0.430958\pi$
$462$	0	0
$463$	−6.86167	−0.318889	−0.159444	−	0.987207i	$-0.550970\pi$
$463$	−6.86167	−0.318889	−0.159444	+	0.987207i	$0.550970\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.1174	1.48621	0.743107	−	0.669172i	$-0.233351\pi$
$467$	32.1174	1.48621	0.743107	+	0.669172i	$0.233351\pi$
$468$	0	0
$469$	−2.34530	−0.108296
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.847956	0.0389891
$474$	0	0
$475$	−1.52435	−0.0699420
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−41.5516	−1.89854	−0.949271	−	0.314458i	$-0.898177\pi$
$479$	−41.5516	−1.89854	−0.949271	+	0.314458i	$0.898177\pi$
$480$	0	0
$481$	6.11341	0.278747
$482$	0	0
$483$	0	0
$484$	0	0
$485$	21.3665	0.970201
$486$	0	0
$487$	−25.1407	−1.13924	−0.569618	−	0.821910i	$-0.692909\pi$
$487$	−25.1407	−1.13924	−0.569618	+	0.821910i	$0.692909\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15.3408	−0.692319	−0.346159	−	0.938176i	$-0.612514\pi$
$491$	−15.3408	−0.692319	−0.346159	+	0.938176i	$0.612514\pi$
$492$	0	0
$493$	19.5239	0.879310
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−19.1831	−0.860480
$498$	0	0
$499$	−4.93509	−0.220925	−0.110462	−	0.993880i	$-0.535233\pi$
$499$	−4.93509	−0.220925	−0.110462	+	0.993880i	$0.535233\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	30.4986	1.35987	0.679934	−	0.733274i	$-0.262009\pi$
$503$	30.4986	1.35987	0.679934	+	0.733274i	$0.262009\pi$
$504$	0	0
$505$	27.9555	1.24400
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.5496	0.689222	0.344611	−	0.938746i	$-0.388011\pi$
$509$	15.5496	0.689222	0.344611	+	0.938746i	$0.388011\pi$
$510$	0	0
$511$	14.5609	0.644134
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.17533	−0.139922
$516$	0	0
$517$	−8.81259	−0.387577
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17.0051	−0.745005	−0.372503	−	0.928031i	$-0.621500\pi$
$521$	−17.0051	−0.745005	−0.372503	+	0.928031i	$0.621500\pi$
$522$	0	0
$523$	−0.951794	−0.0416190	−0.0208095	−	0.999783i	$-0.506624\pi$
$523$	−0.951794	−0.0416190	−0.0208095	+	0.999783i	$0.506624\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−38.6747	−1.68470
$528$	0	0
$529$	−22.8839	−0.994952
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.9029	0.472258
$534$	0	0
$535$	−9.32728	−0.403254
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.41770	−0.147211
$540$	0	0
$541$	30.4906	1.31089	0.655446	−	0.755242i	$-0.272481\pi$
$541$	30.4906	1.31089	0.655446	+	0.755242i	$0.272481\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.4688	0.962460
$546$	0	0
$547$	4.87317	0.208362	0.104181	−	0.994558i	$-0.466778\pi$
$547$	4.87317	0.208362	0.104181	+	0.994558i	$0.466778\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.99275	−0.212698
$552$	0	0
$553$	−19.5423	−0.831024
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.7855	0.541741	0.270871	−	0.962616i	$-0.412688\pi$
$557$	12.7855	0.541741	0.270871	+	0.962616i	$0.412688\pi$
$558$	0	0
$559$	−1.96783	−0.0832304
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.657120	0.0276943	0.0138472	−	0.999904i	$-0.495592\pi$
$563$	0.657120	0.0276943	0.0138472	+	0.999904i	$0.495592\pi$
$564$	0	0
$565$	25.2187	1.06096
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.8890	0.624179	0.312090	−	0.950053i	$-0.398971\pi$
$569$	14.8890	0.624179	0.312090	+	0.950053i	$0.398971\pi$
$570$	0	0
$571$	−15.8204	−0.662062	−0.331031	−	0.943620i	$-0.607396\pi$
$571$	−15.8204	−0.662062	−0.331031	+	0.943620i	$0.607396\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.484734	0.0202148
$576$	0	0
$577$	−18.0455	−0.751244	−0.375622	−	0.926773i	$-0.622571\pi$
$577$	−18.0455	−0.751244	−0.375622	+	0.926773i	$0.622571\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.62012	−0.233162
$582$	0	0
$583$	7.64350	0.316561
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.2678	1.16674	0.583369	−	0.812207i	$-0.301734\pi$
$587$	28.2678	1.16674	0.583369	+	0.812207i	$0.301734\pi$
$588$	0	0
$589$	9.89012	0.407515
$590$	0	0
$591$	0	0
$592$	0	0
$593$	45.0555	1.85021	0.925104	−	0.379715i	$-0.123978\pi$
$593$	45.0555	1.85021	0.925104	+	0.379715i	$0.123978\pi$
$594$	0	0
$595$	13.5630	0.556028
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.86272	0.402980	0.201490	−	0.979491i	$-0.435422\pi$
$599$	9.86272	0.402980	0.201490	+	0.979491i	$0.435422\pi$
$600$	0	0
$601$	9.74584	0.397541	0.198771	−	0.980046i	$-0.436305\pi$
$601$	9.74584	0.397541	0.198771	+	0.980046i	$0.436305\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	19.4729	0.791688
$606$	0	0
$607$	43.3940	1.76131	0.880655	−	0.473759i	$-0.157103\pi$
$607$	43.3940	1.76131	0.880655	+	0.473759i	$0.157103\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	20.4512	0.827366
$612$	0	0
$613$	5.65843	0.228542	0.114271	−	0.993450i	$-0.463547\pi$
$613$	5.65843	0.228542	0.114271	+	0.993450i	$0.463547\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29.7470	−1.19757	−0.598784	−	0.800911i	$-0.704349\pi$
$617$	−29.7470	−1.19757	−0.598784	+	0.800911i	$0.704349\pi$
$618$	0	0
$619$	24.7587	0.995136	0.497568	−	0.867425i	$-0.334227\pi$
$619$	24.7587	0.995136	0.497568	+	0.867425i	$0.334227\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.45388	−0.178441
$624$	0	0
$625$	−15.8638	−0.634552
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.1507	0.524353
$630$	0	0
$631$	−41.5297	−1.65327	−0.826636	−	0.562736i	$-0.809749\pi$
$631$	−41.5297	−1.65327	−0.826636	+	0.562736i	$0.809749\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.25125	0.287757
$636$	0	0
$637$	7.93136	0.314252
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.38360	−0.252137	−0.126069	−	0.992022i	$-0.540236\pi$
$641$	−6.38360	−0.252137	−0.126069	+	0.992022i	$0.540236\pi$
$642$	0	0
$643$	39.8185	1.57029	0.785144	−	0.619314i	$-0.212589\pi$
$643$	39.8185	1.57029	0.785144	+	0.619314i	$0.212589\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29.6552	1.16587	0.582934	−	0.812520i	$-0.301905\pi$
$647$	29.6552	1.16587	0.582934	+	0.812520i	$0.301905\pi$
$648$	0	0
$649$	0.839403	0.0329494
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.9332	−0.466982	−0.233491	−	0.972359i	$-0.575015\pi$
$653$	−11.9332	−0.466982	−0.233491	+	0.972359i	$0.575015\pi$
$654$	0	0
$655$	23.8633	0.932416
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−38.5171	−1.50041	−0.750206	−	0.661205i	$-0.770046\pi$
$659$	−38.5171	−1.50041	−0.750206	+	0.661205i	$0.770046\pi$
$660$	0	0
$661$	4.14983	0.161410	0.0807048	−	0.996738i	$-0.474283\pi$
$661$	4.14983	0.161410	0.0807048	+	0.996738i	$0.474283\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.46840	−0.134499
$666$	0	0
$667$	1.58766	0.0614746
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.497341	−0.0191996
$672$	0	0
$673$	−31.1163	−1.19945	−0.599723	−	0.800208i	$-0.704722\pi$
$673$	−31.1163	−1.19945	−0.599723	+	0.800208i	$0.704722\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.5544	0.520937	0.260469	−	0.965482i	$-0.416123\pi$
$677$	13.5544	0.520937	0.260469	+	0.965482i	$0.416123\pi$
$678$	0	0
$679$	−19.3313	−0.741867
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10.6914	−0.409095	−0.204548	−	0.978857i	$-0.565572\pi$
$683$	−10.6914	−0.409095	−0.204548	+	0.978857i	$0.565572\pi$
$684$	0	0
$685$	3.09274	0.118168
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−17.7381	−0.675767
$690$	0	0
$691$	17.5957	0.669373	0.334686	−	0.942330i	$-0.391370\pi$
$691$	17.5957	0.669373	0.334686	+	0.942330i	$0.391370\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.1981	−0.766158
$696$	0	0
$697$	23.4535	0.888366
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.04395	−0.0771990	−0.0385995	−	0.999255i	$-0.512290\pi$
$701$	−2.04395	−0.0771990	−0.0385995	+	0.999255i	$0.512290\pi$
$702$	0	0
$703$	−3.36297	−0.126837
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−25.2927	−0.951230
$708$	0	0
$709$	5.63968	0.211803	0.105901	−	0.994377i	$-0.466227\pi$
$709$	5.63968	0.211803	0.105901	+	0.994377i	$0.466227\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.14500	−0.117781
$714$	0	0
$715$	3.09274	0.115662
$716$	0	0
$717$	0	0
$718$	0	0
$719$	29.0582	1.08369	0.541844	−	0.840479i	$-0.317726\pi$
$719$	29.0582	1.08369	0.541844	+	0.840479i	$0.317726\pi$
$720$	0	0
$721$	2.87288	0.106992
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.62789	0.246154
$726$	0	0
$727$	−28.8901	−1.07147	−0.535736	−	0.844385i	$-0.679966\pi$
$727$	−28.8901	−1.07147	−0.535736	+	0.844385i	$0.679966\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.23305	−0.156565
$732$	0	0
$733$	−35.1400	−1.29792	−0.648962	−	0.760821i	$-0.724797\pi$
$733$	−35.1400	−1.29792	−0.648962	+	0.760821i	$0.724797\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.15041	−0.0423758
$738$	0	0
$739$	7.42629	0.273180	0.136590	−	0.990628i	$-0.456386\pi$
$739$	7.42629	0.273180	0.136590	+	0.990628i	$0.456386\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.5431	1.23058	0.615289	−	0.788302i	$-0.289039\pi$
$743$	33.5431	1.23058	0.615289	+	0.788302i	$0.289039\pi$
$744$	0	0
$745$	−1.56135	−0.0572034
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.43886	0.308349
$750$	0	0
$751$	16.3345	0.596055	0.298028	−	0.954557i	$-0.403671\pi$
$751$	16.3345	0.596055	0.298028	+	0.954557i	$0.403671\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.74535	−0.0635198
$756$	0	0
$757$	−35.2555	−1.28138	−0.640691	−	0.767798i	$-0.721352\pi$
$757$	−35.2555	−1.28138	−0.640691	+	0.767798i	$0.721352\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.2517	−1.24162	−0.620812	−	0.783959i	$-0.713197\pi$
$761$	−34.2517	−1.24162	−0.620812	+	0.783959i	$0.713197\pi$
$762$	0	0
$763$	−20.3287	−0.735948
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.94798	−0.0703375
$768$	0	0
$769$	11.3833	0.410493	0.205246	−	0.978710i	$-0.434200\pi$
$769$	11.3833	0.410493	0.205246	+	0.978710i	$0.434200\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	54.6916	1.96712	0.983561	−	0.180577i	$-0.0577965\pi$
$773$	54.6916	1.96712	0.983561	+	0.180577i	$0.0577965\pi$
$774$	0	0
$775$	−13.1292	−0.471613
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.99767	−0.214889
$780$	0	0
$781$	−9.40963	−0.336703
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.2024	−0.435523
$786$	0	0
$787$	30.0841	1.07238	0.536192	−	0.844096i	$-0.319862\pi$
$787$	30.0841	1.07238	0.536192	+	0.844096i	$0.319862\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−22.8166	−0.811266
$792$	0	0
$793$	1.15417	0.0409857
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−46.1882	−1.63607	−0.818034	−	0.575169i	$-0.804936\pi$
$797$	−46.1882	−1.63607	−0.818034	+	0.575169i	$0.804936\pi$
$798$	0	0
$799$	43.9930	1.55636
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.14234	0.252048
$804$	0	0
$805$	1.10293	0.0388732
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.3305	−1.34763	−0.673814	−	0.738901i	$-0.735345\pi$
$809$	−38.3305	−1.34763	−0.673814	+	0.738901i	$0.735345\pi$
$810$	0	0
$811$	−39.7774	−1.39677	−0.698387	−	0.715720i	$-0.746099\pi$
$811$	−39.7774	−1.39677	−0.698387	+	0.715720i	$0.746099\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.1949	−0.357111
$816$	0	0
$817$	1.08250	0.0378718
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.8585	0.553464	0.276732	−	0.960947i	$-0.410748\pi$
$821$	15.8585	0.553464	0.276732	+	0.960947i	$0.410748\pi$
$822$	0	0
$823$	33.2123	1.15771	0.578854	−	0.815432i	$-0.303500\pi$
$823$	33.2123	1.15771	0.578854	+	0.815432i	$0.303500\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.92310	0.0668726	0.0334363	−	0.999441i	$-0.489355\pi$
$827$	1.92310	0.0668726	0.0334363	+	0.999441i	$0.489355\pi$
$828$	0	0
$829$	39.8436	1.38383	0.691913	−	0.721981i	$-0.256768\pi$
$829$	39.8436	1.38383	0.691913	+	0.721981i	$0.256768\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	17.0613	0.591140
$834$	0	0
$835$	8.32786	0.288197
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.17026	−0.178497	−0.0892485	−	0.996009i	$-0.528447\pi$
$839$	−5.17026	−0.178497	−0.0892485	+	0.996009i	$0.528447\pi$
$840$	0	0
$841$	−7.29144	−0.251429
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17.4112	0.598964
$846$	0	0
$847$	−17.6181	−0.605366
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.06940	0.0366587
$852$	0	0
$853$	−23.3541	−0.799630	−0.399815	−	0.916596i	$-0.630926\pi$
$853$	−23.3541	−0.799630	−0.399815	+	0.916596i	$0.630926\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.2432	−0.828131	−0.414066	−	0.910247i	$-0.635892\pi$
$857$	−24.2432	−0.828131	−0.414066	+	0.910247i	$0.635892\pi$
$858$	0	0
$859$	−1.01767	−0.0347226	−0.0173613	−	0.999849i	$-0.505527\pi$
$859$	−1.01767	−0.0347226	−0.0173613	+	0.999849i	$0.505527\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.9519	0.849374	0.424687	−	0.905340i	$-0.360384\pi$
$863$	24.9519	0.849374	0.424687	+	0.905340i	$0.360384\pi$
$864$	0	0
$865$	−6.83649	−0.232448
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9.58583	−0.325177
$870$	0	0
$871$	2.66972	0.0904601
$872$	0	0
$873$	0	0
$874$	0	0
$875$	20.7879	0.702760
$876$	0	0
$877$	−48.8396	−1.64920	−0.824599	−	0.565718i	$-0.808599\pi$
$877$	−48.8396	−1.64920	−0.824599	+	0.565718i	$0.808599\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.99092	−0.269221	−0.134610	−	0.990899i	$-0.542978\pi$
$881$	−7.99092	−0.269221	−0.134610	+	0.990899i	$0.542978\pi$
$882$	0	0
$883$	−52.4981	−1.76670	−0.883350	−	0.468713i	$-0.844718\pi$
$883$	−52.4981	−1.76670	−0.883350	+	0.468713i	$0.844718\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.1665	−0.845008	−0.422504	−	0.906361i	$-0.638849\pi$
$887$	−25.1665	−0.845008	−0.422504	+	0.906361i	$0.638849\pi$
$888$	0	0
$889$	−6.56056	−0.220034
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.2501	−0.376472
$894$	0	0
$895$	7.82578	0.261587
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−43.0024	−1.43421
$900$	0	0
$901$	−38.1568	−1.27119
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.2576	−0.540421
$906$	0	0
$907$	−22.6847	−0.753233	−0.376616	−	0.926369i	$-0.622913\pi$
$907$	−22.6847	−0.753233	−0.376616	+	0.926369i	$0.622913\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−38.0169	−1.25955	−0.629777	−	0.776776i	$-0.716854\pi$
$911$	−38.0169	−1.25955	−0.629777	+	0.776776i	$0.716854\pi$
$912$	0	0
$913$	−2.75676	−0.0912355
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21.5903	−0.712975
$918$	0	0
$919$	−0.517972	−0.0170863	−0.00854316	−	0.999964i	$-0.502719\pi$
$919$	−0.517972	−0.0170863	−0.00854316	+	0.999964i	$0.502719\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	21.8367	0.718764
$924$	0	0
$925$	4.46435	0.146787
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7.52034	−0.246734	−0.123367	−	0.992361i	$-0.539369\pi$
$929$	−7.52034	−0.246734	−0.123367	+	0.992361i	$0.539369\pi$
$930$	0	0
$931$	−4.36302	−0.142992
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.65287	0.217572
$936$	0	0
$937$	33.5935	1.09745	0.548727	−	0.836002i	$-0.315113\pi$
$937$	33.5935	1.09745	0.548727	+	0.836002i	$0.315113\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−54.6463	−1.78142	−0.890709	−	0.454574i	$-0.849792\pi$
$941$	−54.6463	−1.78142	−0.890709	+	0.454574i	$0.849792\pi$
$942$	0	0
$943$	1.90722	0.0621077
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.2356	−1.04752	−0.523759	−	0.851867i	$-0.675471\pi$
$947$	−32.2356	−1.04752	−0.523759	+	0.851867i	$0.675471\pi$
$948$	0	0
$949$	−16.5751	−0.538049
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14.5979	−0.472871	−0.236435	−	0.971647i	$-0.575979\pi$
$953$	−14.5979	−0.472871	−0.236435	+	0.971647i	$0.575979\pi$
$954$	0	0
$955$	41.7345	1.35050
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.79816	−0.0903572
$960$	0	0
$961$	54.1831	1.74784
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.18441	−0.166892
$966$	0	0
$967$	5.08791	0.163616	0.0818081	−	0.996648i	$-0.473931\pi$
$967$	5.08791	0.163616	0.0818081	+	0.996648i	$0.473931\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42.3491	−1.35905	−0.679523	−	0.733654i	$-0.737813\pi$
$971$	−42.3491	−1.35905	−0.679523	+	0.733654i	$0.737813\pi$
$972$	0	0
$973$	18.2742	0.585845
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42.9713	−1.37477	−0.687386	−	0.726292i	$-0.741242\pi$
$977$	−42.9713	−1.37477	−0.687386	+	0.726292i	$0.741242\pi$
$978$	0	0
$979$	−2.18470	−0.0698233
$980$	0	0
$981$	0	0
$982$	0	0
$983$	31.9702	1.01969	0.509845	−	0.860266i	$-0.329703\pi$
$983$	31.9702	1.01969	0.509845	+	0.860266i	$0.329703\pi$
$984$	0	0
$985$	−14.4499	−0.460411
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.344228	−0.0109458
$990$	0	0
$991$	−0.143249	−0.00455044	−0.00227522	−	0.999997i	$-0.500724\pi$
$991$	−0.143249	−0.00455044	−0.00227522	+	0.999997i	$0.500724\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−31.4061	−0.995642
$996$	0	0
$997$	11.0116	0.348740	0.174370	−	0.984680i	$-0.444211\pi$
$997$	11.0116	0.348740	0.174370	+	0.984680i	$0.444211\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2124.2.a.i.1.2	✓	4
3.2	odd	2		2124.2.a.j.1.3	yes	4
4.3	odd	2		8496.2.a.bs.1.2		4
12.11	even	2		8496.2.a.bp.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2124.2.a.i.1.2	✓	4	1.1	even	1	trivial
2124.2.a.j.1.3	yes	4	3.2	odd	2
8496.2.a.bp.1.3		4	12.11	even	2
8496.2.a.bs.1.2		4	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:	\( N \)	\(=\)	\( 2124 = 2^{2} \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2124.a (trivial)

\(f(q)\)	\(=\)	\(q-1.89142 q^{5} +1.71126 q^{7} +O(q^{10})\)	\(q-1.89142 q^{5} +1.71126 q^{7} +0.839403 q^{11} -1.94798 q^{13} -4.19035 q^{17} +1.07158 q^{19} -0.340756 q^{23} -1.42253 q^{25} -4.65924 q^{29} +9.22947 q^{31} -3.23672 q^{35} -3.13833 q^{37} -5.59704 q^{41} +1.01019 q^{43} -10.4986 q^{47} -4.07158 q^{49} +9.10587 q^{53} -1.58766 q^{55} +1.00000 q^{59} -0.592494 q^{61} +3.68445 q^{65} -1.37051 q^{67} -11.2099 q^{71} +8.50884 q^{73} +1.43644 q^{77} -11.4198 q^{79} -3.28420 q^{83} +7.92571 q^{85} -2.60268 q^{89} -3.33351 q^{91} -2.02681 q^{95} -11.2965 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^7	\( 4 q + 2 q^{7} - 10 q^{11} - 2 q^{13} - 4 q^{17} - 6 q^{19} - 12 q^{23} + 4 q^{25} - 8 q^{29} - 8 q^{31} - 12 q^{35} + 6 q^{37} - 16 q^{41} - 6 q^{43} - 18 q^{47} - 6 q^{49} - 4 q^{53} - 6 q^{55} + 4 q^{59} - 18 q^{65} + 10 q^{67} - 16 q^{71} - 14 q^{77} + 12 q^{79} - 22 q^{83} - 6 q^{85} + 2 q^{89} + 20 q^{91} - 36 q^{95} + 6 q^{97}+O(q^{100}) \) 4 * q + 2 * q^7 - 10 * q^11 - 2 * q^13 - 4 * q^17 - 6 * q^19 - 12 * q^23 + 4 * q^25 - 8 * q^29 - 8 * q^31 - 12 * q^35 + 6 * q^37 - 16 * q^41 - 6 * q^43 - 18 * q^47 - 6 * q^49 - 4 * q^53 - 6 * q^55 + 4 * q^59 - 18 * q^65 + 10 * q^67 - 16 * q^71 - 14 * q^77 + 12 * q^79 - 22 * q^83 - 6 * q^85 + 2 * q^89 + 20 * q^91 - 36 * q^95 + 6 * q^97

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