LMFDB - Embedded newform 9600.2.a.dc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9600,2,Mod(1,9600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9600 = 2^{7} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9600.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:	$76.6563859404$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
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			1.1
Root			$-3.16228$ of defining polynomial
Character	$\chi$	$=$	9600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.16228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.16228 q^{7} +1.00000 q^{9} -3.16228 q^{11} +1.00000 q^{13} -1.16228 q^{17} -4.16228 q^{19} -4.16228 q^{21} -7.16228 q^{23} +1.00000 q^{27} -5.16228 q^{29} -6.16228 q^{31} -3.16228 q^{33} +6.32456 q^{37} +1.00000 q^{39} +7.16228 q^{41} -10.4868 q^{43} +8.32456 q^{47} +10.3246 q^{49} -1.16228 q^{51} +0.837722 q^{53} -4.16228 q^{57} +8.32456 q^{59} -5.32456 q^{61} -4.16228 q^{63} +12.1623 q^{67} -7.16228 q^{69} -7.48683 q^{71} +10.3246 q^{73} +13.1623 q^{77} +8.00000 q^{79} +1.00000 q^{81} +4.32456 q^{83} -5.16228 q^{87} -4.00000 q^{89} -4.16228 q^{91} -6.16228 q^{93} +15.0000 q^{97} -3.16228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + 2 q^{13} + 4 q^{17} - 2 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{27} - 4 q^{29} - 6 q^{31} + 2 q^{39} + 8 q^{41} - 2 q^{43} + 4 q^{47} + 8 q^{49} + 4 q^{51} + 8 q^{53} - 2 q^{57} + 4 q^{59} + 2 q^{61} - 2 q^{63} + 18 q^{67} - 8 q^{69} + 4 q^{71} + 8 q^{73} + 20 q^{77} + 16 q^{79} + 2 q^{81} - 4 q^{83} - 4 q^{87} - 8 q^{89} - 2 q^{91} - 6 q^{93} + 30 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 + 2 * q^13 + 4 * q^17 - 2 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^27 - 4 * q^29 - 6 * q^31 + 2 * q^39 + 8 * q^41 - 2 * q^43 + 4 * q^47 + 8 * q^49 + 4 * q^51 + 8 * q^53 - 2 * q^57 + 4 * q^59 + 2 * q^61 - 2 * q^63 + 18 * q^67 - 8 * q^69 + 4 * q^71 + 8 * q^73 + 20 * q^77 + 16 * q^79 + 2 * q^81 - 4 * q^83 - 4 * q^87 - 8 * q^89 - 2 * q^91 - 6 * q^93 + 30 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.16228	−1.57319	−0.786597	−	0.617467i	$-0.788159\pi$
$7$	−4.16228	−1.57319	−0.786597	+	0.617467i	$0.788159\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.16228	−0.953463	−0.476731	−	0.879049i	$-0.658179\pi$
$11$	−3.16228	−0.953463	−0.476731	+	0.879049i	$0.658179\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.16228	−0.281894	−0.140947	−	0.990017i	$-0.545015\pi$
$17$	−1.16228	−0.281894	−0.140947	+	0.990017i	$0.545015\pi$
$18$	0	0
$19$	−4.16228	−0.954892	−0.477446	−	0.878661i	$-0.658437\pi$
$19$	−4.16228	−0.954892	−0.477446	+	0.878661i	$0.658437\pi$
$20$	0	0
$21$	−4.16228	−0.908283
$22$	0	0
$23$	−7.16228	−1.49344	−0.746719	−	0.665140i	$-0.768372\pi$
$23$	−7.16228	−1.49344	−0.746719	+	0.665140i	$0.768372\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.16228	−0.958611	−0.479305	−	0.877648i	$-0.659111\pi$
$29$	−5.16228	−0.958611	−0.479305	+	0.877648i	$0.659111\pi$
$30$	0	0
$31$	−6.16228	−1.10678	−0.553389	−	0.832923i	$-0.686666\pi$
$31$	−6.16228	−1.10678	−0.553389	+	0.832923i	$0.686666\pi$
$32$	0	0
$33$	−3.16228	−0.550482
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.32456	1.03975	0.519875	−	0.854242i	$-0.325978\pi$
$37$	6.32456	1.03975	0.519875	+	0.854242i	$0.325978\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	7.16228	1.11856	0.559280	−	0.828979i	$-0.311078\pi$
$41$	7.16228	1.11856	0.559280	+	0.828979i	$0.311078\pi$
$42$	0	0
$43$	−10.4868	−1.59923	−0.799614	−	0.600515i	$-0.794962\pi$
$43$	−10.4868	−1.59923	−0.799614	+	0.600515i	$0.794962\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.32456	1.21426	0.607131	−	0.794602i	$-0.292320\pi$
$47$	8.32456	1.21426	0.607131	+	0.794602i	$0.292320\pi$
$48$	0	0
$49$	10.3246	1.47494
$50$	0	0
$51$	−1.16228	−0.162751
$52$	0	0
$53$	0.837722	0.115070	0.0575350	−	0.998343i	$-0.481676\pi$
$53$	0.837722	0.115070	0.0575350	+	0.998343i	$0.481676\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.16228	−0.551307
$58$	0	0
$59$	8.32456	1.08376	0.541882	−	0.840454i	$-0.317712\pi$
$59$	8.32456	1.08376	0.541882	+	0.840454i	$0.317712\pi$
$60$	0	0
$61$	−5.32456	−0.681739	−0.340870	−	0.940111i	$-0.610722\pi$
$61$	−5.32456	−0.681739	−0.340870	+	0.940111i	$0.610722\pi$
$62$	0	0
$63$	−4.16228	−0.524398
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.1623	1.48586	0.742929	−	0.669370i	$-0.233436\pi$
$67$	12.1623	1.48586	0.742929	+	0.669370i	$0.233436\pi$
$68$	0	0
$69$	−7.16228	−0.862237
$70$	0	0
$71$	−7.48683	−0.888524	−0.444262	−	0.895897i	$-0.646534\pi$
$71$	−7.48683	−0.888524	−0.444262	+	0.895897i	$0.646534\pi$
$72$	0	0
$73$	10.3246	1.20840	0.604199	−	0.796834i	$-0.293493\pi$
$73$	10.3246	1.20840	0.604199	+	0.796834i	$0.293493\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.1623	1.49998
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.32456	0.474682	0.237341	−	0.971426i	$-0.423724\pi$
$83$	4.32456	0.474682	0.237341	+	0.971426i	$0.423724\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.16228	−0.553454
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	−4.16228	−0.436325
$92$	0	0
$93$	−6.16228	−0.638998
$94$	0	0
$95$	0	0
$96$	0	0
$97$	15.0000	1.52302	0.761510	−	0.648154i	$-0.224459\pi$
$97$	15.0000	1.52302	0.761510	+	0.648154i	$0.224459\pi$
$98$	0	0
$99$	−3.16228	−0.317821
$100$	0	0
$101$	9.48683	0.943975	0.471988	−	0.881605i	$-0.343537\pi$
$101$	9.48683	0.943975	0.471988	+	0.881605i	$0.343537\pi$
$102$	0	0
$103$	−1.67544	−0.165086	−0.0825432	−	0.996587i	$-0.526304\pi$
$103$	−1.67544	−0.165086	−0.0825432	+	0.996587i	$0.526304\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.83772	0.661028	0.330514	−	0.943801i	$-0.392778\pi$
$107$	6.83772	0.661028	0.330514	+	0.943801i	$0.392778\pi$
$108$	0	0
$109$	3.32456	0.318435	0.159217	−	0.987244i	$-0.449103\pi$
$109$	3.32456	0.318435	0.159217	+	0.987244i	$0.449103\pi$
$110$	0	0
$111$	6.32456	0.600300
$112$	0	0
$113$	12.6491	1.18993	0.594964	−	0.803752i	$-0.297166\pi$
$113$	12.6491	1.18993	0.594964	+	0.803752i	$0.297166\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	4.83772	0.443473
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	7.16228	0.645801
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.6491	0.944956	0.472478	−	0.881343i	$-0.343360\pi$
$127$	10.6491	0.944956	0.472478	+	0.881343i	$0.343360\pi$
$128$	0	0
$129$	−10.4868	−0.923314
$130$	0	0
$131$	13.1623	1.14999	0.574997	−	0.818156i	$-0.305003\pi$
$131$	13.1623	1.14999	0.574997	+	0.818156i	$0.305003\pi$
$132$	0	0
$133$	17.3246	1.50223
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	8.32456	0.701054
$142$	0	0
$143$	−3.16228	−0.264443
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.3246	0.851555
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−16.1623	−1.31527	−0.657634	−	0.753338i	$-0.728443\pi$
$151$	−16.1623	−1.31527	−0.657634	+	0.753338i	$0.728443\pi$
$152$	0	0
$153$	−1.16228	−0.0939646
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.32456	−0.265328	−0.132664	−	0.991161i	$-0.542353\pi$
$157$	−3.32456	−0.265328	−0.132664	+	0.991161i	$0.542353\pi$
$158$	0	0
$159$	0.837722	0.0664357
$160$	0	0
$161$	29.8114	2.34947
$162$	0	0
$163$	−14.1623	−1.10928	−0.554638	−	0.832092i	$-0.687143\pi$
$163$	−14.1623	−1.10928	−0.554638	+	0.832092i	$0.687143\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.1623	−1.63759	−0.818793	−	0.574089i	$-0.805356\pi$
$167$	−21.1623	−1.63759	−0.818793	+	0.574089i	$0.805356\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−4.16228	−0.318297
$172$	0	0
$173$	−5.16228	−0.392481	−0.196240	−	0.980556i	$-0.562873\pi$
$173$	−5.16228	−0.392481	−0.196240	+	0.980556i	$0.562873\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.32456	0.625712
$178$	0	0
$179$	−12.3246	−0.921181	−0.460590	−	0.887613i	$-0.652362\pi$
$179$	−12.3246	−0.921181	−0.460590	+	0.887613i	$0.652362\pi$
$180$	0	0
$181$	−5.64911	−0.419895	−0.209948	−	0.977713i	$-0.567329\pi$
$181$	−5.64911	−0.419895	−0.209948	+	0.977713i	$0.567329\pi$
$182$	0	0
$183$	−5.32456	−0.393602
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.67544	0.268775
$188$	0	0
$189$	−4.16228	−0.302761
$190$	0	0
$191$	9.48683	0.686443	0.343222	−	0.939254i	$-0.388482\pi$
$191$	9.48683	0.686443	0.343222	+	0.939254i	$0.388482\pi$
$192$	0	0
$193$	−17.3246	−1.24705	−0.623524	−	0.781804i	$-0.714300\pi$
$193$	−17.3246	−1.24705	−0.623524	+	0.781804i	$0.714300\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−27.2982	−1.94492	−0.972459	−	0.233075i	$-0.925121\pi$
$197$	−27.2982	−1.94492	−0.972459	+	0.233075i	$0.925121\pi$
$198$	0	0
$199$	14.4868	1.02694	0.513472	−	0.858106i	$-0.328359\pi$
$199$	14.4868	1.02694	0.513472	+	0.858106i	$0.328359\pi$
$200$	0	0
$201$	12.1623	0.857861
$202$	0	0
$203$	21.4868	1.50808
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.16228	−0.497813
$208$	0	0
$209$	13.1623	0.910454
$210$	0	0
$211$	24.8114	1.70809	0.854043	−	0.520202i	$-0.174143\pi$
$211$	24.8114	1.70809	0.854043	+	0.520202i	$0.174143\pi$
$212$	0	0
$213$	−7.48683	−0.512989
$214$	0	0
$215$	0	0
$216$	0	0
$217$	25.6491	1.74118
$218$	0	0
$219$	10.3246	0.697669
$220$	0	0
$221$	−1.16228	−0.0781833
$222$	0	0
$223$	16.8114	1.12577	0.562887	−	0.826534i	$-0.309691\pi$
$223$	16.8114	1.12577	0.562887	+	0.826534i	$0.309691\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.4868	1.02790	0.513949	−	0.857821i	$-0.328182\pi$
$227$	15.4868	1.02790	0.513949	+	0.857821i	$0.328182\pi$
$228$	0	0
$229$	−10.6754	−0.705453	−0.352727	−	0.935726i	$-0.614745\pi$
$229$	−10.6754	−0.705453	−0.352727	+	0.935726i	$0.614745\pi$
$230$	0	0
$231$	13.1623	0.866014
$232$	0	0
$233$	−18.1359	−1.18813	−0.594063	−	0.804419i	$-0.702477\pi$
$233$	−18.1359	−1.18813	−0.594063	+	0.804419i	$0.702477\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	0.513167	0.0331940	0.0165970	−	0.999862i	$-0.494717\pi$
$239$	0.513167	0.0331940	0.0165970	+	0.999862i	$0.494717\pi$
$240$	0	0
$241$	−26.2982	−1.69402	−0.847009	−	0.531579i	$-0.821599\pi$
$241$	−26.2982	−1.69402	−0.847009	+	0.531579i	$0.821599\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.16228	−0.264839
$248$	0	0
$249$	4.32456	0.274058
$250$	0	0
$251$	28.6491	1.80832	0.904158	−	0.427198i	$-0.140499\pi$
$251$	28.6491	1.80832	0.904158	+	0.427198i	$0.140499\pi$
$252$	0	0
$253$	22.6491	1.42394
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−26.3246	−1.63573
$260$	0	0
$261$	−5.16228	−0.319537
$262$	0	0
$263$	−13.1623	−0.811621	−0.405810	−	0.913957i	$-0.633011\pi$
$263$	−13.1623	−0.811621	−0.405810	+	0.913957i	$0.633011\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.00000	−0.244796
$268$	0	0
$269$	5.16228	0.314750	0.157375	−	0.987539i	$-0.449697\pi$
$269$	5.16228	0.314750	0.157375	+	0.987539i	$0.449697\pi$
$270$	0	0
$271$	−13.6754	−0.830724	−0.415362	−	0.909656i	$-0.636345\pi$
$271$	−13.6754	−0.830724	−0.415362	+	0.909656i	$0.636345\pi$
$272$	0	0
$273$	−4.16228	−0.251913
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.6491	1.06043	0.530216	−	0.847863i	$-0.322111\pi$
$277$	17.6491	1.06043	0.530216	+	0.847863i	$0.322111\pi$
$278$	0	0
$279$	−6.16228	−0.368926
$280$	0	0
$281$	−20.1359	−1.20121	−0.600605	−	0.799546i	$-0.705073\pi$
$281$	−20.1359	−1.20121	−0.600605	+	0.799546i	$0.705073\pi$
$282$	0	0
$283$	8.16228	0.485197	0.242599	−	0.970127i	$-0.422000\pi$
$283$	8.16228	0.485197	0.242599	+	0.970127i	$0.422000\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−29.8114	−1.75971
$288$	0	0
$289$	−15.6491	−0.920536
$290$	0	0
$291$	15.0000	0.879316
$292$	0	0
$293$	22.6491	1.32318	0.661588	−	0.749868i	$-0.269883\pi$
$293$	22.6491	1.32318	0.661588	+	0.749868i	$0.269883\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.16228	−0.183494
$298$	0	0
$299$	−7.16228	−0.414205
$300$	0	0
$301$	43.6491	2.51589
$302$	0	0
$303$	9.48683	0.545004
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.51317	0.200507	0.100254	−	0.994962i	$-0.468035\pi$
$307$	3.51317	0.200507	0.100254	+	0.994962i	$0.468035\pi$
$308$	0	0
$309$	−1.67544	−0.0953127
$310$	0	0
$311$	8.97367	0.508850	0.254425	−	0.967093i	$-0.418114\pi$
$311$	8.97367	0.508850	0.254425	+	0.967093i	$0.418114\pi$
$312$	0	0
$313$	29.9737	1.69421	0.847106	−	0.531424i	$-0.178343\pi$
$313$	29.9737	1.69421	0.847106	+	0.531424i	$0.178343\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.6491	1.38443	0.692216	−	0.721690i	$-0.256635\pi$
$317$	24.6491	1.38443	0.692216	+	0.721690i	$0.256635\pi$
$318$	0	0
$319$	16.3246	0.914000
$320$	0	0
$321$	6.83772	0.381644
$322$	0	0
$323$	4.83772	0.269178
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.32456	0.183848
$328$	0	0
$329$	−34.6491	−1.91027
$330$	0	0
$331$	6.32456	0.347629	0.173814	−	0.984778i	$-0.444391\pi$
$331$	6.32456	0.347629	0.173814	+	0.984778i	$0.444391\pi$
$332$	0	0
$333$	6.32456	0.346583
$334$	0	0
$335$	0	0
$336$	0	0
$337$	31.3246	1.70636	0.853179	−	0.521619i	$-0.174672\pi$
$337$	31.3246	1.70636	0.853179	+	0.521619i	$0.174672\pi$
$338$	0	0
$339$	12.6491	0.687005
$340$	0	0
$341$	19.4868	1.05527
$342$	0	0
$343$	−13.8377	−0.747167
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.6754	0.841502	0.420751	−	0.907176i	$-0.361767\pi$
$347$	15.6754	0.841502	0.420751	+	0.907176i	$0.361767\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−24.1359	−1.28463	−0.642313	−	0.766442i	$-0.722025\pi$
$353$	−24.1359	−1.28463	−0.642313	+	0.766442i	$0.722025\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.83772	0.256039
$358$	0	0
$359$	27.4868	1.45070	0.725350	−	0.688380i	$-0.241678\pi$
$359$	27.4868	1.45070	0.725350	+	0.688380i	$0.241678\pi$
$360$	0	0
$361$	−1.67544	−0.0881813
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−18.8114	−0.981946	−0.490973	−	0.871175i	$-0.663359\pi$
$367$	−18.8114	−0.981946	−0.490973	+	0.871175i	$0.663359\pi$
$368$	0	0
$369$	7.16228	0.372853
$370$	0	0
$371$	−3.48683	−0.181027
$372$	0	0
$373$	21.0000	1.08734	0.543669	−	0.839299i	$-0.317035\pi$
$373$	21.0000	1.08734	0.543669	+	0.839299i	$0.317035\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.16228	−0.265871
$378$	0	0
$379$	−28.8114	−1.47994	−0.739971	−	0.672639i	$-0.765161\pi$
$379$	−28.8114	−1.47994	−0.739971	+	0.672639i	$0.765161\pi$
$380$	0	0
$381$	10.6491	0.545570
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.4868	−0.533076
$388$	0	0
$389$	−6.51317	−0.330231	−0.165115	−	0.986274i	$-0.552800\pi$
$389$	−6.51317	−0.330231	−0.165115	+	0.986274i	$0.552800\pi$
$390$	0	0
$391$	8.32456	0.420991
$392$	0	0
$393$	13.1623	0.663949
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.3509	0.619873	0.309937	−	0.950757i	$-0.399692\pi$
$397$	12.3509	0.619873	0.309937	+	0.950757i	$0.399692\pi$
$398$	0	0
$399$	17.3246	0.867313
$400$	0	0
$401$	23.1623	1.15667	0.578334	−	0.815800i	$-0.303703\pi$
$401$	23.1623	1.15667	0.578334	+	0.815800i	$0.303703\pi$
$402$	0	0
$403$	−6.16228	−0.306965
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.0000	−0.991363
$408$	0	0
$409$	−22.6754	−1.12123	−0.560614	−	0.828077i	$-0.689435\pi$
$409$	−22.6754	−1.12123	−0.560614	+	0.828077i	$0.689435\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	−34.6491	−1.70497
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.00000	0.0979404
$418$	0	0
$419$	34.4605	1.68351	0.841753	−	0.539863i	$-0.181524\pi$
$419$	34.4605	1.68351	0.841753	+	0.539863i	$0.181524\pi$
$420$	0	0
$421$	26.9737	1.31462	0.657308	−	0.753622i	$-0.271695\pi$
$421$	26.9737	1.31462	0.657308	+	0.753622i	$0.271695\pi$
$422$	0	0
$423$	8.32456	0.404754
$424$	0	0
$425$	0	0
$426$	0	0
$427$	22.1623	1.07251
$428$	0	0
$429$	−3.16228	−0.152676
$430$	0	0
$431$	−23.8114	−1.14695	−0.573477	−	0.819222i	$-0.694406\pi$
$431$	−23.8114	−1.14695	−0.573477	+	0.819222i	$0.694406\pi$
$432$	0	0
$433$	13.9737	0.671532	0.335766	−	0.941946i	$-0.391005\pi$
$433$	13.9737	0.671532	0.335766	+	0.941946i	$0.391005\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	29.8114	1.42607
$438$	0	0
$439$	−1.51317	−0.0722195	−0.0361098	−	0.999348i	$-0.511497\pi$
$439$	−1.51317	−0.0722195	−0.0361098	+	0.999348i	$0.511497\pi$
$440$	0	0
$441$	10.3246	0.491645
$442$	0	0
$443$	3.35089	0.159206	0.0796028	−	0.996827i	$-0.474635\pi$
$443$	3.35089	0.159206	0.0796028	+	0.996827i	$0.474635\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	0	0
$451$	−22.6491	−1.06650
$452$	0	0
$453$	−16.1623	−0.759370
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.64911	−0.217476	−0.108738	−	0.994070i	$-0.534681\pi$
$457$	−4.64911	−0.217476	−0.108738	+	0.994070i	$0.534681\pi$
$458$	0	0
$459$	−1.16228	−0.0542505
$460$	0	0
$461$	−7.81139	−0.363813	−0.181906	−	0.983316i	$-0.558227\pi$
$461$	−7.81139	−0.363813	−0.181906	+	0.983316i	$0.558227\pi$
$462$	0	0
$463$	−38.9737	−1.81126	−0.905630	−	0.424069i	$-0.860601\pi$
$463$	−38.9737	−1.81126	−0.905630	+	0.424069i	$0.860601\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.4868	−0.994292	−0.497146	−	0.867667i	$-0.665619\pi$
$467$	−21.4868	−0.994292	−0.497146	+	0.867667i	$0.665619\pi$
$468$	0	0
$469$	−50.6228	−2.33754
$470$	0	0
$471$	−3.32456	−0.153187
$472$	0	0
$473$	33.1623	1.52480
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.837722	0.0383567
$478$	0	0
$479$	12.3246	0.563123	0.281562	−	0.959543i	$-0.409148\pi$
$479$	12.3246	0.563123	0.281562	+	0.959543i	$0.409148\pi$
$480$	0	0
$481$	6.32456	0.288375
$482$	0	0
$483$	29.8114	1.35647
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.1359	0.595246	0.297623	−	0.954683i	$-0.403806\pi$
$487$	13.1359	0.595246	0.297623	+	0.954683i	$0.403806\pi$
$488$	0	0
$489$	−14.1623	−0.640440
$490$	0	0
$491$	−0.649111	−0.0292940	−0.0146470	−	0.999893i	$-0.504662\pi$
$491$	−0.649111	−0.0292940	−0.0146470	+	0.999893i	$0.504662\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	0	0
$495$	0	0
$496$	0	0
$497$	31.1623	1.39782
$498$	0	0
$499$	15.4605	0.692107	0.346053	−	0.938215i	$-0.387522\pi$
$499$	15.4605	0.692107	0.346053	+	0.938215i	$0.387522\pi$
$500$	0	0
$501$	−21.1623	−0.945461
$502$	0	0
$503$	−6.83772	−0.304879	−0.152439	−	0.988313i	$-0.548713\pi$
$503$	−6.83772	−0.304879	−0.152439	+	0.988313i	$0.548713\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−2.64911	−0.117420	−0.0587099	−	0.998275i	$-0.518699\pi$
$509$	−2.64911	−0.117420	−0.0587099	+	0.998275i	$0.518699\pi$
$510$	0	0
$511$	−42.9737	−1.90104
$512$	0	0
$513$	−4.16228	−0.183769
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−26.3246	−1.15775
$518$	0	0
$519$	−5.16228	−0.226599
$520$	0	0
$521$	−25.8114	−1.13082	−0.565409	−	0.824811i	$-0.691282\pi$
$521$	−25.8114	−1.13082	−0.565409	+	0.824811i	$0.691282\pi$
$522$	0	0
$523$	14.8114	0.647657	0.323828	−	0.946116i	$-0.395030\pi$
$523$	14.8114	0.647657	0.323828	+	0.946116i	$0.395030\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.16228	0.311994
$528$	0	0
$529$	28.2982	1.23036
$530$	0	0
$531$	8.32456	0.361255
$532$	0	0
$533$	7.16228	0.310233
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−12.3246	−0.531844
$538$	0	0
$539$	−32.6491	−1.40630
$540$	0	0
$541$	29.0000	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	0	0
$543$	−5.64911	−0.242427
$544$	0	0
$545$	0	0
$546$	0	0
$547$	17.3509	0.741870	0.370935	−	0.928659i	$-0.379037\pi$
$547$	17.3509	0.741870	0.370935	+	0.928659i	$0.379037\pi$
$548$	0	0
$549$	−5.32456	−0.227246
$550$	0	0
$551$	21.4868	0.915370
$552$	0	0
$553$	−33.2982	−1.41598
$554$	0	0
$555$	0	0
$556$	0	0
$557$	46.0000	1.94908	0.974541	−	0.224208i	$-0.0719796\pi$
$557$	46.0000	1.94908	0.974541	+	0.224208i	$0.0719796\pi$
$558$	0	0
$559$	−10.4868	−0.443546
$560$	0	0
$561$	3.67544	0.155177
$562$	0	0
$563$	−15.8114	−0.666371	−0.333185	−	0.942861i	$-0.608123\pi$
$563$	−15.8114	−0.666371	−0.333185	+	0.942861i	$0.608123\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.16228	−0.174799
$568$	0	0
$569$	−18.8377	−0.789718	−0.394859	−	0.918742i	$-0.629207\pi$
$569$	−18.8377	−0.789718	−0.394859	+	0.918742i	$0.629207\pi$
$570$	0	0
$571$	−36.1623	−1.51334	−0.756672	−	0.653795i	$-0.773176\pi$
$571$	−36.1623	−1.51334	−0.756672	+	0.653795i	$0.773176\pi$
$572$	0	0
$573$	9.48683	0.396318
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−12.2982	−0.511982	−0.255991	−	0.966679i	$-0.582402\pi$
$577$	−12.2982	−0.511982	−0.255991	+	0.966679i	$0.582402\pi$
$578$	0	0
$579$	−17.3246	−0.719984
$580$	0	0
$581$	−18.0000	−0.746766
$582$	0	0
$583$	−2.64911	−0.109715
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.8114	1.23045	0.615224	−	0.788352i	$-0.289066\pi$
$587$	29.8114	1.23045	0.615224	+	0.788352i	$0.289066\pi$
$588$	0	0
$589$	25.6491	1.05685
$590$	0	0
$591$	−27.2982	−1.12290
$592$	0	0
$593$	44.4605	1.82577	0.912887	−	0.408213i	$-0.133848\pi$
$593$	44.4605	1.82577	0.912887	+	0.408213i	$0.133848\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.4868	0.592907
$598$	0	0
$599$	−24.6491	−1.00714	−0.503568	−	0.863956i	$-0.667980\pi$
$599$	−24.6491	−1.00714	−0.503568	+	0.863956i	$0.667980\pi$
$600$	0	0
$601$	−0.0263340	−0.00107419	−0.000537094	−	1.00000i	$-0.500171\pi$
$601$	−0.0263340	−0.00107419	−0.000537094		1.00000i	$0.500171\pi$
$602$	0	0
$603$	12.1623	0.495286
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−36.6491	−1.48754	−0.743771	−	0.668435i	$-0.766965\pi$
$607$	−36.6491	−1.48754	−0.743771	+	0.668435i	$0.766965\pi$
$608$	0	0
$609$	21.4868	0.870690
$610$	0	0
$611$	8.32456	0.336775
$612$	0	0
$613$	3.35089	0.135341	0.0676706	−	0.997708i	$-0.478443\pi$
$613$	3.35089	0.135341	0.0676706	+	0.997708i	$0.478443\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.1359	0.569092	0.284546	−	0.958662i	$-0.408157\pi$
$617$	14.1359	0.569092	0.284546	+	0.958662i	$0.408157\pi$
$618$	0	0
$619$	−3.18861	−0.128161	−0.0640806	−	0.997945i	$-0.520411\pi$
$619$	−3.18861	−0.128161	−0.0640806	+	0.997945i	$0.520411\pi$
$620$	0	0
$621$	−7.16228	−0.287412
$622$	0	0
$623$	16.6491	0.667033
$624$	0	0
$625$	0	0
$626$	0	0
$627$	13.1623	0.525651
$628$	0	0
$629$	−7.35089	−0.293099
$630$	0	0
$631$	−44.4868	−1.77099	−0.885496	−	0.464646i	$-0.846182\pi$
$631$	−44.4868	−1.77099	−0.885496	+	0.464646i	$0.846182\pi$
$632$	0	0
$633$	24.8114	0.986164
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.3246	0.409074
$638$	0	0
$639$	−7.48683	−0.296175
$640$	0	0
$641$	44.6491	1.76353	0.881767	−	0.471685i	$-0.156354\pi$
$641$	44.6491	1.76353	0.881767	+	0.471685i	$0.156354\pi$
$642$	0	0
$643$	6.00000	0.236617	0.118308	−	0.992977i	$-0.462253\pi$
$643$	6.00000	0.236617	0.118308	+	0.992977i	$0.462253\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−40.0000	−1.57256	−0.786281	−	0.617869i	$-0.787996\pi$
$647$	−40.0000	−1.57256	−0.786281	+	0.617869i	$0.787996\pi$
$648$	0	0
$649$	−26.3246	−1.03333
$650$	0	0
$651$	25.6491	1.00527
$652$	0	0
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.3246	0.402799
$658$	0	0
$659$	−36.7851	−1.43294	−0.716471	−	0.697617i	$-0.754244\pi$
$659$	−36.7851	−1.43294	−0.716471	+	0.697617i	$0.754244\pi$
$660$	0	0
$661$	3.35089	0.130334	0.0651672	−	0.997874i	$-0.479242\pi$
$661$	3.35089	0.130334	0.0651672	+	0.997874i	$0.479242\pi$
$662$	0	0
$663$	−1.16228	−0.0451391
$664$	0	0
$665$	0	0
$666$	0	0
$667$	36.9737	1.43163
$668$	0	0
$669$	16.8114	0.649966
$670$	0	0
$671$	16.8377	0.650013
$672$	0	0
$673$	−43.6228	−1.68153	−0.840767	−	0.541397i	$-0.817896\pi$
$673$	−43.6228	−1.68153	−0.840767	+	0.541397i	$0.817896\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.4605	1.63189	0.815945	−	0.578130i	$-0.196217\pi$
$677$	42.4605	1.63189	0.815945	+	0.578130i	$0.196217\pi$
$678$	0	0
$679$	−62.4342	−2.39600
$680$	0	0
$681$	15.4868	0.593457
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.6754	−0.407294
$688$	0	0
$689$	0.837722	0.0319147
$690$	0	0
$691$	−14.9737	−0.569625	−0.284813	−	0.958583i	$-0.591931\pi$
$691$	−14.9737	−0.569625	−0.284813	+	0.958583i	$0.591931\pi$
$692$	0	0
$693$	13.1623	0.499994
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.32456	−0.315315
$698$	0	0
$699$	−18.1359	−0.685964
$700$	0	0
$701$	−34.4605	−1.30156	−0.650778	−	0.759268i	$-0.725557\pi$
$701$	−34.4605	−1.30156	−0.650778	+	0.759268i	$0.725557\pi$
$702$	0	0
$703$	−26.3246	−0.992849
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−39.4868	−1.48506
$708$	0	0
$709$	3.97367	0.149234	0.0746171	−	0.997212i	$-0.476227\pi$
$709$	3.97367	0.149234	0.0746171	+	0.997212i	$0.476227\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	44.1359	1.65290
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.513167	0.0191646
$718$	0	0
$719$	12.4605	0.464698	0.232349	−	0.972632i	$-0.425359\pi$
$719$	12.4605	0.464698	0.232349	+	0.972632i	$0.425359\pi$
$720$	0	0
$721$	6.97367	0.259713
$722$	0	0
$723$	−26.2982	−0.978041
$724$	0	0
$725$	0	0
$726$	0	0
$727$	40.4868	1.50157	0.750787	−	0.660545i	$-0.229675\pi$
$727$	40.4868	1.50157	0.750787	+	0.660545i	$0.229675\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.1886	0.450812
$732$	0	0
$733$	−51.2982	−1.89474	−0.947372	−	0.320136i	$-0.896271\pi$
$733$	−51.2982	−1.89474	−0.947372	+	0.320136i	$0.896271\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−38.4605	−1.41671
$738$	0	0
$739$	−45.2982	−1.66632	−0.833161	−	0.553031i	$-0.813471\pi$
$739$	−45.2982	−1.66632	−0.833161	+	0.553031i	$0.813471\pi$
$740$	0	0
$741$	−4.16228	−0.152905
$742$	0	0
$743$	32.6491	1.19778	0.598890	−	0.800831i	$-0.295609\pi$
$743$	32.6491	1.19778	0.598890	+	0.800831i	$0.295609\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.32456	0.158227
$748$	0	0
$749$	−28.4605	−1.03992
$750$	0	0
$751$	−29.3509	−1.07103	−0.535515	−	0.844526i	$-0.679882\pi$
$751$	−29.3509	−1.07103	−0.535515	+	0.844526i	$0.679882\pi$
$752$	0	0
$753$	28.6491	1.04403
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−32.6228	−1.18569	−0.592847	−	0.805315i	$-0.701996\pi$
$757$	−32.6228	−1.18569	−0.592847	+	0.805315i	$0.701996\pi$
$758$	0	0
$759$	22.6491	0.822111
$760$	0	0
$761$	32.6491	1.18353	0.591765	−	0.806111i	$-0.298431\pi$
$761$	32.6491	1.18353	0.591765	+	0.806111i	$0.298431\pi$
$762$	0	0
$763$	−13.8377	−0.500959
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.32456	0.300582
$768$	0	0
$769$	−23.9737	−0.864513	−0.432256	−	0.901751i	$-0.642282\pi$
$769$	−23.9737	−0.864513	−0.432256	+	0.901751i	$0.642282\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	0	0
$773$	12.8377	0.461741	0.230870	−	0.972985i	$-0.425843\pi$
$773$	12.8377	0.461741	0.230870	+	0.972985i	$0.425843\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−26.3246	−0.944388
$778$	0	0
$779$	−29.8114	−1.06810
$780$	0	0
$781$	23.6754	0.847174
$782$	0	0
$783$	−5.16228	−0.184485
$784$	0	0
$785$	0	0
$786$	0	0
$787$	51.7851	1.84594	0.922969	−	0.384875i	$-0.125755\pi$
$787$	51.7851	1.84594	0.922969	+	0.384875i	$0.125755\pi$
$788$	0	0
$789$	−13.1623	−0.468589
$790$	0	0
$791$	−52.6491	−1.87199
$792$	0	0
$793$	−5.32456	−0.189081
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.00000	0.283375	0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000	0.283375	0.141687	+	0.989911i	$0.454747\pi$
$798$	0	0
$799$	−9.67544	−0.342293
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	−32.6491	−1.15216
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.16228	0.181721
$808$	0	0
$809$	−16.1886	−0.569161	−0.284581	−	0.958652i	$-0.591854\pi$
$809$	−16.1886	−0.569161	−0.284581	+	0.958652i	$0.591854\pi$
$810$	0	0
$811$	−5.18861	−0.182197	−0.0910984	−	0.995842i	$-0.529038\pi$
$811$	−5.18861	−0.182197	−0.0910984	+	0.995842i	$0.529038\pi$
$812$	0	0
$813$	−13.6754	−0.479619
$814$	0	0
$815$	0	0
$816$	0	0
$817$	43.6491	1.52709
$818$	0	0
$819$	−4.16228	−0.145442
$820$	0	0
$821$	−39.2982	−1.37152	−0.685759	−	0.727829i	$-0.740529\pi$
$821$	−39.2982	−1.37152	−0.685759	+	0.727829i	$0.740529\pi$
$822$	0	0
$823$	−23.8377	−0.830931	−0.415465	−	0.909609i	$-0.636381\pi$
$823$	−23.8377	−0.830931	−0.415465	+	0.909609i	$0.636381\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	53.4342	1.85809	0.929044	−	0.369969i	$-0.120632\pi$
$827$	53.4342	1.85809	0.929044	+	0.369969i	$0.120632\pi$
$828$	0	0
$829$	44.0000	1.52818	0.764092	−	0.645108i	$-0.223188\pi$
$829$	44.0000	1.52818	0.764092	+	0.645108i	$0.223188\pi$
$830$	0	0
$831$	17.6491	0.612241
$832$	0	0
$833$	−12.0000	−0.415775
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6.16228	−0.212999
$838$	0	0
$839$	32.8377	1.13368	0.566842	−	0.823827i	$-0.308165\pi$
$839$	32.8377	1.13368	0.566842	+	0.823827i	$0.308165\pi$
$840$	0	0
$841$	−2.35089	−0.0810652
$842$	0	0
$843$	−20.1359	−0.693519
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.16228	0.143018
$848$	0	0
$849$	8.16228	0.280129
$850$	0	0
$851$	−45.2982	−1.55280
$852$	0	0
$853$	14.6754	0.502478	0.251239	−	0.967925i	$-0.419162\pi$
$853$	14.6754	0.502478	0.251239	+	0.967925i	$0.419162\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.2982	0.454259	0.227129	−	0.973865i	$-0.427066\pi$
$857$	13.2982	0.454259	0.227129	+	0.973865i	$0.427066\pi$
$858$	0	0
$859$	22.9737	0.783851	0.391926	−	0.919997i	$-0.371809\pi$
$859$	22.9737	0.783851	0.391926	+	0.919997i	$0.371809\pi$
$860$	0	0
$861$	−29.8114	−1.01597
$862$	0	0
$863$	−17.2982	−0.588838	−0.294419	−	0.955676i	$-0.595126\pi$
$863$	−17.2982	−0.588838	−0.294419	+	0.955676i	$0.595126\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−15.6491	−0.531472
$868$	0	0
$869$	−25.2982	−0.858183
$870$	0	0
$871$	12.1623	0.412103
$872$	0	0
$873$	15.0000	0.507673
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.6228	1.50680	0.753402	−	0.657560i	$-0.228411\pi$
$877$	44.6228	1.50680	0.753402	+	0.657560i	$0.228411\pi$
$878$	0	0
$879$	22.6491	0.763936
$880$	0	0
$881$	−13.2982	−0.448028	−0.224014	−	0.974586i	$-0.571916\pi$
$881$	−13.2982	−0.448028	−0.224014	+	0.974586i	$0.571916\pi$
$882$	0	0
$883$	18.4868	0.622131	0.311066	−	0.950388i	$-0.399314\pi$
$883$	18.4868	0.622131	0.311066	+	0.950388i	$0.399314\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.2719	1.41935	0.709676	−	0.704529i	$-0.248842\pi$
$887$	42.2719	1.41935	0.709676	+	0.704529i	$0.248842\pi$
$888$	0	0
$889$	−44.3246	−1.48660
$890$	0	0
$891$	−3.16228	−0.105940
$892$	0	0
$893$	−34.6491	−1.15949
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−7.16228	−0.239141
$898$	0	0
$899$	31.8114	1.06097
$900$	0	0
$901$	−0.973666	−0.0324375
$902$	0	0
$903$	43.6491	1.45255
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	9.48683	0.314658
$910$	0	0
$911$	48.9737	1.62257	0.811285	−	0.584651i	$-0.198769\pi$
$911$	48.9737	1.62257	0.811285	+	0.584651i	$0.198769\pi$
$912$	0	0
$913$	−13.6754	−0.452591
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−54.7851	−1.80916
$918$	0	0
$919$	−18.1623	−0.599118	−0.299559	−	0.954078i	$-0.596840\pi$
$919$	−18.1623	−0.599118	−0.299559	+	0.954078i	$0.596840\pi$
$920$	0	0
$921$	3.51317	0.115763
$922$	0	0
$923$	−7.48683	−0.246432
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.67544	−0.0550288
$928$	0	0
$929$	20.5132	0.673015	0.336508	−	0.941681i	$-0.390754\pi$
$929$	20.5132	0.673015	0.336508	+	0.941681i	$0.390754\pi$
$930$	0	0
$931$	−42.9737	−1.40841
$932$	0	0
$933$	8.97367	0.293785
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−15.3246	−0.500631	−0.250316	−	0.968164i	$-0.580534\pi$
$937$	−15.3246	−0.500631	−0.250316	+	0.968164i	$0.580534\pi$
$938$	0	0
$939$	29.9737	0.978154
$940$	0	0
$941$	43.2982	1.41148	0.705741	−	0.708470i	$-0.250614\pi$
$941$	43.2982	1.41148	0.705741	+	0.708470i	$0.250614\pi$
$942$	0	0
$943$	−51.2982	−1.67050
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.4605	−1.31479	−0.657395	−	0.753546i	$-0.728342\pi$
$947$	−40.4605	−1.31479	−0.657395	+	0.753546i	$0.728342\pi$
$948$	0	0
$949$	10.3246	0.335149
$950$	0	0
$951$	24.6491	0.799302
$952$	0	0
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16.3246	0.527698
$958$	0	0
$959$	−24.9737	−0.806442
$960$	0	0
$961$	6.97367	0.224957
$962$	0	0
$963$	6.83772	0.220343
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17.2982	−0.556273	−0.278137	−	0.960542i	$-0.589717\pi$
$967$	−17.2982	−0.556273	−0.278137	+	0.960542i	$0.589717\pi$
$968$	0	0
$969$	4.83772	0.155410
$970$	0	0
$971$	48.9737	1.57164	0.785820	−	0.618455i	$-0.212241\pi$
$971$	48.9737	1.57164	0.785820	+	0.618455i	$0.212241\pi$
$972$	0	0
$973$	−8.32456	−0.266873
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.2982	0.745376	0.372688	−	0.927957i	$-0.378436\pi$
$977$	23.2982	0.745376	0.372688	+	0.927957i	$0.378436\pi$
$978$	0	0
$979$	12.6491	0.404267
$980$	0	0
$981$	3.32456	0.106145
$982$	0	0
$983$	9.29822	0.296567	0.148284	−	0.988945i	$-0.452625\pi$
$983$	9.29822	0.296567	0.148284	+	0.988945i	$0.452625\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−34.6491	−1.10289
$988$	0	0
$989$	75.1096	2.38835
$990$	0	0
$991$	−3.18861	−0.101290	−0.0506448	−	0.998717i	$-0.516128\pi$
$991$	−3.18861	−0.101290	−0.0506448	+	0.998717i	$0.516128\pi$
$992$	0	0
$993$	6.32456	0.200704
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−8.70178	−0.275588	−0.137794	−	0.990461i	$-0.544001\pi$
$997$	−8.70178	−0.275588	−0.137794	+	0.990461i	$0.544001\pi$
$998$	0	0
$999$	6.32456	0.200100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9600.2.a.dc.1.1	yes	2
4.3	odd	2		9600.2.a.cs.1.2	yes	2
5.4	even	2		9600.2.a.cp.1.2	yes	2
8.3	odd	2		9600.2.a.dj.1.2	yes	2
8.5	even	2		9600.2.a.ch.1.1	✓	2
20.19	odd	2		9600.2.a.cz.1.1	yes	2
40.19	odd	2		9600.2.a.ci.1.1	yes	2
40.29	even	2		9600.2.a.dk.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9600.2.a.ch.1.1	✓	2	8.5	even	2
9600.2.a.ci.1.1	yes	2	40.19	odd	2
9600.2.a.cp.1.2	yes	2	5.4	even	2
9600.2.a.cs.1.2	yes	2	4.3	odd	2
9600.2.a.cz.1.1	yes	2	20.19	odd	2
9600.2.a.dc.1.1	yes	2	1.1	even	1	trivial
9600.2.a.dj.1.2	yes	2	8.3	odd	2
9600.2.a.dk.1.2	yes	2	40.29	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 9600 = 2^{7} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.16228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.16228 q^{7} +1.00000 q^{9} -3.16228 q^{11} +1.00000 q^{13} -1.16228 q^{17} -4.16228 q^{19} -4.16228 q^{21} -7.16228 q^{23} +1.00000 q^{27} -5.16228 q^{29} -6.16228 q^{31} -3.16228 q^{33} +6.32456 q^{37} +1.00000 q^{39} +7.16228 q^{41} -10.4868 q^{43} +8.32456 q^{47} +10.3246 q^{49} -1.16228 q^{51} +0.837722 q^{53} -4.16228 q^{57} +8.32456 q^{59} -5.32456 q^{61} -4.16228 q^{63} +12.1623 q^{67} -7.16228 q^{69} -7.48683 q^{71} +10.3246 q^{73} +13.1623 q^{77} +8.00000 q^{79} +1.00000 q^{81} +4.32456 q^{83} -5.16228 q^{87} -4.00000 q^{89} -4.16228 q^{91} -6.16228 q^{93} +15.0000 q^{97} -3.16228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + 2 q^{13} + 4 q^{17} - 2 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{27} - 4 q^{29} - 6 q^{31} + 2 q^{39} + 8 q^{41} - 2 q^{43} + 4 q^{47} + 8 q^{49} + 4 q^{51} + 8 q^{53} - 2 q^{57} + 4 q^{59} + 2 q^{61} - 2 q^{63} + 18 q^{67} - 8 q^{69} + 4 q^{71} + 8 q^{73} + 20 q^{77} + 16 q^{79} + 2 q^{81} - 4 q^{83} - 4 q^{87} - 8 q^{89} - 2 q^{91} - 6 q^{93} + 30 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 + 2 * q^13 + 4 * q^17 - 2 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^27 - 4 * q^29 - 6 * q^31 + 2 * q^39 + 8 * q^41 - 2 * q^43 + 4 * q^47 + 8 * q^49 + 4 * q^51 + 8 * q^53 - 2 * q^57 + 4 * q^59 + 2 * q^61 - 2 * q^63 + 18 * q^67 - 8 * q^69 + 4 * q^71 + 8 * q^73 + 20 * q^77 + 16 * q^79 + 2 * q^81 - 4 * q^83 - 4 * q^87 - 8 * q^89 - 2 * q^91 - 6 * q^93 + 30 * q^97

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