LMFDB - Embedded newform 3192.2.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3192,2,Mod(1,3192)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3192.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3192.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-4,0,-2,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$25.4882483252$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3192.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.585786 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.82843 q^{11} -5.65685 q^{13} -0.585786 q^{15} -6.24264 q^{17} +1.00000 q^{19} -1.00000 q^{21} +8.48528 q^{23} -4.65685 q^{25} +1.00000 q^{27} +1.75736 q^{29} +3.17157 q^{31} +2.82843 q^{33} +0.585786 q^{35} -6.00000 q^{37} -5.65685 q^{39} -3.17157 q^{41} -3.17157 q^{43} -0.585786 q^{45} -13.4142 q^{47} +1.00000 q^{49} -6.24264 q^{51} +9.07107 q^{53} -1.65685 q^{55} +1.00000 q^{57} -13.6569 q^{59} +2.00000 q^{61} -1.00000 q^{63} +3.31371 q^{65} -14.8284 q^{67} +8.48528 q^{69} -12.7279 q^{71} -3.17157 q^{73} -4.65685 q^{75} -2.82843 q^{77} +13.6569 q^{79} +1.00000 q^{81} -3.75736 q^{83} +3.65685 q^{85} +1.75736 q^{87} -11.1716 q^{89} +5.65685 q^{91} +3.17157 q^{93} -0.585786 q^{95} -11.6569 q^{97} +2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} - 4 q^{17} + 2 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 12 q^{29} + 12 q^{31} + 4 q^{35} - 12 q^{37} - 12 q^{41} - 12 q^{43} - 4 q^{45} - 24 q^{47}+ \cdots - 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^15 - 4 * q^17 + 2 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 + 12 * q^29 + 12 * q^31 + 4 * q^35 - 12 * q^37 - 12 * q^41 - 12 * q^43 - 4 * q^45 - 24 * q^47 + 2 * q^49 - 4 * q^51 + 4 * q^53 + 8 * q^55 + 2 * q^57 - 16 * q^59 + 4 * q^61 - 2 * q^63 - 16 * q^65 - 24 * q^67 - 12 * q^73 + 2 * q^75 + 16 * q^79 + 2 * q^81 - 16 * q^83 - 4 * q^85 + 12 * q^87 - 28 * q^89 + 12 * q^93 - 4 * q^95 - 12 * q^97

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
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	−0.585786	−0.151249
$16$	0	0
$17$	−6.24264	−1.51406	−0.757031	−	0.653379i	$-0.773351\pi$
$17$	−6.24264	−1.51406	−0.757031	+	0.653379i	$0.773351\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	8.48528	1.76930	0.884652	−	0.466252i	$-0.154396\pi$
$23$	8.48528	1.76930	0.884652	+	0.466252i	$0.154396\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.75736	0.326333	0.163167	−	0.986599i	$-0.447829\pi$
$29$	1.75736	0.326333	0.163167	+	0.986599i	$0.447829\pi$
$30$	0	0
$31$	3.17157	0.569631	0.284816	−	0.958582i	$-0.408068\pi$
$31$	3.17157	0.569631	0.284816	+	0.958582i	$0.408068\pi$
$32$	0	0
$33$	2.82843	0.492366
$34$	0	0
$35$	0.585786	0.0990160
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	−5.65685	−0.905822
$40$	0	0
$41$	−3.17157	−0.495316	−0.247658	−	0.968847i	$-0.579661\pi$
$41$	−3.17157	−0.495316	−0.247658	+	0.968847i	$0.579661\pi$
$42$	0	0
$43$	−3.17157	−0.483660	−0.241830	−	0.970319i	$-0.577748\pi$
$43$	−3.17157	−0.483660	−0.241830	+	0.970319i	$0.577748\pi$
$44$	0	0
$45$	−0.585786	−0.0873239
$46$	0	0
$47$	−13.4142	−1.95666	−0.978332	−	0.207042i	$-0.933616\pi$
$47$	−13.4142	−1.95666	−0.978332	+	0.207042i	$0.933616\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.24264	−0.874145
$52$	0	0
$53$	9.07107	1.24601	0.623003	−	0.782219i	$-0.285912\pi$
$53$	9.07107	1.24601	0.623003	+	0.782219i	$0.285912\pi$
$54$	0	0
$55$	−1.65685	−0.223410
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−13.6569	−1.77797	−0.888985	−	0.457935i	$-0.848589\pi$
$59$	−13.6569	−1.77797	−0.888985	+	0.457935i	$0.848589\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.31371	0.411015
$66$	0	0
$67$	−14.8284	−1.81158	−0.905790	−	0.423726i	$-0.860722\pi$
$67$	−14.8284	−1.81158	−0.905790	+	0.423726i	$0.860722\pi$
$68$	0	0
$69$	8.48528	1.02151
$70$	0	0
$71$	−12.7279	−1.51053	−0.755263	−	0.655422i	$-0.772491\pi$
$71$	−12.7279	−1.51053	−0.755263	+	0.655422i	$0.772491\pi$
$72$	0	0
$73$	−3.17157	−0.371205	−0.185602	−	0.982625i	$-0.559424\pi$
$73$	−3.17157	−0.371205	−0.185602	+	0.982625i	$0.559424\pi$
$74$	0	0
$75$	−4.65685	−0.537727
$76$	0	0
$77$	−2.82843	−0.322329
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.75736	−0.412424	−0.206212	−	0.978507i	$-0.566114\pi$
$83$	−3.75736	−0.412424	−0.206212	+	0.978507i	$0.566114\pi$
$84$	0	0
$85$	3.65685	0.396642
$86$	0	0
$87$	1.75736	0.188409
$88$	0	0
$89$	−11.1716	−1.18418	−0.592092	−	0.805870i	$-0.701698\pi$
$89$	−11.1716	−1.18418	−0.592092	+	0.805870i	$0.701698\pi$
$90$	0	0
$91$	5.65685	0.592999
$92$	0	0
$93$	3.17157	0.328877
$94$	0	0
$95$	−0.585786	−0.0601004
$96$	0	0
$97$	−11.6569	−1.18357	−0.591787	−	0.806094i	$-0.701577\pi$
$97$	−11.6569	−1.18357	−0.591787	+	0.806094i	$0.701577\pi$
$98$	0	0
$99$	2.82843	0.284268
$100$	0	0
$101$	−5.07107	−0.504590	−0.252295	−	0.967650i	$-0.581185\pi$
$101$	−5.07107	−0.504590	−0.252295	+	0.967650i	$0.581185\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0.585786	0.0571669
$106$	0	0
$107$	8.72792	0.843760	0.421880	−	0.906652i	$-0.361370\pi$
$107$	8.72792	0.843760	0.421880	+	0.906652i	$0.361370\pi$
$108$	0	0
$109$	12.1421	1.16301	0.581503	−	0.813544i	$-0.302465\pi$
$109$	12.1421	1.16301	0.581503	+	0.813544i	$0.302465\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	11.4142	1.07376	0.536879	−	0.843659i	$-0.319603\pi$
$113$	11.4142	1.07376	0.536879	+	0.843659i	$0.319603\pi$
$114$	0	0
$115$	−4.97056	−0.463507
$116$	0	0
$117$	−5.65685	−0.522976
$118$	0	0
$119$	6.24264	0.572262
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	−3.17157	−0.285971
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	8.48528	0.752947	0.376473	−	0.926427i	$-0.377137\pi$
$127$	8.48528	0.752947	0.376473	+	0.926427i	$0.377137\pi$
$128$	0	0
$129$	−3.17157	−0.279241
$130$	0	0
$131$	−9.89949	−0.864923	−0.432461	−	0.901652i	$-0.642355\pi$
$131$	−9.89949	−0.864923	−0.432461	+	0.901652i	$0.642355\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−0.585786	−0.0504165
$136$	0	0
$137$	−0.343146	−0.0293169	−0.0146585	−	0.999893i	$-0.504666\pi$
$137$	−0.343146	−0.0293169	−0.0146585	+	0.999893i	$0.504666\pi$
$138$	0	0
$139$	4.48528	0.380437	0.190218	−	0.981742i	$-0.439080\pi$
$139$	4.48528	0.380437	0.190218	+	0.981742i	$0.439080\pi$
$140$	0	0
$141$	−13.4142	−1.12968
$142$	0	0
$143$	−16.0000	−1.33799
$144$	0	0
$145$	−1.02944	−0.0854901
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	1.51472	0.124091	0.0620453	−	0.998073i	$-0.480238\pi$
$149$	1.51472	0.124091	0.0620453	+	0.998073i	$0.480238\pi$
$150$	0	0
$151$	−8.48528	−0.690522	−0.345261	−	0.938507i	$-0.612210\pi$
$151$	−8.48528	−0.690522	−0.345261	+	0.938507i	$0.612210\pi$
$152$	0	0
$153$	−6.24264	−0.504688
$154$	0	0
$155$	−1.85786	−0.149227
$156$	0	0
$157$	3.65685	0.291849	0.145924	−	0.989296i	$-0.453384\pi$
$157$	3.65685	0.291849	0.145924	+	0.989296i	$0.453384\pi$
$158$	0	0
$159$	9.07107	0.719382
$160$	0	0
$161$	−8.48528	−0.668734
$162$	0	0
$163$	−18.4853	−1.44788	−0.723939	−	0.689863i	$-0.757671\pi$
$163$	−18.4853	−1.44788	−0.723939	+	0.689863i	$0.757671\pi$
$164$	0	0
$165$	−1.65685	−0.128986
$166$	0	0
$167$	−17.1716	−1.32878	−0.664388	−	0.747388i	$-0.731308\pi$
$167$	−17.1716	−1.32878	−0.664388	+	0.747388i	$0.731308\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−8.82843	−0.671213	−0.335606	−	0.942002i	$-0.608941\pi$
$173$	−8.82843	−0.671213	−0.335606	+	0.942002i	$0.608941\pi$
$174$	0	0
$175$	4.65685	0.352025
$176$	0	0
$177$	−13.6569	−1.02651
$178$	0	0
$179$	−11.0711	−0.827490	−0.413745	−	0.910393i	$-0.635780\pi$
$179$	−11.0711	−0.827490	−0.413745	+	0.910393i	$0.635780\pi$
$180$	0	0
$181$	13.3137	0.989600	0.494800	−	0.869007i	$-0.335241\pi$
$181$	13.3137	0.989600	0.494800	+	0.869007i	$0.335241\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	3.51472	0.258407
$186$	0	0
$187$	−17.6569	−1.29120
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−9.17157	−0.663632	−0.331816	−	0.943344i	$-0.607661\pi$
$191$	−9.17157	−0.663632	−0.331816	+	0.943344i	$0.607661\pi$
$192$	0	0
$193$	−11.6569	−0.839079	−0.419539	−	0.907737i	$-0.637808\pi$
$193$	−11.6569	−0.839079	−0.419539	+	0.907737i	$0.637808\pi$
$194$	0	0
$195$	3.31371	0.237300
$196$	0	0
$197$	−18.9706	−1.35160	−0.675798	−	0.737087i	$-0.736201\pi$
$197$	−18.9706	−1.35160	−0.675798	+	0.737087i	$0.736201\pi$
$198$	0	0
$199$	14.1421	1.00251	0.501255	−	0.865300i	$-0.332872\pi$
$199$	14.1421	1.00251	0.501255	+	0.865300i	$0.332872\pi$
$200$	0	0
$201$	−14.8284	−1.04592
$202$	0	0
$203$	−1.75736	−0.123342
$204$	0	0
$205$	1.85786	0.129759
$206$	0	0
$207$	8.48528	0.589768
$208$	0	0
$209$	2.82843	0.195646
$210$	0	0
$211$	−4.97056	−0.342188	−0.171094	−	0.985255i	$-0.554730\pi$
$211$	−4.97056	−0.342188	−0.171094	+	0.985255i	$0.554730\pi$
$212$	0	0
$213$	−12.7279	−0.872103
$214$	0	0
$215$	1.85786	0.126705
$216$	0	0
$217$	−3.17157	−0.215300
$218$	0	0
$219$	−3.17157	−0.214315
$220$	0	0
$221$	35.3137	2.37546
$222$	0	0
$223$	4.82843	0.323335	0.161668	−	0.986845i	$-0.448313\pi$
$223$	4.82843	0.323335	0.161668	+	0.986845i	$0.448313\pi$
$224$	0	0
$225$	−4.65685	−0.310457
$226$	0	0
$227$	28.4853	1.89063	0.945317	−	0.326152i	$-0.105752\pi$
$227$	28.4853	1.89063	0.945317	+	0.326152i	$0.105752\pi$
$228$	0	0
$229$	−6.48528	−0.428559	−0.214280	−	0.976772i	$-0.568740\pi$
$229$	−6.48528	−0.428559	−0.214280	+	0.976772i	$0.568740\pi$
$230$	0	0
$231$	−2.82843	−0.186097
$232$	0	0
$233$	−6.48528	−0.424865	−0.212432	−	0.977176i	$-0.568139\pi$
$233$	−6.48528	−0.424865	−0.212432	+	0.977176i	$0.568139\pi$
$234$	0	0
$235$	7.85786	0.512591
$236$	0	0
$237$	13.6569	0.887108
$238$	0	0
$239$	−13.6569	−0.883388	−0.441694	−	0.897166i	$-0.645622\pi$
$239$	−13.6569	−0.883388	−0.441694	+	0.897166i	$0.645622\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.585786	−0.0374245
$246$	0	0
$247$	−5.65685	−0.359937
$248$	0	0
$249$	−3.75736	−0.238113
$250$	0	0
$251$	4.24264	0.267793	0.133897	−	0.990995i	$-0.457251\pi$
$251$	4.24264	0.267793	0.133897	+	0.990995i	$0.457251\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	0	0
$255$	3.65685	0.229001
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	1.75736	0.108778
$262$	0	0
$263$	−9.65685	−0.595467	−0.297734	−	0.954649i	$-0.596231\pi$
$263$	−9.65685	−0.595467	−0.297734	+	0.954649i	$0.596231\pi$
$264$	0	0
$265$	−5.31371	−0.326419
$266$	0	0
$267$	−11.1716	−0.683689
$268$	0	0
$269$	−0.343146	−0.0209220	−0.0104610	−	0.999945i	$-0.503330\pi$
$269$	−0.343146	−0.0209220	−0.0104610	+	0.999945i	$0.503330\pi$
$270$	0	0
$271$	16.9706	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$272$	0	0
$273$	5.65685	0.342368
$274$	0	0
$275$	−13.1716	−0.794276
$276$	0	0
$277$	11.3137	0.679775	0.339887	−	0.940466i	$-0.389611\pi$
$277$	11.3137	0.679775	0.339887	+	0.940466i	$0.389611\pi$
$278$	0	0
$279$	3.17157	0.189877
$280$	0	0
$281$	10.2426	0.611025	0.305512	−	0.952188i	$-0.401172\pi$
$281$	10.2426	0.611025	0.305512	+	0.952188i	$0.401172\pi$
$282$	0	0
$283$	−14.1421	−0.840663	−0.420331	−	0.907371i	$-0.638086\pi$
$283$	−14.1421	−0.840663	−0.420331	+	0.907371i	$0.638086\pi$
$284$	0	0
$285$	−0.585786	−0.0346990
$286$	0	0
$287$	3.17157	0.187212
$288$	0	0
$289$	21.9706	1.29239
$290$	0	0
$291$	−11.6569	−0.683337
$292$	0	0
$293$	18.4853	1.07992	0.539961	−	0.841690i	$-0.318439\pi$
$293$	18.4853	1.07992	0.539961	+	0.841690i	$0.318439\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	2.82843	0.164122
$298$	0	0
$299$	−48.0000	−2.77591
$300$	0	0
$301$	3.17157	0.182806
$302$	0	0
$303$	−5.07107	−0.291325
$304$	0	0
$305$	−1.17157	−0.0670841
$306$	0	0
$307$	12.1421	0.692988	0.346494	−	0.938052i	$-0.387372\pi$
$307$	12.1421	0.692988	0.346494	+	0.938052i	$0.387372\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	20.0416	1.13646	0.568228	−	0.822871i	$-0.307629\pi$
$311$	20.0416	1.13646	0.568228	+	0.822871i	$0.307629\pi$
$312$	0	0
$313$	−30.9706	−1.75056	−0.875280	−	0.483617i	$-0.839323\pi$
$313$	−30.9706	−1.75056	−0.875280	+	0.483617i	$0.839323\pi$
$314$	0	0
$315$	0.585786	0.0330053
$316$	0	0
$317$	−15.2132	−0.854459	−0.427229	−	0.904143i	$-0.640510\pi$
$317$	−15.2132	−0.854459	−0.427229	+	0.904143i	$0.640510\pi$
$318$	0	0
$319$	4.97056	0.278298
$320$	0	0
$321$	8.72792	0.487145
$322$	0	0
$323$	−6.24264	−0.347350
$324$	0	0
$325$	26.3431	1.46125
$326$	0	0
$327$	12.1421	0.671462
$328$	0	0
$329$	13.4142	0.739550
$330$	0	0
$331$	14.1421	0.777322	0.388661	−	0.921381i	$-0.372938\pi$
$331$	14.1421	0.777322	0.388661	+	0.921381i	$0.372938\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	8.68629	0.474583
$336$	0	0
$337$	−3.17157	−0.172767	−0.0863833	−	0.996262i	$-0.527531\pi$
$337$	−3.17157	−0.172767	−0.0863833	+	0.996262i	$0.527531\pi$
$338$	0	0
$339$	11.4142	0.619935
$340$	0	0
$341$	8.97056	0.485783
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−4.97056	−0.267606
$346$	0	0
$347$	−12.9706	−0.696296	−0.348148	−	0.937440i	$-0.613189\pi$
$347$	−12.9706	−0.696296	−0.348148	+	0.937440i	$0.613189\pi$
$348$	0	0
$349$	6.68629	0.357909	0.178954	−	0.983857i	$-0.442729\pi$
$349$	6.68629	0.357909	0.178954	+	0.983857i	$0.442729\pi$
$350$	0	0
$351$	−5.65685	−0.301941
$352$	0	0
$353$	19.8995	1.05914	0.529572	−	0.848265i	$-0.322353\pi$
$353$	19.8995	1.05914	0.529572	+	0.848265i	$0.322353\pi$
$354$	0	0
$355$	7.45584	0.395715
$356$	0	0
$357$	6.24264	0.330396
$358$	0	0
$359$	−12.4853	−0.658948	−0.329474	−	0.944165i	$-0.606871\pi$
$359$	−12.4853	−0.658948	−0.329474	+	0.944165i	$0.606871\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−3.00000	−0.157459
$364$	0	0
$365$	1.85786	0.0972451
$366$	0	0
$367$	34.1421	1.78220	0.891102	−	0.453802i	$-0.149933\pi$
$367$	34.1421	1.78220	0.891102	+	0.453802i	$0.149933\pi$
$368$	0	0
$369$	−3.17157	−0.165105
$370$	0	0
$371$	−9.07107	−0.470946
$372$	0	0
$373$	35.6569	1.84624	0.923121	−	0.384510i	$-0.125629\pi$
$373$	35.6569	1.84624	0.923121	+	0.384510i	$0.125629\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	0	0
$377$	−9.94113	−0.511994
$378$	0	0
$379$	9.17157	0.471112	0.235556	−	0.971861i	$-0.424309\pi$
$379$	9.17157	0.471112	0.235556	+	0.971861i	$0.424309\pi$
$380$	0	0
$381$	8.48528	0.434714
$382$	0	0
$383$	−3.51472	−0.179594	−0.0897969	−	0.995960i	$-0.528622\pi$
$383$	−3.51472	−0.179594	−0.0897969	+	0.995960i	$0.528622\pi$
$384$	0	0
$385$	1.65685	0.0844411
$386$	0	0
$387$	−3.17157	−0.161220
$388$	0	0
$389$	−15.4558	−0.783642	−0.391821	−	0.920041i	$-0.628155\pi$
$389$	−15.4558	−0.783642	−0.391821	+	0.920041i	$0.628155\pi$
$390$	0	0
$391$	−52.9706	−2.67884
$392$	0	0
$393$	−9.89949	−0.499363
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−21.3137	−1.06970	−0.534852	−	0.844946i	$-0.679633\pi$
$397$	−21.3137	−1.06970	−0.534852	+	0.844946i	$0.679633\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	24.3848	1.21772	0.608859	−	0.793279i	$-0.291628\pi$
$401$	24.3848	1.21772	0.608859	+	0.793279i	$0.291628\pi$
$402$	0	0
$403$	−17.9411	−0.893711
$404$	0	0
$405$	−0.585786	−0.0291080
$406$	0	0
$407$	−16.9706	−0.841200
$408$	0	0
$409$	26.6274	1.31664	0.658321	−	0.752738i	$-0.271267\pi$
$409$	26.6274	1.31664	0.658321	+	0.752738i	$0.271267\pi$
$410$	0	0
$411$	−0.343146	−0.0169261
$412$	0	0
$413$	13.6569	0.672010
$414$	0	0
$415$	2.20101	0.108043
$416$	0	0
$417$	4.48528	0.219645
$418$	0	0
$419$	20.7279	1.01263	0.506313	−	0.862350i	$-0.331008\pi$
$419$	20.7279	1.01263	0.506313	+	0.862350i	$0.331008\pi$
$420$	0	0
$421$	−18.4853	−0.900917	−0.450459	−	0.892797i	$-0.648740\pi$
$421$	−18.4853	−0.900917	−0.450459	+	0.892797i	$0.648740\pi$
$422$	0	0
$423$	−13.4142	−0.652221
$424$	0	0
$425$	29.0711	1.41015
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−16.0000	−0.772487
$430$	0	0
$431$	−30.5858	−1.47327	−0.736633	−	0.676293i	$-0.763585\pi$
$431$	−30.5858	−1.47327	−0.736633	+	0.676293i	$0.763585\pi$
$432$	0	0
$433$	−40.6274	−1.95243	−0.976215	−	0.216807i	$-0.930436\pi$
$433$	−40.6274	−1.95243	−0.976215	+	0.216807i	$0.930436\pi$
$434$	0	0
$435$	−1.02944	−0.0493577
$436$	0	0
$437$	8.48528	0.405906
$438$	0	0
$439$	−2.20101	−0.105048	−0.0525242	−	0.998620i	$-0.516727\pi$
$439$	−2.20101	−0.105048	−0.0525242	+	0.998620i	$0.516727\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	23.3137	1.10767	0.553834	−	0.832627i	$-0.313164\pi$
$443$	23.3137	1.10767	0.553834	+	0.832627i	$0.313164\pi$
$444$	0	0
$445$	6.54416	0.310223
$446$	0	0
$447$	1.51472	0.0716437
$448$	0	0
$449$	−36.8701	−1.74001	−0.870003	−	0.493047i	$-0.835883\pi$
$449$	−36.8701	−1.74001	−0.870003	+	0.493047i	$0.835883\pi$
$450$	0	0
$451$	−8.97056	−0.422407
$452$	0	0
$453$	−8.48528	−0.398673
$454$	0	0
$455$	−3.31371	−0.155349
$456$	0	0
$457$	−4.68629	−0.219215	−0.109608	−	0.993975i	$-0.534959\pi$
$457$	−4.68629	−0.219215	−0.109608	+	0.993975i	$0.534959\pi$
$458$	0	0
$459$	−6.24264	−0.291382
$460$	0	0
$461$	26.5269	1.23548	0.617741	−	0.786382i	$-0.288048\pi$
$461$	26.5269	1.23548	0.617741	+	0.786382i	$0.288048\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	−1.85786	−0.0861564
$466$	0	0
$467$	−9.89949	−0.458094	−0.229047	−	0.973415i	$-0.573561\pi$
$467$	−9.89949	−0.458094	−0.229047	+	0.973415i	$0.573561\pi$
$468$	0	0
$469$	14.8284	0.684713
$470$	0	0
$471$	3.65685	0.168499
$472$	0	0
$473$	−8.97056	−0.412467
$474$	0	0
$475$	−4.65685	−0.213671
$476$	0	0
$477$	9.07107	0.415336
$478$	0	0
$479$	24.7279	1.12985	0.564924	−	0.825143i	$-0.308906\pi$
$479$	24.7279	1.12985	0.564924	+	0.825143i	$0.308906\pi$
$480$	0	0
$481$	33.9411	1.54758
$482$	0	0
$483$	−8.48528	−0.386094
$484$	0	0
$485$	6.82843	0.310063
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	−18.4853	−0.835933
$490$	0	0
$491$	1.65685	0.0747728	0.0373864	−	0.999301i	$-0.488097\pi$
$491$	1.65685	0.0747728	0.0373864	+	0.999301i	$0.488097\pi$
$492$	0	0
$493$	−10.9706	−0.494089
$494$	0	0
$495$	−1.65685	−0.0744701
$496$	0	0
$497$	12.7279	0.570925
$498$	0	0
$499$	−35.5980	−1.59358	−0.796792	−	0.604253i	$-0.793471\pi$
$499$	−35.5980	−1.59358	−0.796792	+	0.604253i	$0.793471\pi$
$500$	0	0
$501$	−17.1716	−0.767169
$502$	0	0
$503$	−34.1838	−1.52418	−0.762089	−	0.647472i	$-0.775826\pi$
$503$	−34.1838	−1.52418	−0.762089	+	0.647472i	$0.775826\pi$
$504$	0	0
$505$	2.97056	0.132188
$506$	0	0
$507$	19.0000	0.843820
$508$	0	0
$509$	18.9706	0.840855	0.420428	−	0.907326i	$-0.361880\pi$
$509$	18.9706	0.840855	0.420428	+	0.907326i	$0.361880\pi$
$510$	0	0
$511$	3.17157	0.140302
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−4.68629	−0.206503
$516$	0	0
$517$	−37.9411	−1.66865
$518$	0	0
$519$	−8.82843	−0.387525
$520$	0	0
$521$	−5.02944	−0.220344	−0.110172	−	0.993913i	$-0.535140\pi$
$521$	−5.02944	−0.220344	−0.110172	+	0.993913i	$0.535140\pi$
$522$	0	0
$523$	28.1421	1.23057	0.615285	−	0.788305i	$-0.289041\pi$
$523$	28.1421	1.23057	0.615285	+	0.788305i	$0.289041\pi$
$524$	0	0
$525$	4.65685	0.203242
$526$	0	0
$527$	−19.7990	−0.862458
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	−13.6569	−0.592657
$532$	0	0
$533$	17.9411	0.777116
$534$	0	0
$535$	−5.11270	−0.221041
$536$	0	0
$537$	−11.0711	−0.477752
$538$	0	0
$539$	2.82843	0.121829
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	13.3137	0.571346
$544$	0	0
$545$	−7.11270	−0.304675
$546$	0	0
$547$	43.5980	1.86412	0.932058	−	0.362310i	$-0.118012\pi$
$547$	43.5980	1.86412	0.932058	+	0.362310i	$0.118012\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	1.75736	0.0748660
$552$	0	0
$553$	−13.6569	−0.580749
$554$	0	0
$555$	3.51472	0.149191
$556$	0	0
$557$	8.82843	0.374072	0.187036	−	0.982353i	$-0.440112\pi$
$557$	8.82843	0.374072	0.187036	+	0.982353i	$0.440112\pi$
$558$	0	0
$559$	17.9411	0.758829
$560$	0	0
$561$	−17.6569	−0.745473
$562$	0	0
$563$	−7.51472	−0.316708	−0.158354	−	0.987382i	$-0.550619\pi$
$563$	−7.51472	−0.316708	−0.158354	+	0.987382i	$0.550619\pi$
$564$	0	0
$565$	−6.68629	−0.281294
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	46.5269	1.95051	0.975255	−	0.221084i	$-0.0709594\pi$
$569$	46.5269	1.95051	0.975255	+	0.221084i	$0.0709594\pi$
$570$	0	0
$571$	20.9706	0.877591	0.438795	−	0.898587i	$-0.355405\pi$
$571$	20.9706	0.877591	0.438795	+	0.898587i	$0.355405\pi$
$572$	0	0
$573$	−9.17157	−0.383148
$574$	0	0
$575$	−39.5147	−1.64788
$576$	0	0
$577$	15.6569	0.651803	0.325902	−	0.945404i	$-0.394332\pi$
$577$	15.6569	0.651803	0.325902	+	0.945404i	$0.394332\pi$
$578$	0	0
$579$	−11.6569	−0.484442
$580$	0	0
$581$	3.75736	0.155882
$582$	0	0
$583$	25.6569	1.06260
$584$	0	0
$585$	3.31371	0.137005
$586$	0	0
$587$	−15.0711	−0.622050	−0.311025	−	0.950402i	$-0.600672\pi$
$587$	−15.0711	−0.622050	−0.311025	+	0.950402i	$0.600672\pi$
$588$	0	0
$589$	3.17157	0.130682
$590$	0	0
$591$	−18.9706	−0.780345
$592$	0	0
$593$	−11.6152	−0.476980	−0.238490	−	0.971145i	$-0.576652\pi$
$593$	−11.6152	−0.476980	−0.238490	+	0.971145i	$0.576652\pi$
$594$	0	0
$595$	−3.65685	−0.149916
$596$	0	0
$597$	14.1421	0.578799
$598$	0	0
$599$	41.8995	1.71197	0.855983	−	0.517003i	$-0.172952\pi$
$599$	41.8995	1.71197	0.855983	+	0.517003i	$0.172952\pi$
$600$	0	0
$601$	13.3137	0.543077	0.271539	−	0.962428i	$-0.412467\pi$
$601$	13.3137	0.543077	0.271539	+	0.962428i	$0.412467\pi$
$602$	0	0
$603$	−14.8284	−0.603860
$604$	0	0
$605$	1.75736	0.0714468
$606$	0	0
$607$	−41.9411	−1.70234	−0.851169	−	0.524892i	$-0.824106\pi$
$607$	−41.9411	−1.70234	−0.851169	+	0.524892i	$0.824106\pi$
$608$	0	0
$609$	−1.75736	−0.0712118
$610$	0	0
$611$	75.8823	3.06987
$612$	0	0
$613$	44.2843	1.78862	0.894312	−	0.447443i	$-0.147665\pi$
$613$	44.2843	1.78862	0.894312	+	0.447443i	$0.147665\pi$
$614$	0	0
$615$	1.85786	0.0749163
$616$	0	0
$617$	31.9411	1.28590	0.642951	−	0.765908i	$-0.277710\pi$
$617$	31.9411	1.28590	0.642951	+	0.765908i	$0.277710\pi$
$618$	0	0
$619$	28.4853	1.14492	0.572460	−	0.819933i	$-0.305989\pi$
$619$	28.4853	1.14492	0.572460	+	0.819933i	$0.305989\pi$
$620$	0	0
$621$	8.48528	0.340503
$622$	0	0
$623$	11.1716	0.447580
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	2.82843	0.112956
$628$	0	0
$629$	37.4558	1.49346
$630$	0	0
$631$	−35.4558	−1.41147	−0.705737	−	0.708473i	$-0.749384\pi$
$631$	−35.4558	−1.41147	−0.705737	+	0.708473i	$0.749384\pi$
$632$	0	0
$633$	−4.97056	−0.197562
$634$	0	0
$635$	−4.97056	−0.197251
$636$	0	0
$637$	−5.65685	−0.224133
$638$	0	0
$639$	−12.7279	−0.503509
$640$	0	0
$641$	−11.6985	−0.462062	−0.231031	−	0.972946i	$-0.574210\pi$
$641$	−11.6985	−0.462062	−0.231031	+	0.972946i	$0.574210\pi$
$642$	0	0
$643$	26.8284	1.05801	0.529005	−	0.848619i	$-0.322565\pi$
$643$	26.8284	1.05801	0.529005	+	0.848619i	$0.322565\pi$
$644$	0	0
$645$	1.85786	0.0731533
$646$	0	0
$647$	−24.7279	−0.972155	−0.486077	−	0.873916i	$-0.661573\pi$
$647$	−24.7279	−0.972155	−0.486077	+	0.873916i	$0.661573\pi$
$648$	0	0
$649$	−38.6274	−1.51626
$650$	0	0
$651$	−3.17157	−0.124304
$652$	0	0
$653$	−41.5980	−1.62785	−0.813927	−	0.580967i	$-0.802675\pi$
$653$	−41.5980	−1.62785	−0.813927	+	0.580967i	$0.802675\pi$
$654$	0	0
$655$	5.79899	0.226585
$656$	0	0
$657$	−3.17157	−0.123735
$658$	0	0
$659$	7.75736	0.302184	0.151092	−	0.988520i	$-0.451721\pi$
$659$	7.75736	0.302184	0.151092	+	0.988520i	$0.451721\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	0	0
$663$	35.3137	1.37147
$664$	0	0
$665$	0.585786	0.0227158
$666$	0	0
$667$	14.9117	0.577383
$668$	0	0
$669$	4.82843	0.186678
$670$	0	0
$671$	5.65685	0.218380
$672$	0	0
$673$	−1.31371	−0.0506397	−0.0253199	−	0.999679i	$-0.508060\pi$
$673$	−1.31371	−0.0506397	−0.0253199	+	0.999679i	$0.508060\pi$
$674$	0	0
$675$	−4.65685	−0.179242
$676$	0	0
$677$	−27.9411	−1.07386	−0.536932	−	0.843625i	$-0.680417\pi$
$677$	−27.9411	−1.07386	−0.536932	+	0.843625i	$0.680417\pi$
$678$	0	0
$679$	11.6569	0.447349
$680$	0	0
$681$	28.4853	1.09156
$682$	0	0
$683$	−2.10051	−0.0803736	−0.0401868	−	0.999192i	$-0.512795\pi$
$683$	−2.10051	−0.0803736	−0.0401868	+	0.999192i	$0.512795\pi$
$684$	0	0
$685$	0.201010	0.00768020
$686$	0	0
$687$	−6.48528	−0.247429
$688$	0	0
$689$	−51.3137	−1.95490
$690$	0	0
$691$	−4.97056	−0.189089	−0.0945446	−	0.995521i	$-0.530139\pi$
$691$	−4.97056	−0.189089	−0.0945446	+	0.995521i	$0.530139\pi$
$692$	0	0
$693$	−2.82843	−0.107443
$694$	0	0
$695$	−2.62742	−0.0996636
$696$	0	0
$697$	19.7990	0.749940
$698$	0	0
$699$	−6.48528	−0.245296
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	7.85786	0.295944
$706$	0	0
$707$	5.07107	0.190717
$708$	0	0
$709$	9.65685	0.362671	0.181335	−	0.983421i	$-0.441958\pi$
$709$	9.65685	0.362671	0.181335	+	0.983421i	$0.441958\pi$
$710$	0	0
$711$	13.6569	0.512172
$712$	0	0
$713$	26.9117	1.00785
$714$	0	0
$715$	9.37258	0.350515
$716$	0	0
$717$	−13.6569	−0.510025
$718$	0	0
$719$	−30.3848	−1.13316	−0.566580	−	0.824006i	$-0.691734\pi$
$719$	−30.3848	−1.13316	−0.566580	+	0.824006i	$0.691734\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	14.0000	0.520666
$724$	0	0
$725$	−8.18377	−0.303937
$726$	0	0
$727$	18.6274	0.690853	0.345426	−	0.938446i	$-0.387734\pi$
$727$	18.6274	0.690853	0.345426	+	0.938446i	$0.387734\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	19.7990	0.732292
$732$	0	0
$733$	−15.1716	−0.560375	−0.280187	−	0.959945i	$-0.590397\pi$
$733$	−15.1716	−0.560375	−0.280187	+	0.959945i	$0.590397\pi$
$734$	0	0
$735$	−0.585786	−0.0216071
$736$	0	0
$737$	−41.9411	−1.54492
$738$	0	0
$739$	1.79899	0.0661769	0.0330885	−	0.999452i	$-0.489466\pi$
$739$	1.79899	0.0661769	0.0330885	+	0.999452i	$0.489466\pi$
$740$	0	0
$741$	−5.65685	−0.207810
$742$	0	0
$743$	−24.9289	−0.914554	−0.457277	−	0.889324i	$-0.651175\pi$
$743$	−24.9289	−0.914554	−0.457277	+	0.889324i	$0.651175\pi$
$744$	0	0
$745$	−0.887302	−0.0325082
$746$	0	0
$747$	−3.75736	−0.137475
$748$	0	0
$749$	−8.72792	−0.318911
$750$	0	0
$751$	51.7990	1.89017	0.945086	−	0.326822i	$-0.105978\pi$
$751$	51.7990	1.89017	0.945086	+	0.326822i	$0.105978\pi$
$752$	0	0
$753$	4.24264	0.154610
$754$	0	0
$755$	4.97056	0.180897
$756$	0	0
$757$	15.6569	0.569058	0.284529	−	0.958667i	$-0.408163\pi$
$757$	15.6569	0.569058	0.284529	+	0.958667i	$0.408163\pi$
$758$	0	0
$759$	24.0000	0.871145
$760$	0	0
$761$	4.78680	0.173521	0.0867606	−	0.996229i	$-0.472348\pi$
$761$	4.78680	0.173521	0.0867606	+	0.996229i	$0.472348\pi$
$762$	0	0
$763$	−12.1421	−0.439575
$764$	0	0
$765$	3.65685	0.132214
$766$	0	0
$767$	77.2548	2.78951
$768$	0	0
$769$	34.9706	1.26107	0.630535	−	0.776161i	$-0.282835\pi$
$769$	34.9706	1.26107	0.630535	+	0.776161i	$0.282835\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	−30.4853	−1.09648	−0.548240	−	0.836321i	$-0.684702\pi$
$773$	−30.4853	−1.09648	−0.548240	+	0.836321i	$0.684702\pi$
$774$	0	0
$775$	−14.7696	−0.530538
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	−3.17157	−0.113633
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	1.75736	0.0628029
$784$	0	0
$785$	−2.14214	−0.0764561
$786$	0	0
$787$	−12.1421	−0.432820	−0.216410	−	0.976303i	$-0.569435\pi$
$787$	−12.1421	−0.432820	−0.216410	+	0.976303i	$0.569435\pi$
$788$	0	0
$789$	−9.65685	−0.343793
$790$	0	0
$791$	−11.4142	−0.405843
$792$	0	0
$793$	−11.3137	−0.401762
$794$	0	0
$795$	−5.31371	−0.188458
$796$	0	0
$797$	−10.4853	−0.371408	−0.185704	−	0.982606i	$-0.559457\pi$
$797$	−10.4853	−0.371408	−0.185704	+	0.982606i	$0.559457\pi$
$798$	0	0
$799$	83.7401	2.96251
$800$	0	0
$801$	−11.1716	−0.394728
$802$	0	0
$803$	−8.97056	−0.316564
$804$	0	0
$805$	4.97056	0.175189
$806$	0	0
$807$	−0.343146	−0.0120793
$808$	0	0
$809$	6.97056	0.245072	0.122536	−	0.992464i	$-0.460897\pi$
$809$	6.97056	0.245072	0.122536	+	0.992464i	$0.460897\pi$
$810$	0	0
$811$	−11.0294	−0.387296	−0.193648	−	0.981071i	$-0.562032\pi$
$811$	−11.0294	−0.387296	−0.193648	+	0.981071i	$0.562032\pi$
$812$	0	0
$813$	16.9706	0.595184
$814$	0	0
$815$	10.8284	0.379303
$816$	0	0
$817$	−3.17157	−0.110959
$818$	0	0
$819$	5.65685	0.197666
$820$	0	0
$821$	−11.1716	−0.389891	−0.194945	−	0.980814i	$-0.562453\pi$
$821$	−11.1716	−0.389891	−0.194945	+	0.980814i	$0.562453\pi$
$822$	0	0
$823$	15.8579	0.552770	0.276385	−	0.961047i	$-0.410863\pi$
$823$	15.8579	0.552770	0.276385	+	0.961047i	$0.410863\pi$
$824$	0	0
$825$	−13.1716	−0.458575
$826$	0	0
$827$	−2.10051	−0.0730417	−0.0365209	−	0.999333i	$-0.511628\pi$
$827$	−2.10051	−0.0730417	−0.0365209	+	0.999333i	$0.511628\pi$
$828$	0	0
$829$	−6.28427	−0.218262	−0.109131	−	0.994027i	$-0.534807\pi$
$829$	−6.28427	−0.218262	−0.109131	+	0.994027i	$0.534807\pi$
$830$	0	0
$831$	11.3137	0.392468
$832$	0	0
$833$	−6.24264	−0.216295
$834$	0	0
$835$	10.0589	0.348102
$836$	0	0
$837$	3.17157	0.109626
$838$	0	0
$839$	−49.2548	−1.70047	−0.850233	−	0.526407i	$-0.823539\pi$
$839$	−49.2548	−1.70047	−0.850233	+	0.526407i	$0.823539\pi$
$840$	0	0
$841$	−25.9117	−0.893506
$842$	0	0
$843$	10.2426	0.352775
$844$	0	0
$845$	−11.1299	−0.382882
$846$	0	0
$847$	3.00000	0.103081
$848$	0	0
$849$	−14.1421	−0.485357
$850$	0	0
$851$	−50.9117	−1.74523
$852$	0	0
$853$	34.9706	1.19737	0.598685	−	0.800985i	$-0.295690\pi$
$853$	34.9706	1.19737	0.598685	+	0.800985i	$0.295690\pi$
$854$	0	0
$855$	−0.585786	−0.0200335
$856$	0	0
$857$	−9.51472	−0.325017	−0.162508	−	0.986707i	$-0.551958\pi$
$857$	−9.51472	−0.325017	−0.162508	+	0.986707i	$0.551958\pi$
$858$	0	0
$859$	42.3431	1.44473	0.722365	−	0.691512i	$-0.243055\pi$
$859$	42.3431	1.44473	0.722365	+	0.691512i	$0.243055\pi$
$860$	0	0
$861$	3.17157	0.108087
$862$	0	0
$863$	36.2426	1.23371	0.616857	−	0.787075i	$-0.288406\pi$
$863$	36.2426	1.23371	0.616857	+	0.787075i	$0.288406\pi$
$864$	0	0
$865$	5.17157	0.175839
$866$	0	0
$867$	21.9706	0.746159
$868$	0	0
$869$	38.6274	1.31035
$870$	0	0
$871$	83.8823	2.84224
$872$	0	0
$873$	−11.6569	−0.394525
$874$	0	0
$875$	−5.65685	−0.191237
$876$	0	0
$877$	41.3137	1.39506	0.697532	−	0.716553i	$-0.254281\pi$
$877$	41.3137	1.39506	0.697532	+	0.716553i	$0.254281\pi$
$878$	0	0
$879$	18.4853	0.623493
$880$	0	0
$881$	−16.3848	−0.552017	−0.276009	−	0.961155i	$-0.589012\pi$
$881$	−16.3848	−0.552017	−0.276009	+	0.961155i	$0.589012\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	42.4264	1.42454	0.712270	−	0.701906i	$-0.247667\pi$
$887$	42.4264	1.42454	0.712270	+	0.701906i	$0.247667\pi$
$888$	0	0
$889$	−8.48528	−0.284587
$890$	0	0
$891$	2.82843	0.0947559
$892$	0	0
$893$	−13.4142	−0.448890
$894$	0	0
$895$	6.48528	0.216779
$896$	0	0
$897$	−48.0000	−1.60267
$898$	0	0
$899$	5.57359	0.185890
$900$	0	0
$901$	−56.6274	−1.88653
$902$	0	0
$903$	3.17157	0.105543
$904$	0	0
$905$	−7.79899	−0.259247
$906$	0	0
$907$	47.5980	1.58046	0.790232	−	0.612807i	$-0.209960\pi$
$907$	47.5980	1.58046	0.790232	+	0.612807i	$0.209960\pi$
$908$	0	0
$909$	−5.07107	−0.168197
$910$	0	0
$911$	−47.5563	−1.57561	−0.787806	−	0.615923i	$-0.788783\pi$
$911$	−47.5563	−1.57561	−0.787806	+	0.615923i	$0.788783\pi$
$912$	0	0
$913$	−10.6274	−0.351716
$914$	0	0
$915$	−1.17157	−0.0387310
$916$	0	0
$917$	9.89949	0.326910
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	12.1421	0.400097
$922$	0	0
$923$	72.0000	2.36991
$924$	0	0
$925$	27.9411	0.918699
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	0.786797	0.0258140	0.0129070	−	0.999917i	$-0.495891\pi$
$929$	0.786797	0.0258140	0.0129070	+	0.999917i	$0.495891\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	20.0416	0.656134
$934$	0	0
$935$	10.3431	0.338257
$936$	0	0
$937$	58.9706	1.92648	0.963242	−	0.268635i	$-0.0865724\pi$
$937$	58.9706	1.92648	0.963242	+	0.268635i	$0.0865724\pi$
$938$	0	0
$939$	−30.9706	−1.01069
$940$	0	0
$941$	26.2843	0.856843	0.428421	−	0.903579i	$-0.359070\pi$
$941$	26.2843	0.856843	0.428421	+	0.903579i	$0.359070\pi$
$942$	0	0
$943$	−26.9117	−0.876365
$944$	0	0
$945$	0.585786	0.0190556
$946$	0	0
$947$	−12.4853	−0.405717	−0.202859	−	0.979208i	$-0.565023\pi$
$947$	−12.4853	−0.405717	−0.202859	+	0.979208i	$0.565023\pi$
$948$	0	0
$949$	17.9411	0.582394
$950$	0	0
$951$	−15.2132	−0.493322
$952$	0	0
$953$	−1.95837	−0.0634378	−0.0317189	−	0.999497i	$-0.510098\pi$
$953$	−1.95837	−0.0634378	−0.0317189	+	0.999497i	$0.510098\pi$
$954$	0	0
$955$	5.37258	0.173853
$956$	0	0
$957$	4.97056	0.160675
$958$	0	0
$959$	0.343146	0.0110808
$960$	0	0
$961$	−20.9411	−0.675520
$962$	0	0
$963$	8.72792	0.281253
$964$	0	0
$965$	6.82843	0.219815
$966$	0	0
$967$	−19.1716	−0.616516	−0.308258	−	0.951303i	$-0.599746\pi$
$967$	−19.1716	−0.616516	−0.308258	+	0.951303i	$0.599746\pi$
$968$	0	0
$969$	−6.24264	−0.200543
$970$	0	0
$971$	−35.3137	−1.13327	−0.566635	−	0.823969i	$-0.691755\pi$
$971$	−35.3137	−1.13327	−0.566635	+	0.823969i	$0.691755\pi$
$972$	0	0
$973$	−4.48528	−0.143792
$974$	0	0
$975$	26.3431	0.843656
$976$	0	0
$977$	−49.1543	−1.57259	−0.786293	−	0.617854i	$-0.788002\pi$
$977$	−49.1543	−1.57259	−0.786293	+	0.617854i	$0.788002\pi$
$978$	0	0
$979$	−31.5980	−1.00988
$980$	0	0
$981$	12.1421	0.387669
$982$	0	0
$983$	−28.2843	−0.902128	−0.451064	−	0.892492i	$-0.648955\pi$
$983$	−28.2843	−0.902128	−0.451064	+	0.892492i	$0.648955\pi$
$984$	0	0
$985$	11.1127	0.354080
$986$	0	0
$987$	13.4142	0.426979
$988$	0	0
$989$	−26.9117	−0.855742
$990$	0	0
$991$	−32.2843	−1.02554	−0.512772	−	0.858525i	$-0.671381\pi$
$991$	−32.2843	−1.02554	−0.512772	+	0.858525i	$0.671381\pi$
$992$	0	0
$993$	14.1421	0.448787
$994$	0	0
$995$	−8.28427	−0.262629
$996$	0	0
$997$	8.34315	0.264230	0.132115	−	0.991234i	$-0.457823\pi$
$997$	8.34315	0.264230	0.132115	+	0.991234i	$0.457823\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.r.1.2	✓	2
3.2	odd	2		9576.2.a.by.1.1		2
4.3	odd	2		6384.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.r.1.2	✓	2	1.1	even	1	trivial
6384.2.a.bh.1.2		2	4.3	odd	2
9576.2.a.by.1.1		2	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3192.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.585786 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.82843 q^{11} -5.65685 q^{13} -0.585786 q^{15} -6.24264 q^{17} +1.00000 q^{19} -1.00000 q^{21} +8.48528 q^{23} -4.65685 q^{25} +1.00000 q^{27} +1.75736 q^{29} +3.17157 q^{31} +2.82843 q^{33} +0.585786 q^{35} -6.00000 q^{37} -5.65685 q^{39} -3.17157 q^{41} -3.17157 q^{43} -0.585786 q^{45} -13.4142 q^{47} +1.00000 q^{49} -6.24264 q^{51} +9.07107 q^{53} -1.65685 q^{55} +1.00000 q^{57} -13.6569 q^{59} +2.00000 q^{61} -1.00000 q^{63} +3.31371 q^{65} -14.8284 q^{67} +8.48528 q^{69} -12.7279 q^{71} -3.17157 q^{73} -4.65685 q^{75} -2.82843 q^{77} +13.6569 q^{79} +1.00000 q^{81} -3.75736 q^{83} +3.65685 q^{85} +1.75736 q^{87} -11.1716 q^{89} +5.65685 q^{91} +3.17157 q^{93} -0.585786 q^{95} -11.6569 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} - 4 q^{17} + 2 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 12 q^{29} + 12 q^{31} + 4 q^{35} - 12 q^{37} - 12 q^{41} - 12 q^{43} - 4 q^{45} - 24 q^{47}+ \cdots - 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^15 - 4 * q^17 + 2 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 + 12 * q^29 + 12 * q^31 + 4 * q^35 - 12 * q^37 - 12 * q^41 - 12 * q^43 - 4 * q^45 - 24 * q^47 + 2 * q^49 - 4 * q^51 + 4 * q^53 + 8 * q^55 + 2 * q^57 - 16 * q^59 + 4 * q^61 - 2 * q^63 - 16 * q^65 - 24 * q^67 - 12 * q^73 + 2 * q^75 + 16 * q^79 + 2 * q^81 - 16 * q^83 - 4 * q^85 + 12 * q^87 - 28 * q^89 + 12 * q^93 - 4 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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