LMFDB - Embedded newform 2793.2.a.q.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2793,2,Mod(1,2793)] 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("2793.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2793, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2793 = 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2793.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$22.3022172845$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	2793.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} -1.73205 q^{8} +1.00000 q^{9} -3.46410 q^{11} -1.00000 q^{12} +3.46410 q^{13} -5.00000 q^{16} +4.00000 q^{17} +1.73205 q^{18} +1.00000 q^{19} -6.00000 q^{22} -0.535898 q^{23} +1.73205 q^{24} -5.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -4.00000 q^{29} -2.92820 q^{31} -5.19615 q^{32} +3.46410 q^{33} +6.92820 q^{34} +1.00000 q^{36} +4.92820 q^{37} +1.73205 q^{38} -3.46410 q^{39} -10.9282 q^{41} -10.9282 q^{43} -3.46410 q^{44} -0.928203 q^{46} +4.92820 q^{47} +5.00000 q^{48} -8.66025 q^{50} -4.00000 q^{51} +3.46410 q^{52} -4.00000 q^{53} -1.73205 q^{54} -1.00000 q^{57} -6.92820 q^{58} -8.00000 q^{59} +1.07180 q^{61} -5.07180 q^{62} +1.00000 q^{64} +6.00000 q^{66} -6.39230 q^{67} +4.00000 q^{68} +0.535898 q^{69} -3.46410 q^{71} -1.73205 q^{72} -8.00000 q^{73} +8.53590 q^{74} +5.00000 q^{75} +1.00000 q^{76} -6.00000 q^{78} +3.46410 q^{79} +1.00000 q^{81} -18.9282 q^{82} -8.92820 q^{83} -18.9282 q^{86} +4.00000 q^{87} +6.00000 q^{88} -12.0000 q^{89} -0.535898 q^{92} +2.92820 q^{93} +8.53590 q^{94} +5.19615 q^{96} +19.4641 q^{97} -3.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} + 2 q^{9} - 2 q^{12} - 10 q^{16} + 8 q^{17} + 2 q^{19} - 12 q^{22} - 8 q^{23} - 10 q^{25} + 12 q^{26} - 2 q^{27} - 8 q^{29} + 8 q^{31} + 2 q^{36} - 4 q^{37} - 8 q^{41} - 8 q^{43}+ \cdots + 32 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^9 - 2 * q^12 - 10 * q^16 + 8 * q^17 + 2 * q^19 - 12 * q^22 - 8 * q^23 - 10 * q^25 + 12 * q^26 - 2 * q^27 - 8 * q^29 + 8 * q^31 + 2 * q^36 - 4 * q^37 - 8 * q^41 - 8 * q^43 + 12 * q^46 - 4 * q^47 + 10 * q^48 - 8 * q^51 - 8 * q^53 - 2 * q^57 - 16 * q^59 + 16 * q^61 - 24 * q^62 + 2 * q^64 + 12 * q^66 + 8 * q^67 + 8 * q^68 + 8 * q^69 - 16 * q^73 + 24 * q^74 + 10 * q^75 + 2 * q^76 - 12 * q^78 + 2 * q^81 - 24 * q^82 - 4 * q^83 - 24 * q^86 + 8 * q^87 + 12 * q^88 - 24 * q^89 - 8 * q^92 - 8 * q^93 + 24 * q^94 + 32 * 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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.73205	−0.707107
$7$	0	0
$7$	0	0
$8$	−1.73205	−0.612372
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	−1.00000	−0.288675
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	0	0
$16$	−5.00000	−1.25000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	1.73205	0.408248
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−0.535898	−0.111743	−0.0558713	−	0.998438i	$-0.517794\pi$
$23$	−0.535898	−0.111743	−0.0558713	+	0.998438i	$0.517794\pi$
$24$	1.73205	0.353553
$25$	−5.00000	−1.00000
$26$	6.00000	1.17670
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−2.92820	−0.525921	−0.262960	−	0.964807i	$-0.584699\pi$
$31$	−2.92820	−0.525921	−0.262960	+	0.964807i	$0.584699\pi$
$32$	−5.19615	−0.918559
$33$	3.46410	0.603023
$34$	6.92820	1.18818
$35$	0	0
$36$	1.00000	0.166667
$37$	4.92820	0.810192	0.405096	−	0.914274i	$-0.367238\pi$
$37$	4.92820	0.810192	0.405096	+	0.914274i	$0.367238\pi$
$38$	1.73205	0.280976
$39$	−3.46410	−0.554700
$40$	0	0
$41$	−10.9282	−1.70670	−0.853349	−	0.521340i	$-0.825432\pi$
$41$	−10.9282	−1.70670	−0.853349	+	0.521340i	$0.825432\pi$
$42$	0	0
$43$	−10.9282	−1.66654	−0.833268	−	0.552870i	$-0.813533\pi$
$43$	−10.9282	−1.66654	−0.833268	+	0.552870i	$0.813533\pi$
$44$	−3.46410	−0.522233
$45$	0	0
$46$	−0.928203	−0.136856
$47$	4.92820	0.718852	0.359426	−	0.933174i	$-0.382972\pi$
$47$	4.92820	0.718852	0.359426	+	0.933174i	$0.382972\pi$
$48$	5.00000	0.721688
$49$	0	0
$50$	−8.66025	−1.22474
$51$	−4.00000	−0.560112
$52$	3.46410	0.480384
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	−1.73205	−0.235702
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−6.92820	−0.909718
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	1.07180	0.137230	0.0686148	−	0.997643i	$-0.478142\pi$
$61$	1.07180	0.137230	0.0686148	+	0.997643i	$0.478142\pi$
$62$	−5.07180	−0.644119
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	−6.39230	−0.780944	−0.390472	−	0.920615i	$-0.627688\pi$
$67$	−6.39230	−0.780944	−0.390472	+	0.920615i	$0.627688\pi$
$68$	4.00000	0.485071
$69$	0.535898	0.0645146
$70$	0	0
$71$	−3.46410	−0.411113	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410	−0.411113	−0.205557	+	0.978645i	$0.565900\pi$
$72$	−1.73205	−0.204124
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	8.53590	0.992278
$75$	5.00000	0.577350
$76$	1.00000	0.114708
$77$	0	0
$78$	−6.00000	−0.679366
$79$	3.46410	0.389742	0.194871	−	0.980829i	$-0.437571\pi$
$79$	3.46410	0.389742	0.194871	+	0.980829i	$0.437571\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−18.9282	−2.09027
$83$	−8.92820	−0.979998	−0.489999	−	0.871723i	$-0.663003\pi$
$83$	−8.92820	−0.979998	−0.489999	+	0.871723i	$0.663003\pi$
$84$	0	0
$85$	0	0
$86$	−18.9282	−2.04108
$87$	4.00000	0.428845
$88$	6.00000	0.639602
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0	0
$92$	−0.535898	−0.0558713
$93$	2.92820	0.303641
$94$	8.53590	0.880411
$95$	0	0
$96$	5.19615	0.530330
$97$	19.4641	1.97628	0.988140	−	0.153555i	$-0.0490723\pi$
$97$	19.4641	1.97628	0.988140	+	0.153555i	$0.0490723\pi$
$98$	0	0
$99$	−3.46410	−0.348155
$100$	−5.00000	−0.500000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−6.92820	−0.685994
$103$	−2.92820	−0.288524	−0.144262	−	0.989539i	$-0.546081\pi$
$103$	−2.92820	−0.288524	−0.144262	+	0.989539i	$0.546081\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−6.92820	−0.672927
$107$	2.39230	0.231273	0.115636	−	0.993292i	$-0.463109\pi$
$107$	2.39230	0.231273	0.115636	+	0.993292i	$0.463109\pi$
$108$	−1.00000	−0.0962250
$109$	4.92820	0.472036	0.236018	−	0.971749i	$-0.424158\pi$
$109$	4.92820	0.472036	0.236018	+	0.971749i	$0.424158\pi$
$110$	0	0
$111$	−4.92820	−0.467764
$112$	0	0
$113$	−18.9282	−1.78062	−0.890308	−	0.455359i	$-0.849511\pi$
$113$	−18.9282	−1.78062	−0.890308	+	0.455359i	$0.849511\pi$
$114$	−1.73205	−0.162221
$115$	0	0
$116$	−4.00000	−0.371391
$117$	3.46410	0.320256
$118$	−13.8564	−1.27559
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.85641	0.168071
$123$	10.9282	0.985363
$124$	−2.92820	−0.262960
$125$	0	0
$126$	0	0
$127$	−19.4641	−1.72716	−0.863580	−	0.504212i	$-0.831783\pi$
$127$	−19.4641	−1.72716	−0.863580	+	0.504212i	$0.831783\pi$
$128$	12.1244	1.07165
$129$	10.9282	0.962175
$130$	0	0
$131$	−4.92820	−0.430579	−0.215290	−	0.976550i	$-0.569070\pi$
$131$	−4.92820	−0.430579	−0.215290	+	0.976550i	$0.569070\pi$
$132$	3.46410	0.301511
$133$	0	0
$134$	−11.0718	−0.956458
$135$	0	0
$136$	−6.92820	−0.594089
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0.928203	0.0790139
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−4.92820	−0.415030
$142$	−6.00000	−0.503509
$143$	−12.0000	−1.00349
$144$	−5.00000	−0.416667
$145$	0	0
$146$	−13.8564	−1.14676
$147$	0	0
$148$	4.92820	0.405096
$149$	7.07180	0.579344	0.289672	−	0.957126i	$-0.406454\pi$
$149$	7.07180	0.579344	0.289672	+	0.957126i	$0.406454\pi$
$150$	8.66025	0.707107
$151$	10.3923	0.845714	0.422857	−	0.906196i	$-0.361027\pi$
$151$	10.3923	0.845714	0.422857	+	0.906196i	$0.361027\pi$
$152$	−1.73205	−0.140488
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	−3.46410	−0.277350
$157$	16.0000	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$157$	16.0000	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$158$	6.00000	0.477334
$159$	4.00000	0.317221
$160$	0	0
$161$	0	0
$162$	1.73205	0.136083
$163$	10.9282	0.855963	0.427981	−	0.903788i	$-0.359225\pi$
$163$	10.9282	0.855963	0.427981	+	0.903788i	$0.359225\pi$
$164$	−10.9282	−0.853349
$165$	0	0
$166$	−15.4641	−1.20025
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	1.00000	0.0764719
$172$	−10.9282	−0.833268
$173$	17.8564	1.35760	0.678799	−	0.734324i	$-0.262501\pi$
$173$	17.8564	1.35760	0.678799	+	0.734324i	$0.262501\pi$
$174$	6.92820	0.525226
$175$	0	0
$176$	17.3205	1.30558
$177$	8.00000	0.601317
$178$	−20.7846	−1.55787
$179$	−4.53590	−0.339029	−0.169514	−	0.985528i	$-0.554220\pi$
$179$	−4.53590	−0.339029	−0.169514	+	0.985528i	$0.554220\pi$
$180$	0	0
$181$	9.32051	0.692788	0.346394	−	0.938089i	$-0.387406\pi$
$181$	9.32051	0.692788	0.346394	+	0.938089i	$0.387406\pi$
$182$	0	0
$183$	−1.07180	−0.0792295
$184$	0.928203	0.0684280
$185$	0	0
$186$	5.07180	0.371882
$187$	−13.8564	−1.01328
$188$	4.92820	0.359426
$189$	0	0
$190$	0	0
$191$	−7.46410	−0.540083	−0.270042	−	0.962849i	$-0.587037\pi$
$191$	−7.46410	−0.540083	−0.270042	+	0.962849i	$0.587037\pi$
$192$	−1.00000	−0.0721688
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	33.7128	2.42044
$195$	0	0
$196$	0	0
$197$	8.92820	0.636108	0.318054	−	0.948073i	$-0.396971\pi$
$197$	8.92820	0.636108	0.318054	+	0.948073i	$0.396971\pi$
$198$	−6.00000	−0.426401
$199$	−21.8564	−1.54936	−0.774680	−	0.632354i	$-0.782089\pi$
$199$	−21.8564	−1.54936	−0.774680	+	0.632354i	$0.782089\pi$
$200$	8.66025	0.612372
$201$	6.39230	0.450878
$202$	0	0
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−5.07180	−0.353369
$207$	−0.535898	−0.0372475
$208$	−17.3205	−1.20096
$209$	−3.46410	−0.239617
$210$	0	0
$211$	−7.46410	−0.513850	−0.256925	−	0.966431i	$-0.582709\pi$
$211$	−7.46410	−0.513850	−0.256925	+	0.966431i	$0.582709\pi$
$212$	−4.00000	−0.274721
$213$	3.46410	0.237356
$214$	4.14359	0.283250
$215$	0	0
$216$	1.73205	0.117851
$217$	0	0
$218$	8.53590	0.578124
$219$	8.00000	0.540590
$220$	0	0
$221$	13.8564	0.932083
$222$	−8.53590	−0.572892
$223$	26.9282	1.80325	0.901623	−	0.432523i	$-0.142377\pi$
$223$	26.9282	1.80325	0.901623	+	0.432523i	$0.142377\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	−32.7846	−2.18080
$227$	1.85641	0.123214	0.0616070	−	0.998100i	$-0.480377\pi$
$227$	1.85641	0.123214	0.0616070	+	0.998100i	$0.480377\pi$
$228$	−1.00000	−0.0662266
$229$	−6.92820	−0.457829	−0.228914	−	0.973447i	$-0.573518\pi$
$229$	−6.92820	−0.457829	−0.228914	+	0.973447i	$0.573518\pi$
$230$	0	0
$231$	0	0
$232$	6.92820	0.454859
$233$	−19.8564	−1.30084	−0.650418	−	0.759576i	$-0.725406\pi$
$233$	−19.8564	−1.30084	−0.650418	+	0.759576i	$0.725406\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−8.00000	−0.520756
$237$	−3.46410	−0.225018
$238$	0	0
$239$	−8.53590	−0.552141	−0.276071	−	0.961137i	$-0.589032\pi$
$239$	−8.53590	−0.552141	−0.276071	+	0.961137i	$0.589032\pi$
$240$	0	0
$241$	−3.46410	−0.223142	−0.111571	−	0.993756i	$-0.535588\pi$
$241$	−3.46410	−0.223142	−0.111571	+	0.993756i	$0.535588\pi$
$242$	1.73205	0.111340
$243$	−1.00000	−0.0641500
$244$	1.07180	0.0686148
$245$	0	0
$246$	18.9282	1.20682
$247$	3.46410	0.220416
$248$	5.07180	0.322059
$249$	8.92820	0.565802
$250$	0	0
$251$	0.928203	0.0585877	0.0292938	−	0.999571i	$-0.490674\pi$
$251$	0.928203	0.0585877	0.0292938	+	0.999571i	$0.490674\pi$
$252$	0	0
$253$	1.85641	0.116711
$254$	−33.7128	−2.11533
$255$	0	0
$256$	19.0000	1.18750
$257$	17.8564	1.11385	0.556926	−	0.830562i	$-0.311981\pi$
$257$	17.8564	1.11385	0.556926	+	0.830562i	$0.311981\pi$
$258$	18.9282	1.17842
$259$	0	0
$260$	0	0
$261$	−4.00000	−0.247594
$262$	−8.53590	−0.527350
$263$	−0.535898	−0.0330449	−0.0165225	−	0.999863i	$-0.505260\pi$
$263$	−0.535898	−0.0330449	−0.0165225	+	0.999863i	$0.505260\pi$
$264$	−6.00000	−0.369274
$265$	0	0
$266$	0	0
$267$	12.0000	0.734388
$268$	−6.39230	−0.390472
$269$	2.92820	0.178536	0.0892679	−	0.996008i	$-0.471547\pi$
$269$	2.92820	0.178536	0.0892679	+	0.996008i	$0.471547\pi$
$270$	0	0
$271$	1.85641	0.112769	0.0563843	−	0.998409i	$-0.482043\pi$
$271$	1.85641	0.112769	0.0563843	+	0.998409i	$0.482043\pi$
$272$	−20.0000	−1.21268
$273$	0	0
$274$	24.2487	1.46492
$275$	17.3205	1.04447
$276$	0.535898	0.0322573
$277$	29.7128	1.78527	0.892635	−	0.450780i	$-0.148854\pi$
$277$	29.7128	1.78527	0.892635	+	0.450780i	$0.148854\pi$
$278$	−6.92820	−0.415526
$279$	−2.92820	−0.175307
$280$	0	0
$281$	−6.92820	−0.413302	−0.206651	−	0.978415i	$-0.566256\pi$
$281$	−6.92820	−0.413302	−0.206651	+	0.978415i	$0.566256\pi$
$282$	−8.53590	−0.508305
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	−3.46410	−0.205557
$285$	0	0
$286$	−20.7846	−1.22902
$287$	0	0
$288$	−5.19615	−0.306186
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−19.4641	−1.14101
$292$	−8.00000	−0.468165
$293$	−32.7846	−1.91530	−0.957649	−	0.287939i	$-0.907030\pi$
$293$	−32.7846	−1.91530	−0.957649	+	0.287939i	$0.907030\pi$
$294$	0	0
$295$	0	0
$296$	−8.53590	−0.496139
$297$	3.46410	0.201008
$298$	12.2487	0.709549
$299$	−1.85641	−0.107359
$300$	5.00000	0.288675
$301$	0	0
$302$	18.0000	1.03578
$303$	0	0
$304$	−5.00000	−0.286770
$305$	0	0
$306$	6.92820	0.396059
$307$	1.85641	0.105951	0.0529754	−	0.998596i	$-0.483130\pi$
$307$	1.85641	0.105951	0.0529754	+	0.998596i	$0.483130\pi$
$308$	0	0
$309$	2.92820	0.166580
$310$	0	0
$311$	−24.9282	−1.41355	−0.706774	−	0.707439i	$-0.749850\pi$
$311$	−24.9282	−1.41355	−0.706774	+	0.707439i	$0.749850\pi$
$312$	6.00000	0.339683
$313$	−9.07180	−0.512768	−0.256384	−	0.966575i	$-0.582531\pi$
$313$	−9.07180	−0.512768	−0.256384	+	0.966575i	$0.582531\pi$
$314$	27.7128	1.56392
$315$	0	0
$316$	3.46410	0.194871
$317$	28.0000	1.57264	0.786318	−	0.617822i	$-0.211985\pi$
$317$	28.0000	1.57264	0.786318	+	0.617822i	$0.211985\pi$
$318$	6.92820	0.388514
$319$	13.8564	0.775810
$320$	0	0
$321$	−2.39230	−0.133525
$322$	0	0
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	−17.3205	−0.960769
$326$	18.9282	1.04834
$327$	−4.92820	−0.272530
$328$	18.9282	1.04514
$329$	0	0
$330$	0	0
$331$	−36.2487	−1.99241	−0.996205	−	0.0870415i	$-0.972259\pi$
$331$	−36.2487	−1.99241	−0.996205	+	0.0870415i	$0.972259\pi$
$332$	−8.92820	−0.489999
$333$	4.92820	0.270064
$334$	24.0000	1.31322
$335$	0	0
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	−1.73205	−0.0942111
$339$	18.9282	1.02804
$340$	0	0
$341$	10.1436	0.549306
$342$	1.73205	0.0936586
$343$	0	0
$344$	18.9282	1.02054
$345$	0	0
$346$	30.9282	1.66271
$347$	−20.5359	−1.10242	−0.551212	−	0.834365i	$-0.685835\pi$
$347$	−20.5359	−1.10242	−0.551212	+	0.834365i	$0.685835\pi$
$348$	4.00000	0.214423
$349$	22.9282	1.22732	0.613659	−	0.789571i	$-0.289697\pi$
$349$	22.9282	1.22732	0.613659	+	0.789571i	$0.289697\pi$
$350$	0	0
$351$	−3.46410	−0.184900
$352$	18.0000	0.959403
$353$	17.8564	0.950401	0.475200	−	0.879878i	$-0.342376\pi$
$353$	17.8564	0.950401	0.475200	+	0.879878i	$0.342376\pi$
$354$	13.8564	0.736460
$355$	0	0
$356$	−12.0000	−0.635999
$357$	0	0
$358$	−7.85641	−0.415224
$359$	14.3923	0.759597	0.379798	−	0.925069i	$-0.375993\pi$
$359$	14.3923	0.759597	0.379798	+	0.925069i	$0.375993\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	16.1436	0.848488
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	0	0
$366$	−1.85641	−0.0970359
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	2.67949	0.139678
$369$	−10.9282	−0.568900
$370$	0	0
$371$	0	0
$372$	2.92820	0.151820
$373$	18.7846	0.972630	0.486315	−	0.873784i	$-0.338341\pi$
$373$	18.7846	0.972630	0.486315	+	0.873784i	$0.338341\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	−8.53590	−0.440205
$377$	−13.8564	−0.713641
$378$	0	0
$379$	9.60770	0.493514	0.246757	−	0.969077i	$-0.420635\pi$
$379$	9.60770	0.493514	0.246757	+	0.969077i	$0.420635\pi$
$380$	0	0
$381$	19.4641	0.997176
$382$	−12.9282	−0.661464
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	−12.1244	−0.618718
$385$	0	0
$386$	−17.3205	−0.881591
$387$	−10.9282	−0.555512
$388$	19.4641	0.988140
$389$	−19.0718	−0.966978	−0.483489	−	0.875350i	$-0.660631\pi$
$389$	−19.0718	−0.966978	−0.483489	+	0.875350i	$0.660631\pi$
$390$	0	0
$391$	−2.14359	−0.108406
$392$	0	0
$393$	4.92820	0.248595
$394$	15.4641	0.779070
$395$	0	0
$396$	−3.46410	−0.174078
$397$	−28.7846	−1.44466	−0.722329	−	0.691549i	$-0.756928\pi$
$397$	−28.7846	−1.44466	−0.722329	+	0.691549i	$0.756928\pi$
$398$	−37.8564	−1.89757
$399$	0	0
$400$	25.0000	1.25000
$401$	−1.07180	−0.0535230	−0.0267615	−	0.999642i	$-0.508519\pi$
$401$	−1.07180	−0.0535230	−0.0267615	+	0.999642i	$0.508519\pi$
$402$	11.0718	0.552211
$403$	−10.1436	−0.505288
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−17.0718	−0.846218
$408$	6.92820	0.342997
$409$	−9.32051	−0.460869	−0.230435	−	0.973088i	$-0.574015\pi$
$409$	−9.32051	−0.460869	−0.230435	+	0.973088i	$0.574015\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	−2.92820	−0.144262
$413$	0	0
$414$	−0.928203	−0.0456187
$415$	0	0
$416$	−18.0000	−0.882523
$417$	4.00000	0.195881
$418$	−6.00000	−0.293470
$419$	−36.6410	−1.79003	−0.895015	−	0.446035i	$-0.852836\pi$
$419$	−36.6410	−1.79003	−0.895015	+	0.446035i	$0.852836\pi$
$420$	0	0
$421$	4.92820	0.240186	0.120093	−	0.992763i	$-0.461681\pi$
$421$	4.92820	0.240186	0.120093	+	0.992763i	$0.461681\pi$
$422$	−12.9282	−0.629335
$423$	4.92820	0.239617
$424$	6.92820	0.336463
$425$	−20.0000	−0.970143
$426$	6.00000	0.290701
$427$	0	0
$428$	2.39230	0.115636
$429$	12.0000	0.579365
$430$	0	0
$431$	−5.60770	−0.270113	−0.135057	−	0.990838i	$-0.543122\pi$
$431$	−5.60770	−0.270113	−0.135057	+	0.990838i	$0.543122\pi$
$432$	5.00000	0.240563
$433$	25.3205	1.21683	0.608413	−	0.793621i	$-0.291806\pi$
$433$	25.3205	1.21683	0.608413	+	0.793621i	$0.291806\pi$
$434$	0	0
$435$	0	0
$436$	4.92820	0.236018
$437$	−0.535898	−0.0256355
$438$	13.8564	0.662085
$439$	30.6410	1.46242	0.731208	−	0.682155i	$-0.238957\pi$
$439$	30.6410	1.46242	0.731208	+	0.682155i	$0.238957\pi$
$440$	0	0
$441$	0	0
$442$	24.0000	1.14156
$443$	1.32051	0.0627392	0.0313696	−	0.999508i	$-0.490013\pi$
$443$	1.32051	0.0627392	0.0313696	+	0.999508i	$0.490013\pi$
$444$	−4.92820	−0.233882
$445$	0	0
$446$	46.6410	2.20852
$447$	−7.07180	−0.334485
$448$	0	0
$449$	30.9282	1.45959	0.729796	−	0.683665i	$-0.239615\pi$
$449$	30.9282	1.45959	0.729796	+	0.683665i	$0.239615\pi$
$450$	−8.66025	−0.408248
$451$	37.8564	1.78259
$452$	−18.9282	−0.890308
$453$	−10.3923	−0.488273
$454$	3.21539	0.150906
$455$	0	0
$456$	1.73205	0.0811107
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−12.0000	−0.560723
$459$	−4.00000	−0.186704
$460$	0	0
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	0	0
$463$	13.8564	0.643962	0.321981	−	0.946746i	$-0.395651\pi$
$463$	13.8564	0.643962	0.321981	+	0.946746i	$0.395651\pi$
$464$	20.0000	0.928477
$465$	0	0
$466$	−34.3923	−1.59319
$467$	−4.92820	−0.228050	−0.114025	−	0.993478i	$-0.536374\pi$
$467$	−4.92820	−0.228050	−0.114025	+	0.993478i	$0.536374\pi$
$468$	3.46410	0.160128
$469$	0	0
$470$	0	0
$471$	−16.0000	−0.737241
$472$	13.8564	0.637793
$473$	37.8564	1.74064
$474$	−6.00000	−0.275589
$475$	−5.00000	−0.229416
$476$	0	0
$477$	−4.00000	−0.183147
$478$	−14.7846	−0.676232
$479$	−23.0718	−1.05418	−0.527089	−	0.849810i	$-0.676716\pi$
$479$	−23.0718	−1.05418	−0.527089	+	0.849810i	$0.676716\pi$
$480$	0	0
$481$	17.0718	0.778407
$482$	−6.00000	−0.273293
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	−1.73205	−0.0785674
$487$	−18.3923	−0.833435	−0.416717	−	0.909036i	$-0.636820\pi$
$487$	−18.3923	−0.833435	−0.416717	+	0.909036i	$0.636820\pi$
$488$	−1.85641	−0.0840356
$489$	−10.9282	−0.494190
$490$	0	0
$491$	33.3205	1.50373	0.751867	−	0.659315i	$-0.229154\pi$
$491$	33.3205	1.50373	0.751867	+	0.659315i	$0.229154\pi$
$492$	10.9282	0.492681
$493$	−16.0000	−0.720604
$494$	6.00000	0.269953
$495$	0	0
$496$	14.6410	0.657401
$497$	0	0
$498$	15.4641	0.692963
$499$	10.9282	0.489214	0.244607	−	0.969622i	$-0.421341\pi$
$499$	10.9282	0.489214	0.244607	+	0.969622i	$0.421341\pi$
$500$	0	0
$501$	−13.8564	−0.619059
$502$	1.60770	0.0717549
$503$	32.6410	1.45539	0.727695	−	0.685900i	$-0.240591\pi$
$503$	32.6410	1.45539	0.727695	+	0.685900i	$0.240591\pi$
$504$	0	0
$505$	0	0
$506$	3.21539	0.142942
$507$	1.00000	0.0444116
$508$	−19.4641	−0.863580
$509$	1.85641	0.0822838	0.0411419	−	0.999153i	$-0.486900\pi$
$509$	1.85641	0.0822838	0.0411419	+	0.999153i	$0.486900\pi$
$510$	0	0
$511$	0	0
$512$	8.66025	0.382733
$513$	−1.00000	−0.0441511
$514$	30.9282	1.36418
$515$	0	0
$516$	10.9282	0.481087
$517$	−17.0718	−0.750817
$518$	0	0
$519$	−17.8564	−0.783809
$520$	0	0
$521$	−26.9282	−1.17975	−0.589873	−	0.807496i	$-0.700822\pi$
$521$	−26.9282	−1.17975	−0.589873	+	0.807496i	$0.700822\pi$
$522$	−6.92820	−0.303239
$523$	33.8564	1.48044	0.740219	−	0.672366i	$-0.234722\pi$
$523$	33.8564	1.48044	0.740219	+	0.672366i	$0.234722\pi$
$524$	−4.92820	−0.215290
$525$	0	0
$526$	−0.928203	−0.0404716
$527$	−11.7128	−0.510218
$528$	−17.3205	−0.753778
$529$	−22.7128	−0.987514
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−37.8564	−1.63974
$534$	20.7846	0.899438
$535$	0	0
$536$	11.0718	0.478229
$537$	4.53590	0.195738
$538$	5.07180	0.218661
$539$	0	0
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	3.21539	0.138113
$543$	−9.32051	−0.399981
$544$	−20.7846	−0.891133
$545$	0	0
$546$	0	0
$547$	7.46410	0.319142	0.159571	−	0.987186i	$-0.448989\pi$
$547$	7.46410	0.319142	0.159571	+	0.987186i	$0.448989\pi$
$548$	14.0000	0.598050
$549$	1.07180	0.0457432
$550$	30.0000	1.27920
$551$	−4.00000	−0.170406
$552$	−0.928203	−0.0395070
$553$	0	0
$554$	51.4641	2.18650
$555$	0	0
$556$	−4.00000	−0.169638
$557$	3.07180	0.130156	0.0650781	−	0.997880i	$-0.479270\pi$
$557$	3.07180	0.130156	0.0650781	+	0.997880i	$0.479270\pi$
$558$	−5.07180	−0.214706
$559$	−37.8564	−1.60116
$560$	0	0
$561$	13.8564	0.585018
$562$	−12.0000	−0.506189
$563$	−17.8564	−0.752558	−0.376279	−	0.926506i	$-0.622797\pi$
$563$	−17.8564	−0.752558	−0.376279	+	0.926506i	$0.622797\pi$
$564$	−4.92820	−0.207515
$565$	0	0
$566$	13.8564	0.582428
$567$	0	0
$568$	6.00000	0.251754
$569$	24.7846	1.03902	0.519512	−	0.854463i	$-0.326114\pi$
$569$	24.7846	1.03902	0.519512	+	0.854463i	$0.326114\pi$
$570$	0	0
$571$	−17.8564	−0.747267	−0.373634	−	0.927576i	$-0.621888\pi$
$571$	−17.8564	−0.747267	−0.373634	+	0.927576i	$0.621888\pi$
$572$	−12.0000	−0.501745
$573$	7.46410	0.311817
$574$	0	0
$575$	2.67949	0.111743
$576$	1.00000	0.0416667
$577$	28.7846	1.19832	0.599159	−	0.800630i	$-0.295502\pi$
$577$	28.7846	1.19832	0.599159	+	0.800630i	$0.295502\pi$
$578$	−1.73205	−0.0720438
$579$	10.0000	0.415586
$580$	0	0
$581$	0	0
$582$	−33.7128	−1.39744
$583$	13.8564	0.573874
$584$	13.8564	0.573382
$585$	0	0
$586$	−56.7846	−2.34575
$587$	11.0718	0.456982	0.228491	−	0.973546i	$-0.426621\pi$
$587$	11.0718	0.456982	0.228491	+	0.973546i	$0.426621\pi$
$588$	0	0
$589$	−2.92820	−0.120655
$590$	0	0
$591$	−8.92820	−0.367257
$592$	−24.6410	−1.01274
$593$	−15.7128	−0.645248	−0.322624	−	0.946527i	$-0.604565\pi$
$593$	−15.7128	−0.645248	−0.322624	+	0.946527i	$0.604565\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	7.07180	0.289672
$597$	21.8564	0.894523
$598$	−3.21539	−0.131487
$599$	−34.3923	−1.40523	−0.702616	−	0.711569i	$-0.747985\pi$
$599$	−34.3923	−1.40523	−0.702616	+	0.711569i	$0.747985\pi$
$600$	−8.66025	−0.353553
$601$	−2.39230	−0.0975842	−0.0487921	−	0.998809i	$-0.515537\pi$
$601$	−2.39230	−0.0975842	−0.0487921	+	0.998809i	$0.515537\pi$
$602$	0	0
$603$	−6.39230	−0.260315
$604$	10.3923	0.422857
$605$	0	0
$606$	0	0
$607$	24.7846	1.00598	0.502988	−	0.864293i	$-0.332234\pi$
$607$	24.7846	1.00598	0.502988	+	0.864293i	$0.332234\pi$
$608$	−5.19615	−0.210732
$609$	0	0
$610$	0	0
$611$	17.0718	0.690651
$612$	4.00000	0.161690
$613$	−7.85641	−0.317317	−0.158659	−	0.987333i	$-0.550717\pi$
$613$	−7.85641	−0.317317	−0.158659	+	0.987333i	$0.550717\pi$
$614$	3.21539	0.129763
$615$	0	0
$616$	0	0
$617$	−31.8564	−1.28249	−0.641245	−	0.767336i	$-0.721582\pi$
$617$	−31.8564	−1.28249	−0.641245	+	0.767336i	$0.721582\pi$
$618$	5.07180	0.204018
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	0.535898	0.0215049
$622$	−43.1769	−1.73124
$623$	0	0
$624$	17.3205	0.693375
$625$	25.0000	1.00000
$626$	−15.7128	−0.628010
$627$	3.46410	0.138343
$628$	16.0000	0.638470
$629$	19.7128	0.786001
$630$	0	0
$631$	−12.7846	−0.508947	−0.254474	−	0.967080i	$-0.581902\pi$
$631$	−12.7846	−0.508947	−0.254474	+	0.967080i	$0.581902\pi$
$632$	−6.00000	−0.238667
$633$	7.46410	0.296671
$634$	48.4974	1.92608
$635$	0	0
$636$	4.00000	0.158610
$637$	0	0
$638$	24.0000	0.950169
$639$	−3.46410	−0.137038
$640$	0	0
$641$	−26.9282	−1.06360	−0.531800	−	0.846870i	$-0.678484\pi$
$641$	−26.9282	−1.06360	−0.531800	+	0.846870i	$0.678484\pi$
$642$	−4.14359	−0.163535
$643$	−37.5692	−1.48159	−0.740793	−	0.671734i	$-0.765550\pi$
$643$	−37.5692	−1.48159	−0.740793	+	0.671734i	$0.765550\pi$
$644$	0	0
$645$	0	0
$646$	6.92820	0.272587
$647$	42.7846	1.68204	0.841018	−	0.541007i	$-0.181957\pi$
$647$	42.7846	1.68204	0.841018	+	0.541007i	$0.181957\pi$
$648$	−1.73205	−0.0680414
$649$	27.7128	1.08782
$650$	−30.0000	−1.17670
$651$	0	0
$652$	10.9282	0.427981
$653$	32.6410	1.27734	0.638671	−	0.769480i	$-0.279485\pi$
$653$	32.6410	1.27734	0.638671	+	0.769480i	$0.279485\pi$
$654$	−8.53590	−0.333780
$655$	0	0
$656$	54.6410	2.13337
$657$	−8.00000	−0.312110
$658$	0	0
$659$	3.46410	0.134942	0.0674711	−	0.997721i	$-0.478507\pi$
$659$	3.46410	0.134942	0.0674711	+	0.997721i	$0.478507\pi$
$660$	0	0
$661$	−23.1769	−0.901477	−0.450739	−	0.892656i	$-0.648839\pi$
$661$	−23.1769	−0.901477	−0.450739	+	0.892656i	$0.648839\pi$
$662$	−62.7846	−2.44019
$663$	−13.8564	−0.538138
$664$	15.4641	0.600124
$665$	0	0
$666$	8.53590	0.330759
$667$	2.14359	0.0830003
$668$	13.8564	0.536120
$669$	−26.9282	−1.04110
$670$	0	0
$671$	−3.71281	−0.143332
$672$	0	0
$673$	12.1436	0.468101	0.234051	−	0.972224i	$-0.424802\pi$
$673$	12.1436	0.468101	0.234051	+	0.972224i	$0.424802\pi$
$674$	−10.3923	−0.400297
$675$	5.00000	0.192450
$676$	−1.00000	−0.0384615
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	32.7846	1.25909
$679$	0	0
$680$	0	0
$681$	−1.85641	−0.0711377
$682$	17.5692	0.672760
$683$	8.24871	0.315628	0.157814	−	0.987469i	$-0.449555\pi$
$683$	8.24871	0.315628	0.157814	+	0.987469i	$0.449555\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	0	0
$687$	6.92820	0.264327
$688$	54.6410	2.08317
$689$	−13.8564	−0.527887
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	17.8564	0.678799
$693$	0	0
$694$	−35.5692	−1.35019
$695$	0	0
$696$	−6.92820	−0.262613
$697$	−43.7128	−1.65574
$698$	39.7128	1.50315
$699$	19.8564	0.751038
$700$	0	0
$701$	−8.92820	−0.337214	−0.168607	−	0.985683i	$-0.553927\pi$
$701$	−8.92820	−0.337214	−0.168607	+	0.985683i	$0.553927\pi$
$702$	−6.00000	−0.226455
$703$	4.92820	0.185871
$704$	−3.46410	−0.130558
$705$	0	0
$706$	30.9282	1.16400
$707$	0	0
$708$	8.00000	0.300658
$709$	12.1436	0.456062	0.228031	−	0.973654i	$-0.426771\pi$
$709$	12.1436	0.456062	0.228031	+	0.973654i	$0.426771\pi$
$710$	0	0
$711$	3.46410	0.129914
$712$	20.7846	0.778936
$713$	1.56922	0.0587677
$714$	0	0
$715$	0	0
$716$	−4.53590	−0.169514
$717$	8.53590	0.318779
$718$	24.9282	0.930312
$719$	−40.6410	−1.51565	−0.757827	−	0.652455i	$-0.773739\pi$
$719$	−40.6410	−1.51565	−0.757827	+	0.652455i	$0.773739\pi$
$720$	0	0
$721$	0	0
$722$	1.73205	0.0644603
$723$	3.46410	0.128831
$724$	9.32051	0.346394
$725$	20.0000	0.742781
$726$	−1.73205	−0.0642824
$727$	47.7128	1.76957	0.884785	−	0.465999i	$-0.154305\pi$
$727$	47.7128	1.76957	0.884785	+	0.465999i	$0.154305\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−43.7128	−1.61678
$732$	−1.07180	−0.0396147
$733$	50.6410	1.87047	0.935234	−	0.354029i	$-0.115189\pi$
$733$	50.6410	1.87047	0.935234	+	0.354029i	$0.115189\pi$
$734$	6.92820	0.255725
$735$	0	0
$736$	2.78461	0.102642
$737$	22.1436	0.815670
$738$	−18.9282	−0.696757
$739$	−25.8564	−0.951143	−0.475572	−	0.879677i	$-0.657759\pi$
$739$	−25.8564	−0.951143	−0.475572	+	0.879677i	$0.657759\pi$
$740$	0	0
$741$	−3.46410	−0.127257
$742$	0	0
$743$	−24.2487	−0.889599	−0.444799	−	0.895630i	$-0.646725\pi$
$743$	−24.2487	−0.889599	−0.444799	+	0.895630i	$0.646725\pi$
$744$	−5.07180	−0.185941
$745$	0	0
$746$	32.5359	1.19122
$747$	−8.92820	−0.326666
$748$	−13.8564	−0.506640
$749$	0	0
$750$	0	0
$751$	11.4641	0.418331	0.209166	−	0.977880i	$-0.432925\pi$
$751$	11.4641	0.418331	0.209166	+	0.977880i	$0.432925\pi$
$752$	−24.6410	−0.898565
$753$	−0.928203	−0.0338256
$754$	−24.0000	−0.874028
$755$	0	0
$756$	0	0
$757$	−51.8564	−1.88475	−0.942377	−	0.334554i	$-0.891414\pi$
$757$	−51.8564	−1.88475	−0.942377	+	0.334554i	$0.891414\pi$
$758$	16.6410	0.604429
$759$	−1.85641	−0.0673833
$760$	0	0
$761$	−31.7128	−1.14959	−0.574794	−	0.818298i	$-0.694918\pi$
$761$	−31.7128	−1.14959	−0.574794	+	0.818298i	$0.694918\pi$
$762$	33.7128	1.22129
$763$	0	0
$764$	−7.46410	−0.270042
$765$	0	0
$766$	6.92820	0.250326
$767$	−27.7128	−1.00065
$768$	−19.0000	−0.685603
$769$	34.6410	1.24919	0.624593	−	0.780950i	$-0.285265\pi$
$769$	34.6410	1.24919	0.624593	+	0.780950i	$0.285265\pi$
$770$	0	0
$771$	−17.8564	−0.643083
$772$	−10.0000	−0.359908
$773$	7.71281	0.277411	0.138705	−	0.990334i	$-0.455706\pi$
$773$	7.71281	0.277411	0.138705	+	0.990334i	$0.455706\pi$
$774$	−18.9282	−0.680360
$775$	14.6410	0.525921
$776$	−33.7128	−1.21022
$777$	0	0
$778$	−33.0333	−1.18430
$779$	−10.9282	−0.391544
$780$	0	0
$781$	12.0000	0.429394
$782$	−3.71281	−0.132770
$783$	4.00000	0.142948
$784$	0	0
$785$	0	0
$786$	8.53590	0.304465
$787$	−6.14359	−0.218995	−0.109498	−	0.993987i	$-0.534924\pi$
$787$	−6.14359	−0.218995	−0.109498	+	0.993987i	$0.534924\pi$
$788$	8.92820	0.318054
$789$	0.535898	0.0190785
$790$	0	0
$791$	0	0
$792$	6.00000	0.213201
$793$	3.71281	0.131846
$794$	−49.8564	−1.76934
$795$	0	0
$796$	−21.8564	−0.774680
$797$	−24.7846	−0.877916	−0.438958	−	0.898508i	$-0.644652\pi$
$797$	−24.7846	−0.877916	−0.438958	+	0.898508i	$0.644652\pi$
$798$	0	0
$799$	19.7128	0.697389
$800$	25.9808	0.918559
$801$	−12.0000	−0.423999
$802$	−1.85641	−0.0655520
$803$	27.7128	0.977964
$804$	6.39230	0.225439
$805$	0	0
$806$	−17.5692	−0.618849
$807$	−2.92820	−0.103078
$808$	0	0
$809$	−37.7128	−1.32591	−0.662956	−	0.748658i	$-0.730698\pi$
$809$	−37.7128	−1.32591	−0.662956	+	0.748658i	$0.730698\pi$
$810$	0	0
$811$	−7.71281	−0.270833	−0.135417	−	0.990789i	$-0.543237\pi$
$811$	−7.71281	−0.270833	−0.135417	+	0.990789i	$0.543237\pi$
$812$	0	0
$813$	−1.85641	−0.0651070
$814$	−29.5692	−1.03640
$815$	0	0
$816$	20.0000	0.700140
$817$	−10.9282	−0.382329
$818$	−16.1436	−0.564448
$819$	0	0
$820$	0	0
$821$	−2.78461	−0.0971835	−0.0485918	−	0.998819i	$-0.515473\pi$
$821$	−2.78461	−0.0971835	−0.0485918	+	0.998819i	$0.515473\pi$
$822$	−24.2487	−0.845771
$823$	−18.1436	−0.632446	−0.316223	−	0.948685i	$-0.602415\pi$
$823$	−18.1436	−0.632446	−0.316223	+	0.948685i	$0.602415\pi$
$824$	5.07180	0.176684
$825$	−17.3205	−0.603023
$826$	0	0
$827$	2.39230	0.0831886	0.0415943	−	0.999135i	$-0.486756\pi$
$827$	2.39230	0.0831886	0.0415943	+	0.999135i	$0.486756\pi$
$828$	−0.535898	−0.0186238
$829$	32.2487	1.12004	0.560022	−	0.828478i	$-0.310793\pi$
$829$	32.2487	1.12004	0.560022	+	0.828478i	$0.310793\pi$
$830$	0	0
$831$	−29.7128	−1.03073
$832$	3.46410	0.120096
$833$	0	0
$834$	6.92820	0.239904
$835$	0	0
$836$	−3.46410	−0.119808
$837$	2.92820	0.101214
$838$	−63.4641	−2.19233
$839$	−33.8564	−1.16885	−0.584426	−	0.811447i	$-0.698680\pi$
$839$	−33.8564	−1.16885	−0.584426	+	0.811447i	$0.698680\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	8.53590	0.294166
$843$	6.92820	0.238620
$844$	−7.46410	−0.256925
$845$	0	0
$846$	8.53590	0.293470
$847$	0	0
$848$	20.0000	0.686803
$849$	−8.00000	−0.274559
$850$	−34.6410	−1.18818
$851$	−2.64102	−0.0905329
$852$	3.46410	0.118678
$853$	−35.7128	−1.22278	−0.611392	−	0.791328i	$-0.709390\pi$
$853$	−35.7128	−1.22278	−0.611392	+	0.791328i	$0.709390\pi$
$854$	0	0
$855$	0	0
$856$	−4.14359	−0.141625
$857$	−25.8564	−0.883238	−0.441619	−	0.897203i	$-0.645596\pi$
$857$	−25.8564	−0.883238	−0.441619	+	0.897203i	$0.645596\pi$
$858$	20.7846	0.709575
$859$	53.8564	1.83756	0.918778	−	0.394774i	$-0.129177\pi$
$859$	53.8564	1.83756	0.918778	+	0.394774i	$0.129177\pi$
$860$	0	0
$861$	0	0
$862$	−9.71281	−0.330820
$863$	−5.60770	−0.190888	−0.0954441	−	0.995435i	$-0.530427\pi$
$863$	−5.60770	−0.190888	−0.0954441	+	0.995435i	$0.530427\pi$
$864$	5.19615	0.176777
$865$	0	0
$866$	43.8564	1.49030
$867$	1.00000	0.0339618
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−22.1436	−0.750307
$872$	−8.53590	−0.289062
$873$	19.4641	0.658760
$874$	−0.928203	−0.0313969
$875$	0	0
$876$	8.00000	0.270295
$877$	−58.7846	−1.98502	−0.992508	−	0.122183i	$-0.961011\pi$
$877$	−58.7846	−1.98502	−0.992508	+	0.122183i	$0.961011\pi$
$878$	53.0718	1.79109
$879$	32.7846	1.10580
$880$	0	0
$881$	45.5692	1.53527	0.767633	−	0.640890i	$-0.221434\pi$
$881$	45.5692	1.53527	0.767633	+	0.640890i	$0.221434\pi$
$882$	0	0
$883$	24.7846	0.834069	0.417034	−	0.908891i	$-0.363070\pi$
$883$	24.7846	0.834069	0.417034	+	0.908891i	$0.363070\pi$
$884$	13.8564	0.466041
$885$	0	0
$886$	2.28719	0.0768396
$887$	−53.8564	−1.80832	−0.904161	−	0.427193i	$-0.859503\pi$
$887$	−53.8564	−1.80832	−0.904161	+	0.427193i	$0.859503\pi$
$888$	8.53590	0.286446
$889$	0	0
$890$	0	0
$891$	−3.46410	−0.116052
$892$	26.9282	0.901623
$893$	4.92820	0.164916
$894$	−12.2487	−0.409658
$895$	0	0
$896$	0	0
$897$	1.85641	0.0619836
$898$	53.5692	1.78763
$899$	11.7128	0.390644
$900$	−5.00000	−0.166667
$901$	−16.0000	−0.533037
$902$	65.5692	2.18322
$903$	0	0
$904$	32.7846	1.09040
$905$	0	0
$906$	−18.0000	−0.598010
$907$	−10.6795	−0.354607	−0.177303	−	0.984156i	$-0.556737\pi$
$907$	−10.6795	−0.354607	−0.177303	+	0.984156i	$0.556737\pi$
$908$	1.85641	0.0616070
$909$	0	0
$910$	0	0
$911$	55.1769	1.82809	0.914046	−	0.405610i	$-0.132941\pi$
$911$	55.1769	1.82809	0.914046	+	0.405610i	$0.132941\pi$
$912$	5.00000	0.165567
$913$	30.9282	1.02357
$914$	31.1769	1.03124
$915$	0	0
$916$	−6.92820	−0.228914
$917$	0	0
$918$	−6.92820	−0.228665
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−1.85641	−0.0611707
$922$	55.4256	1.82535
$923$	−12.0000	−0.394985
$924$	0	0
$925$	−24.6410	−0.810192
$926$	24.0000	0.788689
$927$	−2.92820	−0.0961748
$928$	20.7846	0.682288
$929$	−44.0000	−1.44359	−0.721797	−	0.692105i	$-0.756683\pi$
$929$	−44.0000	−1.44359	−0.721797	+	0.692105i	$0.756683\pi$
$930$	0	0
$931$	0	0
$932$	−19.8564	−0.650418
$933$	24.9282	0.816113
$934$	−8.53590	−0.279303
$935$	0	0
$936$	−6.00000	−0.196116
$937$	−12.7846	−0.417655	−0.208827	−	0.977952i	$-0.566965\pi$
$937$	−12.7846	−0.417655	−0.208827	+	0.977952i	$0.566965\pi$
$938$	0	0
$939$	9.07180	0.296047
$940$	0	0
$941$	−39.7128	−1.29460	−0.647300	−	0.762235i	$-0.724102\pi$
$941$	−39.7128	−1.29460	−0.647300	+	0.762235i	$0.724102\pi$
$942$	−27.7128	−0.902932
$943$	5.85641	0.190711
$944$	40.0000	1.30189
$945$	0	0
$946$	65.5692	2.13184
$947$	−41.3205	−1.34274	−0.671368	−	0.741124i	$-0.734293\pi$
$947$	−41.3205	−1.34274	−0.671368	+	0.741124i	$0.734293\pi$
$948$	−3.46410	−0.112509
$949$	−27.7128	−0.899596
$950$	−8.66025	−0.280976
$951$	−28.0000	−0.907962
$952$	0	0
$953$	−17.0718	−0.553010	−0.276505	−	0.961013i	$-0.589176\pi$
$953$	−17.0718	−0.553010	−0.276505	+	0.961013i	$0.589176\pi$
$954$	−6.92820	−0.224309
$955$	0	0
$956$	−8.53590	−0.276071
$957$	−13.8564	−0.447914
$958$	−39.9615	−1.29110
$959$	0	0
$960$	0	0
$961$	−22.4256	−0.723407
$962$	29.5692	0.953350
$963$	2.39230	0.0770909
$964$	−3.46410	−0.111571
$965$	0	0
$966$	0	0
$967$	−12.7846	−0.411125	−0.205563	−	0.978644i	$-0.565902\pi$
$967$	−12.7846	−0.411125	−0.205563	+	0.978644i	$0.565902\pi$
$968$	−1.73205	−0.0556702
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−22.1436	−0.710622	−0.355311	−	0.934748i	$-0.615625\pi$
$971$	−22.1436	−0.710622	−0.355311	+	0.934748i	$0.615625\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−31.8564	−1.02075
$975$	17.3205	0.554700
$976$	−5.35898	−0.171537
$977$	−58.6410	−1.87609	−0.938046	−	0.346510i	$-0.887367\pi$
$977$	−58.6410	−1.87609	−0.938046	+	0.346510i	$0.887367\pi$
$978$	−18.9282	−0.605257
$979$	41.5692	1.32856
$980$	0	0
$981$	4.92820	0.157345
$982$	57.7128	1.84169
$983$	−3.71281	−0.118420	−0.0592102	−	0.998246i	$-0.518858\pi$
$983$	−3.71281	−0.118420	−0.0592102	+	0.998246i	$0.518858\pi$
$984$	−18.9282	−0.603409
$985$	0	0
$986$	−27.7128	−0.882556
$987$	0	0
$988$	3.46410	0.110208
$989$	5.85641	0.186223
$990$	0	0
$991$	−53.0333	−1.68466	−0.842329	−	0.538963i	$-0.818816\pi$
$991$	−53.0333	−1.68466	−0.842329	+	0.538963i	$0.818816\pi$
$992$	15.2154	0.483089
$993$	36.2487	1.15032
$994$	0	0
$995$	0	0
$996$	8.92820	0.282901
$997$	2.14359	0.0678883	0.0339441	−	0.999424i	$-0.489193\pi$
$997$	2.14359	0.0678883	0.0339441	+	0.999424i	$0.489193\pi$
$998$	18.9282	0.599162
$999$	−4.92820	−0.155921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2793.2.a.q.1.2	✓	2
3.2	odd	2		8379.2.a.bc.1.1		2
7.6	odd	2		2793.2.a.r.1.2	yes	2
21.20	even	2		8379.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2793.2.a.q.1.2	✓	2	1.1	even	1	trivial
2793.2.a.r.1.2	yes	2	7.6	odd	2
8379.2.a.bc.1.1		2	3.2	odd	2
8379.2.a.bd.1.1		2	21.20	even	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:	\( N \)	\(=\)	\( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2793.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} -1.73205 q^{8} +1.00000 q^{9} -3.46410 q^{11} -1.00000 q^{12} +3.46410 q^{13} -5.00000 q^{16} +4.00000 q^{17} +1.73205 q^{18} +1.00000 q^{19} -6.00000 q^{22} -0.535898 q^{23} +1.73205 q^{24} -5.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -4.00000 q^{29} -2.92820 q^{31} -5.19615 q^{32} +3.46410 q^{33} +6.92820 q^{34} +1.00000 q^{36} +4.92820 q^{37} +1.73205 q^{38} -3.46410 q^{39} -10.9282 q^{41} -10.9282 q^{43} -3.46410 q^{44} -0.928203 q^{46} +4.92820 q^{47} +5.00000 q^{48} -8.66025 q^{50} -4.00000 q^{51} +3.46410 q^{52} -4.00000 q^{53} -1.73205 q^{54} -1.00000 q^{57} -6.92820 q^{58} -8.00000 q^{59} +1.07180 q^{61} -5.07180 q^{62} +1.00000 q^{64} +6.00000 q^{66} -6.39230 q^{67} +4.00000 q^{68} +0.535898 q^{69} -3.46410 q^{71} -1.73205 q^{72} -8.00000 q^{73} +8.53590 q^{74} +5.00000 q^{75} +1.00000 q^{76} -6.00000 q^{78} +3.46410 q^{79} +1.00000 q^{81} -18.9282 q^{82} -8.92820 q^{83} -18.9282 q^{86} +4.00000 q^{87} +6.00000 q^{88} -12.0000 q^{89} -0.535898 q^{92} +2.92820 q^{93} +8.53590 q^{94} +5.19615 q^{96} +19.4641 q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{9} - 2 q^{12} - 10 q^{16} + 8 q^{17} + 2 q^{19} - 12 q^{22} - 8 q^{23} - 10 q^{25} + 12 q^{26} - 2 q^{27} - 8 q^{29} + 8 q^{31} + 2 q^{36} - 4 q^{37} - 8 q^{41} - 8 q^{43}+ \cdots + 32 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^9 - 2 * q^12 - 10 * q^16 + 8 * q^17 + 2 * q^19 - 12 * q^22 - 8 * q^23 - 10 * q^25 + 12 * q^26 - 2 * q^27 - 8 * q^29 + 8 * q^31 + 2 * q^36 - 4 * q^37 - 8 * q^41 - 8 * q^43 + 12 * q^46 - 4 * q^47 + 10 * q^48 - 8 * q^51 - 8 * q^53 - 2 * q^57 - 16 * q^59 + 16 * q^61 - 24 * q^62 + 2 * q^64 + 12 * q^66 + 8 * q^67 + 8 * q^68 + 8 * q^69 - 16 * q^73 + 24 * q^74 + 10 * q^75 + 2 * q^76 - 12 * q^78 + 2 * q^81 - 24 * q^82 - 4 * q^83 - 24 * q^86 + 8 * q^87 + 12 * q^88 - 24 * q^89 - 8 * q^92 - 8 * q^93 + 24 * q^94 + 32 * q^97

Introduction

L-functions

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