LMFDB - Embedded newform 112.16.a.k.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,16,Mod(1,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 16, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.1"); S:= CuspForms(chi, 16); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-3864] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$159.816725712$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 911559x^{4} - 33878904x^{3} + 200237882347x^{2} + 5867703981762x - 7046699346580509 $ x^6 - 2x^5 - 911559x^4 - 33878904x^3 + 200237882347x^2 + 5867703981762*x - 7046699346580509
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{35}\cdot 3^{6}\cdot 5\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$191.407$ of defining polynomial
Character	$\chi$	$=$	112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-658.011 q^{3} -99937.3 q^{5} +823543. q^{7} -1.39159e7 q^{9} +9.86856e7 q^{11} +4.31785e7 q^{13} +6.57598e7 q^{15} -1.63426e9 q^{17} -4.86662e9 q^{19} -5.41900e8 q^{21} +1.52363e10 q^{23} -2.05301e10 q^{25} +1.85986e10 q^{27} +1.67918e11 q^{29} +8.99932e9 q^{31} -6.49362e10 q^{33} -8.23027e10 q^{35} +3.19420e10 q^{37} -2.84119e10 q^{39} -3.04379e11 q^{41} -2.06517e12 q^{43} +1.39072e12 q^{45} +5.53348e12 q^{47} +6.78223e11 q^{49} +1.07536e12 q^{51} -1.68236e12 q^{53} -9.86237e12 q^{55} +3.20229e12 q^{57} +3.20814e13 q^{59} +3.22573e13 q^{61} -1.14604e13 q^{63} -4.31515e12 q^{65} -3.98067e13 q^{67} -1.00256e13 q^{69} -9.64832e13 q^{71} -1.31961e14 q^{73} +1.35090e13 q^{75} +8.12718e13 q^{77} -2.27805e14 q^{79} +1.87440e14 q^{81} +2.11079e14 q^{83} +1.63324e14 q^{85} -1.10492e14 q^{87} +3.32250e14 q^{89} +3.55594e13 q^{91} -5.92165e12 q^{93} +4.86357e14 q^{95} +6.67928e14 q^{97} -1.37330e15 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3864 q^{3} + 168532 q^{5} + 4941258 q^{7} + 27735822 q^{9} - 58040072 q^{11} - 15956444 q^{13} + 705059136 q^{15} - 2879979060 q^{17} - 3435500376 q^{19} - 3182170152 q^{21} - 34508782592 q^{23} + 29232294906 q^{25}+ \cdots - 44\!\cdots\!92 q^{99}+O(q^{100}) $ 6 * q - 3864 * q^3 + 168532 * q^5 + 4941258 * q^7 + 27735822 * q^9 - 58040072 * q^11 - 15956444 * q^13 + 705059136 * q^15 - 2879979060 * q^17 - 3435500376 * q^19 - 3182170152 * q^21 - 34508782592 * q^23 + 29232294906 * q^25 - 102516868944 * q^27 - 46458543420 * q^29 - 300765178336 * q^31 - 457356427584 * q^33 + 138793348876 * q^35 - 509722422412 * q^37 + 437968495008 * q^39 + 1619276439676 * q^41 - 2727781008680 * q^43 + 5316894690660 * q^45 + 1567726443808 * q^47 + 4069338437094 * q^49 - 3110310296208 * q^51 + 2612518893588 * q^53 + 4577322926960 * q^55 - 21550151959824 * q^57 - 5112561397384 * q^59 + 4545346033732 * q^61 + 22841642057346 * q^63 + 57444596600712 * q^65 - 74088461612408 * q^67 + 176631933295104 * q^69 - 51092191255952 * q^71 + 294343628983292 * q^73 - 208088876183496 * q^75 - 47798495015096 * q^77 - 56070305620928 * q^79 + 691991522436870 * q^81 - 984559173278520 * q^83 + 396331720481544 * q^85 - 1024554713548464 * q^87 + 1528383063352572 * q^89 - 13140817761092 * q^91 + 1245617896072608 * q^93 - 2532178187158304 * q^95 + 2255703641701580 * q^97 - 4484026592356392 * q^99

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
$3$	−658.011	−0.173709	−0.0868547	−	0.996221i	$-0.527682\pi$
$3$	−658.011	−0.173709	−0.0868547	+	0.996221i	$0.527682\pi$
$4$	0	0
$5$	−99937.3	−0.572075	−0.286037	−	0.958218i	$-0.592338\pi$
$5$	−99937.3	−0.572075	−0.286037	+	0.958218i	$0.592338\pi$
$6$	0	0
$7$	823543.	0.377964
$7$	823543.	0.377964
$8$	0	0
$9$	−1.39159e7	−0.969825
$10$	0	0
$11$	9.86856e7	1.52689	0.763446	−	0.645871i	$-0.223506\pi$
$11$	9.86856e7	1.52689	0.763446	+	0.645871i	$0.223506\pi$
$12$	0	0
$13$	4.31785e7	0.190850	0.0954251	−	0.995437i	$-0.469579\pi$
$13$	4.31785e7	0.190850	0.0954251	+	0.995437i	$0.469579\pi$
$14$	0	0
$15$	6.57598e7	0.0993748
$16$	0	0
$17$	−1.63426e9	−0.965949	−0.482975	−	0.875634i	$-0.660444\pi$
$17$	−1.63426e9	−0.965949	−0.482975	+	0.875634i	$0.660444\pi$
$18$	0	0
$19$	−4.86662e9	−1.24904	−0.624518	−	0.781010i	$-0.714705\pi$
$19$	−4.86662e9	−1.24904	−0.624518	+	0.781010i	$0.714705\pi$
$20$	0	0
$21$	−5.41900e8	−0.0656560
$22$	0	0
$23$	1.52363e10	0.933082	0.466541	−	0.884500i	$-0.345500\pi$
$23$	1.52363e10	0.933082	0.466541	+	0.884500i	$0.345500\pi$
$24$	0	0
$25$	−2.05301e10	−0.672731
$26$	0	0
$27$	1.85986e10	0.342177
$28$	0	0
$29$	1.67918e11	1.80764	0.903822	−	0.427908i	$-0.140749\pi$
$29$	1.67918e11	1.80764	0.903822	+	0.427908i	$0.140749\pi$
$30$	0	0
$31$	8.99932e9	0.0587485	0.0293743	−	0.999568i	$-0.490649\pi$
$31$	8.99932e9	0.0587485	0.0293743	+	0.999568i	$0.490649\pi$
$32$	0	0
$33$	−6.49362e10	−0.265236
$34$	0	0
$35$	−8.23027e10	−0.216224
$36$	0	0
$37$	3.19420e10	0.0553158	0.0276579	−	0.999617i	$-0.491195\pi$
$37$	3.19420e10	0.0553158	0.0276579	+	0.999617i	$0.491195\pi$
$38$	0	0
$39$	−2.84119e10	−0.0331525
$40$	0	0
$41$	−3.04379e11	−0.244082	−0.122041	−	0.992525i	$-0.538944\pi$
$41$	−3.04379e11	−0.244082	−0.122041	+	0.992525i	$0.538944\pi$
$42$	0	0
$43$	−2.06517e12	−1.15862	−0.579311	−	0.815107i	$-0.696678\pi$
$43$	−2.06517e12	−1.15862	−0.579311	+	0.815107i	$0.696678\pi$
$44$	0	0
$45$	1.39072e12	0.554812
$46$	0	0
$47$	5.53348e12	1.59318	0.796590	−	0.604520i	$-0.206635\pi$
$47$	5.53348e12	1.59318	0.796590	+	0.604520i	$0.206635\pi$
$48$	0	0
$49$	6.78223e11	0.142857
$50$	0	0
$51$	1.07536e12	0.167795
$52$	0	0
$53$	−1.68236e12	−0.196721	−0.0983603	−	0.995151i	$-0.531360\pi$
$53$	−1.68236e12	−0.196721	−0.0983603	+	0.995151i	$0.531360\pi$
$54$	0	0
$55$	−9.86237e12	−0.873496
$56$	0	0
$57$	3.20229e12	0.216970
$58$	0	0
$59$	3.20814e13	1.67828	0.839138	−	0.543918i	$-0.183060\pi$
$59$	3.20814e13	1.67828	0.839138	+	0.543918i	$0.183060\pi$
$60$	0	0
$61$	3.22573e13	1.31418	0.657089	−	0.753813i	$-0.271788\pi$
$61$	3.22573e13	1.31418	0.657089	+	0.753813i	$0.271788\pi$
$62$	0	0
$63$	−1.14604e13	−0.366559
$64$	0	0
$65$	−4.31515e12	−0.109181
$66$	0	0
$67$	−3.98067e13	−0.802409	−0.401204	−	0.915989i	$-0.631408\pi$
$67$	−3.98067e13	−0.802409	−0.401204	+	0.915989i	$0.631408\pi$
$68$	0	0
$69$	−1.00256e13	−0.162085
$70$	0	0
$71$	−9.64832e13	−1.25897	−0.629483	−	0.777014i	$-0.716733\pi$
$71$	−9.64832e13	−1.25897	−0.629483	+	0.777014i	$0.716733\pi$
$72$	0	0
$73$	−1.31961e14	−1.39805	−0.699026	−	0.715096i	$-0.746383\pi$
$73$	−1.31961e14	−1.39805	−0.699026	+	0.715096i	$0.746383\pi$
$74$	0	0
$75$	1.35090e13	0.116860
$76$	0	0
$77$	8.12718e13	0.577111
$78$	0	0
$79$	−2.27805e14	−1.33463	−0.667314	−	0.744777i	$-0.732556\pi$
$79$	−2.27805e14	−1.33463	−0.667314	+	0.744777i	$0.732556\pi$
$80$	0	0
$81$	1.87440e14	0.910386
$82$	0	0
$83$	2.11079e14	0.853806	0.426903	−	0.904297i	$-0.359605\pi$
$83$	2.11079e14	0.853806	0.426903	+	0.904297i	$0.359605\pi$
$84$	0	0
$85$	1.63324e14	0.552595
$86$	0	0
$87$	−1.10492e14	−0.314005
$88$	0	0
$89$	3.32250e14	0.796231	0.398116	−	0.917335i	$-0.369664\pi$
$89$	3.32250e14	0.796231	0.398116	+	0.917335i	$0.369664\pi$
$90$	0	0
$91$	3.55594e13	0.0721346
$92$	0	0
$93$	−5.92165e12	−0.0102052
$94$	0	0
$95$	4.86357e14	0.714542
$96$	0	0
$97$	6.67928e14	0.839347	0.419673	−	0.907675i	$-0.362145\pi$
$97$	6.67928e14	0.839347	0.419673	+	0.907675i	$0.362145\pi$
$98$	0	0
$99$	−1.37330e15	−1.48082
$100$	0	0
$101$	1.67749e15	1.55686	0.778431	−	0.627730i	$-0.216016\pi$
$101$	1.67749e15	1.55686	0.778431	+	0.627730i	$0.216016\pi$
$102$	0	0
$103$	−1.43932e15	−1.15313	−0.576565	−	0.817052i	$-0.695607\pi$
$103$	−1.43932e15	−1.15313	−0.576565	+	0.817052i	$0.695607\pi$
$104$	0	0
$105$	5.41560e13	0.0375601
$106$	0	0
$107$	−1.45184e15	−0.874060	−0.437030	−	0.899447i	$-0.643970\pi$
$107$	−1.45184e15	−0.874060	−0.437030	+	0.899447i	$0.643970\pi$
$108$	0	0
$109$	4.71347e14	0.246969	0.123484	−	0.992347i	$-0.460593\pi$
$109$	4.71347e14	0.246969	0.123484	+	0.992347i	$0.460593\pi$
$110$	0	0
$111$	−2.10182e13	−0.00960888
$112$	0	0
$113$	−4.46072e15	−1.78368	−0.891840	−	0.452351i	$-0.850585\pi$
$113$	−4.46072e15	−1.78368	−0.891840	+	0.452351i	$0.850585\pi$
$114$	0	0
$115$	−1.52267e15	−0.533792
$116$	0	0
$117$	−6.00869e14	−0.185091
$118$	0	0
$119$	−1.34588e15	−0.365095
$120$	0	0
$121$	5.56159e15	1.33140
$122$	0	0
$123$	2.00285e14	0.0423994
$124$	0	0
$125$	5.10157e15	0.956927
$126$	0	0
$127$	1.42075e15	0.236586	0.118293	−	0.992979i	$-0.462258\pi$
$127$	1.42075e15	0.236586	0.118293	+	0.992979i	$0.462258\pi$
$128$	0	0
$129$	1.35890e15	0.201263
$130$	0	0
$131$	−1.35956e16	−1.79417	−0.897085	−	0.441858i	$-0.854320\pi$
$131$	−1.35956e16	−1.79417	−0.897085	+	0.441858i	$0.854320\pi$
$132$	0	0
$133$	−4.00787e15	−0.472091
$134$	0	0
$135$	−1.85869e15	−0.195751
$136$	0	0
$137$	9.72389e15	0.917140	0.458570	−	0.888658i	$-0.348362\pi$
$137$	9.72389e15	0.917140	0.458570	+	0.888658i	$0.348362\pi$
$138$	0	0
$139$	−5.39955e15	−0.456821	−0.228411	−	0.973565i	$-0.573353\pi$
$139$	−5.39955e15	−0.456821	−0.228411	+	0.973565i	$0.573353\pi$
$140$	0	0
$141$	−3.64109e15	−0.276750
$142$	0	0
$143$	4.26110e15	0.291408
$144$	0	0
$145$	−1.67813e16	−1.03411
$146$	0	0
$147$	−4.46278e14	−0.0248156
$148$	0	0
$149$	−3.10552e16	−1.56041	−0.780203	−	0.625527i	$-0.784884\pi$
$149$	−3.10552e16	−1.56041	−0.780203	+	0.625527i	$0.784884\pi$
$150$	0	0
$151$	1.27333e15	0.0578912	0.0289456	−	0.999581i	$-0.490785\pi$
$151$	1.27333e15	0.0578912	0.0289456	+	0.999581i	$0.490785\pi$
$152$	0	0
$153$	2.27423e16	0.936802
$154$	0	0
$155$	−8.99368e14	−0.0336085
$156$	0	0
$157$	−3.02208e16	−1.02579	−0.512895	−	0.858451i	$-0.671427\pi$
$157$	−3.02208e16	−1.02579	−0.512895	+	0.858451i	$0.671427\pi$
$158$	0	0
$159$	1.10701e15	0.0341722
$160$	0	0
$161$	1.25477e16	0.352672
$162$	0	0
$163$	−3.57287e15	−0.0915399	−0.0457699	−	0.998952i	$-0.514574\pi$
$163$	−3.57287e15	−0.0915399	−0.0457699	+	0.998952i	$0.514574\pi$
$164$	0	0
$165$	6.48955e15	0.151735
$166$	0	0
$167$	6.81421e15	0.145560	0.0727799	−	0.997348i	$-0.476813\pi$
$167$	6.81421e15	0.145560	0.0727799	+	0.997348i	$0.476813\pi$
$168$	0	0
$169$	−4.93215e16	−0.963576
$170$	0	0
$171$	6.77235e16	1.21135
$172$	0	0
$173$	−5.80219e16	−0.951143	−0.475572	−	0.879677i	$-0.657759\pi$
$173$	−5.80219e16	−0.951143	−0.475572	+	0.879677i	$0.657759\pi$
$174$	0	0
$175$	−1.69074e16	−0.254268
$176$	0	0
$177$	−2.11099e16	−0.291533
$178$	0	0
$179$	8.67397e16	1.10108	0.550542	−	0.834808i	$-0.314421\pi$
$179$	8.67397e16	1.10108	0.550542	+	0.834808i	$0.314421\pi$
$180$	0	0
$181$	−9.88334e16	−1.15429	−0.577144	−	0.816642i	$-0.695833\pi$
$181$	−9.88334e16	−1.15429	−0.577144	+	0.816642i	$0.695833\pi$
$182$	0	0
$183$	−2.12256e16	−0.228285
$184$	0	0
$185$	−3.19220e15	−0.0316448
$186$	0	0
$187$	−1.61278e17	−1.47490
$188$	0	0
$189$	1.53167e16	0.129331
$190$	0	0
$191$	−1.03183e17	−0.805118	−0.402559	−	0.915394i	$-0.631879\pi$
$191$	−1.03183e17	−0.805118	−0.402559	+	0.915394i	$0.631879\pi$
$192$	0	0
$193$	−1.24304e17	−0.897028	−0.448514	−	0.893776i	$-0.648046\pi$
$193$	−1.24304e17	−0.897028	−0.448514	+	0.893776i	$0.648046\pi$
$194$	0	0
$195$	2.83941e15	0.0189657
$196$	0	0
$197$	−2.72332e17	−1.68501	−0.842503	−	0.538692i	$-0.818919\pi$
$197$	−2.72332e17	−1.68501	−0.842503	+	0.538692i	$0.818919\pi$
$198$	0	0
$199$	−2.83005e16	−0.162329	−0.0811645	−	0.996701i	$-0.525864\pi$
$199$	−2.83005e16	−0.162329	−0.0811645	+	0.996701i	$0.525864\pi$
$200$	0	0
$201$	2.61933e16	0.139386
$202$	0	0
$203$	1.38288e17	0.683225
$204$	0	0
$205$	3.04188e16	0.139633
$206$	0	0
$207$	−2.12027e17	−0.904926
$208$	0	0
$209$	−4.80265e17	−1.90715
$210$	0	0
$211$	−2.28855e17	−0.846140	−0.423070	−	0.906097i	$-0.639048\pi$
$211$	−2.28855e17	−0.846140	−0.423070	+	0.906097i	$0.639048\pi$
$212$	0	0
$213$	6.34870e16	0.218694
$214$	0	0
$215$	2.06387e17	0.662818
$216$	0	0
$217$	7.41133e15	0.0222049
$218$	0	0
$219$	8.68316e16	0.242855
$220$	0	0
$221$	−7.05649e16	−0.184352
$222$	0	0
$223$	−3.95117e17	−0.964804	−0.482402	−	0.875950i	$-0.660236\pi$
$223$	−3.95117e17	−0.964804	−0.482402	+	0.875950i	$0.660236\pi$
$224$	0	0
$225$	2.85696e17	0.652431
$226$	0	0
$227$	−6.90242e17	−1.47505	−0.737527	−	0.675318i	$-0.764006\pi$
$227$	−6.90242e17	−1.47505	−0.737527	+	0.675318i	$0.764006\pi$
$228$	0	0
$229$	−1.38159e17	−0.276448	−0.138224	−	0.990401i	$-0.544139\pi$
$229$	−1.38159e17	−0.276448	−0.138224	+	0.990401i	$0.544139\pi$
$230$	0	0
$231$	−5.34777e16	−0.100250
$232$	0	0
$233$	2.65015e17	0.465694	0.232847	−	0.972513i	$-0.425196\pi$
$233$	2.65015e17	0.465694	0.232847	+	0.972513i	$0.425196\pi$
$234$	0	0
$235$	−5.53001e17	−0.911417
$236$	0	0
$237$	1.49898e17	0.231838
$238$	0	0
$239$	8.93121e16	0.129696	0.0648479	−	0.997895i	$-0.479344\pi$
$239$	8.93121e16	0.129696	0.0648479	+	0.997895i	$0.479344\pi$
$240$	0	0
$241$	−8.52081e17	−1.16239	−0.581197	−	0.813763i	$-0.697415\pi$
$241$	−8.52081e17	−1.16239	−0.581197	+	0.813763i	$0.697415\pi$
$242$	0	0
$243$	−3.90207e17	−0.500320
$244$	0	0
$245$	−6.77798e16	−0.0817249
$246$	0	0
$247$	−2.10133e17	−0.238379
$248$	0	0
$249$	−1.38892e17	−0.148314
$250$	0	0
$251$	−1.15606e18	−1.16260	−0.581298	−	0.813691i	$-0.697455\pi$
$251$	−1.15606e18	−1.16260	−0.581298	+	0.813691i	$0.697455\pi$
$252$	0	0
$253$	1.50360e18	1.42472
$254$	0	0
$255$	−1.07469e17	−0.0959910
$256$	0	0
$257$	8.99709e17	0.757885	0.378943	−	0.925420i	$-0.376288\pi$
$257$	8.99709e17	0.757885	0.378943	+	0.925420i	$0.376288\pi$
$258$	0	0
$259$	2.63056e16	0.0209074
$260$	0	0
$261$	−2.33674e18	−1.75310
$262$	0	0
$263$	−1.45766e18	−1.03273	−0.516365	−	0.856369i	$-0.672715\pi$
$263$	−1.45766e18	−1.03273	−0.516365	+	0.856369i	$0.672715\pi$
$264$	0	0
$265$	1.68130e17	0.112539
$266$	0	0
$267$	−2.18624e17	−0.138313
$268$	0	0
$269$	1.08149e18	0.646965	0.323482	−	0.946234i	$-0.395146\pi$
$269$	1.08149e18	0.646965	0.323482	+	0.946234i	$0.395146\pi$
$270$	0	0
$271$	1.53664e18	0.869567	0.434784	−	0.900535i	$-0.356825\pi$
$271$	1.53664e18	0.869567	0.434784	+	0.900535i	$0.356825\pi$
$272$	0	0
$273$	−2.33984e16	−0.0125305
$274$	0	0
$275$	−2.02603e18	−1.02719
$276$	0	0
$277$	2.47176e18	1.18689	0.593443	−	0.804876i	$-0.297768\pi$
$277$	2.47176e18	1.18689	0.593443	+	0.804876i	$0.297768\pi$
$278$	0	0
$279$	−1.25234e17	−0.0569758
$280$	0	0
$281$	−2.43763e18	−1.05116	−0.525582	−	0.850743i	$-0.676152\pi$
$281$	−2.43763e18	−1.05116	−0.525582	+	0.850743i	$0.676152\pi$
$282$	0	0
$283$	2.21017e18	0.903707	0.451854	−	0.892092i	$-0.350763\pi$
$283$	2.21017e18	0.903707	0.451854	+	0.892092i	$0.350763\pi$
$284$	0	0
$285$	−3.20028e17	−0.124123
$286$	0	0
$287$	−2.50669e17	−0.0922544
$288$	0	0
$289$	−1.91616e17	−0.0669420
$290$	0	0
$291$	−4.39504e17	−0.145803
$292$	0	0
$293$	−2.18203e18	−0.687627	−0.343814	−	0.939038i	$-0.611719\pi$
$293$	−2.18203e18	−0.687627	−0.343814	+	0.939038i	$0.611719\pi$
$294$	0	0
$295$	−3.20613e18	−0.960099
$296$	0	0
$297$	1.83541e18	0.522468
$298$	0	0
$299$	6.57879e17	0.178079
$300$	0	0
$301$	−1.70075e18	−0.437918
$302$	0	0
$303$	−1.10381e18	−0.270442
$304$	0	0
$305$	−3.22371e18	−0.751807
$306$	0	0
$307$	7.40061e18	1.64335	0.821674	−	0.569958i	$-0.193040\pi$
$307$	7.40061e18	1.64335	0.821674	+	0.569958i	$0.193040\pi$
$308$	0	0
$309$	9.47087e17	0.200309
$310$	0	0
$311$	4.91891e17	0.0991211	0.0495605	−	0.998771i	$-0.484218\pi$
$311$	4.91891e17	0.0991211	0.0495605	+	0.998771i	$0.484218\pi$
$312$	0	0
$313$	4.90623e18	0.942249	0.471125	−	0.882067i	$-0.343848\pi$
$313$	4.90623e18	0.942249	0.471125	+	0.882067i	$0.343848\pi$
$314$	0	0
$315$	1.14532e18	0.209699
$316$	0	0
$317$	−3.08075e18	−0.537913	−0.268957	−	0.963152i	$-0.586679\pi$
$317$	−3.08075e18	−0.537913	−0.268957	+	0.963152i	$0.586679\pi$
$318$	0	0
$319$	1.65711e19	2.76008
$320$	0	0
$321$	9.55328e17	0.151832
$322$	0	0
$323$	7.95332e18	1.20651
$324$	0	0
$325$	−8.86460e17	−0.128391
$326$	0	0
$327$	−3.10152e17	−0.0429008
$328$	0	0
$329$	4.55706e18	0.602165
$330$	0	0
$331$	−7.86052e18	−0.992525	−0.496262	−	0.868173i	$-0.665295\pi$
$331$	−7.86052e18	−0.992525	−0.496262	+	0.868173i	$0.665295\pi$
$332$	0	0
$333$	−4.44503e17	−0.0536467
$334$	0	0
$335$	3.97818e18	0.459038
$336$	0	0
$337$	−1.25867e19	−1.38895	−0.694475	−	0.719517i	$-0.744363\pi$
$337$	−1.25867e19	−1.38895	−0.694475	+	0.719517i	$0.744363\pi$
$338$	0	0
$339$	2.93520e18	0.309842
$340$	0	0
$341$	8.88103e17	0.0897027
$342$	0	0
$343$	5.58546e17	0.0539949
$344$	0	0
$345$	1.00193e18	0.0927248
$346$	0	0
$347$	−8.39065e18	−0.743575	−0.371787	−	0.928318i	$-0.621255\pi$
$347$	−8.39065e18	−0.743575	−0.371787	+	0.928318i	$0.621255\pi$
$348$	0	0
$349$	1.43290e19	1.21626	0.608130	−	0.793837i	$-0.291920\pi$
$349$	1.43290e19	1.21626	0.608130	+	0.793837i	$0.291920\pi$
$350$	0	0
$351$	8.03059e17	0.0653046
$352$	0	0
$353$	−2.32305e18	−0.181029	−0.0905147	−	0.995895i	$-0.528851\pi$
$353$	−2.32305e18	−0.181029	−0.0905147	+	0.995895i	$0.528851\pi$
$354$	0	0
$355$	9.64227e18	0.720222
$356$	0	0
$357$	8.85606e17	0.0634204
$358$	0	0
$359$	1.83499e19	1.26016	0.630079	−	0.776531i	$-0.283023\pi$
$359$	1.83499e19	1.26016	0.630079	+	0.776531i	$0.283023\pi$
$360$	0	0
$361$	8.50284e18	0.560093
$362$	0	0
$363$	−3.65959e18	−0.231277
$364$	0	0
$365$	1.31878e19	0.799790
$366$	0	0
$367$	1.43881e19	0.837545	0.418773	−	0.908091i	$-0.362460\pi$
$367$	1.43881e19	0.837545	0.418773	+	0.908091i	$0.362460\pi$
$368$	0	0
$369$	4.23572e18	0.236717
$370$	0	0
$371$	−1.38550e18	−0.0743534
$372$	0	0
$373$	2.77294e18	0.142930	0.0714651	−	0.997443i	$-0.477233\pi$
$373$	2.77294e18	0.142930	0.0714651	+	0.997443i	$0.477233\pi$
$374$	0	0
$375$	−3.35689e18	−0.166227
$376$	0	0
$377$	7.25046e18	0.344989
$378$	0	0
$379$	−1.05312e19	−0.481595	−0.240797	−	0.970575i	$-0.577409\pi$
$379$	−1.05312e19	−0.481595	−0.240797	+	0.970575i	$0.577409\pi$
$380$	0	0
$381$	−9.34869e17	−0.0410973
$382$	0	0
$383$	−2.08518e19	−0.881358	−0.440679	−	0.897665i	$-0.645262\pi$
$383$	−2.08518e19	−0.881358	−0.440679	+	0.897665i	$0.645262\pi$
$384$	0	0
$385$	−8.12209e18	−0.330151
$386$	0	0
$387$	2.87387e19	1.12366
$388$	0	0
$389$	1.02461e19	0.385422	0.192711	−	0.981256i	$-0.438272\pi$
$389$	1.02461e19	0.385422	0.192711	+	0.981256i	$0.438272\pi$
$390$	0	0
$391$	−2.49000e19	−0.901309
$392$	0	0
$393$	8.94605e18	0.311664
$394$	0	0
$395$	2.27662e19	0.763507
$396$	0	0
$397$	−4.28554e18	−0.138381	−0.0691905	−	0.997603i	$-0.522042\pi$
$397$	−4.28554e18	−0.138381	−0.0691905	+	0.997603i	$0.522042\pi$
$398$	0	0
$399$	2.63722e18	0.0820068
$400$	0	0
$401$	3.36371e19	1.00748	0.503739	−	0.863856i	$-0.331957\pi$
$401$	3.36371e19	1.00748	0.503739	+	0.863856i	$0.331957\pi$
$402$	0	0
$403$	3.88577e17	0.0112122
$404$	0	0
$405$	−1.87323e19	−0.520808
$406$	0	0
$407$	3.15222e18	0.0844613
$408$	0	0
$409$	6.90850e19	1.78426	0.892132	−	0.451774i	$-0.149209\pi$
$409$	6.90850e19	1.78426	0.892132	+	0.451774i	$0.149209\pi$
$410$	0	0
$411$	−6.39842e18	−0.159316
$412$	0	0
$413$	2.64204e19	0.634329
$414$	0	0
$415$	−2.10947e19	−0.488441
$416$	0	0
$417$	3.55296e18	0.0793542
$418$	0	0
$419$	−4.75891e19	−1.02542	−0.512711	−	0.858561i	$-0.671359\pi$
$419$	−4.75891e19	−1.02542	−0.512711	+	0.858561i	$0.671359\pi$
$420$	0	0
$421$	2.71257e19	0.563981	0.281991	−	0.959417i	$-0.409005\pi$
$421$	2.71257e19	0.563981	0.281991	+	0.959417i	$0.409005\pi$
$422$	0	0
$423$	−7.70036e19	−1.54511
$424$	0	0
$425$	3.35515e19	0.649824
$426$	0	0
$427$	2.65653e19	0.496712
$428$	0	0
$429$	−2.80385e18	−0.0506203
$430$	0	0
$431$	7.28235e19	1.26967	0.634837	−	0.772646i	$-0.281067\pi$
$431$	7.28235e19	1.26967	0.634837	+	0.772646i	$0.281067\pi$
$432$	0	0
$433$	3.52424e18	0.0593479	0.0296740	−	0.999560i	$-0.490553\pi$
$433$	3.52424e18	0.0593479	0.0296740	+	0.999560i	$0.490553\pi$
$434$	0	0
$435$	1.10423e19	0.179634
$436$	0	0
$437$	−7.41490e19	−1.16545
$438$	0	0
$439$	7.48213e18	0.113643	0.0568213	−	0.998384i	$-0.481903\pi$
$439$	7.48213e18	0.113643	0.0568213	+	0.998384i	$0.481903\pi$
$440$	0	0
$441$	−9.43810e18	−0.138546
$442$	0	0
$443$	−7.37472e18	−0.104645	−0.0523224	−	0.998630i	$-0.516662\pi$
$443$	−7.37472e18	−0.104645	−0.0523224	+	0.998630i	$0.516662\pi$
$444$	0	0
$445$	−3.32041e19	−0.455504
$446$	0	0
$447$	2.04347e19	0.271057
$448$	0	0
$449$	−1.28085e18	−0.0164305	−0.00821526	−	0.999966i	$-0.502615\pi$
$449$	−1.28085e18	−0.0164305	−0.00821526	+	0.999966i	$0.502615\pi$
$450$	0	0
$451$	−3.00378e19	−0.372687
$452$	0	0
$453$	−8.37861e17	−0.0100562
$454$	0	0
$455$	−3.55371e18	−0.0412664
$456$	0	0
$457$	−3.08856e19	−0.347044	−0.173522	−	0.984830i	$-0.555515\pi$
$457$	−3.08856e19	−0.347044	−0.173522	+	0.984830i	$0.555515\pi$
$458$	0	0
$459$	−3.03949e19	−0.330526
$460$	0	0
$461$	−1.04406e20	−1.09893	−0.549465	−	0.835517i	$-0.685168\pi$
$461$	−1.04406e20	−1.09893	−0.549465	+	0.835517i	$0.685168\pi$
$462$	0	0
$463$	−1.04112e20	−1.06083	−0.530414	−	0.847739i	$-0.677964\pi$
$463$	−1.04112e20	−1.06083	−0.530414	+	0.847739i	$0.677964\pi$
$464$	0	0
$465$	5.91794e17	0.00583812
$466$	0	0
$467$	−8.69017e19	−0.830141	−0.415070	−	0.909789i	$-0.636243\pi$
$467$	−8.69017e19	−0.830141	−0.415070	+	0.909789i	$0.636243\pi$
$468$	0	0
$469$	−3.27826e19	−0.303282
$470$	0	0
$471$	1.98856e19	0.178189
$472$	0	0
$473$	−2.03802e20	−1.76909
$474$	0	0
$475$	9.99122e19	0.840265
$476$	0	0
$477$	2.34116e19	0.190785
$478$	0	0
$479$	9.78808e18	0.0773003	0.0386501	−	0.999253i	$-0.487694\pi$
$479$	9.78808e18	0.0773003	0.0386501	+	0.999253i	$0.487694\pi$
$480$	0	0
$481$	1.37921e18	0.0105570
$482$	0	0
$483$	−8.25653e18	−0.0612624
$484$	0	0
$485$	−6.67509e19	−0.480169
$486$	0	0
$487$	−1.85897e20	−1.29660	−0.648300	−	0.761385i	$-0.724520\pi$
$487$	−1.85897e20	−1.29660	−0.648300	+	0.761385i	$0.724520\pi$
$488$	0	0
$489$	2.35099e18	0.0159013
$490$	0	0
$491$	−1.42683e20	−0.935969	−0.467985	−	0.883737i	$-0.655020\pi$
$491$	−1.42683e20	−0.935969	−0.467985	+	0.883737i	$0.655020\pi$
$492$	0	0
$493$	−2.74422e20	−1.74609
$494$	0	0
$495$	1.37244e20	0.847139
$496$	0	0
$497$	−7.94581e19	−0.475844
$498$	0	0
$499$	1.78610e20	1.03789	0.518946	−	0.854807i	$-0.326325\pi$
$499$	1.78610e20	1.03789	0.518946	+	0.854807i	$0.326325\pi$
$500$	0	0
$501$	−4.48382e18	−0.0252851
$502$	0	0
$503$	2.79253e20	1.52840	0.764201	−	0.644978i	$-0.223133\pi$
$503$	2.79253e20	1.52840	0.764201	+	0.644978i	$0.223133\pi$
$504$	0	0
$505$	−1.67644e20	−0.890641
$506$	0	0
$507$	3.24541e19	0.167382
$508$	0	0
$509$	4.60719e19	0.230703	0.115351	−	0.993325i	$-0.463201\pi$
$509$	4.60719e19	0.230703	0.115351	+	0.993325i	$0.463201\pi$
$510$	0	0
$511$	−1.08675e20	−0.528414
$512$	0	0
$513$	−9.05121e19	−0.427392
$514$	0	0
$515$	1.43842e20	0.659676
$516$	0	0
$517$	5.46075e20	2.43261
$518$	0	0
$519$	3.81790e19	0.165223
$520$	0	0
$521$	−5.81618e18	−0.0244543	−0.0122271	−	0.999925i	$-0.503892\pi$
$521$	−5.81618e18	−0.0244543	−0.0122271	+	0.999925i	$0.503892\pi$
$522$	0	0
$523$	−1.20018e20	−0.490325	−0.245162	−	0.969482i	$-0.578841\pi$
$523$	−1.20018e20	−0.490325	−0.245162	+	0.969482i	$0.578841\pi$
$524$	0	0
$525$	1.11253e19	0.0441688
$526$	0	0
$527$	−1.47072e19	−0.0567481
$528$	0	0
$529$	−3.44916e19	−0.129359
$530$	0	0
$531$	−4.46443e20	−1.62763
$532$	0	0
$533$	−1.31426e19	−0.0465831
$534$	0	0
$535$	1.45093e20	0.500027
$536$	0	0
$537$	−5.70757e19	−0.191269
$538$	0	0
$539$	6.69308e19	0.218128
$540$	0	0
$541$	−4.42359e20	−1.40215	−0.701077	−	0.713086i	$-0.747297\pi$
$541$	−4.42359e20	−1.40215	−0.701077	+	0.713086i	$0.747297\pi$
$542$	0	0
$543$	6.50334e19	0.200511
$544$	0	0
$545$	−4.71052e19	−0.141285
$546$	0	0
$547$	8.75594e19	0.255504	0.127752	−	0.991806i	$-0.459224\pi$
$547$	8.75594e19	0.255504	0.127752	+	0.991806i	$0.459224\pi$
$548$	0	0
$549$	−4.48890e20	−1.27452
$550$	0	0
$551$	−8.17194e20	−2.25781
$552$	0	0
$553$	−1.87607e20	−0.504442
$554$	0	0
$555$	2.10050e18	0.00549700
$556$	0	0
$557$	−2.34418e20	−0.597140	−0.298570	−	0.954388i	$-0.596510\pi$
$557$	−2.34418e20	−0.597140	−0.298570	+	0.954388i	$0.596510\pi$
$558$	0	0
$559$	−8.91708e19	−0.221123
$560$	0	0
$561$	1.06123e20	0.256204
$562$	0	0
$563$	−3.21929e20	−0.756740	−0.378370	−	0.925654i	$-0.623515\pi$
$563$	−3.21929e20	−0.756740	−0.378370	+	0.925654i	$0.623515\pi$
$564$	0	0
$565$	4.45793e20	1.02040
$566$	0	0
$567$	1.54365e20	0.344093
$568$	0	0
$569$	6.34954e20	1.37848	0.689240	−	0.724534i	$-0.257945\pi$
$569$	6.34954e20	1.37848	0.689240	+	0.724534i	$0.257945\pi$
$570$	0	0
$571$	1.17643e20	0.248768	0.124384	−	0.992234i	$-0.460305\pi$
$571$	1.17643e20	0.248768	0.124384	+	0.992234i	$0.460305\pi$
$572$	0	0
$573$	6.78958e19	0.139857
$574$	0	0
$575$	−3.12802e20	−0.627713
$576$	0	0
$577$	1.00658e21	1.96803	0.984013	−	0.178095i	$-0.0569935\pi$
$577$	1.00658e21	1.96803	0.984013	+	0.178095i	$0.0569935\pi$
$578$	0	0
$579$	8.17935e19	0.155822
$580$	0	0
$581$	1.73833e20	0.322708
$582$	0	0
$583$	−1.66025e20	−0.300371
$584$	0	0
$585$	6.00493e19	0.105886
$586$	0	0
$587$	6.19103e19	0.106409	0.0532043	−	0.998584i	$-0.483057\pi$
$587$	6.19103e19	0.106409	0.0532043	+	0.998584i	$0.483057\pi$
$588$	0	0
$589$	−4.37962e19	−0.0733791
$590$	0	0
$591$	1.79197e20	0.292701
$592$	0	0
$593$	9.11061e20	1.45090	0.725449	−	0.688275i	$-0.241632\pi$
$593$	9.11061e20	1.45090	0.725449	+	0.688275i	$0.241632\pi$
$594$	0	0
$595$	1.34504e20	0.208861
$596$	0	0
$597$	1.86220e19	0.0281981
$598$	0	0
$599$	2.14013e20	0.316038	0.158019	−	0.987436i	$-0.449489\pi$
$599$	2.14013e20	0.316038	0.158019	+	0.987436i	$0.449489\pi$
$600$	0	0
$601$	−7.77871e20	−1.12034	−0.560169	−	0.828378i	$-0.689264\pi$
$601$	−7.77871e20	−1.12034	−0.560169	+	0.828378i	$0.689264\pi$
$602$	0	0
$603$	5.53948e20	0.778196
$604$	0	0
$605$	−5.55811e20	−0.761661
$606$	0	0
$607$	−9.49483e20	−1.26932	−0.634661	−	0.772790i	$-0.718860\pi$
$607$	−9.49483e20	−1.26932	−0.634661	+	0.772790i	$0.718860\pi$
$608$	0	0
$609$	−9.09949e19	−0.118683
$610$	0	0
$611$	2.38928e20	0.304059
$612$	0	0
$613$	3.19547e20	0.396808	0.198404	−	0.980120i	$-0.436424\pi$
$613$	3.19547e20	0.396808	0.198404	+	0.980120i	$0.436424\pi$
$614$	0	0
$615$	−2.00159e19	−0.0242556
$616$	0	0
$617$	1.22701e21	1.45113	0.725567	−	0.688151i	$-0.241577\pi$
$617$	1.22701e21	1.45113	0.725567	+	0.688151i	$0.241577\pi$
$618$	0	0
$619$	4.36651e20	0.504028	0.252014	−	0.967724i	$-0.418907\pi$
$619$	4.36651e20	0.504028	0.252014	+	0.967724i	$0.418907\pi$
$620$	0	0
$621$	2.83373e20	0.319279
$622$	0	0
$623$	2.73622e20	0.300947
$624$	0	0
$625$	1.16692e20	0.125297
$626$	0	0
$627$	3.16019e20	0.331289
$628$	0	0
$629$	−5.22016e19	−0.0534323
$630$	0	0
$631$	−1.29832e21	−1.29766	−0.648832	−	0.760931i	$-0.724742\pi$
$631$	−1.29832e21	−1.29766	−0.648832	+	0.760931i	$0.724742\pi$
$632$	0	0
$633$	1.50589e20	0.146983
$634$	0	0
$635$	−1.41986e20	−0.135345
$636$	0	0
$637$	2.92847e19	0.0272643
$638$	0	0
$639$	1.34265e21	1.22098
$640$	0	0
$641$	−1.51044e21	−1.34174	−0.670871	−	0.741574i	$-0.734079\pi$
$641$	−1.51044e21	−1.34174	−0.670871	+	0.741574i	$0.734079\pi$
$642$	0	0
$643$	−2.17453e21	−1.88705	−0.943525	−	0.331302i	$-0.892512\pi$
$643$	−2.17453e21	−1.88705	−0.943525	+	0.331302i	$0.892512\pi$
$644$	0	0
$645$	−1.35805e20	−0.115138
$646$	0	0
$647$	2.18233e21	1.80775	0.903877	−	0.427793i	$-0.140709\pi$
$647$	2.18233e21	1.80775	0.903877	+	0.427793i	$0.140709\pi$
$648$	0	0
$649$	3.16597e21	2.56255
$650$	0	0
$651$	−4.87673e18	−0.00385719
$652$	0	0
$653$	−6.63399e20	−0.512774	−0.256387	−	0.966574i	$-0.582532\pi$
$653$	−6.63399e20	−0.512774	−0.256387	+	0.966574i	$0.582532\pi$
$654$	0	0
$655$	1.35871e21	1.02640
$656$	0	0
$657$	1.83636e21	1.35587
$658$	0	0
$659$	−8.84738e20	−0.638520	−0.319260	−	0.947667i	$-0.603434\pi$
$659$	−8.84738e20	−0.638520	−0.319260	+	0.947667i	$0.603434\pi$
$660$	0	0
$661$	−1.29855e21	−0.916108	−0.458054	−	0.888924i	$-0.651453\pi$
$661$	−1.29855e21	−0.916108	−0.458054	+	0.888924i	$0.651453\pi$
$662$	0	0
$663$	4.64325e19	0.0320236
$664$	0	0
$665$	4.00536e20	0.270072
$666$	0	0
$667$	2.55845e21	1.68668
$668$	0	0
$669$	2.59991e20	0.167596
$670$	0	0
$671$	3.18333e21	2.00661
$672$	0	0
$673$	−2.90087e19	−0.0178820	−0.00894100	−	0.999960i	$-0.502846\pi$
$673$	−2.90087e19	−0.0178820	−0.00894100	+	0.999960i	$0.502846\pi$
$674$	0	0
$675$	−3.81831e20	−0.230193
$676$	0	0
$677$	−2.22541e21	−1.31218	−0.656092	−	0.754681i	$-0.727792\pi$
$677$	−2.22541e21	−1.31218	−0.656092	+	0.754681i	$0.727792\pi$
$678$	0	0
$679$	5.50067e20	0.317243
$680$	0	0
$681$	4.54187e20	0.256231
$682$	0	0
$683$	5.76905e20	0.318382	0.159191	−	0.987248i	$-0.449111\pi$
$683$	5.76905e20	0.318382	0.159191	+	0.987248i	$0.449111\pi$
$684$	0	0
$685$	−9.71779e20	−0.524673
$686$	0	0
$687$	9.09101e19	0.0480216
$688$	0	0
$689$	−7.26418e19	−0.0375442
$690$	0	0
$691$	−1.72056e20	−0.0870132	−0.0435066	−	0.999053i	$-0.513853\pi$
$691$	−1.72056e20	−0.0870132	−0.0435066	+	0.999053i	$0.513853\pi$
$692$	0	0
$693$	−1.13097e21	−0.559697
$694$	0	0
$695$	5.39616e20	0.261336
$696$	0	0
$697$	4.97435e20	0.235771
$698$	0	0
$699$	−1.74382e20	−0.0808954
$700$	0	0
$701$	3.56715e21	1.61970	0.809852	−	0.586634i	$-0.199547\pi$
$701$	3.56715e21	1.61970	0.809852	+	0.586634i	$0.199547\pi$
$702$	0	0
$703$	−1.55450e20	−0.0690915
$704$	0	0
$705$	3.63881e20	0.158322
$706$	0	0
$707$	1.38149e21	0.588439
$708$	0	0
$709$	−4.36779e21	−1.82144	−0.910720	−	0.413025i	$-0.864472\pi$
$709$	−4.36779e21	−1.82144	−0.910720	+	0.413025i	$0.864472\pi$
$710$	0	0
$711$	3.17012e21	1.29436
$712$	0	0
$713$	1.37116e20	0.0548172
$714$	0	0
$715$	−4.25843e20	−0.166707
$716$	0	0
$717$	−5.87683e19	−0.0225294
$718$	0	0
$719$	−3.07858e21	−1.15580	−0.577902	−	0.816106i	$-0.696128\pi$
$719$	−3.07858e21	−1.15580	−0.577902	+	0.816106i	$0.696128\pi$
$720$	0	0
$721$	−1.18534e21	−0.435842
$722$	0	0
$723$	5.60679e20	0.201919
$724$	0	0
$725$	−3.44738e21	−1.21606
$726$	0	0
$727$	−3.55890e21	−1.22972	−0.614862	−	0.788635i	$-0.710788\pi$
$727$	−3.55890e21	−1.22972	−0.614862	+	0.788635i	$0.710788\pi$
$728$	0	0
$729$	−2.43280e21	−0.823475
$730$	0	0
$731$	3.37502e21	1.11917
$732$	0	0
$733$	5.10316e21	1.65790	0.828952	−	0.559320i	$-0.188938\pi$
$733$	5.10316e21	1.65790	0.828952	+	0.559320i	$0.188938\pi$
$734$	0	0
$735$	4.45998e19	0.0141964
$736$	0	0
$737$	−3.92835e21	−1.22519
$738$	0	0
$739$	2.92225e20	0.0893068	0.0446534	−	0.999003i	$-0.485782\pi$
$739$	2.92225e20	0.0893068	0.0446534	+	0.999003i	$0.485782\pi$
$740$	0	0
$741$	1.38270e20	0.0414087
$742$	0	0
$743$	−1.26885e21	−0.372386	−0.186193	−	0.982513i	$-0.559615\pi$
$743$	−1.26885e21	−0.372386	−0.186193	+	0.982513i	$0.559615\pi$
$744$	0	0
$745$	3.10357e21	0.892668
$746$	0	0
$747$	−2.93736e21	−0.828043
$748$	0	0
$749$	−1.19565e21	−0.330364
$750$	0	0
$751$	−5.61709e21	−1.52129	−0.760644	−	0.649169i	$-0.775117\pi$
$751$	−5.61709e21	−1.52129	−0.760644	+	0.649169i	$0.775117\pi$
$752$	0	0
$753$	7.60702e20	0.201954
$754$	0	0
$755$	−1.27253e20	−0.0331181
$756$	0	0
$757$	3.05459e20	0.0779353	0.0389677	−	0.999240i	$-0.487593\pi$
$757$	3.05459e20	0.0779353	0.0389677	+	0.999240i	$0.487593\pi$
$758$	0	0
$759$	−9.89384e20	−0.247487
$760$	0	0
$761$	5.46902e21	1.34130	0.670648	−	0.741776i	$-0.266016\pi$
$761$	5.46902e21	1.34130	0.670648	+	0.741776i	$0.266016\pi$
$762$	0	0
$763$	3.88175e20	0.0933455
$764$	0	0
$765$	−2.27280e21	−0.535920
$766$	0	0
$767$	1.38523e21	0.320299
$768$	0	0
$769$	6.99768e21	1.58674	0.793371	−	0.608738i	$-0.208324\pi$
$769$	6.99768e21	1.58674	0.793371	+	0.608738i	$0.208324\pi$
$770$	0	0
$771$	−5.92018e20	−0.131652
$772$	0	0
$773$	2.23177e21	0.486747	0.243374	−	0.969933i	$-0.421746\pi$
$773$	2.23177e21	0.486747	0.243374	+	0.969933i	$0.421746\pi$
$774$	0	0
$775$	−1.84757e20	−0.0395219
$776$	0	0
$777$	−1.73094e19	−0.00363182
$778$	0	0
$779$	1.48130e21	0.304867
$780$	0	0
$781$	−9.52150e21	−1.92231
$782$	0	0
$783$	3.12304e21	0.618535
$784$	0	0
$785$	3.02018e21	0.586828
$786$	0	0
$787$	2.69301e21	0.513367	0.256683	−	0.966496i	$-0.417370\pi$
$787$	2.69301e21	0.513367	0.256683	+	0.966496i	$0.417370\pi$
$788$	0	0
$789$	9.59153e20	0.179395
$790$	0	0
$791$	−3.67360e21	−0.674168
$792$	0	0
$793$	1.39282e21	0.250811
$794$	0	0
$795$	−1.10632e20	−0.0195491
$796$	0	0
$797$	−5.54873e21	−0.962179	−0.481090	−	0.876671i	$-0.659759\pi$
$797$	−5.54873e21	−0.962179	−0.481090	+	0.876671i	$0.659759\pi$
$798$	0	0
$799$	−9.04315e21	−1.53893
$800$	0	0
$801$	−4.62356e21	−0.772205
$802$	0	0
$803$	−1.30226e22	−2.13468
$804$	0	0
$805$	−1.25398e21	−0.201754
$806$	0	0
$807$	−7.11632e20	−0.112384
$808$	0	0
$809$	−3.74521e21	−0.580581	−0.290291	−	0.956939i	$-0.593752\pi$
$809$	−3.74521e21	−0.580581	−0.290291	+	0.956939i	$0.593752\pi$
$810$	0	0
$811$	−1.05009e22	−1.59798	−0.798991	−	0.601343i	$-0.794632\pi$
$811$	−1.05009e22	−1.59798	−0.798991	+	0.601343i	$0.794632\pi$
$812$	0	0
$813$	−1.01113e21	−0.151052
$814$	0	0
$815$	3.57063e20	0.0523676
$816$	0	0
$817$	1.00504e22	1.44716
$818$	0	0
$819$	−4.94842e20	−0.0699579
$820$	0	0
$821$	−6.42001e21	−0.891174	−0.445587	−	0.895239i	$-0.647005\pi$
$821$	−6.42001e21	−0.891174	−0.445587	+	0.895239i	$0.647005\pi$
$822$	0	0
$823$	1.57289e20	0.0214388	0.0107194	−	0.999943i	$-0.496588\pi$
$823$	1.57289e20	0.0214388	0.0107194	+	0.999943i	$0.496588\pi$
$824$	0	0
$825$	1.33315e21	0.178432
$826$	0	0
$827$	−1.86980e21	−0.245756	−0.122878	−	0.992422i	$-0.539212\pi$
$827$	−1.86980e21	−0.245756	−0.122878	+	0.992422i	$0.539212\pi$
$828$	0	0
$829$	−9.87859e21	−1.27508	−0.637538	−	0.770419i	$-0.720047\pi$
$829$	−9.87859e21	−1.27508	−0.637538	+	0.770419i	$0.720047\pi$
$830$	0	0
$831$	−1.62645e21	−0.206173
$832$	0	0
$833$	−1.10839e21	−0.137993
$834$	0	0
$835$	−6.80993e20	−0.0832711
$836$	0	0
$837$	1.67374e20	0.0201024
$838$	0	0
$839$	−3.27415e21	−0.386264	−0.193132	−	0.981173i	$-0.561865\pi$
$839$	−3.27415e21	−0.386264	−0.193132	+	0.981173i	$0.561865\pi$
$840$	0	0
$841$	1.95674e22	2.26758
$842$	0	0
$843$	1.60399e21	0.182597
$844$	0	0
$845$	4.92906e21	0.551237
$846$	0	0
$847$	4.58021e21	0.503223
$848$	0	0
$849$	−1.45432e21	−0.156983
$850$	0	0
$851$	4.86677e20	0.0516142
$852$	0	0
$853$	−2.11428e21	−0.220316	−0.110158	−	0.993914i	$-0.535136\pi$
$853$	−2.11428e21	−0.220316	−0.110158	+	0.993914i	$0.535136\pi$
$854$	0	0
$855$	−6.76810e21	−0.692981
$856$	0	0
$857$	−1.36630e22	−1.37464	−0.687321	−	0.726354i	$-0.741213\pi$
$857$	−1.36630e22	−1.37464	−0.687321	+	0.726354i	$0.741213\pi$
$858$	0	0
$859$	−7.80081e21	−0.771243	−0.385621	−	0.922657i	$-0.626013\pi$
$859$	−7.80081e21	−0.771243	−0.385621	+	0.922657i	$0.626013\pi$
$860$	0	0
$861$	1.64943e20	0.0160255
$862$	0	0
$863$	1.91233e22	1.82592	0.912959	−	0.408052i	$-0.133792\pi$
$863$	1.91233e22	1.82592	0.912959	+	0.408052i	$0.133792\pi$
$864$	0	0
$865$	5.79855e21	0.544125
$866$	0	0
$867$	1.26086e20	0.0116285
$868$	0	0
$869$	−2.24811e22	−2.03783
$870$	0	0
$871$	−1.71880e21	−0.153140
$872$	0	0
$873$	−9.29483e21	−0.814020
$874$	0	0
$875$	4.20136e21	0.361684
$876$	0	0
$877$	1.29282e22	1.09406	0.547031	−	0.837112i	$-0.315758\pi$
$877$	1.29282e22	1.09406	0.547031	+	0.837112i	$0.315758\pi$
$878$	0	0
$879$	1.43580e21	0.119447
$880$	0	0
$881$	1.00464e22	0.821656	0.410828	−	0.911713i	$-0.365240\pi$
$881$	1.00464e22	0.821656	0.410828	+	0.911713i	$0.365240\pi$
$882$	0	0
$883$	1.20567e22	0.969449	0.484724	−	0.874667i	$-0.338920\pi$
$883$	1.20567e22	0.969449	0.484724	+	0.874667i	$0.338920\pi$
$884$	0	0
$885$	2.10967e21	0.166778
$886$	0	0
$887$	−1.21274e22	−0.942626	−0.471313	−	0.881966i	$-0.656220\pi$
$887$	−1.21274e22	−0.942626	−0.471313	+	0.881966i	$0.656220\pi$
$888$	0	0
$889$	1.17005e21	0.0894213
$890$	0	0
$891$	1.84977e22	1.39006
$892$	0	0
$893$	−2.69293e22	−1.98994
$894$	0	0
$895$	−8.66853e21	−0.629902
$896$	0	0
$897$	−4.32892e20	−0.0309340
$898$	0	0
$899$	1.51115e21	0.106196
$900$	0	0
$901$	2.74941e21	0.190022
$902$	0	0
$903$	1.11911e21	0.0760704
$904$	0	0
$905$	9.87714e21	0.660339
$906$	0	0
$907$	−1.76757e21	−0.116231	−0.0581156	−	0.998310i	$-0.518509\pi$
$907$	−1.76757e21	−0.116231	−0.0581156	+	0.998310i	$0.518509\pi$
$908$	0	0
$909$	−2.33439e22	−1.50988
$910$	0	0
$911$	1.09790e22	0.698516	0.349258	−	0.937027i	$-0.386434\pi$
$911$	1.09790e22	0.698516	0.349258	+	0.937027i	$0.386434\pi$
$912$	0	0
$913$	2.08305e22	1.30367
$914$	0	0
$915$	2.12123e21	0.130596
$916$	0	0
$917$	−1.11966e22	−0.678133
$918$	0	0
$919$	1.07271e22	0.639167	0.319584	−	0.947558i	$-0.396457\pi$
$919$	1.07271e22	0.639167	0.319584	+	0.947558i	$0.396457\pi$
$920$	0	0
$921$	−4.86968e21	−0.285465
$922$	0	0
$923$	−4.16600e21	−0.240274
$924$	0	0
$925$	−6.55773e20	−0.0372126
$926$	0	0
$927$	2.00295e22	1.11833
$928$	0	0
$929$	1.10724e22	0.608309	0.304154	−	0.952623i	$-0.401626\pi$
$929$	1.10724e22	0.608309	0.304154	+	0.952623i	$0.401626\pi$
$930$	0	0
$931$	−3.30065e21	−0.178434
$932$	0	0
$933$	−3.23670e20	−0.0172183
$934$	0	0
$935$	1.61177e22	0.843753
$936$	0	0
$937$	−5.64109e21	−0.290614	−0.145307	−	0.989387i	$-0.546417\pi$
$937$	−5.64109e21	−0.290614	−0.145307	+	0.989387i	$0.546417\pi$
$938$	0	0
$939$	−3.22835e21	−0.163678
$940$	0	0
$941$	2.46275e22	1.22885	0.614423	−	0.788976i	$-0.289389\pi$
$941$	2.46275e22	1.22885	0.614423	+	0.788976i	$0.289389\pi$
$942$	0	0
$943$	−4.63760e21	−0.227748
$944$	0	0
$945$	−1.53071e21	−0.0739869
$946$	0	0
$947$	4.79803e21	0.228264	0.114132	−	0.993466i	$-0.463591\pi$
$947$	4.79803e21	0.228264	0.114132	+	0.993466i	$0.463591\pi$
$948$	0	0
$949$	−5.69787e21	−0.266819
$950$	0	0
$951$	2.02717e21	0.0934406
$952$	0	0
$953$	1.92534e22	0.873595	0.436797	−	0.899560i	$-0.356113\pi$
$953$	1.92534e22	0.873595	0.436797	+	0.899560i	$0.356113\pi$
$954$	0	0
$955$	1.03119e22	0.460588
$956$	0	0
$957$	−1.09040e22	−0.479452
$958$	0	0
$959$	8.00804e21	0.346646
$960$	0	0
$961$	−2.33843e22	−0.996549
$962$	0	0
$963$	2.02037e22	0.847685
$964$	0	0
$965$	1.24226e22	0.513167
$966$	0	0
$967$	−3.76415e22	−1.53098	−0.765488	−	0.643450i	$-0.777503\pi$
$967$	−3.76415e22	−1.53098	−0.765488	+	0.643450i	$0.777503\pi$
$968$	0	0
$969$	−5.23337e21	−0.209582
$970$	0	0
$971$	1.24131e22	0.489480	0.244740	−	0.969589i	$-0.421297\pi$
$971$	1.24131e22	0.489480	0.244740	+	0.969589i	$0.421297\pi$
$972$	0	0
$973$	−4.44676e21	−0.172662
$974$	0	0
$975$	5.83300e20	0.0223027
$976$	0	0
$977$	4.32714e22	1.62927	0.814633	−	0.579977i	$-0.196939\pi$
$977$	4.32714e22	1.62927	0.814633	+	0.579977i	$0.196939\pi$
$978$	0	0
$979$	3.27882e22	1.21576
$980$	0	0
$981$	−6.55924e21	−0.239517
$982$	0	0
$983$	1.77525e22	0.638424	0.319212	−	0.947683i	$-0.396582\pi$
$983$	1.77525e22	0.638424	0.319212	+	0.947683i	$0.396582\pi$
$984$	0	0
$985$	2.72161e22	0.963949
$986$	0	0
$987$	−2.99860e21	−0.104602
$988$	0	0
$989$	−3.14654e22	−1.08109
$990$	0	0
$991$	−2.60434e22	−0.881342	−0.440671	−	0.897669i	$-0.645259\pi$
$991$	−2.60434e22	−0.881342	−0.440671	+	0.897669i	$0.645259\pi$
$992$	0	0
$993$	5.17231e21	0.172411
$994$	0	0
$995$	2.82828e21	0.0928643
$996$	0	0
$997$	−6.86864e21	−0.222155	−0.111078	−	0.993812i	$-0.535430\pi$
$997$	−6.86864e21	−0.222155	−0.111078	+	0.993812i	$0.535430\pi$
$998$	0	0
$999$	5.94075e20	0.0189278

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.16.a.k.1.4		6
4.3	odd	2		56.16.a.d.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.16.a.d.1.3	✓	6	4.3	odd	2
112.16.a.k.1.4		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

\(f(q)\)	\(=\)	\(q-658.011 q^{3} -99937.3 q^{5} +823543. q^{7} -1.39159e7 q^{9} +9.86856e7 q^{11} +4.31785e7 q^{13} +6.57598e7 q^{15} -1.63426e9 q^{17} -4.86662e9 q^{19} -5.41900e8 q^{21} +1.52363e10 q^{23} -2.05301e10 q^{25} +1.85986e10 q^{27} +1.67918e11 q^{29} +8.99932e9 q^{31} -6.49362e10 q^{33} -8.23027e10 q^{35} +3.19420e10 q^{37} -2.84119e10 q^{39} -3.04379e11 q^{41} -2.06517e12 q^{43} +1.39072e12 q^{45} +5.53348e12 q^{47} +6.78223e11 q^{49} +1.07536e12 q^{51} -1.68236e12 q^{53} -9.86237e12 q^{55} +3.20229e12 q^{57} +3.20814e13 q^{59} +3.22573e13 q^{61} -1.14604e13 q^{63} -4.31515e12 q^{65} -3.98067e13 q^{67} -1.00256e13 q^{69} -9.64832e13 q^{71} -1.31961e14 q^{73} +1.35090e13 q^{75} +8.12718e13 q^{77} -2.27805e14 q^{79} +1.87440e14 q^{81} +2.11079e14 q^{83} +1.63324e14 q^{85} -1.10492e14 q^{87} +3.32250e14 q^{89} +3.55594e13 q^{91} -5.92165e12 q^{93} +4.86357e14 q^{95} +6.67928e14 q^{97} -1.37330e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3864 q^{3} + 168532 q^{5} + 4941258 q^{7} + 27735822 q^{9} - 58040072 q^{11} - 15956444 q^{13} + 705059136 q^{15} - 2879979060 q^{17} - 3435500376 q^{19} - 3182170152 q^{21} - 34508782592 q^{23} + 29232294906 q^{25}+ \cdots - 44\!\cdots\!92 q^{99}+O(q^{100}) \) 6 * q - 3864 * q^3 + 168532 * q^5 + 4941258 * q^7 + 27735822 * q^9 - 58040072 * q^11 - 15956444 * q^13 + 705059136 * q^15 - 2879979060 * q^17 - 3435500376 * q^19 - 3182170152 * q^21 - 34508782592 * q^23 + 29232294906 * q^25 - 102516868944 * q^27 - 46458543420 * q^29 - 300765178336 * q^31 - 457356427584 * q^33 + 138793348876 * q^35 - 509722422412 * q^37 + 437968495008 * q^39 + 1619276439676 * q^41 - 2727781008680 * q^43 + 5316894690660 * q^45 + 1567726443808 * q^47 + 4069338437094 * q^49 - 3110310296208 * q^51 + 2612518893588 * q^53 + 4577322926960 * q^55 - 21550151959824 * q^57 - 5112561397384 * q^59 + 4545346033732 * q^61 + 22841642057346 * q^63 + 57444596600712 * q^65 - 74088461612408 * q^67 + 176631933295104 * q^69 - 51092191255952 * q^71 + 294343628983292 * q^73 - 208088876183496 * q^75 - 47798495015096 * q^77 - 56070305620928 * q^79 + 691991522436870 * q^81 - 984559173278520 * q^83 + 396331720481544 * q^85 - 1024554713548464 * q^87 + 1528383063352572 * q^89 - 13140817761092 * q^91 + 1245617896072608 * q^93 - 2532178187158304 * q^95 + 2255703641701580 * q^97 - 4484026592356392 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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