LMFDB - Embedded newform 3042.2.b.j.1351.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3042,2,Mod(1351,3042)] 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("3042.1351"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3042, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0,0,0,-4,0,0,0,0,0,4,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$1.58114 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.j.1351.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +4.16228i q^{5} +3.16228i q^{7} -1.00000i q^{8} -4.16228 q^{10} +1.16228i q^{11} -3.16228 q^{14} +1.00000 q^{16} -3.00000 q^{17} +5.16228i q^{19} -4.16228i q^{20} -1.16228 q^{22} +7.16228 q^{23} -12.3246 q^{25} -3.16228i q^{28} +1.83772 q^{29} +6.32456i q^{31} +1.00000i q^{32} -3.00000i q^{34} -13.1623 q^{35} +3.83772i q^{37} -5.16228 q^{38} +4.16228 q^{40} +3.00000i q^{41} -9.16228 q^{43} -1.16228i q^{44} +7.16228i q^{46} -4.83772i q^{47} -3.00000 q^{49} -12.3246i q^{50} +12.4868 q^{53} -4.83772 q^{55} +3.16228 q^{56} +1.83772i q^{58} +2.32456i q^{59} +0.162278 q^{61} -6.32456 q^{62} -1.00000 q^{64} +2.83772i q^{67} +3.00000 q^{68} -13.1623i q^{70} -7.16228i q^{71} -1.00000i q^{73} -3.83772 q^{74} -5.16228i q^{76} -3.67544 q^{77} -4.00000 q^{79} +4.16228i q^{80} -3.00000 q^{82} -3.48683i q^{83} -12.4868i q^{85} -9.16228i q^{86} +1.16228 q^{88} -12.0000i q^{89} -7.16228 q^{92} +4.83772 q^{94} -21.4868 q^{95} +4.00000i q^{97} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{10} + 4 q^{16} - 12 q^{17} + 8 q^{22} + 16 q^{23} - 24 q^{25} + 20 q^{29} - 40 q^{35} - 8 q^{38} + 4 q^{40} - 24 q^{43} - 12 q^{49} + 12 q^{53} - 32 q^{55} - 12 q^{61} - 4 q^{64} + 12 q^{68}+ \cdots - 48 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^10 + 4 * q^16 - 12 * q^17 + 8 * q^22 + 16 * q^23 - 24 * q^25 + 20 * q^29 - 40 * q^35 - 8 * q^38 + 4 * q^40 - 24 * q^43 - 12 * q^49 + 12 * q^53 - 32 * q^55 - 12 * q^61 - 4 * q^64 + 12 * q^68 - 28 * q^74 - 40 * q^77 - 16 * q^79 - 12 * q^82 - 8 * q^88 - 16 * q^92 + 32 * q^94 - 48 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	4.16228i	1.86143i	0.365749	+	0.930714i	$0.380813\pi$
$5$	4.16228i	1.86143i	−0.365749	+	0.930714i	$0.619187\pi$
$6$	0	0
$7$	3.16228i	1.19523i	0.801784	+	0.597614i	$0.203885\pi$
$7$	3.16228i	1.19523i	−0.801784	+	0.597614i	$0.796115\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	−4.16228	−1.31623
$11$	1.16228i	0.350440i	0.984529	+	0.175220i	$0.0560637\pi$
$11$	1.16228i	0.350440i	−0.984529	+	0.175220i	$0.943936\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−3.16228	−0.845154
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	5.16228i	1.18431i	0.805825	+	0.592154i	$0.201722\pi$
$19$	5.16228i	1.18431i	−0.805825	+	0.592154i	$0.798278\pi$
$20$	− 4.16228i	− 0.930714i
$21$	0	0
$22$	−1.16228	−0.247798
$23$	7.16228	1.49344	0.746719	−	0.665140i	$-0.231628\pi$
$23$	7.16228	1.49344	0.746719	+	0.665140i	$0.231628\pi$
$24$	0	0
$25$	−12.3246	−2.46491
$26$	0	0
$27$	0	0
$28$	− 3.16228i	− 0.597614i
$29$	1.83772	0.341256	0.170628	−	0.985335i	$-0.445420\pi$
$29$	1.83772	0.341256	0.170628	+	0.985335i	$0.445420\pi$
$30$	0	0
$31$	6.32456i	1.13592i	0.823055	+	0.567962i	$0.192268\pi$
$31$	6.32456i	1.13592i	−0.823055	+	0.567962i	$0.807732\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 3.00000i	− 0.514496i
$35$	−13.1623	−2.22483
$36$	0	0
$37$	3.83772i	0.630918i	0.948939	+	0.315459i	$0.102158\pi$
$37$	3.83772i	0.630918i	−0.948939	+	0.315459i	$0.897842\pi$
$38$	−5.16228	−0.837432
$39$	0	0
$40$	4.16228	0.658114
$41$	3.00000i	0.468521i	0.972174	+	0.234261i	$0.0752669\pi$
$41$	3.00000i	0.468521i	−0.972174	+	0.234261i	$0.924733\pi$
$42$	0	0
$43$	−9.16228	−1.39723	−0.698617	−	0.715496i	$-0.746201\pi$
$43$	−9.16228	−1.39723	−0.698617	+	0.715496i	$0.746201\pi$
$44$	− 1.16228i	− 0.175220i
$45$	0	0
$46$	7.16228i	1.05602i
$47$	− 4.83772i	− 0.705654i	−0.935689	−	0.352827i	$-0.885220\pi$
$47$	− 4.83772i	− 0.705654i	0.935689	−	0.352827i	$-0.114780\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	− 12.3246i	− 1.74296i
$51$	0	0
$52$	0	0
$53$	12.4868	1.71520	0.857599	−	0.514319i	$-0.171955\pi$
$53$	12.4868	1.71520	0.857599	+	0.514319i	$0.171955\pi$
$54$	0	0
$55$	−4.83772	−0.652318
$56$	3.16228	0.422577
$57$	0	0
$58$	1.83772i	0.241305i
$59$	2.32456i	0.302631i	0.988485	+	0.151316i	$0.0483510\pi$
$59$	2.32456i	0.302631i	−0.988485	+	0.151316i	$0.951649\pi$
$60$	0	0
$61$	0.162278	0.0207775	0.0103888	−	0.999946i	$-0.496693\pi$
$61$	0.162278	0.0207775	0.0103888	+	0.999946i	$0.496693\pi$
$62$	−6.32456	−0.803219
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	2.83772i	0.346683i	0.984862	+	0.173341i	$0.0554564\pi$
$67$	2.83772i	0.346683i	−0.984862	+	0.173341i	$0.944544\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	− 13.1623i	− 1.57319i
$71$	− 7.16228i	− 0.850006i	−0.905192	−	0.425003i	$-0.860273\pi$
$71$	− 7.16228i	− 0.850006i	0.905192	−	0.425003i	$-0.139727\pi$
$72$	0	0
$73$	− 1.00000i	− 0.117041i	−0.998286	−	0.0585206i	$-0.981362\pi$
$73$	− 1.00000i	− 0.117041i	0.998286	−	0.0585206i	$-0.0186383\pi$
$74$	−3.83772	−0.446126
$75$	0	0
$76$	− 5.16228i	− 0.592154i
$77$	−3.67544	−0.418856
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	4.16228i	0.465357i
$81$	0	0
$82$	−3.00000	−0.331295
$83$	− 3.48683i	− 0.382730i	−0.981519	−	0.191365i	$-0.938709\pi$
$83$	− 3.48683i	− 0.382730i	0.981519	−	0.191365i	$-0.0612914\pi$
$84$	0	0
$85$	− 12.4868i	− 1.35439i
$86$	− 9.16228i	− 0.987994i
$87$	0	0
$88$	1.16228	0.123899
$89$	− 12.0000i	− 1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$	− 12.0000i	− 1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$	0	0
$91$	0	0
$92$	−7.16228	−0.746719
$93$	0	0
$94$	4.83772	0.498973
$95$	−21.4868	−2.20450
$96$	0	0
$97$	4.00000i	0.406138i	0.979164	+	0.203069i	$0.0650917\pi$
$97$	4.00000i	0.406138i	−0.979164	+	0.203069i	$0.934908\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	12.3246	1.23246
$101$	7.83772	0.779883	0.389941	−	0.920840i	$-0.372495\pi$
$101$	7.83772	0.779883	0.389941	+	0.920840i	$0.372495\pi$
$102$	0	0
$103$	15.8114	1.55794	0.778971	−	0.627060i	$-0.215742\pi$
$103$	15.8114	1.55794	0.778971	+	0.627060i	$0.215742\pi$
$104$	0	0
$105$	0	0
$106$	12.4868i	1.21283i
$107$	17.8114	1.72189	0.860946	−	0.508696i	$-0.169872\pi$
$107$	17.8114	1.72189	0.860946	+	0.508696i	$0.169872\pi$
$108$	0	0
$109$	− 6.64911i	− 0.636869i	−0.947945	−	0.318435i	$-0.896843\pi$
$109$	− 6.64911i	− 0.636869i	0.947945	−	0.318435i	$-0.103157\pi$
$110$	− 4.83772i	− 0.461259i
$111$	0	0
$112$	3.16228i	0.298807i
$113$	−6.67544	−0.627973	−0.313987	−	0.949427i	$-0.601665\pi$
$113$	−6.67544	−0.627973	−0.313987	+	0.949427i	$0.601665\pi$
$114$	0	0
$115$	29.8114i	2.77993i
$116$	−1.83772	−0.170628
$117$	0	0
$118$	−2.32456	−0.213993
$119$	− 9.48683i	− 0.869657i
$120$	0	0
$121$	9.64911	0.877192
$122$	0.162278i	0.0146919i
$123$	0	0
$124$	− 6.32456i	− 0.567962i
$125$	− 30.4868i	− 2.72683i
$126$	0	0
$127$	18.3246	1.62604	0.813021	−	0.582235i	$-0.197822\pi$
$127$	18.3246	1.62604	0.813021	+	0.582235i	$0.197822\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−16.6491	−1.45464	−0.727320	−	0.686299i	$-0.759234\pi$
$131$	−16.6491	−1.45464	−0.727320	+	0.686299i	$0.759234\pi$
$132$	0	0
$133$	−16.3246	−1.41552
$134$	−2.83772	−0.245142
$135$	0	0
$136$	3.00000i	0.257248i
$137$	3.00000i	0.256307i	0.991754	+	0.128154i	$0.0409051\pi$
$137$	3.00000i	0.256307i	−0.991754	+	0.128154i	$0.959095\pi$
$138$	0	0
$139$	−6.32456	−0.536442	−0.268221	−	0.963357i	$-0.586436\pi$
$139$	−6.32456	−0.536442	−0.268221	+	0.963357i	$0.586436\pi$
$140$	13.1623	1.11242
$141$	0	0
$142$	7.16228	0.601045
$143$	0	0
$144$	0	0
$145$	7.64911i	0.635224i
$146$	1.00000	0.0827606
$147$	0	0
$148$	− 3.83772i	− 0.315459i
$149$	0.486833i	0.0398829i	0.999801	+	0.0199415i	$0.00634798\pi$
$149$	0.486833i	0.0398829i	−0.999801	+	0.0199415i	$0.993652\pi$
$150$	0	0
$151$	0.837722i	0.0681729i	0.999419	+	0.0340864i	$0.0108522\pi$
$151$	0.837722i	0.0681729i	−0.999419	+	0.0340864i	$0.989148\pi$
$152$	5.16228	0.418716
$153$	0	0
$154$	− 3.67544i	− 0.296176i
$155$	−26.3246	−2.11444
$156$	0	0
$157$	−10.4868	−0.836940	−0.418470	−	0.908231i	$-0.637434\pi$
$157$	−10.4868	−0.836940	−0.418470	+	0.908231i	$0.637434\pi$
$158$	− 4.00000i	− 0.318223i
$159$	0	0
$160$	−4.16228	−0.329057
$161$	22.6491i	1.78500i
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	− 3.00000i	− 0.234261i
$165$	0	0
$166$	3.48683	0.270631
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	0	0
$170$	12.4868	0.957696
$171$	0	0
$172$	9.16228	0.698617
$173$	22.6491	1.72198	0.860990	−	0.508622i	$-0.169845\pi$
$173$	22.6491	1.72198	0.860990	+	0.508622i	$0.169845\pi$
$174$	0	0
$175$	− 38.9737i	− 2.94613i
$176$	1.16228i	0.0876100i
$177$	0	0
$178$	12.0000	0.899438
$179$	15.4868	1.15754	0.578770	−	0.815491i	$-0.303533\pi$
$179$	15.4868	1.15754	0.578770	+	0.815491i	$0.303533\pi$
$180$	0	0
$181$	−3.83772	−0.285256	−0.142628	−	0.989776i	$-0.545555\pi$
$181$	−3.83772	−0.285256	−0.142628	+	0.989776i	$0.545555\pi$
$182$	0	0
$183$	0	0
$184$	− 7.16228i	− 0.528010i
$185$	−15.9737	−1.17441
$186$	0	0
$187$	− 3.48683i	− 0.254982i
$188$	4.83772i	0.352827i
$189$	0	0
$190$	− 21.4868i	− 1.55882i
$191$	−14.3246	−1.03649	−0.518244	−	0.855233i	$-0.673414\pi$
$191$	−14.3246	−1.03649	−0.518244	+	0.855233i	$0.673414\pi$
$192$	0	0
$193$	− 19.9737i	− 1.43774i	−0.695147	−	0.718868i	$-0.744661\pi$
$193$	− 19.9737i	− 1.43774i	0.695147	−	0.718868i	$-0.255339\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	3.00000	0.214286
$197$	18.9737i	1.35182i	0.736985	+	0.675909i	$0.236249\pi$
$197$	18.9737i	1.35182i	−0.736985	+	0.675909i	$0.763751\pi$
$198$	0	0
$199$	6.51317	0.461706	0.230853	−	0.972989i	$-0.425848\pi$
$199$	6.51317	0.461706	0.230853	+	0.972989i	$0.425848\pi$
$200$	12.3246i	0.871478i
$201$	0	0
$202$	7.83772i	0.551460i
$203$	5.81139i	0.407879i
$204$	0	0
$205$	−12.4868	−0.872118
$206$	15.8114i	1.10163i
$207$	0	0
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−12.4868	−0.857599
$213$	0	0
$214$	17.8114i	1.21756i
$215$	− 38.1359i	− 2.60085i
$216$	0	0
$217$	−20.0000	−1.35769
$218$	6.64911	0.450335
$219$	0	0
$220$	4.83772	0.326159
$221$	0	0
$222$	0	0
$223$	1.67544i	0.112196i	0.998425	+	0.0560980i	$0.0178659\pi$
$223$	1.67544i	0.112196i	−0.998425	+	0.0560980i	$0.982134\pi$
$224$	−3.16228	−0.211289
$225$	0	0
$226$	− 6.67544i	− 0.444044i
$227$	− 15.4868i	− 1.02790i	−0.857821	−	0.513949i	$-0.828182\pi$
$227$	− 15.4868i	− 1.02790i	0.857821	−	0.513949i	$-0.171818\pi$
$228$	0	0
$229$	22.3246i	1.47525i	0.675212	+	0.737624i	$0.264052\pi$
$229$	22.3246i	1.47525i	−0.675212	+	0.737624i	$0.735948\pi$
$230$	−29.8114	−1.96570
$231$	0	0
$232$	− 1.83772i	− 0.120652i
$233$	−16.6491	−1.09072	−0.545360	−	0.838202i	$-0.683607\pi$
$233$	−16.6491	−1.09072	−0.545360	+	0.838202i	$0.683607\pi$
$234$	0	0
$235$	20.1359	1.31352
$236$	− 2.32456i	− 0.151316i
$237$	0	0
$238$	9.48683	0.614940
$239$	− 21.4868i	− 1.38987i	−0.719074	−	0.694934i	$-0.755434\pi$
$239$	− 21.4868i	− 1.38987i	0.719074	−	0.694934i	$-0.244566\pi$
$240$	0	0
$241$	13.3246i	0.858310i	0.903231	+	0.429155i	$0.141189\pi$
$241$	13.3246i	0.858310i	−0.903231	+	0.429155i	$0.858811\pi$
$242$	9.64911i	0.620268i
$243$	0	0
$244$	−0.162278	−0.0103888
$245$	− 12.4868i	− 0.797754i
$246$	0	0
$247$	0	0
$248$	6.32456	0.401610
$249$	0	0
$250$	30.4868	1.92816
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	8.32456i	0.523360i
$254$	18.3246i	1.14978i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−21.9737	−1.37068	−0.685340	−	0.728223i	$-0.740346\pi$
$257$	−21.9737	−1.37068	−0.685340	+	0.728223i	$0.740346\pi$
$258$	0	0
$259$	−12.1359	−0.754091
$260$	0	0
$261$	0	0
$262$	− 16.6491i	− 1.02859i
$263$	2.51317	0.154969	0.0774843	−	0.996994i	$-0.475311\pi$
$263$	2.51317	0.154969	0.0774843	+	0.996994i	$0.475311\pi$
$264$	0	0
$265$	51.9737i	3.19272i
$266$	− 16.3246i	− 1.00092i
$267$	0	0
$268$	− 2.83772i	− 0.173341i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	− 6.32456i	− 0.384189i	−0.981376	−	0.192095i	$-0.938472\pi$
$271$	− 6.32456i	− 0.384189i	0.981376	−	0.192095i	$-0.0615281\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−3.00000	−0.181237
$275$	− 14.3246i	− 0.863803i
$276$	0	0
$277$	−10.8114	−0.649593	−0.324797	−	0.945784i	$-0.605296\pi$
$277$	−10.8114	−0.649593	−0.324797	+	0.945784i	$0.605296\pi$
$278$	− 6.32456i	− 0.379322i
$279$	0	0
$280$	13.1623i	0.786597i
$281$	0.675445i	0.0402937i	0.999797	+	0.0201468i	$0.00641337\pi$
$281$	0.675445i	0.0402937i	−0.999797	+	0.0201468i	$0.993587\pi$
$282$	0	0
$283$	2.83772	0.168685	0.0843425	−	0.996437i	$-0.473121\pi$
$283$	2.83772	0.168685	0.0843425	+	0.996437i	$0.473121\pi$
$284$	7.16228i	0.425003i
$285$	0	0
$286$	0	0
$287$	−9.48683	−0.559990
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−7.64911	−0.449171
$291$	0	0
$292$	1.00000i	0.0585206i
$293$	− 1.83772i	− 0.107361i	−0.998558	−	0.0536804i	$-0.982905\pi$
$293$	− 1.83772i	− 0.107361i	0.998558	−	0.0536804i	$-0.0170952\pi$
$294$	0	0
$295$	−9.67544	−0.563326
$296$	3.83772	0.223063
$297$	0	0
$298$	−0.486833	−0.0282015
$299$	0	0
$300$	0	0
$301$	− 28.9737i	− 1.67001i
$302$	−0.837722	−0.0482055
$303$	0	0
$304$	5.16228i	0.296077i
$305$	0.675445i	0.0386758i
$306$	0	0
$307$	11.4868i	0.655588i	0.944749	+	0.327794i	$0.106305\pi$
$307$	11.4868i	0.655588i	−0.944749	+	0.327794i	$0.893695\pi$
$308$	3.67544	0.209428
$309$	0	0
$310$	− 26.3246i	− 1.49513i
$311$	21.4868	1.21841	0.609203	−	0.793014i	$-0.291489\pi$
$311$	21.4868	1.21841	0.609203	+	0.793014i	$0.291489\pi$
$312$	0	0
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	− 10.4868i	− 0.591806i
$315$	0	0
$316$	4.00000	0.225018
$317$	12.4868i	0.701330i	0.936501	+	0.350665i	$0.114044\pi$
$317$	12.4868i	0.701330i	−0.936501	+	0.350665i	$0.885956\pi$
$318$	0	0
$319$	2.13594i	0.119590i
$320$	− 4.16228i	− 0.232678i
$321$	0	0
$322$	−22.6491	−1.26219
$323$	− 15.4868i	− 0.861710i
$324$	0	0
$325$	0	0
$326$	16.0000	0.886158
$327$	0	0
$328$	3.00000	0.165647
$329$	15.2982	0.843418
$330$	0	0
$331$	10.9737i	0.603167i	0.953440	+	0.301584i	$0.0975152\pi$
$331$	10.9737i	0.603167i	−0.953440	+	0.301584i	$0.902485\pi$
$332$	3.48683i	0.191365i
$333$	0	0
$334$	12.0000	0.656611
$335$	−11.8114	−0.645325
$336$	0	0
$337$	−11.0000	−0.599208	−0.299604	−	0.954064i	$-0.596855\pi$
$337$	−11.0000	−0.599208	−0.299604	+	0.954064i	$0.596855\pi$
$338$	0	0
$339$	0	0
$340$	12.4868i	0.677194i
$341$	−7.35089	−0.398073
$342$	0	0
$343$	12.6491i	0.682988i
$344$	9.16228i	0.493997i
$345$	0	0
$346$	22.6491i	1.21762i
$347$	15.4868	0.831377	0.415688	−	0.909507i	$-0.363541\pi$
$347$	15.4868	0.831377	0.415688	+	0.909507i	$0.363541\pi$
$348$	0	0
$349$	− 5.35089i	− 0.286427i	−0.989692	−	0.143213i	$-0.954257\pi$
$349$	− 5.35089i	− 0.286427i	0.989692	−	0.143213i	$-0.0457435\pi$
$350$	38.9737	2.08323
$351$	0	0
$352$	−1.16228	−0.0619496
$353$	29.3246i	1.56079i	0.625288	+	0.780394i	$0.284982\pi$
$353$	29.3246i	1.56079i	−0.625288	+	0.780394i	$0.715018\pi$
$354$	0	0
$355$	29.8114	1.58222
$356$	12.0000i	0.635999i
$357$	0	0
$358$	15.4868i	0.818505i
$359$	− 28.4605i	− 1.50209i	−0.660252	−	0.751044i	$-0.729551\pi$
$359$	− 28.4605i	− 1.50209i	0.660252	−	0.751044i	$-0.270449\pi$
$360$	0	0
$361$	−7.64911	−0.402585
$362$	− 3.83772i	− 0.201706i
$363$	0	0
$364$	0	0
$365$	4.16228	0.217864
$366$	0	0
$367$	−25.4868	−1.33040	−0.665201	−	0.746664i	$-0.731654\pi$
$367$	−25.4868	−1.33040	−0.665201	+	0.746664i	$0.731654\pi$
$368$	7.16228	0.373360
$369$	0	0
$370$	− 15.9737i	− 0.830431i
$371$	39.4868i	2.05005i
$372$	0	0
$373$	−16.4868	−0.853656	−0.426828	−	0.904333i	$-0.640369\pi$
$373$	−16.4868	−0.853656	−0.426828	+	0.904333i	$0.640369\pi$
$374$	3.48683	0.180300
$375$	0	0
$376$	−4.83772	−0.249486
$377$	0	0
$378$	0	0
$379$	− 17.6754i	− 0.907927i	−0.891020	−	0.453963i	$-0.850010\pi$
$379$	− 17.6754i	− 0.907927i	0.891020	−	0.453963i	$-0.149990\pi$
$380$	21.4868	1.10225
$381$	0	0
$382$	− 14.3246i	− 0.732908i
$383$	30.9737i	1.58268i	0.611376	+	0.791340i	$0.290616\pi$
$383$	30.9737i	1.58268i	−0.611376	+	0.791340i	$0.709384\pi$
$384$	0	0
$385$	− 15.2982i	− 0.779670i
$386$	19.9737	1.01663
$387$	0	0
$388$	− 4.00000i	− 0.203069i
$389$	12.4868	0.633108	0.316554	−	0.948575i	$-0.397474\pi$
$389$	12.4868	0.633108	0.316554	+	0.948575i	$0.397474\pi$
$390$	0	0
$391$	−21.4868	−1.08664
$392$	3.00000i	0.151523i
$393$	0	0
$394$	−18.9737	−0.955879
$395$	− 16.6491i	− 0.837708i
$396$	0	0
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	6.51317i	0.326476i
$399$	0	0
$400$	−12.3246	−0.616228
$401$	15.0000i	0.749064i	0.927214	+	0.374532i	$0.122197\pi$
$401$	15.0000i	0.749064i	−0.927214	+	0.374532i	$0.877803\pi$
$402$	0	0
$403$	0	0
$404$	−7.83772	−0.389941
$405$	0	0
$406$	−5.81139	−0.288414
$407$	−4.46050	−0.221099
$408$	0	0
$409$	− 14.6754i	− 0.725654i	−0.931857	−	0.362827i	$-0.881812\pi$
$409$	− 14.6754i	− 0.725654i	0.931857	−	0.362827i	$-0.118188\pi$
$410$	− 12.4868i	− 0.616681i
$411$	0	0
$412$	−15.8114	−0.778971
$413$	−7.35089	−0.361714
$414$	0	0
$415$	14.5132	0.712423
$416$	0	0
$417$	0	0
$418$	− 6.00000i	− 0.293470i
$419$	30.9737	1.51316	0.756581	−	0.653900i	$-0.226868\pi$
$419$	30.9737	1.51316	0.756581	+	0.653900i	$0.226868\pi$
$420$	0	0
$421$	23.8377i	1.16178i	0.813982	+	0.580890i	$0.197295\pi$
$421$	23.8377i	1.16178i	−0.813982	+	0.580890i	$0.802705\pi$
$422$	− 4.00000i	− 0.194717i
$423$	0	0
$424$	− 12.4868i	− 0.606414i
$425$	36.9737	1.79349
$426$	0	0
$427$	0.513167i	0.0248339i
$428$	−17.8114	−0.860946
$429$	0	0
$430$	38.1359	1.83908
$431$	− 9.48683i	− 0.456965i	−0.973548	−	0.228482i	$-0.926624\pi$
$431$	− 9.48683i	− 0.456965i	0.973548	−	0.228482i	$-0.0733763\pi$
$432$	0	0
$433$	9.32456	0.448110	0.224055	−	0.974577i	$-0.428071\pi$
$433$	9.32456	0.448110	0.224055	+	0.974577i	$0.428071\pi$
$434$	− 20.0000i	− 0.960031i
$435$	0	0
$436$	6.64911i	0.318435i
$437$	36.9737i	1.76869i
$438$	0	0
$439$	25.4868	1.21642	0.608210	−	0.793776i	$-0.291888\pi$
$439$	25.4868	1.21642	0.608210	+	0.793776i	$0.291888\pi$
$440$	4.83772i	0.230629i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	49.9473	2.36773
$446$	−1.67544	−0.0793346
$447$	0	0
$448$	− 3.16228i	− 0.149404i
$449$	− 7.35089i	− 0.346910i	−0.984842	−	0.173455i	$-0.944507\pi$
$449$	− 7.35089i	− 0.346910i	0.984842	−	0.173455i	$-0.0554931\pi$
$450$	0	0
$451$	−3.48683	−0.164189
$452$	6.67544	0.313987
$453$	0	0
$454$	15.4868	0.726833
$455$	0	0
$456$	0	0
$457$	3.32456i	0.155516i	0.996972	+	0.0777581i	$0.0247762\pi$
$457$	3.32456i	0.155516i	−0.996972	+	0.0777581i	$0.975224\pi$
$458$	−22.3246	−1.04316
$459$	0	0
$460$	− 29.8114i	− 1.38996i
$461$	18.4868i	0.861018i	0.902586	+	0.430509i	$0.141666\pi$
$461$	18.4868i	0.861018i	−0.902586	+	0.430509i	$0.858334\pi$
$462$	0	0
$463$	15.1623i	0.704651i	0.935878	+	0.352325i	$0.114609\pi$
$463$	15.1623i	0.704651i	−0.935878	+	0.352325i	$0.885391\pi$
$464$	1.83772	0.0853141
$465$	0	0
$466$	− 16.6491i	− 0.771255i
$467$	6.18861	0.286375	0.143187	−	0.989696i	$-0.454265\pi$
$467$	6.18861	0.286375	0.143187	+	0.989696i	$0.454265\pi$
$468$	0	0
$469$	−8.97367	−0.414365
$470$	20.1359i	0.928802i
$471$	0	0
$472$	2.32456	0.106996
$473$	− 10.6491i	− 0.489647i
$474$	0	0
$475$	− 63.6228i	− 2.91921i
$476$	9.48683i	0.434828i
$477$	0	0
$478$	21.4868	0.982785
$479$	24.0000i	1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$	24.0000i	1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$	0	0
$481$	0	0
$482$	−13.3246	−0.606917
$483$	0	0
$484$	−9.64911	−0.438596
$485$	−16.6491	−0.755997
$486$	0	0
$487$	13.4868i	0.611147i	0.952169	+	0.305573i	$0.0988481\pi$
$487$	13.4868i	0.611147i	−0.952169	+	0.305573i	$0.901152\pi$
$488$	− 0.162278i	− 0.00734596i
$489$	0	0
$490$	12.4868	0.564098
$491$	5.81139	0.262264	0.131132	−	0.991365i	$-0.458139\pi$
$491$	5.81139	0.262264	0.131132	+	0.991365i	$0.458139\pi$
$492$	0	0
$493$	−5.51317	−0.248301
$494$	0	0
$495$	0	0
$496$	6.32456i	0.283981i
$497$	22.6491	1.01595
$498$	0	0
$499$	− 12.6491i	− 0.566252i	−0.959083	−	0.283126i	$-0.908629\pi$
$499$	− 12.6491i	− 0.566252i	0.959083	−	0.283126i	$-0.0913715\pi$
$500$	30.4868i	1.36341i
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	0.188612	0.00840978	0.00420489	−	0.999991i	$-0.498662\pi$
$503$	0.188612	0.00840978	0.00420489	+	0.999991i	$0.498662\pi$
$504$	0	0
$505$	32.6228i	1.45169i
$506$	−8.32456	−0.370072
$507$	0	0
$508$	−18.3246	−0.813021
$509$	− 28.1623i	− 1.24827i	−0.781316	−	0.624136i	$-0.785451\pi$
$509$	− 28.1623i	− 1.24827i	0.781316	−	0.624136i	$-0.214549\pi$
$510$	0	0
$511$	3.16228	0.139891
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 21.9737i	− 0.969217i
$515$	65.8114i	2.90000i
$516$	0	0
$517$	5.62278	0.247289
$518$	− 12.1359i	− 0.533223i
$519$	0	0
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	−5.16228	−0.225731	−0.112865	−	0.993610i	$-0.536003\pi$
$523$	−5.16228	−0.225731	−0.112865	+	0.993610i	$0.536003\pi$
$524$	16.6491	0.727320
$525$	0	0
$526$	2.51317i	0.109579i
$527$	− 18.9737i	− 0.826506i
$528$	0	0
$529$	28.2982	1.23036
$530$	−51.9737	−2.25759
$531$	0	0
$532$	16.3246	0.707759
$533$	0	0
$534$	0	0
$535$	74.1359i	3.20518i
$536$	2.83772	0.122571
$537$	0	0
$538$	− 6.00000i	− 0.258678i
$539$	− 3.48683i	− 0.150189i
$540$	0	0
$541$	− 15.5132i	− 0.666963i	−0.942757	−	0.333482i	$-0.891777\pi$
$541$	− 15.5132i	− 0.666963i	0.942757	−	0.333482i	$-0.108223\pi$
$542$	6.32456	0.271663
$543$	0	0
$544$	− 3.00000i	− 0.128624i
$545$	27.6754	1.18549
$546$	0	0
$547$	−33.8114	−1.44567	−0.722835	−	0.691020i	$-0.757161\pi$
$547$	−33.8114	−1.44567	−0.722835	+	0.691020i	$0.757161\pi$
$548$	− 3.00000i	− 0.128154i
$549$	0	0
$550$	14.3246	0.610801
$551$	9.48683i	0.404153i
$552$	0	0
$553$	− 12.6491i	− 0.537895i
$554$	− 10.8114i	− 0.459332i
$555$	0	0
$556$	6.32456	0.268221
$557$	18.4868i	0.783312i	0.920112	+	0.391656i	$0.128098\pi$
$557$	18.4868i	0.783312i	−0.920112	+	0.391656i	$0.871902\pi$
$558$	0	0
$559$	0	0
$560$	−13.1623	−0.556208
$561$	0	0
$562$	−0.675445	−0.0284919
$563$	40.6491	1.71316	0.856578	−	0.516018i	$-0.172586\pi$
$563$	40.6491	1.71316	0.856578	+	0.516018i	$0.172586\pi$
$564$	0	0
$565$	− 27.7851i	− 1.16893i
$566$	2.83772i	0.119278i
$567$	0	0
$568$	−7.16228	−0.300522
$569$	27.2982	1.14440	0.572200	−	0.820114i	$-0.306090\pi$
$569$	27.2982	1.14440	0.572200	+	0.820114i	$0.306090\pi$
$570$	0	0
$571$	36.1359	1.51224	0.756121	−	0.654432i	$-0.227092\pi$
$571$	36.1359	1.51224	0.756121	+	0.654432i	$0.227092\pi$
$572$	0	0
$573$	0	0
$574$	− 9.48683i	− 0.395973i
$575$	−88.2719	−3.68119
$576$	0	0
$577$	− 0.350889i	− 0.0146077i	−0.999973	−	0.00730386i	$-0.997675\pi$
$577$	− 0.350889i	− 0.0146077i	0.999973	−	0.00730386i	$-0.00232491\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	− 7.64911i	− 0.317612i
$581$	11.0263	0.457449
$582$	0	0
$583$	14.5132i	0.601074i
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	1.83772	0.0759156
$587$	28.6491i	1.18248i	0.806497	+	0.591238i	$0.201360\pi$
$587$	28.6491i	1.18248i	−0.806497	+	0.591238i	$0.798640\pi$
$588$	0	0
$589$	−32.6491	−1.34528
$590$	− 9.67544i	− 0.398332i
$591$	0	0
$592$	3.83772i	0.157729i
$593$	− 0.675445i	− 0.0277372i	−0.999904	−	0.0138686i	$-0.995585\pi$
$593$	− 0.675445i	− 0.0277372i	0.999904	−	0.0138686i	$-0.00441465\pi$
$594$	0	0
$595$	39.4868	1.61880
$596$	− 0.486833i	− 0.0199415i
$597$	0	0
$598$	0	0
$599$	23.6228	0.965200	0.482600	−	0.875841i	$-0.339692\pi$
$599$	23.6228	0.965200	0.482600	+	0.875841i	$0.339692\pi$
$600$	0	0
$601$	−34.2982	−1.39905	−0.699527	−	0.714606i	$-0.746606\pi$
$601$	−34.2982	−1.39905	−0.699527	+	0.714606i	$0.746606\pi$
$602$	28.9737	1.18088
$603$	0	0
$604$	− 0.837722i	− 0.0340864i
$605$	40.1623i	1.63283i
$606$	0	0
$607$	17.2982	0.702113	0.351057	−	0.936354i	$-0.385822\pi$
$607$	17.2982	0.702113	0.351057	+	0.936354i	$0.385822\pi$
$608$	−5.16228	−0.209358
$609$	0	0
$610$	−0.675445	−0.0273480
$611$	0	0
$612$	0	0
$613$	− 20.4868i	− 0.827455i	−0.910401	−	0.413728i	$-0.864227\pi$
$613$	− 20.4868i	− 0.827455i	0.910401	−	0.413728i	$-0.135773\pi$
$614$	−11.4868	−0.463571
$615$	0	0
$616$	3.67544i	0.148088i
$617$	− 26.6228i	− 1.07179i	−0.844284	−	0.535896i	$-0.819974\pi$
$617$	− 26.6228i	− 1.07179i	0.844284	−	0.535896i	$-0.180026\pi$
$618$	0	0
$619$	24.6491i	0.990731i	0.868685	+	0.495366i	$0.164966\pi$
$619$	24.6491i	0.990731i	−0.868685	+	0.495366i	$0.835034\pi$
$620$	26.3246	1.05722
$621$	0	0
$622$	21.4868i	0.861544i
$623$	37.9473	1.52033
$624$	0	0
$625$	65.2719	2.61088
$626$	− 4.00000i	− 0.159872i
$627$	0	0
$628$	10.4868	0.418470
$629$	− 11.5132i	− 0.459060i
$630$	0	0
$631$	− 25.2982i	− 1.00711i	−0.863964	−	0.503553i	$-0.832026\pi$
$631$	− 25.2982i	− 1.00711i	0.863964	−	0.503553i	$-0.167974\pi$
$632$	4.00000i	0.159111i
$633$	0	0
$634$	−12.4868	−0.495915
$635$	76.2719i	3.02676i
$636$	0	0
$637$	0	0
$638$	−2.13594	−0.0845628
$639$	0	0
$640$	4.16228	0.164528
$641$	16.3509	0.645821	0.322911	−	0.946429i	$-0.395339\pi$
$641$	16.3509	0.645821	0.322911	+	0.946429i	$0.395339\pi$
$642$	0	0
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	− 22.6491i	− 0.892500i
$645$	0	0
$646$	15.4868	0.609321
$647$	−40.6491	−1.59808	−0.799041	−	0.601277i	$-0.794659\pi$
$647$	−40.6491	−1.59808	−0.799041	+	0.601277i	$0.794659\pi$
$648$	0	0
$649$	−2.70178	−0.106054
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	− 69.2982i	− 2.70771i
$656$	3.00000i	0.117130i
$657$	0	0
$658$	15.2982i	0.596387i
$659$	5.02633	0.195798	0.0978991	−	0.995196i	$-0.468788\pi$
$659$	5.02633	0.195798	0.0978991	+	0.995196i	$0.468788\pi$
$660$	0	0
$661$	26.4868i	1.03022i	0.857125	+	0.515109i	$0.172249\pi$
$661$	26.4868i	1.03022i	−0.857125	+	0.515109i	$0.827751\pi$
$662$	−10.9737	−0.426504
$663$	0	0
$664$	−3.48683	−0.135315
$665$	− 67.9473i	− 2.63488i
$666$	0	0
$667$	13.1623	0.509645
$668$	12.0000i	0.464294i
$669$	0	0
$670$	− 11.8114i	− 0.456314i
$671$	0.188612i	0.00728127i
$672$	0	0
$673$	−14.6754	−0.565697	−0.282848	−	0.959165i	$-0.591279\pi$
$673$	−14.6754	−0.565697	−0.282848	+	0.959165i	$0.591279\pi$
$674$	− 11.0000i	− 0.423704i
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−12.6491	−0.485428
$680$	−12.4868	−0.478848
$681$	0	0
$682$	− 7.35089i	− 0.281480i
$683$	35.6228i	1.36307i	0.731787	+	0.681534i	$0.238687\pi$
$683$	35.6228i	1.36307i	−0.731787	+	0.681534i	$0.761313\pi$
$684$	0	0
$685$	−12.4868	−0.477097
$686$	−12.6491	−0.482945
$687$	0	0
$688$	−9.16228	−0.349309
$689$	0	0
$690$	0	0
$691$	− 9.16228i	− 0.348549i	−0.984697	−	0.174275i	$-0.944242\pi$
$691$	− 9.16228i	− 0.348549i	0.984697	−	0.174275i	$-0.0557581\pi$
$692$	−22.6491	−0.860990
$693$	0	0
$694$	15.4868i	0.587872i
$695$	− 26.3246i	− 0.998547i
$696$	0	0
$697$	− 9.00000i	− 0.340899i
$698$	5.35089	0.202534
$699$	0	0
$700$	38.9737i	1.47307i
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−19.8114	−0.747201
$704$	− 1.16228i	− 0.0438050i
$705$	0	0
$706$	−29.3246	−1.10364
$707$	24.7851i	0.932138i
$708$	0	0
$709$	45.4605i	1.70730i	0.520843	+	0.853652i	$0.325618\pi$
$709$	45.4605i	1.70730i	−0.520843	+	0.853652i	$0.674382\pi$
$710$	29.8114i	1.11880i
$711$	0	0
$712$	−12.0000	−0.449719
$713$	45.2982i	1.69643i
$714$	0	0
$715$	0	0
$716$	−15.4868	−0.578770
$717$	0	0
$718$	28.4605	1.06214
$719$	−38.3246	−1.42926	−0.714632	−	0.699500i	$-0.753406\pi$
$719$	−38.3246	−1.42926	−0.714632	+	0.699500i	$0.753406\pi$
$720$	0	0
$721$	50.0000i	1.86210i
$722$	− 7.64911i	− 0.284670i
$723$	0	0
$724$	3.83772	0.142628
$725$	−22.6491	−0.841167
$726$	0	0
$727$	8.83772	0.327773	0.163886	−	0.986479i	$-0.447597\pi$
$727$	8.83772	0.327773	0.163886	+	0.986479i	$0.447597\pi$
$728$	0	0
$729$	0	0
$730$	4.16228i	0.154053i
$731$	27.4868	1.01664
$732$	0	0
$733$	− 21.4605i	− 0.792662i	−0.918108	−	0.396331i	$-0.870283\pi$
$733$	− 21.4605i	− 0.792662i	0.918108	−	0.396331i	$-0.129717\pi$
$734$	− 25.4868i	− 0.940736i
$735$	0	0
$736$	7.16228i	0.264005i
$737$	−3.29822	−0.121492
$738$	0	0
$739$	− 39.6228i	− 1.45755i	−0.684755	−	0.728774i	$-0.740091\pi$
$739$	− 39.6228i	− 1.45755i	0.684755	−	0.728774i	$-0.259909\pi$
$740$	15.9737	0.587204
$741$	0	0
$742$	−39.4868	−1.44961
$743$	2.32456i	0.0852797i	0.999091	+	0.0426398i	$0.0135768\pi$
$743$	2.32456i	0.0852797i	−0.999091	+	0.0426398i	$0.986423\pi$
$744$	0	0
$745$	−2.02633	−0.0742391
$746$	− 16.4868i	− 0.603626i
$747$	0	0
$748$	3.48683i	0.127491i
$749$	56.3246i	2.05805i
$750$	0	0
$751$	−38.7851	−1.41529	−0.707643	−	0.706570i	$-0.750242\pi$
$751$	−38.7851	−1.41529	−0.707643	+	0.706570i	$0.750242\pi$
$752$	− 4.83772i	− 0.176414i
$753$	0	0
$754$	0	0
$755$	−3.48683	−0.126899
$756$	0	0
$757$	−26.6491	−0.968578	−0.484289	−	0.874908i	$-0.660922\pi$
$757$	−26.6491	−0.968578	−0.484289	+	0.874908i	$0.660922\pi$
$758$	17.6754	0.642001
$759$	0	0
$760$	21.4868i	0.779409i
$761$	12.0000i	0.435000i	0.976060	+	0.217500i	$0.0697902\pi$
$761$	12.0000i	0.435000i	−0.976060	+	0.217500i	$0.930210\pi$
$762$	0	0
$763$	21.0263	0.761204
$764$	14.3246	0.518244
$765$	0	0
$766$	−30.9737	−1.11912
$767$	0	0
$768$	0	0
$769$	− 3.35089i	− 0.120836i	−0.998173	−	0.0604181i	$-0.980757\pi$
$769$	− 3.35089i	− 0.120836i	0.998173	−	0.0604181i	$-0.0192434\pi$
$770$	15.2982	0.551310
$771$	0	0
$772$	19.9737i	0.718868i
$773$	22.6491i	0.814632i	0.913287	+	0.407316i	$0.133535\pi$
$773$	22.6491i	0.814632i	−0.913287	+	0.407316i	$0.866465\pi$
$774$	0	0
$775$	− 77.9473i	− 2.79995i
$776$	4.00000	0.143592
$777$	0	0
$778$	12.4868i	0.447675i
$779$	−15.4868	−0.554873
$780$	0	0
$781$	8.32456	0.297876
$782$	− 21.4868i	− 0.768368i
$783$	0	0
$784$	−3.00000	−0.107143
$785$	− 43.6491i	− 1.55790i
$786$	0	0
$787$	− 9.02633i	− 0.321754i	−0.986974	−	0.160877i	$-0.948568\pi$
$787$	− 9.02633i	− 0.321754i	0.986974	−	0.160877i	$-0.0514323\pi$
$788$	− 18.9737i	− 0.675909i
$789$	0	0
$790$	16.6491	0.592349
$791$	− 21.1096i	− 0.750571i
$792$	0	0
$793$	0	0
$794$	−26.0000	−0.922705
$795$	0	0
$796$	−6.51317	−0.230853
$797$	−18.9737	−0.672082	−0.336041	−	0.941847i	$-0.609088\pi$
$797$	−18.9737	−0.672082	−0.336041	+	0.941847i	$0.609088\pi$
$798$	0	0
$799$	14.5132i	0.513439i
$800$	− 12.3246i	− 0.435739i
$801$	0	0
$802$	−15.0000	−0.529668
$803$	1.16228	0.0410159
$804$	0	0
$805$	−94.2719	−3.32265
$806$	0	0
$807$	0	0
$808$	− 7.83772i	− 0.275730i
$809$	−3.00000	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$810$	0	0
$811$	− 1.02633i	− 0.0360395i	−0.999838	−	0.0180197i	$-0.994264\pi$
$811$	− 1.02633i	− 0.0360395i	0.999838	−	0.0180197i	$-0.00573617\pi$
$812$	− 5.81139i	− 0.203940i
$813$	0	0
$814$	− 4.46050i	− 0.156340i
$815$	66.5964	2.33277
$816$	0	0
$817$	− 47.2982i	− 1.65476i
$818$	14.6754	0.513115
$819$	0	0
$820$	12.4868	0.436059
$821$	10.6491i	0.371657i	0.982582	+	0.185828i	$0.0594968\pi$
$821$	10.6491i	0.371657i	−0.982582	+	0.185828i	$0.940503\pi$
$822$	0	0
$823$	−53.2982	−1.85786	−0.928930	−	0.370256i	$-0.879270\pi$
$823$	−53.2982	−1.85786	−0.928930	+	0.370256i	$0.879270\pi$
$824$	− 15.8114i	− 0.550816i
$825$	0	0
$826$	− 7.35089i	− 0.255770i
$827$	6.97367i	0.242498i	0.992622	+	0.121249i	$0.0386900\pi$
$827$	6.97367i	0.242498i	−0.992622	+	0.121249i	$0.961310\pi$
$828$	0	0
$829$	−12.1623	−0.422413	−0.211207	−	0.977441i	$-0.567739\pi$
$829$	−12.1623	−0.422413	−0.211207	+	0.977441i	$0.567739\pi$
$830$	14.5132i	0.503759i
$831$	0	0
$832$	0	0
$833$	9.00000	0.311832
$834$	0	0
$835$	49.9473	1.72850
$836$	6.00000	0.207514
$837$	0	0
$838$	30.9737i	1.06997i
$839$	− 4.64911i	− 0.160505i	−0.996775	−	0.0802526i	$-0.974427\pi$
$839$	− 4.64911i	− 0.160505i	0.996775	−	0.0802526i	$-0.0255727\pi$
$840$	0	0
$841$	−25.6228	−0.883544
$842$	−23.8377	−0.821502
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	30.5132i	1.04844i
$848$	12.4868	0.428800
$849$	0	0
$850$	36.9737i	1.26819i
$851$	27.4868i	0.942236i
$852$	0	0
$853$	55.1359i	1.88782i	0.330205	+	0.943909i	$0.392882\pi$
$853$	55.1359i	1.88782i	−0.330205	+	0.943909i	$0.607118\pi$
$854$	−0.513167	−0.0175602
$855$	0	0
$856$	− 17.8114i	− 0.608781i
$857$	−25.6491	−0.876157	−0.438078	−	0.898937i	$-0.644341\pi$
$857$	−25.6491	−0.876157	−0.438078	+	0.898937i	$0.644341\pi$
$858$	0	0
$859$	28.5132	0.972857	0.486428	−	0.873720i	$-0.338299\pi$
$859$	28.5132	0.972857	0.486428	+	0.873720i	$0.338299\pi$
$860$	38.1359i	1.30042i
$861$	0	0
$862$	9.48683	0.323123
$863$	47.8114i	1.62752i	0.581202	+	0.813759i	$0.302583\pi$
$863$	47.8114i	1.62752i	−0.581202	+	0.813759i	$0.697417\pi$
$864$	0	0
$865$	94.2719i	3.20534i
$866$	9.32456i	0.316861i
$867$	0	0
$868$	20.0000	0.678844
$869$	− 4.64911i	− 0.157710i
$870$	0	0
$871$	0	0
$872$	−6.64911	−0.225167
$873$	0	0
$874$	−36.9737	−1.25065
$875$	96.4078	3.25918
$876$	0	0
$877$	45.1359i	1.52413i	0.647499	+	0.762066i	$0.275815\pi$
$877$	45.1359i	1.52413i	−0.647499	+	0.762066i	$0.724185\pi$
$878$	25.4868i	0.860139i
$879$	0	0
$880$	−4.83772	−0.163080
$881$	9.97367	0.336021	0.168011	−	0.985785i	$-0.446266\pi$
$881$	9.97367	0.336021	0.168011	+	0.985785i	$0.446266\pi$
$882$	0	0
$883$	11.3509	0.381988	0.190994	−	0.981591i	$-0.438829\pi$
$883$	11.3509	0.381988	0.190994	+	0.981591i	$0.438829\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38.3246	−1.28681	−0.643406	−	0.765525i	$-0.722479\pi$
$887$	−38.3246	−1.28681	−0.643406	+	0.765525i	$0.722479\pi$
$888$	0	0
$889$	57.9473i	1.94349i
$890$	49.9473i	1.67424i
$891$	0	0
$892$	− 1.67544i	− 0.0560980i
$893$	24.9737	0.835712
$894$	0	0
$895$	64.4605i	2.15468i
$896$	3.16228	0.105644
$897$	0	0
$898$	7.35089	0.245302
$899$	11.6228i	0.387641i
$900$	0	0
$901$	−37.4605	−1.24799
$902$	− 3.48683i	− 0.116099i
$903$	0	0
$904$	6.67544i	0.222022i
$905$	− 15.9737i	− 0.530983i
$906$	0	0
$907$	−36.2719	−1.20439	−0.602194	−	0.798350i	$-0.705707\pi$
$907$	−36.2719	−1.20439	−0.602194	+	0.798350i	$0.705707\pi$
$908$	15.4868i	0.513949i
$909$	0	0
$910$	0	0
$911$	49.9473	1.65483	0.827414	−	0.561592i	$-0.189811\pi$
$911$	49.9473	1.65483	0.827414	+	0.561592i	$0.189811\pi$
$912$	0	0
$913$	4.05267	0.134124
$914$	−3.32456	−0.109967
$915$	0	0
$916$	− 22.3246i	− 0.737624i
$917$	− 52.6491i	− 1.73863i
$918$	0	0
$919$	3.35089	0.110536	0.0552678	−	0.998472i	$-0.482399\pi$
$919$	3.35089	0.110536	0.0552678	+	0.998472i	$0.482399\pi$
$920$	29.8114	0.982852
$921$	0	0
$922$	−18.4868	−0.608831
$923$	0	0
$924$	0	0
$925$	− 47.2982i	− 1.55516i
$926$	−15.1623	−0.498263
$927$	0	0
$928$	1.83772i	0.0603262i
$929$	28.9473i	0.949731i	0.880058	+	0.474866i	$0.157503\pi$
$929$	28.9473i	0.949731i	−0.880058	+	0.474866i	$0.842497\pi$
$930$	0	0
$931$	− 15.4868i	− 0.507560i
$932$	16.6491	0.545360
$933$	0	0
$934$	6.18861i	0.202498i
$935$	14.5132	0.474631
$936$	0	0
$937$	14.2982	0.467103	0.233551	−	0.972344i	$-0.424965\pi$
$937$	14.2982	0.467103	0.233551	+	0.972344i	$0.424965\pi$
$938$	− 8.97367i	− 0.293001i
$939$	0	0
$940$	−20.1359	−0.656762
$941$	− 48.5964i	− 1.58420i	−0.610392	−	0.792099i	$-0.708988\pi$
$941$	− 48.5964i	− 1.58420i	0.610392	−	0.792099i	$-0.291012\pi$
$942$	0	0
$943$	21.4868i	0.699708i
$944$	2.32456i	0.0756578i
$945$	0	0
$946$	10.6491	0.346232
$947$	− 18.9737i	− 0.616561i	−0.951295	−	0.308281i	$-0.900246\pi$
$947$	− 18.9737i	− 0.616561i	0.951295	−	0.308281i	$-0.0997536\pi$
$948$	0	0
$949$	0	0
$950$	63.6228	2.06420
$951$	0	0
$952$	−9.48683	−0.307470
$953$	−27.2982	−0.884276	−0.442138	−	0.896947i	$-0.645780\pi$
$953$	−27.2982	−0.884276	−0.442138	+	0.896947i	$0.645780\pi$
$954$	0	0
$955$	− 59.6228i	− 1.92935i
$956$	21.4868i	0.694934i
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−9.48683	−0.306346
$960$	0	0
$961$	−9.00000	−0.290323
$962$	0	0
$963$	0	0
$964$	− 13.3246i	− 0.429155i
$965$	83.1359	2.67624
$966$	0	0
$967$	− 34.5132i	− 1.10987i	−0.831894	−	0.554934i	$-0.812743\pi$
$967$	− 34.5132i	− 1.10987i	0.831894	−	0.554934i	$-0.187257\pi$
$968$	− 9.64911i	− 0.310134i
$969$	0	0
$970$	− 16.6491i	− 0.534571i
$971$	18.9737	0.608894	0.304447	−	0.952529i	$-0.401528\pi$
$971$	18.9737	0.608894	0.304447	+	0.952529i	$0.401528\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	−13.4868	−0.432146
$975$	0	0
$976$	0.162278	0.00519438
$977$	21.0000i	0.671850i	0.941889	+	0.335925i	$0.109049\pi$
$977$	21.0000i	0.671850i	−0.941889	+	0.335925i	$0.890951\pi$
$978$	0	0
$979$	13.9473	0.445759
$980$	12.4868i	0.398877i
$981$	0	0
$982$	5.81139i	0.185449i
$983$	− 23.6228i	− 0.753450i	−0.926325	−	0.376725i	$-0.877050\pi$
$983$	− 23.6228i	− 0.753450i	0.926325	−	0.376725i	$-0.122950\pi$
$984$	0	0
$985$	−78.9737	−2.51631
$986$	− 5.51317i	− 0.175575i
$987$	0	0
$988$	0	0
$989$	−65.6228	−2.08668
$990$	0	0
$991$	26.7851	0.850855	0.425428	−	0.904992i	$-0.360124\pi$
$991$	26.7851	0.850855	0.425428	+	0.904992i	$0.360124\pi$
$992$	−6.32456	−0.200805
$993$	0	0
$994$	22.6491i	0.718386i
$995$	27.1096i	0.859432i
$996$	0	0
$997$	−59.4605	−1.88313	−0.941566	−	0.336827i	$-0.890646\pi$
$997$	−59.4605	−1.88313	−0.941566	+	0.336827i	$0.890646\pi$
$998$	12.6491	0.400401
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.j.1351.4		4
3.2	odd	2		3042.2.b.k.1351.1		4
13.5	odd	4		3042.2.a.x.1.2		2
13.7	odd	12		234.2.h.e.55.1	yes	4
13.8	odd	4		3042.2.a.r.1.1		2
13.11	odd	12		234.2.h.e.217.1	yes	4
13.12	even	2	inner	3042.2.b.j.1351.1		4
39.5	even	4		3042.2.a.q.1.1		2
39.8	even	4		3042.2.a.w.1.2		2
39.11	even	12		234.2.h.d.217.2	yes	4
39.20	even	12		234.2.h.d.55.2	✓	4
39.38	odd	2		3042.2.b.k.1351.4		4
52.7	even	12		1872.2.t.n.289.1		4
52.11	even	12		1872.2.t.n.1153.1		4
156.11	odd	12		1872.2.t.p.1153.2		4
156.59	odd	12		1872.2.t.p.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.h.d.55.2	✓	4	39.20	even	12
234.2.h.d.217.2	yes	4	39.11	even	12
234.2.h.e.55.1	yes	4	13.7	odd	12
234.2.h.e.217.1	yes	4	13.11	odd	12
1872.2.t.n.289.1		4	52.7	even	12
1872.2.t.n.1153.1		4	52.11	even	12
1872.2.t.p.289.2		4	156.59	odd	12
1872.2.t.p.1153.2		4	156.11	odd	12
3042.2.a.q.1.1		2	39.5	even	4
3042.2.a.r.1.1		2	13.8	odd	4
3042.2.a.w.1.2		2	39.8	even	4
3042.2.a.x.1.2		2	13.5	odd	4
3042.2.b.j.1351.1		4	13.12	even	2	inner
3042.2.b.j.1351.4		4	1.1	even	1	trivial
3042.2.b.k.1351.1		4	3.2	odd	2
3042.2.b.k.1351.4		4	39.38	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +4.16228i q^{5} +3.16228i q^{7} -1.00000i q^{8} -4.16228 q^{10} +1.16228i q^{11} -3.16228 q^{14} +1.00000 q^{16} -3.00000 q^{17} +5.16228i q^{19} -4.16228i q^{20} -1.16228 q^{22} +7.16228 q^{23} -12.3246 q^{25} -3.16228i q^{28} +1.83772 q^{29} +6.32456i q^{31} +1.00000i q^{32} -3.00000i q^{34} -13.1623 q^{35} +3.83772i q^{37} -5.16228 q^{38} +4.16228 q^{40} +3.00000i q^{41} -9.16228 q^{43} -1.16228i q^{44} +7.16228i q^{46} -4.83772i q^{47} -3.00000 q^{49} -12.3246i q^{50} +12.4868 q^{53} -4.83772 q^{55} +3.16228 q^{56} +1.83772i q^{58} +2.32456i q^{59} +0.162278 q^{61} -6.32456 q^{62} -1.00000 q^{64} +2.83772i q^{67} +3.00000 q^{68} -13.1623i q^{70} -7.16228i q^{71} -1.00000i q^{73} -3.83772 q^{74} -5.16228i q^{76} -3.67544 q^{77} -4.00000 q^{79} +4.16228i q^{80} -3.00000 q^{82} -3.48683i q^{83} -12.4868i q^{85} -9.16228i q^{86} +1.16228 q^{88} -12.0000i q^{89} -7.16228 q^{92} +4.83772 q^{94} -21.4868 q^{95} +4.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{10} + 4 q^{16} - 12 q^{17} + 8 q^{22} + 16 q^{23} - 24 q^{25} + 20 q^{29} - 40 q^{35} - 8 q^{38} + 4 q^{40} - 24 q^{43} - 12 q^{49} + 12 q^{53} - 32 q^{55} - 12 q^{61} - 4 q^{64} + 12 q^{68}+ \cdots - 48 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^10 + 4 * q^16 - 12 * q^17 + 8 * q^22 + 16 * q^23 - 24 * q^25 + 20 * q^29 - 40 * q^35 - 8 * q^38 + 4 * q^40 - 24 * q^43 - 12 * q^49 + 12 * q^53 - 32 * q^55 - 12 * q^61 - 4 * q^64 + 12 * q^68 - 28 * q^74 - 40 * q^77 - 16 * q^79 - 12 * q^82 - 8 * q^88 - 16 * q^92 + 32 * q^94 - 48 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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