LMFDB - Embedded newform 1368.2.s.j.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1368,2,Mod(505,1368)] 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("1368.505"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1368, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.s (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{18})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			577.1
Root			$-0.766044 + 0.642788i$ of defining polynomial
Character	$\chi$	$=$	1368.577
Dual form			1368.2.s.j.505.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.87939 - 3.25519i) q^{5} +4.75877 q^{7} +4.36959 q^{11} +(-0.500000 + 0.866025i) q^{13} +(3.06418 + 5.30731i) q^{17} +(0.694593 - 4.30320i) q^{19} +(-1.87939 + 3.25519i) q^{23} +(-4.56418 + 7.90539i) q^{25} +(2.69459 - 4.66717i) q^{29} +7.36959 q^{31} +(-8.94356 - 15.4907i) q^{35} -7.12836 q^{37} +(-1.75877 - 3.04628i) q^{41} +(0.379385 + 0.657115i) q^{43} +(-3.00000 + 5.19615i) q^{47} +15.6459 q^{49} +(2.57398 - 4.45826i) q^{53} +(-8.21213 - 14.2238i) q^{55} +(4.63816 + 8.03352i) q^{59} +(4.25877 - 7.37641i) q^{61} +3.75877 q^{65} +(-0.684793 + 1.18610i) q^{67} +(-5.82295 - 10.0856i) q^{71} +(-2.25877 - 3.91231i) q^{73} +20.7939 q^{77} +(-7.07398 - 12.2525i) q^{79} -4.90673 q^{83} +(11.5175 - 19.9490i) q^{85} +(4.18479 - 7.24827i) q^{89} +(-2.37939 + 4.12122i) q^{91} +(-15.3131 + 5.82634i) q^{95} +(5.82295 + 10.0856i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{7} + 12 q^{11} - 3 q^{13} - 9 q^{25} + 12 q^{29} + 30 q^{31} - 24 q^{35} - 6 q^{37} + 12 q^{41} - 9 q^{43} - 18 q^{47} + 12 q^{49} - 6 q^{59} + 3 q^{61} + 3 q^{67} + 6 q^{71} + 9 q^{73} + 12 q^{77}+ \cdots - 6 q^{97}+O(q^{100}) $ 6 * q + 6 * q^7 + 12 * q^11 - 3 * q^13 - 9 * q^25 + 12 * q^29 + 30 * q^31 - 24 * q^35 - 6 * q^37 + 12 * q^41 - 9 * q^43 - 18 * q^47 + 12 * q^49 - 6 * q^59 + 3 * q^61 + 3 * q^67 + 6 * q^71 + 9 * q^73 + 12 * q^77 - 27 * q^79 + 24 * q^83 + 24 * q^85 + 18 * q^89 - 3 * q^91 - 48 * q^95 - 6 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.87939	−	3.25519i	−0.840487	−	1.45577i	−0.889484	−	0.456967i	$-0.848936\pi$
$5$	−1.87939	−	3.25519i	−0.840487	−	1.45577i	0.0489972	−	0.998799i	$-0.484397\pi$
$6$		0			0
$7$	4.75877			1.79865			0.899323	−	0.437285i	$-0.144060\pi$
$7$	4.75877			1.79865			0.899323	+	0.437285i	$0.144060\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.36959			1.31748			0.658740	−	0.752371i	$-0.271090\pi$
$11$	4.36959			1.31748			0.658740	+	0.752371i	$0.271090\pi$
$12$		0			0
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	−0.926995	−	0.375073i	$-0.877618\pi$
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	0.788320	+	0.615265i	$0.210951\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.06418	+	5.30731i	0.743172	+	1.28721i	0.951044	+	0.309056i	$0.100013\pi$
$17$	3.06418	+	5.30731i	0.743172	+	1.28721i	−0.207872	+	0.978156i	$0.566654\pi$
$18$		0			0
$19$	0.694593	−	4.30320i	0.159350	−	0.987222i
$19$	0.694593	−	4.30320i	0.159350	−	0.987222i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.87939	+	3.25519i	−0.391879	+	0.678754i	−0.992697	−	0.120631i	$-0.961508\pi$
$23$	−1.87939	+	3.25519i	−0.391879	+	0.678754i	0.600818	+	0.799385i	$0.294841\pi$
$24$		0			0
$25$	−4.56418	+	7.90539i	−0.912836	+	1.58108i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.69459	−	4.66717i	0.500373	−	0.866672i	−0.499627	−	0.866241i	$-0.666529\pi$
$29$	2.69459	−	4.66717i	0.500373	−	0.866672i	1.00000		0.000431109i	$-0.000137226\pi$
$30$		0			0
$31$	7.36959			1.32362			0.661808	−	0.749673i	$-0.269789\pi$
$31$	7.36959			1.32362			0.661808	+	0.749673i	$0.269789\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.94356	−	15.4907i	−1.51174	−	2.61841i
$36$		0			0
$37$	−7.12836			−1.17189			−0.585947	−	0.810349i	$-0.699277\pi$
$37$	−7.12836			−1.17189			−0.585947	+	0.810349i	$0.699277\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.75877	−	3.04628i	−0.274674	−	0.475749i	0.695379	−	0.718643i	$-0.255237\pi$
$41$	−1.75877	−	3.04628i	−0.274674	−	0.475749i	−0.970053	+	0.242894i	$0.921903\pi$
$42$		0			0
$43$	0.379385	+	0.657115i	0.0578557	+	0.100209i	0.893503	−	0.449058i	$-0.148240\pi$
$43$	0.379385	+	0.657115i	0.0578557	+	0.100209i	−0.835647	+	0.549267i	$0.814907\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	$-0.977503\pi$
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	0.559908	+	0.828554i	$0.310836\pi$
$48$		0			0
$49$	15.6459			2.23513
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.57398	−	4.45826i	0.353563	−	0.612389i	−0.633308	−	0.773900i	$-0.718303\pi$
$53$	2.57398	−	4.45826i	0.353563	−	0.612389i	0.986871	+	0.161511i	$0.0516366\pi$
$54$		0			0
$55$	−8.21213	−	14.2238i	−1.10732	−	1.91794i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.63816	+	8.03352i	0.603836	+	1.04588i	0.992234	+	0.124384i	$0.0396953\pi$
$59$	4.63816	+	8.03352i	0.603836	+	1.04588i	−0.388398	+	0.921492i	$0.626971\pi$
$60$		0			0
$61$	4.25877	−	7.37641i	0.545280	−	0.944452i	−0.453310	−	0.891353i	$-0.649757\pi$
$61$	4.25877	−	7.37641i	0.545280	−	0.944452i	0.998589	−	0.0530990i	$-0.0169099\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.75877			0.466218
$66$		0			0
$67$	−0.684793	+	1.18610i	−0.0836607	+	0.144905i	−0.904820	−	0.425795i	$-0.859995\pi$
$67$	−0.684793	+	1.18610i	−0.0836607	+	0.144905i	0.821159	+	0.570699i	$0.193328\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5.82295	−	10.0856i	−0.691057	−	1.19695i	−0.971492	−	0.237073i	$-0.923812\pi$
$71$	−5.82295	−	10.0856i	−0.691057	−	1.19695i	0.280435	−	0.959873i	$-0.409521\pi$
$72$		0			0
$73$	−2.25877	−	3.91231i	−0.264369	−	0.457901i	0.703029	−	0.711161i	$-0.251830\pi$
$73$	−2.25877	−	3.91231i	−0.264369	−	0.457901i	−0.967398	+	0.253260i	$0.918497\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	20.7939			2.36968
$78$		0			0
$79$	−7.07398	−	12.2525i	−0.795885	−	1.37851i	−0.922276	−	0.386532i	$-0.873673\pi$
$79$	−7.07398	−	12.2525i	−0.795885	−	1.37851i	0.126391	−	0.991980i	$-0.459661\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.90673			−0.538583			−0.269292	−	0.963059i	$-0.586790\pi$
$83$	−4.90673			−0.538583			−0.269292	+	0.963059i	$0.586790\pi$
$84$		0			0
$85$	11.5175	−	19.9490i	1.24925	−	2.16377i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.18479	−	7.24827i	0.443587	−	0.768315i	−0.554365	−	0.832273i	$-0.687039\pi$
$89$	4.18479	−	7.24827i	0.443587	−	0.768315i	0.997953	+	0.0639579i	$0.0203723\pi$
$90$		0			0
$91$	−2.37939	+	4.12122i	−0.249427	+	0.432021i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−15.3131	+	5.82634i	−1.57110	+	0.597770i
$96$		0			0
$97$	5.82295	+	10.0856i	0.591231	+	1.02404i	0.994067	+	0.108770i	$0.0346911\pi$
$97$	5.82295	+	10.0856i	0.591231	+	1.02404i	−0.402836	+	0.915272i	$0.631976\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.06418	+	3.57526i	−0.205393	+	0.355752i	−0.950258	−	0.311464i	$-0.899181\pi$
$101$	−2.06418	+	3.57526i	−0.205393	+	0.355752i	0.744865	+	0.667216i	$0.232514\pi$
$102$		0			0
$103$	−6.75877			−0.665961			−0.332981	−	0.942934i	$-0.608054\pi$
$103$	−6.75877			−0.665961			−0.332981	+	0.942934i	$0.608054\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.12836			0.399103			0.199552	−	0.979887i	$-0.436051\pi$
$107$	4.12836			0.399103			0.199552	+	0.979887i	$0.436051\pi$
$108$		0			0
$109$	0.0641778	+	0.111159i	0.00614712	+	0.0106471i	0.869083	−	0.494667i	$-0.164710\pi$
$109$	0.0641778	+	0.111159i	0.00614712	+	0.0106471i	−0.862935	+	0.505314i	$0.831377\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.40879			0.226600			0.113300	−	0.993561i	$-0.463858\pi$
$113$	2.40879			0.226600			0.113300	+	0.993561i	$0.463858\pi$
$114$		0			0
$115$	14.1284			1.31748
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.5817	+	25.2563i	1.33670	+	2.31524i
$120$		0			0
$121$	8.09327			0.735752
$122$		0			0
$123$		0			0
$124$		0			0
$125$	15.5175			1.38793
$126$		0			0
$127$	−0.241230	+	0.417822i	−0.0214057	+	0.0370757i	−0.876530	−	0.481348i	$-0.840147\pi$
$127$	−0.241230	+	0.417822i	−0.0214057	+	0.0370757i	0.855124	+	0.518423i	$0.173481\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.69459	−	2.93512i	−0.148057	−	0.256443i	0.782452	−	0.622711i	$-0.213969\pi$
$131$	−1.69459	−	2.93512i	−0.148057	−	0.256443i	−0.930509	+	0.366268i	$0.880635\pi$
$132$		0			0
$133$	3.30541	−	20.4779i	0.286615	−	1.77566i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.21213	+	15.9559i	−0.787046	+	1.36320i	0.140724	+	0.990049i	$0.455057\pi$
$137$	−9.21213	+	15.9559i	−0.787046	+	1.36320i	−0.927769	+	0.373154i	$0.878276\pi$
$138$		0			0
$139$	−0.00980018	+	0.0169744i	−0.000831240	+	0.00143975i	−0.866441	−	0.499280i	$-0.833598\pi$
$139$	−0.00980018	+	0.0169744i	−0.000831240	+	0.00143975i	0.865609	+	0.500720i	$0.166931\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.18479	+	3.78417i	−0.182702	+	0.316448i
$144$		0			0
$145$	−20.2567			−1.68223
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.12061	+	5.40506i	0.255651	+	0.442800i	0.965072	−	0.261985i	$-0.0843770\pi$
$149$	3.12061	+	5.40506i	0.255651	+	0.442800i	−0.709421	+	0.704785i	$0.751044\pi$
$150$		0			0
$151$	0.739170			0.0601528			0.0300764	−	0.999548i	$-0.490425\pi$
$151$	0.739170			0.0601528			0.0300764	+	0.999548i	$0.490425\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−13.8503	−	23.9894i	−1.11248	−	1.92688i
$156$		0			0
$157$	3.32295	+	5.75552i	0.265200	+	0.459340i	0.967616	−	0.252427i	$-0.0812286\pi$
$157$	3.32295	+	5.75552i	0.265200	+	0.459340i	−0.702416	+	0.711767i	$0.747895\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.94356	+	15.4907i	−0.704852	+	1.22084i
$162$		0			0
$163$	−11.3696			−0.890535			−0.445267	−	0.895398i	$-0.646891\pi$
$163$	−11.3696			−0.890535			−0.445267	+	0.895398i	$0.646891\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.49020	−	2.58110i	0.115315	−	0.199732i	−0.802591	−	0.596530i	$-0.796546\pi$
$167$	1.49020	−	2.58110i	0.115315	−	0.199732i	0.917906	+	0.396799i	$0.129879\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.69459	−	4.66717i	−0.204866	−	0.354838i	0.745224	−	0.666814i	$-0.232343\pi$
$173$	−2.69459	−	4.66717i	−0.204866	−	0.354838i	−0.950090	+	0.311976i	$0.899009\pi$
$174$		0			0
$175$	−21.7199	+	37.6199i	−1.64187	+	2.84380i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.24123			−0.317004			−0.158502	−	0.987359i	$-0.550666\pi$
$179$	−4.24123			−0.317004			−0.158502	+	0.987359i	$0.550666\pi$
$180$		0			0
$181$	−6.30541	+	10.9213i	−0.468677	+	0.811773i	−0.999359	−	0.0357984i	$-0.988603\pi$
$181$	−6.30541	+	10.9213i	−0.468677	+	0.811773i	0.530682	+	0.847571i	$0.321936\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	13.3969	+	23.2042i	0.984962	+	1.70600i
$186$		0			0
$187$	13.3892	+	23.1907i	0.979114	+	1.69588i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.6263			1.63718			0.818591	−	0.574377i	$-0.194756\pi$
$191$	22.6263			1.63718			0.818591	+	0.574377i	$0.194756\pi$
$192$		0			0
$193$	−12.3229	−	21.3440i	−0.887025	−	1.53637i	−0.843375	−	0.537326i	$-0.819435\pi$
$193$	−12.3229	−	21.3440i	−0.887025	−	1.53637i	−0.0436505	−	0.999047i	$-0.513899\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−22.6655			−1.61485			−0.807425	−	0.589970i	$-0.799139\pi$
$197$	−22.6655			−1.61485			−0.807425	+	0.589970i	$0.799139\pi$
$198$		0			0
$199$	−6.96110	+	12.0570i	−0.493460	+	0.854697i	−0.999972	−	0.00753584i	$-0.997601\pi$
$199$	−6.96110	+	12.0570i	−0.493460	+	0.854697i	0.506512	+	0.862233i	$0.330935\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.8229	−	22.2100i	0.899995	−	1.55884i
$204$		0			0
$205$	−6.61081	+	11.4503i	−0.461719	+	0.799721i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.03508	−	18.8032i	0.209941	−	1.30064i
$210$		0			0
$211$	1.00980	+	1.74903i	0.0695175	+	0.120408i	0.898689	−	0.438586i	$-0.144521\pi$
$211$	1.00980	+	1.74903i	0.0695175	+	0.120408i	−0.829172	+	0.558994i	$0.811187\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.42602	−	2.46994i	0.0972539	−	0.168449i
$216$		0			0
$217$	35.0702			2.38072
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.12836			−0.412238
$222$		0			0
$223$	−14.1827	−	24.5652i	−0.949746	−	1.64501i	−0.745958	−	0.665993i	$-0.768008\pi$
$223$	−14.1827	−	24.5652i	−0.949746	−	1.64501i	−0.203789	−	0.979015i	$-0.565325\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.1830			−1.33960			−0.669798	−	0.742544i	$-0.733619\pi$
$227$	−20.1830			−1.33960			−0.669798	+	0.742544i	$0.733619\pi$
$228$		0			0
$229$	11.7784			0.778337			0.389168	−	0.921167i	$-0.372762\pi$
$229$	11.7784			0.778337			0.389168	+	0.921167i	$0.372762\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.0641778	−	0.111159i	−0.00420443	−	0.00728228i	0.863916	−	0.503637i	$-0.168005\pi$
$233$	−0.0641778	−	0.111159i	−0.00420443	−	0.00728228i	−0.868120	+	0.496354i	$0.834672\pi$
$234$		0			0
$235$	22.5526			1.47117
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.10876			−0.201089			−0.100544	−	0.994933i	$-0.532058\pi$
$239$	−3.10876			−0.201089			−0.100544	+	0.994933i	$0.532058\pi$
$240$		0			0
$241$	−5.64796	+	9.78255i	−0.363817	+	0.630149i	−0.988586	−	0.150661i	$-0.951860\pi$
$241$	−5.64796	+	9.78255i	−0.363817	+	0.630149i	0.624769	+	0.780810i	$0.285193\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−29.4047	−	50.9304i	−1.87860	−	3.25382i
$246$		0			0
$247$	3.37939	+	2.75314i	0.215025	+	0.175178i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.758770	+	1.31423i	−0.0478932	+	0.0829534i	−0.888978	−	0.457950i	$-0.848584\pi$
$251$	−0.758770	+	1.31423i	−0.0478932	+	0.0829534i	0.841085	+	0.540903i	$0.181917\pi$
$252$		0			0
$253$	−8.21213	+	14.2238i	−0.516292	+	0.894245i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.09152	+	10.5508i	−0.379979	+	0.658142i	−0.991059	−	0.133427i	$-0.957402\pi$
$257$	−6.09152	+	10.5508i	−0.379979	+	0.658142i	0.611080	+	0.791569i	$0.290735\pi$
$258$		0			0
$259$	−33.9222			−2.10782
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.69459	−	16.7915i	−0.597794	−	1.03541i	−0.993146	−	0.116880i	$-0.962711\pi$
$263$	−9.69459	−	16.7915i	−0.597794	−	1.03541i	0.395352	−	0.918530i	$-0.370623\pi$
$264$		0			0
$265$	−19.3500			−1.18866
$266$		0			0
$267$		0			0
$268$		0			0
$269$	11.7023	+	20.2690i	0.713504	+	1.23582i	0.963534	+	0.267587i	$0.0862262\pi$
$269$	11.7023	+	20.2690i	0.713504	+	1.23582i	−0.250030	+	0.968238i	$0.580440\pi$
$270$		0			0
$271$	−0.482459	−	0.835644i	−0.0293073	−	0.0507617i	0.851000	−	0.525166i	$-0.175997\pi$
$271$	−0.482459	−	0.835644i	−0.0293073	−	0.0507617i	−0.880307	+	0.474404i	$0.842663\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−19.9436	+	34.5433i	−1.20264	+	2.08304i
$276$		0			0
$277$	15.3892			0.924647			0.462323	−	0.886711i	$-0.347016\pi$
$277$	15.3892			0.924647			0.462323	+	0.886711i	$0.347016\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.24897	−	7.35943i	0.253472	−	0.439027i	−0.711007	−	0.703185i	$-0.751761\pi$
$281$	4.24897	−	7.35943i	0.253472	−	0.439027i	0.964479	+	0.264158i	$0.0850940\pi$
$282$		0			0
$283$	5.51754	+	9.55666i	0.327984	+	0.568085i	0.982112	−	0.188299i	$-0.0602975\pi$
$283$	5.51754	+	9.55666i	0.327984	+	0.568085i	−0.654128	+	0.756384i	$0.726964\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.36959	−	14.4965i	−0.494041	−	0.855704i
$288$		0			0
$289$	−10.2784	+	17.8027i	−0.604610	+	1.04722i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.03508			0.0604701			0.0302351	−	0.999543i	$-0.490374\pi$
$293$	1.03508			0.0604701			0.0302351	+	0.999543i	$0.490374\pi$
$294$		0			0
$295$	17.4338	−	30.1962i	1.01503	−	1.75809i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.87939	−	3.25519i	−0.108688	−	0.188253i
$300$		0			0
$301$	1.80541	+	3.12706i	0.104062	+	0.180241i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−32.0155			−1.83320
$306$		0			0
$307$	−7.43376	−	12.8757i	−0.424267	−	0.734852i	0.572084	−	0.820195i	$-0.306135\pi$
$307$	−7.43376	−	12.8757i	−0.424267	−	0.734852i	−0.996352	+	0.0853423i	$0.972802\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	16.0547			0.910378			0.455189	−	0.890395i	$-0.349572\pi$
$311$	16.0547			0.910378			0.455189	+	0.890395i	$0.349572\pi$
$312$		0			0
$313$	−17.5817	+	30.4524i	−0.993777	+	1.72127i	−0.400428	+	0.916328i	$0.631138\pi$
$313$	−17.5817	+	30.4524i	−0.993777	+	1.72127i	−0.593350	+	0.804945i	$0.702195\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.26857	−	16.0536i	0.520575	−	0.901662i	−0.479139	−	0.877739i	$-0.659051\pi$
$317$	9.26857	−	16.0536i	0.520575	−	0.901662i	0.999714	−	0.0239230i	$-0.00761564\pi$
$318$		0			0
$319$	11.7743	−	20.3936i	0.659232	−	1.14182i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	24.9668	−	9.49935i	1.38919	−	0.528558i
$324$		0			0
$325$	−4.56418	−	7.90539i	−0.253175	−	0.438512i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−14.2763	+	24.7273i	−0.787079	+	1.36326i
$330$		0			0
$331$	−8.75877			−0.481426			−0.240713	−	0.970596i	$-0.577381\pi$
$331$	−8.75877			−0.481426			−0.240713	+	0.970596i	$0.577381\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.14796			0.281263
$336$		0			0
$337$	10.6925	+	18.5200i	0.582459	+	1.00885i	0.995187	+	0.0979947i	$0.0312428\pi$
$337$	10.6925	+	18.5200i	0.582459	+	1.00885i	−0.412728	+	0.910855i	$0.635424\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	32.2020			1.74384
$342$		0			0
$343$	41.1438			2.22156
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.9632	+	20.7208i	0.642216	+	1.11235i	0.984937	+	0.172914i	$0.0553182\pi$
$347$	11.9632	+	20.7208i	0.642216	+	1.11235i	−0.342721	+	0.939437i	$0.611348\pi$
$348$		0			0
$349$	−23.0702			−1.23492			−0.617459	−	0.786603i	$-0.711838\pi$
$349$	−23.0702			−1.23492			−0.617459	+	0.786603i	$0.711838\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	28.4979			1.51679			0.758396	−	0.651794i	$-0.225983\pi$
$353$	28.4979			1.51679			0.758396	+	0.651794i	$0.225983\pi$
$354$		0			0
$355$	−21.8871	+	37.9096i	−1.16165	+	2.01203i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.48246	+	4.29975i	0.131019	+	0.226932i	0.924070	−	0.382224i	$-0.124842\pi$
$359$	2.48246	+	4.29975i	0.131019	+	0.226932i	−0.793051	+	0.609156i	$0.791508\pi$
$360$		0			0
$361$	−18.0351	−	5.97794i	−0.949215	−	0.314629i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.49020	+	14.7055i	−0.444397	+	0.769719i
$366$		0			0
$367$	−9.42396	+	16.3228i	−0.491927	+	0.852042i	−0.999957	−	0.00929712i	$-0.997041\pi$
$367$	−9.42396	+	16.3228i	−0.491927	+	0.852042i	0.508030	+	0.861339i	$0.330374\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.2490	−	21.2158i	0.635935	−	1.10147i
$372$		0			0
$373$	35.9026			1.85897			0.929483	−	0.368864i	$-0.120253\pi$
$373$	35.9026			1.85897			0.929483	+	0.368864i	$0.120253\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.69459	+	4.66717i	0.138779	+	0.240372i
$378$		0			0
$379$	−12.2763			−0.630592			−0.315296	−	0.948993i	$-0.602104\pi$
$379$	−12.2763			−0.630592			−0.315296	+	0.948993i	$0.602104\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.20439	+	12.4784i	0.368127	+	0.637615i	0.989273	−	0.146080i	$-0.0466657\pi$
$383$	7.20439	+	12.4784i	0.368127	+	0.637615i	−0.621145	+	0.783695i	$0.713332\pi$
$384$		0			0
$385$	−39.0797	−	67.6880i	−1.99168	−	3.44970i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.18479	−	5.51622i	0.161475	−	0.279684i	−0.773923	−	0.633280i	$-0.781708\pi$
$389$	3.18479	−	5.51622i	0.161475	−	0.279684i	0.935398	+	0.353597i	$0.115041\pi$
$390$		0			0
$391$	−23.0351			−1.16493
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−26.5895	+	46.0543i	−1.33786	+	2.31724i
$396$		0			0
$397$	5.27837	+	9.14241i	0.264914	+	0.458844i	0.967541	−	0.252714i	$-0.0813231\pi$
$397$	5.27837	+	9.14241i	0.264914	+	0.458844i	−0.702627	+	0.711558i	$0.747990\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10.7219	+	18.5709i	0.535428	+	0.927388i	0.999143	+	0.0414036i	$0.0131829\pi$
$401$	10.7219	+	18.5709i	0.535428	+	0.927388i	−0.463715	+	0.885985i	$0.653484\pi$
$402$		0			0
$403$	−3.68479	+	6.38225i	−0.183553	+	0.317922i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−31.1480			−1.54395
$408$		0			0
$409$	−3.32501	+	5.75908i	−0.164411	+	0.284768i	−0.936446	−	0.350812i	$-0.885906\pi$
$409$	−3.32501	+	5.75908i	−0.164411	+	0.284768i	0.772035	+	0.635580i	$0.219239\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	22.0719	+	38.2297i	1.08609	+	1.88116i
$414$		0			0
$415$	9.22163	+	15.9723i	0.452672	+	0.784051i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.84793			−0.285690			−0.142845	−	0.989745i	$-0.545625\pi$
$419$	−5.84793			−0.285690			−0.142845	+	0.989745i	$0.545625\pi$
$420$		0			0
$421$	−0.714193	−	1.23702i	−0.0348076	−	0.0602886i	0.848097	−	0.529841i	$-0.177749\pi$
$421$	−0.714193	−	1.23702i	−0.0348076	−	0.0602886i	−0.882904	+	0.469553i	$0.844415\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−55.9418			−2.71358
$426$		0			0
$427$	20.2665	−	35.1026i	0.980765	−	1.69874i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.389185	−	0.674089i	0.0187464	−	0.0324697i	−0.856500	−	0.516147i	$-0.827366\pi$
$431$	0.389185	−	0.674089i	0.0187464	−	0.0324697i	0.875246	+	0.483677i	$0.160699\pi$
$432$		0			0
$433$	−13.3425	+	23.1100i	−0.641202	+	1.11059i	0.343963	+	0.938983i	$0.388231\pi$
$433$	−13.3425	+	23.1100i	−0.641202	+	1.11059i	−0.985165	+	0.171611i	$0.945103\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.7023	+	10.3484i	0.607635	+	0.495031i
$438$		0			0
$439$	14.2665	+	24.7103i	0.680903	+	1.17936i	0.974706	+	0.223493i	$0.0717460\pi$
$439$	14.2665	+	24.7103i	0.680903	+	1.17936i	−0.293802	+	0.955866i	$0.594921\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.82295	+	15.2818i	−0.419191	+	0.726060i	−0.995858	−	0.0909191i	$-0.971020\pi$
$443$	−8.82295	+	15.2818i	−0.419191	+	0.726060i	0.576667	+	0.816979i	$0.304353\pi$
$444$		0			0
$445$	−31.4593			−1.49132
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.38919			0.0655597			0.0327799	−	0.999463i	$-0.489564\pi$
$449$	1.38919			0.0655597			0.0327799	+	0.999463i	$0.489564\pi$
$450$		0			0
$451$	−7.68510	−	13.3110i	−0.361877	−	0.626790i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	17.8871			0.838561
$456$		0			0
$457$	24.6851			1.15472			0.577360	−	0.816490i	$-0.304083\pi$
$457$	24.6851			1.15472			0.577360	+	0.816490i	$0.304083\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.795607	+	1.37803i	0.0370551	+	0.0641813i	0.883958	−	0.467566i	$-0.154869\pi$
$461$	0.795607	+	1.37803i	0.0370551	+	0.0641813i	−0.846903	+	0.531747i	$0.821536\pi$
$462$		0			0
$463$	−21.7547			−1.01102			−0.505512	−	0.862819i	$-0.668696\pi$
$463$	−21.7547			−1.01102			−0.505512	+	0.862819i	$0.668696\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−15.5175			−0.718066			−0.359033	−	0.933325i	$-0.616893\pi$
$467$	−15.5175			−0.718066			−0.359033	+	0.933325i	$0.616893\pi$
$468$		0			0
$469$	−3.25877	+	5.64436i	−0.150476	+	0.260632i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.65776	+	2.87132i	0.0762237	+	0.132023i
$474$		0			0
$475$	30.8482	+	25.1316i	1.41541	+	1.15312i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.82295	+	10.0856i	−0.266057	+	0.460825i	−0.967840	−	0.251566i	$-0.919054\pi$
$479$	−5.82295	+	10.0856i	−0.266057	+	0.460825i	0.701783	+	0.712391i	$0.252388\pi$
$480$		0			0
$481$	3.56418	−	6.17334i	0.162513	−	0.281480i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	21.8871	−	37.9096i	0.993843	−	1.72139i
$486$		0			0
$487$	38.5526			1.74699			0.873493	−	0.486837i	$-0.161849\pi$
$487$	38.5526			1.74699			0.873493	+	0.486837i	$0.161849\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.82295	+	6.62154i	0.172527	+	0.298826i	0.939303	−	0.343089i	$-0.111473\pi$
$491$	3.82295	+	6.62154i	0.172527	+	0.298826i	−0.766776	+	0.641915i	$0.778140\pi$
$492$		0			0
$493$	33.0268			1.48745
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−27.7101	−	47.9953i	−1.24297	−	2.15288i
$498$		0			0
$499$	−15.3152	−	26.5267i	−0.685603	−	1.18750i	−0.973247	−	0.229762i	$-0.926205\pi$
$499$	−15.3152	−	26.5267i	−0.685603	−	1.18750i	0.287644	−	0.957737i	$-0.407128\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.69459	+	4.66717i	−0.120146	+	0.208099i	−0.919825	−	0.392329i	$-0.871670\pi$
$503$	−2.69459	+	4.66717i	−0.120146	+	0.208099i	0.799679	+	0.600428i	$0.205003\pi$
$504$		0			0
$505$	15.5175			0.690522
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.12836	+	1.95437i	−0.0500135	+	0.0866259i	−0.889948	−	0.456061i	$-0.849260\pi$
$509$	−1.12836	+	1.95437i	−0.0500135	+	0.0866259i	0.839935	+	0.542687i	$0.182593\pi$
$510$		0			0
$511$	−10.7490	−	18.6178i	−0.475506	−	0.823601i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.7023	+	22.0011i	0.559732	+	0.969484i
$516$		0			0
$517$	−13.1088	+	22.7050i	−0.576522	+	0.998566i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−37.2763			−1.63310			−0.816552	−	0.577271i	$-0.804118\pi$
$521$	−37.2763			−1.63310			−0.816552	+	0.577271i	$0.804118\pi$
$522$		0			0
$523$	0.186852	−	0.323637i	0.00817046	−	0.0141517i	−0.861911	−	0.507059i	$-0.830733\pi$
$523$	0.186852	−	0.323637i	0.00817046	−	0.0141517i	0.870082	+	0.492907i	$0.164066\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	22.5817	+	39.1127i	0.983675	+	1.70378i
$528$		0			0
$529$	4.43582	+	7.68307i	0.192862	+	0.334046i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.51754			0.152362
$534$		0			0
$535$	−7.75877	−	13.4386i	−0.335441	−	0.581001i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	68.3661			2.94474
$540$		0			0
$541$	9.54458	−	16.5317i	0.410353	−	0.710753i	−0.584575	−	0.811340i	$-0.698739\pi$
$541$	9.54458	−	16.5317i	0.410353	−	0.710753i	0.994928	+	0.100587i	$0.0320720\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.241230	−	0.417822i	0.0103331	−	0.0178975i
$546$		0			0
$547$	−21.8773	+	37.8926i	−0.935407	+	1.62017i	−0.161500	+	0.986873i	$0.551633\pi$
$547$	−21.8773	+	37.8926i	−0.935407	+	1.62017i	−0.773907	+	0.633300i	$0.781700\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18.2121	−	14.8372i	−0.775863	−	0.632084i
$552$		0			0
$553$	−33.6634	−	58.3068i	−1.43151	−	2.47946i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.9513	−	27.6285i	0.675878	−	1.17066i	−0.300333	−	0.953834i	$-0.597098\pi$
$557$	15.9513	−	27.6285i	0.675878	−	1.17066i	0.976211	−	0.216821i	$-0.0695689\pi$
$558$		0			0
$559$	−0.758770			−0.0320926
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3.87164			0.163170			0.0815852	−	0.996666i	$-0.474002\pi$
$563$	3.87164			0.163170			0.0815852	+	0.996666i	$0.474002\pi$
$564$		0			0
$565$	−4.52704	−	7.84106i	−0.190454	−	0.329876i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	32.9959			1.38326			0.691630	−	0.722252i	$-0.256893\pi$
$569$	32.9959			1.38326			0.691630	+	0.722252i	$0.256893\pi$
$570$		0			0
$571$	−24.1789			−1.01186			−0.505928	−	0.862576i	$-0.668850\pi$
$571$	−24.1789			−1.01186			−0.505928	+	0.862576i	$0.668850\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−17.1557	−	29.7145i	−0.715442	−	1.23918i
$576$		0			0
$577$	−3.16344			−0.131696			−0.0658478	−	0.997830i	$-0.520975\pi$
$577$	−3.16344			−0.131696			−0.0658478	+	0.997830i	$0.520975\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−23.3500			−0.968721
$582$		0			0
$583$	11.2472	−	19.4807i	0.465812	−	0.806810i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3.00774	−	5.20956i	−0.124143	−	0.215022i	0.797255	−	0.603643i	$-0.206285\pi$
$587$	−3.00774	−	5.20956i	−0.124143	−	0.215022i	−0.921398	+	0.388621i	$0.872951\pi$
$588$		0			0
$589$	5.11886	−	31.7128i	0.210919	−	1.30670i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11.4611	−	19.8512i	0.470651	−	0.815192i	−0.528785	−	0.848756i	$-0.677352\pi$
$593$	11.4611	−	19.8512i	0.470651	−	0.815192i	0.999437	+	0.0335639i	$0.0106857\pi$
$594$		0			0
$595$	54.8093	−	94.9326i	2.24696	−	3.89186i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	17.2199	−	29.8257i	0.703585	−	1.21864i	−0.263615	−	0.964628i	$-0.584915\pi$
$599$	17.2199	−	29.8257i	0.703585	−	1.21864i	0.967200	−	0.254017i	$-0.0817519\pi$
$600$		0			0
$601$	24.1634			0.985647			0.492824	−	0.870129i	$-0.335965\pi$
$601$	24.1634			0.985647			0.492824	+	0.870129i	$0.335965\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−15.2104	−	26.3451i	−0.618390	−	1.07108i
$606$		0			0
$607$	−6.84793			−0.277949			−0.138974	−	0.990296i	$-0.544381\pi$
$607$	−6.84793			−0.277949			−0.138974	+	0.990296i	$0.544381\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.00000	−	5.19615i	−0.121367	−	0.210214i
$612$		0			0
$613$	−8.47296	−	14.6756i	−0.342220	−	0.592742i	0.642625	−	0.766181i	$-0.277845\pi$
$613$	−8.47296	−	14.6756i	−0.342220	−	0.592742i	−0.984845	+	0.173439i	$0.944512\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.1361	−	36.6088i	0.850907	−	1.47381i	−0.0294834	−	0.999565i	$-0.509386\pi$
$617$	21.1361	−	36.6088i	0.850907	−	1.47381i	0.880391	−	0.474249i	$-0.157280\pi$
$618$		0			0
$619$	4.59121			0.184536			0.0922682	−	0.995734i	$-0.470588\pi$
$619$	4.59121			0.184536			0.0922682	+	0.995734i	$0.470588\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	19.9145	−	34.4929i	0.797856	−	1.38193i
$624$		0			0
$625$	−6.34255	−	10.9856i	−0.253702	−	0.439425i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−21.8425	−	37.8324i	−0.870919	−	1.50848i
$630$		0			0
$631$	18.7841	−	32.5349i	0.747781	−	1.29520i	−0.201103	−	0.979570i	$-0.564452\pi$
$631$	18.7841	−	32.5349i	0.747781	−	1.29520i	0.948884	−	0.315625i	$-0.102214\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.81345			0.0719647
$636$		0			0
$637$	−7.82295	+	13.5497i	−0.309956	+	0.536860i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.75877	−	4.77833i	−0.108965	−	0.188733i	0.806386	−	0.591389i	$-0.201420\pi$
$641$	−2.75877	−	4.77833i	−0.108965	−	0.188733i	−0.915351	+	0.402656i	$0.868087\pi$
$642$		0			0
$643$	17.5915	+	30.4694i	0.693742	+	1.20160i	0.970603	+	0.240686i	$0.0773724\pi$
$643$	17.5915	+	30.4694i	0.693742	+	1.20160i	−0.276861	+	0.960910i	$0.589294\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.57573			0.140577			0.0702883	−	0.997527i	$-0.477608\pi$
$647$	3.57573			0.140577			0.0702883	+	0.997527i	$0.477608\pi$
$648$		0			0
$649$	20.2668	+	35.1032i	0.795542	+	1.37792i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−32.6418			−1.27737			−0.638686	−	0.769468i	$-0.720522\pi$
$653$	−32.6418			−1.27737			−0.638686	+	0.769468i	$0.720522\pi$
$654$		0			0
$655$	−6.36959	+	11.0324i	−0.248880	+	0.431073i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	24.4807	−	42.4018i	0.953633	−	1.65174i	0.216167	−	0.976356i	$-0.430645\pi$
$659$	24.4807	−	42.4018i	0.953633	−	1.65174i	0.737466	−	0.675384i	$-0.236022\pi$
$660$		0			0
$661$	−17.7392	+	30.7251i	−0.689974	+	1.19507i	0.281872	+	0.959452i	$0.409045\pi$
$661$	−17.7392	+	30.7251i	−0.689974	+	1.19507i	−0.971846	+	0.235618i	$0.924289\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−72.8718	+	27.7262i	−2.82585	+	1.07518i
$666$		0			0
$667$	10.1284	+	17.5428i	0.392171	+	0.679261i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	18.6091	−	32.2318i	0.718395	−	1.24430i
$672$		0			0
$673$	2.68510			0.103503			0.0517514	−	0.998660i	$-0.483520\pi$
$673$	2.68510			0.103503			0.0517514	+	0.998660i	$0.483520\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.3351			−0.704676			−0.352338	−	0.935873i	$-0.614613\pi$
$677$	−18.3351			−0.704676			−0.352338	+	0.935873i	$0.614613\pi$
$678$		0			0
$679$	27.7101	+	47.9953i	1.06342	+	1.84189i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.07367			−0.155875			−0.0779374	−	0.996958i	$-0.524833\pi$
$683$	−4.07367			−0.155875			−0.0779374	+	0.996958i	$0.524833\pi$
$684$		0			0
$685$	69.2526			2.64601
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.57398	+	4.45826i	0.0980608	+	0.169846i
$690$		0			0
$691$	43.5485			1.65666			0.828332	−	0.560238i	$-0.189290\pi$
$691$	43.5485			1.65666			0.828332	+	0.560238i	$0.189290\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.0736733			0.00279459
$696$		0			0
$697$	10.7784	−	18.6687i	0.408260	−	0.707127i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	15.6536	+	27.1129i	0.591230	+	1.02404i	0.994067	+	0.108768i	$0.0346907\pi$
$701$	15.6536	+	27.1129i	0.591230	+	1.02404i	−0.402837	+	0.915272i	$0.631976\pi$
$702$		0			0
$703$	−4.95130	+	30.6747i	−0.186742	+	1.15692i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.82295	+	17.0138i	−0.369430	+	0.639872i
$708$		0			0
$709$	−11.9047	+	20.6195i	−0.447089	+	0.774381i	−0.998195	−	0.0600541i	$-0.980873\pi$
$709$	−11.9047	+	20.6195i	−0.447089	+	0.774381i	0.551106	+	0.834435i	$0.314206\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.8503	+	23.9894i	−0.518697	+	0.898410i
$714$		0			0
$715$	16.4243			0.614233
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15.8949	+	27.5307i	0.592779	+	1.02672i	0.993856	+	0.110678i	$0.0353024\pi$
$719$	15.8949	+	27.5307i	0.592779	+	1.02672i	−0.401078	+	0.916044i	$0.631364\pi$
$720$		0			0
$721$	−32.1634			−1.19783
$722$		0			0
$723$		0			0
$724$		0			0
$725$	24.5972	+	42.6036i	0.913517	+	1.58226i
$726$		0			0
$727$	3.22193	+	5.58055i	0.119495	+	0.206971i	0.919568	−	0.392932i	$-0.128539\pi$
$727$	3.22193	+	5.58055i	0.119495	+	0.206971i	−0.800073	+	0.599903i	$0.795206\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.32501	+	4.02703i	−0.0859935	+	0.148945i
$732$		0			0
$733$	−33.9728			−1.25481			−0.627406	−	0.778692i	$-0.715884\pi$
$733$	−33.9728			−1.25481			−0.627406	+	0.778692i	$0.715884\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.99226	+	5.18274i	−0.110221	+	0.190909i
$738$		0			0
$739$	−11.4632	−	19.8548i	−0.421679	−	0.730370i	0.574425	−	0.818557i	$-0.305226\pi$
$739$	−11.4632	−	19.8548i	−0.421679	−	0.730370i	−0.996104	+	0.0881876i	$0.971893\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.98276	−	6.89835i	−0.146113	−	0.253076i	0.783674	−	0.621172i	$-0.213343\pi$
$743$	−3.98276	−	6.89835i	−0.146113	−	0.253076i	−0.929788	+	0.368096i	$0.880010\pi$
$744$		0			0
$745$	11.7297	−	20.3164i	0.429742	−	0.744335i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	19.6459			0.717845
$750$		0			0
$751$	11.3794	−	19.7097i	0.415240	−	0.719216i	−0.580214	−	0.814464i	$-0.697031\pi$
$751$	11.3794	−	19.7097i	0.415240	−	0.719216i	0.995454	+	0.0952479i	$0.0303644\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.38919	−	2.40614i	−0.0505576	−	0.0875684i
$756$		0			0
$757$	−2.38713	−	4.13462i	−0.0867616	−	0.150275i	0.819379	−	0.573252i	$-0.194318\pi$
$757$	−2.38713	−	4.13462i	−0.0867616	−	0.150275i	−0.906140	+	0.422977i	$0.860985\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−28.2222			−1.02306			−0.511528	−	0.859267i	$-0.670920\pi$
$761$	−28.2222			−1.02306			−0.511528	+	0.859267i	$0.670920\pi$
$762$		0			0
$763$	0.305407	+	0.528981i	0.0110565	+	0.0191504i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.27631			−0.334948
$768$		0			0
$769$	−12.1108	+	20.9765i	−0.436727	+	0.756434i	−0.997435	−	0.0715802i	$-0.977196\pi$
$769$	−12.1108	+	20.9765i	−0.436727	+	0.756434i	0.560708	+	0.828014i	$0.310529\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	11.9804	−	20.7507i	0.430905	−	0.746349i	−0.566046	−	0.824373i	$-0.691528\pi$
$773$	11.9804	−	20.7507i	0.430905	−	0.746349i	0.996951	+	0.0780239i	$0.0248611\pi$
$774$		0			0
$775$	−33.6361	+	58.2594i	−1.20824	+	2.09274i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−14.3304	+	5.45242i	−0.513439	+	0.195353i
$780$		0			0
$781$	−25.4439	−	44.0701i	−0.910453	−	1.57695i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.4902	−	21.6337i	0.445794	−	0.772138i
$786$		0			0
$787$	−28.8871			−1.02971			−0.514857	−	0.857276i	$-0.672155\pi$
$787$	−28.8871			−1.02971			−0.514857	+	0.857276i	$0.672155\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	11.4629			0.407572
$792$		0			0
$793$	4.25877	+	7.37641i	0.151233	+	0.261944i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.61493			0.269735			0.134867	−	0.990864i	$-0.456939\pi$
$797$	7.61493			0.269735			0.134867	+	0.990864i	$0.456939\pi$
$798$		0			0
$799$	−36.7701			−1.30083
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9.86989	−	17.0952i	−0.348301	−	0.603275i
$804$		0			0
$805$	67.2336			2.36967
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.5681			−0.793452			−0.396726	−	0.917937i	$-0.629854\pi$
$809$	−22.5681			−0.793452			−0.396726	+	0.917937i	$0.629854\pi$
$810$		0			0
$811$	12.3696	−	21.4247i	0.434355	−	0.752325i	−0.562888	−	0.826533i	$-0.690310\pi$
$811$	12.3696	−	21.4247i	0.434355	−	0.752325i	0.997243	+	0.0742086i	$0.0236431\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	21.3678	+	37.0102i	0.748482	+	1.29641i
$816$		0			0
$817$	3.09121	−	1.17614i	0.108148	−	0.0411481i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.30541	+	16.1174i	−0.324761	+	0.562502i	−0.981464	−	0.191647i	$-0.938617\pi$
$821$	−9.30541	+	16.1174i	−0.324761	+	0.562502i	0.656703	+	0.754149i	$0.271951\pi$
$822$		0			0
$823$	10.9804	−	19.0186i	0.382753	−	0.662947i	−0.608702	−	0.793399i	$-0.708309\pi$
$823$	10.9804	−	19.0186i	0.382753	−	0.662947i	0.991455	+	0.130452i	$0.0416428\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.9513	+	22.4323i	−0.450361	+	0.780048i	−0.998408	−	0.0563993i	$-0.982038\pi$
$827$	−12.9513	+	22.4323i	−0.450361	+	0.780048i	0.548047	+	0.836447i	$0.315371\pi$
$828$		0			0
$829$	−43.0310			−1.49453			−0.747264	−	0.664528i	$-0.768633\pi$
$829$	−43.0310			−1.49453			−0.747264	+	0.664528i	$0.768633\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	47.9418	+	83.0376i	1.66109	+	2.87708i
$834$		0			0
$835$	−11.2026			−0.387683
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4.16519	+	7.21432i	0.143798	+	0.249066i	0.928924	−	0.370270i	$-0.120735\pi$
$839$	4.16519	+	7.21432i	0.143798	+	0.249066i	−0.785126	+	0.619337i	$0.787402\pi$
$840$		0			0
$841$	−0.0216598	−	0.0375158i	−0.000746888	−	0.00129365i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	22.5526	−	39.0623i	0.775834	−	1.34378i
$846$		0			0
$847$	38.5140			1.32336
$848$		0			0
$849$		0			0
$850$		0			0
$851$	13.3969	−	23.2042i	0.459241	−	0.795428i
$852$		0			0
$853$	21.6925	+	37.5726i	0.742738	+	1.28646i	0.951244	+	0.308439i	$0.0998066\pi$
$853$	21.6925	+	37.5726i	0.742738	+	1.28646i	−0.208506	+	0.978021i	$0.566860\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	17.2567	+	29.8895i	0.589478	+	1.02101i	0.994301	+	0.106611i	$0.0339999\pi$
$857$	17.2567	+	29.8895i	0.589478	+	1.02101i	−0.404823	+	0.914395i	$0.632667\pi$
$858$		0			0
$859$	9.08946	−	15.7434i	0.310128	−	0.537158i	−0.668262	−	0.743926i	$-0.732961\pi$
$859$	9.08946	−	15.7434i	0.310128	−	0.537158i	0.978390	+	0.206768i	$0.0662946\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−44.3269			−1.50890			−0.754452	−	0.656355i	$-0.772097\pi$
$863$	−44.3269			−1.50890			−0.754452	+	0.656355i	$0.772097\pi$
$864$		0			0
$865$	−10.1284	+	17.5428i	−0.344374	+	0.596474i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−30.9103	−	53.5383i	−1.04856	−	1.81616i
$870$		0			0
$871$	−0.684793	−	1.18610i	−0.0232033	−	0.0401893i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	73.8444			2.49640
$876$		0			0
$877$	−5.95336	−	10.3115i	−0.201031	−	0.348196i	0.747830	−	0.663890i	$-0.231096\pi$
$877$	−5.95336	−	10.3115i	−0.201031	−	0.348196i	−0.948861	+	0.315695i	$0.897762\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−28.9804			−0.976374			−0.488187	−	0.872739i	$-0.662342\pi$
$881$	−28.9804			−0.976374			−0.488187	+	0.872739i	$0.662342\pi$
$882$		0			0
$883$	−2.23143	+	3.86495i	−0.0750936	+	0.130066i	−0.901127	−	0.433555i	$-0.857259\pi$
$883$	−2.23143	+	3.86495i	−0.0750936	+	0.130066i	0.826033	+	0.563621i	$0.190592\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14.1557	+	24.5184i	−0.475302	+	0.823247i	−0.999600	−	0.0282880i	$-0.990994\pi$
$887$	−14.1557	+	24.5184i	−0.475302	+	0.823247i	0.524298	+	0.851535i	$0.324328\pi$
$888$		0			0
$889$	−1.14796	+	1.98832i	−0.0385012	+	0.0666860i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	20.2763	+	16.5188i	0.678521	+	0.552781i
$894$		0			0
$895$	7.97090	+	13.8060i	0.266438	+	0.461484i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	19.8580	−	34.3951i	0.662302	−	1.14714i
$900$		0			0
$901$	31.5485			1.05103
$902$		0			0
$903$		0			0
$904$		0			0
$905$	47.4012			1.57567
$906$		0			0
$907$	−10.2567	−	17.7651i	−0.340569	−	0.589882i	0.643970	−	0.765051i	$-0.277286\pi$
$907$	−10.2567	−	17.7651i	−0.340569	−	0.589882i	−0.984538	+	0.175169i	$0.943953\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−21.9608			−0.727594			−0.363797	−	0.931478i	$-0.618520\pi$
$911$	−21.9608			−0.727594			−0.363797	+	0.931478i	$0.618520\pi$
$912$		0			0
$913$	−21.4404			−0.709572
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.06418	−	13.9676i	−0.266303	−	0.461250i
$918$		0			0
$919$	−22.0898			−0.728674			−0.364337	−	0.931267i	$-0.618704\pi$
$919$	−22.0898			−0.728674			−0.364337	+	0.931267i	$0.618704\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11.6459			0.383329
$924$		0			0
$925$	32.5351	−	56.3524i	1.06975	−	1.85286i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	26.2645	+	45.4914i	0.861709	+	1.49252i	0.870278	+	0.492560i	$0.163939\pi$
$929$	26.2645	+	45.4914i	0.861709	+	1.49252i	−0.00856978	+	0.999963i	$0.502728\pi$
$930$		0			0
$931$	10.8675	−	67.3274i	0.356169	−	2.20657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	50.3269	−	87.1687i	1.64586	−	2.85072i
$936$		0			0
$937$	12.9688	−	22.4627i	0.423674	−	0.733824i	−0.572622	−	0.819820i	$-0.694074\pi$
$937$	12.9688	−	22.4627i	0.423674	−	0.733824i	0.996296	+	0.0859953i	$0.0274070\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8.90673	−	15.4269i	0.290351	−	0.502903i	−0.683542	−	0.729911i	$-0.739561\pi$
$941$	8.90673	−	15.4269i	0.290351	−	0.502903i	0.973893	+	0.227009i	$0.0728946\pi$
$942$		0			0
$943$	13.2216			0.430555
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.0446	−	29.5221i	−0.553874	−	0.959339i	−0.997990	−	0.0633691i	$-0.979815\pi$
$947$	−17.0446	−	29.5221i	−0.553874	−	0.959339i	0.444116	−	0.895969i	$-0.353518\pi$
$948$		0			0
$949$	4.51754			0.146646
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−6.34224	−	10.9851i	−0.205445	−	0.355842i	0.744829	−	0.667255i	$-0.232531\pi$
$953$	−6.34224	−	10.9851i	−0.205445	−	0.355842i	−0.950275	+	0.311413i	$0.899198\pi$
$954$		0			0
$955$	−42.5235	−	73.6529i	−1.37603	−	2.38335i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−43.8384	+	75.9304i	−1.41562	+	2.45192i
$960$		0			0
$961$	23.3108			0.751961
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−46.3191	+	80.2271i	−1.49107	+	2.58260i
$966$		0			0
$967$	23.6266	+	40.9225i	0.759780	+	1.31598i	0.942962	+	0.332899i	$0.108027\pi$
$967$	23.6266	+	40.9225i	0.759780	+	1.31598i	−0.183182	+	0.983079i	$0.558640\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−17.6810	−	30.6244i	−0.567410	−	0.982782i	−0.996821	−	0.0796733i	$-0.974612\pi$
$971$	−17.6810	−	30.6244i	−0.567410	−	0.982782i	0.429411	−	0.903109i	$-0.358721\pi$
$972$		0			0
$973$	−0.0466368	+	0.0807773i	−0.00149511	+	0.00258960i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−45.0660			−1.44179			−0.720895	−	0.693044i	$-0.756269\pi$
$977$	−45.0660			−1.44179			−0.720895	+	0.693044i	$0.756269\pi$
$978$		0			0
$979$	18.2858	−	31.6719i	0.584417	−	1.01224i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13.0915	−	22.6752i	−0.417555	−	0.723226i	0.578138	−	0.815939i	$-0.303779\pi$
$983$	−13.0915	−	22.6752i	−0.417555	−	0.723226i	−0.995693	+	0.0927130i	$0.970446\pi$
$984$		0			0
$985$	42.5972	+	73.7805i	1.35726	+	2.35084i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.85204			−0.0906897
$990$		0			0
$991$	10.6070	+	18.3719i	0.336942	+	0.583601i	0.983856	−	0.178962i	$-0.0572739\pi$
$991$	10.6070	+	18.3719i	0.336942	+	0.583601i	−0.646914	+	0.762563i	$0.723941\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	52.3304			1.65898
$996$		0			0
$997$	27.1364	−	47.0016i	0.859418	−	1.48856i	−0.0130661	−	0.999915i	$-0.504159\pi$
$997$	27.1364	−	47.0016i	0.859418	−	1.48856i	0.872485	−	0.488642i	$-0.162507\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.s.j.577.1		6
3.2	odd	2		456.2.q.f.121.3	yes	6
4.3	odd	2		2736.2.s.x.577.1		6
12.11	even	2		912.2.q.k.577.3		6
19.11	even	3	inner	1368.2.s.j.505.1		6
57.11	odd	6		456.2.q.f.49.3	✓	6
57.26	odd	6		8664.2.a.x.1.1		3
57.50	even	6		8664.2.a.z.1.1		3
76.11	odd	6		2736.2.s.x.1873.1		6
228.11	even	6		912.2.q.k.49.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.q.f.49.3	✓	6	57.11	odd	6
456.2.q.f.121.3	yes	6	3.2	odd	2
912.2.q.k.49.3		6	228.11	even	6
912.2.q.k.577.3		6	12.11	even	2
1368.2.s.j.505.1		6	19.11	even	3	inner
1368.2.s.j.577.1		6	1.1	even	1	trivial
2736.2.s.x.577.1		6	4.3	odd	2
2736.2.s.x.1873.1		6	76.11	odd	6
8664.2.a.x.1.1		3	57.26	odd	6
8664.2.a.z.1.1		3	57.50	even	6

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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.87939 - 3.25519i) q^{5} +4.75877 q^{7} +4.36959 q^{11} +(-0.500000 + 0.866025i) q^{13} +(3.06418 + 5.30731i) q^{17} +(0.694593 - 4.30320i) q^{19} +(-1.87939 + 3.25519i) q^{23} +(-4.56418 + 7.90539i) q^{25} +(2.69459 - 4.66717i) q^{29} +7.36959 q^{31} +(-8.94356 - 15.4907i) q^{35} -7.12836 q^{37} +(-1.75877 - 3.04628i) q^{41} +(0.379385 + 0.657115i) q^{43} +(-3.00000 + 5.19615i) q^{47} +15.6459 q^{49} +(2.57398 - 4.45826i) q^{53} +(-8.21213 - 14.2238i) q^{55} +(4.63816 + 8.03352i) q^{59} +(4.25877 - 7.37641i) q^{61} +3.75877 q^{65} +(-0.684793 + 1.18610i) q^{67} +(-5.82295 - 10.0856i) q^{71} +(-2.25877 - 3.91231i) q^{73} +20.7939 q^{77} +(-7.07398 - 12.2525i) q^{79} -4.90673 q^{83} +(11.5175 - 19.9490i) q^{85} +(4.18479 - 7.24827i) q^{89} +(-2.37939 + 4.12122i) q^{91} +(-15.3131 + 5.82634i) q^{95} +(5.82295 + 10.0856i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{7} + 12 q^{11} - 3 q^{13} - 9 q^{25} + 12 q^{29} + 30 q^{31} - 24 q^{35} - 6 q^{37} + 12 q^{41} - 9 q^{43} - 18 q^{47} + 12 q^{49} - 6 q^{59} + 3 q^{61} + 3 q^{67} + 6 q^{71} + 9 q^{73} + 12 q^{77}+ \cdots - 6 q^{97}+O(q^{100}) \) 6 * q + 6 * q^7 + 12 * q^11 - 3 * q^13 - 9 * q^25 + 12 * q^29 + 30 * q^31 - 24 * q^35 - 6 * q^37 + 12 * q^41 - 9 * q^43 - 18 * q^47 + 12 * q^49 - 6 * q^59 + 3 * q^61 + 3 * q^67 + 6 * q^71 + 9 * q^73 + 12 * q^77 - 27 * q^79 + 24 * q^83 + 24 * q^85 + 18 * q^89 - 3 * q^91 - 48 * q^95 - 6 * q^97

Introduction

L-functions

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