LMFDB - Embedded newform 2268.2.i.l.865.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(865,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 4])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.865"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$2268 = 2^{2} \cdot 3^{4} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2268.i (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-2,0,1,0,0,0,5,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.310217769.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1$ x^8 + 4x^6 - 2x^5 + 15x^4 - 4x^3 + 5*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$3^{3}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.1
Root			$0.346911 - 0.600868i$ of defining polynomial
Character	$\chi$	$=$	2268.865
Dual form			2268.2.i.l.2053.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00677 + 3.47583i) q^{5} +(-1.89234 - 1.84906i) q^{7} +(0.885571 + 1.53385i) q^{11} +(-0.114429 - 0.198197i) q^{13} +(3.04751 - 5.27844i) q^{17} +(-3.27792 - 5.67752i) q^{19} +(0.769592 - 1.33297i) q^{23} +(-5.55428 - 9.62030i) q^{25} +(0.271142 - 0.469632i) q^{29} +4.55273 q^{31} +(10.2245 - 2.86683i) q^{35} +(1.54073 + 2.66863i) q^{37} +(4.43985 + 7.69005i) q^{41} +(-2.12120 + 3.67403i) q^{43} -0.757595 q^{47} +(0.161936 + 6.99813i) q^{49} +(-3.19645 + 5.53641i) q^{53} -7.10856 q^{55} +5.17648 q^{59} +12.5015 q^{61} +0.918533 q^{65} -6.18803 q^{67} +13.9068 q^{71} +(-5.08824 + 8.81309i) q^{73} +(1.16039 - 4.54006i) q^{77} +11.5829 q^{79} +(8.66871 - 15.0146i) q^{83} +(12.2313 + 21.1853i) q^{85} +(-5.04596 - 8.73985i) q^{89} +(-0.149939 + 0.586643i) q^{91} +26.3121 q^{95} +(4.91267 - 8.50899i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 2 q^{5} + q^{7} + 5 q^{11} - 3 q^{13} + 2 q^{17} - 8 q^{19} + 2 q^{23} - 8 q^{25} - 2 q^{29} + 11 q^{35} + 4 q^{37} - 3 q^{41} - 5 q^{43} - 30 q^{47} - 19 q^{49} - 24 q^{53} + 16 q^{55} - 20 q^{59}+ \cdots - 19 q^{97}+O(q^{100})$ 8 * q - 2 * q^5 + q^7 + 5 * q^11 - 3 * q^13 + 2 * q^17 - 8 * q^19 + 2 * q^23 - 8 * q^25 - 2 * q^29 + 11 * q^35 + 4 * q^37 - 3 * q^41 - 5 * q^43 - 30 * q^47 - 19 * q^49 - 24 * q^53 + 16 * q^55 - 20 * q^59 + 24 * q^61 + 24 * q^65 + 14 * q^67 - 22 * q^71 - 10 * q^73 - 11 * q^77 + 35 * q^83 + 13 * q^85 - 18 * q^89 - 9 * q^91 + 20 * q^95 - 19 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$ .

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−2.00677	+	3.47583i	−0.897456	+	1.55444i	−0.0667218	+	0.997772i	$0.521254\pi$
$5$	−2.00677	+	3.47583i	−0.897456	+	1.55444i	−0.830735	+	0.556669i	$0.812079\pi$
$6$		0			0
$7$	−1.89234	−	1.84906i	−0.715239	−	0.698880i
$7$	−1.89234	−	1.84906i	−0.715239	−	0.698880i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.885571	+	1.53385i	0.267010	+	0.462474i	0.968088	−	0.250610i	$-0.0806311\pi$
$11$	0.885571	+	1.53385i	0.267010	+	0.462474i	−0.701079	+	0.713084i	$0.747298\pi$
$12$		0			0
$13$	−0.114429	−	0.198197i	−0.0317369	−	0.0549699i	0.849721	−	0.527233i	$-0.176771\pi$
$13$	−0.114429	−	0.198197i	−0.0317369	−	0.0549699i	−0.881458	+	0.472263i	$0.843437\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.04751	−	5.27844i	0.739129	−	1.28021i	−0.213759	−	0.976886i	$-0.568571\pi$
$17$	3.04751	−	5.27844i	0.739129	−	1.28021i	0.952888	−	0.303323i	$-0.0980960\pi$
$18$		0			0
$19$	−3.27792	−	5.67752i	−0.752005	−	1.30251i	−0.946849	−	0.321677i	$-0.895753\pi$
$19$	−3.27792	−	5.67752i	−0.752005	−	1.30251i	0.194844	−	0.980834i	$-0.437580\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.769592	−	1.33297i	0.160471	−	0.277944i	−0.774567	−	0.632492i	$-0.782032\pi$
$23$	0.769592	−	1.33297i	0.160471	−	0.277944i	0.935038	+	0.354548i	$0.115365\pi$
$24$		0			0
$25$	−5.55428	−	9.62030i	−1.11086	−	1.92406i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.271142	−	0.469632i	0.0503498	−	0.0872084i	−0.839752	−	0.542970i	$-0.817300\pi$
$29$	0.271142	−	0.469632i	0.0503498	−	0.0872084i	0.890102	+	0.455762i	$0.150633\pi$
$30$		0			0
$31$	4.55273			0.817695			0.408847	−	0.912603i	$-0.365931\pi$
$31$	4.55273			0.817695			0.408847	+	0.912603i	$0.365931\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	10.2245	−	2.86683i	1.72826	−	0.484582i
$36$		0			0
$37$	1.54073	+	2.66863i	0.253295	+	0.438720i	0.964431	−	0.264335i	$-0.0851523\pi$
$37$	1.54073	+	2.66863i	0.253295	+	0.438720i	−0.711136	+	0.703054i	$0.751819\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.43985	+	7.69005i	0.693388	+	1.20098i	0.970721	+	0.240210i	$0.0772163\pi$
$41$	4.43985	+	7.69005i	0.693388	+	1.20098i	−0.277333	+	0.960774i	$0.589450\pi$
$42$		0			0
$43$	−2.12120	+	3.67403i	−0.323480	+	0.560284i	−0.981204	−	0.192975i	$-0.938186\pi$
$43$	−2.12120	+	3.67403i	−0.323480	+	0.560284i	0.657723	+	0.753260i	$0.271520\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.757595			−0.110507			−0.0552533	−	0.998472i	$-0.517597\pi$
$47$	−0.757595			−0.110507			−0.0552533	+	0.998472i	$0.517597\pi$
$48$		0			0
$49$	0.161936	+	6.99813i	0.0231338	+	0.999732i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.19645	+	5.53641i	−0.439066	+	0.760485i	−0.997618	−	0.0689852i	$-0.978024\pi$
$53$	−3.19645	+	5.53641i	−0.439066	+	0.760485i	0.558552	+	0.829470i	$0.311357\pi$
$54$		0			0
$55$	−7.10856			−0.958518
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.17648			0.673921			0.336960	−	0.941519i	$-0.390601\pi$
$59$	5.17648			0.673921			0.336960	+	0.941519i	$0.390601\pi$
$60$		0			0
$61$	12.5015			1.60065			0.800324	−	0.599568i	$-0.204661\pi$
$61$	12.5015			1.60065			0.800324	+	0.599568i	$0.204661\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.918533			0.113930
$66$		0			0
$67$	−6.18803			−0.755989			−0.377994	−	0.925808i	$-0.623386\pi$
$67$	−6.18803			−0.755989			−0.377994	+	0.925808i	$0.623386\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	13.9068			1.65043			0.825217	−	0.564816i	$-0.191053\pi$
$71$	13.9068			1.65043			0.825217	+	0.564816i	$0.191053\pi$
$72$		0			0
$73$	−5.08824	+	8.81309i	−0.595534	+	1.03149i	0.397938	+	0.917412i	$0.369726\pi$
$73$	−5.08824	+	8.81309i	−0.595534	+	1.03149i	−0.993471	+	0.114082i	$0.963607\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.16039	−	4.54006i	0.132238	−	0.517387i
$78$		0			0
$79$	11.5829			1.30318			0.651590	−	0.758571i	$-0.274102\pi$
$79$	11.5829			1.30318			0.651590	+	0.758571i	$0.274102\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.66871	−	15.0146i	0.951515	−	1.64807i	0.209365	−	0.977838i	$-0.432860\pi$
$83$	8.66871	−	15.0146i	0.951515	−	1.64807i	0.742150	−	0.670234i	$-0.233806\pi$
$84$		0			0
$85$	12.2313	+	21.1853i	1.32667	+	2.29786i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.04596	−	8.73985i	−0.534870	−	0.926423i	−0.999170	−	0.0407443i	$-0.987027\pi$
$89$	−5.04596	−	8.73985i	−0.534870	−	0.926423i	0.464299	−	0.885678i	$-0.346306\pi$
$90$		0			0
$91$	−0.149939	+	0.586643i	−0.0157179	+	0.0614969i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	26.3121			2.69957
$96$		0			0
$97$	4.91267	−	8.50899i	0.498806	−	0.863957i	−0.501193	−	0.865335i	$-0.667105\pi$
$97$	4.91267	−	8.50899i	0.498806	−	0.863957i	0.999999	+	0.00137861i	$0.000438824\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.730408	+	1.26510i	0.0726783	+	0.125883i	0.900074	−	0.435736i	$-0.143512\pi$
$101$	0.730408	+	1.26510i	0.0726783	+	0.125883i	−0.827396	+	0.561619i	$0.810179\pi$
$102$		0			0
$103$	8.68226	−	15.0381i	0.855488	−	1.48175i	−0.0207031	−	0.999786i	$-0.506590\pi$
$103$	8.68226	−	15.0381i	0.855488	−	1.48175i	0.876191	−	0.481963i	$-0.160076\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.97436	+	6.88380i	0.384216	+	0.665482i	0.991660	−	0.128881i	$-0.0411384\pi$
$107$	3.97436	+	6.88380i	0.384216	+	0.665482i	−0.607444	+	0.794363i	$0.707805\pi$
$108$		0			0
$109$	−8.83065	+	15.2951i	−0.845823	+	1.46501i	0.0390825	+	0.999236i	$0.487556\pi$
$109$	−8.83065	+	15.2951i	−0.845823	+	1.46501i	−0.884905	+	0.465772i	$0.845777\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.97381	+	8.61490i	0.467897	+	0.810421i	0.999327	−	0.0366810i	$-0.0116785\pi$
$113$	4.97381	+	8.61490i	0.467897	+	0.810421i	−0.531430	+	0.847102i	$0.678345\pi$
$114$		0			0
$115$	3.08879	+	5.34995i	0.288031	+	0.498885i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−15.5271	+	4.35359i	−1.42337	+	0.399093i
$120$		0			0
$121$	3.93153	−	6.80961i	0.357412	−	0.619055i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.5170			2.19287
$126$		0			0
$127$	−11.4189			−1.01326			−0.506631	−	0.862163i	$-0.669109\pi$
$127$	−11.4189			−1.01326			−0.506631	+	0.862163i	$0.669109\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.16249	+	2.01349i	−0.101567	+	0.175919i	−0.912330	−	0.409455i	$-0.865719\pi$
$131$	−1.16249	+	2.01349i	−0.101567	+	0.175919i	0.810763	+	0.585374i	$0.199052\pi$
$132$		0			0
$133$	−4.29514	+	16.8049i	−0.372436	+	1.45717i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.12366	−	15.8026i	−0.779487	−	1.35011i	−0.932238	−	0.361846i	$-0.882147\pi$
$137$	−9.12366	−	15.8026i	−0.779487	−	1.35011i	0.152751	−	0.988265i	$-0.451187\pi$
$138$		0			0
$139$	−5.08202	−	8.80232i	−0.431051	−	0.746603i	0.565913	−	0.824465i	$-0.308524\pi$
$139$	−5.08202	−	8.80232i	−0.431051	−	0.746603i	−0.996964	+	0.0778624i	$0.975191\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.202670	−	0.351035i	0.0169481	−	0.0293550i
$144$		0			0
$145$	1.08824	+	1.88489i	0.0903735	+	0.156531i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.18226	+	14.1721i	−0.670317	+	1.16102i	0.307498	+	0.951549i	$0.400508\pi$
$149$	−8.18226	+	14.1721i	−0.670317	+	1.16102i	−0.977814	+	0.209474i	$0.932825\pi$
$150$		0			0
$151$	−11.6483	−	20.1754i	−0.947925	−	1.64185i	−0.749786	−	0.661680i	$-0.769844\pi$
$151$	−11.6483	−	20.1754i	−0.947925	−	1.64185i	−0.198139	−	0.980174i	$-0.563490\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.13630	+	15.8245i	−0.733845	+	1.27106i
$156$		0			0
$157$	13.3541			1.06577			0.532885	−	0.846187i	$-0.321108\pi$
$157$	13.3541			1.06577			0.532885	+	0.846187i	$0.321108\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.92108	+	1.09942i	−0.309024	+	0.0866464i
$162$		0			0
$163$	−1.31865	−	2.28397i	−0.103285	−	0.178894i	0.809751	−	0.586773i	$-0.199602\pi$
$163$	−1.31865	−	2.28397i	−0.103285	−	0.178894i	−0.913036	+	0.407879i	$0.866269\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.75073	+	6.49645i	0.290240	+	0.502711i	0.973866	−	0.227122i	$-0.0729315\pi$
$167$	3.75073	+	6.49645i	0.290240	+	0.502711i	−0.683626	+	0.729832i	$0.739598\pi$
$168$		0			0
$169$	6.47381	−	11.2130i	0.497986	−	0.862536i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.04374			0.383469			0.191734	−	0.981447i	$-0.438589\pi$
$173$	5.04374			0.383469			0.191734	+	0.981447i	$0.438589\pi$
$174$		0			0
$175$	−7.27792	+	28.4751i	−0.550159	+	2.15252i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.35938	−	2.35452i	0.101605	−	0.175985i	−0.810741	−	0.585405i	$-0.800936\pi$
$179$	1.35938	−	2.35452i	0.101605	−	0.175985i	0.912346	+	0.409420i	$0.134269\pi$
$180$		0			0
$181$	16.5527			1.23035			0.615177	−	0.788389i	$-0.289084\pi$
$181$	16.5527			1.23035			0.615177	+	0.788389i	$0.289084\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.3676			−0.909285
$186$		0			0
$187$	10.7951			0.789419
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.0426			1.59495			0.797475	−	0.603352i	$-0.206169\pi$
$191$	22.0426			1.59495			0.797475	+	0.603352i	$0.206169\pi$
$192$		0			0
$193$	−12.7470			−0.917547			−0.458773	−	0.888553i	$-0.651711\pi$
$193$	−12.7470			−0.917547			−0.458773	+	0.888553i	$0.651711\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.35097			0.666229			0.333114	−	0.942886i	$-0.391900\pi$
$197$	9.35097			0.666229			0.333114	+	0.942886i	$0.391900\pi$
$198$		0			0
$199$	7.48059	−	12.9568i	0.530285	−	0.918480i	−0.469091	−	0.883150i	$-0.655418\pi$
$199$	7.48059	−	12.9568i	0.530285	−	0.918480i	0.999376	−	0.0353302i	$-0.0112483\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.38147	+	0.387347i	−0.0969603	+	0.0271864i
$204$		0			0
$205$	−35.6391			−2.48914
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.80565	−	10.0557i	0.401585	−	0.695566i
$210$		0			0
$211$	4.70167	+	8.14353i	0.323676	+	0.560624i	0.981244	−	0.192772i	$-0.0617477\pi$
$211$	4.70167	+	8.14353i	0.323676	+	0.560624i	−0.657567	+	0.753396i	$0.728414\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.51355	−	14.7459i	−0.580619	−	1.00566i
$216$		0			0
$217$	−8.61534	−	8.41828i	−0.584847	−	0.571470i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.39489			−0.0938307
$222$		0			0
$223$	−0.877799	+	1.52039i	−0.0587818	+	0.101813i	−0.893919	−	0.448229i	$-0.852055\pi$
$223$	−0.877799	+	1.52039i	−0.0587818	+	0.101813i	0.835137	+	0.550042i	$0.185388\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.1823	+	19.3682i	0.742192	+	1.28552i	0.951495	+	0.307664i	$0.0995473\pi$
$227$	11.1823	+	19.3682i	0.742192	+	1.28552i	−0.209303	+	0.977851i	$0.567119\pi$
$228$		0			0
$229$	4.45340	−	7.71351i	0.294289	−	0.509723i	−0.680530	−	0.732720i	$-0.738251\pi$
$229$	4.45340	−	7.71351i	0.294289	−	0.509723i	0.974819	+	0.222997i	$0.0715839\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.91267	−	5.04488i	−0.190815	−	0.330501i	0.754706	−	0.656064i	$-0.227780\pi$
$233$	−2.91267	−	5.04488i	−0.190815	−	0.330501i	−0.945521	+	0.325562i	$0.894446\pi$
$234$		0			0
$235$	1.52032	−	2.63327i	0.0991748	−	0.171776i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.92475	−	6.79787i	−0.253871	−	0.439718i	0.710717	−	0.703478i	$-0.248371\pi$
$239$	−3.92475	−	6.79787i	−0.253871	−	0.439718i	−0.964588	+	0.263760i	$0.915037\pi$
$240$		0			0
$241$	−12.9957	−	22.5092i	−0.837126	−	1.44994i	−0.892288	−	0.451467i	$-0.850901\pi$
$241$	−12.9957	−	22.5092i	−0.837126	−	1.44994i	0.0551625	−	0.998477i	$-0.482432\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−24.6493	−	13.4808i	−1.57479	−	0.861256i
$246$		0			0
$247$	−0.750177	+	1.29935i	−0.0477326	+	0.0826754i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.64593			0.419488			0.209744	−	0.977756i	$-0.432737\pi$
$251$	6.64593			0.419488			0.209744	+	0.977756i	$0.432737\pi$
$252$		0			0
$253$	2.72611			0.171389
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.5621	+	18.2940i	−0.658843	+	1.14115i	0.322073	+	0.946715i	$0.395620\pi$
$257$	−10.5621	+	18.2940i	−0.658843	+	1.14115i	−0.980916	+	0.194434i	$0.937713\pi$
$258$		0			0
$259$	2.01886	−	7.89888i	0.125446	−	0.490812i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	13.0966	+	22.6839i	0.807569	+	1.39875i	0.914543	+	0.404488i	$0.132550\pi$
$263$	13.0966	+	22.6839i	0.807569	+	1.39875i	−0.106974	+	0.994262i	$0.534116\pi$
$264$		0			0
$265$	−12.8291	−	22.2206i	−0.788085	−	1.36500i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.13475	−	14.0898i	0.495984	−	0.859070i	−0.504005	−	0.863701i	$-0.668141\pi$
$269$	8.13475	−	14.0898i	0.495984	−	0.859070i	0.999989	+	0.00463056i	$0.00147396\pi$
$270$		0			0
$271$	−5.55950	−	9.62934i	−0.337716	−	0.584941i	0.646287	−	0.763095i	$-0.276321\pi$
$271$	−5.55950	−	9.62934i	−0.337716	−	0.584941i	−0.984003	+	0.178154i	$0.942988\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.83742	−	17.0389i	0.593219	−	1.02749i
$276$		0			0
$277$	−0.600882	−	1.04076i	−0.0361035	−	0.0625331i	0.847409	−	0.530941i	$-0.178161\pi$
$277$	−0.600882	−	1.04076i	−0.0361035	−	0.0625331i	−0.883513	+	0.468407i	$0.844828\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.23141	+	7.32901i	−0.252425	+	0.437212i	−0.964193	−	0.265202i	$-0.914561\pi$
$281$	−4.23141	+	7.32901i	−0.252425	+	0.437212i	0.711768	+	0.702414i	$0.247895\pi$
$282$		0			0
$283$	−0.552731			−0.0328564			−0.0164282	−	0.999865i	$-0.505229\pi$
$283$	−0.552731			−0.0328564			−0.0164282	+	0.999865i	$0.505229\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.81765	−	22.7618i	0.343405	−	1.34359i
$288$		0			0
$289$	−10.0746	−	17.4497i	−0.592624	−	1.02645i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−13.6921	−	23.7155i	−0.799903	−	1.38547i	−0.919679	−	0.392671i	$-0.871551\pi$
$293$	−13.6921	−	23.7155i	−0.799903	−	1.38547i	0.119776	−	0.992801i	$-0.461782\pi$
$294$		0			0
$295$	−10.3880	+	17.9926i	−0.604814	+	1.04757i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.352255			−0.0203714
$300$		0			0
$301$	10.8076	−	3.03030i	0.622937	−	0.174663i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−25.0876	+	43.4530i	−1.43651	+	2.48811i
$306$		0			0
$307$	15.4899			0.884056			0.442028	−	0.897001i	$-0.354259\pi$
$307$	15.4899			0.884056			0.442028	+	0.897001i	$0.354259\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	17.7710			1.00770			0.503849	−	0.863792i	$-0.331917\pi$
$311$	17.7710			1.00770			0.503849	+	0.863792i	$0.331917\pi$
$312$		0			0
$313$	16.6897			0.943356			0.471678	−	0.881771i	$-0.343648\pi$
$313$	16.6897			0.943356			0.471678	+	0.881771i	$0.343648\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.55383			0.536597			0.268298	−	0.963336i	$-0.413539\pi$
$317$	9.55383			0.536597			0.268298	+	0.963336i	$0.413539\pi$
$318$		0			0
$319$	0.960462			0.0537755
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−39.9579			−2.22332
$324$		0			0
$325$	−1.27114	+	2.20168i	−0.0705103	+	0.122127i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.43363	+	1.40084i	0.0790386	+	0.0772308i
$330$		0			0
$331$	24.8815			1.36761			0.683806	−	0.729664i	$-0.260323\pi$
$331$	24.8815			1.36761			0.683806	+	0.729664i	$0.260323\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.4180	−	21.5086i	0.678467	−	1.17514i
$336$		0			0
$337$	−10.2668	−	17.7827i	−0.559270	−	0.968683i	−0.997558	−	0.0698488i	$-0.977748\pi$
$337$	−10.2668	−	17.7827i	−0.559270	−	0.968683i	0.438288	−	0.898835i	$-0.355585\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.03177	+	6.98322i	0.218332	+	0.378163i
$342$		0			0
$343$	12.6335	−	13.5423i	0.682147	−	0.731215i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.9716			−0.696353			−0.348176	−	0.937429i	$-0.613199\pi$
$347$	−12.9716			−0.696353			−0.348176	+	0.937429i	$0.613199\pi$
$348$		0			0
$349$	−8.38657	+	14.5260i	−0.448923	+	0.777557i	−0.998316	−	0.0580068i	$-0.981526\pi$
$349$	−8.38657	+	14.5260i	−0.448923	+	0.777557i	0.549393	+	0.835564i	$0.314859\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.94973	−	12.0373i	−0.369897	−	0.640680i	0.619652	−	0.784876i	$-0.287274\pi$
$353$	−6.94973	−	12.0373i	−0.369897	−	0.640680i	−0.989549	+	0.144197i	$0.953940\pi$
$354$		0			0
$355$	−27.9078	+	48.3377i	−1.48119	+	2.56550i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.88247	+	3.26053i	0.0993530	+	0.172084i	0.911417	−	0.411484i	$-0.134989\pi$
$359$	1.88247	+	3.26053i	0.0993530	+	0.172084i	−0.812064	+	0.583568i	$0.801656\pi$
$360$		0			0
$361$	−11.9895	+	20.7664i	−0.631024	+	1.09297i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−20.4219	−	35.3718i	−1.06893	−	1.85144i
$366$		0			0
$367$	17.6283	+	30.5332i	0.920191	+	1.59382i	0.799118	+	0.601175i	$0.205300\pi$
$367$	17.6283	+	30.5332i	0.920191	+	1.59382i	0.121074	+	0.992644i	$0.461366\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	16.2860	−	4.56636i	0.845524	−	0.237074i
$372$		0			0
$373$	8.26683	−	14.3186i	0.428040	−	0.741387i	−0.568659	−	0.822573i	$-0.692538\pi$
$373$	8.26683	−	14.3186i	0.428040	−	0.741387i	0.996699	+	0.0811864i	$0.0258709\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.124106			−0.00639179
$378$		0			0
$379$	6.51811			0.334813			0.167406	−	0.985888i	$-0.446461\pi$
$379$	6.51811			0.334813			0.167406	+	0.985888i	$0.446461\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.44663	−	5.96973i	0.176114	−	0.305039i	−0.764432	−	0.644704i	$-0.776980\pi$
$383$	3.44663	−	5.96973i	0.176114	−	0.305039i	0.940546	+	0.339665i	$0.110314\pi$
$384$		0			0
$385$	13.4518	+	13.1442i	0.685570	+	0.669889i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.8516	+	23.9917i	0.702305	+	1.21643i	0.967656	+	0.252275i	$0.0811788\pi$
$389$	13.8516	+	23.9917i	0.702305	+	1.21643i	−0.265351	+	0.964152i	$0.585488\pi$
$390$		0			0
$391$	−4.69067	−	8.12448i	−0.237218	−	0.410873i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−23.2443	+	40.2603i	−1.16955	+	2.02572i
$396$		0			0
$397$	−3.75604	−	6.50566i	−0.188510	−	0.326510i	0.756243	−	0.654290i	$-0.227033\pi$
$397$	−3.75604	−	6.50566i	−0.188510	−	0.326510i	−0.944754	+	0.327781i	$0.893699\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.428407	−	0.742022i	0.0213936	−	0.0370548i	−0.855130	−	0.518413i	$-0.826523\pi$
$401$	0.428407	−	0.742022i	0.0213936	−	0.0370548i	0.876524	+	0.481358i	$0.159856\pi$
$402$		0			0
$403$	−0.520965	−	0.902337i	−0.0259511	−	0.0449486i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.72886	+	4.72652i	−0.135264	+	0.234285i
$408$		0			0
$409$	1.49544			0.0739448			0.0369724	−	0.999316i	$-0.488229\pi$
$409$	1.49544			0.0739448			0.0369724	+	0.999316i	$0.488229\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.79569	−	9.57164i	−0.482014	−	0.470990i
$414$		0			0
$415$	34.7923	+	60.2620i	1.70789	+	2.95814i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.97436	+	12.0800i	0.340720	+	0.590144i	0.984567	−	0.175010i	$-0.0559959\pi$
$419$	6.97436	+	12.0800i	0.340720	+	0.590144i	−0.643847	+	0.765155i	$0.722663\pi$
$420$		0			0
$421$	−5.86405	+	10.1568i	−0.285797	+	0.495014i	−0.972802	−	0.231638i	$-0.925591\pi$
$421$	−5.86405	+	10.1568i	−0.285797	+	0.495014i	0.687005	+	0.726652i	$0.258925\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−67.7068			−3.28426
$426$		0			0
$427$	−23.6571	−	23.1160i	−1.14485	−	1.11866i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.69844	+	15.0661i	−0.418989	+	0.725711i	−0.995838	−	0.0911402i	$-0.970949\pi$
$431$	−8.69844	+	15.0661i	−0.418989	+	0.725711i	0.576849	+	0.816851i	$0.304282\pi$
$432$		0			0
$433$	9.99800			0.480473			0.240237	−	0.970714i	$-0.422775\pi$
$433$	9.99800			0.480473			0.240237	+	0.970714i	$0.422775\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.0906			−0.482700
$438$		0			0
$439$	5.62575			0.268502			0.134251	−	0.990947i	$-0.457137\pi$
$439$	5.62575			0.268502			0.134251	+	0.990947i	$0.457137\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−29.1250			−1.38377			−0.691886	−	0.722007i	$-0.743220\pi$
$443$	−29.1250			−1.38377			−0.691886	+	0.722007i	$0.743220\pi$
$444$		0			0
$445$	40.5044			1.92009
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−17.9088			−0.845168			−0.422584	−	0.906324i	$-0.638877\pi$
$449$	−17.9088			−0.845168			−0.422584	+	0.906324i	$0.638877\pi$
$450$		0			0
$451$	−7.86361	+	13.6202i	−0.370283	+	0.641349i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.73818	−	1.69842i	−0.0814871	−	0.0796234i
$456$		0			0
$457$	−7.03554			−0.329109			−0.164554	−	0.986368i	$-0.552619\pi$
$457$	−7.03554			−0.329109			−0.164554	+	0.986368i	$0.552619\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.02409	−	12.1661i	0.327144	−	0.566631i	−0.654800	−	0.755802i	$-0.727247\pi$
$461$	7.02409	−	12.1661i	0.327144	−	0.566631i	0.981944	+	0.189172i	$0.0605803\pi$
$462$		0			0
$463$	−5.64208	−	9.77236i	−0.262209	−	0.454160i	0.704619	−	0.709586i	$-0.251118\pi$
$463$	−5.64208	−	9.77236i	−0.262209	−	0.454160i	−0.966829	+	0.255425i	$0.917784\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.8813	−	32.7033i	−0.873720	−	1.51333i	−0.858120	−	0.513450i	$-0.828367\pi$
$467$	−18.8813	−	32.7033i	−0.873720	−	1.51333i	−0.0156005	−	0.999878i	$-0.504966\pi$
$468$		0			0
$469$	11.7099	+	11.4421i	0.540712	+	0.528345i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.51390			−0.345490
$474$		0			0
$475$	−36.4129	+	63.0690i	−1.67074	+	2.89381i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.61956	−	16.6616i	−0.439529	−	0.761286i	0.558124	−	0.829757i	$-0.311521\pi$
$479$	−9.61956	−	16.6616i	−0.439529	−	0.761286i	−0.997653	+	0.0684710i	$0.978188\pi$
$480$		0			0
$481$	0.352609	−	0.610737i	0.0160776	−	0.0278472i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	19.7172	+	34.1512i	0.895313	+	1.55073i
$486$		0			0
$487$	7.06196	−	12.2317i	0.320008	−	0.554270i	−0.660481	−	0.750842i	$-0.729648\pi$
$487$	7.06196	−	12.2317i	0.320008	−	0.554270i	0.980489	+	0.196572i	$0.0629811\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.9592	+	22.4459i	0.584839	+	1.01297i	0.994895	+	0.100911i	$0.0321758\pi$
$491$	12.9592	+	22.4459i	0.584839	+	1.01297i	−0.410056	+	0.912060i	$0.634491\pi$
$492$		0			0
$493$	−1.65261	−	2.86241i	−0.0744300	−	0.128917i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−26.3165	−	25.7145i	−1.18045	−	1.15345i
$498$		0			0
$499$	8.43208	−	14.6048i	0.377472	−	0.653800i	−0.613222	−	0.789911i	$-0.710127\pi$
$499$	8.43208	−	14.6048i	0.377472	−	0.653800i	0.990694	+	0.136110i	$0.0434602\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−7.12392			−0.317640			−0.158820	−	0.987308i	$-0.550769\pi$
$503$	−7.12392			−0.317640			−0.158820	+	0.987308i	$0.550769\pi$
$504$		0			0
$505$	−5.86306			−0.260903
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.79201	−	6.56796i	0.168078	−	0.291120i	−0.769666	−	0.638447i	$-0.779577\pi$
$509$	3.79201	−	6.56796i	0.168078	−	0.291120i	0.937744	+	0.347327i	$0.112911\pi$
$510$		0			0
$511$	25.9247	−	7.26893i	1.14684	−	0.321559i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	34.8467	+	60.3562i	1.53553	+	2.65961i
$516$		0			0
$517$	−0.670904	−	1.16204i	−0.0295063	−	0.0511064i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.6691	+	23.6755i	−0.598853	+	1.03724i	0.394138	+	0.919051i	$0.371043\pi$
$521$	−13.6691	+	23.6755i	−0.598853	+	1.03724i	−0.992991	+	0.118192i	$0.962290\pi$
$522$		0			0
$523$	−1.12652	−	1.95119i	−0.0492592	−	0.0853194i	0.840344	−	0.542053i	$-0.182353\pi$
$523$	−1.12652	−	1.95119i	−0.0492592	−	0.0853194i	−0.889604	+	0.456733i	$0.849019\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	13.8745	−	24.0313i	0.604382	−	1.04682i
$528$		0			0
$529$	10.3155	+	17.8669i	0.448498	+	0.776822i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.01610	−	1.75993i	0.0440120	−	0.0762310i
$534$		0			0
$535$	−31.9026			−1.37927
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−10.5907	+	6.44572i	−0.456174	+	0.277637i
$540$		0			0
$541$	−9.33120	−	16.1621i	−0.401180	−	0.694863i	0.592689	−	0.805431i	$-0.298066\pi$
$541$	−9.33120	−	16.1621i	−0.401180	−	0.694863i	−0.993869	+	0.110568i	$0.964733\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−35.4422	−	61.3877i	−1.51818	−	2.62956i
$546$		0			0
$547$	19.8979	−	34.4642i	0.850774	−	1.47358i	−0.0297373	−	0.999558i	$-0.509467\pi$
$547$	19.8979	−	34.4642i	0.850774	−	1.47358i	0.880511	−	0.474026i	$-0.157200\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.55512			−0.151453
$552$		0			0
$553$	−21.9189	−	21.4176i	−0.932086	−	0.910767i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.86525	+	6.69481i	−0.163776	+	0.283668i	−0.936220	−	0.351415i	$-0.885701\pi$
$557$	−3.86525	+	6.69481i	−0.163776	+	0.283668i	0.772444	+	0.635083i	$0.219034\pi$
$558$		0			0
$559$	0.970909			0.0410651
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−23.8631			−1.00571			−0.502854	−	0.864371i	$-0.667717\pi$
$563$	−23.8631			−1.00571			−0.502854	+	0.864371i	$0.667717\pi$
$564$		0			0
$565$	−39.9253			−1.67967
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.5132			0.650347			0.325173	−	0.945654i	$-0.394577\pi$
$569$	15.5132			0.650347			0.325173	+	0.945654i	$0.394577\pi$
$570$		0			0
$571$	31.5934			1.32214			0.661071	−	0.750324i	$-0.270102\pi$
$571$	31.5934			1.32214			0.661071	+	0.750324i	$0.270102\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−17.0981			−0.713041
$576$		0			0
$577$	6.33842	−	10.9785i	0.263872	−	0.457039i	−0.703396	−	0.710799i	$-0.748334\pi$
$577$	6.33842	−	10.9785i	0.263872	−	0.457039i	0.967267	+	0.253759i	$0.0816671\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−44.1672	+	12.3839i	−1.83236	+	0.513771i
$582$		0			0
$583$	−11.3227			−0.468939
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.8041	−	23.9094i	0.569756	−	0.986847i	−0.426833	−	0.904330i	$-0.640371\pi$
$587$	13.8041	−	23.9094i	0.569756	−	0.986847i	0.996590	−	0.0825166i	$-0.0262958\pi$
$588$		0			0
$589$	−14.9235	−	25.8482i	−0.614911	−	1.06506i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11.4142	−	19.7700i	−0.468726	−	0.811857i	0.530635	−	0.847600i	$-0.321953\pi$
$593$	−11.4142	−	19.7700i	−0.468726	−	0.811857i	−0.999361	+	0.0357436i	$0.988620\pi$
$594$		0			0
$595$	16.0270	−	62.7063i	0.657043	−	2.57071i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−34.8105			−1.42232			−0.711159	−	0.703031i	$-0.751829\pi$
$599$	−34.8105			−1.42232			−0.711159	+	0.703031i	$0.751829\pi$
$600$		0			0
$601$	−0.892988	+	1.54670i	−0.0364257	+	0.0630912i	−0.883664	−	0.468123i	$-0.844931\pi$
$601$	−0.892988	+	1.54670i	−0.0364257	+	0.0630912i	0.847238	+	0.531214i	$0.178264\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.7794	+	27.3307i	0.641523	+	1.11115i
$606$		0			0
$607$	−8.63997	+	14.9649i	−0.350686	+	0.607405i	−0.986370	−	0.164544i	$-0.947385\pi$
$607$	−8.63997	+	14.9649i	−0.350686	+	0.607405i	0.635684	+	0.771949i	$0.280718\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.0866908	+	0.150153i	0.00350714	+	0.00607454i
$612$		0			0
$613$	1.73973	−	3.01330i	0.0702671	−	0.121706i	−0.828751	−	0.559617i	$-0.810948\pi$
$613$	1.73973	−	3.01330i	0.0702671	−	0.121706i	0.899018	+	0.437911i	$0.144282\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	23.8562	+	41.3201i	0.960414	+	1.66349i	0.721461	+	0.692455i	$0.243471\pi$
$617$	23.8562	+	41.3201i	0.960414	+	1.66349i	0.238953	+	0.971031i	$0.423196\pi$
$618$		0			0
$619$	1.35261	+	2.34279i	0.0543660	+	0.0941646i	0.891928	−	0.452178i	$-0.149353\pi$
$619$	1.35261	+	2.34279i	0.0543660	+	0.0941646i	−0.837562	+	0.546343i	$0.816020\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.61185	+	25.8691i	−0.264898	+	1.03642i
$624$		0			0
$625$	−21.4287	+	37.1155i	−0.857147	+	1.48462i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	18.7816			0.748871
$630$		0			0
$631$	1.45861			0.0580663			0.0290332	−	0.999578i	$-0.490757\pi$
$631$	1.45861			0.0580663			0.0290332	+	0.999578i	$0.490757\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	22.9151	−	39.6902i	0.909359	−	1.57506i
$636$		0			0
$637$	1.36848	−	0.832884i	0.0542210	−	0.0330001i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	11.8155	+	20.4651i	0.466686	+	0.808324i	0.999276	−	0.0380495i	$-0.0121144\pi$
$641$	11.8155	+	20.4651i	0.466686	+	0.808324i	−0.532590	+	0.846374i	$0.678781\pi$
$642$		0			0
$643$	−4.12220	−	7.13986i	−0.162564	−	0.281569i	0.773224	−	0.634133i	$-0.218643\pi$
$643$	−4.12220	−	7.13986i	−0.162564	−	0.281569i	−0.935787	+	0.352565i	$0.885310\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.10644	−	3.64846i	0.0828127	−	0.143436i	−0.821644	−	0.570000i	$-0.806943\pi$
$647$	2.10644	−	3.64846i	0.0828127	−	0.143436i	0.904457	+	0.426565i	$0.140276\pi$
$648$		0			0
$649$	4.58414	+	7.93997i	0.179943	+	0.311671i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.6534	+	34.0407i	−0.769097	+	1.33211i	0.168956	+	0.985624i	$0.445960\pi$
$653$	−19.6534	+	34.0407i	−0.769097	+	1.33211i	−0.938053	+	0.346491i	$0.887373\pi$
$654$		0			0
$655$	−4.66570	−	8.08123i	−0.182304	−	0.315760i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.5076	−	32.0561i	0.720953	−	1.24873i	−0.239665	−	0.970856i	$-0.577038\pi$
$659$	18.5076	−	32.0561i	0.720953	−	1.24873i	0.960618	−	0.277872i	$-0.0896291\pi$
$660$		0			0
$661$	−6.03954			−0.234911			−0.117455	−	0.993078i	$-0.537474\pi$
$661$	−6.03954			−0.234911			−0.117455	+	0.993078i	$0.537474\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−49.7916	−	48.6528i	−1.93084	−	1.88667i
$666$		0			0
$667$	−0.417337	−	0.722849i	−0.0161594	−	0.0279888i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	11.0709	+	19.1754i	0.427388	+	0.740259i
$672$		0			0
$673$	17.4194	−	30.1714i	0.671470	−	1.16302i	−0.306017	−	0.952026i	$-0.598997\pi$
$673$	17.4194	−	30.1714i	0.671470	−	1.16302i	0.977487	−	0.210994i	$-0.0676701\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−27.1024			−1.04163			−0.520814	−	0.853670i	$-0.674372\pi$
$677$	−27.1024			−1.04163			−0.520814	+	0.853670i	$0.674372\pi$
$678$		0			0
$679$	−25.0301	+	7.01811i	−0.960567	+	0.269330i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.0557	−	19.1491i	0.423036	−	0.732720i	−0.573199	−	0.819416i	$-0.694298\pi$
$683$	11.0557	−	19.1491i	0.423036	−	0.732720i	0.996235	+	0.0866964i	$0.0276310\pi$
$684$		0			0
$685$	73.2365			2.79822
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.46307			0.0557384
$690$		0			0
$691$	−40.5186			−1.54140			−0.770701	−	0.637197i	$-0.780094\pi$
$691$	−40.5186			−1.54140			−0.770701	+	0.637197i	$0.780094\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	40.7938			1.54740
$696$		0			0
$697$	54.1219			2.05001
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−31.5703			−1.19239			−0.596197	−	0.802838i	$-0.703322\pi$
$701$	−31.5703			−1.19239			−0.596197	+	0.802838i	$0.703322\pi$
$702$		0			0
$703$	10.1008	−	17.4951i	0.380959	−	0.659839i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.957073	−	3.74458i	0.0359944	−	0.140830i
$708$		0			0
$709$	−20.0708			−0.753776			−0.376888	−	0.926259i	$-0.623006\pi$
$709$	−20.0708			−0.753776			−0.376888	+	0.926259i	$0.623006\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.50374	−	6.06866i	0.131216	−	0.227273i
$714$		0			0
$715$	0.813426	+	1.40889i	0.0304204	+	0.0526897i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7.08603	+	12.2734i	0.264264	+	0.457719i	0.967371	−	0.253366i	$-0.0815376\pi$
$719$	7.08603	+	12.2734i	0.264264	+	0.457719i	−0.703106	+	0.711085i	$0.748204\pi$
$720$		0			0
$721$	−44.2362	+	12.4033i	−1.64744	+	0.461921i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.02399			−0.223726
$726$		0			0
$727$	−12.5067	+	21.6622i	−0.463847	+	0.803407i	−0.999149	−	0.0412542i	$-0.986865\pi$
$727$	−12.5067	+	21.6622i	−0.463847	+	0.803407i	0.535302	+	0.844661i	$0.320198\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.9288	+	22.3933i	0.478188	+	0.828245i
$732$		0			0
$733$	−18.3515	+	31.7858i	−0.677829	+	1.17403i	0.297805	+	0.954627i	$0.403746\pi$
$733$	−18.3515	+	31.7858i	−0.677829	+	1.17403i	−0.975633	+	0.219407i	$0.929588\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.47994	−	9.49154i	−0.201856	−	0.349625i
$738$		0			0
$739$	5.70944	−	9.88905i	0.210025	−	0.363774i	−0.741697	−	0.670735i	$-0.765979\pi$
$739$	5.70944	−	9.88905i	0.210025	−	0.363774i	0.951722	+	0.306961i	$0.0993120\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−22.3645	−	38.7365i	−0.820474	−	1.42110i	−0.905329	−	0.424710i	$-0.860376\pi$
$743$	−22.3645	−	38.7365i	−0.820474	−	1.42110i	0.0848549	−	0.996393i	$-0.472957\pi$
$744$		0			0
$745$	−32.8399	−	56.8803i	−1.20316	−	2.08393i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.20771	−	20.3754i	0.190286	−	0.744500i
$750$		0			0
$751$	15.6654	−	27.1333i	0.571638	−	0.990107i	−0.424760	−	0.905306i	$-0.639641\pi$
$751$	15.6654	−	27.1333i	0.571638	−	0.990107i	0.996398	−	0.0848004i	$-0.0270253\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	93.5020			3.40289
$756$		0			0
$757$	−32.8242			−1.19302			−0.596508	−	0.802607i	$-0.703446\pi$
$757$	−32.8242			−1.19302			−0.596508	+	0.802607i	$0.703446\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.70422	−	4.68385i	0.0980279	−	0.169789i	−0.812840	−	0.582486i	$-0.802080\pi$
$761$	2.70422	−	4.68385i	0.0980279	−	0.169789i	0.910868	+	0.412697i	$0.135413\pi$
$762$		0			0
$763$	44.9923	−	12.6152i	1.62883	−	0.456702i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.592340	−	1.02596i	−0.0213882	−	0.0370454i
$768$		0			0
$769$	−1.54660	−	2.67879i	−0.0557718	−	0.0965997i	0.836792	−	0.547522i	$-0.184429\pi$
$769$	−1.54660	−	2.67879i	−0.0557718	−	0.0965997i	−0.892563	+	0.450922i	$0.851095\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−11.0166	+	19.0812i	−0.396238	+	0.686304i	−0.993258	−	0.115922i	$-0.963018\pi$
$773$	−11.0166	+	19.0812i	−0.396238	+	0.686304i	0.597020	+	0.802226i	$0.296351\pi$
$774$		0			0
$775$	−25.2871	−	43.7986i	−0.908341	−	1.57329i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	29.1069	−	50.4147i	1.04286	−	1.80629i
$780$		0			0
$781$	12.3155	+	21.3310i	0.440682	+	0.763283i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−26.7986	+	46.4165i	−0.956483	+	1.65668i
$786$		0			0
$787$	−35.6704			−1.27151			−0.635756	−	0.771890i	$-0.719312\pi$
$787$	−35.6704			−1.27151			−0.635756	+	0.771890i	$0.719312\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.51731	−	25.4992i	0.231729	−	0.906649i
$792$		0			0
$793$	−1.43053	−	2.47775i	−0.0507996	−	0.0879875i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.95927	−	13.7859i	−0.281932	−	0.488320i	0.689929	−	0.723877i	$-0.257642\pi$
$797$	−7.95927	−	13.7859i	−0.281932	−	0.488320i	−0.971860	+	0.235557i	$0.924308\pi$
$798$		0			0
$799$	−2.30878	+	3.99892i	−0.0816786	+	0.141471i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−18.0240			−0.636053
$804$		0			0
$805$	4.04732	−	15.8353i	0.142649	−	0.558121i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	7.52433	−	13.0325i	0.264541	−	0.458199i	−0.702902	−	0.711287i	$-0.748113\pi$
$809$	7.52433	−	13.0325i	0.264541	−	0.458199i	0.967443	+	0.253088i	$0.0814461\pi$
$810$		0			0
$811$	−11.8358			−0.415610			−0.207805	−	0.978170i	$-0.566632\pi$
$811$	−11.8358			−0.415610			−0.207805	+	0.978170i	$0.566632\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.5849			0.370774
$816$		0			0
$817$	27.8125			0.973036
$818$		0			0
$819$		0			0
$820$		0			0
$821$	48.9514			1.70842			0.854208	−	0.519931i	$-0.174042\pi$
$821$	48.9514			1.70842			0.854208	+	0.519931i	$0.174042\pi$
$822$		0			0
$823$	13.3157			0.464155			0.232078	−	0.972697i	$-0.425448\pi$
$823$	13.3157			0.464155			0.232078	+	0.972697i	$0.425448\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−27.3154			−0.949850			−0.474925	−	0.880026i	$-0.657525\pi$
$827$	−27.3154			−0.949850			−0.474925	+	0.880026i	$0.657525\pi$
$828$		0			0
$829$	−5.36003	+	9.28384i	−0.186161	+	0.322441i	−0.943967	−	0.330039i	$-0.892938\pi$
$829$	−5.36003	+	9.28384i	−0.186161	+	0.322441i	0.757806	+	0.652480i	$0.226271\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	37.4327	+	20.4721i	1.29697	+	0.709315i
$834$		0			0
$835$	−30.1075			−1.04191
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8.96082	−	15.5206i	0.309362	−	0.535830i	−0.668861	−	0.743387i	$-0.733218\pi$
$839$	8.96082	−	15.5206i	0.309362	−	0.535830i	0.978223	+	0.207557i	$0.0665513\pi$
$840$		0			0
$841$	14.3530	+	24.8601i	0.494930	+	0.857244i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	25.9830	+	45.0038i	0.893841	+	1.54818i
$846$		0			0
$847$	−20.0312	+	5.61648i	−0.688280	+	0.192985i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.74294			0.162586
$852$		0			0
$853$	−5.13174	+	8.88844i	−0.175707	+	0.304334i	−0.940406	−	0.340054i	$-0.889555\pi$
$853$	−5.13174	+	8.88844i	−0.175707	+	0.304334i	0.764698	+	0.644388i	$0.222888\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−16.3185	−	28.2644i	−0.557428	−	0.965494i	−0.997710	−	0.0676344i	$-0.978455\pi$
$857$	−16.3185	−	28.2644i	−0.557428	−	0.965494i	0.440282	−	0.897860i	$-0.354878\pi$
$858$		0			0
$859$	−2.71212	+	4.69753i	−0.0925363	+	0.160278i	−0.908578	−	0.417716i	$-0.862831\pi$
$859$	−2.71212	+	4.69753i	−0.0925363	+	0.160278i	0.816041	+	0.577993i	$0.196164\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.27005	+	16.0562i	0.315556	+	0.546559i	0.979556	−	0.201174i	$-0.0644756\pi$
$863$	9.27005	+	16.0562i	0.315556	+	0.546559i	−0.663999	+	0.747733i	$0.731142\pi$
$864$		0			0
$865$	−10.1216	+	17.5312i	−0.344146	+	0.596079i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.2575	+	17.7665i	0.347962	+	0.602688i
$870$		0			0
$871$	0.708091	+	1.22645i	0.0239927	+	0.0415566i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−46.3946	−	45.3335i	−1.56842	−	1.53255i
$876$		0			0
$877$	−8.40689	+	14.5612i	−0.283881	+	0.491695i	−0.972337	−	0.233582i	$-0.924955\pi$
$877$	−8.40689	+	14.5612i	−0.283881	+	0.491695i	0.688457	+	0.725278i	$0.258289\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.86926			−0.130359			−0.0651793	−	0.997874i	$-0.520762\pi$
$881$	−3.86926			−0.130359			−0.0651793	+	0.997874i	$0.520762\pi$
$882$		0			0
$883$	34.9696			1.17682			0.588411	−	0.808562i	$-0.299754\pi$
$883$	34.9696			1.17682			0.588411	+	0.808562i	$0.299754\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.7381	+	32.4553i	−0.629163	+	1.08974i	0.358556	+	0.933508i	$0.383269\pi$
$887$	−18.7381	+	32.4553i	−0.629163	+	1.08974i	−0.987720	+	0.156235i	$0.950064\pi$
$888$		0			0
$889$	21.6085	+	21.1142i	0.724725	+	0.708149i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.48333	+	4.30126i	0.0831015	+	0.143936i
$894$		0			0
$895$	5.45595	+	9.44998i	0.182372	+	0.315878i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.23444	−	2.13811i	0.0411708	−	0.0713098i
$900$		0			0
$901$	19.4824	+	33.7445i	0.649053	+	1.12419i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−33.2176	+	57.5345i	−1.10419	+	1.91251i
$906$		0			0
$907$	19.3273	+	33.4759i	0.641753	+	1.11155i	0.985041	+	0.172319i	$0.0551261\pi$
$907$	19.3273	+	33.4759i	0.641753	+	1.11155i	−0.343288	+	0.939230i	$0.611541\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−25.4700	+	44.1153i	−0.843857	+	1.46160i	0.0427526	+	0.999086i	$0.486387\pi$
$911$	−25.4700	+	44.1153i	−0.843857	+	1.46160i	−0.886610	+	0.462518i	$0.846946\pi$
$912$		0			0
$913$	30.7070			1.01625
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.92289	−	1.66070i	0.195591	−	0.0548412i
$918$		0			0
$919$	−3.71044	−	6.42667i	−0.122396	−	0.211996i	0.798316	−	0.602239i	$-0.205725\pi$
$919$	−3.71044	−	6.42667i	−0.122396	−	0.211996i	−0.920712	+	0.390242i	$0.872391\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.59134	−	2.75628i	−0.0523796	−	0.0907242i
$924$		0			0
$925$	17.1153	−	29.6446i	0.562749	−	0.974709i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25.9829			−0.852472			−0.426236	−	0.904612i	$-0.640161\pi$
$929$	−25.9829			−0.852472			−0.426236	+	0.904612i	$0.640161\pi$
$930$		0			0
$931$	39.2012	−	23.8587i	1.28477	−	0.781936i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−21.6634	+	37.5221i	−0.708469	+	1.22710i
$936$		0			0
$937$	25.0972			0.819891			0.409945	−	0.912110i	$-0.365548\pi$
$937$	25.0972			0.819891			0.409945	+	0.912110i	$0.365548\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	51.3308			1.67334			0.836668	−	0.547711i	$-0.184501\pi$
$941$	51.3308			1.67334			0.836668	+	0.547711i	$0.184501\pi$
$942$		0			0
$943$	13.6675			0.445075
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−44.9808			−1.46168			−0.730839	−	0.682549i	$-0.760871\pi$
$947$	−44.9808			−1.46168			−0.730839	+	0.682549i	$0.760871\pi$
$948$		0			0
$949$	2.32897			0.0756016
$950$		0			0
$951$		0			0
$952$		0			0
$953$	11.6027			0.375847			0.187924	−	0.982184i	$-0.439824\pi$
$953$	11.6027			0.375847			0.187924	+	0.982184i	$0.439824\pi$
$954$		0			0
$955$	−44.2346	+	76.6166i	−1.43140	+	2.47925i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−11.9550	+	46.7743i	−0.386046	+	1.51042i
$960$		0			0
$961$	−10.2726			−0.331376
$962$		0			0
$963$		0			0
$964$		0			0
$965$	25.5803	−	44.3063i	0.823458	−	1.42627i
$966$		0			0
$967$	−9.52123	−	16.4912i	−0.306182	−	0.530323i	0.671342	−	0.741148i	$-0.265718\pi$
$967$	−9.52123	−	16.4912i	−0.306182	−	0.530323i	−0.977524	+	0.210825i	$0.932385\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.39580	+	14.5420i	0.269434	+	0.466673i	0.968716	−	0.248173i	$-0.0798300\pi$
$971$	8.39580	+	14.5420i	0.269434	+	0.466673i	−0.699282	+	0.714846i	$0.746497\pi$
$972$		0			0
$973$	−6.65910	+	26.0540i	−0.213481	+	0.835252i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	8.66019			0.277064			0.138532	−	0.990358i	$-0.455762\pi$
$977$	8.66019			0.277064			0.138532	+	0.990358i	$0.455762\pi$
$978$		0			0
$979$	8.93711	−	15.4795i	0.285631	−	0.494728i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	7.34851	+	12.7280i	0.234381	+	0.405960i	0.959093	−	0.283092i	$-0.0913603\pi$
$983$	7.34851	+	12.7280i	0.234381	+	0.405960i	−0.724712	+	0.689052i	$0.758027\pi$
$984$		0			0
$985$	−18.7653	+	32.5024i	−0.597911	+	1.03561i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.26492	+	5.65501i	0.103818	+	0.179819i
$990$		0			0
$991$	−20.9319	+	36.2551i	−0.664923	+	1.15168i	0.314383	+	0.949296i	$0.398202\pi$
$991$	−20.9319	+	36.2551i	−0.664923	+	1.15168i	−0.979306	+	0.202384i	$0.935131\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	30.0237	+	52.0025i	0.951815	+	1.64859i
$996$		0			0
$997$	7.77891	+	13.4735i	0.246361	+	0.426709i	0.962513	−	0.271235i	$-0.0874319\pi$
$997$	7.77891	+	13.4735i	0.246361	+	0.426709i	−0.716153	+	0.697944i	$0.754099\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.i.l.865.1		8
3.2	odd	2		2268.2.i.m.865.4		8
7.2	even	3		2268.2.l.m.541.4		8
9.2	odd	6		2268.2.k.d.1621.4	yes	8
9.4	even	3		2268.2.l.m.109.4		8
9.5	odd	6		2268.2.l.l.109.1		8
9.7	even	3		2268.2.k.c.1621.1	yes	8
21.2	odd	6		2268.2.l.l.541.1		8
63.2	odd	6		2268.2.k.d.1297.4	yes	8
63.16	even	3		2268.2.k.c.1297.1	✓	8
63.23	odd	6		2268.2.i.m.2053.4		8
63.58	even	3	inner	2268.2.i.l.2053.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.i.l.865.1		8	1.1	even	1	trivial
2268.2.i.l.2053.1		8	63.58	even	3	inner
2268.2.i.m.865.4		8	3.2	odd	2
2268.2.i.m.2053.4		8	63.23	odd	6
2268.2.k.c.1297.1	✓	8	63.16	even	3
2268.2.k.c.1621.1	yes	8	9.7	even	3
2268.2.k.d.1297.4	yes	8	63.2	odd	6
2268.2.k.d.1621.4	yes	8	9.2	odd	6
2268.2.l.l.109.1		8	9.5	odd	6
2268.2.l.l.541.1		8	21.2	odd	6
2268.2.l.m.109.4		8	9.4	even	3
2268.2.l.m.541.4		8	7.2	even	3

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2268.2.i.l.865.1

Level	$2268$
Weight	$2$
Character	2268.865
Analytic conductor	$18.110$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$