LMFDB - Embedded newform 1071.2.f.a.883.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1071,2,Mod(883,1071)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1071, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1071.883");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1071 = 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1071.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$8.55197805648$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 357)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			883.6
Root			$-0.854638 + 0.854638i$ of defining polynomial
Character	$\chi$	$=$	1071.883
Dual form			1071.2.f.a.883.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17009 q^{2} -0.630898 q^{4} +0.460811i q^{5} -1.00000i q^{7} -3.07838 q^{8} +O(q^{10})$	\(q+1.17009 q^{2} -0.630898 q^{4} +0.460811i q^{5} -1.00000i q^{7} -3.07838 q^{8} +0.539189i q^{10} -0.709275i q^{11} +3.87936 q^{13} -1.17009i q^{14} -2.34017 q^{16} +(2.53919 - 3.24846i) q^{17} +4.17009 q^{19} -0.290725i q^{20} -0.829914i q^{22} -7.34017i q^{23} +4.78765 q^{25} +4.53919 q^{26} +0.630898i q^{28} -1.70928i q^{29} +4.87936i q^{31} +3.41855 q^{32} +(2.97107 - 3.80098i) q^{34} +0.460811 q^{35} -1.90829i q^{37} +4.87936 q^{38} -1.41855i q^{40} -4.17009i q^{41} +8.75872 q^{43} +0.447480i q^{44} -8.58864i q^{46} -6.72261 q^{47} -1.00000 q^{49} +5.60197 q^{50} -2.44748 q^{52} -4.58864 q^{53} +0.326842 q^{55} +3.07838i q^{56} -2.00000i q^{58} +3.21953 q^{59} -8.48133i q^{61} +5.70928i q^{62} +8.68035 q^{64} +1.78765i q^{65} -6.81432 q^{67} +(-1.60197 + 2.04945i) q^{68} +0.539189 q^{70} +10.7298i q^{71} -5.23513i q^{73} -2.23287i q^{74} -2.63090 q^{76} -0.709275 q^{77} +14.1639i q^{79} -1.07838i q^{80} -4.87936i q^{82} +0.496928 q^{83} +(1.49693 + 1.17009i) q^{85} +10.2485 q^{86} +2.18342i q^{88} -3.60197 q^{89} -3.87936i q^{91} +4.63090i q^{92} -7.86603 q^{94} +1.92162i q^{95} -11.0784i q^{97} -1.17009 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} + 4 q^{4} - 12 q^{8}+O(q^{10}) $ 6 * q - 4 * q^2 + 4 * q^4 - 12 * q^8	$ 6 q - 4 q^{2} + 4 q^{4} - 12 q^{8} - 2 q^{13} + 8 q^{16} + 12 q^{17} + 14 q^{19} + 8 q^{25} + 24 q^{26} - 8 q^{32} - 12 q^{34} + 6 q^{35} + 4 q^{38} + 2 q^{43} - 28 q^{47} - 6 q^{49} - 4 q^{50} - 16 q^{52} + 12 q^{53} - 22 q^{55} - 28 q^{59} + 8 q^{64} - 24 q^{67} + 28 q^{68} - 8 q^{76} + 10 q^{77} - 32 q^{83} - 26 q^{85} + 44 q^{86} + 16 q^{89} - 20 q^{94} + 4 q^{98}+O(q^{100}) $ 6 * q - 4 * q^2 + 4 * q^4 - 12 * q^8 - 2 * q^13 + 8 * q^16 + 12 * q^17 + 14 * q^19 + 8 * q^25 + 24 * q^26 - 8 * q^32 - 12 * q^34 + 6 * q^35 + 4 * q^38 + 2 * q^43 - 28 * q^47 - 6 * q^49 - 4 * q^50 - 16 * q^52 + 12 * q^53 - 22 * q^55 - 28 * q^59 + 8 * q^64 - 24 * q^67 + 28 * q^68 - 8 * q^76 + 10 * q^77 - 32 * q^83 - 26 * q^85 + 44 * q^86 + 16 * q^89 - 20 * q^94 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1071\mathbb{Z}\right)^\times$.

$n$	$190$	$596$	$766$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.17009			0.827376			0.413688	−	0.910419i	$-0.364240\pi$
$2$	1.17009			0.827376			0.413688	+	0.910419i	$0.364240\pi$
$3$		0			0
$3$		0			0
$4$	−0.630898			−0.315449
$5$			0.460811i			0.206081i	0.994677	+	0.103041i	$0.0328571\pi$
$5$			0.460811i			0.206081i	−0.994677	+	0.103041i	$0.967143\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$	−3.07838			−1.08837
$9$		0			0
$10$			0.539189i			0.170506i
$11$		−	0.709275i		−	0.213855i	−0.994267	−	0.106927i	$-0.965899\pi$
$11$		−	0.709275i		−	0.213855i	0.994267	−	0.106927i	$-0.0341012\pi$
$12$		0			0
$13$	3.87936			1.07594			0.537971	−	0.842964i	$-0.319191\pi$
$13$	3.87936			1.07594			0.537971	+	0.842964i	$0.319191\pi$
$14$		−	1.17009i		−	0.312719i
$15$		0			0
$16$	−2.34017			−0.585043
$17$	2.53919	−	3.24846i	0.615844	−	0.787868i
$17$	2.53919	−	3.24846i	0.615844	−	0.787868i
$18$		0			0
$19$	4.17009			0.956683			0.478342	−	0.878174i	$-0.341238\pi$
$19$	4.17009			0.956683			0.478342	+	0.878174i	$0.341238\pi$
$20$		−	0.290725i		−	0.0650080i
$21$		0			0
$22$		−	0.829914i		−	0.176938i
$23$		−	7.34017i		−	1.53053i	−0.643714	−	0.765266i	$-0.722607\pi$
$23$		−	7.34017i		−	1.53053i	0.643714	−	0.765266i	$-0.277393\pi$
$24$		0			0
$25$	4.78765			0.957531
$26$	4.53919			0.890208
$27$		0			0
$28$			0.630898i			0.119228i
$29$		−	1.70928i		−	0.317404i	−0.987326	−	0.158702i	$-0.949269\pi$
$29$		−	1.70928i		−	0.317404i	0.987326	−	0.158702i	$-0.0507310\pi$
$30$		0			0
$31$			4.87936i			0.876359i	0.898887	+	0.438180i	$0.144377\pi$
$31$			4.87936i			0.876359i	−0.898887	+	0.438180i	$0.855623\pi$
$32$	3.41855			0.604320
$33$		0			0
$34$	2.97107	−	3.80098i	0.509534	−	0.651863i
$35$	0.460811			0.0778913
$36$		0			0
$37$		−	1.90829i		−	0.313721i	−0.987621	−	0.156861i	$-0.949863\pi$
$37$		−	1.90829i		−	0.313721i	0.987621	−	0.156861i	$-0.0501373\pi$
$38$	4.87936			0.791537
$39$		0			0
$40$		−	1.41855i		−	0.224293i
$41$		−	4.17009i		−	0.651258i	−0.945498	−	0.325629i	$-0.894424\pi$
$41$		−	4.17009i		−	0.651258i	0.945498	−	0.325629i	$-0.105576\pi$
$42$		0			0
$43$	8.75872			1.33569			0.667846	−	0.744299i	$-0.267216\pi$
$43$	8.75872			1.33569			0.667846	+	0.744299i	$0.267216\pi$
$44$			0.447480i			0.0674602i
$45$		0			0
$46$		−	8.58864i		−	1.26633i
$47$	−6.72261			−0.980593			−0.490296	−	0.871556i	$-0.663111\pi$
$47$	−6.72261			−0.980593			−0.490296	+	0.871556i	$0.663111\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	5.60197			0.792238
$51$		0			0
$52$	−2.44748			−0.339404
$53$	−4.58864			−0.630298			−0.315149	−	0.949042i	$-0.602054\pi$
$53$	−4.58864			−0.630298			−0.315149	+	0.949042i	$0.602054\pi$
$54$		0			0
$55$	0.326842			0.0440714
$56$			3.07838i			0.411366i
$57$		0			0
$58$		−	2.00000i		−	0.262613i
$59$	3.21953			0.419148			0.209574	−	0.977793i	$-0.432792\pi$
$59$	3.21953			0.419148			0.209574	+	0.977793i	$0.432792\pi$
$60$		0			0
$61$		−	8.48133i		−	1.08592i	−0.839758	−	0.542962i	$-0.817303\pi$
$61$		−	8.48133i		−	1.08592i	0.839758	−	0.542962i	$-0.182697\pi$
$62$			5.70928i			0.725079i
$63$		0			0
$64$	8.68035			1.08504
$65$			1.78765i			0.221731i
$66$		0			0
$67$	−6.81432			−0.832501			−0.416251	−	0.909250i	$-0.636656\pi$
$67$	−6.81432			−0.832501			−0.416251	+	0.909250i	$0.636656\pi$
$68$	−1.60197	+	2.04945i	−0.194267	+	0.248532i
$69$		0			0
$70$	0.539189			0.0644454
$71$			10.7298i			1.27339i	0.771115	+	0.636696i	$0.219699\pi$
$71$			10.7298i			1.27339i	−0.771115	+	0.636696i	$0.780301\pi$
$72$		0			0
$73$		−	5.23513i		−	0.612726i	−0.951915	−	0.306363i	$-0.900888\pi$
$73$		−	5.23513i		−	0.612726i	0.951915	−	0.306363i	$-0.0991121\pi$
$74$		−	2.23287i		−	0.259565i
$75$		0			0
$76$	−2.63090			−0.301785
$77$	−0.709275			−0.0808294
$78$		0			0
$79$			14.1639i			1.59357i	0.604264	+	0.796784i	$0.293467\pi$
$79$			14.1639i			1.59357i	−0.604264	+	0.796784i	$0.706533\pi$
$80$		−	1.07838i		−	0.120566i
$81$		0			0
$82$		−	4.87936i		−	0.538835i
$83$	0.496928			0.0545450			0.0272725	−	0.999628i	$-0.491318\pi$
$83$	0.496928			0.0545450			0.0272725	+	0.999628i	$0.491318\pi$
$84$		0			0
$85$	1.49693	+	1.17009i	0.162365	+	0.126914i
$86$	10.2485			1.10512
$87$		0			0
$88$			2.18342i			0.232753i
$89$	−3.60197			−0.381808			−0.190904	−	0.981609i	$-0.561142\pi$
$89$	−3.60197			−0.381808			−0.190904	+	0.981609i	$0.561142\pi$
$90$		0			0
$91$		−	3.87936i		−	0.406668i
$92$			4.63090i			0.482804i
$93$		0			0
$94$	−7.86603			−0.811319
$95$			1.92162i			0.197154i
$96$		0			0
$97$		−	11.0784i		−	1.12484i	−0.826852	−	0.562419i	$-0.809871\pi$
$97$		−	11.0784i		−	1.12484i	0.826852	−	0.562419i	$-0.190129\pi$
$98$	−1.17009			−0.118197
$99$		0			0
$100$	−3.02052			−0.302052
$101$	14.0566			1.39869			0.699344	−	0.714785i	$-0.253476\pi$
$101$	14.0566			1.39869			0.699344	+	0.714785i	$0.253476\pi$
$102$		0			0
$103$	−17.0566			−1.68064			−0.840320	−	0.542090i	$-0.817633\pi$
$103$	−17.0566			−1.68064			−0.840320	+	0.542090i	$0.817633\pi$
$104$	−11.9421			−1.17102
$105$		0			0
$106$	−5.36910			−0.521493
$107$		−	0.973338i		−	0.0940961i	−0.998893	−	0.0470481i	$-0.985019\pi$
$107$		−	0.973338i		−	0.0940961i	0.998893	−	0.0470481i	$-0.0149814\pi$
$108$		0			0
$109$		−	2.40522i		−	0.230378i	−0.993344	−	0.115189i	$-0.963253\pi$
$109$		−	2.40522i		−	0.230378i	0.993344	−	0.115189i	$-0.0367474\pi$
$110$	0.382433			0.0364636
$111$		0			0
$112$			2.34017i			0.221126i
$113$			16.9916i			1.59843i	0.601042	+	0.799217i	$0.294752\pi$
$113$			16.9916i			1.59843i	−0.601042	+	0.799217i	$0.705248\pi$
$114$		0			0
$115$	3.38243			0.315414
$116$			1.07838i			0.100125i
$117$		0			0
$118$	3.76713			0.346793
$119$	−3.24846	−	2.53919i	−0.297786	−	0.232767i
$120$		0			0
$121$	10.4969			0.954266
$122$		−	9.92389i		−	0.898467i
$123$		0			0
$124$		−	3.07838i		−	0.276446i
$125$			4.51026i			0.403410i
$126$		0			0
$127$	4.21235			0.373785			0.186893	−	0.982380i	$-0.440158\pi$
$127$	4.21235			0.373785			0.186893	+	0.982380i	$0.440158\pi$
$128$	3.31965			0.293419
$129$		0			0
$130$			2.09171i			0.183455i
$131$		−	16.0361i		−	1.40108i	−0.713612	−	0.700541i	$-0.752942\pi$
$131$		−	16.0361i		−	1.40108i	0.713612	−	0.700541i	$-0.247058\pi$
$132$		0			0
$133$		−	4.17009i		−	0.361592i
$134$	−7.97334			−0.688791
$135$		0			0
$136$	−7.81658	+	10.0000i	−0.670266	+	0.857493i
$137$	6.43188			0.549513			0.274756	−	0.961514i	$-0.411403\pi$
$137$	6.43188			0.549513			0.274756	+	0.961514i	$0.411403\pi$
$138$		0			0
$139$			12.4391i			1.05507i	0.849534	+	0.527534i	$0.176883\pi$
$139$			12.4391i			1.05507i	−0.849534	+	0.527534i	$0.823117\pi$
$140$	−0.290725			−0.0245707
$141$		0			0
$142$			12.5548i			1.05357i
$143$		−	2.75154i		−	0.230095i
$144$		0			0
$145$	0.787653			0.0654110
$146$		−	6.12556i		−	0.506955i
$147$		0			0
$148$			1.20394i			0.0989630i
$149$	−14.1256			−1.15721			−0.578605	−	0.815608i	$-0.696403\pi$
$149$	−14.1256			−1.15721			−0.578605	+	0.815608i	$0.696403\pi$
$150$		0			0
$151$	16.2907			1.32572			0.662860	−	0.748743i	$-0.269342\pi$
$151$	16.2907			1.32572			0.662860	+	0.748743i	$0.269342\pi$
$152$	−12.8371			−1.04123
$153$		0			0
$154$	−0.829914			−0.0668763
$155$	−2.24846			−0.180601
$156$		0			0
$157$	−19.7721			−1.57798			−0.788991	−	0.614405i	$-0.789396\pi$
$157$	−19.7721			−1.57798			−0.788991	+	0.614405i	$0.789396\pi$
$158$			16.5730i			1.31848i
$159$		0			0
$160$			1.57531i			0.124539i
$161$	−7.34017			−0.578487
$162$		0			0
$163$		−	5.10504i		−	0.399858i	−0.979810	−	0.199929i	$-0.935929\pi$
$163$		−	5.10504i		−	0.399858i	0.979810	−	0.199929i	$-0.0640711\pi$
$164$			2.63090i			0.205439i
$165$		0			0
$166$	0.581449			0.0451292
$167$			1.64650i			0.127410i	0.997969	+	0.0637048i	$0.0202916\pi$
$167$			1.64650i			0.127410i	−0.997969	+	0.0637048i	$0.979708\pi$
$168$		0			0
$169$	2.04945			0.157650
$170$	1.75154	+	1.36910i	0.134337	+	0.105005i
$171$		0			0
$172$	−5.52586			−0.421343
$173$		−	16.7165i		−	1.27093i	−0.772130	−	0.635465i	$-0.780809\pi$
$173$		−	16.7165i		−	1.27093i	0.772130	−	0.635465i	$-0.219191\pi$
$174$		0			0
$175$		−	4.78765i		−	0.361913i
$176$			1.65983i			0.125114i
$177$		0			0
$178$	−4.21461			−0.315899
$179$	−24.0144			−1.79492			−0.897459	−	0.441097i	$-0.854589\pi$
$179$	−24.0144			−1.79492			−0.897459	+	0.441097i	$0.854589\pi$
$180$		0			0
$181$			19.5174i			1.45072i	0.688369	+	0.725360i	$0.258327\pi$
$181$			19.5174i			1.45072i	−0.688369	+	0.725360i	$0.741673\pi$
$182$		−	4.53919i		−	0.336467i
$183$		0			0
$184$			22.5958i			1.66579i
$185$	0.879362			0.0646520
$186$		0			0
$187$	−2.30406	−	1.80098i	−0.168489	−	0.131701i
$188$	4.24128			0.309327
$189$		0			0
$190$			2.24846i			0.163121i
$191$	10.8371			0.784145			0.392073	−	0.919934i	$-0.371758\pi$
$191$	10.8371			0.784145			0.392073	+	0.919934i	$0.371758\pi$
$192$		0			0
$193$		−	7.72487i		−	0.556049i	−0.960574	−	0.278024i	$-0.910320\pi$
$193$		−	7.72487i		−	0.556049i	0.960574	−	0.278024i	$-0.0896796\pi$
$194$		−	12.9627i		−	0.930665i
$195$		0			0
$196$	0.630898			0.0450641
$197$			13.2062i			0.940903i	0.882426	+	0.470452i	$0.155909\pi$
$197$			13.2062i			0.940903i	−0.882426	+	0.470452i	$0.844091\pi$
$198$		0			0
$199$			17.5597i			1.24477i	0.782709	+	0.622387i	$0.213837\pi$
$199$			17.5597i			1.24477i	−0.782709	+	0.622387i	$0.786163\pi$
$200$	−14.7382			−1.04215
$201$		0			0
$202$	16.4475			1.15724
$203$	−1.70928			−0.119968
$204$		0			0
$205$	1.92162			0.134212
$206$	−19.9577			−1.39052
$207$		0			0
$208$	−9.07838			−0.629472
$209$		−	2.95774i		−	0.204591i
$210$		0			0
$211$			4.14957i			0.285668i	0.989747	+	0.142834i	$0.0456215\pi$
$211$			4.14957i			0.285668i	−0.989747	+	0.142834i	$0.954378\pi$
$212$	2.89496			0.198827
$213$		0			0
$214$		−	1.13889i		−	0.0778529i
$215$			4.03612i			0.275261i
$216$		0			0
$217$	4.87936			0.331233
$218$		−	2.81432i		−	0.190609i
$219$		0			0
$220$	−0.206204			−0.0139023
$221$	9.85043	−	12.6020i	0.662612	−	0.847700i
$222$		0			0
$223$	−3.69594			−0.247499			−0.123749	−	0.992314i	$-0.539492\pi$
$223$	−3.69594			−0.247499			−0.123749	+	0.992314i	$0.539492\pi$
$224$		−	3.41855i		−	0.228412i
$225$		0			0
$226$			19.8816i			1.32251i
$227$			11.2751i			0.748356i	0.927357	+	0.374178i	$0.122075\pi$
$227$			11.2751i			0.748356i	−0.927357	+	0.374178i	$0.877925\pi$
$228$		0			0
$229$	−9.96493			−0.658501			−0.329250	−	0.944243i	$-0.606796\pi$
$229$	−9.96493			−0.658501			−0.329250	+	0.944243i	$0.606796\pi$
$230$	3.95774			0.260966
$231$		0			0
$232$			5.26180i			0.345454i
$233$			25.9093i			1.69738i	0.528893	+	0.848689i	$0.322607\pi$
$233$			25.9093i			1.69738i	−0.528893	+	0.848689i	$0.677393\pi$
$234$		0			0
$235$		−	3.09785i		−	0.202082i
$236$	−2.03120			−0.132220
$237$		0			0
$238$	−3.80098	−	2.97107i	−0.246381	−	0.192586i
$239$	19.4524			1.25827			0.629136	−	0.777296i	$-0.283409\pi$
$239$	19.4524			1.25827			0.629136	+	0.777296i	$0.283409\pi$
$240$		0			0
$241$		−	21.9421i		−	1.41342i	−0.707505	−	0.706709i	$-0.750179\pi$
$241$		−	21.9421i		−	1.41342i	0.707505	−	0.706709i	$-0.249821\pi$
$242$	12.2823			0.789537
$243$		0			0
$244$			5.35085i			0.342553i
$245$		−	0.460811i		−	0.0294401i
$246$		0			0
$247$	16.1773			1.02934
$248$		−	15.0205i		−	0.953804i
$249$		0			0
$250$			5.27739i			0.333772i
$251$	−7.00946			−0.442433			−0.221216	−	0.975225i	$-0.571003\pi$
$251$	−7.00946			−0.442433			−0.221216	+	0.975225i	$0.571003\pi$
$252$		0			0
$253$	−5.20620			−0.327311
$254$	4.92881			0.309261
$255$		0			0
$256$	−13.4764			−0.842276
$257$	9.71646			0.606096			0.303048	−	0.952975i	$-0.401996\pi$
$257$	9.71646			0.606096			0.303048	+	0.952975i	$0.401996\pi$
$258$		0			0
$259$	−1.90829			−0.118575
$260$		−	1.12783i		−	0.0699448i
$261$		0			0
$262$		−	18.7636i		−	1.15922i
$263$	−5.17727			−0.319244			−0.159622	−	0.987178i	$-0.551028\pi$
$263$	−5.17727			−0.319244			−0.159622	+	0.987178i	$0.551028\pi$
$264$		0			0
$265$		−	2.11450i		−	0.129892i
$266$		−	4.87936i		−	0.299173i
$267$		0			0
$268$	4.29914			0.262611
$269$		−	1.58864i		−	0.0968609i	−0.998827	−	0.0484305i	$-0.984578\pi$
$269$		−	1.58864i		−	0.0968609i	0.998827	−	0.0484305i	$-0.0154219\pi$
$270$		0			0
$271$	−19.9867			−1.21410			−0.607052	−	0.794662i	$-0.707648\pi$
$271$	−19.9867			−1.21410			−0.607052	+	0.794662i	$0.707648\pi$
$272$	−5.94214	+	7.60197i	−0.360295	+	0.460937i
$273$		0			0
$274$	7.52586			0.454654
$275$		−	3.39576i		−	0.204772i
$276$		0			0
$277$		−	18.6947i		−	1.12326i	−0.827390	−	0.561628i	$-0.810175\pi$
$277$		−	18.6947i		−	1.12326i	0.827390	−	0.561628i	$-0.189825\pi$
$278$			14.5548i			0.872938i
$279$		0			0
$280$	−1.41855			−0.0847746
$281$	22.4969			1.34205			0.671027	−	0.741433i	$-0.265853\pi$
$281$	22.4969			1.34205			0.671027	+	0.741433i	$0.265853\pi$
$282$		0			0
$283$			2.06892i			0.122985i	0.998108	+	0.0614923i	$0.0195860\pi$
$283$			2.06892i			0.122985i	−0.998108	+	0.0614923i	$0.980414\pi$
$284$		−	6.76940i		−	0.401690i
$285$		0			0
$286$		−	3.21953i		−	0.190375i
$287$	−4.17009			−0.246152
$288$		0			0
$289$	−4.10504	−	16.4969i	−0.241473	−	0.970408i
$290$	0.921622			0.0541195
$291$		0			0
$292$			3.30283i			0.193284i
$293$	29.6020			1.72937			0.864683	−	0.502318i	$-0.167519\pi$
$293$	29.6020			1.72937			0.864683	+	0.502318i	$0.167519\pi$
$294$		0			0
$295$			1.48360i			0.0863784i
$296$			5.87444i			0.341445i
$297$		0			0
$298$	−16.5281			−0.957449
$299$		−	28.4752i		−	1.64676i
$300$		0			0
$301$		−	8.75872i		−	0.504844i
$302$	19.0616			1.09687
$303$		0			0
$304$	−9.75872			−0.559701
$305$	3.90829			0.223788
$306$		0			0
$307$	−3.28846			−0.187682			−0.0938411	−	0.995587i	$-0.529915\pi$
$307$	−3.28846			−0.187682			−0.0938411	+	0.995587i	$0.529915\pi$
$308$	0.447480			0.0254975
$309$		0			0
$310$	−2.63090			−0.149425
$311$			19.2351i			1.09072i	0.838201	+	0.545362i	$0.183608\pi$
$311$			19.2351i			1.09072i	−0.838201	+	0.545362i	$0.816392\pi$
$312$		0			0
$313$			8.74927i			0.494538i	0.968947	+	0.247269i	$0.0795331\pi$
$313$			8.74927i			0.494538i	−0.968947	+	0.247269i	$0.920467\pi$
$314$	−23.1350			−1.30558
$315$		0			0
$316$		−	8.93600i		−	0.502689i
$317$			18.2823i			1.02684i	0.858138	+	0.513419i	$0.171621\pi$
$317$			18.2823i			1.02684i	−0.858138	+	0.513419i	$0.828379\pi$
$318$		0			0
$319$	−1.21235			−0.0678784
$320$			4.00000i			0.223607i
$321$		0			0
$322$	−8.58864			−0.478626
$323$	10.5886	−	13.5464i	0.589168	−	0.753741i
$324$		0			0
$325$	18.5730			1.03025
$326$		−	5.97334i		−	0.330833i
$327$		0			0
$328$			12.8371i			0.708810i
$329$			6.72261i			0.370629i
$330$		0			0
$331$	27.7298			1.52417			0.762084	−	0.647479i	$-0.224176\pi$
$331$	27.7298			1.52417			0.762084	+	0.647479i	$0.224176\pi$
$332$	−0.313511			−0.0172062
$333$		0			0
$334$			1.92654i			0.105416i
$335$		−	3.14011i		−	0.171563i
$336$		0			0
$337$			22.6297i			1.23272i	0.787466	+	0.616358i	$0.211393\pi$
$337$			22.6297i			1.23272i	−0.787466	+	0.616358i	$0.788607\pi$
$338$	2.39803			0.130436
$339$		0			0
$340$	−0.944409	−	0.738205i	−0.0512177	−	0.0400348i
$341$	3.46081			0.187413
$342$		0			0
$343$			1.00000i			0.0539949i
$344$	−26.9627			−1.45373
$345$		0			0
$346$		−	19.5597i		−	1.05154i
$347$			1.00227i			0.0538045i	0.999638	+	0.0269023i	$0.00856429\pi$
$347$			1.00227i			0.0538045i	−0.999638	+	0.0269023i	$0.991436\pi$
$348$		0			0
$349$	−10.4569			−0.559747			−0.279873	−	0.960037i	$-0.590293\pi$
$349$	−10.4569			−0.559747			−0.279873	+	0.960037i	$0.590293\pi$
$350$		−	5.60197i		−	0.299438i
$351$		0			0
$352$		−	2.42469i		−	0.129237i
$353$	−19.1617			−1.01987			−0.509937	−	0.860212i	$-0.670331\pi$
$353$	−19.1617			−1.01987			−0.509937	+	0.860212i	$0.670331\pi$
$354$		0			0
$355$	−4.94441			−0.262422
$356$	2.27247			0.120441
$357$		0			0
$358$	−28.0989			−1.48507
$359$	−4.67316			−0.246640			−0.123320	−	0.992367i	$-0.539354\pi$
$359$	−4.67316			−0.246640			−0.123320	+	0.992367i	$0.539354\pi$
$360$		0			0
$361$	−1.61038			−0.0847568
$362$			22.8371i			1.20029i
$363$		0			0
$364$			2.44748i			0.128283i
$365$	2.41241			0.126271
$366$		0			0
$367$		−	33.1917i		−	1.73259i	−0.499533	−	0.866295i	$-0.666495\pi$
$367$		−	33.1917i		−	1.73259i	0.499533	−	0.866295i	$-0.333505\pi$
$368$			17.1773i			0.895427i
$369$		0			0
$370$	1.02893			0.0534915
$371$			4.58864i			0.238230i
$372$		0			0
$373$	−5.02893			−0.260388			−0.130194	−	0.991489i	$-0.541560\pi$
$373$	−5.02893			−0.260388			−0.130194	+	0.991489i	$0.541560\pi$
$374$	−2.69594	−	2.10731i	−0.139404	−	0.108966i
$375$		0			0
$376$	20.6947			1.06725
$377$		−	6.63090i		−	0.341509i
$378$		0			0
$379$			31.6742i			1.62699i	0.581569	+	0.813497i	$0.302439\pi$
$379$			31.6742i			1.62699i	−0.581569	+	0.813497i	$0.697561\pi$
$380$		−	1.21235i		−	0.0621921i
$381$		0			0
$382$	12.6803			0.648783
$383$	7.93108			0.405259			0.202630	−	0.979255i	$-0.435051\pi$
$383$	7.93108			0.405259			0.202630	+	0.979255i	$0.435051\pi$
$384$		0			0
$385$		−	0.326842i		−	0.0166574i
$386$		−	9.03877i		−	0.460061i
$387$		0			0
$388$			6.98932i			0.354829i
$389$	−16.5503			−0.839131			−0.419566	−	0.907725i	$-0.637818\pi$
$389$	−16.5503			−0.839131			−0.419566	+	0.907725i	$0.637818\pi$
$390$		0			0
$391$	−23.8443	−	18.6381i	−1.20586	−	0.942569i
$392$	3.07838			0.155482
$393$		0			0
$394$			15.4524i			0.778481i
$395$	−6.52690			−0.328404
$396$		0			0
$397$		−	26.3701i		−	1.32348i	−0.749734	−	0.661740i	$-0.769818\pi$
$397$		−	26.3701i		−	1.32348i	0.749734	−	0.661740i	$-0.230182\pi$
$398$			20.5464i			1.02990i
$399$		0			0
$400$	−11.2039			−0.560197
$401$		−	26.2534i		−	1.31103i	−0.755182	−	0.655516i	$-0.772451\pi$
$401$		−	26.2534i		−	1.31103i	0.755182	−	0.655516i	$-0.227549\pi$
$402$		0			0
$403$			18.9288i			0.942911i
$404$	−8.86830			−0.441214
$405$		0			0
$406$	−2.00000			−0.0992583
$407$	−1.35350			−0.0670907
$408$		0			0
$409$	15.2713			0.755115			0.377557	−	0.925986i	$-0.376764\pi$
$409$	15.2713			0.755115			0.377557	+	0.925986i	$0.376764\pi$
$410$	2.24846			0.111044
$411$		0			0
$412$	10.7610			0.530156
$413$		−	3.21953i		−	0.158423i
$414$		0			0
$415$			0.228990i			0.0112407i
$416$	13.2618			0.650213
$417$		0			0
$418$		−	3.46081i		−	0.169274i
$419$			9.57918i			0.467974i	0.972240	+	0.233987i	$0.0751773\pi$
$419$			9.57918i			0.467974i	−0.972240	+	0.233987i	$0.924823\pi$
$420$		0			0
$421$	33.6453			1.63977			0.819885	−	0.572528i	$-0.194037\pi$
$421$	33.6453			1.63977			0.819885	+	0.572528i	$0.194037\pi$
$422$			4.85535i			0.236355i
$423$		0			0
$424$	14.1256			0.685998
$425$	12.1568	−	15.5525i	0.589689	−	0.754408i
$426$		0			0
$427$	−8.48133			−0.410440
$428$			0.614077i			0.0296825i
$429$		0			0
$430$			4.72261i			0.227744i
$431$			15.6332i			0.753023i	0.926412	+	0.376512i	$0.122876\pi$
$431$			15.6332i			0.753023i	−0.926412	+	0.376512i	$0.877124\pi$
$432$		0			0
$433$	−10.4524			−0.502310			−0.251155	−	0.967947i	$-0.580810\pi$
$433$	−10.4524			−0.502310			−0.251155	+	0.967947i	$0.580810\pi$
$434$	5.70928			0.274054
$435$		0			0
$436$			1.51745i			0.0726725i
$437$		−	30.6092i		−	1.46423i
$438$		0			0
$439$		−	13.6586i		−	0.651890i	−0.945389	−	0.325945i	$-0.894318\pi$
$439$		−	13.6586i		−	0.651890i	0.945389	−	0.325945i	$-0.105682\pi$
$440$	−1.00614			−0.0479660
$441$		0			0
$442$	11.5259	−	14.7454i	0.548229	−	0.701367i
$443$	−15.4836			−0.735648			−0.367824	−	0.929895i	$-0.619897\pi$
$443$	−15.4836			−0.735648			−0.367824	+	0.929895i	$0.619897\pi$
$444$		0			0
$445$		−	1.65983i		−	0.0786833i
$446$	−4.32457			−0.204775
$447$		0			0
$448$		−	8.68035i		−	0.410108i
$449$			16.9132i			0.798184i	0.916911	+	0.399092i	$0.130675\pi$
$449$			16.9132i			0.798184i	−0.916911	+	0.399092i	$0.869325\pi$
$450$		0			0
$451$	−2.95774			−0.139275
$452$		−	10.7200i		−	0.504224i
$453$		0			0
$454$			13.1929i			0.619172i
$455$	1.78765			0.0838065
$456$		0			0
$457$	7.98545			0.373543			0.186772	−	0.982403i	$-0.440198\pi$
$457$	7.98545			0.373543			0.186772	+	0.982403i	$0.440198\pi$
$458$	−11.6598			−0.544828
$459$		0			0
$460$	−2.13397			−0.0994968
$461$	28.4235			1.32381			0.661907	−	0.749586i	$-0.269748\pi$
$461$	28.4235			1.32381			0.661907	+	0.749586i	$0.269748\pi$
$462$		0			0
$463$	2.89884			0.134720			0.0673602	−	0.997729i	$-0.478542\pi$
$463$	2.89884			0.134720			0.0673602	+	0.997729i	$0.478542\pi$
$464$			4.00000i			0.185695i
$465$		0			0
$466$			30.3162i			1.40437i
$467$	0.993857			0.0459902			0.0229951	−	0.999736i	$-0.492680\pi$
$467$	0.993857			0.0459902			0.0229951	+	0.999736i	$0.492680\pi$
$468$		0			0
$469$			6.81432i			0.314656i
$470$		−	3.62475i		−	0.167197i
$471$		0			0
$472$	−9.91094			−0.456188
$473$		−	6.21235i		−	0.285644i
$474$		0			0
$475$	19.9649			0.916054
$476$	2.04945	+	1.60197i	0.0939363	+	0.0734261i
$477$		0			0
$478$	22.7610			1.04106
$479$			17.9288i			0.819188i	0.912268	+	0.409594i	$0.134330\pi$
$479$			17.9288i			0.819188i	−0.912268	+	0.409594i	$0.865670\pi$
$480$		0			0
$481$		−	7.40295i		−	0.337546i
$482$		−	25.6742i		−	1.16943i
$483$		0			0
$484$	−6.62249			−0.301022
$485$	5.10504			0.231808
$486$		0			0
$487$			12.0410i			0.545632i	0.962066	+	0.272816i	$0.0879549\pi$
$487$			12.0410i			0.545632i	−0.962066	+	0.272816i	$0.912045\pi$
$488$			26.1087i			1.18189i
$489$		0			0
$490$		−	0.539189i		−	0.0243581i
$491$	−19.5753			−0.883421			−0.441711	−	0.897158i	$-0.645628\pi$
$491$	−19.5753			−0.883421			−0.441711	+	0.897158i	$0.645628\pi$
$492$		0			0
$493$	−5.55252	−	4.34017i	−0.250073	−	0.195472i
$494$	18.9288			0.851647
$495$		0			0
$496$		−	11.4186i		−	0.512708i
$497$	10.7298			0.481297
$498$		0			0
$499$			29.9493i			1.34072i	0.742038	+	0.670358i	$0.233859\pi$
$499$			29.9493i			1.34072i	−0.742038	+	0.670358i	$0.766141\pi$
$500$		−	2.84551i		−	0.127255i
$501$		0			0
$502$	−8.20167			−0.366058
$503$		−	13.9204i		−	0.620680i	−0.950626	−	0.310340i	$-0.899557\pi$
$503$		−	13.9204i		−	0.620680i	0.950626	−	0.310340i	$-0.100443\pi$
$504$		0			0
$505$			6.47745i			0.288243i
$506$	−6.09171			−0.270809
$507$		0			0
$508$	−2.65756			−0.117910
$509$	−24.5380			−1.08763			−0.543813	−	0.839206i	$-0.683020\pi$
$509$	−24.5380			−1.08763			−0.543813	+	0.839206i	$0.683020\pi$
$510$		0			0
$511$	−5.23513			−0.231589
$512$	−22.4079			−0.990297
$513$		0			0
$514$	11.3691			0.501470
$515$		−	7.85989i		−	0.346348i
$516$		0			0
$517$			4.76818i			0.209704i
$518$	−2.23287			−0.0981065
$519$		0			0
$520$		−	5.50307i		−	0.241326i
$521$			14.9350i			0.654312i	0.944970	+	0.327156i	$0.106090\pi$
$521$			14.9350i			0.654312i	−0.944970	+	0.327156i	$0.893910\pi$
$522$		0			0
$523$	2.21008			0.0966400			0.0483200	−	0.998832i	$-0.484613\pi$
$523$	2.21008			0.0966400			0.0483200	+	0.998832i	$0.484613\pi$
$524$			10.1171i			0.441970i
$525$		0			0
$526$	−6.05786			−0.264135
$527$	15.8504	+	12.3896i	0.690456	+	0.539700i
$528$		0			0
$529$	−30.8781			−1.34253
$530$		−	2.47414i		−	0.107470i
$531$		0			0
$532$			2.63090i			0.114064i
$533$		−	16.1773i		−	0.700716i
$534$		0			0
$535$	0.448525			0.0193914
$536$	20.9770			0.906070
$537$		0			0
$538$		−	1.85884i		−	0.0801404i
$539$			0.709275i			0.0305507i
$540$		0			0
$541$		−	9.75872i		−	0.419560i	−0.977749	−	0.209780i	$-0.932725\pi$
$541$		−	9.75872i		−	0.419560i	0.977749	−	0.209780i	$-0.0672748\pi$
$542$	−23.3861			−1.00452
$543$		0			0
$544$	8.68035	−	11.1050i	0.372167	−	0.476125i
$545$	1.10835			0.0474766
$546$		0			0
$547$		−	4.61530i		−	0.197336i	−0.995120	−	0.0986680i	$-0.968542\pi$
$547$		−	4.61530i		−	0.197336i	0.995120	−	0.0986680i	$-0.0314582\pi$
$548$	−4.05786			−0.173343
$549$		0			0
$550$		−	3.97334i		−	0.169424i
$551$		−	7.12783i		−	0.303656i
$552$		0			0
$553$	14.1639			0.602312
$554$		−	21.8744i		−	0.929356i
$555$		0			0
$556$		−	7.84778i		−	0.332820i
$557$	13.4836			0.571318			0.285659	−	0.958331i	$-0.407787\pi$
$557$	13.4836			0.571318			0.285659	+	0.958331i	$0.407787\pi$
$558$		0			0
$559$	33.9783			1.43713
$560$	−1.07838			−0.0455698
$561$		0			0
$562$	26.3234			1.11038
$563$	0.779243			0.0328412			0.0164206	−	0.999865i	$-0.494773\pi$
$563$	0.779243			0.0328412			0.0164206	+	0.999865i	$0.494773\pi$
$564$		0			0
$565$	−7.82991			−0.329407
$566$			2.42082i			0.101755i
$567$		0			0
$568$		−	33.0304i		−	1.38592i
$569$	18.8638			0.790810			0.395405	−	0.918507i	$-0.370604\pi$
$569$	18.8638			0.790810			0.395405	+	0.918507i	$0.370604\pi$
$570$		0			0
$571$		−	39.6697i		−	1.66012i	−0.557671	−	0.830062i	$-0.688305\pi$
$571$		−	39.6697i		−	1.66012i	0.557671	−	0.830062i	$-0.311695\pi$
$572$			1.73594i			0.0725832i
$573$		0			0
$574$	−4.87936			−0.203661
$575$		−	35.1422i		−	1.46553i
$576$		0			0
$577$	34.2072			1.42407			0.712033	−	0.702146i	$-0.247775\pi$
$577$	34.2072			1.42407			0.712033	+	0.702146i	$0.247775\pi$
$578$	−4.80325	−	19.3028i	−0.199789	−	0.802892i
$579$		0			0
$580$	−0.496928			−0.0206338
$581$		−	0.496928i		−	0.0206161i
$582$		0			0
$583$			3.25461i			0.134792i
$584$			16.1157i			0.666873i
$585$		0			0
$586$	34.6369			1.43084
$587$	−26.1990			−1.08135			−0.540675	−	0.841232i	$-0.681831\pi$
$587$	−26.1990			−1.08135			−0.540675	+	0.841232i	$0.681831\pi$
$588$		0			0
$589$			20.3474i			0.838398i
$590$			1.73594i			0.0714674i
$591$		0			0
$592$			4.46573i			0.183540i
$593$	0.749268			0.0307688			0.0153844	−	0.999882i	$-0.495103\pi$
$593$	0.749268			0.0307688			0.0153844	+	0.999882i	$0.495103\pi$
$594$		0			0
$595$	1.17009	−	1.49693i	0.0479689	−	0.0613681i
$596$	8.91178			0.365041
$597$		0			0
$598$		−	33.3184i		−	1.36249i
$599$	−5.67420			−0.231842			−0.115921	−	0.993258i	$-0.536982\pi$
$599$	−5.67420			−0.231842			−0.115921	+	0.993258i	$0.536982\pi$
$600$		0			0
$601$		−	2.97826i		−	0.121486i	−0.998153	−	0.0607428i	$-0.980653\pi$
$601$		−	2.97826i		−	0.121486i	0.998153	−	0.0607428i	$-0.0193470\pi$
$602$		−	10.2485i		−	0.417696i
$603$		0			0
$604$	−10.2778			−0.418197
$605$			4.83710i			0.196656i
$606$		0			0
$607$		−	33.3328i		−	1.35294i	−0.736472	−	0.676468i	$-0.763510\pi$
$607$		−	33.3328i		−	1.35294i	0.736472	−	0.676468i	$-0.236490\pi$
$608$	14.2557			0.578143
$609$		0			0
$610$	4.57304			0.185157
$611$	−26.0794			−1.05506
$612$		0			0
$613$	9.65368			0.389909			0.194954	−	0.980812i	$-0.437544\pi$
$613$	9.65368			0.389909			0.194954	+	0.980812i	$0.437544\pi$
$614$	−3.84778			−0.155284
$615$		0			0
$616$	2.18342			0.0879724
$617$			15.7815i			0.635340i	0.948201	+	0.317670i	$0.102900\pi$
$617$			15.7815i			0.635340i	−0.948201	+	0.317670i	$0.897100\pi$
$618$		0			0
$619$			22.0566i			0.886531i	0.896390	+	0.443266i	$0.146180\pi$
$619$			22.0566i			0.886531i	−0.896390	+	0.443266i	$0.853820\pi$
$620$	1.41855			0.0569704
$621$		0			0
$622$			22.5068i			0.902439i
$623$			3.60197i			0.144310i
$624$		0			0
$625$	21.8599			0.874396
$626$			10.2374i			0.409169i
$627$		0			0
$628$	12.4741			0.497772
$629$	−6.19902	−	4.84551i	−0.247171	−	0.193203i
$630$		0			0
$631$	−39.4801			−1.57168			−0.785839	−	0.618431i	$-0.787769\pi$
$631$	−39.4801			−1.57168			−0.785839	+	0.618431i	$0.787769\pi$
$632$		−	43.6020i		−	1.73439i
$633$		0			0
$634$			21.3919i			0.849580i
$635$			1.94110i			0.0770301i
$636$		0			0
$637$	−3.87936			−0.153706
$638$	−1.41855			−0.0561610
$639$		0			0
$640$			1.52973i			0.0604680i
$641$		−	45.2122i		−	1.78577i	−0.450281	−	0.892887i	$-0.648676\pi$
$641$		−	45.2122i		−	1.78577i	0.450281	−	0.892887i	$-0.351324\pi$
$642$		0			0
$643$			30.3090i			1.19527i	0.801769	+	0.597635i	$0.203893\pi$
$643$			30.3090i			1.19527i	−0.801769	+	0.597635i	$0.796107\pi$
$644$	4.63090			0.182483
$645$		0			0
$646$	12.3896	−	15.8504i	0.487463	−	0.623627i
$647$	−35.8576			−1.40971			−0.704854	−	0.709352i	$-0.748988\pi$
$647$	−35.8576			−1.40971			−0.704854	+	0.709352i	$0.748988\pi$
$648$		0			0
$649$		−	2.28354i		−	0.0896367i
$650$	21.7321			0.852402
$651$		0			0
$652$			3.22076i			0.126135i
$653$			38.4452i			1.50448i	0.658891	+	0.752239i	$0.271026\pi$
$653$			38.4452i			1.50448i	−0.658891	+	0.752239i	$0.728974\pi$
$654$		0			0
$655$	7.38962			0.288736
$656$			9.75872i			0.381014i
$657$		0			0
$658$			7.86603i			0.306650i
$659$	25.7392			1.00266			0.501329	−	0.865257i	$-0.332844\pi$
$659$	25.7392			1.00266			0.501329	+	0.865257i	$0.332844\pi$
$660$		0			0
$661$	16.0589			0.624619			0.312309	−	0.949980i	$-0.398897\pi$
$661$	16.0589			0.624619			0.312309	+	0.949980i	$0.398897\pi$
$662$	32.4463			1.26106
$663$		0			0
$664$	−1.52973			−0.0593652
$665$	1.92162			0.0745173
$666$		0			0
$667$	−12.5464			−0.485798
$668$		−	1.03877i		−	0.0401912i
$669$		0			0
$670$		−	3.67420i		−	0.141947i
$671$	−6.01560			−0.232230
$672$		0			0
$673$		−	32.2388i		−	1.24272i	−0.783527	−	0.621358i	$-0.786581\pi$
$673$		−	32.2388i		−	1.24272i	0.783527	−	0.621358i	$-0.213419\pi$
$674$			26.4787i			1.01992i
$675$		0			0
$676$	−1.29299			−0.0497305
$677$			35.5718i			1.36714i	0.729887	+	0.683568i	$0.239573\pi$
$677$			35.5718i			1.36714i	−0.729887	+	0.683568i	$0.760427\pi$
$678$		0			0
$679$	−11.0784			−0.425149
$680$	−4.60811	−	3.60197i	−0.176713	−	0.138129i
$681$		0			0
$682$	4.04945			0.155061
$683$			9.67647i			0.370260i	0.982714	+	0.185130i	$0.0592706\pi$
$683$			9.67647i			0.370260i	−0.982714	+	0.185130i	$0.940729\pi$
$684$		0			0
$685$			2.96388i			0.113244i
$686$			1.17009i			0.0446741i
$687$		0			0
$688$	−20.4969			−0.781438
$689$	−17.8010			−0.678163
$690$		0			0
$691$			29.7887i			1.13322i	0.823988	+	0.566608i	$0.191745\pi$
$691$			29.7887i			1.13322i	−0.823988	+	0.566608i	$0.808255\pi$
$692$			10.5464i			0.400913i
$693$		0			0
$694$			1.17274i			0.0445166i
$695$	−5.73206			−0.217429
$696$		0			0
$697$	−13.5464	−	10.5886i	−0.513106	−	0.401073i
$698$	−12.2355			−0.463121
$699$		0			0
$700$			3.02052i			0.114165i
$701$	4.39803			0.166111			0.0830557	−	0.996545i	$-0.473532\pi$
$701$	4.39803			0.166111			0.0830557	+	0.996545i	$0.473532\pi$
$702$		0			0
$703$		−	7.95774i		−	0.300132i
$704$		−	6.15676i		−	0.232041i
$705$		0			0
$706$	−22.4208			−0.843819
$707$		−	14.0566i		−	0.528654i
$708$		0			0
$709$		−	17.3607i		−	0.651994i	−0.945371	−	0.325997i	$-0.894300\pi$
$709$		−	17.3607i		−	0.651994i	0.945371	−	0.325997i	$-0.105700\pi$
$710$	−5.78539			−0.217122
$711$		0			0
$712$	11.0882			0.415549
$713$	35.8154			1.34130
$714$		0			0
$715$	1.26794			0.0474182
$716$	15.1506			0.566205
$717$		0			0
$718$	−5.46800			−0.204064
$719$			24.9265i			0.929603i	0.885415	+	0.464802i	$0.153874\pi$
$719$			24.9265i			0.929603i	−0.885415	+	0.464802i	$0.846126\pi$
$720$		0			0
$721$			17.0566i			0.635222i
$722$	−1.88428			−0.0701257
$723$		0			0
$724$		−	12.3135i		−	0.457628i
$725$		−	8.18342i		−	0.303924i
$726$		0			0
$727$	−20.7115			−0.768149			−0.384074	−	0.923302i	$-0.625479\pi$
$727$	−20.7115			−0.768149			−0.384074	+	0.923302i	$0.625479\pi$
$728$			11.9421i			0.442605i
$729$		0			0
$730$	2.82273			0.104474
$731$	22.2401	−	28.4524i	0.822578	−	1.05235i
$732$		0			0
$733$	−2.81044			−0.103806			−0.0519030	−	0.998652i	$-0.516529\pi$
$733$	−2.81044			−0.103806			−0.0519030	+	0.998652i	$0.516529\pi$
$734$		−	38.8371i		−	1.43350i
$735$		0			0
$736$		−	25.0928i		−	0.924931i
$737$			4.83323i			0.178034i
$738$		0			0
$739$	−41.2388			−1.51699			−0.758497	−	0.651676i	$-0.774066\pi$
$739$	−41.2388			−1.51699			−0.758497	+	0.651676i	$0.774066\pi$
$740$	−0.554787			−0.0203944
$741$		0			0
$742$			5.36910i			0.197106i
$743$		−	52.1750i		−	1.91412i	−0.289898	−	0.957058i	$-0.593621\pi$
$743$		−	52.1750i		−	1.91412i	0.289898	−	0.957058i	$-0.406379\pi$
$744$		0			0
$745$		−	6.50921i		−	0.238479i
$746$	−5.88428			−0.215439
$747$		0			0
$748$	1.45362	+	1.13624i	0.0531497	+	0.0415449i
$749$	−0.973338			−0.0355650
$750$		0			0
$751$			53.0544i			1.93598i	0.250986	+	0.967991i	$0.419245\pi$
$751$			53.0544i			1.93598i	−0.250986	+	0.967991i	$0.580755\pi$
$752$	15.7321			0.573689
$753$		0			0
$754$		−	7.75872i		−	0.282556i
$755$			7.50695i			0.273206i
$756$		0			0
$757$	34.7548			1.26319			0.631593	−	0.775300i	$-0.282401\pi$
$757$	34.7548			1.26319			0.631593	+	0.775300i	$0.282401\pi$
$758$			37.0616i			1.34614i
$759$		0			0
$760$		−	5.91548i		−	0.214577i
$761$	−1.60650			−0.0582357			−0.0291178	−	0.999576i	$-0.509270\pi$
$761$	−1.60650			−0.0582357			−0.0291178	+	0.999576i	$0.509270\pi$
$762$		0			0
$763$	−2.40522			−0.0870748
$764$	−6.83710			−0.247358
$765$		0			0
$766$	9.28005			0.335302
$767$	12.4897			0.450978
$768$		0			0
$769$	40.9676			1.47733			0.738664	−	0.674073i	$-0.235457\pi$
$769$	40.9676			1.47733			0.738664	+	0.674073i	$0.235457\pi$
$770$		−	0.382433i		−	0.0137819i
$771$		0			0
$772$			4.87360i			0.175405i
$773$	6.66475			0.239714			0.119857	−	0.992791i	$-0.461756\pi$
$773$	6.66475			0.239714			0.119857	+	0.992791i	$0.461756\pi$
$774$		0			0
$775$			23.3607i			0.839141i
$776$			34.1034i			1.22424i
$777$		0			0
$778$	−19.3652			−0.694277
$779$		−	17.3896i		−	0.623048i
$780$		0			0
$781$	7.61038			0.272321
$782$	−27.8999	−	21.8082i	−0.997698	−	0.779859i
$783$		0			0
$784$	2.34017			0.0835776
$785$		−	9.11118i		−	0.325192i
$786$		0			0
$787$		−	12.5080i		−	0.445862i	−0.974834	−	0.222931i	$-0.928438\pi$
$787$		−	12.5080i		−	0.445862i	0.974834	−	0.222931i	$-0.0715624\pi$
$788$		−	8.33176i		−	0.296807i
$789$		0			0
$790$	−7.63704			−0.271714
$791$	16.9916			0.604151
$792$		0			0
$793$		−	32.9021i		−	1.16839i
$794$		−	30.8554i		−	1.09502i
$795$		0			0
$796$		−	11.0784i		−	0.392663i
$797$	−0.707008			−0.0250435			−0.0125218	−	0.999922i	$-0.503986\pi$
$797$	−0.707008			−0.0250435			−0.0125218	+	0.999922i	$0.503986\pi$
$798$		0			0
$799$	−17.0700	+	21.8381i	−0.603892	+	0.772578i
$800$	16.3668			0.578655
$801$		0			0
$802$		−	30.7187i		−	1.08472i
$803$	−3.71315			−0.131034
$804$		0			0
$805$		−	3.38243i		−	0.119215i
$806$			22.1483i			0.780142i
$807$		0			0
$808$	−43.2716			−1.52229
$809$			8.20006i			0.288299i	0.989556	+	0.144149i	$0.0460446\pi$
$809$			8.20006i			0.288299i	−0.989556	+	0.144149i	$0.953955\pi$
$810$		0			0
$811$		−	49.6697i		−	1.74414i	−0.489383	−	0.872069i	$-0.662778\pi$
$811$		−	49.6697i		−	1.74414i	0.489383	−	0.872069i	$-0.337222\pi$
$812$	1.07838			0.0378436
$813$		0			0
$814$	−1.58372			−0.0555092
$815$	2.35246			0.0824030
$816$		0			0
$817$	36.5246			1.27784
$818$	17.8687			0.624764
$819$		0			0
$820$	−1.21235			−0.0423370
$821$			3.29914i			0.115141i	0.998341	+	0.0575703i	$0.0183353\pi$
$821$			3.29914i			0.115141i	−0.998341	+	0.0575703i	$0.981665\pi$
$822$		0			0
$823$			6.60092i			0.230094i	0.993360	+	0.115047i	$0.0367018\pi$
$823$			6.60092i			0.230094i	−0.993360	+	0.115047i	$0.963298\pi$
$824$	52.5068			1.82916
$825$		0			0
$826$		−	3.76713i		−	0.131075i
$827$			36.9383i			1.28447i	0.766508	+	0.642235i	$0.221993\pi$
$827$			36.9383i			1.28447i	−0.766508	+	0.642235i	$0.778007\pi$
$828$		0			0
$829$	−38.0288			−1.32079			−0.660397	−	0.750917i	$-0.729612\pi$
$829$	−38.0288			−1.32079			−0.660397	+	0.750917i	$0.729612\pi$
$830$			0.267938i			0.00930027i
$831$		0			0
$832$	33.6742			1.16744
$833$	−2.53919	+	3.24846i	−0.0879777	+	0.112553i
$834$		0			0
$835$	−0.758724			−0.0262567
$836$			1.86603i			0.0645380i
$837$		0			0
$838$			11.2085i			0.387190i
$839$			51.0893i			1.76380i	0.471439	+	0.881899i	$0.343735\pi$
$839$			51.0893i			1.76380i	−0.471439	+	0.881899i	$0.656265\pi$
$840$		0			0
$841$	26.0784			0.899254
$842$	39.3679			1.35671
$843$		0			0
$844$		−	2.61795i		−	0.0901136i
$845$			0.944409i			0.0324886i
$846$		0			0
$847$		−	10.4969i		−	0.360679i
$848$	10.7382			0.368751
$849$		0			0
$850$	14.2245	−	18.1978i	0.487895	−	0.624179i
$851$	−14.0072			−0.480160
$852$		0			0
$853$			56.9081i			1.94850i	0.225478	+	0.974248i	$0.427606\pi$
$853$			56.9081i			1.94850i	−0.225478	+	0.974248i	$0.572394\pi$
$854$	−9.92389			−0.339589
$855$		0			0
$856$			2.99630i			0.102411i
$857$			11.7731i			0.402161i	0.979575	+	0.201081i	$0.0644454\pi$
$857$			11.7731i			0.402161i	−0.979575	+	0.201081i	$0.935555\pi$
$858$		0			0
$859$	−19.0349			−0.649462			−0.324731	−	0.945806i	$-0.605274\pi$
$859$	−19.0349			−0.649462			−0.324731	+	0.945806i	$0.605274\pi$
$860$		−	2.54638i		−	0.0868307i
$861$		0			0
$862$			18.2922i			0.623033i
$863$	11.2858			0.384173			0.192087	−	0.981378i	$-0.438475\pi$
$863$	11.2858			0.384173			0.192087	+	0.981378i	$0.438475\pi$
$864$		0			0
$865$	7.70313			0.261914
$866$	−12.2302			−0.415600
$867$		0			0
$868$	−3.07838			−0.104487
$869$	10.0461			0.340792
$870$		0			0
$871$	−26.4352			−0.895722
$872$			7.40417i			0.250737i
$873$		0			0
$874$		−	35.8154i		−	1.21147i
$875$	4.51026			0.152475
$876$		0			0
$877$			15.4257i			0.520890i	0.965489	+	0.260445i	$0.0838693\pi$
$877$			15.4257i			0.520890i	−0.965489	+	0.260445i	$0.916131\pi$
$878$		−	15.9817i		−	0.539358i
$879$		0			0
$880$	−0.764867			−0.0257837
$881$		−	30.1301i		−	1.01511i	−0.861620	−	0.507554i	$-0.830550\pi$
$881$		−	30.1301i		−	1.01511i	0.861620	−	0.507554i	$-0.169450\pi$
$882$		0			0
$883$	−37.1217			−1.24924			−0.624622	−	0.780927i	$-0.714747\pi$
$883$	−37.1217			−1.24924			−0.624622	+	0.780927i	$0.714747\pi$
$884$	−6.21461	+	7.95055i	−0.209020	+	0.267406i
$885$		0			0
$886$	−18.1171			−0.608657
$887$			17.8033i			0.597775i	0.954288	+	0.298887i	$0.0966155\pi$
$887$			17.8033i			0.597775i	−0.954288	+	0.298887i	$0.903385\pi$
$888$		0			0
$889$		−	4.21235i		−	0.141278i
$890$		−	1.94214i		−	0.0651007i
$891$		0			0
$892$	2.33176			0.0780732
$893$	−28.0338			−0.938117
$894$		0			0
$895$		−	11.0661i		−	0.369899i
$896$		−	3.31965i		−	0.110902i
$897$		0			0
$898$			19.7899i			0.660398i
$899$	8.34017			0.278160
$900$		0			0
$901$	−11.6514	+	14.9060i	−0.388165	+	0.496592i
$902$	−3.46081			−0.115232
$903$		0			0
$904$		−	52.3065i		−	1.73969i
$905$	−8.99386			−0.298966
$906$		0			0
$907$			19.8576i			0.659361i	0.944092	+	0.329681i	$0.106941\pi$
$907$			19.8576i			0.659361i	−0.944092	+	0.329681i	$0.893059\pi$
$908$		−	7.11345i		−	0.236068i
$909$		0			0
$910$	2.09171			0.0693395
$911$			37.1483i			1.23078i	0.788223	+	0.615390i	$0.211001\pi$
$911$			37.1483i			1.23078i	−0.788223	+	0.615390i	$0.788999\pi$
$912$		0			0
$913$		−	0.352459i		−	0.0116647i
$914$	9.34366			0.309061
$915$		0			0
$916$	6.28685			0.207723
$917$	−16.0361			−0.529559
$918$		0			0
$919$	−29.4268			−0.970700			−0.485350	−	0.874320i	$-0.661308\pi$
$919$	−29.4268			−0.970700			−0.485350	+	0.874320i	$0.661308\pi$
$920$	−10.4124			−0.343287
$921$		0			0
$922$	33.2579			1.09529
$923$			41.6248i			1.37010i
$924$		0			0
$925$		−	9.13624i		−	0.300398i
$926$	3.39189			0.111464
$927$		0			0
$928$		−	5.84324i		−	0.191814i
$929$		−	37.7864i		−	1.23973i	−0.784707	−	0.619866i	$-0.787187\pi$
$929$		−	37.7864i		−	1.23973i	0.784707	−	0.619866i	$-0.212813\pi$
$930$		0			0
$931$	−4.17009			−0.136669
$932$		−	16.3461i		−	0.535436i
$933$		0			0
$934$	1.16290			0.0380512
$935$	0.829914	−	1.06173i	0.0271411	−	0.0347224i
$936$		0			0
$937$	−37.5981			−1.22828			−0.614138	−	0.789199i	$-0.710496\pi$
$937$	−37.5981			−1.22828			−0.614138	+	0.789199i	$0.710496\pi$
$938$			7.97334i			0.260339i
$939$		0			0
$940$			1.95443i			0.0637464i
$941$			8.49239i			0.276844i	0.990373	+	0.138422i	$0.0442030\pi$
$941$			8.49239i			0.276844i	−0.990373	+	0.138422i	$0.955797\pi$
$942$		0			0
$943$	−30.6092			−0.996771
$944$	−7.53427			−0.245220
$945$		0			0
$946$		−	7.26898i		−	0.236335i
$947$		−	29.0166i		−	0.942914i	−0.881889	−	0.471457i	$-0.843728\pi$
$947$		−	29.0166i		−	0.942914i	0.881889	−	0.471457i	$-0.156272\pi$
$948$		0			0
$949$		−	20.3090i		−	0.659257i
$950$	23.3607			0.757921
$951$		0			0
$952$	10.0000	+	7.81658i	0.324102	+	0.253337i
$953$	−52.5574			−1.70250			−0.851251	−	0.524758i	$-0.824156\pi$
$953$	−52.5574			−1.70250			−0.851251	+	0.524758i	$0.824156\pi$
$954$		0			0
$955$			4.99386i			0.161597i
$956$	−12.2725			−0.396920
$957$		0			0
$958$			20.9783i			0.677777i
$959$		−	6.43188i		−	0.207696i
$960$		0			0
$961$	7.19183			0.231994
$962$		−	8.66209i		−	0.279277i
$963$		0			0
$964$			13.8432i			0.445861i
$965$	3.55971			0.114591
$966$		0			0
$967$	−11.1795			−0.359510			−0.179755	−	0.983711i	$-0.557530\pi$
$967$	−11.1795			−0.359510			−0.179755	+	0.983711i	$0.557530\pi$
$968$	−32.3135			−1.03860
$969$		0			0
$970$	5.97334			0.191792
$971$	15.7275			0.504720			0.252360	−	0.967633i	$-0.418793\pi$
$971$	15.7275			0.504720			0.252360	+	0.967633i	$0.418793\pi$
$972$		0			0
$973$	12.4391			0.398778
$974$			14.0891i			0.451442i
$975$		0			0
$976$			19.8478i			0.635312i
$977$	−45.9688			−1.47067			−0.735336	−	0.677703i	$-0.762976\pi$
$977$	−45.9688			−1.47067			−0.735336	+	0.677703i	$0.762976\pi$
$978$		0			0
$979$			2.55479i			0.0816514i
$980$			0.290725i			0.00928686i
$981$		0			0
$982$	−22.9048			−0.730922
$983$			14.6670i			0.467805i	0.972260	+	0.233903i	$0.0751497\pi$
$983$			14.6670i			0.467805i	−0.972260	+	0.233903i	$0.924850\pi$
$984$		0			0
$985$	−6.08557			−0.193902
$986$	−6.49693	−	5.07838i	−0.206904	−	0.161728i
$987$		0			0
$988$	−10.2062			−0.324703
$989$		−	64.2905i		−	2.04432i
$990$		0			0
$991$			17.5343i			0.556994i	0.960437	+	0.278497i	$0.0898363\pi$
$991$			17.5343i			0.556994i	−0.960437	+	0.278497i	$0.910164\pi$
$992$			16.6803i			0.529602i
$993$		0			0
$994$	12.5548			0.398214
$995$	−8.09171			−0.256524
$996$		0			0
$997$			57.1506i			1.80998i	0.425435	+	0.904989i	$0.360121\pi$
$997$			57.1506i			1.80998i	−0.425435	+	0.904989i	$0.639879\pi$
$998$			35.0433i			1.10928i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1071.2.f.a.883.6		6
3.2	odd	2		357.2.f.a.169.2	yes	6
17.16	even	2	inner	1071.2.f.a.883.5		6
51.38	odd	4		6069.2.a.m.1.3		3
51.47	odd	4		6069.2.a.k.1.3		3
51.50	odd	2		357.2.f.a.169.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.f.a.169.1	✓	6	51.50	odd	2
357.2.f.a.169.2	yes	6	3.2	odd	2
1071.2.f.a.883.5		6	17.16	even	2	inner
1071.2.f.a.883.6		6	1.1	even	1	trivial
6069.2.a.k.1.3		3	51.47	odd	4
6069.2.a.m.1.3		3	51.38	odd	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1071 = 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1071.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.17009 q^{2} -0.630898 q^{4} +0.460811i q^{5} -1.00000i q^{7} -3.07838 q^{8} +O(q^{10})\)	\(q+1.17009 q^{2} -0.630898 q^{4} +0.460811i q^{5} -1.00000i q^{7} -3.07838 q^{8} +0.539189i q^{10} -0.709275i q^{11} +3.87936 q^{13} -1.17009i q^{14} -2.34017 q^{16} +(2.53919 - 3.24846i) q^{17} +4.17009 q^{19} -0.290725i q^{20} -0.829914i q^{22} -7.34017i q^{23} +4.78765 q^{25} +4.53919 q^{26} +0.630898i q^{28} -1.70928i q^{29} +4.87936i q^{31} +3.41855 q^{32} +(2.97107 - 3.80098i) q^{34} +0.460811 q^{35} -1.90829i q^{37} +4.87936 q^{38} -1.41855i q^{40} -4.17009i q^{41} +8.75872 q^{43} +0.447480i q^{44} -8.58864i q^{46} -6.72261 q^{47} -1.00000 q^{49} +5.60197 q^{50} -2.44748 q^{52} -4.58864 q^{53} +0.326842 q^{55} +3.07838i q^{56} -2.00000i q^{58} +3.21953 q^{59} -8.48133i q^{61} +5.70928i q^{62} +8.68035 q^{64} +1.78765i q^{65} -6.81432 q^{67} +(-1.60197 + 2.04945i) q^{68} +0.539189 q^{70} +10.7298i q^{71} -5.23513i q^{73} -2.23287i q^{74} -2.63090 q^{76} -0.709275 q^{77} +14.1639i q^{79} -1.07838i q^{80} -4.87936i q^{82} +0.496928 q^{83} +(1.49693 + 1.17009i) q^{85} +10.2485 q^{86} +2.18342i q^{88} -3.60197 q^{89} -3.87936i q^{91} +4.63090i q^{92} -7.86603 q^{94} +1.92162i q^{95} -11.0784i q^{97} -1.17009 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 4 q^{4} - 12 q^{8}+O(q^{10}) \) 6 * q - 4 * q^2 + 4 * q^4 - 12 * q^8	\( 6 q - 4 q^{2} + 4 q^{4} - 12 q^{8} - 2 q^{13} + 8 q^{16} + 12 q^{17} + 14 q^{19} + 8 q^{25} + 24 q^{26} - 8 q^{32} - 12 q^{34} + 6 q^{35} + 4 q^{38} + 2 q^{43} - 28 q^{47} - 6 q^{49} - 4 q^{50} - 16 q^{52} + 12 q^{53} - 22 q^{55} - 28 q^{59} + 8 q^{64} - 24 q^{67} + 28 q^{68} - 8 q^{76} + 10 q^{77} - 32 q^{83} - 26 q^{85} + 44 q^{86} + 16 q^{89} - 20 q^{94} + 4 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 + 4 * q^4 - 12 * q^8 - 2 * q^13 + 8 * q^16 + 12 * q^17 + 14 * q^19 + 8 * q^25 + 24 * q^26 - 8 * q^32 - 12 * q^34 + 6 * q^35 + 4 * q^38 + 2 * q^43 - 28 * q^47 - 6 * q^49 - 4 * q^50 - 16 * q^52 + 12 * q^53 - 22 * q^55 - 28 * q^59 + 8 * q^64 - 24 * q^67 + 28 * q^68 - 8 * q^76 + 10 * q^77 - 32 * q^83 - 26 * q^85 + 44 * q^86 + 16 * q^89 - 20 * q^94 + 4 * q^98

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