LMFDB - Embedded newform 1600.2.o.f.607.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1600,2,Mod(543,1600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1600.543"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.o (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,-112] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(39)] == traces)

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

Self dual:	no
Analytic conductor:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.12745506816.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			607.1
Root			$1.54779 + 1.54779i$ of defining polynomial
Character	$\chi$	$=$	1600.607
Dual form			1600.2.o.f.543.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.87083 + 1.87083i) q^{3} +(-2.44949 + 2.44949i) q^{7} -4.00000i q^{9} +4.58258 q^{11} +(3.74166 + 3.74166i) q^{13} +(-1.22474 - 1.22474i) q^{17} +4.58258i q^{19} -9.16515i q^{21} +(4.89898 + 4.89898i) q^{23} +(1.87083 + 1.87083i) q^{27} +9.16515 q^{29} +4.00000i q^{31} +(-8.57321 + 8.57321i) q^{33} +(3.74166 - 3.74166i) q^{37} -14.0000 q^{39} -9.00000 q^{41} +(-3.74166 + 3.74166i) q^{43} +(2.44949 - 2.44949i) q^{47} -5.00000i q^{49} +4.58258 q^{51} +(-8.57321 - 8.57321i) q^{57} +9.16515i q^{61} +(9.79796 + 9.79796i) q^{63} +(-1.87083 - 1.87083i) q^{67} -18.3303 q^{69} -6.00000i q^{71} +(-6.12372 + 6.12372i) q^{73} +(-11.2250 + 11.2250i) q^{77} -10.0000 q^{79} +5.00000 q^{81} +(5.61249 - 5.61249i) q^{83} +(-17.1464 + 17.1464i) q^{87} -15.0000i q^{89} -18.3303 q^{91} +(-7.48331 - 7.48331i) q^{93} +(-4.89898 - 4.89898i) q^{97} -18.3303i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 112 q^{39} - 72 q^{41} - 80 q^{79} + 40 q^{81}+O(q^{100}) $ 8 * q - 112 * q^39 - 72 * q^41 - 80 * q^79 + 40 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.87083	+	1.87083i	−1.08012	+	1.08012i	−0.0836263	+	0.996497i	$0.526650\pi$
$3$	−1.87083	+	1.87083i	−1.08012	+	1.08012i	−0.996497	+	0.0836263i	$0.973350\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.44949	+	2.44949i	−0.925820	+	0.925820i	−0.997433	−	0.0716124i	$-0.977186\pi$
$7$	−2.44949	+	2.44949i	−0.925820	+	0.925820i	0.0716124	+	0.997433i	$0.477186\pi$
$8$		0			0
$9$		−	4.00000i		−	1.33333i
$10$		0			0
$11$	4.58258			1.38170			0.690849	−	0.722999i	$-0.257237\pi$
$11$	4.58258			1.38170			0.690849	+	0.722999i	$0.257237\pi$
$12$		0			0
$13$	3.74166	+	3.74166i	1.03775	+	1.03775i	0.999259	+	0.0384901i	$0.0122548\pi$
$13$	3.74166	+	3.74166i	1.03775	+	1.03775i	0.0384901	+	0.999259i	$0.487745\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.22474	−	1.22474i	−0.297044	−	0.297044i	0.542811	−	0.839855i	$-0.317360\pi$
$17$	−1.22474	−	1.22474i	−0.297044	−	0.297044i	−0.839855	+	0.542811i	$0.817360\pi$
$18$		0			0
$19$			4.58258i			1.05131i	0.850696	+	0.525657i	$0.176181\pi$
$19$			4.58258i			1.05131i	−0.850696	+	0.525657i	$0.823819\pi$
$20$		0			0
$21$		−	9.16515i		−	2.00000i
$22$		0			0
$23$	4.89898	+	4.89898i	1.02151	+	1.02151i	0.999764	+	0.0217443i	$0.00692196\pi$
$23$	4.89898	+	4.89898i	1.02151	+	1.02151i	0.0217443	+	0.999764i	$0.493078\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.87083	+	1.87083i	0.360041	+	0.360041i
$28$		0			0
$29$	9.16515			1.70193			0.850963	−	0.525226i	$-0.176019\pi$
$29$	9.16515			1.70193			0.850963	+	0.525226i	$0.176019\pi$
$30$		0			0
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		0			0
$33$	−8.57321	+	8.57321i	−1.49241	+	1.49241i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.74166	−	3.74166i	0.615125	−	0.615125i	−0.329152	−	0.944277i	$-0.606763\pi$
$37$	3.74166	−	3.74166i	0.615125	−	0.615125i	0.944277	+	0.329152i	$0.106763\pi$
$38$		0			0
$39$	−14.0000			−2.24179
$40$		0			0
$41$	−9.00000			−1.40556			−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000			−1.40556			−0.702782	+	0.711405i	$0.748059\pi$
$42$		0			0
$43$	−3.74166	+	3.74166i	−0.570597	+	0.570597i	−0.932295	−	0.361698i	$-0.882197\pi$
$43$	−3.74166	+	3.74166i	−0.570597	+	0.570597i	0.361698	+	0.932295i	$0.382197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.44949	−	2.44949i	0.357295	−	0.357295i	−0.505520	−	0.862815i	$-0.668699\pi$
$47$	2.44949	−	2.44949i	0.357295	−	0.357295i	0.862815	+	0.505520i	$0.168699\pi$
$48$		0			0
$49$		−	5.00000i		−	0.714286i
$50$		0			0
$51$	4.58258			0.641689
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−8.57321	−	8.57321i	−1.13555	−	1.13555i
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$			9.16515i			1.17348i	0.809776	+	0.586739i	$0.199588\pi$
$61$			9.16515i			1.17348i	−0.809776	+	0.586739i	$0.800412\pi$
$62$		0			0
$63$	9.79796	+	9.79796i	1.23443	+	1.23443i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.87083	−	1.87083i	−0.228558	−	0.228558i	0.583532	−	0.812090i	$-0.301670\pi$
$67$	−1.87083	−	1.87083i	−0.228558	−	0.228558i	−0.812090	+	0.583532i	$0.801670\pi$
$68$		0			0
$69$	−18.3303			−2.20671
$70$		0			0
$71$		−	6.00000i		−	0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$		−	6.00000i		−	0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$		0			0
$73$	−6.12372	+	6.12372i	−0.716728	+	0.716728i	−0.967934	−	0.251206i	$-0.919173\pi$
$73$	−6.12372	+	6.12372i	−0.716728	+	0.716728i	0.251206	+	0.967934i	$0.419173\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−11.2250	+	11.2250i	−1.27920	+	1.27920i
$78$		0			0
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	5.61249	−	5.61249i	0.616050	−	0.616050i	−0.328466	−	0.944516i	$-0.606531\pi$
$83$	5.61249	−	5.61249i	0.616050	−	0.616050i	0.944516	+	0.328466i	$0.106531\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−17.1464	+	17.1464i	−1.83829	+	1.83829i
$88$		0			0
$89$		−	15.0000i		−	1.59000i	−0.606612	−	0.794998i	$-0.707472\pi$
$89$		−	15.0000i		−	1.59000i	0.606612	−	0.794998i	$-0.292528\pi$
$90$		0			0
$91$	−18.3303			−1.92154
$92$		0			0
$93$	−7.48331	−	7.48331i	−0.775984	−	0.775984i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.89898	−	4.89898i	−0.497416	−	0.497416i	0.413217	−	0.910633i	$-0.364405\pi$
$97$	−4.89898	−	4.89898i	−0.497416	−	0.497416i	−0.910633	+	0.413217i	$0.864405\pi$
$98$		0			0
$99$		−	18.3303i		−	1.84226i
$100$		0			0
$101$			9.16515i			0.911967i	0.889988	+	0.455983i	$0.150712\pi$
$101$			9.16515i			0.911967i	−0.889988	+	0.455983i	$0.849288\pi$
$102$		0			0
$103$	−2.44949	−	2.44949i	−0.241355	−	0.241355i	0.576055	−	0.817411i	$-0.304591\pi$
$103$	−2.44949	−	2.44949i	−0.241355	−	0.241355i	−0.817411	+	0.576055i	$0.804591\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.61249	−	5.61249i	−0.542580	−	0.542580i	0.381705	−	0.924284i	$-0.375337\pi$
$107$	−5.61249	−	5.61249i	−0.542580	−	0.542580i	−0.924284	+	0.381705i	$0.875337\pi$
$108$		0			0
$109$	9.16515			0.877862			0.438931	−	0.898521i	$-0.355357\pi$
$109$	9.16515			0.877862			0.438931	+	0.898521i	$0.355357\pi$
$110$		0			0
$111$			14.0000i			1.32882i
$112$		0			0
$113$	−1.22474	+	1.22474i	−0.115214	+	0.115214i	−0.762363	−	0.647149i	$-0.775961\pi$
$113$	−1.22474	+	1.22474i	−0.115214	+	0.115214i	0.647149	+	0.762363i	$0.275961\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	14.9666	−	14.9666i	1.38367	−	1.38367i
$118$		0			0
$119$	6.00000			0.550019
$120$		0			0
$121$	10.0000			0.909091
$122$		0			0
$123$	16.8375	−	16.8375i	1.51818	−	1.51818i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$		0			0
$129$		−	14.0000i		−	1.23263i
$130$		0			0
$131$	−18.3303			−1.60153			−0.800763	−	0.598981i	$-0.795572\pi$
$131$	−18.3303			−1.60153			−0.800763	+	0.598981i	$0.795572\pi$
$132$		0			0
$133$	−11.2250	−	11.2250i	−0.973329	−	0.973329i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.0227	+	11.0227i	0.941733	+	0.941733i	0.998394	−	0.0566604i	$-0.0180452\pi$
$137$	11.0227	+	11.0227i	0.941733	+	0.941733i	−0.0566604	+	0.998394i	$0.518045\pi$
$138$		0			0
$139$			4.58258i			0.388689i	0.980933	+	0.194344i	$0.0622580\pi$
$139$			4.58258i			0.388689i	−0.980933	+	0.194344i	$0.937742\pi$
$140$		0			0
$141$			9.16515i			0.771845i
$142$		0			0
$143$	17.1464	+	17.1464i	1.43386	+	1.43386i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	9.35414	+	9.35414i	0.771517	+	0.771517i
$148$		0			0
$149$	−9.16515			−0.750838			−0.375419	−	0.926855i	$-0.622501\pi$
$149$	−9.16515			−0.750838			−0.375419	+	0.926855i	$0.622501\pi$
$150$		0			0
$151$		−	10.0000i		−	0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$151$		−	10.0000i		−	0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$152$		0			0
$153$	−4.89898	+	4.89898i	−0.396059	+	0.396059i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−14.9666	+	14.9666i	−1.19447	+	1.19447i	−0.218668	+	0.975799i	$0.570171\pi$
$157$	−14.9666	+	14.9666i	−1.19447	+	1.19447i	−0.975799	+	0.218668i	$0.929829\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−24.0000			−1.89146
$162$		0			0
$163$	1.87083	−	1.87083i	0.146535	−	0.146535i	−0.630033	−	0.776568i	$-0.716959\pi$
$163$	1.87083	−	1.87083i	0.146535	−	0.146535i	0.776568	+	0.630033i	$0.216959\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.34847	−	7.34847i	0.568642	−	0.568642i	−0.363106	−	0.931748i	$-0.618284\pi$
$167$	7.34847	−	7.34847i	0.568642	−	0.568642i	0.931748	+	0.363106i	$0.118284\pi$
$168$		0			0
$169$			15.0000i			1.15385i
$170$		0			0
$171$	18.3303			1.40175
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			13.7477i			1.02755i	0.857924	+	0.513777i	$0.171754\pi$
$179$			13.7477i			1.02755i	−0.857924	+	0.513777i	$0.828246\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−17.1464	−	17.1464i	−1.26750	−	1.26750i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−5.61249	−	5.61249i	−0.410426	−	0.410426i
$188$		0			0
$189$	−9.16515			−0.666667
$190$		0			0
$191$			24.0000i			1.73658i	0.496058	+	0.868290i	$0.334780\pi$
$191$			24.0000i			1.73658i	−0.496058	+	0.868290i	$0.665220\pi$
$192$		0			0
$193$	6.12372	−	6.12372i	0.440795	−	0.440795i	−0.451484	−	0.892279i	$-0.649105\pi$
$193$	6.12372	−	6.12372i	0.440795	−	0.440795i	0.892279	+	0.451484i	$0.149105\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$	7.00000			0.493742
$202$		0			0
$203$	−22.4499	+	22.4499i	−1.57568	+	1.57568i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	19.5959	−	19.5959i	1.36201	−	1.36201i
$208$		0			0
$209$			21.0000i			1.45260i
$210$		0			0
$211$	13.7477			0.946433			0.473216	−	0.880946i	$-0.343093\pi$
$211$	13.7477			0.946433			0.473216	+	0.880946i	$0.343093\pi$
$212$		0			0
$213$	11.2250	+	11.2250i	0.769122	+	0.769122i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−9.79796	−	9.79796i	−0.665129	−	0.665129i
$218$		0			0
$219$		−	22.9129i		−	1.54831i
$220$		0			0
$221$		−	9.16515i		−	0.616515i
$222$		0			0
$223$	−4.89898	−	4.89898i	−0.328060	−	0.328060i	0.523788	−	0.851848i	$-0.324518\pi$
$223$	−4.89898	−	4.89898i	−0.328060	−	0.328060i	−0.851848	+	0.523788i	$0.824518\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.2250	−	11.2250i	−0.745028	−	0.745028i	0.228513	−	0.973541i	$-0.426614\pi$
$227$	−11.2250	−	11.2250i	−0.745028	−	0.745028i	−0.973541	+	0.228513i	$0.926614\pi$
$228$		0			0
$229$	18.3303			1.21130			0.605650	−	0.795731i	$-0.292913\pi$
$229$	18.3303			1.21130			0.605650	+	0.795731i	$0.292913\pi$
$230$		0			0
$231$		−	42.0000i		−	2.76340i
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	18.7083	−	18.7083i	1.21523	−	1.21523i
$238$		0			0
$239$	6.00000			0.388108			0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000			0.388108			0.194054	+	0.980991i	$0.437836\pi$
$240$		0			0
$241$	7.00000			0.450910			0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000			0.450910			0.225455	+	0.974254i	$0.427613\pi$
$242$		0			0
$243$	−14.9666	+	14.9666i	−0.960110	+	0.960110i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−17.1464	+	17.1464i	−1.09100	+	1.09100i
$248$		0			0
$249$			21.0000i			1.33082i
$250$		0			0
$251$	−4.58258			−0.289250			−0.144625	−	0.989487i	$-0.546198\pi$
$251$	−4.58258			−0.289250			−0.144625	+	0.989487i	$0.546198\pi$
$252$		0			0
$253$	22.4499	+	22.4499i	1.41142	+	1.41142i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.89898	+	4.89898i	0.305590	+	0.305590i	0.843196	−	0.537606i	$-0.180671\pi$
$257$	4.89898	+	4.89898i	0.305590	+	0.305590i	−0.537606	+	0.843196i	$0.680671\pi$
$258$		0			0
$259$			18.3303i			1.13899i
$260$		0			0
$261$		−	36.6606i		−	2.26923i
$262$		0			0
$263$	−14.6969	−	14.6969i	−0.906252	−	0.906252i	0.0897154	−	0.995967i	$-0.471404\pi$
$263$	−14.6969	−	14.6969i	−0.906252	−	0.906252i	−0.995967	+	0.0897154i	$0.971404\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	28.0624	+	28.0624i	1.71739	+	1.71739i
$268$		0			0
$269$	18.3303			1.11762			0.558809	−	0.829296i	$-0.311258\pi$
$269$	18.3303			1.11762			0.558809	+	0.829296i	$0.311258\pi$
$270$		0			0
$271$		−	8.00000i		−	0.485965i	−0.970031	−	0.242983i	$-0.921874\pi$
$271$		−	8.00000i		−	0.485965i	0.970031	−	0.242983i	$-0.0781258\pi$
$272$		0			0
$273$	34.2929	−	34.2929i	2.07550	−	2.07550i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−18.7083	+	18.7083i	−1.12407	+	1.12407i	−0.132949	+	0.991123i	$0.542445\pi$
$277$	−18.7083	+	18.7083i	−1.12407	+	1.12407i	−0.991123	+	0.132949i	$0.957555\pi$
$278$		0			0
$279$	16.0000			0.957895
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	9.35414	−	9.35414i	0.556046	−	0.556046i	−0.372133	−	0.928179i	$-0.621374\pi$
$283$	9.35414	−	9.35414i	0.556046	−	0.556046i	0.928179	+	0.372133i	$0.121374\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	22.0454	−	22.0454i	1.30130	−	1.30130i
$288$		0			0
$289$		−	14.0000i		−	0.823529i
$290$		0			0
$291$	18.3303			1.07454
$292$		0			0
$293$	11.2250	+	11.2250i	0.655770	+	0.655770i	0.954376	−	0.298606i	$-0.0965218\pi$
$293$	11.2250	+	11.2250i	0.655770	+	0.655770i	−0.298606	+	0.954376i	$0.596522\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	8.57321	+	8.57321i	0.497468	+	0.497468i
$298$		0			0
$299$			36.6606i			2.12014i
$300$		0			0
$301$		−	18.3303i		−	1.05654i
$302$		0			0
$303$	−17.1464	−	17.1464i	−0.985037	−	0.985037i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−9.35414	−	9.35414i	−0.533869	−	0.533869i	0.387852	−	0.921722i	$-0.373217\pi$
$307$	−9.35414	−	9.35414i	−0.533869	−	0.533869i	−0.921722	+	0.387852i	$0.873217\pi$
$308$		0			0
$309$	9.16515			0.521387
$310$		0			0
$311$			18.0000i			1.02069i	0.859971	+	0.510343i	$0.170482\pi$
$311$			18.0000i			1.02069i	−0.859971	+	0.510343i	$0.829518\pi$
$312$		0			0
$313$	14.6969	−	14.6969i	0.830720	−	0.830720i	−0.156895	−	0.987615i	$-0.550148\pi$
$313$	14.6969	−	14.6969i	0.830720	−	0.830720i	0.987615	+	0.156895i	$0.0501485\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	22.4499	−	22.4499i	1.26091	−	1.26091i	0.310264	−	0.950650i	$-0.399583\pi$
$317$	22.4499	−	22.4499i	1.26091	−	1.26091i	0.950650	−	0.310264i	$-0.100417\pi$
$318$		0			0
$319$	42.0000			2.35155
$320$		0			0
$321$	21.0000			1.17211
$322$		0			0
$323$	5.61249	−	5.61249i	0.312287	−	0.312287i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−17.1464	+	17.1464i	−0.948200	+	0.948200i
$328$		0			0
$329$			12.0000i			0.661581i
$330$		0			0
$331$	32.0780			1.76317			0.881584	−	0.472027i	$-0.156478\pi$
$331$	32.0780			1.76317			0.881584	+	0.472027i	$0.156478\pi$
$332$		0			0
$333$	−14.9666	−	14.9666i	−0.820166	−	0.820166i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	18.3712	+	18.3712i	1.00074	+	1.00074i	1.00000		0.000741840i	$0.000236135\pi$
$337$	18.3712	+	18.3712i	1.00074	+	1.00074i	0.000741840		1.00000i	$0.499764\pi$
$338$		0			0
$339$		−	4.58258i		−	0.248891i
$340$		0			0
$341$			18.3303i			0.992642i
$342$		0			0
$343$	−4.89898	−	4.89898i	−0.264520	−	0.264520i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.61249	−	5.61249i	−0.301294	−	0.301294i	0.540226	−	0.841520i	$-0.318339\pi$
$347$	−5.61249	−	5.61249i	−0.301294	−	0.301294i	−0.841520	+	0.540226i	$0.818339\pi$
$348$		0			0
$349$	−9.16515			−0.490599			−0.245300	−	0.969447i	$-0.578886\pi$
$349$	−9.16515			−0.490599			−0.245300	+	0.969447i	$0.578886\pi$
$350$		0			0
$351$			14.0000i			0.747265i
$352$		0			0
$353$	−24.4949	+	24.4949i	−1.30373	+	1.30373i	−0.377875	+	0.925856i	$0.623345\pi$
$353$	−24.4949	+	24.4949i	−1.30373	+	1.30373i	−0.925856	+	0.377875i	$0.876655\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−11.2250	+	11.2250i	−0.594089	+	0.594089i
$358$		0			0
$359$	6.00000			0.316668			0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000			0.316668			0.158334	+	0.987386i	$0.449388\pi$
$360$		0			0
$361$	−2.00000			−0.105263
$362$		0			0
$363$	−18.7083	+	18.7083i	−0.981930	+	0.981930i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	14.6969	−	14.6969i	0.767174	−	0.767174i	−0.210434	−	0.977608i	$-0.567488\pi$
$367$	14.6969	−	14.6969i	0.767174	−	0.767174i	0.977608	+	0.210434i	$0.0674877\pi$
$368$		0			0
$369$			36.0000i			1.87409i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−14.9666	−	14.9666i	−0.774943	−	0.774943i	0.204023	−	0.978966i	$-0.434598\pi$
$373$	−14.9666	−	14.9666i	−0.774943	−	0.774943i	−0.978966	+	0.204023i	$0.934598\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	34.2929	+	34.2929i	1.76617	+	1.76617i
$378$		0			0
$379$		−	13.7477i		−	0.706173i	−0.935591	−	0.353087i	$-0.885132\pi$
$379$		−	13.7477i		−	0.706173i	0.935591	−	0.353087i	$-0.114868\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.44949	+	2.44949i	0.125163	+	0.125163i	0.766914	−	0.641750i	$-0.221792\pi$
$383$	2.44949	+	2.44949i	0.125163	+	0.125163i	−0.641750	+	0.766914i	$0.721792\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	14.9666	+	14.9666i	0.760797	+	0.760797i
$388$		0			0
$389$	18.3303			0.929383			0.464692	−	0.885473i	$-0.346165\pi$
$389$	18.3303			0.929383			0.464692	+	0.885473i	$0.346165\pi$
$390$		0			0
$391$		−	12.0000i		−	0.606866i
$392$		0			0
$393$	34.2929	−	34.2929i	1.72985	−	1.72985i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7.48331	−	7.48331i	0.375577	−	0.375577i	−0.493927	−	0.869504i	$-0.664439\pi$
$397$	7.48331	−	7.48331i	0.375577	−	0.375577i	0.869504	+	0.493927i	$0.164439\pi$
$398$		0			0
$399$	42.0000			2.10263
$400$		0			0
$401$	−15.0000			−0.749064			−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000			−0.749064			−0.374532	+	0.927214i	$0.622197\pi$
$402$		0			0
$403$	−14.9666	+	14.9666i	−0.745541	+	0.745541i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	17.1464	−	17.1464i	0.849917	−	0.849917i
$408$		0			0
$409$		−	17.0000i		−	0.840596i	−0.907386	−	0.420298i	$-0.861926\pi$
$409$		−	17.0000i		−	0.840596i	0.907386	−	0.420298i	$-0.138074\pi$
$410$		0			0
$411$	−41.2432			−2.03438
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−8.57321	−	8.57321i	−0.419832	−	0.419832i
$418$		0			0
$419$		−	22.9129i		−	1.11937i	−0.828706	−	0.559684i	$-0.810923\pi$
$419$		−	22.9129i		−	1.11937i	0.828706	−	0.559684i	$-0.189077\pi$
$420$		0			0
$421$			36.6606i			1.78673i	0.449333	+	0.893364i	$0.351662\pi$
$421$			36.6606i			1.78673i	−0.449333	+	0.893364i	$0.648338\pi$
$422$		0			0
$423$	−9.79796	−	9.79796i	−0.476393	−	0.476393i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−22.4499	−	22.4499i	−1.08643	−	1.08643i
$428$		0			0
$429$	−64.1561			−3.09748
$430$		0			0
$431$			30.0000i			1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$			30.0000i			1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$		0			0
$433$	13.4722	−	13.4722i	0.647432	−	0.647432i	−0.304939	−	0.952372i	$-0.598636\pi$
$433$	13.4722	−	13.4722i	0.647432	−	0.647432i	0.952372	+	0.304939i	$0.0986362\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−22.4499	+	22.4499i	−1.07393	+	1.07393i
$438$		0			0
$439$	−22.0000			−1.05000			−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000			−1.05000			−0.525001	+	0.851101i	$0.675935\pi$
$440$		0			0
$441$	−20.0000			−0.952381
$442$		0			0
$443$	28.0624	−	28.0624i	1.33329	−	1.33329i	0.430874	−	0.902412i	$-0.358205\pi$
$443$	28.0624	−	28.0624i	1.33329	−	1.33329i	0.902412	−	0.430874i	$-0.141795\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	17.1464	−	17.1464i	0.810998	−	0.810998i
$448$		0			0
$449$			27.0000i			1.27421i	0.770778	+	0.637104i	$0.219868\pi$
$449$			27.0000i			1.27421i	−0.770778	+	0.637104i	$0.780132\pi$
$450$		0			0
$451$	−41.2432			−1.94207
$452$		0			0
$453$	18.7083	+	18.7083i	0.878992	+	0.878992i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−25.7196	−	25.7196i	−1.20311	−	1.20311i	−0.973214	−	0.229900i	$-0.926160\pi$
$457$	−25.7196	−	25.7196i	−1.20311	−	1.20311i	−0.229900	−	0.973214i	$-0.573840\pi$
$458$		0			0
$459$		−	4.58258i		−	0.213896i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−9.79796	−	9.79796i	−0.455350	−	0.455350i	0.441776	−	0.897126i	$-0.354349\pi$
$463$	−9.79796	−	9.79796i	−0.455350	−	0.455350i	−0.897126	+	0.441776i	$0.854349\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.2250	+	11.2250i	0.519430	+	0.519430i	0.917399	−	0.397969i	$-0.130285\pi$
$467$	11.2250	+	11.2250i	0.519430	+	0.519430i	−0.397969	+	0.917399i	$0.630285\pi$
$468$		0			0
$469$	9.16515			0.423207
$470$		0			0
$471$		−	56.0000i		−	2.58034i
$472$		0			0
$473$	−17.1464	+	17.1464i	−0.788394	+	0.788394i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$	28.0000			1.27669
$482$		0			0
$483$	44.8999	−	44.8999i	2.04302	−	2.04302i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−14.6969	+	14.6969i	−0.665982	+	0.665982i	−0.956783	−	0.290802i	$-0.906078\pi$
$487$	−14.6969	+	14.6969i	−0.665982	+	0.665982i	0.290802	+	0.956783i	$0.406078\pi$
$488$		0			0
$489$			7.00000i			0.316551i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	−11.2250	−	11.2250i	−0.505547	−	0.505547i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	14.6969	+	14.6969i	0.659248	+	0.659248i
$498$		0			0
$499$		−	18.3303i		−	0.820577i	−0.911956	−	0.410289i	$-0.865428\pi$
$499$		−	18.3303i		−	0.820577i	0.911956	−	0.410289i	$-0.134572\pi$
$500$		0			0
$501$			27.4955i			1.22841i
$502$		0			0
$503$	−9.79796	−	9.79796i	−0.436869	−	0.436869i	0.454088	−	0.890957i	$-0.349965\pi$
$503$	−9.79796	−	9.79796i	−0.436869	−	0.436869i	−0.890957	+	0.454088i	$0.849965\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−28.0624	−	28.0624i	−1.24630	−	1.24630i
$508$		0			0
$509$	−18.3303			−0.812476			−0.406238	−	0.913767i	$-0.633160\pi$
$509$	−18.3303			−0.812476			−0.406238	+	0.913767i	$0.633160\pi$
$510$		0			0
$511$		−	30.0000i		−	1.32712i
$512$		0			0
$513$	−8.57321	+	8.57321i	−0.378517	+	0.378517i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	11.2250	−	11.2250i	0.493674	−	0.493674i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$	9.35414	−	9.35414i	0.409028	−	0.409028i	−0.472371	−	0.881400i	$-0.656602\pi$
$523$	9.35414	−	9.35414i	0.409028	−	0.409028i	0.881400	+	0.472371i	$0.156602\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.89898	−	4.89898i	0.213403	−	0.213403i
$528$		0			0
$529$			25.0000i			1.08696i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−33.6749	−	33.6749i	−1.45862	−	1.45862i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−25.7196	−	25.7196i	−1.10988	−	1.10988i
$538$		0			0
$539$		−	22.9129i		−	0.986928i
$540$		0			0
$541$		−	18.3303i		−	0.788081i	−0.919093	−	0.394041i	$-0.871077\pi$
$541$		−	18.3303i		−	0.788081i	0.919093	−	0.394041i	$-0.128923\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	13.0958	+	13.0958i	0.559936	+	0.559936i	0.929289	−	0.369353i	$-0.120421\pi$
$547$	13.0958	+	13.0958i	0.559936	+	0.559936i	−0.369353	+	0.929289i	$0.620421\pi$
$548$		0			0
$549$	36.6606			1.56464
$550$		0			0
$551$			42.0000i			1.78926i
$552$		0			0
$553$	24.4949	−	24.4949i	1.04163	−	1.04163i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11.2250	+	11.2250i	−0.475617	+	0.475617i	−0.903727	−	0.428110i	$-0.859180\pi$
$557$	−11.2250	+	11.2250i	−0.475617	+	0.475617i	0.428110	+	0.903727i	$0.359180\pi$
$558$		0			0
$559$	−28.0000			−1.18427
$560$		0			0
$561$	21.0000			0.886621
$562$		0			0
$563$	−11.2250	+	11.2250i	−0.473076	+	0.473076i	−0.902909	−	0.429832i	$-0.858573\pi$
$563$	−11.2250	+	11.2250i	−0.473076	+	0.473076i	0.429832	+	0.902909i	$0.358573\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−12.2474	+	12.2474i	−0.514344	+	0.514344i
$568$		0			0
$569$		−	3.00000i		−	0.125767i	−0.998021	−	0.0628833i	$-0.979970\pi$
$569$		−	3.00000i		−	0.125767i	0.998021	−	0.0628833i	$-0.0200296\pi$
$570$		0			0
$571$	−36.6606			−1.53420			−0.767099	−	0.641528i	$-0.778301\pi$
$571$	−36.6606			−1.53420			−0.767099	+	0.641528i	$0.778301\pi$
$572$		0			0
$573$	−44.8999	−	44.8999i	−1.87572	−	1.87572i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−28.1691	−	28.1691i	−1.17270	−	1.17270i	−0.981564	−	0.191132i	$-0.938784\pi$
$577$	−28.1691	−	28.1691i	−1.17270	−	1.17270i	−0.191132	−	0.981564i	$-0.561216\pi$
$578$		0			0
$579$			22.9129i			0.952227i
$580$		0			0
$581$			27.4955i			1.14070i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	28.0624	+	28.0624i	1.15826	+	1.15826i	0.984850	+	0.173411i	$0.0554789\pi$
$587$	28.0624	+	28.0624i	1.15826	+	1.15826i	0.173411	+	0.984850i	$0.444521\pi$
$588$		0			0
$589$	−18.3303			−0.755287
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.12372	+	6.12372i	−0.251471	+	0.251471i	−0.821574	−	0.570102i	$-0.806904\pi$
$593$	−6.12372	+	6.12372i	−0.251471	+	0.251471i	0.570102	+	0.821574i	$0.306904\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	7.48331	−	7.48331i	0.306272	−	0.306272i
$598$		0			0
$599$	30.0000			1.22577			0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000			1.22577			0.612883	+	0.790173i	$0.290010\pi$
$600$		0			0
$601$	−35.0000			−1.42768			−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000			−1.42768			−0.713840	+	0.700309i	$0.753046\pi$
$602$		0			0
$603$	−7.48331	+	7.48331i	−0.304744	+	0.304744i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−19.5959	+	19.5959i	−0.795374	+	0.795374i	−0.982362	−	0.186988i	$-0.940127\pi$
$607$	−19.5959	+	19.5959i	−0.795374	+	0.795374i	0.186988	+	0.982362i	$0.440127\pi$
$608$		0			0
$609$		−	84.0000i		−	3.40385i
$610$		0			0
$611$	18.3303			0.741565
$612$		0			0
$613$	7.48331	+	7.48331i	0.302248	+	0.302248i	0.841893	−	0.539645i	$-0.181441\pi$
$613$	7.48331	+	7.48331i	0.302248	+	0.302248i	−0.539645	+	0.841893i	$0.681441\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.4949	+	24.4949i	0.986127	+	0.986127i	0.999905	−	0.0137776i	$-0.00438570\pi$
$617$	24.4949	+	24.4949i	0.986127	+	0.986127i	−0.0137776	+	0.999905i	$0.504386\pi$
$618$		0			0
$619$			36.6606i			1.47351i	0.676158	+	0.736757i	$0.263644\pi$
$619$			36.6606i			1.47351i	−0.676158	+	0.736757i	$0.736356\pi$
$620$		0			0
$621$			18.3303i			0.735570i
$622$		0			0
$623$	36.7423	+	36.7423i	1.47205	+	1.47205i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−39.2874	−	39.2874i	−1.56899	−	1.56899i
$628$		0			0
$629$	−9.16515			−0.365439
$630$		0			0
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$		0			0
$633$	−25.7196	+	25.7196i	−1.02226	+	1.02226i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	18.7083	−	18.7083i	0.741249	−	0.741249i
$638$		0			0
$639$	−24.0000			−0.949425
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$	−3.74166	+	3.74166i	−0.147557	+	0.147557i	−0.777026	−	0.629469i	$-0.783272\pi$
$643$	−3.74166	+	3.74166i	−0.147557	+	0.147557i	0.629469	+	0.777026i	$0.283272\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.6969	−	14.6969i	0.577796	−	0.577796i	−0.356499	−	0.934296i	$-0.616030\pi$
$647$	14.6969	−	14.6969i	0.577796	−	0.577796i	0.934296	+	0.356499i	$0.116030\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	36.6606			1.43684
$652$		0			0
$653$	33.6749	+	33.6749i	1.31780	+	1.31780i	0.915513	+	0.402288i	$0.131785\pi$
$653$	33.6749	+	33.6749i	1.31780	+	1.31780i	0.402288	+	0.915513i	$0.368215\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	24.4949	+	24.4949i	0.955637	+	0.955637i
$658$		0			0
$659$		−	13.7477i		−	0.535535i	−0.963484	−	0.267768i	$-0.913714\pi$
$659$		−	13.7477i		−	0.535535i	0.963484	−	0.267768i	$-0.0862860\pi$
$660$		0			0
$661$			9.16515i			0.356483i	0.983987	+	0.178242i	$0.0570408\pi$
$661$			9.16515i			0.356483i	−0.983987	+	0.178242i	$0.942959\pi$
$662$		0			0
$663$	17.1464	+	17.1464i	0.665912	+	0.665912i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	44.8999	+	44.8999i	1.73853	+	1.73853i
$668$		0			0
$669$	18.3303			0.708690
$670$		0			0
$671$			42.0000i			1.62139i
$672$		0			0
$673$	−24.4949	+	24.4949i	−0.944209	+	0.944209i	−0.998524	−	0.0543149i	$-0.982703\pi$
$673$	−24.4949	+	24.4949i	−0.944209	+	0.944209i	0.0543149	+	0.998524i	$0.482703\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.4499	+	22.4499i	−0.862821	+	0.862821i	−0.991665	−	0.128843i	$-0.958874\pi$
$677$	−22.4499	+	22.4499i	−0.862821	+	0.862821i	0.128843	+	0.991665i	$0.458874\pi$
$678$		0			0
$679$	24.0000			0.921035
$680$		0			0
$681$	42.0000			1.60944
$682$		0			0
$683$	−5.61249	+	5.61249i	−0.214756	+	0.214756i	−0.806284	−	0.591528i	$-0.798525\pi$
$683$	−5.61249	+	5.61249i	−0.214756	+	0.214756i	0.591528	+	0.806284i	$0.298525\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−34.2929	+	34.2929i	−1.30835	+	1.30835i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	13.7477			0.522988			0.261494	−	0.965205i	$-0.415785\pi$
$691$	13.7477			0.522988			0.261494	+	0.965205i	$0.415785\pi$
$692$		0			0
$693$	44.8999	+	44.8999i	1.70561	+	1.70561i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	11.0227	+	11.0227i	0.417515	+	0.417515i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	45.8258i		−	1.73081i	−0.501069	−	0.865407i	$-0.667060\pi$
$701$		−	45.8258i		−	1.73081i	0.501069	−	0.865407i	$-0.332940\pi$
$702$		0			0
$703$	17.1464	+	17.1464i	0.646690	+	0.646690i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.4499	−	22.4499i	−0.844317	−	0.844317i
$708$		0			0
$709$	−36.6606			−1.37682			−0.688409	−	0.725323i	$-0.741691\pi$
$709$	−36.6606			−1.37682			−0.688409	+	0.725323i	$0.741691\pi$
$710$		0			0
$711$			40.0000i			1.50012i
$712$		0			0
$713$	−19.5959	+	19.5959i	−0.733873	+	0.733873i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−11.2250	+	11.2250i	−0.419204	+	0.419204i
$718$		0			0
$719$	−6.00000			−0.223762			−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000			−0.223762			−0.111881	+	0.993722i	$0.535688\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$		0			0
$723$	−13.0958	+	13.0958i	−0.487038	+	0.487038i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	12.2474	−	12.2474i	0.454233	−	0.454233i	−0.442524	−	0.896757i	$-0.645917\pi$
$727$	12.2474	−	12.2474i	0.454233	−	0.454233i	0.896757	+	0.442524i	$0.145917\pi$
$728$		0			0
$729$		−	41.0000i		−	1.51852i
$730$		0			0
$731$	9.16515			0.338985
$732$		0			0
$733$	18.7083	+	18.7083i	0.691006	+	0.691006i	0.962453	−	0.271447i	$-0.0875022\pi$
$733$	18.7083	+	18.7083i	0.691006	+	0.691006i	−0.271447	+	0.962453i	$0.587502\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.57321	−	8.57321i	−0.315798	−	0.315798i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		−	64.1561i		−	2.35683i
$742$		0			0
$743$	9.79796	+	9.79796i	0.359452	+	0.359452i	0.863611	−	0.504159i	$-0.168197\pi$
$743$	9.79796	+	9.79796i	0.359452	+	0.359452i	−0.504159	+	0.863611i	$0.668197\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−22.4499	−	22.4499i	−0.821401	−	0.821401i
$748$		0			0
$749$	27.4955			1.00466
$750$		0			0
$751$			22.0000i			0.802791i	0.915905	+	0.401396i	$0.131475\pi$
$751$			22.0000i			0.802791i	−0.915905	+	0.401396i	$0.868525\pi$
$752$		0			0
$753$	8.57321	−	8.57321i	0.312425	−	0.312425i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	7.48331	−	7.48331i	0.271986	−	0.271986i	−0.557913	−	0.829899i	$-0.688398\pi$
$757$	7.48331	−	7.48331i	0.271986	−	0.271986i	0.829899	+	0.557913i	$0.188398\pi$
$758$		0			0
$759$	−84.0000			−3.04901
$760$		0			0
$761$	−27.0000			−0.978749			−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000			−0.978749			−0.489375	+	0.872074i	$0.662775\pi$
$762$		0			0
$763$	−22.4499	+	22.4499i	−0.812743	+	0.812743i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	35.0000i		−	1.26213i	−0.775729	−	0.631066i	$-0.782618\pi$
$769$		−	35.0000i		−	1.26213i	0.775729	−	0.631066i	$-0.217382\pi$
$770$		0			0
$771$	−18.3303			−0.660150
$772$		0			0
$773$	−11.2250	−	11.2250i	−0.403734	−	0.403734i	0.475813	−	0.879547i	$-0.342154\pi$
$773$	−11.2250	−	11.2250i	−0.403734	−	0.403734i	−0.879547	+	0.475813i	$0.842154\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−34.2929	−	34.2929i	−1.23025	−	1.23025i
$778$		0			0
$779$		−	41.2432i		−	1.47769i
$780$		0			0
$781$		−	27.4955i		−	0.983865i
$782$		0			0
$783$	17.1464	+	17.1464i	0.612763	+	0.612763i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	18.7083	+	18.7083i	0.666878	+	0.666878i	0.956992	−	0.290114i	$-0.0936931\pi$
$787$	18.7083	+	18.7083i	0.666878	+	0.666878i	−0.290114	+	0.956992i	$0.593693\pi$
$788$		0			0
$789$	54.9909			1.95773
$790$		0			0
$791$		−	6.00000i		−	0.213335i
$792$		0			0
$793$	−34.2929	+	34.2929i	−1.21778	+	1.21778i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22.4499	−	22.4499i	0.795218	−	0.795218i	−0.187119	−	0.982337i	$-0.559915\pi$
$797$	22.4499	−	22.4499i	0.795218	−	0.795218i	0.982337	+	0.187119i	$0.0599151\pi$
$798$		0			0
$799$	−6.00000			−0.212265
$800$		0			0
$801$	−60.0000			−2.12000
$802$		0			0
$803$	−28.0624	+	28.0624i	−0.990302	+	0.990302i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−34.2929	+	34.2929i	−1.20717	+	1.20717i
$808$		0			0
$809$		−	18.0000i		−	0.632846i	−0.948618	−	0.316423i	$-0.897518\pi$
$809$		−	18.0000i		−	0.632846i	0.948618	−	0.316423i	$-0.102482\pi$
$810$		0			0
$811$	36.6606			1.28733			0.643664	−	0.765308i	$-0.277413\pi$
$811$	36.6606			1.28733			0.643664	+	0.765308i	$0.277413\pi$
$812$		0			0
$813$	14.9666	+	14.9666i	0.524903	+	0.524903i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−17.1464	−	17.1464i	−0.599878	−	0.599878i
$818$		0			0
$819$			73.3212i			2.56205i
$820$		0			0
$821$		−	54.9909i		−	1.91920i	−0.281375	−	0.959598i	$-0.590790\pi$
$821$		−	54.9909i		−	1.91920i	0.281375	−	0.959598i	$-0.409210\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5.61249	−	5.61249i	−0.195165	−	0.195165i	0.602758	−	0.797924i	$-0.294068\pi$
$827$	−5.61249	−	5.61249i	−0.195165	−	0.195165i	−0.797924	+	0.602758i	$0.794068\pi$
$828$		0			0
$829$	45.8258			1.59159			0.795797	−	0.605563i	$-0.207052\pi$
$829$	45.8258			1.59159			0.795797	+	0.605563i	$0.207052\pi$
$830$		0			0
$831$		−	70.0000i		−	2.42827i
$832$		0			0
$833$	−6.12372	+	6.12372i	−0.212174	+	0.212174i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−7.48331	+	7.48331i	−0.258661	+	0.258661i
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	55.0000			1.89655
$842$		0			0
$843$	−11.2250	+	11.2250i	−0.386609	+	0.386609i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−24.4949	+	24.4949i	−0.841655	+	0.841655i
$848$		0			0
$849$			35.0000i			1.20120i
$850$		0			0
$851$	36.6606			1.25671
$852$		0			0
$853$	7.48331	+	7.48331i	0.256224	+	0.256224i	0.823516	−	0.567293i	$-0.192009\pi$
$853$	7.48331	+	7.48331i	0.256224	+	0.256224i	−0.567293	+	0.823516i	$0.692009\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−11.0227	−	11.0227i	−0.376528	−	0.376528i	0.493320	−	0.869848i	$-0.335783\pi$
$857$	−11.0227	−	11.0227i	−0.376528	−	0.376528i	−0.869848	+	0.493320i	$0.835783\pi$
$858$		0			0
$859$			4.58258i			0.156355i	0.996939	+	0.0781777i	$0.0249102\pi$
$859$			4.58258i			0.156355i	−0.996939	+	0.0781777i	$0.975090\pi$
$860$		0			0
$861$			82.4864i			2.81113i
$862$		0			0
$863$	12.2474	+	12.2474i	0.416908	+	0.416908i	0.884137	−	0.467229i	$-0.154747\pi$
$863$	12.2474	+	12.2474i	0.416908	+	0.416908i	−0.467229	+	0.884137i	$0.654747\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	26.1916	+	26.1916i	0.889513	+	0.889513i
$868$		0			0
$869$	−45.8258			−1.55453
$870$		0			0
$871$		−	14.0000i		−	0.474372i
$872$		0			0
$873$	−19.5959	+	19.5959i	−0.663221	+	0.663221i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	14.9666	−	14.9666i	0.505387	−	0.505387i	−0.407720	−	0.913107i	$-0.633676\pi$
$877$	14.9666	−	14.9666i	0.505387	−	0.505387i	0.913107	+	0.407720i	$0.133676\pi$
$878$		0			0
$879$	−42.0000			−1.41662
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$	20.5791	−	20.5791i	0.692542	−	0.692542i	−0.270248	−	0.962791i	$-0.587106\pi$
$883$	20.5791	−	20.5791i	0.692542	−	0.692542i	0.962791	+	0.270248i	$0.0871058\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	39.1918	−	39.1918i	1.31593	−	1.31593i	0.398968	−	0.916965i	$-0.369368\pi$
$887$	39.1918	−	39.1918i	1.31593	−	1.31593i	0.916965	−	0.398968i	$-0.130632\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	22.9129			0.767610
$892$		0			0
$893$	11.2250	+	11.2250i	0.375629	+	0.375629i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−68.5857	−	68.5857i	−2.29001	−	2.29001i
$898$		0			0
$899$			36.6606i			1.22270i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	34.2929	+	34.2929i	1.14119	+	1.14119i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.74166	+	3.74166i	0.124240	+	0.124240i	0.766493	−	0.642253i	$-0.222000\pi$
$907$	3.74166	+	3.74166i	0.124240	+	0.124240i	−0.642253	+	0.766493i	$0.722000\pi$
$908$		0			0
$909$	36.6606			1.21596
$910$		0			0
$911$		−	36.0000i		−	1.19273i	−0.802712	−	0.596367i	$-0.796610\pi$
$911$		−	36.0000i		−	1.19273i	0.802712	−	0.596367i	$-0.203390\pi$
$912$		0			0
$913$	25.7196	−	25.7196i	0.851196	−	0.851196i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	44.8999	−	44.8999i	1.48272	−	1.48272i
$918$		0			0
$919$	−52.0000			−1.71532			−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000			−1.71532			−0.857661	+	0.514216i	$0.828083\pi$
$920$		0			0
$921$	35.0000			1.15329
$922$		0			0
$923$	22.4499	−	22.4499i	0.738949	−	0.738949i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−9.79796	+	9.79796i	−0.321807	+	0.321807i
$928$		0			0
$929$		−	42.0000i		−	1.37798i	−0.724773	−	0.688988i	$-0.758055\pi$
$929$		−	42.0000i		−	1.37798i	0.724773	−	0.688988i	$-0.241945\pi$
$930$		0			0
$931$	22.9129			0.750939
$932$		0			0
$933$	−33.6749	−	33.6749i	−1.10247	−	1.10247i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.22474	−	1.22474i	−0.0400107	−	0.0400107i	0.686818	−	0.726829i	$-0.259007\pi$
$937$	−1.22474	−	1.22474i	−0.0400107	−	0.0400107i	−0.726829	+	0.686818i	$0.759007\pi$
$938$		0			0
$939$			54.9909i			1.79456i
$940$		0			0
$941$			9.16515i			0.298775i	0.988779	+	0.149388i	$0.0477302\pi$
$941$			9.16515i			0.298775i	−0.988779	+	0.149388i	$0.952270\pi$
$942$		0			0
$943$	−44.0908	−	44.0908i	−1.43579	−	1.43579i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33.6749	+	33.6749i	1.09429	+	1.09429i	0.995065	+	0.0992225i	$0.0316355\pi$
$947$	33.6749	+	33.6749i	1.09429	+	1.09429i	0.0992225	+	0.995065i	$0.468364\pi$
$948$		0			0
$949$	−45.8258			−1.48757
$950$		0			0
$951$			84.0000i			2.72389i
$952$		0			0
$953$	−8.57321	+	8.57321i	−0.277714	+	0.277714i	−0.832196	−	0.554482i	$-0.812916\pi$
$953$	−8.57321	+	8.57321i	−0.277714	+	0.277714i	0.554482	+	0.832196i	$0.312916\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−78.5748	+	78.5748i	−2.53996	+	2.53996i
$958$		0			0
$959$	−54.0000			−1.74375
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$	−22.4499	+	22.4499i	−0.723439	+	0.723439i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−24.4949	+	24.4949i	−0.787703	+	0.787703i	−0.981117	−	0.193414i	$-0.938044\pi$
$967$	−24.4949	+	24.4949i	−0.787703	+	0.787703i	0.193414	+	0.981117i	$0.438044\pi$
$968$		0			0
$969$			21.0000i			0.674617i
$970$		0			0
$971$	−22.9129			−0.735309			−0.367655	−	0.929962i	$-0.619839\pi$
$971$	−22.9129			−0.735309			−0.367655	+	0.929962i	$0.619839\pi$
$972$		0			0
$973$	−11.2250	−	11.2250i	−0.359856	−	0.359856i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28.1691	+	28.1691i	0.901210	+	0.901210i	0.995541	−	0.0943306i	$-0.0300711\pi$
$977$	28.1691	+	28.1691i	0.901210	+	0.901210i	−0.0943306	+	0.995541i	$0.530071\pi$
$978$		0			0
$979$		−	68.7386i		−	2.19690i
$980$		0			0
$981$		−	36.6606i		−	1.17048i
$982$		0			0
$983$	31.8434	+	31.8434i	1.01565	+	1.01565i	0.999876	+	0.0157700i	$0.00501996\pi$
$983$	31.8434	+	31.8434i	1.01565	+	1.01565i	0.0157700	+	0.999876i	$0.494980\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−22.4499	−	22.4499i	−0.714590	−	0.714590i
$988$		0			0
$989$	−36.6606			−1.16574
$990$		0			0
$991$			2.00000i			0.0635321i	0.999495	+	0.0317660i	$0.0101131\pi$
$991$			2.00000i			0.0635321i	−0.999495	+	0.0317660i	$0.989887\pi$
$992$		0			0
$993$	−60.0125	+	60.0125i	−1.90444	+	1.90444i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	7.48331	−	7.48331i	0.236999	−	0.236999i	−0.578607	−	0.815606i	$-0.696404\pi$
$997$	7.48331	−	7.48331i	0.236999	−	0.236999i	0.815606	+	0.578607i	$0.196404\pi$
$998$		0			0
$999$	14.0000			0.442940

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.o.f.607.1	yes	8
4.3	odd	2		1600.2.o.k.607.4	yes	8
5.2	odd	4		1600.2.o.k.543.3	yes	8
5.3	odd	4		1600.2.o.k.543.2	yes	8
5.4	even	2	inner	1600.2.o.f.607.4	yes	8
8.3	odd	2		1600.2.o.k.607.2	yes	8
8.5	even	2	inner	1600.2.o.f.607.3	yes	8
20.3	even	4	inner	1600.2.o.f.543.3	yes	8
20.7	even	4	inner	1600.2.o.f.543.2	yes	8
20.19	odd	2		1600.2.o.k.607.1	yes	8
40.3	even	4	inner	1600.2.o.f.543.1	✓	8
40.13	odd	4		1600.2.o.k.543.4	yes	8
40.19	odd	2		1600.2.o.k.607.3	yes	8
40.27	even	4	inner	1600.2.o.f.543.4	yes	8
40.29	even	2	inner	1600.2.o.f.607.2	yes	8
40.37	odd	4		1600.2.o.k.543.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1600.2.o.f.543.1	✓	8	40.3	even	4	inner
1600.2.o.f.543.2	yes	8	20.7	even	4	inner
1600.2.o.f.543.3	yes	8	20.3	even	4	inner
1600.2.o.f.543.4	yes	8	40.27	even	4	inner
1600.2.o.f.607.1	yes	8	1.1	even	1	trivial
1600.2.o.f.607.2	yes	8	40.29	even	2	inner
1600.2.o.f.607.3	yes	8	8.5	even	2	inner
1600.2.o.f.607.4	yes	8	5.4	even	2	inner
1600.2.o.k.543.1	yes	8	40.37	odd	4
1600.2.o.k.543.2	yes	8	5.3	odd	4
1600.2.o.k.543.3	yes	8	5.2	odd	4
1600.2.o.k.543.4	yes	8	40.13	odd	4
1600.2.o.k.607.1	yes	8	20.19	odd	2
1600.2.o.k.607.2	yes	8	8.3	odd	2
1600.2.o.k.607.3	yes	8	40.19	odd	2
1600.2.o.k.607.4	yes	8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.o (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.87083 + 1.87083i) q^{3} +(-2.44949 + 2.44949i) q^{7} -4.00000i q^{9} +4.58258 q^{11} +(3.74166 + 3.74166i) q^{13} +(-1.22474 - 1.22474i) q^{17} +4.58258i q^{19} -9.16515i q^{21} +(4.89898 + 4.89898i) q^{23} +(1.87083 + 1.87083i) q^{27} +9.16515 q^{29} +4.00000i q^{31} +(-8.57321 + 8.57321i) q^{33} +(3.74166 - 3.74166i) q^{37} -14.0000 q^{39} -9.00000 q^{41} +(-3.74166 + 3.74166i) q^{43} +(2.44949 - 2.44949i) q^{47} -5.00000i q^{49} +4.58258 q^{51} +(-8.57321 - 8.57321i) q^{57} +9.16515i q^{61} +(9.79796 + 9.79796i) q^{63} +(-1.87083 - 1.87083i) q^{67} -18.3303 q^{69} -6.00000i q^{71} +(-6.12372 + 6.12372i) q^{73} +(-11.2250 + 11.2250i) q^{77} -10.0000 q^{79} +5.00000 q^{81} +(5.61249 - 5.61249i) q^{83} +(-17.1464 + 17.1464i) q^{87} -15.0000i q^{89} -18.3303 q^{91} +(-7.48331 - 7.48331i) q^{93} +(-4.89898 - 4.89898i) q^{97} -18.3303i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 112 q^{39} - 72 q^{41} - 80 q^{79} + 40 q^{81}+O(q^{100}) \) 8 * q - 112 * q^39 - 72 * q^41 - 80 * q^79 + 40 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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