LMFDB - Embedded newform 3300.2.x.d.1693.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3300,2,Mod(1693,3300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3300, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3300.1693"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3300.x (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,0,0,-24,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$26.3506326670$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1693.8
Character	$\chi$	$=$	3300.1693
Dual form			3300.2.x.d.1957.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{3} +(0.414559 + 0.414559i) q^{7} +1.00000i q^{9} +(-3.18030 + 0.941114i) q^{11} +(-3.69090 + 3.69090i) q^{13} +(-0.270937 - 0.270937i) q^{17} -7.48511 q^{19} -0.586275i q^{21} +(1.83571 + 1.83571i) q^{23} +(0.707107 - 0.707107i) q^{27} +5.90866 q^{29} -1.59609 q^{31} +(2.91428 + 1.58334i) q^{33} +(6.85097 - 6.85097i) q^{37} +5.21972 q^{39} -9.62710i q^{41} +(8.26564 - 8.26564i) q^{43} +(2.66191 - 2.66191i) q^{47} -6.65628i q^{49} +0.383162i q^{51} +(5.53290 + 5.53290i) q^{53} +(5.29277 + 5.29277i) q^{57} -1.32814i q^{59} +4.30471i q^{61} +(-0.414559 + 0.414559i) q^{63} +(7.04044 - 7.04044i) q^{67} -2.59609i q^{69} -13.1528 q^{71} +(-1.83275 + 1.83275i) q^{73} +(-1.70857 - 0.928275i) q^{77} +7.46438 q^{79} -1.00000 q^{81} +(-9.58191 + 9.58191i) q^{83} +(-4.17806 - 4.17806i) q^{87} -16.4810i q^{89} -3.06019 q^{91} +(1.12861 + 1.12861i) q^{93} +(8.42684 - 8.42684i) q^{97} +(-0.941114 - 3.18030i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 24 q^{11} + 16 q^{31} - 32 q^{71} - 32 q^{81} + 80 q^{91}+O(q^{100}) $ 32 * q - 24 * q^11 + 16 * q^31 - 32 * q^71 - 32 * q^81 + 80 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3300\mathbb{Z}\right)^\times$.

$n$	$1201$	$1651$	$2201$	$2377$
$\chi(n)$	$-1$	$1$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.414559	+	0.414559i	0.156689	+	0.156689i	0.781098	−	0.624409i	$-0.214660\pi$
$7$	0.414559	+	0.414559i	0.156689	+	0.156689i	−0.624409	+	0.781098i	$0.714660\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	−3.18030	+	0.941114i	−0.958896	+	0.283756i
$11$	−3.18030	+	0.941114i	−0.958896	+	0.283756i
$12$		0			0
$13$	−3.69090	+	3.69090i	−1.02367	+	1.02367i	−0.0239586	+	0.999713i	$0.507627\pi$
$13$	−3.69090	+	3.69090i	−1.02367	+	1.02367i	−0.999713	+	0.0239586i	$0.992373\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.270937	−	0.270937i	−0.0657118	−	0.0657118i	0.673487	−	0.739199i	$-0.264796\pi$
$17$	−0.270937	−	0.270937i	−0.0657118	−	0.0657118i	−0.739199	+	0.673487i	$0.764796\pi$
$18$		0			0
$19$	−7.48511			−1.71720			−0.858601	−	0.512644i	$-0.828666\pi$
$19$	−7.48511			−1.71720			−0.858601	+	0.512644i	$0.828666\pi$
$20$		0			0
$21$		−	0.586275i		−	0.127936i
$22$		0			0
$23$	1.83571	+	1.83571i	0.382773	+	0.382773i	0.872100	−	0.489328i	$-0.162758\pi$
$23$	1.83571	+	1.83571i	0.382773	+	0.382773i	−0.489328	+	0.872100i	$0.662758\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.707107	−	0.707107i	0.136083	−	0.136083i
$28$		0			0
$29$	5.90866			1.09721			0.548606	−	0.836081i	$-0.315159\pi$
$29$	5.90866			1.09721			0.548606	+	0.836081i	$0.315159\pi$
$30$		0			0
$31$	−1.59609			−0.286666			−0.143333	−	0.989675i	$-0.545782\pi$
$31$	−1.59609			−0.286666			−0.143333	+	0.989675i	$0.545782\pi$
$32$		0			0
$33$	2.91428	+	1.58334i	0.507311	+	0.275625i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.85097	−	6.85097i	1.12629	−	1.12629i	0.135518	−	0.990775i	$-0.456730\pi$
$37$	6.85097	−	6.85097i	1.12629	−	1.12629i	0.990775	−	0.135518i	$-0.0432700\pi$
$38$		0			0
$39$	5.21972			0.835824
$40$		0			0
$41$		−	9.62710i		−	1.50350i	−0.659448	−	0.751750i	$-0.729210\pi$
$41$		−	9.62710i		−	1.50350i	0.659448	−	0.751750i	$-0.270790\pi$
$42$		0			0
$43$	8.26564	−	8.26564i	1.26050	−	1.26050i	0.309646	−	0.950852i	$-0.399790\pi$
$43$	8.26564	−	8.26564i	1.26050	−	1.26050i	0.950852	−	0.309646i	$-0.100210\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.66191	−	2.66191i	0.388279	−	0.388279i	−0.485794	−	0.874073i	$-0.661469\pi$
$47$	2.66191	−	2.66191i	0.388279	−	0.388279i	0.874073	+	0.485794i	$0.161469\pi$
$48$		0			0
$49$		−	6.65628i		−	0.950897i
$50$		0			0
$51$			0.383162i			0.0536534i
$52$		0			0
$53$	5.53290	+	5.53290i	0.760002	+	0.760002i	0.976322	−	0.216320i	$-0.0694056\pi$
$53$	5.53290	+	5.53290i	0.760002	+	0.760002i	−0.216320	+	0.976322i	$0.569406\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	5.29277	+	5.29277i	0.701045	+	0.701045i
$58$		0			0
$59$		−	1.32814i		−	0.172909i	−0.996256	−	0.0864546i	$-0.972446\pi$
$59$		−	1.32814i		−	0.172909i	0.996256	−	0.0864546i	$-0.0275538\pi$
$60$		0			0
$61$			4.30471i			0.551161i	0.961278	+	0.275581i	$0.0888701\pi$
$61$			4.30471i			0.551161i	−0.961278	+	0.275581i	$0.911130\pi$
$62$		0			0
$63$	−0.414559	+	0.414559i	−0.0522295	+	0.0522295i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	7.04044	−	7.04044i	0.860127	−	0.860127i	−0.131226	−	0.991353i	$-0.541891\pi$
$67$	7.04044	−	7.04044i	0.860127	−	0.860127i	0.991353	+	0.131226i	$0.0418912\pi$
$68$		0			0
$69$		−	2.59609i		−	0.312533i
$70$		0			0
$71$	−13.1528			−1.56096			−0.780478	−	0.625184i	$-0.785024\pi$
$71$	−13.1528			−1.56096			−0.780478	+	0.625184i	$0.785024\pi$
$72$		0			0
$73$	−1.83275	+	1.83275i	−0.214508	+	0.214508i	−0.806179	−	0.591672i	$-0.798468\pi$
$73$	−1.83275	+	1.83275i	−0.214508	+	0.214508i	0.591672	+	0.806179i	$0.298468\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.70857	−	0.928275i	−0.194709	−	0.105787i
$78$		0			0
$79$	7.46438			0.839808			0.419904	−	0.907569i	$-0.362064\pi$
$79$	7.46438			0.839808			0.419904	+	0.907569i	$0.362064\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	−9.58191	+	9.58191i	−1.05175	+	1.05175i	−0.0531655	+	0.998586i	$0.516931\pi$
$83$	−9.58191	+	9.58191i	−1.05175	+	1.05175i	−0.998586	+	0.0531655i	$0.983069\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−4.17806	−	4.17806i	−0.447935	−	0.447935i
$88$		0			0
$89$		−	16.4810i		−	1.74698i	−0.486842	−	0.873490i	$-0.661851\pi$
$89$		−	16.4810i		−	1.74698i	0.486842	−	0.873490i	$-0.338149\pi$
$90$		0			0
$91$	−3.06019			−0.320795
$92$		0			0
$93$	1.12861	+	1.12861i	0.117031	+	0.117031i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	8.42684	−	8.42684i	0.855616	−	0.855616i	−0.135202	−	0.990818i	$-0.543168\pi$
$97$	8.42684	−	8.42684i	0.855616	−	0.855616i	0.990818	+	0.135202i	$0.0431684\pi$
$98$		0			0
$99$	−0.941114	−	3.18030i	−0.0945855	−	0.319632i
$100$		0			0
$101$			14.2145i			1.41440i	0.707015	+	0.707199i	$0.250042\pi$
$101$			14.2145i			1.41440i	−0.707015	+	0.707199i	$0.749958\pi$
$102$		0			0
$103$	3.86370	+	3.86370i	0.380702	+	0.380702i	0.871355	−	0.490653i	$-0.163242\pi$
$103$	3.86370	+	3.86370i	0.380702	+	0.380702i	−0.490653	+	0.871355i	$0.663242\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.4804	+	10.4804i	1.01318	+	1.01318i	0.999912	+	0.0132671i	$0.00422318\pi$
$107$	10.4804	+	10.4804i	1.01318	+	1.01318i	0.0132671	+	0.999912i	$0.495777\pi$
$108$		0			0
$109$	12.4496			1.19246			0.596229	−	0.802815i	$-0.296665\pi$
$109$	12.4496			1.19246			0.596229	+	0.802815i	$0.296665\pi$
$110$		0			0
$111$	−9.68874			−0.919615
$112$		0			0
$113$	0.633919	+	0.633919i	0.0596341	+	0.0596341i	0.736295	−	0.676661i	$-0.236574\pi$
$113$	0.633919	+	0.633919i	0.0596341	+	0.0596341i	−0.676661	+	0.736295i	$0.736574\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.69090	−	3.69090i	−0.341224	−	0.341224i
$118$		0			0
$119$		−	0.224638i		−	0.0205926i
$120$		0			0
$121$	9.22861	−	5.98605i	0.838965	−	0.544186i
$122$		0			0
$123$	−6.80739	+	6.80739i	−0.613801	+	0.613801i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	12.3815	+	12.3815i	1.09868	+	1.09868i	0.994566	+	0.104111i	$0.0331998\pi$
$127$	12.3815	+	12.3815i	1.09868	+	1.09868i	0.104111	+	0.994566i	$0.466800\pi$
$128$		0			0
$129$	−11.6894			−1.02919
$130$		0			0
$131$		−	10.9944i		−	0.960581i	−0.877109	−	0.480291i	$-0.840531\pi$
$131$		−	10.9944i		−	0.960581i	0.877109	−	0.480291i	$-0.159469\pi$
$132$		0			0
$133$	−3.10302	−	3.10302i	−0.269066	−	0.269066i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.70499	−	7.70499i	0.658281	−	0.658281i	−0.296692	−	0.954973i	$-0.595883\pi$
$137$	7.70499	−	7.70499i	0.658281	−	0.658281i	0.954973	+	0.296692i	$0.0958834\pi$
$138$		0			0
$139$	1.17255			0.0994544			0.0497272	−	0.998763i	$-0.484165\pi$
$139$	1.17255			0.0994544			0.0497272	+	0.998763i	$0.484165\pi$
$140$		0			0
$141$	−3.76451			−0.317029
$142$		0			0
$143$	8.26461	−	15.2117i	0.691122	−	1.27207i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−4.70670	+	4.70670i	−0.388202	+	0.388202i
$148$		0			0
$149$	−22.1035			−1.81079			−0.905396	−	0.424568i	$-0.860426\pi$
$149$	−22.1035			−1.81079			−0.905396	+	0.424568i	$0.860426\pi$
$150$		0			0
$151$		−	5.12161i		−	0.416790i	−0.978045	−	0.208395i	$-0.933176\pi$
$151$		−	5.12161i		−	0.416790i	0.978045	−	0.208395i	$-0.0668240\pi$
$152$		0			0
$153$	0.270937	−	0.270937i	0.0219039	−	0.0219039i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	4.01974	−	4.01974i	0.320810	−	0.320810i	−0.528268	−	0.849078i	$-0.677158\pi$
$157$	4.01974	−	4.01974i	0.320810	−	0.320810i	0.849078	+	0.528268i	$0.177158\pi$
$158$		0			0
$159$		−	7.82470i		−	0.620539i
$160$		0			0
$161$			1.52202i			0.119952i
$162$		0			0
$163$	−3.17674	−	3.17674i	−0.248821	−	0.248821i	0.571665	−	0.820487i	$-0.306298\pi$
$163$	−3.17674	−	3.17674i	−0.248821	−	0.248821i	−0.820487	+	0.571665i	$0.806298\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.3018	−	14.3018i	−1.10671	−	1.10671i	−0.993580	−	0.113129i	$-0.963913\pi$
$167$	−14.3018	−	14.3018i	−1.10671	−	1.10671i	−0.113129	−	0.993580i	$-0.536087\pi$
$168$		0			0
$169$		−	14.2455i		−	1.09581i
$170$		0			0
$171$		−	7.48511i		−	0.572401i
$172$		0			0
$173$	−3.86706	+	3.86706i	−0.294007	+	0.294007i	−0.838661	−	0.544654i	$-0.816661\pi$
$173$	−3.86706	+	3.86706i	−0.294007	+	0.294007i	0.544654	+	0.838661i	$0.316661\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.939137	+	0.939137i	−0.0705899	+	0.0705899i
$178$		0			0
$179$		−	6.76054i		−	0.505306i	−0.967557	−	0.252653i	$-0.918697\pi$
$179$		−	6.76054i		−	0.505306i	0.967557	−	0.252653i	$-0.0813031\pi$
$180$		0			0
$181$	6.79300			0.504919			0.252460	−	0.967607i	$-0.418760\pi$
$181$	6.79300			0.504919			0.252460	+	0.967607i	$0.418760\pi$
$182$		0			0
$183$	3.04389	−	3.04389i	0.225011	−	0.225011i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.11664	+	0.606677i	0.0816569	+	0.0443646i
$188$		0			0
$189$	0.586275			0.0426452
$190$		0			0
$191$	−15.2415			−1.10284			−0.551419	−	0.834229i	$-0.685913\pi$
$191$	−15.2415			−1.10284			−0.551419	+	0.834229i	$0.685913\pi$
$192$		0			0
$193$	2.35567	−	2.35567i	0.169565	−	0.169565i	−0.617223	−	0.786788i	$-0.711742\pi$
$193$	2.35567	−	2.35567i	0.169565	−	0.169565i	0.786788	+	0.617223i	$0.211742\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.2095	+	10.2095i	0.727395	+	0.727395i	0.970100	−	0.242705i	$-0.0780349\pi$
$197$	10.2095	+	10.2095i	0.727395	+	0.727395i	−0.242705	+	0.970100i	$0.578035\pi$
$198$		0			0
$199$		−	13.9884i		−	0.991611i	−0.868434	−	0.495806i	$-0.834873\pi$
$199$		−	13.9884i		−	0.991611i	0.868434	−	0.495806i	$-0.165127\pi$
$200$		0			0
$201$	−9.95669			−0.702291
$202$		0			0
$203$	2.44949	+	2.44949i	0.171920	+	0.171920i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.83571	+	1.83571i	−0.127591	+	0.127591i
$208$		0			0
$209$	23.8049	−	7.04434i	1.64662	−	0.487267i
$210$		0			0
$211$		−	5.27637i		−	0.363240i	−0.983369	−	0.181620i	$-0.941866\pi$
$211$		−	5.27637i		−	0.363240i	0.983369	−	0.181620i	$-0.0581341\pi$
$212$		0			0
$213$	9.30046	+	9.30046i	0.637257	+	0.637257i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.661673	−	0.661673i	−0.0449173	−	0.0449173i
$218$		0			0
$219$	2.59190			0.175145
$220$		0			0
$221$	2.00000			0.134535
$222$		0			0
$223$	−15.8031	−	15.8031i	−1.05826	−	1.05826i	−0.998195	−	0.0600608i	$-0.980871\pi$
$223$	−15.8031	−	15.8031i	−1.05826	−	1.05826i	−0.0600608	−	0.998195i	$-0.519129\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.42120	−	8.42120i	−0.558934	−	0.558934i	0.370070	−	0.929004i	$-0.379334\pi$
$227$	−8.42120	−	8.42120i	−0.558934	−	0.558934i	−0.929004	+	0.370070i	$0.879334\pi$
$228$		0			0
$229$			16.2001i			1.07053i	0.844683	+	0.535267i	$0.179789\pi$
$229$			16.2001i			1.07053i	−0.844683	+	0.535267i	$0.820211\pi$
$230$		0			0
$231$	0.551751	+	1.86453i	0.0363026	+	0.122677i
$232$		0			0
$233$	14.8600	−	14.8600i	0.973512	−	0.973512i	−0.0261458	−	0.999658i	$-0.508323\pi$
$233$	14.8600	−	14.8600i	0.973512	−	0.973512i	0.999658	+	0.0261458i	$0.00832341\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−5.27811	−	5.27811i	−0.342850	−	0.342850i
$238$		0			0
$239$	27.0451			1.74940			0.874701	−	0.484664i	$-0.161058\pi$
$239$	27.0451			1.74940			0.874701	+	0.484664i	$0.161058\pi$
$240$		0			0
$241$			18.8060i			1.21140i	0.795693	+	0.605700i	$0.207107\pi$
$241$			18.8060i			1.21140i	−0.795693	+	0.605700i	$0.792893\pi$
$242$		0			0
$243$	0.707107	+	0.707107i	0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	27.6268	−	27.6268i	1.75785	−	1.75785i
$248$		0			0
$249$	13.5509			0.858751
$250$		0			0
$251$	−6.62458			−0.418140			−0.209070	−	0.977901i	$-0.567044\pi$
$251$	−6.62458			−0.418140			−0.209070	+	0.977901i	$0.567044\pi$
$252$		0			0
$253$	−7.56573	−	4.11050i	−0.475653	−	0.258425i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.91968	−	7.91968i	0.494016	−	0.494016i	−0.415553	−	0.909569i	$-0.636412\pi$
$257$	7.91968	−	7.91968i	0.494016	−	0.494016i	0.909569	+	0.415553i	$0.136412\pi$
$258$		0			0
$259$	5.68027			0.352955
$260$		0			0
$261$			5.90866i			0.365737i
$262$		0			0
$263$	−11.7744	+	11.7744i	−0.726042	+	0.726042i	−0.969829	−	0.243786i	$-0.921610\pi$
$263$	−11.7744	+	11.7744i	−0.726042	+	0.726042i	0.243786	+	0.969829i	$0.421610\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−11.6538	+	11.6538i	−0.713202	+	0.713202i
$268$		0			0
$269$		−	26.0417i		−	1.58779i	−0.608054	−	0.793895i	$-0.708050\pi$
$269$		−	26.0417i		−	1.58779i	0.608054	−	0.793895i	$-0.291950\pi$
$270$		0			0
$271$			0.424631i			0.0257945i	0.999917	+	0.0128972i	$0.00410543\pi$
$271$			0.424631i			0.0257945i	−0.999917	+	0.0128972i	$0.995895\pi$
$272$		0			0
$273$	2.16388	+	2.16388i	0.130964	+	0.130964i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	14.3122	+	14.3122i	0.859937	+	0.859937i	0.991330	−	0.131393i	$-0.0419450\pi$
$277$	14.3122	+	14.3122i	0.859937	+	0.859937i	−0.131393	+	0.991330i	$0.541945\pi$
$278$		0			0
$279$		−	1.59609i		−	0.0955554i
$280$		0			0
$281$		−	6.67044i		−	0.397925i	−0.980007	−	0.198962i	$-0.936243\pi$
$281$		−	6.67044i		−	0.397925i	0.980007	−	0.198962i	$-0.0637572\pi$
$282$		0			0
$283$	−12.0839	+	12.0839i	−0.718310	+	0.718310i	−0.968259	−	0.249949i	$-0.919586\pi$
$283$	−12.0839	+	12.0839i	−0.718310	+	0.718310i	0.249949	+	0.968259i	$0.419586\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.99100	−	3.99100i	0.235581	−	0.235581i
$288$		0			0
$289$		−	16.8532i		−	0.991364i
$290$		0			0
$291$	−11.9174			−0.698607
$292$		0			0
$293$	−3.79769	+	3.79769i	−0.221863	+	0.221863i	−0.809283	−	0.587419i	$-0.800144\pi$
$293$	−3.79769	+	3.79769i	−0.221863	+	0.221863i	0.587419	+	0.809283i	$0.300144\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.58334	+	2.91428i	−0.0918749	+	0.169104i
$298$		0			0
$299$	−13.5509			−0.783667
$300$		0			0
$301$	6.85319			0.395011
$302$		0			0
$303$	10.0512	−	10.0512i	0.577426	−	0.577426i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−12.6952	−	12.6952i	−0.724552	−	0.724552i	0.244977	−	0.969529i	$-0.421220\pi$
$307$	−12.6952	−	12.6952i	−0.724552	−	0.724552i	−0.969529	+	0.244977i	$0.921220\pi$
$308$		0			0
$309$		−	5.46410i		−	0.310842i
$310$		0			0
$311$	−4.18530			−0.237327			−0.118663	−	0.992935i	$-0.537861\pi$
$311$	−4.18530			−0.237327			−0.118663	+	0.992935i	$0.537861\pi$
$312$		0			0
$313$	−6.77445	−	6.77445i	−0.382914	−	0.382914i	0.489237	−	0.872151i	$-0.337275\pi$
$313$	−6.77445	−	6.77445i	−0.382914	−	0.382914i	−0.872151	+	0.489237i	$0.837275\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.9829	+	12.9829i	−0.729193	+	0.729193i	−0.970459	−	0.241266i	$-0.922437\pi$
$317$	−12.9829	+	12.9829i	−0.729193	+	0.729193i	0.241266	+	0.970459i	$0.422437\pi$
$318$		0			0
$319$	−18.7913	+	5.56072i	−1.05211	+	0.311341i
$320$		0			0
$321$		−	14.8215i		−	0.827257i
$322$		0			0
$323$	2.02799	+	2.02799i	0.112840	+	0.112840i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−8.80321	−	8.80321i	−0.486819	−	0.486819i
$328$		0			0
$329$	2.20704			0.121678
$330$		0			0
$331$	−2.86329			−0.157380			−0.0786902	−	0.996899i	$-0.525074\pi$
$331$	−2.86329			−0.157380			−0.0786902	+	0.996899i	$0.525074\pi$
$332$		0			0
$333$	6.85097	+	6.85097i	0.375431	+	0.375431i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	10.4657	+	10.4657i	0.570105	+	0.570105i	0.932158	−	0.362052i	$-0.117924\pi$
$337$	10.4657	+	10.4657i	0.570105	+	0.570105i	−0.362052	+	0.932158i	$0.617924\pi$
$338$		0			0
$339$		−	0.896498i		−	0.0486911i
$340$		0			0
$341$	5.07604	−	1.50210i	0.274883	−	0.0813434i
$342$		0			0
$343$	5.66133	−	5.66133i	0.305683	−	0.305683i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.86157	+	9.86157i	0.529397	+	0.529397i	0.920392	−	0.390996i	$-0.127869\pi$
$347$	9.86157	+	9.86157i	0.529397	+	0.529397i	−0.390996	+	0.920392i	$0.627869\pi$
$348$		0			0
$349$	−1.17255			−0.0627652			−0.0313826	−	0.999507i	$-0.509991\pi$
$349$	−1.17255			−0.0627652			−0.0313826	+	0.999507i	$0.509991\pi$
$350$		0			0
$351$			5.21972i			0.278608i
$352$		0			0
$353$	9.37418	+	9.37418i	0.498937	+	0.498937i	0.911107	−	0.412170i	$-0.135229\pi$
$353$	9.37418	+	9.37418i	0.498937	+	0.498937i	−0.412170	+	0.911107i	$0.635229\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−0.158843	+	0.158843i	−0.00840688	+	0.00840688i
$358$		0			0
$359$	−24.6646			−1.30175			−0.650874	−	0.759186i	$-0.725597\pi$
$359$	−24.6646			−1.30175			−0.650874	+	0.759186i	$0.725597\pi$
$360$		0			0
$361$	37.0269			1.94878
$362$		0			0
$363$	−10.7584	−	2.29284i	−0.564669	−	0.120343i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	12.5901	−	12.5901i	0.657200	−	0.657200i	−0.297517	−	0.954717i	$-0.596158\pi$
$367$	12.5901	−	12.5901i	0.657200	−	0.657200i	0.954717	+	0.297517i	$0.0961583\pi$
$368$		0			0
$369$	9.62710			0.501167
$370$		0			0
$371$			4.58743i			0.238167i
$372$		0			0
$373$	7.18503	−	7.18503i	0.372027	−	0.372027i	−0.496188	−	0.868215i	$-0.665267\pi$
$373$	7.18503	−	7.18503i	0.372027	−	0.372027i	0.868215	+	0.496188i	$0.165267\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−21.8083	+	21.8083i	−1.12318	+	1.12318i
$378$		0			0
$379$			7.65306i			0.393111i	0.980493	+	0.196556i	$0.0629757\pi$
$379$			7.65306i			0.393111i	−0.980493	+	0.196556i	$0.937024\pi$
$380$		0			0
$381$		−	17.5100i		−	0.897066i
$382$		0			0
$383$	10.0070	+	10.0070i	0.511336	+	0.511336i	0.914936	−	0.403600i	$-0.132241\pi$
$383$	10.0070	+	10.0070i	0.511336	+	0.511336i	−0.403600	+	0.914936i	$0.632241\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	8.26564	+	8.26564i	0.420166	+	0.420166i
$388$		0			0
$389$			16.3606i			0.829515i	0.909932	+	0.414758i	$0.136134\pi$
$389$			16.3606i			0.829515i	−0.909932	+	0.414758i	$0.863866\pi$
$390$		0			0
$391$		−	0.994724i		−	0.0503053i
$392$		0			0
$393$	−7.77418	+	7.77418i	−0.392156	+	0.392156i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	14.5756	−	14.5756i	0.731527	−	0.731527i	−0.239395	−	0.970922i	$-0.576949\pi$
$397$	14.5756	−	14.5756i	0.731527	−	0.731527i	0.970922	+	0.239395i	$0.0769492\pi$
$398$		0			0
$399$			4.38833i			0.219691i
$400$		0			0
$401$	−27.5607			−1.37632			−0.688158	−	0.725560i	$-0.741581\pi$
$401$	−27.5607			−1.37632			−0.688158	+	0.725560i	$0.741581\pi$
$402$		0			0
$403$	5.89101	−	5.89101i	0.293452	−	0.293452i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−15.3406	+	28.2357i	−0.760406	+	1.39959i
$408$		0			0
$409$	−8.18246			−0.404596			−0.202298	−	0.979324i	$-0.564841\pi$
$409$	−8.18246			−0.404596			−0.202298	+	0.979324i	$0.564841\pi$
$410$		0			0
$411$	−10.8965			−0.537485
$412$		0			0
$413$	0.550593	−	0.550593i	0.0270929	−	0.0270929i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.829118	−	0.829118i	−0.0406021	−	0.0406021i
$418$		0			0
$419$		−	2.31920i		−	0.113301i	−0.998394	−	0.0566503i	$-0.981958\pi$
$419$		−	2.31920i		−	0.113301i	0.998394	−	0.0566503i	$-0.0180420\pi$
$420$		0			0
$421$	37.0871			1.80751			0.903757	−	0.428047i	$-0.140798\pi$
$421$	37.0871			1.80751			0.903757	+	0.428047i	$0.140798\pi$
$422$		0			0
$423$	2.66191	+	2.66191i	0.129426	+	0.129426i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.78456	+	1.78456i	−0.0863607	+	0.0863607i
$428$		0			0
$429$	−16.6003	+	4.91235i	−0.801469	+	0.237171i
$430$		0			0
$431$		−	27.6459i		−	1.33166i	−0.746105	−	0.665828i	$-0.768078\pi$
$431$		−	27.6459i		−	1.33166i	0.746105	−	0.665828i	$-0.231922\pi$
$432$		0			0
$433$	−4.01974	−	4.01974i	−0.193176	−	0.193176i	0.603891	−	0.797067i	$-0.293616\pi$
$433$	−4.01974	−	4.01974i	−0.193176	−	0.193176i	−0.797067	+	0.603891i	$0.793616\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−13.7405	−	13.7405i	−0.657298	−	0.657298i
$438$		0			0
$439$	−15.7389			−0.751176			−0.375588	−	0.926787i	$-0.622559\pi$
$439$	−15.7389			−0.751176			−0.375588	+	0.926787i	$0.622559\pi$
$440$		0			0
$441$	6.65628			0.316966
$442$		0			0
$443$	−25.8900	−	25.8900i	−1.23007	−	1.23007i	−0.963935	−	0.266137i	$-0.914252\pi$
$443$	−25.8900	−	25.8900i	−1.23007	−	1.23007i	−0.266137	−	0.963935i	$-0.585748\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	15.6296	+	15.6296i	0.739253	+	0.739253i
$448$		0			0
$449$			30.2487i			1.42752i	0.700388	+	0.713762i	$0.253010\pi$
$449$			30.2487i			1.42752i	−0.700388	+	0.713762i	$0.746990\pi$
$450$		0			0
$451$	9.06019	+	30.6171i	0.426628	+	1.44170i
$452$		0			0
$453$	−3.62152	+	3.62152i	−0.170154	+	0.170154i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−26.0331	−	26.0331i	−1.21778	−	1.21778i	−0.968409	−	0.249366i	$-0.919778\pi$
$457$	−26.0331	−	26.0331i	−1.21778	−	1.21778i	−0.249366	−	0.968409i	$-0.580222\pi$
$458$		0			0
$459$	−0.383162			−0.0178845
$460$		0			0
$461$		−	5.33250i		−	0.248359i	−0.992260	−	0.124180i	$-0.960370\pi$
$461$		−	5.33250i		−	0.248359i	0.992260	−	0.124180i	$-0.0396299\pi$
$462$		0			0
$463$	−5.32382	−	5.32382i	−0.247419	−	0.247419i	0.572492	−	0.819911i	$-0.305977\pi$
$463$	−5.32382	−	5.32382i	−0.247419	−	0.247419i	−0.819911	+	0.572492i	$0.805977\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−15.9035	+	15.9035i	−0.735927	+	0.735927i	−0.971787	−	0.235860i	$-0.924209\pi$
$467$	−15.9035	+	15.9035i	−0.735927	+	0.735927i	0.235860	+	0.971787i	$0.424209\pi$
$468$		0			0
$469$	5.83736			0.269544
$470$		0			0
$471$	−5.68477			−0.261940
$472$		0			0
$473$	−18.5083	+	34.0661i	−0.851012	+	1.56636i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−5.53290	+	5.53290i	−0.253334	+	0.253334i
$478$		0			0
$479$	13.2428			0.605077			0.302539	−	0.953137i	$-0.402166\pi$
$479$	13.2428			0.605077			0.302539	+	0.953137i	$0.402166\pi$
$480$		0			0
$481$			50.5725i			2.30591i
$482$		0			0
$483$	1.07623	−	1.07623i	0.0489703	−	0.0489703i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	9.42868	−	9.42868i	0.427254	−	0.427254i	−0.460438	−	0.887692i	$-0.652308\pi$
$487$	9.42868	−	9.42868i	0.427254	−	0.427254i	0.887692	+	0.460438i	$0.152308\pi$
$488$		0			0
$489$			4.49259i			0.203162i
$490$		0			0
$491$			9.62092i			0.434186i	0.976151	+	0.217093i	$0.0696575\pi$
$491$			9.62092i			0.434186i	−0.976151	+	0.217093i	$0.930343\pi$
$492$		0			0
$493$	−1.60087	−	1.60087i	−0.0720997	−	0.0720997i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.45263	−	5.45263i	−0.244584	−	0.244584i
$498$		0			0
$499$			40.8869i			1.83035i	0.403055	+	0.915176i	$0.367948\pi$
$499$			40.8869i			1.83035i	−0.403055	+	0.915176i	$0.632052\pi$
$500$		0			0
$501$			20.2259i			0.903625i
$502$		0			0
$503$	11.7420	−	11.7420i	0.523548	−	0.523548i	−0.395093	−	0.918641i	$-0.629288\pi$
$503$	11.7420	−	11.7420i	0.523548	−	0.523548i	0.918641	+	0.395093i	$0.129288\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−10.0731	+	10.0731i	−0.447361	+	0.447361i
$508$		0			0
$509$			14.0100i			0.620982i	0.950576	+	0.310491i	$0.100494\pi$
$509$			14.0100i			0.620982i	−0.950576	+	0.310491i	$0.899506\pi$
$510$		0			0
$511$	−1.51957			−0.0672217
$512$		0			0
$513$	−5.29277	+	5.29277i	−0.233682	+	0.233682i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−5.96051	+	10.9708i	−0.262143	+	0.482497i
$518$		0			0
$519$	5.46885			0.240056
$520$		0			0
$521$	34.2977			1.50261			0.751306	−	0.659954i	$-0.229424\pi$
$521$	34.2977			1.50261			0.751306	+	0.659954i	$0.229424\pi$
$522$		0			0
$523$	20.2265	−	20.2265i	0.884445	−	0.884445i	−0.109538	−	0.993983i	$-0.534937\pi$
$523$	20.2265	−	20.2265i	0.884445	−	0.884445i	0.993983	+	0.109538i	$0.0349370\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.432439	+	0.432439i	0.0188373	+	0.0188373i
$528$		0			0
$529$		−	16.2603i		−	0.706970i
$530$		0			0
$531$	1.32814			0.0576364
$532$		0			0
$533$	35.5327	+	35.5327i	1.53909	+	1.53909i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−4.78042	+	4.78042i	−0.206290	+	0.206290i
$538$		0			0
$539$	6.26432	+	21.1690i	0.269823	+	0.911812i
$540$		0			0
$541$		−	2.83923i		−	0.122068i	−0.998136	−	0.0610340i	$-0.980560\pi$
$541$		−	2.83923i		−	0.122068i	0.998136	−	0.0610340i	$-0.0194398\pi$
$542$		0			0
$543$	−4.80337	−	4.80337i	−0.206132	−	0.206132i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−4.35694	−	4.35694i	−0.186289	−	0.186289i	0.607800	−	0.794090i	$-0.292052\pi$
$547$	−4.35694	−	4.35694i	−0.186289	−	0.186289i	−0.794090	+	0.607800i	$0.792052\pi$
$548$		0			0
$549$	−4.30471			−0.183720
$550$		0			0
$551$	−44.2270			−1.88413
$552$		0			0
$553$	3.09442	+	3.09442i	0.131588	+	0.131588i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.7856	−	21.7856i	−0.923087	−	0.923087i	0.0741594	−	0.997246i	$-0.476373\pi$
$557$	−21.7856	−	21.7856i	−0.923087	−	0.923087i	−0.997246	+	0.0741594i	$0.976373\pi$
$558$		0			0
$559$			61.0153i			2.58067i
$560$		0			0
$561$	−0.360599	−	1.21857i	−0.0152245	−	0.0514481i
$562$		0			0
$563$	9.35660	−	9.35660i	0.394334	−	0.394334i	−0.481895	−	0.876229i	$-0.660051\pi$
$563$	9.35660	−	9.35660i	0.394334	−	0.394334i	0.876229	+	0.481895i	$0.160051\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.414559	−	0.414559i	−0.0174098	−	0.0174098i
$568$		0			0
$569$	9.93055			0.416310			0.208155	−	0.978096i	$-0.433254\pi$
$569$	9.93055			0.416310			0.208155	+	0.978096i	$0.433254\pi$
$570$		0			0
$571$		−	19.7483i		−	0.826442i	−0.910631	−	0.413221i	$-0.864404\pi$
$571$		−	19.7483i		−	0.826442i	0.910631	−	0.413221i	$-0.135596\pi$
$572$		0			0
$573$	10.7774	+	10.7774i	0.450232	+	0.450232i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	13.9068	−	13.9068i	0.578946	−	0.578946i	−0.355667	−	0.934613i	$-0.615746\pi$
$577$	13.9068	−	13.9068i	0.578946	−	0.578946i	0.934613	+	0.355667i	$0.115746\pi$
$578$		0			0
$579$	−3.33142			−0.138449
$580$		0			0
$581$	−7.94453			−0.329595
$582$		0			0
$583$	−22.8034	−	12.3892i	−0.944419	−	0.513108i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.91411	−	5.91411i	0.244102	−	0.244102i	−0.574443	−	0.818545i	$-0.694781\pi$
$587$	5.91411	−	5.91411i	0.244102	−	0.244102i	0.818545	+	0.574443i	$0.194781\pi$
$588$		0			0
$589$	11.9469			0.492264
$590$		0			0
$591$		−	14.4384i		−	0.593915i
$592$		0			0
$593$	−9.50924	+	9.50924i	−0.390498	+	0.390498i	−0.874865	−	0.484367i	$-0.839050\pi$
$593$	−9.50924	+	9.50924i	−0.390498	+	0.390498i	0.484367	+	0.874865i	$0.339050\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−9.89129	+	9.89129i	−0.404824	+	0.404824i
$598$		0			0
$599$		−	15.0310i		−	0.614148i	−0.951686	−	0.307074i	$-0.900650\pi$
$599$		−	15.0310i		−	0.614148i	0.951686	−	0.307074i	$-0.0993499\pi$
$600$		0			0
$601$		−	35.2583i		−	1.43822i	−0.694899	−	0.719108i	$-0.744551\pi$
$601$		−	35.2583i		−	1.43822i	0.694899	−	0.719108i	$-0.255449\pi$
$602$		0			0
$603$	7.04044	+	7.04044i	0.286709	+	0.286709i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	13.3633	+	13.3633i	0.542399	+	0.542399i	0.924232	−	0.381832i	$-0.124707\pi$
$607$	13.3633	+	13.3633i	0.542399	+	0.542399i	−0.381832	+	0.924232i	$0.624707\pi$
$608$		0			0
$609$		−	3.46410i		−	0.140372i
$610$		0			0
$611$			19.6497i			0.794941i
$612$		0			0
$613$	29.8946	−	29.8946i	1.20743	−	1.20743i	0.235573	−	0.971857i	$-0.424303\pi$
$613$	29.8946	−	29.8946i	1.20743	−	1.20743i	0.971857	−	0.235573i	$-0.0756968\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.89362	−	1.89362i	0.0762344	−	0.0762344i	−0.667961	−	0.744196i	$-0.732833\pi$
$617$	1.89362	−	1.89362i	0.0762344	−	0.0762344i	0.744196	+	0.667961i	$0.232833\pi$
$618$		0			0
$619$			47.2006i			1.89715i	0.316550	+	0.948576i	$0.397475\pi$
$619$			47.2006i			1.89715i	−0.316550	+	0.948576i	$0.602525\pi$
$620$		0			0
$621$	2.59609			0.104178
$622$		0			0
$623$	6.83234	−	6.83234i	0.273732	−	0.273732i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−21.8137	−	11.8515i	−0.871155	−	0.473303i
$628$		0			0
$629$	−3.71236			−0.148021
$630$		0			0
$631$	2.58921			0.103075			0.0515374	−	0.998671i	$-0.483588\pi$
$631$	2.58921			0.103075			0.0515374	+	0.998671i	$0.483588\pi$
$632$		0			0
$633$	−3.73096	+	3.73096i	−0.148292	+	0.148292i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	24.5677	+	24.5677i	0.973407	+	0.973407i
$638$		0			0
$639$		−	13.1528i		−	0.520318i
$640$		0			0
$641$	36.0734			1.42481			0.712407	−	0.701766i	$-0.247605\pi$
$641$	36.0734			1.42481			0.712407	+	0.701766i	$0.247605\pi$
$642$		0			0
$643$	−6.02606	−	6.02606i	−0.237645	−	0.237645i	0.578229	−	0.815874i	$-0.303744\pi$
$643$	−6.02606	−	6.02606i	−0.237645	−	0.237645i	−0.815874	+	0.578229i	$0.803744\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21.9065	+	21.9065i	−0.861233	+	0.861233i	−0.991481	−	0.130249i	$-0.958422\pi$
$647$	−21.9065	+	21.9065i	−0.861233	+	0.861233i	0.130249	+	0.991481i	$0.458422\pi$
$648$		0			0
$649$	1.24993	+	4.22389i	0.0490641	+	0.165802i
$650$		0			0
$651$			0.935748i			0.0366748i
$652$		0			0
$653$	18.4714	+	18.4714i	0.722840	+	0.722840i	0.969183	−	0.246343i	$-0.0792289\pi$
$653$	18.4714	+	18.4714i	0.722840	+	0.722840i	−0.246343	+	0.969183i	$0.579229\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.83275	−	1.83275i	−0.0715025	−	0.0715025i
$658$		0			0
$659$	−28.9779			−1.12882			−0.564409	−	0.825495i	$-0.690896\pi$
$659$	−28.9779			−1.12882			−0.564409	+	0.825495i	$0.690896\pi$
$660$		0			0
$661$	36.4228			1.41668			0.708342	−	0.705869i	$-0.249443\pi$
$661$	36.4228			1.41668			0.708342	+	0.705869i	$0.249443\pi$
$662$		0			0
$663$	−1.41421	−	1.41421i	−0.0549235	−	0.0549235i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	10.8466	+	10.8466i	0.419982	+	0.419982i
$668$		0			0
$669$			22.3490i			0.864062i
$670$		0			0
$671$	−4.05122	−	13.6903i	−0.156396	−	0.528507i
$672$		0			0
$673$	−4.95639	+	4.95639i	−0.191055	+	0.191055i	−0.796152	−	0.605097i	$-0.793134\pi$
$673$	−4.95639	+	4.95639i	−0.191055	+	0.191055i	0.605097	+	0.796152i	$0.293134\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.24446	+	2.24446i	0.0862616	+	0.0862616i	0.748921	−	0.662659i	$-0.230572\pi$
$677$	2.24446	+	2.24446i	0.0862616	+	0.0862616i	−0.662659	+	0.748921i	$0.730572\pi$
$678$		0			0
$679$	6.98684			0.268130
$680$		0			0
$681$			11.9094i			0.456368i
$682$		0			0
$683$	−31.0552	−	31.0552i	−1.18829	−	1.18829i	−0.977539	−	0.210754i	$-0.932408\pi$
$683$	−31.0552	−	31.0552i	−1.18829	−	1.18829i	−0.210754	−	0.977539i	$-0.567592\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	11.4552	−	11.4552i	0.437044	−	0.437044i
$688$		0			0
$689$	−40.8428			−1.55599
$690$		0			0
$691$	33.4409			1.27215			0.636076	−	0.771627i	$-0.280557\pi$
$691$	33.4409			1.27215			0.636076	+	0.771627i	$0.280557\pi$
$692$		0			0
$693$	0.928275	−	1.70857i	0.0352622	−	0.0649032i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.60833	+	2.60833i	−0.0987977	+	0.0987977i
$698$		0			0
$699$	−21.0152			−0.794869
$700$		0			0
$701$		−	17.8718i		−	0.675007i	−0.941324	−	0.337503i	$-0.890418\pi$
$701$		−	17.8718i		−	0.675007i	0.941324	−	0.337503i	$-0.109582\pi$
$702$		0			0
$703$	−51.2803	+	51.2803i	−1.93407	+	1.93407i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.89276	+	5.89276i	−0.221620	+	0.221620i
$708$		0			0
$709$		−	32.9847i		−	1.23877i	−0.785088	−	0.619384i	$-0.787382\pi$
$709$		−	32.9847i		−	1.23877i	0.785088	−	0.619384i	$-0.212618\pi$
$710$		0			0
$711$			7.46438i			0.279936i
$712$		0			0
$713$	−2.92996	−	2.92996i	−0.109728	−	0.109728i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−19.1238	−	19.1238i	−0.714190	−	0.714190i
$718$		0			0
$719$		−	14.5221i		−	0.541584i	−0.962638	−	0.270792i	$-0.912714\pi$
$719$		−	14.5221i		−	0.541584i	0.962638	−	0.270792i	$-0.0872856\pi$
$720$		0			0
$721$			3.20347i			0.119303i
$722$		0			0
$723$	13.2978	−	13.2978i	0.494552	−	0.494552i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	11.9951	−	11.9951i	0.444872	−	0.444872i	−0.448774	−	0.893646i	$-0.648139\pi$
$727$	11.9951	−	11.9951i	0.444872	−	0.444872i	0.893646	+	0.448774i	$0.148139\pi$
$728$		0			0
$729$		−	1.00000i		−	0.0370370i
$730$		0			0
$731$	−4.47893			−0.165659
$732$		0			0
$733$	−0.254628	+	0.254628i	−0.00940492	+	0.00940492i	−0.711794	−	0.702389i	$-0.752117\pi$
$733$	−0.254628	+	0.254628i	−0.00940492	+	0.00940492i	0.702389	+	0.711794i	$0.252117\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−15.7649	+	29.0166i	−0.580706	+	1.06884i
$738$		0			0
$739$	0.441437			0.0162385			0.00811926	−	0.999967i	$-0.497416\pi$
$739$	0.441437			0.0162385			0.00811926	+	0.999967i	$0.497416\pi$
$740$		0			0
$741$	−39.0702			−1.43528
$742$		0			0
$743$	−4.34828	+	4.34828i	−0.159523	+	0.159523i	−0.782355	−	0.622832i	$-0.785982\pi$
$743$	−4.34828	+	4.34828i	−0.159523	+	0.159523i	0.622832	+	0.782355i	$0.285982\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−9.58191	−	9.58191i	−0.350584	−	0.350584i
$748$		0			0
$749$			8.68949i			0.317507i
$750$		0			0
$751$	0.807820			0.0294778			0.0147389	−	0.999891i	$-0.495308\pi$
$751$	0.807820			0.0294778			0.0147389	+	0.999891i	$0.495308\pi$
$752$		0			0
$753$	4.68428	+	4.68428i	0.170705	+	0.170705i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−25.6578	+	25.6578i	−0.932548	+	0.932548i	−0.997865	−	0.0653167i	$-0.979194\pi$
$757$	−25.6578	+	25.6578i	−0.932548	+	0.932548i	0.0653167	+	0.997865i	$0.479194\pi$
$758$		0			0
$759$	2.44322	+	8.25634i	0.0886831	+	0.299686i
$760$		0			0
$761$			45.8975i			1.66378i	0.554938	+	0.831892i	$0.312742\pi$
$761$			45.8975i			1.66378i	−0.554938	+	0.831892i	$0.687258\pi$
$762$		0			0
$763$	5.16110	+	5.16110i	0.186844	+	0.186844i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.90204	+	4.90204i	0.177002	+	0.177002i
$768$		0			0
$769$	−25.9544			−0.935938			−0.467969	−	0.883745i	$-0.655014\pi$
$769$	−25.9544			−0.935938			−0.467969	+	0.883745i	$0.655014\pi$
$770$		0			0
$771$	−11.2001			−0.403363
$772$		0			0
$773$	5.68034	+	5.68034i	0.204308	+	0.204308i	0.801843	−	0.597535i	$-0.203853\pi$
$773$	5.68034	+	5.68034i	0.204308	+	0.204308i	−0.597535	+	0.801843i	$0.703853\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−4.01655	−	4.01655i	−0.144093	−	0.144093i
$778$		0			0
$779$			72.0599i			2.58181i
$780$		0			0
$781$	41.8300	−	12.3783i	1.49679	−	0.442931i
$782$		0			0
$783$	4.17806	−	4.17806i	0.149312	−	0.149312i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1.59271	−	1.59271i	−0.0567740	−	0.0567740i	0.678150	−	0.734924i	$-0.262782\pi$
$787$	−1.59271	−	1.59271i	−0.0567740	−	0.0567740i	−0.734924	+	0.678150i	$0.762782\pi$
$788$		0			0
$789$	16.6516			0.592811
$790$		0			0
$791$			0.525594i			0.0186880i
$792$		0			0
$793$	−15.8882	−	15.8882i	−0.564208	−	0.564208i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.96575	−	8.96575i	0.317583	−	0.317583i	−0.530255	−	0.847838i	$-0.677904\pi$
$797$	8.96575	−	8.96575i	0.317583	−	0.317583i	0.847838	+	0.530255i	$0.177904\pi$
$798$		0			0
$799$	−1.44242			−0.0510291
$800$		0			0
$801$	16.4810			0.582327
$802$		0			0
$803$	4.10388	−	7.55353i	0.144823	−	0.266558i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−18.4143	+	18.4143i	−0.648213	+	0.648213i
$808$		0			0
$809$	29.0087			1.01989			0.509946	−	0.860206i	$-0.329665\pi$
$809$	29.0087			1.01989			0.509946	+	0.860206i	$0.329665\pi$
$810$		0			0
$811$		−	46.3995i		−	1.62931i	−0.579948	−	0.814654i	$-0.696927\pi$
$811$		−	46.3995i		−	1.62931i	0.579948	−	0.814654i	$-0.303073\pi$
$812$		0			0
$813$	0.300259	−	0.300259i	0.0105306	−	0.0105306i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−61.8692	+	61.8692i	−2.16453	+	2.16453i
$818$		0			0
$819$		−	3.06019i		−	0.106932i
$820$		0			0
$821$			26.9824i			0.941691i	0.882216	+	0.470846i	$0.156051\pi$
$821$			26.9824i			0.941691i	−0.882216	+	0.470846i	$0.843949\pi$
$822$		0			0
$823$	9.68221	+	9.68221i	0.337501	+	0.337501i	0.855426	−	0.517925i	$-0.173295\pi$
$823$	9.68221	+	9.68221i	0.337501	+	0.337501i	−0.517925	+	0.855426i	$0.673295\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.3525	−	31.3525i	−1.09023	−	1.09023i	−0.995503	−	0.0947309i	$-0.969801\pi$
$827$	−31.3525	−	31.3525i	−1.09023	−	1.09023i	−0.0947309	−	0.995503i	$-0.530199\pi$
$828$		0			0
$829$			10.4572i			0.363194i	0.983373	+	0.181597i	$0.0581267\pi$
$829$			10.4572i			0.363194i	−0.983373	+	0.181597i	$0.941873\pi$
$830$		0			0
$831$		−	20.2405i		−	0.702136i
$832$		0			0
$833$	−1.80343	+	1.80343i	−0.0624852	+	0.0624852i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.12861	+	1.12861i	−0.0390103	+	0.0390103i
$838$		0			0
$839$			14.9837i			0.517294i	0.965972	+	0.258647i	$0.0832766\pi$
$839$			14.9837i			0.517294i	−0.965972	+	0.258647i	$0.916723\pi$
$840$		0			0
$841$	5.91231			0.203873
$842$		0			0
$843$	−4.71671	+	4.71671i	−0.162452	+	0.162452i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	6.30737	+	1.34423i	0.216724	+	0.0461884i
$848$		0			0
$849$	17.0891			0.586498
$850$		0			0
$851$	25.1528			0.862228
$852$		0			0
$853$	−26.6297	+	26.6297i	−0.911782	+	0.911782i	−0.996412	−	0.0846302i	$-0.973029\pi$
$853$	−26.6297	+	26.6297i	−0.911782	+	0.911782i	0.0846302	+	0.996412i	$0.473029\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.998060	+	0.998060i	0.0340931	+	0.0340931i	0.723948	−	0.689855i	$-0.242326\pi$
$857$	0.998060	+	0.998060i	0.0340931	+	0.0340931i	−0.689855	+	0.723948i	$0.742326\pi$
$858$		0			0
$859$			9.43445i			0.321899i	0.986963	+	0.160950i	$0.0514557\pi$
$859$			9.43445i			0.321899i	−0.986963	+	0.160950i	$0.948544\pi$
$860$		0			0
$861$	−5.64413			−0.192351
$862$		0			0
$863$	−25.0854	−	25.0854i	−0.853918	−	0.853918i	0.136695	−	0.990613i	$-0.456352\pi$
$863$	−25.0854	−	25.0854i	−0.853918	−	0.853918i	−0.990613	+	0.136695i	$0.956352\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−11.9170	+	11.9170i	−0.404723	+	0.404723i
$868$		0			0
$869$	−23.7390	+	7.02483i	−0.805289	+	0.238301i
$870$		0			0
$871$			51.9711i			1.76098i
$872$		0			0
$873$	8.42684	+	8.42684i	0.285205	+	0.285205i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.10600	−	1.10600i	−0.0373468	−	0.0373468i	0.688187	−	0.725534i	$-0.258407\pi$
$877$	−1.10600	−	1.10600i	−0.0373468	−	0.0373468i	−0.725534	+	0.688187i	$0.758407\pi$
$878$		0			0
$879$	5.37074			0.181151
$880$		0			0
$881$	3.96829			0.133695			0.0668476	−	0.997763i	$-0.478706\pi$
$881$	3.96829			0.133695			0.0668476	+	0.997763i	$0.478706\pi$
$882$		0			0
$883$	−9.02587	−	9.02587i	−0.303745	−	0.303745i	0.538732	−	0.842477i	$-0.318903\pi$
$883$	−9.02587	−	9.02587i	−0.303745	−	0.303745i	−0.842477	+	0.538732i	$0.818903\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.3848	+	18.3848i	0.617300	+	0.617300i	0.944838	−	0.327538i	$-0.106219\pi$
$887$	18.3848	+	18.3848i	0.617300	+	0.617300i	−0.327538	+	0.944838i	$0.606219\pi$
$888$		0			0
$889$			10.2657i			0.344300i
$890$		0			0
$891$	3.18030	−	0.941114i	0.106544	−	0.0315285i
$892$		0			0
$893$	−19.9247	+	19.9247i	−0.666754	+	0.666754i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	9.58191	+	9.58191i	0.319931	+	0.319931i
$898$		0			0
$899$	−9.43076			−0.314533
$900$		0			0
$901$		−	2.99813i		−	0.0998822i
$902$		0			0
$903$	−4.84593	−	4.84593i	−0.161263	−	0.161263i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	29.4439	−	29.4439i	0.977668	−	0.977668i	−0.0220876	−	0.999756i	$-0.507031\pi$
$907$	29.4439	−	29.4439i	0.977668	−	0.977668i	0.999756	+	0.0220876i	$0.00703129\pi$
$908$		0			0
$909$	−14.2145			−0.471466
$910$		0			0
$911$	29.4016			0.974117			0.487058	−	0.873369i	$-0.338070\pi$
$911$	29.4016			0.974117			0.487058	+	0.873369i	$0.338070\pi$
$912$		0			0
$913$	21.4557	−	39.4910i	0.710079	−	1.30696i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.55781	−	4.55781i	0.150512	−	0.150512i
$918$		0			0
$919$	−34.1762			−1.12737			−0.563684	−	0.825991i	$-0.690616\pi$
$919$	−34.1762			−1.12737			−0.563684	+	0.825991i	$0.690616\pi$
$920$		0			0
$921$			17.9537i			0.591594i
$922$		0			0
$923$	48.5458	−	48.5458i	1.59791	−	1.59791i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−3.86370	+	3.86370i	−0.126901	+	0.126901i
$928$		0			0
$929$		−	20.1600i		−	0.661429i	−0.943731	−	0.330715i	$-0.892710\pi$
$929$		−	20.1600i		−	0.661429i	0.943731	−	0.330715i	$-0.107290\pi$
$930$		0			0
$931$			49.8230i			1.63288i
$932$		0			0
$933$	2.95945	+	2.95945i	0.0968882	+	0.0968882i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−24.5487	−	24.5487i	−0.801972	−	0.801972i	0.181432	−	0.983404i	$-0.441927\pi$
$937$	−24.5487	−	24.5487i	−0.801972	−	0.801972i	−0.983404	+	0.181432i	$0.941927\pi$
$938$		0			0
$939$			9.58051i			0.312648i
$940$		0			0
$941$		−	17.4702i		−	0.569512i	−0.958600	−	0.284756i	$-0.908087\pi$
$941$		−	17.4702i		−	0.569512i	0.958600	−	0.284756i	$-0.0919126\pi$
$942$		0			0
$943$	17.6726	−	17.6726i	0.575499	−	0.575499i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	3.87677	−	3.87677i	0.125978	−	0.125978i	−0.641307	−	0.767285i	$-0.721607\pi$
$947$	3.87677	−	3.87677i	0.125978	−	0.125978i	0.767285	+	0.641307i	$0.221607\pi$
$948$		0			0
$949$		−	13.5290i		−	0.439171i
$950$		0			0
$951$	18.3606			0.595383
$952$		0			0
$953$	−19.8977	+	19.8977i	−0.644549	+	0.644549i	−0.951670	−	0.307122i	$-0.900634\pi$
$953$	−19.8977	+	19.8977i	−0.644549	+	0.644549i	0.307122	+	0.951670i	$0.400634\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	17.2195	+	9.35545i	0.556627	+	0.302419i
$958$		0			0
$959$	6.38834			0.206290
$960$		0			0
$961$	−28.4525			−0.917822
$962$		0			0
$963$	−10.4804	+	10.4804i	−0.337726	+	0.337726i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.26156	−	1.26156i	−0.0405689	−	0.0405689i	0.686531	−	0.727100i	$-0.259132\pi$
$967$	−1.26156	−	1.26156i	−0.0405689	−	0.0405689i	−0.727100	+	0.686531i	$0.759132\pi$
$968$		0			0
$969$		−	2.86801i		−	0.0921338i
$970$		0			0
$971$	32.4966			1.04286			0.521432	−	0.853293i	$-0.325398\pi$
$971$	32.4966			1.04286			0.521432	+	0.853293i	$0.325398\pi$
$972$		0			0
$973$	0.486091	+	0.486091i	0.0155834	+	0.0155834i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−26.3045	+	26.3045i	−0.841554	+	0.841554i	−0.989061	−	0.147507i	$-0.952875\pi$
$977$	−26.3045	+	26.3045i	−0.841554	+	0.841554i	0.147507	+	0.989061i	$0.452875\pi$
$978$		0			0
$979$	15.5105	+	52.4145i	0.495717	+	1.67517i
$980$		0			0
$981$			12.4496i			0.397486i
$982$		0			0
$983$	−15.6284	−	15.6284i	−0.498469	−	0.498469i	0.412492	−	0.910961i	$-0.364658\pi$
$983$	−15.6284	−	15.6284i	−0.498469	−	0.498469i	−0.910961	+	0.412492i	$0.864658\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.56061	−	1.56061i	−0.0496748	−	0.0496748i
$988$		0			0
$989$	30.3467			0.964968
$990$		0			0
$991$	−37.3089			−1.18516			−0.592578	−	0.805513i	$-0.701890\pi$
$991$	−37.3089			−1.18516			−0.592578	+	0.805513i	$0.701890\pi$
$992$		0			0
$993$	2.02465	+	2.02465i	0.0642503	+	0.0642503i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−27.4191	−	27.4191i	−0.868371	−	0.868371i	0.123921	−	0.992292i	$-0.460453\pi$
$997$	−27.4191	−	27.4191i	−0.868371	−	0.868371i	−0.992292	+	0.123921i	$0.960453\pi$
$998$		0			0
$999$		−	9.68874i		−	0.306538i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3300.2.x.d.1693.8	yes	32
5.2	odd	4	inner	3300.2.x.d.1957.7	yes	32
5.3	odd	4	inner	3300.2.x.d.1957.12	yes	32
5.4	even	2	inner	3300.2.x.d.1693.11	yes	32
11.10	odd	2	inner	3300.2.x.d.1693.7	✓	32
55.32	even	4	inner	3300.2.x.d.1957.8	yes	32
55.43	even	4	inner	3300.2.x.d.1957.11	yes	32
55.54	odd	2	inner	3300.2.x.d.1693.12	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3300.2.x.d.1693.7	✓	32	11.10	odd	2	inner
3300.2.x.d.1693.8	yes	32	1.1	even	1	trivial
3300.2.x.d.1693.11	yes	32	5.4	even	2	inner
3300.2.x.d.1693.12	yes	32	55.54	odd	2	inner
3300.2.x.d.1957.7	yes	32	5.2	odd	4	inner
3300.2.x.d.1957.8	yes	32	55.32	even	4	inner
3300.2.x.d.1957.11	yes	32	55.43	even	4	inner
3300.2.x.d.1957.12	yes	32	5.3	odd	4	inner

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:	\( N \)	\(=\)	\( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3300.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{3} +(0.414559 + 0.414559i) q^{7} +1.00000i q^{9} +(-3.18030 + 0.941114i) q^{11} +(-3.69090 + 3.69090i) q^{13} +(-0.270937 - 0.270937i) q^{17} -7.48511 q^{19} -0.586275i q^{21} +(1.83571 + 1.83571i) q^{23} +(0.707107 - 0.707107i) q^{27} +5.90866 q^{29} -1.59609 q^{31} +(2.91428 + 1.58334i) q^{33} +(6.85097 - 6.85097i) q^{37} +5.21972 q^{39} -9.62710i q^{41} +(8.26564 - 8.26564i) q^{43} +(2.66191 - 2.66191i) q^{47} -6.65628i q^{49} +0.383162i q^{51} +(5.53290 + 5.53290i) q^{53} +(5.29277 + 5.29277i) q^{57} -1.32814i q^{59} +4.30471i q^{61} +(-0.414559 + 0.414559i) q^{63} +(7.04044 - 7.04044i) q^{67} -2.59609i q^{69} -13.1528 q^{71} +(-1.83275 + 1.83275i) q^{73} +(-1.70857 - 0.928275i) q^{77} +7.46438 q^{79} -1.00000 q^{81} +(-9.58191 + 9.58191i) q^{83} +(-4.17806 - 4.17806i) q^{87} -16.4810i q^{89} -3.06019 q^{91} +(1.12861 + 1.12861i) q^{93} +(8.42684 - 8.42684i) q^{97} +(-0.941114 - 3.18030i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 24 q^{11} + 16 q^{31} - 32 q^{71} - 32 q^{81} + 80 q^{91}+O(q^{100}) \) 32 * q - 24 * q^11 + 16 * q^31 - 32 * q^71 - 32 * q^81 + 80 * q^91

Introduction

L-functions

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