LMFDB - Embedded newform 2700.2.bf.f.557.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2700,2,Mod(557,2700)] 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("2700.557"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 10, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.bf (of order $12$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,0,0,12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$8$ over $\Q(\zeta_{12})$
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			557.8
Character	$\chi$	$=$	2700.557
Dual form			2700.2.bf.f.1493.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.876983 - 3.27295i) q^{7} +(0.419659 - 0.242290i) q^{11} +(0.693644 + 2.58871i) q^{13} +(-4.14993 + 4.14993i) q^{17} +3.85410i q^{19} +(3.50358 - 0.938781i) q^{23} +(3.96149 + 6.86149i) q^{29} +(1.19381 - 2.06773i) q^{31} +(-1.09958 - 1.09958i) q^{37} +(10.1637 + 5.86804i) q^{41} +(9.85458 + 2.64053i) q^{43} +(-1.10685 - 0.296580i) q^{47} +(-3.88090 - 2.24064i) q^{49} +(-7.94909 - 7.94909i) q^{53} +(3.17605 - 5.50108i) q^{59} +(3.04683 + 5.27726i) q^{61} +(7.53469 - 2.01891i) q^{67} +0.601711i q^{71} +(8.84189 - 8.84189i) q^{73} +(-0.424969 - 1.58601i) q^{77} +(-12.1047 + 6.98866i) q^{79} +(3.89739 - 14.5453i) q^{83} -13.7208 q^{89} +9.08103 q^{91} +(0.0357970 - 0.133596i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 12 q^{11} + 8 q^{31} - 60 q^{41} + 52 q^{61} + 120 q^{91}+O(q^{100}) $ 32 * q + 12 * q^11 + 8 * q^31 - 60 * q^41 + 52 * q^61 + 120 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{1}{4}\right)$

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			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.876983	−	3.27295i	0.331468	−	1.23706i	−0.576179	−	0.817324i	$-0.695457\pi$
$7$	0.876983	−	3.27295i	0.331468	−	1.23706i	0.907647	−	0.419734i	$-0.137876\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.419659	−	0.242290i	0.126532	−	0.0730533i	−0.435398	−	0.900238i	$-0.643392\pi$
$11$	0.419659	−	0.242290i	0.126532	−	0.0730533i	0.561930	+	0.827185i	$0.310059\pi$
$12$		0			0
$13$	0.693644	+	2.58871i	0.192382	+	0.717980i	0.992929	+	0.118710i	$0.0378758\pi$
$13$	0.693644	+	2.58871i	0.192382	+	0.717980i	−0.800547	+	0.599270i	$0.795458\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.14993	+	4.14993i	−1.00651	+	1.00651i	−0.00652688	+	0.999979i	$0.502078\pi$
$17$	−4.14993	+	4.14993i	−1.00651	+	1.00651i	−0.999979	+	0.00652688i	$0.997922\pi$
$18$		0			0
$19$			3.85410i			0.884192i	0.896968	+	0.442096i	$0.145765\pi$
$19$			3.85410i			0.884192i	−0.896968	+	0.442096i	$0.854235\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.50358	−	0.938781i	0.730546	−	0.195749i	0.125674	−	0.992072i	$-0.459891\pi$
$23$	3.50358	−	0.938781i	0.730546	−	0.195749i	0.604873	+	0.796322i	$0.293224\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.96149	+	6.86149i	0.735629	+	1.27415i	0.954447	+	0.298382i	$0.0964468\pi$
$29$	3.96149	+	6.86149i	0.735629	+	1.27415i	−0.218817	+	0.975766i	$0.570220\pi$
$30$		0			0
$31$	1.19381	−	2.06773i	0.214414	−	0.371376i	−0.738677	−	0.674059i	$-0.764549\pi$
$31$	1.19381	−	2.06773i	0.214414	−	0.371376i	0.953091	+	0.302683i	$0.0978825\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.09958	−	1.09958i	−0.180771	−	0.180771i	0.610921	−	0.791692i	$-0.290799\pi$
$37$	−1.09958	−	1.09958i	−0.180771	−	0.180771i	−0.791692	+	0.610921i	$0.790799\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.1637	+	5.86804i	1.58731	+	0.916434i	0.993748	+	0.111646i	$0.0356122\pi$
$41$	10.1637	+	5.86804i	1.58731	+	0.916434i	0.593562	+	0.804788i	$0.297721\pi$
$42$		0			0
$43$	9.85458	+	2.64053i	1.50281	+	0.402677i	0.914040	−	0.405624i	$-0.132946\pi$
$43$	9.85458	+	2.64053i	1.50281	+	0.402677i	0.588770	+	0.808301i	$0.299613\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.10685	−	0.296580i	−0.161451	−	0.0432607i	0.177188	−	0.984177i	$-0.443300\pi$
$47$	−1.10685	−	0.296580i	−0.161451	−	0.0432607i	−0.338639	+	0.940916i	$0.609967\pi$
$48$		0			0
$49$	−3.88090	−	2.24064i	−0.554414	−	0.320091i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.94909	−	7.94909i	−1.09189	−	1.09189i	−0.995327	−	0.0965652i	$-0.969214\pi$
$53$	−7.94909	−	7.94909i	−1.09189	−	1.09189i	−0.0965652	−	0.995327i	$-0.530786\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.17605	−	5.50108i	0.413486	−	0.716179i	−0.581782	−	0.813345i	$-0.697644\pi$
$59$	3.17605	−	5.50108i	0.413486	−	0.716179i	0.995268	+	0.0971654i	$0.0309776\pi$
$60$		0			0
$61$	3.04683	+	5.27726i	0.390107	+	0.675684i	0.992463	−	0.122543i	$-0.0391048\pi$
$61$	3.04683	+	5.27726i	0.390107	+	0.675684i	−0.602357	+	0.798227i	$0.705771\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	7.53469	−	2.01891i	0.920509	−	0.246650i	0.232706	−	0.972547i	$-0.425242\pi$
$67$	7.53469	−	2.01891i	0.920509	−	0.246650i	0.687803	+	0.725897i	$0.258575\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			0.601711i			0.0714100i	0.999362	+	0.0357050i	$0.0113677\pi$
$71$			0.601711i			0.0714100i	−0.999362	+	0.0357050i	$0.988632\pi$
$72$		0			0
$73$	8.84189	−	8.84189i	1.03487	−	1.03487i	0.0354953	−	0.999370i	$-0.488699\pi$
$73$	8.84189	−	8.84189i	1.03487	−	1.03487i	0.999370	−	0.0354953i	$-0.0113009\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.424969	−	1.58601i	−0.0484297	−	0.180742i
$78$		0			0
$79$	−12.1047	+	6.98866i	−1.36189	+	0.786286i	−0.989875	−	0.141941i	$-0.954666\pi$
$79$	−12.1047	+	6.98866i	−1.36189	+	0.786286i	−0.372013	+	0.928228i	$0.621332\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.89739	−	14.5453i	0.427794	−	1.59655i	−0.329950	−	0.943998i	$-0.607032\pi$
$83$	3.89739	−	14.5453i	0.427794	−	1.59655i	0.757744	−	0.652551i	$-0.226301\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.7208			−1.45441			−0.727203	−	0.686423i	$-0.759180\pi$
$89$	−13.7208			−1.45441			−0.727203	+	0.686423i	$0.759180\pi$
$90$		0			0
$91$	9.08103			0.951951
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.0357970	−	0.133596i	0.00363463	−	0.0135646i	−0.964085	−	0.265595i	$-0.914432\pi$
$97$	0.0357970	−	0.133596i	0.00363463	−	0.0135646i	0.967719	+	0.252030i	$0.0810983\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	13.1416	−	7.58729i	1.30763	−	0.754963i	0.325934	−	0.945393i	$-0.394321\pi$
$101$	13.1416	−	7.58729i	1.30763	−	0.754963i	0.981701	+	0.190429i	$0.0609879\pi$
$102$		0			0
$103$	0.0520016	+	0.194073i	0.00512387	+	0.0191225i	0.968440	−	0.249245i	$-0.0801825\pi$
$103$	0.0520016	+	0.194073i	0.00512387	+	0.0191225i	−0.963317	+	0.268368i	$0.913516\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.11849	−	8.11849i	0.784844	−	0.784844i	−0.195800	−	0.980644i	$-0.562730\pi$
$107$	8.11849	−	8.11849i	0.784844	−	0.784844i	0.980644	+	0.195800i	$0.0627304\pi$
$108$		0			0
$109$		−	5.49606i		−	0.526427i	−0.964738	−	0.263213i	$-0.915218\pi$
$109$		−	5.49606i		−	0.526427i	0.964738	−	0.263213i	$-0.0847823\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.4044	−	4.12758i	1.44912	−	0.388290i	0.553402	−	0.832914i	$-0.313329\pi$
$113$	15.4044	−	4.12758i	1.44912	−	0.388290i	0.895718	+	0.444624i	$0.146663\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.94307	+	17.2219i	0.911480	+	1.57873i
$120$		0			0
$121$	−5.38259	+	9.32292i	−0.489326	+	0.847538i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−0.697687	−	0.697687i	−0.0619097	−	0.0619097i	0.675474	−	0.737384i	$-0.263939\pi$
$127$	−0.697687	−	0.697687i	−0.0619097	−	0.0619097i	−0.737384	+	0.675474i	$0.763939\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.161760	−	0.0933923i	−0.0141330	−	0.00815972i	0.492917	−	0.870076i	$-0.335931\pi$
$131$	−0.161760	−	0.0933923i	−0.0141330	−	0.00815972i	−0.507050	+	0.861917i	$0.669264\pi$
$132$		0			0
$133$	12.6143	+	3.37998i	1.09380	+	0.293082i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.7939	−	2.89222i	−0.922187	−	0.247099i	−0.233667	−	0.972317i	$-0.575073\pi$
$137$	−10.7939	−	2.89222i	−0.922187	−	0.247099i	−0.688520	+	0.725217i	$0.741739\pi$
$138$		0			0
$139$	7.19491	+	4.15398i	0.610264	+	0.352336i	0.773069	−	0.634322i	$-0.218721\pi$
$139$	7.19491	+	4.15398i	0.610264	+	0.352336i	−0.162805	+	0.986658i	$0.552054\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.918314	+	0.918314i	0.0767933	+	0.0767933i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.09023	+	7.08448i	−0.335084	+	0.580383i	−0.983501	−	0.180902i	$-0.942098\pi$
$149$	−4.09023	+	7.08448i	−0.335084	+	0.580383i	0.648417	+	0.761286i	$0.275432\pi$
$150$		0			0
$151$	10.5366	+	18.2499i	0.857454	+	1.48515i	0.874349	+	0.485297i	$0.161288\pi$
$151$	10.5366	+	18.2499i	0.857454	+	1.48515i	−0.0168953	+	0.999857i	$0.505378\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−12.4202	+	3.32798i	−0.991239	+	0.265602i	−0.717771	−	0.696279i	$-0.754838\pi$
$157$	−12.4202	+	3.32798i	−0.991239	+	0.265602i	−0.273468	+	0.961881i	$0.588171\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	12.2903i		−	0.968613i
$162$		0			0
$163$	−1.18932	+	1.18932i	−0.0931550	+	0.0931550i	−0.752149	−	0.658994i	$-0.770982\pi$
$163$	−1.18932	+	1.18932i	−0.0931550	+	0.0931550i	0.658994	+	0.752149i	$0.270982\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.83369	+	10.5755i	0.219277	+	0.818355i	0.984617	+	0.174728i	$0.0559045\pi$
$167$	2.83369	+	10.5755i	0.219277	+	0.818355i	−0.765339	+	0.643627i	$0.777429\pi$
$168$		0			0
$169$	5.03803	−	2.90871i	0.387541	−	0.223747i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.72513	+	17.6344i	−0.359245	+	1.34072i	0.515813	+	0.856701i	$0.327490\pi$
$173$	−4.72513	+	17.6344i	−0.359245	+	1.34072i	−0.875058	+	0.484019i	$0.839177\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.45364			0.407624			0.203812	−	0.979010i	$-0.434667\pi$
$179$	5.45364			0.407624			0.203812	+	0.979010i	$0.434667\pi$
$180$		0			0
$181$	19.0584			1.41660			0.708298	−	0.705913i	$-0.249463\pi$
$181$	19.0584			1.41660			0.708298	+	0.705913i	$0.249463\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.736068	+	2.74704i	−0.0538266	+	0.200884i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	21.2673	−	12.2787i	1.53885	−	0.888456i	0.539944	−	0.841701i	$-0.318445\pi$
$191$	21.2673	−	12.2787i	1.53885	−	0.888456i	0.998906	−	0.0467550i	$-0.0148880\pi$
$192$		0			0
$193$	0.493662	+	1.84237i	0.0355346	+	0.132617i	0.981414	−	0.191900i	$-0.0614650\pi$
$193$	0.493662	+	1.84237i	0.0355346	+	0.132617i	−0.945880	+	0.324517i	$0.894798\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.35836	−	5.35836i	0.381767	−	0.381767i	−0.489971	−	0.871739i	$-0.662993\pi$
$197$	5.35836	−	5.35836i	0.381767	−	0.381767i	0.871739	+	0.489971i	$0.162993\pi$
$198$		0			0
$199$			27.7151i			1.96467i	0.187127	+	0.982336i	$0.440082\pi$
$199$			27.7151i			1.96467i	−0.187127	+	0.982336i	$0.559918\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	25.9315	−	6.94831i	1.82003	−	0.487676i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.933812	+	1.61741i	0.0645931	+	0.111879i
$210$		0			0
$211$	−7.24803	+	12.5540i	−0.498975	+	0.864250i	−0.999999	−	0.00118335i	$-0.999623\pi$
$211$	−7.24803	+	12.5540i	−0.498975	+	0.864250i	0.501024	+	0.865433i	$0.332957\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−5.72063	−	5.72063i	−0.388342	−	0.388342i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−13.6215	−	7.86441i	−0.916285	−	0.529017i
$222$		0			0
$223$	−19.5772	−	5.24569i	−1.31099	−	0.351278i	−0.465392	−	0.885105i	$-0.654087\pi$
$223$	−19.5772	−	5.24569i	−1.31099	−	0.351278i	−0.845594	+	0.533827i	$0.820753\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.46194	−	2.26737i	−0.561639	−	0.150491i	−0.0331795	−	0.999449i	$-0.510563\pi$
$227$	−8.46194	−	2.26737i	−0.561639	−	0.150491i	−0.528459	+	0.848959i	$0.677230\pi$
$228$		0			0
$229$	−10.2098	−	5.89462i	−0.674681	−	0.389527i	0.123167	−	0.992386i	$-0.460695\pi$
$229$	−10.2098	−	5.89462i	−0.674681	−	0.389527i	−0.797848	+	0.602859i	$0.794028\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.56964	−	9.56964i	−0.626928	−	0.626928i	0.320366	−	0.947294i	$-0.396194\pi$
$233$	−9.56964	−	9.56964i	−0.626928	−	0.626928i	−0.947294	+	0.320366i	$0.896194\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.25108	−	12.5592i	0.469034	−	0.812390i	−0.530340	−	0.847785i	$-0.677936\pi$
$239$	7.25108	−	12.5592i	0.469034	−	0.812390i	0.999373	+	0.0353952i	$0.0112690\pi$
$240$		0			0
$241$	6.52918	+	11.3089i	0.420582	+	0.728469i	0.995996	−	0.0893931i	$-0.0284927\pi$
$241$	6.52918	+	11.3089i	0.420582	+	0.728469i	−0.575415	+	0.817862i	$0.695159\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−9.97717	+	2.67337i	−0.634832	+	0.170103i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			14.8303i			0.936083i	0.883706	+	0.468042i	$0.155040\pi$
$251$			14.8303i			0.936083i	−0.883706	+	0.468042i	$0.844960\pi$
$252$		0			0
$253$	1.24285	−	1.24285i	0.0781374	−	0.0781374i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.06717	−	3.98273i	−0.0665682	−	0.248436i	0.924621	−	0.380888i	$-0.124382\pi$
$257$	−1.06717	−	3.98273i	−0.0665682	−	0.248436i	−0.991189	+	0.132452i	$0.957715\pi$
$258$		0			0
$259$	−4.56320	+	2.63456i	−0.283543	+	0.163704i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.62595	+	9.80018i	−0.161923	+	0.604305i	0.836490	+	0.547983i	$0.184604\pi$
$263$	−2.62595	+	9.80018i	−0.161923	+	0.604305i	−0.998413	+	0.0563224i	$0.982063\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.90783			−0.360207			−0.180103	−	0.983648i	$-0.557643\pi$
$269$	−5.90783			−0.360207			−0.180103	+	0.983648i	$0.557643\pi$
$270$		0			0
$271$	−28.2338			−1.71508			−0.857541	−	0.514415i	$-0.828009\pi$
$271$	−28.2338			−1.71508			−0.857541	+	0.514415i	$0.828009\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.87868	+	7.01132i	−0.112879	+	0.421269i	−0.999119	−	0.0419558i	$-0.986641\pi$
$277$	−1.87868	+	7.01132i	−0.112879	+	0.421269i	0.886241	+	0.463225i	$0.153308\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.97782	−	1.71925i	0.177642	−	0.102562i	−0.408542	−	0.912739i	$-0.633963\pi$
$281$	2.97782	−	1.71925i	0.177642	−	0.102562i	0.586184	+	0.810178i	$0.300629\pi$
$282$		0			0
$283$	1.26988	+	4.73926i	0.0754866	+	0.281720i	0.993343	−	0.115192i	$-0.0367484\pi$
$283$	1.26988	+	4.73926i	0.0754866	+	0.281720i	−0.917857	+	0.396912i	$0.870082\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	28.1192	−	28.1192i	1.65982	−	1.65982i
$288$		0			0
$289$		−	17.4438i		−	1.02611i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.0724	−	3.50273i	0.763696	−	0.204632i	0.144111	−	0.989562i	$-0.453968\pi$
$293$	13.0724	−	3.50273i	0.763696	−	0.204632i	0.619585	+	0.784930i	$0.287301\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.86047	+	8.41858i	0.281088	+	0.486859i
$300$		0			0
$301$	17.2846	−	29.9378i	0.996268	−	1.72559i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	18.5221	+	18.5221i	1.05711	+	1.05711i	0.998267	+	0.0588446i	$0.0187417\pi$
$307$	18.5221	+	18.5221i	1.05711	+	1.05711i	0.0588446	+	0.998267i	$0.481258\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.13958	+	1.81264i	0.178030	+	0.102785i	0.586367	−	0.810046i	$-0.300558\pi$
$311$	3.13958	+	1.81264i	0.178030	+	0.102785i	−0.408337	+	0.912831i	$0.633891\pi$
$312$		0			0
$313$	13.7155	+	3.67507i	0.775248	+	0.207727i	0.624689	−	0.780874i	$-0.285226\pi$
$313$	13.7155	+	3.67507i	0.775248	+	0.207727i	0.150559	+	0.988601i	$0.451893\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.2621	−	3.82151i	−0.801038	−	0.214637i	−0.164998	−	0.986294i	$-0.552762\pi$
$317$	−14.2621	−	3.82151i	−0.801038	−	0.214637i	−0.636040	+	0.771656i	$0.719428\pi$
$318$		0			0
$319$	3.32495	+	1.91966i	0.186161	+	0.107480i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−15.9942	−	15.9942i	−0.889944	−	0.889944i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.94138	+	3.36257i	−0.107032	+	0.185385i
$330$		0			0
$331$	−10.6313	−	18.4140i	−0.584350	−	1.01212i	−0.994956	−	0.100311i	$-0.968016\pi$
$331$	−10.6313	−	18.4140i	−0.584350	−	1.01212i	0.410606	−	0.911813i	$-0.365317\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	12.7121	−	3.40620i	0.692473	−	0.185548i	0.104616	−	0.994513i	$-0.466639\pi$
$337$	12.7121	−	3.40620i	0.692473	−	0.185548i	0.587857	+	0.808965i	$0.299972\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	1.15699i		−	0.0626546i
$342$		0			0
$343$	6.03478	−	6.03478i	0.325847	−	0.325847i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.38277	+	27.5529i	0.396328	+	1.47912i	0.819506	+	0.573070i	$0.194248\pi$
$347$	7.38277	+	27.5529i	0.396328	+	1.47912i	−0.423178	+	0.906046i	$0.639086\pi$
$348$		0			0
$349$	2.40088	−	1.38615i	0.128516	−	0.0741988i	−0.434364	−	0.900738i	$-0.643027\pi$
$349$	2.40088	−	1.38615i	0.128516	−	0.0741988i	0.562880	+	0.826539i	$0.309693\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.66469	+	24.8730i	−0.354726	+	1.32385i	0.526103	+	0.850421i	$0.323653\pi$
$353$	−6.66469	+	24.8730i	−0.354726	+	1.32385i	−0.880829	+	0.473434i	$0.843014\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.82143			−0.465577			−0.232789	−	0.972527i	$-0.574785\pi$
$359$	−8.82143			−0.465577			−0.232789	+	0.972527i	$0.574785\pi$
$360$		0			0
$361$	4.14590			0.218205
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.08035	−	4.03193i	0.0563939	−	0.210465i	−0.931979	−	0.362511i	$-0.881920\pi$
$367$	1.08035	−	4.03193i	0.0563939	−	0.210465i	0.988373	+	0.152046i	$0.0485862\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−32.9882	+	19.0457i	−1.71266	+	0.988805i
$372$		0			0
$373$	−4.49526	−	16.7765i	−0.232756	−	0.868656i	−0.979148	−	0.203149i	$-0.934882\pi$
$373$	−4.49526	−	16.7765i	−0.232756	−	0.868656i	0.746392	−	0.665506i	$-0.231784\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.0146	+	15.0146i	−0.773290	+	0.773290i
$378$		0			0
$379$		−	10.8196i		−	0.555764i	−0.960615	−	0.277882i	$-0.910368\pi$
$379$		−	10.8196i		−	0.555764i	0.960615	−	0.277882i	$-0.0896325\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.7173	+	4.74733i	−0.905310	+	0.242577i	−0.681295	−	0.732009i	$-0.738583\pi$
$383$	−17.7173	+	4.74733i	−0.905310	+	0.242577i	−0.224015	+	0.974586i	$0.571916\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.5841	−	23.5283i	−0.688740	−	1.19293i	−0.972246	−	0.233962i	$-0.924831\pi$
$389$	−13.5841	−	23.5283i	−0.688740	−	1.19293i	0.283506	−	0.958970i	$-0.408502\pi$
$390$		0			0
$391$	−10.6437	+	18.4355i	−0.538276	+	0.932322i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−12.7467	−	12.7467i	−0.639741	−	0.639741i	0.310751	−	0.950491i	$-0.399420\pi$
$397$	−12.7467	−	12.7467i	−0.639741	−	0.639741i	−0.950491	+	0.310751i	$0.899420\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.37845	−	2.52790i	−0.218649	−	0.126237i	0.386675	−	0.922216i	$-0.373623\pi$
$401$	−4.37845	−	2.52790i	−0.218649	−	0.126237i	−0.605325	+	0.795979i	$0.706957\pi$
$402$		0			0
$403$	6.18085	+	1.65615i	0.307890	+	0.0824988i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.727869	−	0.195032i	−0.0360791	−	0.00966738i
$408$		0			0
$409$	−2.02932	−	1.17163i	−0.100344	−	0.0579334i	0.448988	−	0.893538i	$-0.351784\pi$
$409$	−2.02932	−	1.17163i	−0.100344	−	0.0579334i	−0.549332	+	0.835604i	$0.685118\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−15.2194	−	15.2194i	−0.748897	−	0.748897i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	15.2094	−	26.3435i	0.743028	−	1.28696i	−0.208083	−	0.978111i	$-0.566722\pi$
$419$	15.2094	−	26.3435i	0.743028	−	1.28696i	0.951110	−	0.308851i	$-0.0999444\pi$
$420$		0			0
$421$	−15.3759	−	26.6318i	−0.749375	−	1.29796i	−0.948123	−	0.317905i	$-0.897021\pi$
$421$	−15.3759	−	26.6318i	−0.749375	−	1.29796i	0.198747	−	0.980051i	$-0.436313\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	19.9442	−	5.34404i	0.965168	−	0.258616i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			17.0019i			0.818952i	0.912321	+	0.409476i	$0.134288\pi$
$431$			17.0019i			0.818952i	−0.912321	+	0.409476i	$0.865712\pi$
$432$		0			0
$433$	−22.0907	+	22.0907i	−1.06161	+	1.06161i	−0.0636393	+	0.997973i	$0.520271\pi$
$433$	−22.0907	+	22.0907i	−1.06161	+	1.06161i	−0.997973	+	0.0636393i	$0.979729\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.61816	+	13.5031i	0.173080	+	0.645943i
$438$		0			0
$439$	22.0369	−	12.7230i	1.05176	−	0.607235i	0.128620	−	0.991694i	$-0.458945\pi$
$439$	22.0369	−	12.7230i	1.05176	−	0.607235i	0.923142	+	0.384459i	$0.125612\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.59048	+	5.93574i	−0.0755658	+	0.282015i	−0.993361	−	0.115039i	$-0.963301\pi$
$443$	−1.59048	+	5.93574i	−0.0755658	+	0.282015i	0.917795	+	0.397054i	$0.129967\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−41.0686			−1.93815			−0.969073	−	0.246773i	$-0.920630\pi$
$449$	−41.0686			−1.93815			−0.969073	+	0.246773i	$0.920630\pi$
$450$		0			0
$451$	5.68708			0.267794
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	7.06463	−	26.3655i	0.330469	−	1.23333i	−0.578229	−	0.815875i	$-0.696256\pi$
$457$	7.06463	−	26.3655i	0.330469	−	1.23333i	0.908698	−	0.417454i	$-0.137077\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.54121	+	5.50862i	−0.444378	+	0.256562i	−0.705453	−	0.708757i	$-0.749256\pi$
$461$	−9.54121	+	5.50862i	−0.444378	+	0.256562i	0.261075	+	0.965319i	$0.415923\pi$
$462$		0			0
$463$	−3.71367	−	13.8596i	−0.172589	−	0.644110i	−0.996950	−	0.0780455i	$-0.975132\pi$
$463$	−3.71367	−	13.8596i	−0.172589	−	0.644110i	0.824361	−	0.566064i	$-0.191535\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.87179	+	3.87179i	−0.179165	+	0.179165i	−0.790992	−	0.611827i	$-0.790435\pi$
$467$	−3.87179	+	3.87179i	−0.179165	+	0.179165i	0.611827	+	0.790992i	$0.290435\pi$
$468$		0			0
$469$		−	26.4312i		−	1.22048i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.77534	−	1.27955i	0.219570	−	0.0588337i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18.5611	+	32.1489i	0.848080	+	1.46892i	0.882919	+	0.469526i	$0.155575\pi$
$479$	18.5611	+	32.1489i	0.848080	+	1.46892i	−0.0348384	+	0.999393i	$0.511092\pi$
$480$		0			0
$481$	2.08379	−	3.60923i	0.0950126	−	0.164567i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−5.92813	−	5.92813i	−0.268629	−	0.268629i	0.559919	−	0.828548i	$-0.310832\pi$
$487$	−5.92813	−	5.92813i	−0.268629	−	0.268629i	−0.828548	+	0.559919i	$0.810832\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.14174	+	2.96859i	0.232044	+	0.133970i	0.611514	−	0.791233i	$-0.290561\pi$
$491$	5.14174	+	2.96859i	0.232044	+	0.133970i	−0.379471	+	0.925204i	$0.623894\pi$
$492$		0			0
$493$	−44.9146	−	12.0348i	−2.02285	−	0.542021i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.96937	+	0.527690i	0.0883382	+	0.0236701i
$498$		0			0
$499$	12.6479	+	7.30225i	0.566196	+	0.326894i	0.755629	−	0.655000i	$-0.227331\pi$
$499$	12.6479	+	7.30225i	0.566196	+	0.326894i	−0.189432	+	0.981894i	$0.560665\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.216785	−	0.216785i	−0.00966595	−	0.00966595i	0.702257	−	0.711923i	$-0.252176\pi$
$503$	−0.216785	−	0.216785i	−0.00966595	−	0.00966595i	−0.711923	+	0.702257i	$0.752176\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.5139	−	23.4067i	0.598993	−	1.03749i	−0.393977	−	0.919120i	$-0.628901\pi$
$509$	13.5139	−	23.4067i	0.598993	−	1.03749i	0.992970	−	0.118366i	$-0.0377655\pi$
$510$		0			0
$511$	−21.1848	−	36.6932i	−0.937162	−	1.62321i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−0.536359	+	0.143717i	−0.0235891	+	0.00632067i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			18.7999i			0.823637i	0.911266	+	0.411818i	$0.135106\pi$
$521$			18.7999i			0.823637i	−0.911266	+	0.411818i	$0.864894\pi$
$522$		0			0
$523$	−5.36486	+	5.36486i	−0.234589	+	0.234589i	−0.814605	−	0.580016i	$-0.803046\pi$
$523$	−5.36486	+	5.36486i	−0.234589	+	0.234589i	0.580016	+	0.814605i	$0.303046\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.62673	+	13.5352i	0.157983	+	0.589601i
$528$		0			0
$529$	−8.52484	+	4.92182i	−0.370645	+	0.213992i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−8.14066	+	30.3814i	−0.352611	+	1.31596i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.17154			−0.0935348
$540$		0			0
$541$	−39.2086			−1.68571			−0.842854	−	0.538142i	$-0.819126\pi$
$541$	−39.2086			−1.68571			−0.842854	+	0.538142i	$0.819126\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−8.34694	+	31.1512i	−0.356890	+	1.33193i	0.521200	+	0.853434i	$0.325484\pi$
$547$	−8.34694	+	31.1512i	−0.356890	+	1.33193i	−0.878090	+	0.478496i	$0.841182\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−26.4449	+	15.2680i	−1.12659	+	0.650437i
$552$		0			0
$553$	12.2579	+	45.7470i	0.521258	+	1.94536i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.5285	+	25.5285i	−1.08168	+	1.08168i	−0.0853228	+	0.996353i	$0.527192\pi$
$557$	−25.5285	+	25.5285i	−1.08168	+	1.08168i	−0.996353	+	0.0853228i	$0.972808\pi$
$558$		0			0
$559$			27.3423i			1.15645i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.0320	+	7.51116i	−1.18141	+	0.316558i	−0.795485	−	0.605973i	$-0.792784\pi$
$563$	−28.0320	+	7.51116i	−1.18141	+	0.316558i	−0.385924	+	0.922530i	$0.626117\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.44549	+	11.1639i	0.270209	+	0.468016i	0.968915	−	0.247393i	$-0.0795739\pi$
$569$	6.44549	+	11.1639i	0.270209	+	0.468016i	−0.698706	+	0.715409i	$0.746241\pi$
$570$		0			0
$571$	20.9266	−	36.2459i	0.875749	−	1.51684i	0.0197865	−	0.999804i	$-0.493701\pi$
$571$	20.9266	−	36.2459i	0.875749	−	1.51684i	0.855963	−	0.517038i	$-0.172965\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.05491	−	1.05491i	−0.0439164	−	0.0439164i	0.684808	−	0.728724i	$-0.259886\pi$
$577$	−1.05491	−	1.05491i	−0.0439164	−	0.0439164i	−0.728724	+	0.684808i	$0.759886\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−44.1879	−	25.5119i	−1.83322	−	1.05841i
$582$		0			0
$583$	−5.26190	−	1.40992i	−0.217926	−	0.0583930i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.645811	+	0.173044i	0.0266555	+	0.00714231i	0.272122	−	0.962263i	$-0.412275\pi$
$587$	0.645811	+	0.173044i	0.0266555	+	0.00714231i	−0.245467	+	0.969405i	$0.578941\pi$
$588$		0			0
$589$	7.96926	+	4.60105i	0.328367	+	0.189583i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22.6736	+	22.6736i	0.931092	+	0.931092i	0.997774	−	0.0666826i	$-0.0212415\pi$
$593$	22.6736	+	22.6736i	0.931092	+	0.931092i	−0.0666826	+	0.997774i	$0.521242\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	19.7958	−	34.2874i	0.808835	−	1.40094i	−0.104836	−	0.994490i	$-0.533432\pi$
$599$	19.7958	−	34.2874i	0.808835	−	1.40094i	0.913671	−	0.406454i	$-0.133235\pi$
$600$		0			0
$601$	−2.27721	−	3.94424i	−0.0928893	−	0.160889i	0.815836	−	0.578283i	$-0.196277\pi$
$601$	−2.27721	−	3.94424i	−0.0928893	−	0.160889i	−0.908726	+	0.417394i	$0.862944\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	9.84249	−	2.63729i	0.399494	−	0.107044i	−0.0534774	−	0.998569i	$-0.517031\pi$
$607$	9.84249	−	2.63729i	0.399494	−	0.107044i	0.452972	+	0.891525i	$0.350364\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	3.07105i		−	0.124241i
$612$		0			0
$613$	−14.0188	+	14.0188i	−0.566214	+	0.566214i	−0.931066	−	0.364852i	$-0.881120\pi$
$613$	−14.0188	+	14.0188i	−0.566214	+	0.566214i	0.364852	+	0.931066i	$0.381120\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.91474	−	14.6100i	−0.157601	−	0.588176i	−0.998869	−	0.0475568i	$-0.984856\pi$
$617$	−3.91474	−	14.6100i	−0.157601	−	0.588176i	0.841267	−	0.540620i	$-0.181810\pi$
$618$		0			0
$619$	1.46721	−	0.847096i	0.0589723	−	0.0340477i	−0.470224	−	0.882547i	$-0.655827\pi$
$619$	1.46721	−	0.847096i	0.0589723	−	0.0340477i	0.529196	+	0.848499i	$0.322494\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.0329	+	44.9075i	−0.482090	+	1.79918i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9.12639			0.363893
$630$		0			0
$631$	−10.0584			−0.400417			−0.200209	−	0.979753i	$-0.564162\pi$
$631$	−10.0584			−0.400417			−0.200209	+	0.979753i	$0.564162\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.10841	−	11.6007i	0.123160	−	0.459638i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.1025	−	11.0288i	0.754504	−	0.435613i	−0.0728150	−	0.997345i	$-0.523198\pi$
$641$	19.1025	−	11.0288i	0.754504	−	0.435613i	0.827319	+	0.561732i	$0.189865\pi$
$642$		0			0
$643$	0.0363262	+	0.135571i	0.00143256	+	0.00534640i	0.966638	−	0.256145i	$-0.0824524\pi$
$643$	0.0363262	+	0.135571i	0.00143256	+	0.00534640i	−0.965206	+	0.261491i	$0.915786\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	21.4519	−	21.4519i	0.843360	−	0.843360i	−0.145934	−	0.989294i	$-0.546619\pi$
$647$	21.4519	−	21.4519i	0.843360	−	0.843360i	0.989294	+	0.145934i	$0.0466188\pi$
$648$		0			0
$649$		−	3.07810i		−	0.120826i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.04126	−	2.42260i	0.353812	−	0.0948036i	−0.0775353	−	0.996990i	$-0.524705\pi$
$653$	9.04126	−	2.42260i	0.353812	−	0.0948036i	0.431347	+	0.902186i	$0.358038\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.63337	+	8.02524i	0.180491	+	0.312619i	0.942048	−	0.335479i	$-0.108898\pi$
$659$	4.63337	+	8.02524i	0.180491	+	0.312619i	−0.761557	+	0.648098i	$0.775565\pi$
$660$		0			0
$661$	4.88507	−	8.46119i	0.190007	−	0.329102i	−0.755245	−	0.655442i	$-0.772482\pi$
$661$	4.88507	−	8.46119i	0.190007	−	0.329102i	0.945252	+	0.326340i	$0.105816\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	20.3208	+	20.3208i	0.786825	+	0.786825i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.55726	+	1.47643i	0.0987219	+	0.0569971i
$672$		0			0
$673$	−14.6169	−	3.91660i	−0.563442	−	0.150974i	−0.0341547	−	0.999417i	$-0.510874\pi$
$673$	−14.6169	−	3.91660i	−0.563442	−	0.150974i	−0.529287	+	0.848443i	$0.677541\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11.7341	+	3.14415i	0.450979	+	0.120839i	0.477157	−	0.878818i	$-0.341667\pi$
$677$	11.7341	+	3.14415i	0.450979	+	0.120839i	−0.0261785	+	0.999657i	$0.508334\pi$
$678$		0			0
$679$	−0.405859	−	0.234323i	−0.0155755	−	0.00899249i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.290114	+	0.290114i	0.0111009	+	0.0111009i	0.712635	−	0.701535i	$-0.247501\pi$
$683$	0.290114	+	0.290114i	0.0111009	+	0.0111009i	−0.701535	+	0.712635i	$0.747501\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	15.0641	−	26.0918i	0.573896	−	0.994017i
$690$		0			0
$691$	2.86257	+	4.95812i	0.108897	+	0.188616i	0.915324	−	0.402719i	$-0.131935\pi$
$691$	2.86257	+	4.95812i	0.108897	+	0.188616i	−0.806426	+	0.591334i	$0.798601\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−66.5308	+	17.8269i	−2.52003	+	0.675241i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			12.5871i			0.475407i	0.971338	+	0.237703i	$0.0763946\pi$
$701$			12.5871i			0.475407i	−0.971338	+	0.237703i	$0.923605\pi$
$702$		0			0
$703$	4.23791	−	4.23791i	0.159836	−	0.159836i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−13.3078	−	49.6656i	−0.500493	−	1.86787i
$708$		0			0
$709$	33.5532	−	19.3719i	1.26012	−	0.727529i	0.287020	−	0.957925i	$-0.407335\pi$
$709$	33.5532	−	19.3719i	1.26012	−	0.727529i	0.973097	+	0.230396i	$0.0740021\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.24145	−	8.36519i	0.0839428	−	0.313279i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−18.3617			−0.684776			−0.342388	−	0.939559i	$-0.611236\pi$
$719$	−18.3617			−0.684776			−0.342388	+	0.939559i	$0.611236\pi$
$720$		0			0
$721$	0.680794			0.0253541
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.988143	+	3.68780i	−0.0366482	+	0.136773i	−0.981826	−	0.189783i	$-0.939222\pi$
$727$	−0.988143	+	3.68780i	−0.0366482	+	0.136773i	0.945178	+	0.326556i	$0.105888\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−51.8538	+	29.9378i	−1.91788	+	1.10729i
$732$		0			0
$733$	−13.1700	−	49.1511i	−0.486444	−	1.81544i	−0.573466	−	0.819229i	$-0.694402\pi$
$733$	−13.1700	−	49.1511i	−0.486444	−	1.81544i	0.0870218	−	0.996206i	$-0.472265\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.67284	−	2.67284i	0.0984553	−	0.0984553i
$738$		0			0
$739$		−	16.4016i		−	0.603344i	−0.953412	−	0.301672i	$-0.902455\pi$
$739$		−	16.4016i		−	0.603344i	0.953412	−	0.301672i	$-0.0975447\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.9128	+	4.26383i	−0.583785	+	0.156425i	−0.538611	−	0.842555i	$-0.681051\pi$
$743$	−15.9128	+	4.26383i	−0.583785	+	0.156425i	−0.0451735	+	0.998979i	$0.514384\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−19.4516	−	33.6911i	−0.710746	−	1.23105i
$750$		0			0
$751$	−14.4753	+	25.0720i	−0.528213	+	0.914892i	0.471246	+	0.882002i	$0.343804\pi$
$751$	−14.4753	+	25.0720i	−0.528213	+	0.914892i	−0.999459	+	0.0328898i	$0.989529\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−5.26561	−	5.26561i	−0.191382	−	0.191382i	0.604911	−	0.796293i	$-0.293209\pi$
$757$	−5.26561	−	5.26561i	−0.191382	−	0.191382i	−0.796293	+	0.604911i	$0.793209\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.02110	+	2.89893i	0.182015	+	0.105086i	0.588239	−	0.808687i	$-0.299821\pi$
$761$	5.02110	+	2.89893i	0.182015	+	0.105086i	−0.406224	+	0.913773i	$0.633155\pi$
$762$		0			0
$763$	−17.9883	−	4.81995i	−0.651220	−	0.174494i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	16.4438	+	4.40609i	0.593750	+	0.159095i
$768$		0			0
$769$	−32.9790	−	19.0404i	−1.18925	−	0.686615i	−0.231115	−	0.972926i	$-0.574237\pi$
$769$	−32.9790	−	19.0404i	−1.18925	−	0.686615i	−0.958136	+	0.286312i	$0.907571\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.2745	−	19.2745i	−0.693257	−	0.693257i	0.269690	−	0.962947i	$-0.413079\pi$
$773$	−19.2745	−	19.2745i	−0.693257	−	0.693257i	−0.962947	+	0.269690i	$0.913079\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−22.6160	+	39.1721i	−0.810303	+	1.40349i
$780$		0			0
$781$	0.145789	+	0.252514i	0.00521673	+	0.00903564i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	34.3267	−	9.19780i	1.22361	−	0.327866i	0.411524	−	0.911399i	$-0.364997\pi$
$787$	34.3267	−	9.19780i	1.22361	−	0.327866i	0.812089	+	0.583533i	$0.198330\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	54.0374i		−	1.92135i
$792$		0			0
$793$	−11.5479	+	11.5479i	−0.410078	+	0.410078i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.6949	+	39.9138i	0.378831	+	1.41382i	0.847665	+	0.530531i	$0.178008\pi$
$797$	10.6949	+	39.9138i	0.378831	+	1.41382i	−0.468834	+	0.883286i	$0.655326\pi$
$798$		0			0
$799$	5.82415	−	3.36257i	0.206043	−	0.118959i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.56828	−	5.85289i	0.0553433	−	0.206544i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−41.5139			−1.45955			−0.729776	−	0.683687i	$-0.760375\pi$
$809$	−41.5139			−1.45955			−0.729776	+	0.683687i	$0.760375\pi$
$810$		0			0
$811$	−42.8314			−1.50401			−0.752007	−	0.659155i	$-0.770914\pi$
$811$	−42.8314			−1.50401			−0.752007	+	0.659155i	$0.770914\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−10.1769	+	37.9806i	−0.356043	+	1.32877i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	26.8095	−	15.4785i	0.935659	−	0.540203i	0.0470622	−	0.998892i	$-0.485014\pi$
$821$	26.8095	−	15.4785i	0.935659	−	0.540203i	0.888597	+	0.458689i	$0.151681\pi$
$822$		0			0
$823$	−6.06293	−	22.6272i	−0.211341	−	0.788734i	−0.987423	−	0.158102i	$-0.949463\pi$
$823$	−6.06293	−	22.6272i	−0.211341	−	0.788734i	0.776082	−	0.630632i	$-0.217204\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22.0194	−	22.0194i	0.765690	−	0.765690i	−0.211654	−	0.977345i	$-0.567885\pi$
$827$	22.0194	−	22.0194i	0.765690	−	0.765690i	0.977345	+	0.211654i	$0.0678851\pi$
$828$		0			0
$829$			6.65597i			0.231171i	0.993298	+	0.115586i	$0.0368745\pi$
$829$			6.65597i			0.231171i	−0.993298	+	0.115586i	$0.963126\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	25.4039	−	6.80696i	0.880194	−	0.235847i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−11.8643	−	20.5496i	−0.409601	−	0.709450i	0.585244	−	0.810857i	$-0.300999\pi$
$839$	−11.8643	−	20.5496i	−0.409601	−	0.709450i	−0.994845	+	0.101407i	$0.967665\pi$
$840$		0			0
$841$	−16.8867	+	29.2487i	−0.582301	+	1.00858i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	25.7930	+	25.7930i	0.886257	+	0.886257i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.88475	−	2.82021i	−0.167447	−	0.0966756i
$852$		0			0
$853$	−18.4649	−	4.94766i	−0.632227	−	0.169405i	−0.0715468	−	0.997437i	$-0.522794\pi$
$853$	−18.4649	−	4.94766i	−0.632227	−	0.169405i	−0.560680	+	0.828033i	$0.689460\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	30.8026	+	8.25353i	1.05220	+	0.281935i	0.743159	−	0.669115i	$-0.233327\pi$
$857$	30.8026	+	8.25353i	1.05220	+	0.281935i	0.309037	+	0.951050i	$0.399993\pi$
$858$		0			0
$859$	14.2177	+	8.20859i	0.485101	+	0.280073i	0.722540	−	0.691329i	$-0.242974\pi$
$859$	14.2177	+	8.20859i	0.485101	+	0.280073i	−0.237439	+	0.971403i	$0.576308\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.10066	+	9.10066i	0.309790	+	0.309790i	0.844828	−	0.535038i	$-0.179703\pi$
$863$	9.10066	+	9.10066i	0.309790	+	0.309790i	−0.535038	+	0.844828i	$0.679703\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.38657	+	5.86571i	−0.114882	+	0.198981i
$870$		0			0
$871$	10.4528	+	18.1048i	0.354179	+	0.613456i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14.1411	+	3.78909i	−0.477510	+	0.127948i	−0.489543	−	0.871979i	$-0.662836\pi$
$877$	−14.1411	+	3.78909i	−0.477510	+	0.127948i	0.0120331	+	0.999928i	$0.496170\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			17.7146i			0.596820i	0.954438	+	0.298410i	$0.0964563\pi$
$881$			17.7146i			0.596820i	−0.954438	+	0.298410i	$0.903544\pi$
$882$		0			0
$883$	8.62634	−	8.62634i	0.290299	−	0.290299i	−0.546899	−	0.837198i	$-0.684192\pi$
$883$	8.62634	−	8.62634i	0.290299	−	0.290299i	0.837198	+	0.546899i	$0.184192\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.907087	+	3.38530i	0.0304570	+	0.113667i	0.979481	−	0.201537i	$-0.0645937\pi$
$887$	0.907087	+	3.38530i	0.0304570	+	0.113667i	−0.949024	+	0.315204i	$0.897927\pi$
$888$		0			0
$889$	−2.89535	+	1.67163i	−0.0971069	+	0.0560647i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.14305	−	4.26592i	0.0382507	−	0.142754i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	18.9170			0.630917
$900$		0			0
$901$	65.9763			2.19799
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	5.03909	−	18.8061i	0.167320	−	0.624447i	−0.830413	−	0.557149i	$-0.811895\pi$
$907$	5.03909	−	18.8061i	0.167320	−	0.624447i	0.997733	−	0.0672984i	$-0.0214380\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.48726	−	0.858672i	0.0492753	−	0.0284491i	−0.475160	−	0.879899i	$-0.657610\pi$
$911$	1.48726	−	0.858672i	0.0492753	−	0.0284491i	0.524435	+	0.851450i	$0.324276\pi$
$912$		0			0
$913$	−1.88860	−	7.04835i	−0.0625035	−	0.233266i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.447529	+	0.447529i	−0.0147787	+	0.0147787i
$918$		0			0
$919$		−	43.0970i		−	1.42164i	−0.703375	−	0.710819i	$-0.748324\pi$
$919$		−	43.0970i		−	1.42164i	0.703375	−	0.710819i	$-0.251676\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.55766	+	0.417373i	−0.0512709	+	0.0137380i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.64047	+	6.30549i	0.119440	+	0.206876i	0.919546	−	0.392983i	$-0.128557\pi$
$929$	3.64047	+	6.30549i	0.119440	+	0.206876i	−0.800106	+	0.599859i	$0.795223\pi$
$930$		0			0
$931$	8.63564	−	14.9574i	0.283022	−	0.490208i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−10.9183	−	10.9183i	−0.356685	−	0.356685i	0.505904	−	0.862590i	$-0.331159\pi$
$937$	−10.9183	−	10.9183i	−0.356685	−	0.356685i	−0.862590	+	0.505904i	$0.831159\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−35.4650	−	20.4757i	−1.15613	−	0.667490i	−0.205754	−	0.978604i	$-0.565965\pi$
$941$	−35.4650	−	20.4757i	−1.15613	−	0.667490i	−0.950373	+	0.311114i	$0.899298\pi$
$942$		0			0
$943$	41.1183	+	11.0176i	1.33900	+	0.358783i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−37.4169	−	10.0258i	−1.21588	−	0.325795i	−0.406816	−	0.913510i	$-0.633361\pi$
$947$	−37.4169	−	10.0258i	−1.21588	−	0.325795i	−0.809068	+	0.587715i	$0.800028\pi$
$948$		0			0
$949$	29.0222	+	16.7560i	0.942102	+	0.543923i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.57092	+	9.57092i	0.310032	+	0.310032i	0.844922	−	0.534890i	$-0.179647\pi$
$953$	9.57092	+	9.57092i	0.310032	+	0.310032i	−0.534890	+	0.844922i	$0.679647\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.9322	+	32.7915i	−0.611352	+	1.05889i
$960$		0			0
$961$	12.6497	+	21.9098i	0.408053	+	0.706769i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−32.6676	+	8.75325i	−1.05052	+	0.281485i	−0.742466	−	0.669884i	$-0.766344\pi$
$967$	−32.6676	+	8.75325i	−1.05052	+	0.281485i	−0.308052	+	0.951369i	$0.599677\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	30.6923i		−	0.984964i	−0.870323	−	0.492482i	$-0.836090\pi$
$971$		−	30.6923i		−	0.984964i	0.870323	−	0.492482i	$-0.163910\pi$
$972$		0			0
$973$	19.9056	−	19.9056i	0.638143	−	0.638143i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.67329	−	6.24481i	−0.0535334	−	0.199789i	0.933980	−	0.357326i	$-0.116311\pi$
$977$	−1.67329	−	6.24481i	−0.0535334	−	0.199789i	−0.987513	+	0.157537i	$0.949645\pi$
$978$		0			0
$979$	−5.75807	+	3.32443i	−0.184029	+	0.106249i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.51459	−	5.65254i	0.0483081	−	0.180288i	−0.937556	−	0.347834i	$-0.886917\pi$
$983$	1.51459	−	5.65254i	0.0483081	−	0.180288i	0.985864	+	0.167545i	$0.0535841\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	37.0052			1.17670
$990$		0			0
$991$	−3.05991			−0.0972011			−0.0486006	−	0.998818i	$-0.515476\pi$
$991$	−3.05991			−0.0972011			−0.0486006	+	0.998818i	$0.515476\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.556025	−	2.07511i	0.0176095	−	0.0657195i	−0.956562	−	0.291529i	$-0.905836\pi$
$997$	0.556025	−	2.07511i	0.0176095	−	0.0657195i	0.974171	+	0.225809i	$0.0725027\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.bf.f.557.8		32
3.2	odd	2		900.2.be.f.257.8	yes	32
5.2	odd	4	inner	2700.2.bf.f.2393.8		32
5.3	odd	4	inner	2700.2.bf.f.2393.1		32
5.4	even	2	inner	2700.2.bf.f.557.1		32
9.2	odd	6	inner	2700.2.bf.f.2357.1		32
9.7	even	3		900.2.be.f.857.5	yes	32
15.2	even	4		900.2.be.f.293.4	yes	32
15.8	even	4		900.2.be.f.293.5	yes	32
15.14	odd	2		900.2.be.f.257.1	✓	32
45.2	even	12	inner	2700.2.bf.f.1493.1		32
45.7	odd	12		900.2.be.f.893.1	yes	32
45.29	odd	6	inner	2700.2.bf.f.2357.8		32
45.34	even	6		900.2.be.f.857.4	yes	32
45.38	even	12	inner	2700.2.bf.f.1493.8		32
45.43	odd	12		900.2.be.f.893.8	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.be.f.257.1	✓	32	15.14	odd	2
900.2.be.f.257.8	yes	32	3.2	odd	2
900.2.be.f.293.4	yes	32	15.2	even	4
900.2.be.f.293.5	yes	32	15.8	even	4
900.2.be.f.857.4	yes	32	45.34	even	6
900.2.be.f.857.5	yes	32	9.7	even	3
900.2.be.f.893.1	yes	32	45.7	odd	12
900.2.be.f.893.8	yes	32	45.43	odd	12
2700.2.bf.f.557.1		32	5.4	even	2	inner
2700.2.bf.f.557.8		32	1.1	even	1	trivial
2700.2.bf.f.1493.1		32	45.2	even	12	inner
2700.2.bf.f.1493.8		32	45.38	even	12	inner
2700.2.bf.f.2357.1		32	9.2	odd	6	inner
2700.2.bf.f.2357.8		32	45.29	odd	6	inner
2700.2.bf.f.2393.1		32	5.3	odd	4	inner
2700.2.bf.f.2393.8		32	5.2	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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.bf (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.876983 - 3.27295i) q^{7} +(0.419659 - 0.242290i) q^{11} +(0.693644 + 2.58871i) q^{13} +(-4.14993 + 4.14993i) q^{17} +3.85410i q^{19} +(3.50358 - 0.938781i) q^{23} +(3.96149 + 6.86149i) q^{29} +(1.19381 - 2.06773i) q^{31} +(-1.09958 - 1.09958i) q^{37} +(10.1637 + 5.86804i) q^{41} +(9.85458 + 2.64053i) q^{43} +(-1.10685 - 0.296580i) q^{47} +(-3.88090 - 2.24064i) q^{49} +(-7.94909 - 7.94909i) q^{53} +(3.17605 - 5.50108i) q^{59} +(3.04683 + 5.27726i) q^{61} +(7.53469 - 2.01891i) q^{67} +0.601711i q^{71} +(8.84189 - 8.84189i) q^{73} +(-0.424969 - 1.58601i) q^{77} +(-12.1047 + 6.98866i) q^{79} +(3.89739 - 14.5453i) q^{83} -13.7208 q^{89} +9.08103 q^{91} +(0.0357970 - 0.133596i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 12 q^{11} + 8 q^{31} - 60 q^{41} + 52 q^{61} + 120 q^{91}+O(q^{100}) \) 32 * q + 12 * q^11 + 8 * q^31 - 60 * q^41 + 52 * q^61 + 120 * q^91

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