LMFDB - Embedded newform 2700.2.i.f.1801.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2700,2,Mod(901,2700)] 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("2700.901"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 4x^{10} + 10x^{8} + 6x^{6} + 90x^{4} + 324x^{2} + 729 $ x^12 + 4x^10 + 10x^8 + 6x^6 + 90x^4 + 324*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1801.1
Root			$-0.602950 - 1.62372i$ of defining polynomial
Character	$\chi$	$=$	2700.1801
Dual form			2700.2.i.f.901.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.88482 + 3.26460i) q^{7} +(-1.77290 + 3.07076i) q^{11} +(0.501753 + 0.869061i) q^{13} +1.56023 q^{17} +7.21013 q^{19} +(-3.09072 - 5.35328i) q^{23} +(-1.50000 + 2.59808i) q^{29} +(2.89142 + 5.00809i) q^{31} -0.851576 q^{37} +(-1.55926 - 2.70072i) q^{41} +(-1.35333 + 2.34403i) q^{43} +(-4.87434 + 8.44260i) q^{47} +(-3.60506 - 6.24415i) q^{49} -11.2494 q^{53} +(-4.83216 - 8.36955i) q^{59} +(4.10506 - 7.11018i) q^{61} +(-1.45903 - 2.52711i) q^{67} -7.21013 q^{71} -16.7817 q^{73} +(-6.68319 - 11.5756i) q^{77} +(-5.49649 + 9.52020i) q^{79} +(-5.09773 + 8.82952i) q^{83} +5.33567 q^{89} -3.78285 q^{91} +(-1.93105 + 3.34467i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{11} - 18 q^{29} + 6 q^{31} - 14 q^{41} - 34 q^{59} + 6 q^{61} + 6 q^{79} + 112 q^{89} + 12 q^{91}+O(q^{100}) $ 12 * q - 2 * q^11 - 18 * q^29 + 6 * q^31 - 14 * q^41 - 34 * q^59 + 6 * q^61 + 6 * q^79 + 112 * q^89 + 12 * q^91

Character values

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

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.88482	+	3.26460i	−0.712394	+	1.23390i	0.251563	+	0.967841i	$0.419056\pi$
$7$	−1.88482	+	3.26460i	−0.712394	+	1.23390i	−0.963956	+	0.266061i	$0.914278\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.77290	+	3.07076i	−0.534550	+	0.925868i	0.464635	+	0.885502i	$0.346186\pi$
$11$	−1.77290	+	3.07076i	−0.534550	+	0.925868i	−0.999185	+	0.0403654i	$0.987148\pi$
$12$		0			0
$13$	0.501753	+	0.869061i	0.139161	+	0.241034i	0.927179	−	0.374618i	$-0.122226\pi$
$13$	0.501753	+	0.869061i	0.139161	+	0.241034i	−0.788018	+	0.615652i	$0.788893\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.56023			0.378410			0.189205	−	0.981938i	$-0.439409\pi$
$17$	1.56023			0.378410			0.189205	+	0.981938i	$0.439409\pi$
$18$		0			0
$19$	7.21013			1.65412			0.827058	−	0.562116i	$-0.190013\pi$
$19$	7.21013			1.65412			0.827058	+	0.562116i	$0.190013\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.09072	−	5.35328i	−0.644459	−	1.11624i	−0.984426	−	0.175799i	$-0.943749\pi$
$23$	−3.09072	−	5.35328i	−0.644459	−	1.11624i	0.339967	−	0.940437i	$-0.389584\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.50000	+	2.59808i	−0.278543	+	0.482451i	−0.971023	−	0.238987i	$-0.923185\pi$
$29$	−1.50000	+	2.59808i	−0.278543	+	0.482451i	0.692480	+	0.721437i	$0.256518\pi$
$30$		0			0
$31$	2.89142	+	5.00809i	0.519315	+	0.899480i	0.999748	+	0.0224486i	$0.00714621\pi$
$31$	2.89142	+	5.00809i	0.519315	+	0.899480i	−0.480433	+	0.877031i	$0.659520\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.851576			−0.139998			−0.0699991	−	0.997547i	$-0.522300\pi$
$37$	−0.851576			−0.139998			−0.0699991	+	0.997547i	$0.522300\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.55926	−	2.70072i	−0.243516	−	0.421782i	0.718198	−	0.695839i	$-0.244967\pi$
$41$	−1.55926	−	2.70072i	−0.243516	−	0.421782i	−0.961713	+	0.274058i	$0.911634\pi$
$42$		0			0
$43$	−1.35333	+	2.34403i	−0.206381	+	0.357462i	−0.950572	−	0.310505i	$-0.899502\pi$
$43$	−1.35333	+	2.34403i	−0.206381	+	0.357462i	0.744191	+	0.667967i	$0.232835\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.87434	+	8.44260i	−0.710995	+	1.23148i	0.253489	+	0.967338i	$0.418422\pi$
$47$	−4.87434	+	8.44260i	−0.710995	+	1.23148i	−0.964484	+	0.264141i	$0.914912\pi$
$48$		0			0
$49$	−3.60506	−	6.24415i	−0.515009	−	0.892022i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−11.2494			−1.54523			−0.772614	−	0.634876i	$-0.781051\pi$
$53$	−11.2494			−1.54523			−0.772614	+	0.634876i	$0.781051\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.83216	−	8.36955i	−0.629094	−	1.08962i	−0.987734	−	0.156147i	$-0.950093\pi$
$59$	−4.83216	−	8.36955i	−0.629094	−	1.08962i	0.358640	−	0.933476i	$-0.383241\pi$
$60$		0			0
$61$	4.10506	−	7.11018i	0.525600	−	0.910365i	−0.473956	−	0.880549i	$-0.657174\pi$
$61$	4.10506	−	7.11018i	0.525600	−	0.910365i	0.999555	−	0.0298166i	$-0.00949232\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.45903	−	2.52711i	−0.178249	−	0.308736i	0.763032	−	0.646361i	$-0.223710\pi$
$67$	−1.45903	−	2.52711i	−0.178249	−	0.308736i	−0.941281	+	0.337625i	$0.890376\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.21013			−0.855685			−0.427842	−	0.903853i	$-0.640726\pi$
$71$	−7.21013			−0.855685			−0.427842	+	0.903853i	$0.640726\pi$
$72$		0			0
$73$	−16.7817			−1.96415			−0.982074	−	0.188498i	$-0.939638\pi$
$73$	−16.7817			−1.96415			−0.982074	+	0.188498i	$0.939638\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.68319	−	11.5756i	−0.761620	−	1.31916i
$78$		0			0
$79$	−5.49649	+	9.52020i	−0.618403	+	1.07111i	0.371374	+	0.928483i	$0.378887\pi$
$79$	−5.49649	+	9.52020i	−0.618403	+	1.07111i	−0.989777	+	0.142622i	$0.954447\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.09773	+	8.82952i	−0.559548	+	0.969166i	0.437986	+	0.898982i	$0.355692\pi$
$83$	−5.09773	+	8.82952i	−0.559548	+	0.969166i	−0.997534	+	0.0701843i	$0.977641\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.33567			0.565580			0.282790	−	0.959182i	$-0.408740\pi$
$89$	5.33567			0.565580			0.282790	+	0.959182i	$0.408740\pi$
$90$		0			0
$91$	−3.78285			−0.396550
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.93105	+	3.34467i	−0.196068	+	0.339600i	−0.947250	−	0.320495i	$-0.896151\pi$
$97$	−1.93105	+	3.34467i	−0.196068	+	0.339600i	0.751182	+	0.660095i	$0.229484\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.37797	−	4.11876i	0.236616	−	0.409832i	−0.723125	−	0.690717i	$-0.757295\pi$
$101$	2.37797	−	4.11876i	0.236616	−	0.409832i	0.959741	+	0.280886i	$0.0906282\pi$
$102$		0			0
$103$	9.04452	+	15.6656i	0.891183	+	1.54357i	0.838458	+	0.544966i	$0.183457\pi$
$103$	9.04452	+	15.6656i	0.891183	+	1.54357i	0.0527251	+	0.998609i	$0.483209\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.61770			−0.349737			−0.174868	−	0.984592i	$-0.555950\pi$
$107$	−3.61770			−0.349737			−0.174868	+	0.984592i	$0.555950\pi$
$108$		0			0
$109$	−9.63741			−0.923096			−0.461548	−	0.887115i	$-0.652706\pi$
$109$	−9.63741			−0.923096			−0.461548	+	0.887115i	$0.652706\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.77912	−	3.08152i	−0.167365	−	0.289885i	0.770127	−	0.637890i	$-0.220193\pi$
$113$	−1.77912	−	3.08152i	−0.167365	−	0.289885i	−0.937493	+	0.348005i	$0.886859\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.94074	+	5.09351i	−0.269577	+	0.466921i
$120$		0			0
$121$	−0.786360	−	1.36201i	−0.0714872	−	0.123820i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	6.78015			0.601641			0.300820	−	0.953681i	$-0.402739\pi$
$127$	6.78015			0.601641			0.300820	+	0.953681i	$0.402739\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.71364	−	2.96811i	−0.149721	−	0.259325i	0.781403	−	0.624027i	$-0.214504\pi$
$131$	−1.71364	−	2.96811i	−0.149721	−	0.259325i	−0.931124	+	0.364702i	$0.881171\pi$
$132$		0			0
$133$	−13.5898	+	23.5382i	−1.17838	+	2.04102i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.46330	−	12.9268i	0.637633	−	1.10441i	−0.348318	−	0.937376i	$-0.613247\pi$
$137$	7.46330	−	12.9268i	0.637633	−	1.10441i	0.985951	−	0.167036i	$-0.0534196\pi$
$138$		0			0
$139$	2.71364	+	4.70016i	0.230168	+	0.398663i	0.957857	−	0.287244i	$-0.0927391\pi$
$139$	2.71364	+	4.70016i	0.230168	+	0.398663i	−0.727689	+	0.685907i	$0.759406\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.55823			−0.297554
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.76939	+	4.79672i	0.226877	+	0.392963i	0.956881	−	0.290480i	$-0.0938150\pi$
$149$	2.76939	+	4.79672i	0.226877	+	0.392963i	−0.730004	+	0.683443i	$0.760482\pi$
$150$		0			0
$151$	−0.108576	+	0.188059i	−0.00883580	+	0.0153041i	−0.870409	−	0.492328i	$-0.836146\pi$
$151$	−0.108576	+	0.188059i	−0.00883580	+	0.0153041i	0.861574	+	0.507633i	$0.169479\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.07947	+	1.86970i	0.0861511	+	0.149218i	0.905881	−	0.423532i	$-0.139210\pi$
$157$	1.07947	+	1.86970i	0.0861511	+	0.149218i	−0.819730	+	0.572750i	$0.805877\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	23.3017			1.83643
$162$		0			0
$163$	−8.39084			−0.657221			−0.328611	−	0.944465i	$-0.606580\pi$
$163$	−8.39084			−0.657221			−0.328611	+	0.944465i	$0.606580\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.31060	+	4.00208i	0.178800	+	0.309691i	0.941470	−	0.337097i	$-0.109445\pi$
$167$	2.31060	+	4.00208i	0.178800	+	0.309691i	−0.762670	+	0.646788i	$0.776112\pi$
$168$		0			0
$169$	5.99649	−	10.3862i	0.461268	−	0.798940i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.725146	+	1.25599i	−0.0551318	+	0.0954911i	−0.892274	−	0.451494i	$-0.850891\pi$
$173$	−0.725146	+	1.25599i	−0.0551318	+	0.0954911i	0.837142	+	0.546985i	$0.184225\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	19.4472			1.45355			0.726775	−	0.686876i	$-0.241018\pi$
$179$	19.4472			1.45355			0.726775	+	0.686876i	$0.241018\pi$
$180$		0			0
$181$	−10.4273			−0.775054			−0.387527	−	0.921858i	$-0.626671\pi$
$181$	−10.4273			−0.775054			−0.387527	+	0.921858i	$0.626671\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.76613	+	4.79107i	−0.202279	+	0.350358i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.89142	+	3.27604i	−0.136859	+	0.237046i	−0.926306	−	0.376772i	$-0.877034\pi$
$191$	−1.89142	+	3.27604i	−0.136859	+	0.237046i	0.789447	+	0.613818i	$0.210367\pi$
$192$		0			0
$193$	−7.03751	−	12.1893i	−0.506571	−	0.877407i	−0.999971	−	0.00760445i	$-0.997579\pi$
$193$	−7.03751	−	12.1893i	−0.506571	−	0.877407i	0.493400	−	0.869803i	$-0.335754\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−21.5044			−1.53212			−0.766061	−	0.642768i	$-0.777786\pi$
$197$	−21.5044			−1.53212			−0.766061	+	0.642768i	$0.777786\pi$
$198$		0			0
$199$	7.21013			0.511112			0.255556	−	0.966794i	$-0.417741\pi$
$199$	7.21013			0.511112			0.255556	+	0.966794i	$0.417741\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.65445	−	9.79379i	−0.396864	−	0.687389i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−12.7828	+	22.1405i	−0.884208	+	1.53149i
$210$		0			0
$211$	−1.89142	−	3.27604i	−0.130211	−	0.225532i	0.793547	−	0.608509i	$-0.208232\pi$
$211$	−1.89142	−	3.27604i	−0.130211	−	0.225532i	−0.923758	+	0.382977i	$0.874899\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−21.7992			−1.47983
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.782848	+	1.35593i	0.0526600	+	0.0912099i
$222$		0			0
$223$	1.45903	−	2.52711i	0.0977038	−	0.169228i	−0.813030	−	0.582222i	$-0.802184\pi$
$223$	1.45903	−	2.52711i	0.0977038	−	0.169228i	0.910734	+	0.412994i	$0.135517\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.55923	+	4.43272i	−0.169862	+	0.294210i	−0.938371	−	0.345629i	$-0.887666\pi$
$227$	−2.55923	+	4.43272i	−0.169862	+	0.294210i	0.768509	+	0.639839i	$0.220999\pi$
$228$		0			0
$229$	4.71013	+	8.15818i	0.311254	+	0.539108i	0.978634	−	0.205609i	$-0.0659177\pi$
$229$	4.71013	+	8.15818i	0.311254	+	0.539108i	−0.667380	+	0.744717i	$0.732584\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−26.1850			−1.71544			−0.857719	−	0.514118i	$-0.828119\pi$
$233$	−26.1850			−1.71544			−0.857719	+	0.514118i	$0.828119\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.3780	−	19.7072i	−0.735979	−	1.27475i	−0.954292	−	0.298875i	$-0.903389\pi$
$239$	−11.3780	−	19.7072i	−0.735979	−	1.27475i	0.218313	−	0.975879i	$-0.429945\pi$
$240$		0			0
$241$	−8.31519	+	14.4023i	−0.535629	+	0.927736i	0.463504	+	0.886095i	$0.346592\pi$
$241$	−8.31519	+	14.4023i	−0.535629	+	0.927736i	−0.999133	+	0.0416412i	$0.986741\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.61770	+	6.26604i	0.230189	+	0.398699i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	20.7758			1.31136			0.655679	−	0.755040i	$-0.272382\pi$
$251$	20.7758			1.31136			0.655679	+	0.755040i	$0.272382\pi$
$252$		0			0
$253$	21.9181			1.37798
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.52978	+	16.5061i	0.594451	+	1.02962i	0.993624	+	0.112744i	$0.0359641\pi$
$257$	9.52978	+	16.5061i	0.594451	+	1.02962i	−0.399173	+	0.916876i	$0.630703\pi$
$258$		0			0
$259$	1.60506	−	2.78005i	0.0997338	−	0.172744i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.06899	−	7.04770i	0.250905	−	0.434580i	−0.712870	−	0.701296i	$-0.752605\pi$
$263$	4.06899	−	7.04770i	0.250905	−	0.434580i	0.963775	+	0.266716i	$0.0859386\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	25.6374			1.56314			0.781570	−	0.623817i	$-0.214419\pi$
$269$	25.6374			1.56314			0.781570	+	0.623817i	$0.214419\pi$
$270$		0			0
$271$	−3.21013			−0.195001			−0.0975007	−	0.995235i	$-0.531085\pi$
$271$	−3.21013			−0.195001			−0.0975007	+	0.995235i	$0.531085\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.349823	+	0.605911i	−0.0210188	+	0.0364057i	−0.876343	−	0.481687i	$-0.840024\pi$
$277$	−0.349823	+	0.605911i	−0.0210188	+	0.0364057i	0.855325	+	0.518092i	$0.173358\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−15.3152	+	26.5267i	−0.913628	+	1.58245i	−0.104730	+	0.994501i	$0.533398\pi$
$281$	−15.3152	+	26.5267i	−0.913628	+	1.58245i	−0.808898	+	0.587949i	$0.799936\pi$
$282$		0			0
$283$	−4.80287	−	8.31882i	−0.285501	−	0.494502i	0.687229	−	0.726440i	$-0.258827\pi$
$283$	−4.80287	−	8.31882i	−0.285501	−	0.494502i	−0.972731	+	0.231938i	$0.925493\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	11.7557			0.693916
$288$		0			0
$289$	−14.5657			−0.856806
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.50526	+	2.60718i	0.0879381	+	0.152313i	0.906639	−	0.421906i	$-0.138639\pi$
$293$	1.50526	+	2.60718i	0.0879381	+	0.152313i	−0.818701	+	0.574220i	$0.805306\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.10155	−	5.37205i	0.179367	−	0.310673i
$300$		0			0
$301$	−5.10155	−	8.83615i	−0.294049	−	0.509307i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	14.8671			0.848512			0.424256	−	0.905542i	$-0.360536\pi$
$307$	14.8671			0.848512			0.424256	+	0.905542i	$0.360536\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.53586	+	2.66018i	0.0870905	+	0.150845i	0.906280	−	0.422678i	$-0.138910\pi$
$311$	1.53586	+	2.66018i	0.0870905	+	0.150845i	−0.819190	+	0.573523i	$0.805576\pi$
$312$		0			0
$313$	11.5068	−	19.9303i	0.650402	−	1.12653i	−0.332623	−	0.943060i	$-0.607934\pi$
$313$	11.5068	−	19.9303i	0.650402	−	1.12653i	0.983025	−	0.183470i	$-0.0587330\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.00450177	+	0.00779729i	−0.000252844	+	0.000437939i	−0.866152	−	0.499781i	$-0.833414\pi$
$317$	−0.00450177	+	0.00779729i	−0.000252844	+	0.000437939i	0.865899	+	0.500219i	$0.166747\pi$
$318$		0			0
$319$	−5.31870	−	9.21227i	−0.297790	−	0.515788i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	11.2494			0.625935
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−18.3745	−	31.8255i	−1.01302	−	1.75460i
$330$		0			0
$331$	−4.49649	+	7.78815i	−0.247149	+	0.428075i	−0.962734	−	0.270451i	$-0.912827\pi$
$331$	−4.49649	+	7.78815i	−0.247149	+	0.428075i	0.715584	+	0.698526i	$0.246160\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.63069	−	4.55649i	−0.143303	−	0.248208i	0.785436	−	0.618943i	$-0.212439\pi$
$337$	−2.63069	−	4.55649i	−0.143303	−	0.248208i	−0.928739	+	0.370735i	$0.879106\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−20.5048			−1.11040
$342$		0			0
$343$	0.792107			0.0427698
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.7477	+	18.6155i	0.576965	+	0.999333i	0.995825	+	0.0912825i	$0.0290966\pi$
$347$	10.7477	+	18.6155i	0.576965	+	0.999333i	−0.418860	+	0.908051i	$0.637570\pi$
$348$		0			0
$349$	6.28285	−	10.8822i	0.336313	−	0.582511i	−0.647423	−	0.762131i	$-0.724153\pi$
$349$	6.28285	−	10.8822i	0.336313	−	0.582511i	0.983736	+	0.179619i	$0.0574866\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.5313	−	21.7048i	0.666973	−	1.15523i	−0.311773	−	0.950157i	$-0.600923\pi$
$353$	12.5313	−	21.7048i	0.666973	−	1.15523i	0.978746	−	0.205075i	$-0.0657439\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−23.1961			−1.22424			−0.612121	−	0.790764i	$-0.709684\pi$
$359$	−23.1961			−1.22424			−0.612121	+	0.790764i	$0.709684\pi$
$360$		0			0
$361$	32.9860			1.73610
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	8.89259	−	15.4024i	0.464190	−	0.804000i	−0.534975	−	0.844868i	$-0.679679\pi$
$367$	8.89259	−	15.4024i	0.464190	−	0.804000i	0.999165	+	0.0408679i	$0.0130123\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	21.2031	−	36.7249i	1.10081	−	1.90666i
$372$		0			0
$373$	6.45979	+	11.1887i	0.334475	+	0.579328i	0.983384	−	0.181538i	$-0.0581075\pi$
$373$	6.45979	+	11.1887i	0.334475	+	0.579328i	−0.648908	+	0.760866i	$0.724774\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.01052			−0.155050
$378$		0			0
$379$	−5.56570			−0.285891			−0.142945	−	0.989731i	$-0.545657\pi$
$379$	−5.56570			−0.285891			−0.142945	+	0.989731i	$0.545657\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.8716	+	25.7584i	0.759905	+	1.31619i	0.942899	+	0.333080i	$0.108088\pi$
$383$	14.8716	+	25.7584i	0.759905	+	1.31619i	−0.182994	+	0.983114i	$0.558579\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.77641	+	6.54094i	−0.191472	+	0.331639i	−0.945738	−	0.324930i	$-0.894659\pi$
$389$	−3.77641	+	6.54094i	−0.191472	+	0.331639i	0.754266	+	0.656568i	$0.227993\pi$
$390$		0			0
$391$	−4.82222	−	8.35232i	−0.243870	−	0.422395i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−30.1571			−1.51354			−0.756770	−	0.653682i	$-0.773224\pi$
$397$	−30.1571			−1.51354			−0.756770	+	0.653682i	$0.773224\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.8322	−	18.7619i	−0.540932	−	0.936922i	−0.998851	−	0.0479282i	$-0.984738\pi$
$401$	−10.8322	−	18.7619i	−0.540932	−	0.936922i	0.457918	−	0.888994i	$-0.348595\pi$
$402$		0			0
$403$	−2.90156	+	5.02565i	−0.144537	+	0.250345i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.50976	−	2.61498i	0.0748360	−	0.129620i
$408$		0			0
$409$	4.06921	+	7.04807i	0.201209	+	0.348505i	0.948918	−	0.315522i	$-0.102180\pi$
$409$	4.06921	+	7.04807i	0.201209	+	0.348505i	−0.747709	+	0.664027i	$0.768846\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	36.4310			1.79265
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.64736	+	11.5136i	0.324745	+	0.562474i	0.981461	−	0.191664i	$-0.0613883\pi$
$419$	6.64736	+	11.5136i	0.324745	+	0.562474i	−0.656716	+	0.754138i	$0.728055\pi$
$420$		0			0
$421$	−10.0692	+	17.4404i	−0.490743	+	0.849992i	−0.999943	−	0.0106561i	$-0.996608\pi$
$421$	−10.0692	+	17.4404i	−0.490743	+	0.849992i	0.509200	+	0.860648i	$0.329941\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	15.4746	+	26.8028i	0.748868	+	1.29708i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−25.8027			−1.24287			−0.621437	−	0.783464i	$-0.713451\pi$
$431$	−25.8027			−1.24287			−0.621437	+	0.783464i	$0.713451\pi$
$432$		0			0
$433$	6.38383			0.306787			0.153394	−	0.988165i	$-0.450980\pi$
$433$	6.38383			0.306787			0.153394	+	0.988165i	$0.450980\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−22.2845	−	38.5978i	−1.06601	−	1.84638i
$438$		0			0
$439$	−5.49649	+	9.52020i	−0.262333	+	0.454374i	−0.966861	−	0.255302i	$-0.917825\pi$
$439$	−5.49649	+	9.52020i	−0.262333	+	0.454374i	0.704528	+	0.709676i	$0.251159\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.6160	+	21.8515i	−0.599404	+	1.03820i	0.393505	+	0.919322i	$0.371262\pi$
$443$	−12.6160	+	21.8515i	−0.599404	+	1.03820i	−0.992909	+	0.118876i	$0.962071\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	19.5657			0.923362			0.461681	−	0.887046i	$-0.347246\pi$
$449$	19.5657			0.923362			0.461681	+	0.887046i	$0.347246\pi$
$450$		0			0
$451$	11.0577			0.520685
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.26788	−	5.66013i	0.152865	−	0.264770i	−0.779415	−	0.626508i	$-0.784483\pi$
$457$	3.26788	−	5.66013i	0.152865	−	0.264770i	0.932280	+	0.361739i	$0.117817\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.7101	+	30.6748i	−0.824843	+	1.42867i	0.0771964	+	0.997016i	$0.475403\pi$
$461$	−17.7101	+	30.6748i	−0.824843	+	1.42867i	−0.902039	+	0.431654i	$0.857930\pi$
$462$		0			0
$463$	−8.13348	−	14.0876i	−0.377995	−	0.654706i	0.612776	−	0.790257i	$-0.290053\pi$
$463$	−8.13348	−	14.0876i	−0.377995	−	0.654706i	−0.990770	+	0.135551i	$0.956720\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15.8112			0.731654			0.365827	−	0.930683i	$-0.380786\pi$
$467$	15.8112			0.731654			0.365827	+	0.930683i	$0.380786\pi$
$468$		0			0
$469$	11.0000			0.507933
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.79864	−	8.31148i	−0.220642	−	0.382162i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.3709	+	28.3553i	−0.748007	+	1.29559i	0.200769	+	0.979639i	$0.435656\pi$
$479$	−16.3709	+	28.3553i	−0.748007	+	1.29559i	−0.948776	+	0.315948i	$0.897677\pi$
$480$		0			0
$481$	−0.427281	−	0.740072i	−0.0194823	−	0.0337444i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	36.4220			1.65044			0.825218	−	0.564814i	$-0.191052\pi$
$487$	36.4220			1.65044			0.825218	+	0.564814i	$0.191052\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.71364	−	2.96811i	−0.0773355	−	0.133949i	0.824764	−	0.565477i	$-0.191308\pi$
$491$	−1.71364	−	2.96811i	−0.0773355	−	0.133949i	−0.902100	+	0.431528i	$0.857975\pi$
$492$		0			0
$493$	−2.34034	+	4.05359i	−0.105404	+	0.182564i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13.5898	−	23.5382i	0.609584	−	1.05583i
$498$		0			0
$499$	−1.71364	−	2.96811i	−0.0767131	−	0.132871i	0.825117	−	0.564962i	$-0.191109\pi$
$499$	−1.71364	−	2.96811i	−0.0767131	−	0.132871i	−0.901830	+	0.432091i	$0.857776\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	17.4219			0.776802			0.388401	−	0.921490i	$-0.373027\pi$
$503$	17.4219			0.776802			0.388401	+	0.921490i	$0.373027\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.07272	−	12.2503i	−0.313493	−	0.542985i	0.665623	−	0.746288i	$-0.268166\pi$
$509$	−7.07272	−	12.2503i	−0.313493	−	0.542985i	−0.979116	+	0.203303i	$0.934832\pi$
$510$		0			0
$511$	31.6304	−	54.7854i	1.39925	−	2.42356i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−17.2834	−	29.9358i	−0.760125	−	1.31657i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	20.2300			0.886293			0.443147	−	0.896449i	$-0.353862\pi$
$521$	20.2300			0.886293			0.443147	+	0.896449i	$0.353862\pi$
$522$		0			0
$523$	36.3295			1.58858			0.794289	−	0.607540i	$-0.207844\pi$
$523$	36.3295			1.58858			0.794289	+	0.607540i	$0.207844\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.51127	+	7.81376i	0.196514	+	0.340373i
$528$		0			0
$529$	−7.60506	+	13.1724i	−0.330655	+	0.572711i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.56473	−	2.71019i	0.0677759	−	0.117391i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	25.5657			1.10119
$540$		0			0
$541$	24.8475			1.06828			0.534140	−	0.845396i	$-0.320636\pi$
$541$	24.8475			1.06828			0.534140	+	0.845396i	$0.320636\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	16.0523	−	27.8034i	0.686347	−	1.18879i	−0.286665	−	0.958031i	$-0.592547\pi$
$547$	16.0523	−	27.8034i	0.686347	−	1.18879i	0.973012	−	0.230757i	$-0.0741201\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.8152	+	18.7325i	−0.460743	+	0.798030i
$552$		0			0
$553$	−20.7197	−	35.8876i	−0.881093	−	1.52610i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	17.3894			0.736813			0.368406	−	0.929665i	$-0.379904\pi$
$557$	17.3894			0.736813			0.368406	+	0.929665i	$0.379904\pi$
$558$		0			0
$559$	−2.71615			−0.114881
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.6700	+	23.6771i	0.576120	+	0.997870i	0.995919	+	0.0902525i	$0.0287674\pi$
$563$	13.6700	+	23.6771i	0.576120	+	0.997870i	−0.419799	+	0.907617i	$0.637899\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.59512	+	13.1551i	−0.318404	+	0.551492i	−0.980155	−	0.198232i	$-0.936480\pi$
$569$	−7.59512	+	13.1551i	−0.318404	+	0.551492i	0.661751	+	0.749723i	$0.269813\pi$
$570$		0			0
$571$	19.3510	+	33.5170i	0.809816	+	1.40264i	0.912991	+	0.407980i	$0.133767\pi$
$571$	19.3510	+	33.5170i	0.809816	+	1.40264i	−0.103175	+	0.994663i	$0.532900\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	3.82910			0.159408			0.0797038	−	0.996819i	$-0.474603\pi$
$577$	3.82910			0.159408			0.0797038	+	0.996819i	$0.474603\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−19.2166	−	33.2841i	−0.797237	−	1.38086i
$582$		0			0
$583$	19.9441	−	34.5442i	0.826001	−	1.43068i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.3421	+	21.3772i	−0.509415	+	0.882332i	0.490526	+	0.871427i	$0.336805\pi$
$587$	−12.3421	+	21.3772i	−0.509415	+	0.882332i	−0.999941	+	0.0109053i	$0.996529\pi$
$588$		0			0
$589$	20.8475	+	36.1090i	0.859008	+	1.48785i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	38.4290			1.57809			0.789044	−	0.614336i	$-0.210576\pi$
$593$	38.4290			1.57809			0.789044	+	0.614336i	$0.210576\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.71364	+	13.3604i	0.315171	+	0.545892i	0.979474	−	0.201572i	$-0.0646049\pi$
$599$	7.71364	+	13.3604i	0.315171	+	0.545892i	−0.664303	+	0.747463i	$0.731272\pi$
$600$		0			0
$601$	0.286360	−	0.495989i	0.0116809	−	0.0202318i	−0.860126	−	0.510082i	$-0.829615\pi$
$601$	0.286360	−	0.495989i	0.0116809	−	0.0202318i	0.871807	+	0.489850i	$0.162948\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−4.52901	−	7.84448i	−0.183827	−	0.318398i	0.759354	−	0.650678i	$-0.225515\pi$
$607$	−4.52901	−	7.84448i	−0.183827	−	0.318398i	−0.943181	+	0.332280i	$0.892182\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.78285			−0.395772
$612$		0			0
$613$	−6.99155			−0.282386			−0.141193	−	0.989982i	$-0.545094\pi$
$613$	−6.99155			−0.282386			−0.141193	+	0.989982i	$0.545094\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9.91710	−	17.1769i	−0.399247	−	0.691517i	0.594386	−	0.804180i	$-0.297395\pi$
$617$	−9.91710	−	17.1769i	−0.399247	−	0.691517i	−0.993633	+	0.112663i	$0.964062\pi$
$618$		0			0
$619$	14.1016	−	24.4246i	0.566789	−	0.981708i	−0.430092	−	0.902785i	$-0.641519\pi$
$619$	14.1016	−	24.4246i	0.566789	−	0.981708i	0.996881	−	0.0789225i	$-0.0251480\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.0568	+	17.4188i	−0.402916	+	0.697871i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.32865			−0.0529768
$630$		0			0
$631$	−7.27482			−0.289606			−0.144803	−	0.989461i	$-0.546255\pi$
$631$	−7.27482			−0.289606			−0.144803	+	0.989461i	$0.546255\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.61770	−	6.26604i	0.143339	−	0.248270i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.43372	−	11.1435i	0.254116	−	0.440143i	−0.710539	−	0.703658i	$-0.751549\pi$
$641$	6.43372	−	11.1435i	0.254116	−	0.440143i	0.964655	+	0.263515i	$0.0848820\pi$
$642$		0			0
$643$	12.6160	+	21.8515i	0.497526	+	0.861741i	0.999996	−	0.00285403i	$-0.000908467\pi$
$643$	12.6160	+	21.8515i	0.497526	+	0.861741i	−0.502470	+	0.864595i	$0.667575\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	23.5618			0.926311			0.463156	−	0.886277i	$-0.346717\pi$
$647$	23.5618			0.926311			0.463156	+	0.886277i	$0.346717\pi$
$648$		0			0
$649$	34.2678			1.34513
$650$		0			0
$651$		0			0
$652$		0			0
$653$	8.69020	+	15.0519i	0.340074	+	0.589025i	0.984446	−	0.175687i	$-0.0562146\pi$
$653$	8.69020	+	15.0519i	0.340074	+	0.589025i	−0.644372	+	0.764712i	$0.722881\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−7.83216	+	13.5657i	−0.305098	+	0.528445i	−0.977283	−	0.211938i	$-0.932022\pi$
$659$	−7.83216	+	13.5657i	−0.305098	+	0.528445i	0.672185	+	0.740383i	$0.265356\pi$
$660$		0			0
$661$	−11.7136	−	20.2886i	−0.455608	−	0.789136i	0.543115	−	0.839658i	$-0.317245\pi$
$661$	−11.7136	−	20.2886i	−0.455608	−	0.789136i	−0.998723	+	0.0505224i	$0.983911\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	18.5443			0.718038
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.5557	+	25.2113i	0.561918	+	0.973271i
$672$		0			0
$673$	14.3654	−	24.8816i	0.553745	−	0.959114i	−0.444255	−	0.895900i	$-0.646532\pi$
$673$	14.3654	−	24.8816i	0.553745	−	0.959114i	0.998000	−	0.0632136i	$-0.0201350\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−25.1770	+	43.6079i	−0.967632	+	1.67599i	−0.265260	+	0.964177i	$0.585458\pi$
$677$	−25.1770	+	43.6079i	−0.967632	+	1.67599i	−0.702372	+	0.711810i	$0.747876\pi$
$678$		0			0
$679$	−7.27934	−	12.6082i	−0.279355	−	0.483858i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.5953			0.443681			0.221841	−	0.975083i	$-0.428794\pi$
$683$	11.5953			0.443681			0.221841	+	0.975083i	$0.428794\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.64443	−	9.77644i	−0.215036	−	0.372453i
$690$		0			0
$691$	−22.9238	+	39.7051i	−0.872061	+	1.51045i	−0.0122006	+	0.999926i	$0.503884\pi$
$691$	−22.9238	+	39.7051i	−0.872061	+	1.51045i	−0.859861	+	0.510529i	$0.829450\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.43280	−	4.21373i	−0.0921489	−	0.159606i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	38.1115			1.43945			0.719726	−	0.694259i	$-0.244268\pi$
$701$	38.1115			1.43945			0.719726	+	0.694259i	$0.244268\pi$
$702$		0			0
$703$	−6.13997			−0.231573
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.96406	+	15.5262i	0.337128	+	0.583923i
$708$		0			0
$709$	1.14443	−	1.98222i	0.0429801	−	0.0744437i	−0.843735	−	0.536760i	$-0.819648\pi$
$709$	1.14443	−	1.98222i	0.0429801	−	0.0744437i	0.886715	+	0.462316i	$0.152981\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	17.8731	−	30.9572i	0.669355	−	1.15936i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	16.6035			0.619205			0.309602	−	0.950866i	$-0.399804\pi$
$719$	16.6035			0.619205			0.309602	+	0.950866i	$0.399804\pi$
$720$		0			0
$721$	−68.1891			−2.53949
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−19.3037	+	33.4350i	−0.715934	+	1.24003i	0.246664	+	0.969101i	$0.420666\pi$
$727$	−19.3037	+	33.4350i	−0.715934	+	1.24003i	−0.962598	+	0.270933i	$0.912668\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.11150	+	3.65722i	−0.0780966	+	0.135267i
$732$		0			0
$733$	16.9501	+	29.3584i	0.626066	+	1.08438i	0.988334	+	0.152305i	$0.0486695\pi$
$733$	16.9501	+	29.3584i	0.626066	+	1.08438i	−0.362267	+	0.932074i	$0.617997\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.3469			0.381131
$738$		0			0
$739$	−36.8405			−1.35520			−0.677600	−	0.735431i	$-0.736980\pi$
$739$	−36.8405			−1.35520			−0.677600	+	0.735431i	$0.736980\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.71392	−	9.89680i	−0.209623	−	0.363078i	0.741973	−	0.670430i	$-0.233890\pi$
$743$	−5.71392	−	9.89680i	−0.209623	−	0.363078i	−0.951596	+	0.307352i	$0.900557\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.81870	−	11.8103i	0.249150	−	0.431541i
$750$		0			0
$751$	8.10155	+	14.0323i	0.295630	+	0.512046i	0.975131	−	0.221628i	$-0.0711372\pi$
$751$	8.10155	+	14.0323i	0.295630	+	0.512046i	−0.679501	+	0.733674i	$0.737804\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−8.08698			−0.293926			−0.146963	−	0.989142i	$-0.546950\pi$
$757$	−8.08698			−0.293926			−0.146963	+	0.989142i	$0.546950\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10.7031	+	18.5383i	0.387987	+	0.672014i	0.992179	−	0.124826i	$-0.0398371\pi$
$761$	10.7031	+	18.5383i	0.387987	+	0.672014i	−0.604191	+	0.796839i	$0.706504\pi$
$762$		0			0
$763$	18.1647	−	31.4623i	0.657608	−	1.13901i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.84910	−	8.39889i	0.175091	−	0.303266i
$768$		0			0
$769$	−25.7031	−	44.5191i	−0.926878	−	1.60540i	−0.788513	−	0.615018i	$-0.789149\pi$
$769$	−25.7031	−	44.5191i	−0.926878	−	1.60540i	−0.138364	−	0.990381i	$-0.544185\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−11.2425	−	19.4725i	−0.402803	−	0.697676i
$780$		0			0
$781$	12.7828	−	22.1405i	0.457406	−	0.792251i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−2.35683	−	4.08216i	−0.0840121	−	0.145513i	0.820958	−	0.570989i	$-0.193440\pi$
$787$	−2.35683	−	4.08216i	−0.0840121	−	0.145513i	−0.904970	+	0.425476i	$0.860107\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	13.4132			0.476920
$792$		0			0
$793$	8.23891			0.292572
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.13196	+	8.88882i	0.181784	+	0.314858i	0.942488	−	0.334240i	$-0.108480\pi$
$797$	5.13196	+	8.88882i	0.181784	+	0.314858i	−0.760704	+	0.649098i	$0.775146\pi$
$798$		0			0
$799$	−7.60506	+	13.1724i	−0.269048	+	0.466005i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	29.7523	−	51.5324i	1.04993	−	1.81854i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−21.9860			−0.772985			−0.386492	−	0.922293i	$-0.626313\pi$
$809$	−21.9860			−0.772985			−0.386492	+	0.922293i	$0.626313\pi$
$810$		0			0
$811$	−25.6951			−0.902276			−0.451138	−	0.892454i	$-0.648982\pi$
$811$	−25.6951			−0.902276			−0.451138	+	0.892454i	$0.648982\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−9.75767	+	16.9008i	−0.341378	+	0.591284i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12.1967	−	21.1253i	0.425667	−	0.737276i	−0.570816	−	0.821078i	$-0.693373\pi$
$821$	12.1967	−	21.1253i	0.425667	−	0.737276i	0.996482	+	0.0838018i	$0.0267063\pi$
$822$		0			0
$823$	−1.30710	−	2.26396i	−0.0455626	−	0.0789167i	0.842345	−	0.538939i	$-0.181175\pi$
$823$	−1.30710	−	2.26396i	−0.0455626	−	0.0789167i	−0.887907	+	0.460022i	$0.847841\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.0684725			0.00238102			0.00119051	−	0.999999i	$-0.499621\pi$
$827$	0.0684725			0.00238102			0.00119051	+	0.999999i	$0.499621\pi$
$828$		0			0
$829$	−24.7828			−0.860744			−0.430372	−	0.902652i	$-0.641618\pi$
$829$	−24.7828			−0.860744			−0.430372	+	0.902652i	$0.641618\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.62471	−	9.74229i	−0.194885	−	0.337550i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22.6204	−	39.1797i	0.780944	−	1.35264i	−0.150448	−	0.988618i	$-0.548072\pi$
$839$	22.6204	−	39.1797i	0.780944	−	1.35264i	0.931392	−	0.364017i	$-0.118595\pi$
$840$		0			0
$841$	10.0000	+	17.3205i	0.344828	+	0.597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	5.92857			0.203708
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.63198	+	4.55872i	0.0902231	+	0.156271i
$852$		0			0
$853$	10.1075	−	17.5067i	0.346074	−	0.599418i	−0.639474	−	0.768813i	$-0.720848\pi$
$853$	10.1075	−	17.5067i	0.346074	−	0.599418i	0.985548	+	0.169395i	$0.0541812\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	13.4213	−	23.2464i	0.458464	−	0.794083i	−0.540416	−	0.841398i	$-0.681733\pi$
$857$	13.4213	−	23.2464i	0.458464	−	0.794083i	0.998880	+	0.0473147i	$0.0150664\pi$
$858$		0			0
$859$	−1.68130	−	2.91209i	−0.0573651	−	0.0993592i	0.835917	−	0.548856i	$-0.184937\pi$
$859$	−1.68130	−	2.91209i	−0.0573651	−	0.0993592i	−0.893282	+	0.449497i	$0.851603\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−18.4524			−0.628126			−0.314063	−	0.949402i	$-0.601690\pi$
$863$	−18.4524			−0.628126			−0.314063	+	0.949402i	$0.601690\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−19.4895	−	33.7567i	−0.661135	−	1.14512i
$870$		0			0
$871$	1.46414	−	2.53597i	0.0496106	−	0.0859281i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	11.5993	+	20.0905i	0.391679	+	0.678408i	0.992671	−	0.120847i	$-0.0385611\pi$
$877$	11.5993	+	20.0905i	0.391679	+	0.678408i	−0.600992	+	0.799255i	$0.705228\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.34854			−0.0791244			−0.0395622	−	0.999217i	$-0.512596\pi$
$881$	−2.34854			−0.0791244			−0.0395622	+	0.999217i	$0.512596\pi$
$882$		0			0
$883$	16.3264			0.549428			0.274714	−	0.961526i	$-0.411417\pi$
$883$	16.3264			0.549428			0.274714	+	0.961526i	$0.411417\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13.4213	+	23.2464i	0.450645	+	0.780539i	0.998426	−	0.0560821i	$-0.0178609\pi$
$887$	13.4213	+	23.2464i	0.450645	+	0.780539i	−0.547782	+	0.836621i	$0.684528\pi$
$888$		0			0
$889$	−12.7793	+	22.1345i	−0.428605	+	0.742366i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−35.1446	+	60.8722i	−1.17607	+	2.03701i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−17.3485			−0.578606
$900$		0			0
$901$	−17.5516			−0.584730
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−3.92130	+	6.79188i	−0.130204	+	0.225521i	−0.923755	−	0.382983i	$-0.874897\pi$
$907$	−3.92130	+	6.79188i	−0.130204	+	0.225521i	0.793551	+	0.608504i	$0.208230\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.28636	−	2.22804i	0.0426190	−	0.0738183i	−0.843929	−	0.536455i	$-0.819763\pi$
$911$	1.28636	−	2.22804i	0.0426190	−	0.0738183i	0.886548	+	0.462637i	$0.153097\pi$
$912$		0			0
$913$	−18.0755	−	31.3078i	−0.598213	−	1.03614i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12.9196			0.426642
$918$		0			0
$919$	42.4062			1.39885			0.699426	−	0.714705i	$-0.253439\pi$
$919$	42.4062			1.39885			0.699426	+	0.714705i	$0.253439\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.61770	−	6.26604i	−0.119078	−	0.206249i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	25.9167	−	44.8891i	0.850301	−	1.47276i	−0.0306361	−	0.999531i	$-0.509753\pi$
$929$	25.9167	−	44.8891i	0.850301	−	1.47276i	0.880937	−	0.473234i	$-0.156913\pi$
$930$		0			0
$931$	−25.9930	−	45.0212i	−0.851885	−	1.47551i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−16.9666			−0.554275			−0.277137	−	0.960830i	$-0.589386\pi$
$937$	−16.9666			−0.554275			−0.277137	+	0.960830i	$0.589386\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	2.82865	+	4.89937i	0.0922114	+	0.159715i	0.908441	−	0.418012i	$-0.137273\pi$
$941$	2.82865	+	4.89937i	0.0922114	+	0.159715i	−0.816230	+	0.577727i	$0.803940\pi$
$942$		0			0
$943$	−9.63847	+	16.6943i	−0.313872	+	0.543642i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.6930	−	30.6452i	0.574945	−	0.995834i	−0.421103	−	0.907013i	$-0.638357\pi$
$947$	17.6930	−	30.6452i	0.574945	−	0.995834i	0.996048	−	0.0888210i	$-0.0283099\pi$
$948$		0			0
$949$	−8.42026	−	14.5843i	−0.273333	−	0.473427i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.66116			0.0538101			0.0269051	−	0.999638i	$-0.491435\pi$
$953$	1.66116			0.0538101			0.0269051	+	0.999638i	$0.491435\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	28.1339	+	48.7293i	0.908491	+	1.57355i
$960$		0			0
$961$	−1.22066	+	2.11425i	−0.0393763	+	0.0682017i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−10.3681	−	17.9581i	−0.333416	−	0.577494i	0.649763	−	0.760137i	$-0.274868\pi$
$967$	−10.3681	−	17.9581i	−0.333416	−	0.577494i	−0.983179	+	0.182643i	$0.941535\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20.9193			−0.671331			−0.335665	−	0.941981i	$-0.608961\pi$
$971$	−20.9193			−0.671331			−0.335665	+	0.941981i	$0.608961\pi$
$972$		0			0
$973$	−20.4589			−0.655881
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.44579	+	2.50418i	0.0462549	+	0.0801159i	0.888226	−	0.459407i	$-0.151938\pi$
$977$	1.44579	+	2.50418i	0.0462549	+	0.0801159i	−0.841971	+	0.539523i	$0.818605\pi$
$978$		0			0
$979$	−9.45963	+	16.3846i	−0.302331	+	0.523653i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.15019	−	5.45628i	0.100475	−	0.174028i	−0.811405	−	0.584484i	$-0.801297\pi$
$983$	3.15019	−	5.45628i	0.100475	−	0.174028i	0.911881	+	0.410456i	$0.134630\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16.7310			0.532016
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−4.78963	+	8.29589i	−0.151689	+	0.262733i	−0.931849	−	0.362847i	$-0.881805\pi$
$997$	−4.78963	+	8.29589i	−0.151689	+	0.262733i	0.780159	+	0.625581i	$0.215138\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.i.f.1801.1		12
3.2	odd	2		900.2.i.f.601.3		12
5.2	odd	4		540.2.r.a.289.4		12
5.3	odd	4		540.2.r.a.289.6		12
5.4	even	2	inner	2700.2.i.f.1801.6		12
9.2	odd	6		8100.2.a.bc.1.6		6
9.4	even	3	inner	2700.2.i.f.901.1		12
9.5	odd	6		900.2.i.f.301.3		12
9.7	even	3		8100.2.a.bd.1.6		6
15.2	even	4		180.2.r.a.169.1	yes	12
15.8	even	4		180.2.r.a.169.6	yes	12
15.14	odd	2		900.2.i.f.601.4		12
20.3	even	4		2160.2.by.e.289.6		12
20.7	even	4		2160.2.by.e.289.4		12
45.2	even	12		1620.2.d.c.649.6		6
45.4	even	6	inner	2700.2.i.f.901.6		12
45.7	odd	12		1620.2.d.d.649.1		6
45.13	odd	12		540.2.r.a.469.4		12
45.14	odd	6		900.2.i.f.301.4		12
45.22	odd	12		540.2.r.a.469.6		12
45.23	even	12		180.2.r.a.49.1	✓	12
45.29	odd	6		8100.2.a.bc.1.1		6
45.32	even	12		180.2.r.a.49.6	yes	12
45.34	even	6		8100.2.a.bd.1.1		6
45.38	even	12		1620.2.d.c.649.5		6
45.43	odd	12		1620.2.d.d.649.2		6
60.23	odd	4		720.2.by.e.529.1		12
60.47	odd	4		720.2.by.e.529.6		12
180.23	odd	12		720.2.by.e.49.6		12
180.67	even	12		2160.2.by.e.1009.6		12
180.103	even	12		2160.2.by.e.1009.4		12
180.167	odd	12		720.2.by.e.49.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.r.a.49.1	✓	12	45.23	even	12
180.2.r.a.49.6	yes	12	45.32	even	12
180.2.r.a.169.1	yes	12	15.2	even	4
180.2.r.a.169.6	yes	12	15.8	even	4
540.2.r.a.289.4		12	5.2	odd	4
540.2.r.a.289.6		12	5.3	odd	4
540.2.r.a.469.4		12	45.13	odd	12
540.2.r.a.469.6		12	45.22	odd	12
720.2.by.e.49.1		12	180.167	odd	12
720.2.by.e.49.6		12	180.23	odd	12
720.2.by.e.529.1		12	60.23	odd	4
720.2.by.e.529.6		12	60.47	odd	4
900.2.i.f.301.3		12	9.5	odd	6
900.2.i.f.301.4		12	45.14	odd	6
900.2.i.f.601.3		12	3.2	odd	2
900.2.i.f.601.4		12	15.14	odd	2
1620.2.d.c.649.5		6	45.38	even	12
1620.2.d.c.649.6		6	45.2	even	12
1620.2.d.d.649.1		6	45.7	odd	12
1620.2.d.d.649.2		6	45.43	odd	12
2160.2.by.e.289.4		12	20.7	even	4
2160.2.by.e.289.6		12	20.3	even	4
2160.2.by.e.1009.4		12	180.103	even	12
2160.2.by.e.1009.6		12	180.67	even	12
2700.2.i.f.901.1		12	9.4	even	3	inner
2700.2.i.f.901.6		12	45.4	even	6	inner
2700.2.i.f.1801.1		12	1.1	even	1	trivial
2700.2.i.f.1801.6		12	5.4	even	2	inner
8100.2.a.bc.1.1		6	45.29	odd	6
8100.2.a.bc.1.6		6	9.2	odd	6
8100.2.a.bd.1.1		6	45.34	even	6
8100.2.a.bd.1.6		6	9.7	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.88482 + 3.26460i) q^{7} +(-1.77290 + 3.07076i) q^{11} +(0.501753 + 0.869061i) q^{13} +1.56023 q^{17} +7.21013 q^{19} +(-3.09072 - 5.35328i) q^{23} +(-1.50000 + 2.59808i) q^{29} +(2.89142 + 5.00809i) q^{31} -0.851576 q^{37} +(-1.55926 - 2.70072i) q^{41} +(-1.35333 + 2.34403i) q^{43} +(-4.87434 + 8.44260i) q^{47} +(-3.60506 - 6.24415i) q^{49} -11.2494 q^{53} +(-4.83216 - 8.36955i) q^{59} +(4.10506 - 7.11018i) q^{61} +(-1.45903 - 2.52711i) q^{67} -7.21013 q^{71} -16.7817 q^{73} +(-6.68319 - 11.5756i) q^{77} +(-5.49649 + 9.52020i) q^{79} +(-5.09773 + 8.82952i) q^{83} +5.33567 q^{89} -3.78285 q^{91} +(-1.93105 + 3.34467i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{11} - 18 q^{29} + 6 q^{31} - 14 q^{41} - 34 q^{59} + 6 q^{61} + 6 q^{79} + 112 q^{89} + 12 q^{91}+O(q^{100}) \) 12 * q - 2 * q^11 - 18 * q^29 + 6 * q^31 - 14 * q^41 - 34 * q^59 + 6 * q^61 + 6 * q^79 + 112 * q^89 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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