LMFDB - Embedded newform 2700.3.u.c.2249.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,3,Mod(449,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2700.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2700.u (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$73.5696713773$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2249.7
Character	$\chi$	$=$	2700.2249
Dual form			2700.3.u.c.449.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.02985 + 0.594587i) q^{7} +O(q^{10})$	$q+(1.02985 + 0.594587i) q^{7} +(-3.63473 - 2.09851i) q^{11} +(12.3948 - 7.15614i) q^{13} +18.7074 q^{17} -33.8986 q^{19} +(-6.74989 - 11.6912i) q^{23} +(30.6315 + 17.6851i) q^{29} +(4.97816 + 8.62242i) q^{31} -19.3047i q^{37} +(55.9648 - 32.3113i) q^{41} +(-35.9872 - 20.7772i) q^{43} +(-33.6057 + 58.2068i) q^{47} +(-23.7929 - 41.2106i) q^{49} -30.0712 q^{53} +(3.66554 - 2.11630i) q^{59} +(43.8210 - 75.9002i) q^{61} +(-31.0106 + 17.9040i) q^{67} -24.5173i q^{71} -11.9305i q^{73} +(-2.49549 - 4.32232i) q^{77} +(-19.0858 + 33.0575i) q^{79} +(-12.0489 + 20.8692i) q^{83} +44.4494i q^{89} +17.0198 q^{91} +(-55.3688 - 31.9672i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 96 q^{11} - 144 q^{19} + 300 q^{29} - 24 q^{31} - 180 q^{41} - 96 q^{49} + 96 q^{59} - 156 q^{61} - 240 q^{79} + 240 q^{91}+O(q^{100}) $ 24 * q - 96 * q^11 - 144 * q^19 + 300 * q^29 - 24 * q^31 - 180 * q^41 - 96 * q^49 + 96 * q^59 - 156 * q^61 - 240 * q^79 + 240 * q^91

Display 100 coefficients 10 coefficients

Character values

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

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

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
$7$	1.02985	+	0.594587i	0.147122	+	0.0849410i	0.571754	−	0.820425i	$-0.306263\pi$
$7$	1.02985	+	0.594587i	0.147122	+	0.0849410i	−0.424632	+	0.905366i	$0.639596\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.63473	−	2.09851i	−0.330430	−	0.190774i	0.325602	−	0.945507i	$-0.394433\pi$
$11$	−3.63473	−	2.09851i	−0.330430	−	0.190774i	−0.656032	+	0.754733i	$0.727766\pi$
$12$		0			0
$13$	12.3948	−	7.15614i	0.953447	−	0.550473i	0.0592967	−	0.998240i	$-0.481114\pi$
$13$	12.3948	−	7.15614i	0.953447	−	0.550473i	0.894150	+	0.447768i	$0.147781\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	18.7074			1.10044			0.550218	−	0.835021i	$-0.314545\pi$
$17$	18.7074			1.10044			0.550218	+	0.835021i	$0.314545\pi$
$18$		0			0
$19$	−33.8986			−1.78414			−0.892069	−	0.451899i	$-0.850747\pi$
$19$	−33.8986			−1.78414			−0.892069	+	0.451899i	$0.850747\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.74989	−	11.6912i	−0.293474	−	0.508311i	0.681155	−	0.732139i	$-0.261478\pi$
$23$	−6.74989	−	11.6912i	−0.293474	−	0.508311i	−0.974629	+	0.223828i	$0.928145\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	30.6315	+	17.6851i	1.05626	+	0.609831i	0.924395	−	0.381437i	$-0.124571\pi$
$29$	30.6315	+	17.6851i	1.05626	+	0.609831i	0.131863	+	0.991268i	$0.457904\pi$
$30$		0			0
$31$	4.97816	+	8.62242i	0.160586	+	0.278143i	0.935079	−	0.354440i	$-0.115328\pi$
$31$	4.97816	+	8.62242i	0.160586	+	0.278143i	−0.774493	+	0.632582i	$0.781995\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	19.3047i		−	0.521750i	−0.965373	−	0.260875i	$-0.915989\pi$
$37$		−	19.3047i		−	0.521750i	0.965373	−	0.260875i	$-0.0840110\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	55.9648	−	32.3113i	1.36499	−	0.788080i	0.374711	−	0.927142i	$-0.377742\pi$
$41$	55.9648	−	32.3113i	1.36499	−	0.788080i	0.990284	+	0.139062i	$0.0444088\pi$
$42$		0			0
$43$	−35.9872	−	20.7772i	−0.836911	−	0.483191i	0.0193018	−	0.999814i	$-0.493856\pi$
$43$	−35.9872	−	20.7772i	−0.836911	−	0.483191i	−0.856213	+	0.516623i	$0.827189\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−33.6057	+	58.2068i	−0.715015	+	1.23844i	0.247939	+	0.968776i	$0.420247\pi$
$47$	−33.6057	+	58.2068i	−0.715015	+	1.23844i	−0.962954	+	0.269667i	$0.913087\pi$
$48$		0			0
$49$	−23.7929	−	41.2106i	−0.485570	−	0.841032i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−30.0712			−0.567382			−0.283691	−	0.958916i	$-0.591559\pi$
$53$	−30.0712			−0.567382			−0.283691	+	0.958916i	$0.591559\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.66554	−	2.11630i	0.0621278	−	0.0358695i	−0.468614	−	0.883403i	$-0.655247\pi$
$59$	3.66554	−	2.11630i	0.0621278	−	0.0358695i	0.530742	+	0.847533i	$0.321913\pi$
$60$		0			0
$61$	43.8210	−	75.9002i	0.718377	−	1.24427i	−0.243266	−	0.969960i	$-0.578219\pi$
$61$	43.8210	−	75.9002i	0.718377	−	1.24427i	0.961643	−	0.274305i	$-0.0884479\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−31.0106	+	17.9040i	−0.462845	+	0.267224i	−0.713240	−	0.700920i	$-0.752773\pi$
$67$	−31.0106	+	17.9040i	−0.462845	+	0.267224i	0.250395	+	0.968144i	$0.419440\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	24.5173i		−	0.345314i	−0.984982	−	0.172657i	$-0.944765\pi$
$71$		−	24.5173i		−	0.345314i	0.984982	−	0.172657i	$-0.0552352\pi$
$72$		0			0
$73$		−	11.9305i		−	0.163432i	−0.996656	−	0.0817159i	$-0.973960\pi$
$73$		−	11.9305i		−	0.163432i	0.996656	−	0.0817159i	$-0.0260400\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.49549	−	4.32232i	−0.0324090	−	0.0561340i
$78$		0			0
$79$	−19.0858	+	33.0575i	−0.241592	+	0.418449i	−0.961168	−	0.275964i	$-0.911003\pi$
$79$	−19.0858	+	33.0575i	−0.241592	+	0.418449i	0.719576	+	0.694414i	$0.244336\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.0489	+	20.8692i	−0.145167	+	0.251436i	−0.929435	−	0.368985i	$-0.879705\pi$
$83$	−12.0489	+	20.8692i	−0.145167	+	0.251436i	0.784268	+	0.620422i	$0.213039\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			44.4494i			0.499431i	0.968319	+	0.249716i	$0.0803371\pi$
$89$			44.4494i			0.499431i	−0.968319	+	0.249716i	$0.919663\pi$
$90$		0			0
$91$	17.0198			0.187031
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−55.3688	−	31.9672i	−0.570812	−	0.329558i	0.186662	−	0.982424i	$-0.440233\pi$
$97$	−55.3688	−	31.9672i	−0.570812	−	0.329558i	−0.757474	+	0.652866i	$0.773567\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.3591	−	5.98084i	−0.102566	−	0.0592163i	0.447840	−	0.894114i	$-0.352193\pi$
$101$	−10.3591	−	5.98084i	−0.102566	−	0.0592163i	−0.550405	+	0.834898i	$0.685527\pi$
$102$		0			0
$103$	142.001	−	81.9841i	1.37865	−	0.795963i	0.386651	−	0.922226i	$-0.373632\pi$
$103$	142.001	−	81.9841i	1.37865	−	0.795963i	0.991997	+	0.126264i	$0.0402985\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−102.401			−0.957019			−0.478509	−	0.878082i	$-0.658823\pi$
$107$	−102.401			−0.957019			−0.478509	+	0.878082i	$0.658823\pi$
$108$		0			0
$109$	56.4485			0.517876			0.258938	−	0.965894i	$-0.416627\pi$
$109$	56.4485			0.517876			0.258938	+	0.965894i	$0.416627\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	86.2022	+	149.307i	0.762852	+	1.32130i	0.941375	+	0.337362i	$0.109535\pi$
$113$	86.2022	+	149.307i	0.762852	+	1.32130i	−0.178523	+	0.983936i	$0.557132\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	19.2659	+	11.1232i	0.161899	+	0.0934722i
$120$		0			0
$121$	−51.6925	−	89.5341i	−0.427211	−	0.739951i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	195.853i		−	1.54215i	−0.636746	−	0.771073i	$-0.719720\pi$
$127$		−	195.853i		−	1.54215i	0.636746	−	0.771073i	$-0.280280\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	98.5249	−	56.8834i	0.752099	−	0.434224i	−0.0743530	−	0.997232i	$-0.523689\pi$
$131$	98.5249	−	56.8834i	0.752099	−	0.434224i	0.826452	+	0.563008i	$0.190356\pi$
$132$		0			0
$133$	−34.9107	−	20.1557i	−0.262486	−	0.151546i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	90.5130	−	156.773i	0.660679	−	1.14433i	−0.319759	−	0.947499i	$-0.603602\pi$
$137$	90.5130	−	156.773i	0.660679	−	1.14433i	0.980438	−	0.196830i	$-0.0630648\pi$
$138$		0			0
$139$	−114.555	−	198.414i	−0.824134	−	1.42744i	−0.902579	−	0.430524i	$-0.858329\pi$
$139$	−114.555	−	198.414i	−0.824134	−	1.42744i	0.0784454	−	0.996918i	$-0.475004\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−60.0689			−0.420063
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	233.844	−	135.010i	1.56942	−	0.906107i	0.573187	−	0.819425i	$-0.305707\pi$
$149$	233.844	−	135.010i	1.56942	−	0.906107i	0.996236	−	0.0866818i	$-0.0276263\pi$
$150$		0			0
$151$	81.5705	−	141.284i	0.540202	−	0.935658i	−0.458690	−	0.888596i	$-0.651681\pi$
$151$	81.5705	−	141.284i	0.540202	−	0.935658i	0.998892	−	0.0470612i	$-0.0149856\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−116.567	+	67.2998i	−0.742463	+	0.428661i	−0.822964	−	0.568094i	$-0.807681\pi$
$157$	−116.567	+	67.2998i	−0.742463	+	0.428661i	0.0805014	+	0.996754i	$0.474348\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	16.0536i		−	0.0997118i
$162$		0			0
$163$			159.658i			0.979500i	0.871863	+	0.489750i	$0.162912\pi$
$163$			159.658i			0.979500i	−0.871863	+	0.489750i	$0.837088\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−55.5770	−	96.2622i	−0.332796	−	0.576420i	0.650263	−	0.759709i	$-0.274659\pi$
$167$	−55.5770	−	96.2622i	−0.332796	−	0.576420i	−0.983059	+	0.183289i	$0.941326\pi$
$168$		0			0
$169$	17.9208	−	31.0397i	0.106040	−	0.183667i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	73.6755	−	127.610i	0.425870	−	0.737629i	−0.570631	−	0.821207i	$-0.693301\pi$
$173$	73.6755	−	127.610i	0.425870	−	0.737629i	0.996501	+	0.0835778i	$0.0266347\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	253.022i		−	1.41353i	−0.707449	−	0.706765i	$-0.750154\pi$
$179$		−	253.022i		−	1.41353i	0.707449	−	0.706765i	$-0.249846\pi$
$180$		0			0
$181$	−212.017			−1.17136			−0.585682	−	0.810541i	$-0.699173\pi$
$181$	−212.017			−1.17136			−0.585682	+	0.810541i	$0.699173\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−67.9963	−	39.2577i	−0.363617	−	0.209934i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	287.027	+	165.715i	1.50276	+	0.867619i	0.999995	+	0.00319548i	$0.00101715\pi$
$191$	287.027	+	165.715i	1.50276	+	0.867619i	0.502765	+	0.864423i	$0.332316\pi$
$192$		0			0
$193$	−288.004	+	166.279i	−1.49225	+	0.861549i	−0.999961	−	0.00888358i	$-0.997172\pi$
$193$	−288.004	+	166.279i	−1.49225	+	0.861549i	−0.492287	+	0.870433i	$0.663839\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−321.219			−1.63056			−0.815278	−	0.579070i	$-0.803416\pi$
$197$	−321.219			−1.63056			−0.815278	+	0.579070i	$0.803416\pi$
$198$		0			0
$199$	208.913			1.04981			0.524907	−	0.851160i	$-0.324100\pi$
$199$	208.913			1.04981			0.524907	+	0.851160i	$0.324100\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	21.0307	+	36.4262i	0.103599	+	0.179439i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	123.212	+	71.1366i	0.589532	+	0.340366i
$210$		0			0
$211$	−110.304	−	191.053i	−0.522769	−	0.905463i	−0.999649	−	0.0264945i	$-0.991566\pi$
$211$	−110.304	−	191.053i	−0.522769	−	0.905463i	0.476880	−	0.878969i	$-0.341768\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			11.8398i			0.0545613i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	231.875	−	133.873i	1.04921	−	0.605760i
$222$		0			0
$223$	248.767	+	143.626i	1.11555	+	0.644062i	0.940261	−	0.340455i	$-0.110581\pi$
$223$	248.767	+	143.626i	1.11555	+	0.644062i	0.175287	+	0.984517i	$0.443915\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	130.758	−	226.479i	0.576024	−	0.997704i	−0.419905	−	0.907568i	$-0.637937\pi$
$227$	130.758	−	226.479i	0.576024	−	0.997704i	0.995930	−	0.0901356i	$-0.0287301\pi$
$228$		0			0
$229$	99.2244	+	171.862i	0.433294	+	0.750488i	0.997155	−	0.0753822i	$-0.0240177\pi$
$229$	99.2244	+	171.862i	0.433294	+	0.750488i	−0.563860	+	0.825870i	$0.690684\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	182.063			0.781387			0.390693	−	0.920521i	$-0.372235\pi$
$233$	182.063			0.781387			0.390693	+	0.920521i	$0.372235\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−210.133	+	121.320i	−0.879218	+	0.507617i	−0.870400	−	0.492344i	$-0.836140\pi$
$239$	−210.133	+	121.320i	−0.879218	+	0.507617i	−0.00881748	+	0.999961i	$0.502807\pi$
$240$		0			0
$241$	158.882	−	275.192i	0.659263	−	1.14188i	−0.321544	−	0.946895i	$-0.604202\pi$
$241$	158.882	−	275.192i	0.659263	−	1.14188i	0.980807	−	0.194983i	$-0.0624650\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−420.167	+	242.583i	−1.70108	+	0.982119i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	64.0280i		−	0.255092i	−0.991833	−	0.127546i	$-0.959290\pi$
$251$		−	64.0280i		−	0.255092i	0.991833	−	0.127546i	$-0.0407100\pi$
$252$		0			0
$253$			56.6589i			0.223948i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−232.101	−	402.010i	−0.903116	−	1.56424i	−0.823426	−	0.567423i	$-0.807940\pi$
$257$	−232.101	−	402.010i	−0.903116	−	1.56424i	−0.0796896	−	0.996820i	$-0.525393\pi$
$258$		0			0
$259$	11.4784	−	19.8811i	0.0443180	−	0.0767610i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	28.0482	−	48.5809i	0.106647	−	0.184718i	−0.807763	−	0.589508i	$-0.799322\pi$
$263$	28.0482	−	48.5809i	0.106647	−	0.184718i	0.914410	+	0.404789i	$0.132655\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			29.8910i			0.111119i	0.998455	+	0.0555594i	$0.0176942\pi$
$269$			29.8910i			0.111119i	−0.998455	+	0.0555594i	$0.982306\pi$
$270$		0			0
$271$	−166.826			−0.615594			−0.307797	−	0.951452i	$-0.599592\pi$
$271$	−166.826			−0.615594			−0.307797	+	0.951452i	$0.599592\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−170.443	−	98.4054i	−0.615318	−	0.355254i	0.159726	−	0.987161i	$-0.448939\pi$
$277$	−170.443	−	98.4054i	−0.615318	−	0.355254i	−0.775044	+	0.631907i	$0.782272\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−288.196	−	166.390i	−1.02561	−	0.592136i	−0.109886	−	0.993944i	$-0.535049\pi$
$281$	−288.196	−	166.390i	−1.02561	−	0.592136i	−0.915724	+	0.401808i	$0.868382\pi$
$282$		0			0
$283$	404.376	−	233.467i	1.42889	−	0.824971i	0.431858	−	0.901942i	$-0.357858\pi$
$283$	404.376	−	233.467i	1.42889	−	0.824971i	0.997033	+	0.0769712i	$0.0245249\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	76.8474			0.267761
$288$		0			0
$289$	60.9675			0.210960
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.7616	−	18.6397i	−0.0367291	−	0.0636167i	0.847077	−	0.531471i	$-0.178361\pi$
$293$	−10.7616	−	18.6397i	−0.0367291	−	0.0636167i	−0.883806	+	0.467854i	$0.845027\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−167.327	−	96.6064i	−0.559623	−	0.323098i
$300$		0			0
$301$	−24.7077	−	42.7950i	−0.0820854	−	0.142176i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			452.833i			1.47503i	0.675332	+	0.737514i	$0.264000\pi$
$307$			452.833i			1.47503i	−0.675332	+	0.737514i	$0.736000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	488.695	−	282.148i	1.57137	−	0.907229i	0.575364	−	0.817897i	$-0.304860\pi$
$311$	488.695	−	282.148i	1.57137	−	0.907229i	0.996002	−	0.0893313i	$-0.0284730\pi$
$312$		0			0
$313$	−399.515	−	230.660i	−1.27641	−	0.736933i	−0.300220	−	0.953870i	$-0.597060\pi$
$313$	−399.515	−	230.660i	−1.27641	−	0.736933i	−0.976186	+	0.216937i	$0.930393\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	168.380	−	291.643i	0.531167	−	0.920008i	−0.468171	−	0.883638i	$-0.655087\pi$
$317$	168.380	−	291.643i	0.531167	−	0.920008i	0.999338	−	0.0363706i	$-0.0115797\pi$
$318$		0			0
$319$	−74.2247	−	128.561i	−0.232679	−	0.403012i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−634.156			−1.96333
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−69.2180	+	39.9630i	−0.210389	+	0.121468i
$330$		0			0
$331$	210.896	−	365.283i	0.637149	−	1.10357i	−0.348907	−	0.937158i	$-0.613447\pi$
$331$	210.896	−	365.283i	0.637149	−	1.10357i	0.986055	−	0.166417i	$-0.0532197\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−102.557	+	59.2115i	−0.304325	+	0.175702i	−0.644384	−	0.764702i	$-0.722886\pi$
$337$	−102.557	+	59.2115i	−0.304325	+	0.175702i	0.340059	+	0.940404i	$0.389553\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	41.7869i		−	0.122542i
$342$		0			0
$343$		−	114.857i		−	0.334861i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−85.0001	−	147.224i	−0.244957	−	0.424278i	0.717163	−	0.696906i	$-0.245441\pi$
$347$	−85.0001	−	147.224i	−0.244957	−	0.424278i	−0.962120	+	0.272628i	$0.912107\pi$
$348$		0			0
$349$	310.127	−	537.156i	0.888617	−	1.53913i	0.0471063	−	0.998890i	$-0.485000\pi$
$349$	310.127	−	537.156i	0.888617	−	1.53913i	0.841511	−	0.540240i	$-0.181667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	204.704	−	354.557i	0.579897	−	1.00441i	−0.415593	−	0.909551i	$-0.636426\pi$
$353$	204.704	−	354.557i	0.579897	−	1.00441i	0.995491	−	0.0948611i	$-0.0302407\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	138.441i		−	0.385630i	−0.981235	−	0.192815i	$-0.938238\pi$
$359$		−	138.441i		−	0.385630i	0.981235	−	0.192815i	$-0.0617618\pi$
$360$		0			0
$361$	788.116			2.18315
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−68.4781	−	39.5358i	−0.186589	−	0.107727i	0.403796	−	0.914849i	$-0.367691\pi$
$367$	−68.4781	−	39.5358i	−0.186589	−	0.107727i	−0.590385	+	0.807122i	$0.701024\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−30.9690	−	17.8800i	−0.0834744	−	0.0481940i
$372$		0			0
$373$	−89.8672	+	51.8849i	−0.240931	+	0.139102i	−0.615605	−	0.788055i	$-0.711088\pi$
$373$	−89.8672	+	51.8849i	−0.240931	+	0.139102i	0.374674	+	0.927157i	$0.377755\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	506.228			1.34278
$378$		0			0
$379$	−315.487			−0.832419			−0.416210	−	0.909269i	$-0.636642\pi$
$379$	−315.487			−0.832419			−0.416210	+	0.909269i	$0.636642\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	207.092	+	358.695i	0.540711	+	0.936540i	0.998863	+	0.0476657i	$0.0151782\pi$
$383$	207.092	+	358.695i	0.540711	+	0.936540i	−0.458152	+	0.888874i	$0.651488\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−59.9057	−	34.5866i	−0.153999	−	0.0889115i	0.421020	−	0.907051i	$-0.361672\pi$
$389$	−59.9057	−	34.5866i	−0.153999	−	0.0889115i	−0.575019	+	0.818140i	$0.695006\pi$
$390$		0			0
$391$	−126.273	−	218.711i	−0.322949	−	0.559364i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			347.995i			0.876562i	0.898838	+	0.438281i	$0.144413\pi$
$397$			347.995i			0.876562i	−0.898838	+	0.438281i	$0.855587\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	635.313	−	366.798i	1.58432	−	0.914709i	0.590104	−	0.807327i	$-0.299087\pi$
$401$	635.313	−	366.798i	1.58432	−	0.914709i	0.994218	−	0.107382i	$-0.0342467\pi$
$402$		0			0
$403$	123.407	+	71.2488i	0.306220	+	0.176796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−40.5112	+	70.1675i	−0.0995361	+	0.172402i
$408$		0			0
$409$	47.0727	+	81.5324i	0.115092	+	0.199346i	0.917817	−	0.397005i	$-0.129950\pi$
$409$	47.0727	+	81.5324i	0.115092	+	0.199346i	−0.802724	+	0.596350i	$0.796617\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.03330			0.0121872
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−90.6043	+	52.3104i	−0.216239	+	0.124846i	−0.604208	−	0.796827i	$-0.706510\pi$
$419$	−90.6043	+	52.3104i	−0.216239	+	0.124846i	0.387968	+	0.921673i	$0.373177\pi$
$420$		0			0
$421$	−276.542	+	478.984i	−0.656868	+	1.13773i	0.324553	+	0.945867i	$0.394786\pi$
$421$	−276.542	+	478.984i	−0.656868	+	1.13773i	−0.981422	+	0.191862i	$0.938547\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	90.2585	−	52.1108i	0.211378	−	0.122039i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			469.242i			1.08873i	0.838849	+	0.544364i	$0.183229\pi$
$431$			469.242i			1.08873i	−0.838849	+	0.544364i	$0.816771\pi$
$432$		0			0
$433$		−	426.644i		−	0.985321i	−0.870222	−	0.492660i	$-0.836024\pi$
$433$		−	426.644i		−	0.985321i	0.870222	−	0.492660i	$-0.163976\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	228.812	+	396.314i	0.523598	+	0.906898i
$438$		0			0
$439$	−286.573	+	496.359i	−0.652786	+	1.13066i	0.329658	+	0.944100i	$0.393067\pi$
$439$	−286.573	+	496.359i	−0.652786	+	1.13066i	−0.982444	+	0.186558i	$0.940267\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−52.2986	+	90.5838i	−0.118055	+	0.204478i	−0.918997	−	0.394264i	$-0.870999\pi$
$443$	−52.2986	+	90.5838i	−0.118055	+	0.204478i	0.800942	+	0.598743i	$0.204333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			401.878i			0.895051i	0.894271	+	0.447526i	$0.147695\pi$
$449$			401.878i			0.895051i	−0.894271	+	0.447526i	$0.852305\pi$
$450$		0			0
$451$	−271.222			−0.601379
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−83.4142	−	48.1592i	−0.182526	−	0.105381i	0.405953	−	0.913894i	$-0.366940\pi$
$457$	−83.4142	−	48.1592i	−0.182526	−	0.105381i	−0.588479	+	0.808513i	$0.700273\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	64.4588	+	37.2153i	0.139824	+	0.0807274i	0.568280	−	0.822835i	$-0.307609\pi$
$461$	64.4588	+	37.2153i	0.139824	+	0.0807274i	−0.428456	+	0.903563i	$0.640942\pi$
$462$		0			0
$463$	−344.558	+	198.930i	−0.744185	+	0.429655i	−0.823589	−	0.567187i	$-0.808032\pi$
$463$	−344.558	+	198.930i	−0.744185	+	0.429655i	0.0794042	+	0.996843i	$0.474698\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−643.763			−1.37851			−0.689254	−	0.724520i	$-0.742062\pi$
$467$	−643.763			−1.37851			−0.689254	+	0.724520i	$0.742062\pi$
$468$		0			0
$469$	−42.5819			−0.0907931
$470$		0			0
$471$		0			0
$472$		0			0
$473$	87.2024	+	151.039i	0.184360	+	0.319321i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−152.770	−	88.2019i	−0.318936	−	0.184138i	0.331982	−	0.943286i	$-0.392283\pi$
$479$	−152.770	−	88.2019i	−0.318936	−	0.184138i	−0.650918	+	0.759148i	$0.725616\pi$
$480$		0			0
$481$	−138.148	−	239.279i	−0.287209	−	0.497461i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	494.859i		−	1.01614i	−0.861317	−	0.508069i	$-0.830360\pi$
$487$		−	494.859i		−	1.01614i	0.861317	−	0.508069i	$-0.169640\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−159.777	+	92.2472i	−0.325411	+	0.187876i	−0.653802	−	0.756666i	$-0.726827\pi$
$491$	−159.777	+	92.2472i	−0.325411	+	0.187876i	0.328391	+	0.944542i	$0.393494\pi$
$492$		0			0
$493$	573.036	+	330.842i	1.16234	+	0.671080i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	14.5777	−	25.2492i	0.0293313	−	0.0508033i
$498$		0			0
$499$	162.276	+	281.071i	0.325203	+	0.563269i	0.981553	−	0.191188i	$-0.0612339\pi$
$499$	162.276	+	281.071i	0.325203	+	0.563269i	−0.656350	+	0.754456i	$0.727901\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−626.131			−1.24479			−0.622397	−	0.782702i	$-0.713841\pi$
$503$	−626.131			−1.24479			−0.622397	+	0.782702i	$0.713841\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	387.634	−	223.800i	0.761559	−	0.439686i	−0.0682960	−	0.997665i	$-0.521756\pi$
$509$	387.634	−	223.800i	0.761559	−	0.439686i	0.829855	+	0.557979i	$0.188423\pi$
$510$		0			0
$511$	7.09374	−	12.2867i	0.0138821	−	0.0240444i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	244.295	−	141.044i	0.472524	−	0.272812i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			181.821i			0.348984i	0.984659	+	0.174492i	$0.0558284\pi$
$521$			181.821i			0.348984i	−0.984659	+	0.174492i	$0.944172\pi$
$522$		0			0
$523$			293.719i			0.561604i	0.959766	+	0.280802i	$0.0906004\pi$
$523$			293.719i			0.561604i	−0.959766	+	0.280802i	$0.909400\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	93.1285	+	161.303i	0.176714	+	0.306078i
$528$		0			0
$529$	173.378	−	300.299i	0.327746	−	0.567673i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	462.448	−	800.984i	0.867633	−	1.50278i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			199.719i			0.370536i
$540$		0			0
$541$	343.593			0.635107			0.317553	−	0.948240i	$-0.397139\pi$
$541$	343.593			0.635107			0.317553	+	0.948240i	$0.397139\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	35.6434	+	20.5788i	0.0651617	+	0.0376211i	0.532227	−	0.846602i	$-0.321355\pi$
$547$	35.6434	+	20.5788i	0.0651617	+	0.0376211i	−0.467065	+	0.884223i	$0.654689\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1038.36	−	599.500i	−1.88451	−	1.08802i
$552$		0			0
$553$	−39.3111	+	22.6963i	−0.0710870	+	0.0410421i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−976.675			−1.75346			−0.876728	−	0.480986i	$-0.840279\pi$
$557$	−976.675			−1.75346			−0.876728	+	0.480986i	$0.840279\pi$
$558$		0			0
$559$	−594.739			−1.06393
$560$		0			0
$561$		0			0
$562$		0			0
$563$	221.313	+	383.325i	0.393096	+	0.680862i	0.992856	−	0.119318i	$-0.0380708\pi$
$563$	221.313	+	383.325i	0.393096	+	0.680862i	−0.599760	+	0.800180i	$0.704737\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	681.591	+	393.517i	1.19788	+	0.691594i	0.960081	−	0.279721i	$-0.0902421\pi$
$569$	681.591	+	393.517i	1.19788	+	0.691594i	0.237795	+	0.971315i	$0.423575\pi$
$570$		0			0
$571$	3.41373	+	5.91275i	0.00597851	+	0.0103551i	0.868999	−	0.494814i	$-0.164764\pi$
$571$	3.41373	+	5.91275i	0.00597851	+	0.0103551i	−0.863021	+	0.505169i	$0.831430\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	45.4328i		−	0.0787396i	−0.999225	−	0.0393698i	$-0.987465\pi$
$577$		−	45.4328i		−	0.0787396i	0.999225	−	0.0393698i	$-0.0125350\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24.8171	+	14.3282i	−0.0427145	+	0.0246612i
$582$		0			0
$583$	109.301	+	63.1048i	0.187480	+	0.108241i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	49.6244	−	85.9520i	0.0845391	−	0.146426i	−0.820656	−	0.571423i	$-0.806392\pi$
$587$	49.6244	−	85.9520i	0.0845391	−	0.146426i	0.905195	+	0.424997i	$0.139725\pi$
$588$		0			0
$589$	−168.753	−	292.288i	−0.286507	−	0.496245i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−226.897			−0.382626			−0.191313	−	0.981529i	$-0.561274\pi$
$593$	−226.897			−0.382626			−0.191313	+	0.981529i	$0.561274\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	652.129	−	376.507i	1.08870	−	0.628559i	0.155467	−	0.987841i	$-0.450312\pi$
$599$	652.129	−	376.507i	1.08870	−	0.628559i	0.933229	+	0.359282i	$0.116978\pi$
$600$		0			0
$601$	−342.525	+	593.270i	−0.569924	+	0.987138i	0.426648	+	0.904418i	$0.359694\pi$
$601$	−342.525	+	593.270i	−0.569924	+	0.987138i	−0.996573	+	0.0827206i	$0.973639\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	787.174	−	454.475i	1.29683	−	0.748724i	0.316973	−	0.948435i	$-0.397334\pi$
$607$	787.174	−	454.475i	1.29683	−	0.748724i	0.979855	+	0.199711i	$0.0640003\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			961.949i			1.57438i
$612$		0			0
$613$		−	514.234i		−	0.838881i	−0.907783	−	0.419441i	$-0.862226\pi$
$613$		−	514.234i		−	0.838881i	0.907783	−	0.419441i	$-0.137774\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	575.133	+	996.159i	0.932144	+	1.61452i	0.779650	+	0.626216i	$0.215397\pi$
$617$	575.133	+	996.159i	0.932144	+	1.61452i	0.152494	+	0.988304i	$0.451270\pi$
$618$		0			0
$619$	−69.9937	+	121.233i	−0.113076	+	0.195853i	−0.917009	−	0.398867i	$-0.869404\pi$
$619$	−69.9937	+	121.233i	−0.113076	+	0.195853i	0.803933	+	0.594719i	$0.202737\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−26.4290	+	45.7764i	−0.0424222	+	0.0734774i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	361.142i		−	0.574153i
$630$		0			0
$631$	−369.971			−0.586325			−0.293163	−	0.956063i	$-0.594708\pi$
$631$	−369.971			−0.586325			−0.293163	+	0.956063i	$0.594708\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−589.818	−	340.531i	−0.925930	−	0.534586i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−67.0156	−	38.6914i	−0.104548	−	0.0603611i	0.446814	−	0.894627i	$-0.352559\pi$
$641$	−67.0156	−	38.6914i	−0.104548	−	0.0603611i	−0.551363	+	0.834266i	$0.685892\pi$
$642$		0			0
$643$	−226.733	+	130.904i	−0.352617	+	0.203583i	−0.665837	−	0.746097i	$-0.731925\pi$
$643$	−226.733	+	130.904i	−0.352617	+	0.203583i	0.313220	+	0.949680i	$0.398592\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	617.428			0.954293			0.477147	−	0.878824i	$-0.341671\pi$
$647$	617.428			0.954293			0.477147	+	0.878824i	$0.341671\pi$
$648$		0			0
$649$	−17.7643			−0.0273718
$650$		0			0
$651$		0			0
$652$		0			0
$653$	337.239	+	584.115i	0.516446	+	0.894510i	0.999818	+	0.0190951i	$0.00607852\pi$
$653$	337.239	+	584.115i	0.516446	+	0.894510i	−0.483372	+	0.875415i	$0.660588\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	332.973	+	192.242i	0.505271	+	0.291718i	0.730888	−	0.682498i	$-0.239106\pi$
$659$	332.973	+	192.242i	0.505271	+	0.291718i	−0.225617	+	0.974216i	$0.572440\pi$
$660$		0			0
$661$	151.713	+	262.775i	0.229520	+	0.397541i	0.957666	−	0.287882i	$-0.0929509\pi$
$661$	151.713	+	262.775i	0.229520	+	0.397541i	−0.728146	+	0.685422i	$0.759618\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	477.490i		−	0.715877i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−318.554	+	183.918i	−0.474746	+	0.274095i
$672$		0			0
$673$	−485.081	−	280.062i	−0.720775	−	0.416139i	0.0942631	−	0.995547i	$-0.469951\pi$
$673$	−485.081	−	280.062i	−0.720775	−	0.416139i	−0.815038	+	0.579408i	$0.803284\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−508.282	+	880.370i	−0.750786	+	1.30040i	0.196656	+	0.980472i	$0.436992\pi$
$677$	−508.282	+	880.370i	−0.750786	+	1.30040i	−0.947442	+	0.319927i	$0.896342\pi$
$678$		0			0
$679$	−38.0145	−	65.8431i	−0.0559860	−	0.0969707i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−88.7713			−0.129973			−0.0649863	−	0.997886i	$-0.520700\pi$
$683$	−88.7713			−0.129973			−0.0649863	+	0.997886i	$0.520700\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−372.727	+	215.194i	−0.540968	+	0.312328i
$690$		0			0
$691$	−403.626	+	699.101i	−0.584119	+	1.01172i	0.410866	+	0.911696i	$0.365226\pi$
$691$	−403.626	+	699.101i	−0.584119	+	1.01172i	−0.994985	+	0.100027i	$0.968107\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1046.96	−	604.460i	1.50209	−	0.867232i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			201.018i			0.286758i	0.989668	+	0.143379i	$0.0457968\pi$
$701$			201.018i			0.286758i	−0.989668	+	0.143379i	$0.954203\pi$
$702$		0			0
$703$			654.404i			0.930874i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.11226	−	12.3188i	−0.0100598	−	0.0174240i
$708$		0			0
$709$	195.660	−	338.894i	0.275967	−	0.477989i	−0.694412	−	0.719578i	$-0.744335\pi$
$709$	195.660	−	338.894i	0.275967	−	0.477989i	0.970379	+	0.241589i	$0.0776687\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	67.2041	−	116.401i	0.0942554	−	0.163255i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1241.75i		−	1.72705i	−0.504308	−	0.863524i	$-0.668253\pi$
$719$		−	1241.75i		−	1.72705i	0.504308	−	0.863524i	$-0.331747\pi$
$720$		0			0
$721$	194.987			0.270439
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	173.571	+	100.212i	0.238750	+	0.137843i	0.614602	−	0.788837i	$-0.289317\pi$
$727$	173.571	+	100.212i	0.238750	+	0.137843i	−0.375852	+	0.926680i	$0.622650\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−673.227	−	388.688i	−0.920968	−	0.531721i
$732$		0			0
$733$	104.548	−	60.3607i	0.142630	−	0.0823475i	−0.426987	−	0.904258i	$-0.640425\pi$
$733$	104.548	−	60.3607i	0.142630	−	0.0823475i	0.569617	+	0.821910i	$0.307092\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	150.287			0.203917
$738$		0			0
$739$	471.461			0.637972			0.318986	−	0.947759i	$-0.396658\pi$
$739$	471.461			0.637972			0.318986	+	0.947759i	$0.396658\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	651.339	+	1128.15i	0.876634	+	1.51837i	0.855012	+	0.518608i	$0.173550\pi$
$743$	651.339	+	1128.15i	0.876634	+	1.51837i	0.0216220	+	0.999766i	$0.493117\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−105.458	−	60.8863i	−0.140799	−	0.0812901i
$750$		0			0
$751$	439.726	+	761.629i	0.585521	+	1.01415i	0.994810	+	0.101748i	$0.0324435\pi$
$751$	439.726	+	761.629i	0.585521	+	1.01415i	−0.409289	+	0.912405i	$0.634223\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	35.1814i		−	0.0464748i	−0.999730	−	0.0232374i	$-0.992603\pi$
$757$		−	35.1814i		−	0.0464748i	0.999730	−	0.0232374i	$-0.00739736\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1102.94	+	636.785i	−1.44933	+	0.836774i	−0.998442	−	0.0558046i	$-0.982228\pi$
$761$	−1102.94	+	636.785i	−1.44933	+	0.836774i	−0.450893	+	0.892578i	$0.648894\pi$
$762$		0			0
$763$	58.1337	+	33.5635i	0.0761910	+	0.0439889i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	30.2891	−	52.4623i	0.0394903	−	0.0683993i
$768$		0			0
$769$	379.742	+	657.733i	0.493813	+	0.855310i	0.999975	−	0.00712921i	$-0.00226932\pi$
$769$	379.742	+	657.733i	0.493813	+	0.855310i	−0.506161	+	0.862439i	$0.668936\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−165.240			−0.213765			−0.106883	−	0.994272i	$-0.534087\pi$
$773$	−165.240			−0.213765			−0.106883	+	0.994272i	$0.534087\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1897.13	+	1095.31i	−2.43534	+	1.40604i
$780$		0			0
$781$	−51.4497	+	89.1135i	−0.0658767	+	0.114102i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	498.401	−	287.752i	0.633293	−	0.365632i	−0.148733	−	0.988877i	$-0.547520\pi$
$787$	498.401	−	287.752i	0.633293	−	0.365632i	0.782026	+	0.623246i	$0.214186\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			205.019i			0.259190i
$792$		0			0
$793$		−	1254.36i		−	1.58179i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	303.808	+	526.210i	0.381189	+	0.660238i	0.991232	−	0.132129i	$-0.0421814\pi$
$797$	303.808	+	526.210i	0.381189	+	0.660238i	−0.610044	+	0.792368i	$0.708848\pi$
$798$		0			0
$799$	−628.676	+	1088.90i	−0.786828	+	1.36283i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−25.0363	+	43.3642i	−0.0311785	+	0.0540027i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			962.401i			1.18962i	0.803867	+	0.594809i	$0.202772\pi$
$809$			962.401i			1.18962i	−0.803867	+	0.594809i	$0.797228\pi$
$810$		0			0
$811$	−140.096			−0.172745			−0.0863723	−	0.996263i	$-0.527527\pi$
$811$	−140.096			−0.172745			−0.0863723	+	0.996263i	$0.527527\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1219.92	+	704.319i	1.49317	+	0.862079i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	280.294	+	161.828i	0.341406	+	0.197111i	0.660894	−	0.750480i	$-0.270177\pi$
$821$	280.294	+	161.828i	0.341406	+	0.197111i	−0.319488	+	0.947590i	$0.603511\pi$
$822$		0			0
$823$	167.327	−	96.6061i	0.203313	−	0.117383i	−0.394887	−	0.918730i	$-0.629216\pi$
$823$	167.327	−	96.6061i	0.203313	−	0.117383i	0.598200	+	0.801347i	$0.295883\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	15.7916			0.0190951			0.00954754	−	0.999954i	$-0.496961\pi$
$827$	15.7916			0.0190951			0.00954754	+	0.999954i	$0.496961\pi$
$828$		0			0
$829$	265.576			0.320357			0.160178	−	0.987088i	$-0.448793\pi$
$829$	265.576			0.320357			0.160178	+	0.987088i	$0.448793\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−445.104	−	770.943i	−0.534339	−	0.925502i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−761.943	−	439.908i	−0.908156	−	0.524324i	−0.0283189	−	0.999599i	$-0.509015\pi$
$839$	−761.943	−	439.908i	−0.908156	−	0.524324i	−0.879838	+	0.475275i	$0.842349\pi$
$840$		0			0
$841$	205.025	+	355.114i	0.243787	+	0.422252i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	122.943i		−	0.145151i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−225.695	+	130.305i	−0.265211	+	0.153120i
$852$		0			0
$853$	176.795	+	102.073i	0.207263	+	0.119663i	0.600039	−	0.799971i	$-0.295152\pi$
$853$	176.795	+	102.073i	0.207263	+	0.119663i	−0.392776	+	0.919634i	$0.628485\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	106.729	−	184.859i	0.124537	−	0.215705i	−0.797015	−	0.603960i	$-0.793589\pi$
$857$	106.729	−	184.859i	0.124537	−	0.215705i	0.921552	+	0.388255i	$0.126922\pi$
$858$		0			0
$859$	323.677	+	560.625i	0.376806	+	0.652648i	0.990596	−	0.136822i	$-0.0436889\pi$
$859$	323.677	+	560.625i	0.376806	+	0.652648i	−0.613789	+	0.789470i	$0.710356\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−699.772			−0.810859			−0.405430	−	0.914126i	$-0.632878\pi$
$863$	−699.772			−0.810859			−0.405430	+	0.914126i	$0.632878\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	138.743	−	80.1033i	0.159658	−	0.0921787i
$870$		0			0
$871$	−256.247	+	443.833i	−0.294199	+	0.509567i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1234.05	+	712.480i	−1.40713	+	0.812406i	−0.995110	−	0.0987694i	$-0.968509\pi$
$877$	−1234.05	+	712.480i	−1.40713	+	0.812406i	−0.412018	+	0.911176i	$0.635176\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1598.97i			1.81495i	0.420102	+	0.907477i	$0.361994\pi$
$881$			1598.97i			1.81495i	−0.420102	+	0.907477i	$0.638006\pi$
$882$		0			0
$883$			1117.03i			1.26504i	0.774542	+	0.632522i	$0.217980\pi$
$883$			1117.03i			1.26504i	−0.774542	+	0.632522i	$0.782020\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	583.003	+	1009.79i	0.657275	+	1.13843i	0.981318	+	0.192391i	$0.0616242\pi$
$887$	583.003	+	1009.79i	0.657275	+	1.13843i	−0.324044	+	0.946042i	$0.605042\pi$
$888$		0			0
$889$	116.451	−	201.700i	0.130992	−	0.226884i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1139.19	−	1973.13i	1.27569	−	2.20955i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			352.157i			0.391721i
$900$		0			0
$901$	−562.555			−0.624367
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1526.36	+	881.243i	1.68286	+	0.971602i	0.959742	+	0.280882i	$0.0906271\pi$
$907$	1526.36	+	881.243i	1.68286	+	0.971602i	0.723122	+	0.690720i	$0.242706\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1449.90	−	837.098i	−1.59154	−	0.918878i	−0.993043	−	0.117755i	$-0.962430\pi$
$911$	−1449.90	−	837.098i	−1.59154	−	0.918878i	−0.598501	−	0.801122i	$-0.704237\pi$
$912$		0			0
$913$	87.5885	−	50.5693i	0.0959349	−	0.0553880i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	135.289			0.147534
$918$		0			0
$919$	−1136.29			−1.23644			−0.618221	−	0.786004i	$-0.712146\pi$
$919$	−1136.29			−1.23644			−0.618221	+	0.786004i	$0.712146\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−175.449	−	303.887i	−0.190086	−	0.329238i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	932.936	+	538.631i	1.00424	+	0.579796i	0.909499	−	0.415706i	$-0.136465\pi$
$929$	932.936	+	538.631i	1.00424	+	0.579796i	0.0947376	+	0.995502i	$0.469799\pi$
$930$		0			0
$931$	806.548	+	1396.98i	0.866324	+	1.50052i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			1063.87i			1.13540i	0.823234	+	0.567702i	$0.192167\pi$
$937$			1063.87i			1.13540i	−0.823234	+	0.567702i	$0.807833\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	994.783	−	574.338i	1.05716	−	0.610349i	0.132511	−	0.991182i	$-0.457696\pi$
$941$	994.783	−	574.338i	1.05716	−	0.610349i	0.924644	+	0.380833i	$0.124363\pi$
$942$		0			0
$943$	−755.513	−	436.195i	−0.801180	−	0.462561i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−57.9461	+	100.366i	−0.0611891	+	0.105983i	−0.894997	−	0.446072i	$-0.852823\pi$
$947$	−57.9461	+	100.366i	−0.0611891	+	0.105983i	0.833808	+	0.552054i	$0.186156\pi$
$948$		0			0
$949$	−85.3766	−	147.877i	−0.0899648	−	0.155824i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1854.05			−1.94549			−0.972744	−	0.231881i	$-0.925512\pi$
$953$	−1854.05			−1.94549			−0.972744	+	0.231881i	$0.925512\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	186.431	−	107.636i	0.194401	−	0.112237i
$960$		0			0
$961$	430.936	−	746.403i	0.448424	−	0.776694i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−790.590	+	456.447i	−0.817569	+	0.472024i	−0.849578	−	0.527464i	$-0.823143\pi$
$967$	−790.590	+	456.447i	−0.817569	+	0.472024i	0.0320081	+	0.999488i	$0.489810\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1732.32i		−	1.78405i	−0.451982	−	0.892027i	$-0.649283\pi$
$971$		−	1732.32i		−	1.78405i	0.451982	−	0.892027i	$-0.350717\pi$
$972$		0			0
$973$		−	272.451i		−	0.280011i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	115.673	+	200.352i	0.118396	+	0.205069i	0.919132	−	0.393949i	$-0.128891\pi$
$977$	115.673	+	200.352i	0.118396	+	0.205069i	−0.800736	+	0.599018i	$0.795558\pi$
$978$		0			0
$979$	93.2774	−	161.561i	0.0952783	−	0.165027i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−389.601	+	674.808i	−0.396338	+	0.686478i	−0.993271	−	0.115813i	$-0.963053\pi$
$983$	−389.601	+	674.808i	−0.396338	+	0.686478i	0.596933	+	0.802291i	$0.296386\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			560.976i			0.567215i
$990$		0			0
$991$	222.470			0.224490			0.112245	−	0.993681i	$-0.464196\pi$
$991$	222.470			0.224490			0.112245	+	0.993681i	$0.464196\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	392.676	+	226.712i	0.393858	+	0.227394i	0.683830	−	0.729641i	$-0.260313\pi$
$997$	392.676	+	226.712i	0.393858	+	0.227394i	−0.289972	+	0.957035i	$0.593646\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.3.u.c.2249.7		24
3.2	odd	2		900.3.u.c.749.4		24
5.2	odd	4		2700.3.p.c.1601.3		12
5.3	odd	4		540.3.o.b.521.5		12
5.4	even	2	inner	2700.3.u.c.2249.6		24
9.4	even	3		900.3.u.c.149.9		24
9.5	odd	6	inner	2700.3.u.c.449.6		24
15.2	even	4		900.3.p.c.101.5		12
15.8	even	4		180.3.o.b.101.2	yes	12
15.14	odd	2		900.3.u.c.749.9		24
20.3	even	4		2160.3.bs.b.1601.5		12
45.4	even	6		900.3.u.c.149.4		24
45.13	odd	12		180.3.o.b.41.2	✓	12
45.14	odd	6	inner	2700.3.u.c.449.7		24
45.22	odd	12		900.3.p.c.401.5		12
45.23	even	12		540.3.o.b.341.5		12
45.32	even	12		2700.3.p.c.2501.3		12
45.38	even	12		1620.3.g.b.161.3		12
45.43	odd	12		1620.3.g.b.161.9		12
60.23	odd	4		720.3.bs.b.641.5		12
180.23	odd	12		2160.3.bs.b.881.5		12
180.103	even	12		720.3.bs.b.401.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.o.b.41.2	✓	12	45.13	odd	12
180.3.o.b.101.2	yes	12	15.8	even	4
540.3.o.b.341.5		12	45.23	even	12
540.3.o.b.521.5		12	5.3	odd	4
720.3.bs.b.401.5		12	180.103	even	12
720.3.bs.b.641.5		12	60.23	odd	4
900.3.p.c.101.5		12	15.2	even	4
900.3.p.c.401.5		12	45.22	odd	12
900.3.u.c.149.4		24	45.4	even	6
900.3.u.c.149.9		24	9.4	even	3
900.3.u.c.749.4		24	3.2	odd	2
900.3.u.c.749.9		24	15.14	odd	2
1620.3.g.b.161.3		12	45.38	even	12
1620.3.g.b.161.9		12	45.43	odd	12
2160.3.bs.b.881.5		12	180.23	odd	12
2160.3.bs.b.1601.5		12	20.3	even	4
2700.3.p.c.1601.3		12	5.2	odd	4
2700.3.p.c.2501.3		12	45.32	even	12
2700.3.u.c.449.6		24	9.5	odd	6	inner
2700.3.u.c.449.7		24	45.14	odd	6	inner
2700.3.u.c.2249.6		24	5.4	even	2	inner
2700.3.u.c.2249.7		24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2700.u (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.02985 + 0.594587i) q^{7} +O(q^{10})\)	\(q+(1.02985 + 0.594587i) q^{7} +(-3.63473 - 2.09851i) q^{11} +(12.3948 - 7.15614i) q^{13} +18.7074 q^{17} -33.8986 q^{19} +(-6.74989 - 11.6912i) q^{23} +(30.6315 + 17.6851i) q^{29} +(4.97816 + 8.62242i) q^{31} -19.3047i q^{37} +(55.9648 - 32.3113i) q^{41} +(-35.9872 - 20.7772i) q^{43} +(-33.6057 + 58.2068i) q^{47} +(-23.7929 - 41.2106i) q^{49} -30.0712 q^{53} +(3.66554 - 2.11630i) q^{59} +(43.8210 - 75.9002i) q^{61} +(-31.0106 + 17.9040i) q^{67} -24.5173i q^{71} -11.9305i q^{73} +(-2.49549 - 4.32232i) q^{77} +(-19.0858 + 33.0575i) q^{79} +(-12.0489 + 20.8692i) q^{83} +44.4494i q^{89} +17.0198 q^{91} +(-55.3688 - 31.9672i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 96 q^{11} - 144 q^{19} + 300 q^{29} - 24 q^{31} - 180 q^{41} - 96 q^{49} + 96 q^{59} - 156 q^{61} - 240 q^{79} + 240 q^{91}+O(q^{100}) \) 24 * q - 96 * q^11 - 144 * q^19 + 300 * q^29 - 24 * q^31 - 180 * q^41 - 96 * q^49 + 96 * q^59 - 156 * q^61 - 240 * q^79 + 240 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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