LMFDB - Embedded newform 1728.3.o.i.1279.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(127,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 4]))

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

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

chi := DirichletCharacter("1728.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.o (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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1279.6
Character	$\chi$	$=$	1728.1279
Dual form			1728.3.o.i.127.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.25241 + 2.16924i) q^{5} +(-0.842935 + 0.486669i) q^{7} +O(q^{10})$	$q+(-1.25241 + 2.16924i) q^{5} +(-0.842935 + 0.486669i) q^{7} +(-3.01493 + 1.74067i) q^{11} +(7.51352 - 13.0138i) q^{13} -19.3227 q^{17} -13.9609i q^{19} +(33.5924 + 19.3946i) q^{23} +(9.36292 + 16.2170i) q^{25} +(6.95537 + 12.0470i) q^{29} +(-12.1511 - 7.01542i) q^{31} -2.43804i q^{35} -28.1331 q^{37} +(-18.4281 + 31.9183i) q^{41} +(-36.2892 + 20.9516i) q^{43} +(10.7892 - 6.22913i) q^{47} +(-24.0263 + 41.6148i) q^{49} +16.6482 q^{53} -8.72017i q^{55} +(-11.3836 - 6.57234i) q^{59} +(-41.5302 - 71.9324i) q^{61} +(18.8201 + 32.5973i) q^{65} +(-95.1188 - 54.9169i) q^{67} +96.4767i q^{71} -40.1094 q^{73} +(1.69426 - 2.93455i) q^{77} +(117.038 - 67.5719i) q^{79} +(52.4342 - 30.2729i) q^{83} +(24.2000 - 41.9156i) q^{85} -138.741 q^{89} +14.6264i q^{91} +(30.2846 + 17.4848i) q^{95} +(-79.4667 - 137.640i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 24 q^{13} + 24 q^{17} - 108 q^{25} - 24 q^{29} - 96 q^{37} + 60 q^{41} - 132 q^{49} + 96 q^{53} + 336 q^{61} - 216 q^{65} + 696 q^{73} - 24 q^{77} + 528 q^{85} + 240 q^{89} - 444 q^{97}+O(q^{100}) $ 24 * q + 24 * q^13 + 24 * q^17 - 108 * q^25 - 24 * q^29 - 96 * q^37 + 60 * q^41 - 132 * q^49 + 96 * q^53 + 336 * q^61 - 216 * q^65 + 696 * q^73 - 24 * q^77 + 528 * q^85 + 240 * q^89 - 444 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{3}\right)$

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
$5$	−1.25241	+	2.16924i	−0.250483	+	0.433849i	−0.963659	−	0.267136i	$-0.913923\pi$
$5$	−1.25241	+	2.16924i	−0.250483	+	0.433849i	0.713176	+	0.700985i	$0.247256\pi$
$6$		0			0
$7$	−0.842935	+	0.486669i	−0.120419	+	0.0695241i	−0.559000	−	0.829168i	$-0.688815\pi$
$7$	−0.842935	+	0.486669i	−0.120419	+	0.0695241i	0.438580	+	0.898692i	$0.355481\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.01493	+	1.74067i	−0.274085	+	0.158243i	−0.630742	−	0.775992i	$-0.717250\pi$
$11$	−3.01493	+	1.74067i	−0.274085	+	0.158243i	0.356658	+	0.934235i	$0.383916\pi$
$12$		0			0
$13$	7.51352	−	13.0138i	0.577963	−	1.00106i	−0.417750	−	0.908562i	$-0.637181\pi$
$13$	7.51352	−	13.0138i	0.577963	−	1.00106i	0.995713	−	0.0924992i	$-0.0294856\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−19.3227			−1.13663			−0.568314	−	0.822812i	$-0.692404\pi$
$17$	−19.3227			−1.13663			−0.568314	+	0.822812i	$0.692404\pi$
$18$		0			0
$19$		−	13.9609i		−	0.734783i	−0.930066	−	0.367392i	$-0.880251\pi$
$19$		−	13.9609i		−	0.734783i	0.930066	−	0.367392i	$-0.119749\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	33.5924	+	19.3946i	1.46054	+	0.843243i	0.999036	−	0.0438947i	$-0.0139766\pi$
$23$	33.5924	+	19.3946i	1.46054	+	0.843243i	0.461504	+	0.887138i	$0.347310\pi$
$24$		0			0
$25$	9.36292	+	16.2170i	0.374517	+	0.648682i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.95537	+	12.0470i	0.239840	+	0.415415i	0.960668	−	0.277698i	$-0.0895716\pi$
$29$	6.95537	+	12.0470i	0.239840	+	0.415415i	−0.720828	+	0.693114i	$0.756238\pi$
$30$		0			0
$31$	−12.1511	−	7.01542i	−0.391970	−	0.226304i	0.291043	−	0.956710i	$-0.405998\pi$
$31$	−12.1511	−	7.01542i	−0.391970	−	0.226304i	−0.683013	+	0.730406i	$0.739331\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	2.43804i		−	0.0696584i
$36$		0			0
$37$	−28.1331			−0.760354			−0.380177	−	0.924914i	$-0.624137\pi$
$37$	−28.1331			−0.760354			−0.380177	+	0.924914i	$0.624137\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−18.4281	+	31.9183i	−0.449465	+	0.778496i	−0.998351	−	0.0574009i	$-0.981719\pi$
$41$	−18.4281	+	31.9183i	−0.449465	+	0.778496i	0.548886	+	0.835897i	$0.315052\pi$
$42$		0			0
$43$	−36.2892	+	20.9516i	−0.843935	+	0.487246i	−0.858600	−	0.512646i	$-0.828665\pi$
$43$	−36.2892	+	20.9516i	−0.843935	+	0.487246i	0.0146648	+	0.999892i	$0.495332\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	10.7892	−	6.22913i	0.229557	−	0.132535i	−0.380811	−	0.924653i	$-0.624355\pi$
$47$	10.7892	−	6.22913i	0.229557	−	0.132535i	0.610368	+	0.792118i	$0.291022\pi$
$48$		0			0
$49$	−24.0263	+	41.6148i	−0.490333	+	0.849281i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	16.6482			0.314116			0.157058	−	0.987589i	$-0.449799\pi$
$53$	16.6482			0.314116			0.157058	+	0.987589i	$0.449799\pi$
$54$		0			0
$55$		−	8.72017i		−	0.158548i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.3836	−	6.57234i	−0.192943	−	0.111396i	0.400417	−	0.916333i	$-0.368865\pi$
$59$	−11.3836	−	6.57234i	−0.192943	−	0.111396i	−0.593359	+	0.804938i	$0.702199\pi$
$60$		0			0
$61$	−41.5302	−	71.9324i	−0.680823	−	1.17922i	−0.974730	−	0.223386i	$-0.928289\pi$
$61$	−41.5302	−	71.9324i	−0.680823	−	1.17922i	0.293907	−	0.955834i	$-0.405044\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	18.8201	+	32.5973i	0.289540	+	0.501497i
$66$		0			0
$67$	−95.1188	−	54.9169i	−1.41968	−	0.819655i	−0.423413	−	0.905937i	$-0.639168\pi$
$67$	−95.1188	−	54.9169i	−1.41968	−	0.819655i	−0.996271	+	0.0862815i	$0.972502\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			96.4767i			1.35883i	0.733756	+	0.679413i	$0.237766\pi$
$71$			96.4767i			1.35883i	−0.733756	+	0.679413i	$0.762234\pi$
$72$		0			0
$73$	−40.1094			−0.549444			−0.274722	−	0.961524i	$-0.588586\pi$
$73$	−40.1094			−0.549444			−0.274722	+	0.961524i	$0.588586\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.69426	−	2.93455i	0.0220034	−	0.0381110i
$78$		0			0
$79$	117.038	−	67.5719i	1.48149	−	0.855340i	0.481713	−	0.876329i	$-0.340015\pi$
$79$	117.038	−	67.5719i	1.48149	−	0.855340i	0.999780	+	0.0209890i	$0.00668151\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	52.4342	−	30.2729i	0.631738	−	0.364734i	−0.149687	−	0.988733i	$-0.547827\pi$
$83$	52.4342	−	30.2729i	0.631738	−	0.364734i	0.781425	+	0.624000i	$0.214493\pi$
$84$		0			0
$85$	24.2000	−	41.9156i	0.284706	−	0.493125i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−138.741			−1.55889			−0.779444	−	0.626472i	$-0.784498\pi$
$89$	−138.741			−1.55889			−0.779444	+	0.626472i	$0.784498\pi$
$90$		0			0
$91$			14.6264i			0.160729i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	30.2846	+	17.4848i	0.318785	+	0.184051i
$96$		0			0
$97$	−79.4667	−	137.640i	−0.819244	−	1.41897i	−0.906240	−	0.422764i	$-0.861060\pi$
$97$	−79.4667	−	137.640i	−0.819244	−	1.41897i	0.0869954	−	0.996209i	$-0.472273\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−51.6687	−	89.4929i	−0.511572	−	0.886068i	−0.999910	−	0.0134136i	$-0.995730\pi$
$101$	−51.6687	−	89.4929i	−0.511572	−	0.886068i	0.488338	−	0.872654i	$-0.337603\pi$
$102$		0			0
$103$	−16.5726	−	9.56821i	−0.160899	−	0.0928952i	0.417388	−	0.908728i	$-0.362946\pi$
$103$	−16.5726	−	9.56821i	−0.160899	−	0.0928952i	−0.578288	+	0.815833i	$0.696279\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	14.1915i		−	0.132630i	−0.997799	−	0.0663152i	$-0.978876\pi$
$107$		−	14.1915i		−	0.132630i	0.997799	−	0.0663152i	$-0.0211243\pi$
$108$		0			0
$109$	−168.973			−1.55021			−0.775104	−	0.631833i	$-0.782303\pi$
$109$	−168.973			−1.55021			−0.775104	+	0.631833i	$0.782303\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−109.790	+	190.163i	−0.971597	+	1.68286i	−0.280860	+	0.959749i	$0.590620\pi$
$113$	−109.790	+	190.163i	−0.971597	+	1.68286i	−0.690737	+	0.723107i	$0.742714\pi$
$114$		0			0
$115$	−84.1433	+	48.5801i	−0.731681	+	0.422436i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	16.2878	−	9.40374i	0.136872	−	0.0790231i
$120$		0			0
$121$	−54.4401	+	94.2931i	−0.449918	+	0.779282i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−109.526			−0.876206
$126$		0			0
$127$			158.301i			1.24647i	0.782036	+	0.623233i	$0.214181\pi$
$127$			158.301i			1.24647i	−0.782036	+	0.623233i	$0.785819\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−154.026	−	88.9267i	−1.17577	−	0.678830i	−0.220736	−	0.975334i	$-0.570846\pi$
$131$	−154.026	−	88.9267i	−1.17577	−	0.678830i	−0.955032	+	0.296504i	$0.904179\pi$
$132$		0			0
$133$	6.79433	+	11.7681i	0.0510852	+	0.0884821i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−31.8904	−	55.2358i	−0.232776	−	0.403181i	0.725848	−	0.687855i	$-0.241448\pi$
$137$	−31.8904	−	55.2358i	−0.232776	−	0.403181i	−0.958624	+	0.284675i	$0.908114\pi$
$138$		0			0
$139$	140.171	+	80.9275i	1.00842	+	0.582212i	0.910729	−	0.413005i	$-0.135521\pi$
$139$	140.171	+	80.9275i	1.00842	+	0.582212i	0.0976920	+	0.995217i	$0.468854\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			52.3143i			0.365834i
$144$		0			0
$145$	−34.8440			−0.240303
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−46.4133	+	80.3902i	−0.311499	+	0.539532i	−0.978687	−	0.205357i	$-0.934164\pi$
$149$	−46.4133	+	80.3902i	−0.311499	+	0.539532i	0.667188	+	0.744889i	$0.267498\pi$
$150$		0			0
$151$	85.2221	−	49.2030i	0.564385	−	0.325848i	−0.190519	−	0.981684i	$-0.561017\pi$
$151$	85.2221	−	49.2030i	0.564385	−	0.325848i	0.754903	+	0.655836i	$0.227684\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	30.4363	−	17.5724i	0.196363	−	0.113370i
$156$		0			0
$157$	76.1683	−	131.927i	0.485149	−	0.840302i	−0.514706	−	0.857367i	$-0.672099\pi$
$157$	76.1683	−	131.927i	0.485149	−	0.840302i	0.999854	+	0.0170647i	$0.00543213\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−37.7550			−0.234503
$162$		0			0
$163$		−	107.979i		−	0.662450i	−0.943552	−	0.331225i	$-0.892538\pi$
$163$		−	107.979i		−	0.662450i	0.943552	−	0.331225i	$-0.107462\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−268.150	−	154.816i	−1.60569	−	0.927043i	−0.990320	−	0.138802i	$-0.955675\pi$
$167$	−268.150	−	154.816i	−1.60569	−	0.927043i	−0.615366	−	0.788241i	$-0.710992\pi$
$168$		0			0
$169$	−28.4060	−	49.2006i	−0.168083	−	0.291128i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	137.931	+	238.903i	0.797288	+	1.38094i	0.921376	+	0.388672i	$0.127066\pi$
$173$	137.931	+	238.903i	0.797288	+	1.38094i	−0.124089	+	0.992271i	$0.539601\pi$
$174$		0			0
$175$	−15.7847	−	9.11328i	−0.0901981	−	0.0520759i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			66.7842i			0.373096i	0.982446	+	0.186548i	$0.0597300\pi$
$179$			66.7842i			0.373096i	−0.982446	+	0.186548i	$0.940270\pi$
$180$		0			0
$181$	−217.288			−1.20049			−0.600244	−	0.799817i	$-0.704930\pi$
$181$	−217.288			−1.20049			−0.600244	+	0.799817i	$0.704930\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	35.2343	−	61.0276i	0.190456	−	0.329879i
$186$		0			0
$187$	58.2565	−	33.6344i	0.311532	−	0.179863i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	217.435	−	125.536i	1.13841	−	0.657258i	0.192370	−	0.981322i	$-0.438383\pi$
$191$	217.435	−	125.536i	1.13841	−	0.657258i	0.946035	+	0.324064i	$0.105049\pi$
$192$		0			0
$193$	−20.4473	+	35.4158i	−0.105945	+	0.183501i	−0.914124	−	0.405435i	$-0.867120\pi$
$193$	−20.4473	+	35.4158i	−0.105945	+	0.183501i	0.808179	+	0.588937i	$0.200453\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−189.163			−0.960219			−0.480109	−	0.877209i	$-0.659403\pi$
$197$	−189.163			−0.960219			−0.480109	+	0.877209i	$0.659403\pi$
$198$		0			0
$199$			76.3625i			0.383731i	0.981421	+	0.191866i	$0.0614538\pi$
$199$			76.3625i			0.383731i	−0.981421	+	0.191866i	$0.938546\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−11.7258	−	6.76992i	−0.0577628	−	0.0333494i
$204$		0			0
$205$	−46.1591	−	79.9500i	−0.225167	−	0.390000i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	24.3013	+	42.0911i	0.116274	+	0.201393i
$210$		0			0
$211$	301.301	+	173.956i	1.42797	+	0.824436i	0.996960	−	0.0779128i	$-0.0248256\pi$
$211$	301.301	+	173.956i	1.42797	+	0.824436i	0.431006	+	0.902349i	$0.358159\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	104.960i		−	0.488187i
$216$		0			0
$217$	13.6567			0.0629343
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−145.181	+	251.461i	−0.656929	+	1.13783i
$222$		0			0
$223$	−160.807	+	92.8418i	−0.721106	+	0.416331i	−0.815160	−	0.579236i	$-0.803351\pi$
$223$	−160.807	+	92.8418i	−0.721106	+	0.416331i	0.0940534	+	0.995567i	$0.470018\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−123.810	+	71.4817i	−0.545418	+	0.314897i	−0.747272	−	0.664518i	$-0.768637\pi$
$227$	−123.810	+	71.4817i	−0.545418	+	0.314897i	0.201854	+	0.979416i	$0.435303\pi$
$228$		0			0
$229$	−89.9704	+	155.833i	−0.392884	+	0.680495i	−0.992829	−	0.119547i	$-0.961856\pi$
$229$	−89.9704	+	155.833i	−0.392884	+	0.680495i	0.599945	+	0.800041i	$0.295189\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.00836			−0.0129114			−0.00645570	−	0.999979i	$-0.502055\pi$
$233$	−3.00836			−0.0129114			−0.00645570	+	0.999979i	$0.502055\pi$
$234$		0			0
$235$			31.2058i			0.132791i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−91.9748	−	53.1017i	−0.384832	−	0.222183i	0.295087	−	0.955470i	$-0.404651\pi$
$239$	−91.9748	−	53.1017i	−0.384832	−	0.222183i	−0.679918	+	0.733288i	$0.737985\pi$
$240$		0			0
$241$	−218.501	−	378.454i	−0.906642	−	1.57035i	−0.818698	−	0.574225i	$-0.805303\pi$
$241$	−218.501	−	378.454i	−0.906642	−	1.57035i	−0.0879447	−	0.996125i	$-0.528030\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−60.1818	−	104.238i	−0.245640	−	0.425461i
$246$		0			0
$247$	−181.684	−	104.895i	−0.735563	−	0.424678i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			224.126i			0.892931i	0.894801	+	0.446466i	$0.147318\pi$
$251$			224.126i			0.892931i	−0.894801	+	0.446466i	$0.852682\pi$
$252$		0			0
$253$	−135.038			−0.533749
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.46397	−	16.3921i	0.0368248	−	0.0637824i	−0.847026	−	0.531552i	$-0.821609\pi$
$257$	9.46397	−	16.3921i	0.0368248	−	0.0637824i	0.883850	+	0.467770i	$0.154942\pi$
$258$		0			0
$259$	23.7144	−	13.6915i	0.0915613	−	0.0528630i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−185.185	+	106.917i	−0.704126	+	0.406527i	−0.808882	−	0.587971i	$-0.799927\pi$
$263$	−185.185	+	106.917i	−0.704126	+	0.406527i	0.104756	+	0.994498i	$0.466594\pi$
$264$		0			0
$265$	−20.8504	+	36.1140i	−0.0786808	+	0.136279i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−73.4182			−0.272930			−0.136465	−	0.990645i	$-0.543574\pi$
$269$	−73.4182			−0.272930			−0.136465	+	0.990645i	$0.543574\pi$
$270$		0			0
$271$			95.2505i			0.351478i	0.984437	+	0.175739i	$0.0562315\pi$
$271$			95.2505i			0.351478i	−0.984437	+	0.175739i	$0.943769\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−56.4571	−	32.5955i	−0.205299	−	0.118529i
$276$		0			0
$277$	166.006	+	287.531i	0.599299	+	1.03802i	0.992925	+	0.118746i	$0.0378874\pi$
$277$	166.006	+	287.531i	0.599299	+	1.03802i	−0.393625	+	0.919271i	$0.628779\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−37.0286	−	64.1353i	−0.131774	−	0.228240i	0.792586	−	0.609760i	$-0.208734\pi$
$281$	−37.0286	−	64.1353i	−0.131774	−	0.228240i	−0.924361	+	0.381520i	$0.875401\pi$
$282$		0			0
$283$	179.846	+	103.834i	0.635499	+	0.366905i	0.782879	−	0.622175i	$-0.213751\pi$
$283$	179.846	+	103.834i	0.635499	+	0.366905i	−0.147380	+	0.989080i	$0.547084\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	35.8735i		−	0.124995i
$288$		0			0
$289$	84.3657			0.291923
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−87.3353	+	151.269i	−0.298073	+	0.516277i	−0.975695	−	0.219133i	$-0.929677\pi$
$293$	−87.3353	+	151.269i	−0.298073	+	0.516277i	0.677622	+	0.735410i	$0.263010\pi$
$294$		0			0
$295$	28.5140	−	16.4626i	0.0966577	−	0.0558054i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	504.795	−	291.443i	1.68828	−	0.974727i
$300$		0			0
$301$	20.3930	−	35.3217i	0.0677507	−	0.117348i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	208.052			0.682138
$306$		0			0
$307$		−	485.954i		−	1.58291i	−0.611227	−	0.791456i	$-0.709324\pi$
$307$		−	485.954i		−	1.58291i	0.611227	−	0.791456i	$-0.290676\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−220.636	−	127.384i	−0.709440	−	0.409595i	0.101414	−	0.994844i	$-0.467663\pi$
$311$	−220.636	−	127.384i	−0.709440	−	0.409595i	−0.810853	+	0.585249i	$0.800997\pi$
$312$		0			0
$313$	−71.0799	−	123.114i	−0.227092	−	0.393336i	0.729853	−	0.683604i	$-0.239589\pi$
$313$	−71.0799	−	123.114i	−0.227092	−	0.393336i	−0.956945	+	0.290269i	$0.906255\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−58.8218	−	101.882i	−0.185558	−	0.321395i	0.758207	−	0.652014i	$-0.226076\pi$
$317$	−58.8218	−	101.882i	−0.185558	−	0.321395i	−0.943764	+	0.330619i	$0.892742\pi$
$318$		0			0
$319$	−41.9399	−	24.2140i	−0.131473	−	0.0759060i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			269.762i			0.835175i
$324$		0			0
$325$	281.394			0.865827
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.06305	+	10.5015i	−0.0184287	+	0.0319195i
$330$		0			0
$331$	−190.553	+	110.016i	−0.575690	+	0.332375i	−0.759419	−	0.650602i	$-0.774516\pi$
$331$	−190.553	+	110.016i	−0.575690	+	0.332375i	0.183729	+	0.982977i	$0.441183\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	238.256	−	137.557i	0.711213	−	0.410619i
$336$		0			0
$337$	110.267	−	190.989i	0.327203	−	0.566733i	−0.654753	−	0.755843i	$-0.727227\pi$
$337$	110.267	−	190.989i	0.327203	−	0.566733i	0.981956	+	0.189111i	$0.0605605\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	48.8462			0.143244
$342$		0			0
$343$		−	94.4650i		−	0.275408i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	360.081	+	207.893i	1.03770	+	0.599115i	0.919180	−	0.393837i	$-0.128852\pi$
$347$	360.081	+	207.893i	1.03770	+	0.599115i	0.118518	+	0.992952i	$0.462186\pi$
$348$		0			0
$349$	296.154	+	512.954i	0.848579	+	1.46978i	0.882477	+	0.470356i	$0.155875\pi$
$349$	296.154	+	512.954i	0.848579	+	1.46978i	−0.0338977	+	0.999425i	$0.510792\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	144.258	+	249.862i	0.408663	+	0.707826i	0.994740	−	0.102430i	$-0.0326617\pi$
$353$	144.258	+	249.862i	0.408663	+	0.707826i	−0.586077	+	0.810255i	$0.699328\pi$
$354$		0			0
$355$	−209.281	−	120.829i	−0.589525	−	0.340363i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			317.550i			0.884541i	0.896882	+	0.442270i	$0.145827\pi$
$359$			317.550i			0.884541i	−0.896882	+	0.442270i	$0.854173\pi$
$360$		0			0
$361$	166.094			0.460094
$362$		0			0
$363$		0			0
$364$		0			0
$365$	50.2336	−	87.0072i	0.137626	−	0.238376i
$366$		0			0
$367$	154.752	−	89.3462i	0.421668	−	0.243450i	−0.274123	−	0.961695i	$-0.588387\pi$
$367$	154.752	−	89.3462i	0.421668	−	0.243450i	0.695791	+	0.718245i	$0.255054\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−14.0333	+	8.10215i	−0.0378257	+	0.0218387i
$372$		0			0
$373$	11.3286	−	19.6218i	0.0303717	−	0.0526053i	−0.850440	−	0.526072i	$-0.823664\pi$
$373$	11.3286	−	19.6218i	0.0303717	−	0.0526053i	0.880812	+	0.473467i	$0.156998\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	209.037			0.554475
$378$		0			0
$379$			156.270i			0.412321i	0.978518	+	0.206160i	$0.0660969\pi$
$379$			156.270i			0.412321i	−0.978518	+	0.206160i	$0.933903\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−28.7614	−	16.6054i	−0.0750951	−	0.0433562i	0.461982	−	0.886889i	$-0.347138\pi$
$383$	−28.7614	−	16.6054i	−0.0750951	−	0.0433562i	−0.537077	+	0.843533i	$0.680472\pi$
$384$		0			0
$385$	4.24383	+	7.35054i	0.0110229	+	0.0190923i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−85.0289	−	147.274i	−0.218583	−	0.378598i	0.735792	−	0.677208i	$-0.236810\pi$
$389$	−85.0289	−	147.274i	−0.218583	−	0.378598i	−0.954375	+	0.298610i	$0.903477\pi$
$390$		0			0
$391$	−649.095	−	374.755i	−1.66009	−	0.958454i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			338.512i			0.856992i
$396$		0			0
$397$	−340.573			−0.857866			−0.428933	−	0.903336i	$-0.641110\pi$
$397$	−340.573			−0.857866			−0.428933	+	0.903336i	$0.641110\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	236.842	−	410.223i	0.590630	−	1.02300i	−0.403518	−	0.914972i	$-0.632213\pi$
$401$	236.842	−	410.223i	0.590630	−	1.02300i	0.994148	−	0.108029i	$-0.0344539\pi$
$402$		0			0
$403$	−182.595	+	105.421i	−0.453088	+	0.261591i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	84.8194	−	48.9705i	0.208401	−	0.120321i
$408$		0			0
$409$	20.6558	−	35.7768i	0.0505031	−	0.0874740i	−0.839669	−	0.543099i	$-0.817251\pi$
$409$	20.6558	−	35.7768i	0.0505031	−	0.0874740i	0.890172	+	0.455625i	$0.150584\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.7942			0.0309787
$414$		0			0
$415$			151.657i			0.365438i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−431.309	−	249.016i	−1.02938	−	0.594310i	−0.112570	−	0.993644i	$-0.535908\pi$
$419$	−431.309	−	249.016i	−1.02938	−	0.594310i	−0.916806	+	0.399333i	$0.869242\pi$
$420$		0			0
$421$	−61.3780	−	106.310i	−0.145791	−	0.252517i	0.783877	−	0.620916i	$-0.213239\pi$
$421$	−61.3780	−	106.310i	−0.145791	−	0.252517i	−0.929668	+	0.368399i	$0.879906\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−180.917	−	313.357i	−0.425686	−	0.737310i
$426$		0			0
$427$	70.0145	+	40.4229i	0.163968	+	0.0946672i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			512.109i			1.18819i	0.804395	+	0.594094i	$0.202489\pi$
$431$			512.109i			1.18819i	−0.804395	+	0.594094i	$0.797511\pi$
$432$		0			0
$433$	35.8368			0.0827640			0.0413820	−	0.999143i	$-0.486824\pi$
$433$	35.8368			0.0827640			0.0413820	+	0.999143i	$0.486824\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	270.766	−	468.980i	0.619601	−	1.07318i
$438$		0			0
$439$	−215.796	+	124.590i	−0.491562	+	0.283803i	−0.725222	−	0.688515i	$-0.758263\pi$
$439$	−215.796	+	124.590i	−0.491562	+	0.283803i	0.233660	+	0.972318i	$0.424930\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	477.985	−	275.965i	1.07897	−	0.622946i	0.148354	−	0.988934i	$-0.452602\pi$
$443$	477.985	−	275.965i	1.07897	−	0.622946i	0.930619	+	0.365989i	$0.119269\pi$
$444$		0			0
$445$	173.761	−	300.963i	0.390475	−	0.676322i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−344.049			−0.766256			−0.383128	−	0.923695i	$-0.625153\pi$
$449$	−344.049			−0.766256			−0.383128	+	0.923695i	$0.625153\pi$
$450$		0			0
$451$		−	128.309i		−	0.284499i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−31.7282	−	18.3183i	−0.0697323	−	0.0402600i
$456$		0			0
$457$	193.531	+	335.206i	0.423481	+	0.733491i	0.996277	−	0.0862068i	$-0.0274746\pi$
$457$	193.531	+	335.206i	0.423481	+	0.733491i	−0.572796	+	0.819698i	$0.694141\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	46.4011	+	80.3691i	0.100653	+	0.174337i	0.911954	−	0.410293i	$-0.134573\pi$
$461$	46.4011	+	80.3691i	0.100653	+	0.174337i	−0.811301	+	0.584629i	$0.801240\pi$
$462$		0			0
$463$	639.918	+	369.457i	1.38211	+	0.797962i	0.992409	−	0.122979i	$-0.0392447\pi$
$463$	639.918	+	369.457i	1.38211	+	0.797962i	0.389702	+	0.920941i	$0.372578\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			423.746i			0.907379i	0.891160	+	0.453689i	$0.149892\pi$
$467$			423.746i			0.907379i	−0.891160	+	0.453689i	$0.850108\pi$
$468$		0			0
$469$	106.905			0.227943
$470$		0			0
$471$		0			0
$472$		0			0
$473$	72.9397	−	126.335i	0.154206	−	0.267093i
$474$		0			0
$475$	226.404	−	130.715i	0.476641	−	0.275189i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	652.039	−	376.455i	1.36125	−	0.785918i	0.371459	−	0.928449i	$-0.378857\pi$
$479$	652.039	−	376.455i	1.36125	−	0.785918i	0.989790	+	0.142531i	$0.0455241\pi$
$480$		0			0
$481$	−211.379	+	366.118i	−0.439457	+	0.761161i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	398.101			0.820827
$486$		0			0
$487$		−	703.286i		−	1.44412i	−0.691831	−	0.722060i	$-0.743195\pi$
$487$		−	703.286i		−	1.44412i	0.691831	−	0.722060i	$-0.256805\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−364.667	−	210.540i	−0.742702	−	0.428799i	0.0803490	−	0.996767i	$-0.474397\pi$
$491$	−364.667	−	210.540i	−0.742702	−	0.428799i	−0.823051	+	0.567968i	$0.807730\pi$
$492$		0			0
$493$	−134.396	−	232.781i	−0.272609	−	0.472173i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−46.9522	−	81.3236i	−0.0944712	−	0.163629i
$498$		0			0
$499$	−587.432	−	339.154i	−1.17722	−	0.679668i	−0.221850	−	0.975081i	$-0.571209\pi$
$499$	−587.432	−	339.154i	−1.17722	−	0.679668i	−0.955370	+	0.295413i	$0.904543\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			0.891657i			0.00177268i	1.00000		0.000886339i	$0.000282130\pi$
$503$			0.891657i			0.00177268i	−1.00000		0.000886339i	$0.999718\pi$
$504$		0			0
$505$	258.843			0.512560
$506$		0			0
$507$		0			0
$508$		0			0
$509$	287.777	−	498.444i	0.565377	−	0.979261i	−0.431638	−	0.902047i	$-0.642064\pi$
$509$	287.777	−	498.444i	0.565377	−	0.979261i	0.997015	−	0.0772141i	$-0.0246025\pi$
$510$		0			0
$511$	33.8097	−	19.5200i	0.0661637	−	0.0381996i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	41.5116	−	23.9667i	0.0806050	−	0.0465373i
$516$		0			0
$517$	−21.6858	+	37.5608i	−0.0419454	+	0.0726515i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	735.526			1.41176			0.705879	−	0.708332i	$-0.250552\pi$
$521$	735.526			1.41176			0.705879	+	0.708332i	$0.250552\pi$
$522$		0			0
$523$			112.072i			0.214286i	0.994244	+	0.107143i	$0.0341703\pi$
$523$			112.072i			0.214286i	−0.994244	+	0.107143i	$0.965830\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	234.791	+	135.557i	0.445524	+	0.257223i
$528$		0			0
$529$	487.801	+	844.896i	0.922119	+	1.59716i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	276.919	+	479.638i	0.519548	+	0.899884i
$534$		0			0
$535$	30.7847	+	17.7736i	0.0575416	+	0.0332216i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	167.288i		−	0.310367i
$540$		0			0
$541$	878.760			1.62433			0.812163	−	0.583431i	$-0.198290\pi$
$541$	878.760			1.62433			0.812163	+	0.583431i	$0.198290\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	211.624	−	366.543i	0.388301	−	0.672557i
$546$		0			0
$547$	−101.275	+	58.4709i	−0.185146	+	0.106894i	−0.589708	−	0.807617i	$-0.700757\pi$
$547$	−101.275	+	58.4709i	−0.185146	+	0.106894i	0.404562	+	0.914510i	$0.367424\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	168.187	−	97.1030i	0.305240	−	0.176231i
$552$		0			0
$553$	−65.7702	+	113.917i	−0.118934	+	0.205999i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	983.235			1.76523			0.882616	−	0.470094i	$-0.155780\pi$
$557$	983.235			1.76523			0.882616	+	0.470094i	$0.155780\pi$
$558$		0			0
$559$			629.681i			1.12644i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−721.045	−	416.296i	−1.28072	−	0.739424i	−0.303740	−	0.952755i	$-0.598235\pi$
$563$	−721.045	−	416.296i	−1.28072	−	0.739424i	−0.976980	+	0.213331i	$0.931569\pi$
$564$		0			0
$565$	−275.006	−	476.325i	−0.486737	−	0.843053i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	151.574	+	262.534i	0.266386	+	0.461395i	0.967926	−	0.251236i	$-0.0808370\pi$
$569$	151.574	+	262.534i	0.266386	+	0.461395i	−0.701540	+	0.712631i	$0.747504\pi$
$570$		0			0
$571$	630.357	+	363.937i	1.10395	+	0.637368i	0.937257	−	0.348640i	$-0.113357\pi$
$571$	630.357	+	363.937i	1.10395	+	0.637368i	0.166697	+	0.986008i	$0.446690\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			726.360i			1.26323i
$576$		0			0
$577$	694.935			1.20439			0.602197	−	0.798348i	$-0.294292\pi$
$577$	694.935			1.20439			0.602197	+	0.798348i	$0.294292\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−29.4658	+	51.0362i	−0.0507156	+	0.0878420i
$582$		0			0
$583$	−50.1931	+	28.9790i	−0.0860945	+	0.0497067i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.3821	+	12.9223i	−0.0381297	+	0.0220142i	−0.518944	−	0.854808i	$-0.673675\pi$
$587$	−22.3821	+	12.9223i	−0.0381297	+	0.0220142i	0.480814	+	0.876823i	$0.340341\pi$
$588$		0			0
$589$	−97.9415	+	169.640i	−0.166284	+	0.288013i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−441.443			−0.744424			−0.372212	−	0.928148i	$-0.621400\pi$
$593$	−441.443			−0.744424			−0.372212	+	0.928148i	$0.621400\pi$
$594$		0			0
$595$			47.1095i			0.0791757i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−700.138	−	404.225i	−1.16884	−	0.674833i	−0.215437	−	0.976518i	$-0.569118\pi$
$599$	−700.138	−	404.225i	−1.16884	−	0.674833i	−0.953408	+	0.301685i	$0.902451\pi$
$600$		0			0
$601$	239.504	+	414.833i	0.398509	+	0.690238i	0.993542	−	0.113463i	$-0.0361944\pi$
$601$	239.504	+	414.833i	0.398509	+	0.690238i	−0.595033	+	0.803701i	$0.702861\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−136.363	−	236.188i	−0.225394	−	0.390393i
$606$		0			0
$607$	797.010	+	460.154i	1.31303	+	0.758079i	0.982597	−	0.185750i	$-0.0594714\pi$
$607$	797.010	+	460.154i	1.31303	+	0.758079i	0.330434	+	0.943829i	$0.392805\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	187.211i		−	0.306401i
$612$		0			0
$613$	931.251			1.51917			0.759585	−	0.650408i	$-0.225402\pi$
$613$	931.251			1.51917			0.759585	+	0.650408i	$0.225402\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−480.293	+	831.891i	−0.778432	+	1.34828i	0.154413	+	0.988006i	$0.450651\pi$
$617$	−480.293	+	831.891i	−0.778432	+	1.34828i	−0.932845	+	0.360278i	$0.882682\pi$
$618$		0			0
$619$	−248.873	+	143.687i	−0.402057	+	0.232128i	−0.687371	−	0.726306i	$-0.741235\pi$
$619$	−248.873	+	143.687i	−0.402057	+	0.232128i	0.285314	+	0.958434i	$0.407902\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	116.950	−	67.5210i	0.187720	−	0.108380i
$624$		0			0
$625$	−96.9014	+	167.838i	−0.155042	+	0.268541i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	543.607			0.864240
$630$		0			0
$631$		−	422.954i		−	0.670291i	−0.942166	−	0.335146i	$-0.891214\pi$
$631$		−	422.954i		−	0.670291i	0.942166	−	0.335146i	$-0.108786\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−343.394	−	198.259i	−0.540778	−	0.312218i
$636$		0			0
$637$	361.044	+	625.347i	0.566788	+	0.981706i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25.3956	+	43.9865i	0.0396188	+	0.0686217i	0.885155	−	0.465297i	$-0.154052\pi$
$641$	25.3956	+	43.9865i	0.0396188	+	0.0686217i	−0.845536	+	0.533918i	$0.820719\pi$
$642$		0			0
$643$	−722.106	−	416.908i	−1.12303	−	0.648380i	−0.180855	−	0.983510i	$-0.557887\pi$
$643$	−722.106	−	416.908i	−1.12303	−	0.648380i	−0.942172	+	0.335130i	$0.891220\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			826.902i			1.27806i	0.769184	+	0.639028i	$0.220663\pi$
$647$			826.902i			1.27806i	−0.769184	+	0.639028i	$0.779337\pi$
$648$		0			0
$649$	45.7611			0.0705102
$650$		0			0
$651$		0			0
$652$		0			0
$653$	587.256	−	1017.16i	0.899320	−	1.55767i	0.0709551	−	0.997480i	$-0.477395\pi$
$653$	587.256	−	1017.16i	0.899320	−	1.55767i	0.828365	−	0.560189i	$-0.189271\pi$
$654$		0			0
$655$	385.807	−	222.746i	0.589019	−	0.340070i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−575.835	+	332.458i	−0.873801	+	0.504489i	−0.868609	−	0.495497i	$-0.834986\pi$
$659$	−575.835	+	332.458i	−0.873801	+	0.504489i	−0.00519133	+	0.999987i	$0.501652\pi$
$660$		0			0
$661$	165.356	−	286.404i	0.250160	−	0.433290i	−0.713410	−	0.700747i	$-0.752850\pi$
$661$	165.356	−	286.404i	0.250160	−	0.433290i	0.963570	+	0.267457i	$0.0861835\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−34.0372			−0.0511838
$666$		0			0
$667$			539.586i			0.808975i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	250.421	+	144.581i	0.373206	+	0.215471i
$672$		0			0
$673$	256.452	+	444.187i	0.381057	+	0.660011i	0.991214	−	0.132270i	$-0.0422267\pi$
$673$	256.452	+	444.187i	0.381057	+	0.660011i	−0.610156	+	0.792281i	$0.708893\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−221.500	−	383.649i	−0.327179	−	0.566690i	0.654772	−	0.755826i	$-0.272765\pi$
$677$	−221.500	−	383.649i	−0.327179	−	0.566690i	−0.981951	+	0.189136i	$0.939431\pi$
$678$		0			0
$679$	133.971	+	77.3479i	0.197306	+	0.113914i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1159.07i		−	1.69703i	−0.529169	−	0.848516i	$-0.677496\pi$
$683$		−	1159.07i		−	1.69703i	0.529169	−	0.848516i	$-0.322504\pi$
$684$		0			0
$685$	159.760			0.233226
$686$		0			0
$687$		0			0
$688$		0			0
$689$	125.086	−	216.656i	0.181548	−	0.314450i
$690$		0			0
$691$	338.634	−	195.510i	0.490064	−	0.282938i	−0.234537	−	0.972107i	$-0.575357\pi$
$691$	338.634	−	195.510i	0.490064	−	0.282938i	0.724601	+	0.689169i	$0.242024\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−351.103	+	202.709i	−0.505184	+	0.291668i
$696$		0			0
$697$	356.079	−	616.748i	0.510874	−	0.884860i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1328.18			−1.89469			−0.947344	−	0.320217i	$-0.896244\pi$
$701$	−1328.18			−1.89469			−0.947344	+	0.320217i	$0.896244\pi$
$702$		0			0
$703$			392.763i			0.558695i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	87.1068	+	50.2911i	0.123206	+	0.0711331i
$708$		0			0
$709$	8.13480	+	14.0899i	0.0114736	+	0.0198729i	0.871705	−	0.490031i	$-0.163014\pi$
$709$	8.13480	+	14.0899i	0.0114736	+	0.0198729i	−0.860232	+	0.509904i	$0.829681\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−272.123	−	471.330i	−0.381659	−	0.661052i
$714$		0			0
$715$	−113.482	−	65.5191i	−0.158717	−	0.0916352i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1356.78i		−	1.88704i	−0.331316	−	0.943520i	$-0.607493\pi$
$719$		−	1356.78i		−	1.88704i	0.331316	−	0.943520i	$-0.392507\pi$
$720$		0			0
$721$	18.6262			0.0258338
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−130.245	+	225.591i	−0.179648	+	0.311160i
$726$		0			0
$727$	782.541	−	451.800i	1.07640	−	0.621458i	0.146475	−	0.989214i	$-0.453207\pi$
$727$	782.541	−	451.800i	1.07640	−	0.621458i	0.929922	+	0.367756i	$0.119874\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	701.204	−	404.841i	0.959240	−	0.553818i
$732$		0			0
$733$	−96.9820	+	167.978i	−0.132308	+	0.229165i	−0.924566	−	0.381022i	$-0.875572\pi$
$733$	−96.9820	+	167.978i	−0.132308	+	0.229165i	0.792258	+	0.610187i	$0.208906\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	382.369			0.518818
$738$		0			0
$739$			947.887i			1.28266i	0.767265	+	0.641331i	$0.221617\pi$
$739$			947.887i			1.28266i	−0.767265	+	0.641331i	$0.778383\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−612.087	−	353.389i	−0.823805	−	0.475624i	0.0279221	−	0.999610i	$-0.491111\pi$
$743$	−612.087	−	353.389i	−0.823805	−	0.475624i	−0.851727	+	0.523986i	$0.824444\pi$
$744$		0			0
$745$	−116.257	−	201.364i	−0.156050	−	0.270287i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.90654	+	11.9625i	0.00922101	+	0.0159713i
$750$		0			0
$751$	−23.5131	−	13.5753i	−0.0313091	−	0.0180763i	0.484264	−	0.874922i	$-0.339088\pi$
$751$	−23.5131	−	13.5753i	−0.0313091	−	0.0180763i	−0.515573	+	0.856846i	$0.672421\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			246.490i			0.326477i
$756$		0			0
$757$	−727.176			−0.960602			−0.480301	−	0.877104i	$-0.659473\pi$
$757$	−727.176			−0.960602			−0.480301	+	0.877104i	$0.659473\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−145.954	+	252.799i	−0.191792	+	0.332194i	−0.945844	−	0.324621i	$-0.894763\pi$
$761$	−145.954	+	252.799i	−0.191792	+	0.332194i	0.754052	+	0.656815i	$0.228097\pi$
$762$		0			0
$763$	142.433	−	82.2338i	0.186675	−	0.107777i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−171.062	+	98.7628i	−0.223028	+	0.128765i
$768$		0			0
$769$	232.025	−	401.879i	0.301723	−	0.522600i	−0.674803	−	0.737998i	$-0.735771\pi$
$769$	232.025	−	401.879i	0.301723	−	0.522600i	0.976526	+	0.215398i	$0.0691048\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	846.362			1.09491			0.547453	−	0.836837i	$-0.315598\pi$
$773$	846.362			1.09491			0.547453	+	0.836837i	$0.315598\pi$
$774$		0			0
$775$		−	262.739i		−	0.339018i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	445.608	+	257.272i	0.572026	+	0.330259i
$780$		0			0
$781$	−167.934	−	290.871i	−0.215025	−	0.372433i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	190.789	+	330.456i	0.243043	+	0.420963i
$786$		0			0
$787$	−558.431	−	322.410i	−0.709569	−	0.409670i	0.101332	−	0.994853i	$-0.467689\pi$
$787$	−558.431	−	322.410i	−0.709569	−	0.409670i	−0.810901	+	0.585183i	$0.801023\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	213.726i		−	0.270198i
$792$		0			0
$793$	−1248.15			−1.57396
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.26487	−	3.92287i	0.00284175	−	0.00492205i	−0.864601	−	0.502459i	$-0.832429\pi$
$797$	2.26487	−	3.92287i	0.00284175	−	0.00492205i	0.867443	+	0.497537i	$0.165762\pi$
$798$		0			0
$799$	−208.476	+	120.364i	−0.260921	+	0.150643i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	120.927	−	69.8174i	0.150594	−	0.0869456i
$804$		0			0
$805$	47.2849	−	81.8998i	0.0587390	−	0.101739i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	748.596			0.925335			0.462668	−	0.886532i	$-0.346892\pi$
$809$	748.596			0.925335			0.462668	+	0.886532i	$0.346892\pi$
$810$		0			0
$811$			817.917i			1.00853i	0.863549	+	0.504265i	$0.168236\pi$
$811$			817.917i			1.00853i	−0.863549	+	0.504265i	$0.831764\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	234.234	+	135.235i	0.287403	+	0.165932i
$816$		0			0
$817$	292.503	+	506.629i	0.358020	+	0.620109i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.9045	+	44.8679i	0.0315524	+	0.0546503i	0.881370	−	0.472426i	$-0.156622\pi$
$821$	25.9045	+	44.8679i	0.0315524	+	0.0546503i	−0.849818	+	0.527076i	$0.823288\pi$
$822$		0			0
$823$	653.931	+	377.547i	0.794570	+	0.458745i	0.841569	−	0.540150i	$-0.181632\pi$
$823$	653.931	+	377.547i	0.794570	+	0.458745i	−0.0469992	+	0.998895i	$0.514966\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1215.15i			1.46935i	0.678421	+	0.734673i	$0.262664\pi$
$827$			1215.15i			1.46935i	−0.678421	+	0.734673i	$0.737336\pi$
$828$		0			0
$829$	831.072			1.00250			0.501250	−	0.865303i	$-0.332874\pi$
$829$	831.072			1.00250			0.501250	+	0.865303i	$0.332874\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	464.252	−	804.109i	0.557326	−	0.965317i
$834$		0			0
$835$	671.669	−	387.788i	0.804394	−	0.464417i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−856.055	+	494.243i	−1.02033	+	0.589086i	−0.914199	−	0.405266i	$-0.867179\pi$
$839$	−856.055	+	494.243i	−1.02033	+	0.589086i	−0.106129	+	0.994352i	$0.533846\pi$
$840$		0			0
$841$	323.746	−	560.744i	0.384953	−	0.666759i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	142.304			0.168407
$846$		0			0
$847$		−	105.977i		−	0.125121i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−945.059	−	545.630i	−1.11053	−	0.641164i
$852$		0			0
$853$	−171.904	−	297.747i	−0.201529	−	0.349058i	0.747492	−	0.664270i	$-0.231258\pi$
$853$	−171.904	−	297.747i	−0.201529	−	0.349058i	−0.949021	+	0.315212i	$0.897924\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	278.107	+	481.696i	0.324513	+	0.562072i	0.981414	−	0.191904i	$-0.0614664\pi$
$857$	278.107	+	481.696i	0.324513	+	0.562072i	−0.656901	+	0.753977i	$0.728133\pi$
$858$		0			0
$859$	−227.344	−	131.257i	−0.264661	−	0.152802i	0.361798	−	0.932257i	$-0.382163\pi$
$859$	−227.344	−	131.257i	−0.264661	−	0.152802i	−0.626459	+	0.779455i	$0.715496\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			601.806i			0.697342i	0.937245	+	0.348671i	$0.113367\pi$
$863$			601.806i			0.697342i	−0.937245	+	0.348671i	$0.886633\pi$
$864$		0			0
$865$	−690.986			−0.798827
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−235.241	+	407.449i	−0.270703	+	0.468871i
$870$		0			0
$871$	−1429.35	+	825.238i	−1.64105	+	0.947461i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	92.3231	−	53.3027i	0.105512	−	0.0609174i
$876$		0			0
$877$	−803.626	+	1391.92i	−0.916335	+	1.58714i	−0.111399	+	0.993776i	$0.535533\pi$
$877$	−803.626	+	1391.92i	−0.916335	+	1.58714i	−0.804935	+	0.593363i	$0.797800\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−553.611			−0.628389			−0.314194	−	0.949359i	$-0.601734\pi$
$881$	−553.611			−0.628389			−0.314194	+	0.949359i	$0.601734\pi$
$882$		0			0
$883$		−	890.278i		−	1.00824i	−0.863633	−	0.504121i	$-0.831816\pi$
$883$		−	890.278i		−	1.00824i	0.863633	−	0.504121i	$-0.168184\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	63.4815	+	36.6511i	0.0715688	+	0.0413203i	0.535357	−	0.844626i	$-0.320177\pi$
$887$	63.4815	+	36.6511i	0.0715688	+	0.0413203i	−0.463789	+	0.885946i	$0.653510\pi$
$888$		0			0
$889$	−77.0402	−	133.438i	−0.0866594	−	0.150099i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−86.9642	−	150.626i	−0.0973843	−	0.168675i
$894$		0			0
$895$	−144.871	−	83.6415i	−0.161867	−	0.0934542i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	195.179i		−	0.217107i
$900$		0			0
$901$	−321.687			−0.357034
$902$		0			0
$903$		0			0
$904$		0			0
$905$	272.135	−	471.351i	0.300702	−	0.520830i
$906$		0			0
$907$	831.864	−	480.277i	0.917159	−	0.529522i	0.0344316	−	0.999407i	$-0.489038\pi$
$907$	831.864	−	480.277i	0.917159	−	0.529522i	0.882728	+	0.469885i	$0.155705\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1574.75	−	909.184i	1.72860	−	0.998007i	0.832746	−	0.553655i	$-0.186767\pi$
$911$	1574.75	−	909.184i	1.72860	−	0.998007i	0.895852	−	0.444352i	$-0.146566\pi$
$912$		0			0
$913$	−105.390	+	182.542i	−0.115433	+	0.199936i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	173.111			0.188780
$918$		0			0
$919$		−	345.325i		−	0.375762i	−0.982192	−	0.187881i	$-0.939838\pi$
$919$		−	345.325i		−	0.375762i	0.982192	−	0.187881i	$-0.0601619\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1255.53	+	724.879i	1.36027	+	0.785351i
$924$		0			0
$925$	−263.408	−	456.236i	−0.284765	−	0.493228i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−360.436	−	624.293i	−0.387982	−	0.672005i	0.604196	−	0.796836i	$-0.293495\pi$
$929$	−360.436	−	624.293i	−0.387982	−	0.672005i	−0.992178	+	0.124831i	$0.960161\pi$
$930$		0			0
$931$	580.979	+	335.428i	0.624038	+	0.360288i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			168.497i			0.180211i
$936$		0			0
$937$	−277.841			−0.296522			−0.148261	−	0.988948i	$-0.547368\pi$
$937$	−277.841			−0.296522			−0.148261	+	0.988948i	$0.547368\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−313.560	+	543.101i	−0.333220	+	0.577153i	−0.983141	−	0.182848i	$-0.941468\pi$
$941$	−313.560	+	543.101i	−0.333220	+	0.577153i	0.649922	+	0.760001i	$0.274802\pi$
$942$		0			0
$943$	−1238.09	+	714.810i	−1.31292	+	0.758017i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	72.1510	−	41.6564i	0.0761890	−	0.0439877i	−0.461422	−	0.887181i	$-0.652660\pi$
$947$	72.1510	−	41.6564i	0.0761890	−	0.0439877i	0.537611	+	0.843193i	$0.319327\pi$
$948$		0			0
$949$	−301.363	+	521.976i	−0.317559	+	0.550027i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−341.249			−0.358079			−0.179039	−	0.983842i	$-0.557299\pi$
$953$	−341.249			−0.358079			−0.179039	+	0.983842i	$0.557299\pi$
$954$		0			0
$955$			628.894i			0.658528i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	53.7630	+	31.0401i	0.0560616	+	0.0323672i
$960$		0			0
$961$	−382.068	−	661.761i	−0.397573	−	0.688617i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−51.2170	−	88.7104i	−0.0530746	−	0.0919279i
$966$		0			0
$967$	−1556.33	−	898.550i	−1.60945	−	0.929214i	−0.989494	−	0.144573i	$-0.953819\pi$
$967$	−1556.33	−	898.550i	−1.60945	−	0.929214i	−0.619951	−	0.784640i	$-0.712848\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1706.81i			1.75779i	0.477019	+	0.878893i	$0.341717\pi$
$971$			1706.81i			1.75779i	−0.477019	+	0.878893i	$0.658283\pi$
$972$		0			0
$973$	−157.540			−0.161911
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−181.062	+	313.609i	−0.185325	+	0.320992i	−0.943686	−	0.330843i	$-0.892667\pi$
$977$	−181.062	+	313.609i	−0.185325	+	0.320992i	0.758361	+	0.651835i	$0.226000\pi$
$978$		0			0
$979$	418.295	−	241.503i	0.427267	−	0.246683i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	900.133	−	519.692i	0.915700	−	0.528680i	0.0334391	−	0.999441i	$-0.489354\pi$
$983$	900.133	−	519.692i	0.915700	−	0.528680i	0.882261	+	0.470761i	$0.156021\pi$
$984$		0			0
$985$	236.910	−	410.341i	0.240518	−	0.416590i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1625.39			−1.64347
$990$		0			0
$991$		−	1717.76i		−	1.73336i	−0.498863	−	0.866681i	$-0.666249\pi$
$991$		−	1717.76i		−	1.73336i	0.498863	−	0.866681i	$-0.333751\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−165.649	−	95.6375i	−0.166481	−	0.0961181i
$996$		0			0
$997$	−77.3349	−	133.948i	−0.0775676	−	0.134351i	0.824632	−	0.565669i	$-0.191382\pi$
$997$	−77.3349	−	133.948i	−0.0775676	−	0.134351i	−0.902200	+	0.431318i	$0.858049\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.o.i.1279.6		24
3.2	odd	2		576.3.o.i.319.3		24
4.3	odd	2	inner	1728.3.o.i.1279.5		24
8.3	odd	2		864.3.o.c.415.8		24
8.5	even	2		864.3.o.c.415.7		24
9.2	odd	6		576.3.o.i.511.10		24
9.7	even	3	inner	1728.3.o.i.127.5		24
12.11	even	2		576.3.o.i.319.10		24
24.5	odd	2		288.3.o.c.31.10	yes	24
24.11	even	2		288.3.o.c.31.3	✓	24
36.7	odd	6	inner	1728.3.o.i.127.6		24
36.11	even	6		576.3.o.i.511.3		24
72.5	odd	6		2592.3.g.k.2431.7		12
72.11	even	6		288.3.o.c.223.10	yes	24
72.13	even	6		2592.3.g.l.2431.5		12
72.29	odd	6		288.3.o.c.223.3	yes	24
72.43	odd	6		864.3.o.c.127.7		24
72.59	even	6		2592.3.g.k.2431.8		12
72.61	even	6		864.3.o.c.127.8		24
72.67	odd	6		2592.3.g.l.2431.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.o.c.31.3	✓	24	24.11	even	2
288.3.o.c.31.10	yes	24	24.5	odd	2
288.3.o.c.223.3	yes	24	72.29	odd	6
288.3.o.c.223.10	yes	24	72.11	even	6
576.3.o.i.319.3		24	3.2	odd	2
576.3.o.i.319.10		24	12.11	even	2
576.3.o.i.511.3		24	36.11	even	6
576.3.o.i.511.10		24	9.2	odd	6
864.3.o.c.127.7		24	72.43	odd	6
864.3.o.c.127.8		24	72.61	even	6
864.3.o.c.415.7		24	8.5	even	2
864.3.o.c.415.8		24	8.3	odd	2
1728.3.o.i.127.5		24	9.7	even	3	inner
1728.3.o.i.127.6		24	36.7	odd	6	inner
1728.3.o.i.1279.5		24	4.3	odd	2	inner
1728.3.o.i.1279.6		24	1.1	even	1	trivial
2592.3.g.k.2431.7		12	72.5	odd	6
2592.3.g.k.2431.8		12	72.59	even	6
2592.3.g.l.2431.5		12	72.13	even	6
2592.3.g.l.2431.6		12	72.67	odd	6

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.25241 + 2.16924i) q^{5} +(-0.842935 + 0.486669i) q^{7} +O(q^{10})\)	\(q+(-1.25241 + 2.16924i) q^{5} +(-0.842935 + 0.486669i) q^{7} +(-3.01493 + 1.74067i) q^{11} +(7.51352 - 13.0138i) q^{13} -19.3227 q^{17} -13.9609i q^{19} +(33.5924 + 19.3946i) q^{23} +(9.36292 + 16.2170i) q^{25} +(6.95537 + 12.0470i) q^{29} +(-12.1511 - 7.01542i) q^{31} -2.43804i q^{35} -28.1331 q^{37} +(-18.4281 + 31.9183i) q^{41} +(-36.2892 + 20.9516i) q^{43} +(10.7892 - 6.22913i) q^{47} +(-24.0263 + 41.6148i) q^{49} +16.6482 q^{53} -8.72017i q^{55} +(-11.3836 - 6.57234i) q^{59} +(-41.5302 - 71.9324i) q^{61} +(18.8201 + 32.5973i) q^{65} +(-95.1188 - 54.9169i) q^{67} +96.4767i q^{71} -40.1094 q^{73} +(1.69426 - 2.93455i) q^{77} +(117.038 - 67.5719i) q^{79} +(52.4342 - 30.2729i) q^{83} +(24.2000 - 41.9156i) q^{85} -138.741 q^{89} +14.6264i q^{91} +(30.2846 + 17.4848i) q^{95} +(-79.4667 - 137.640i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 24 q^{13} + 24 q^{17} - 108 q^{25} - 24 q^{29} - 96 q^{37} + 60 q^{41} - 132 q^{49} + 96 q^{53} + 336 q^{61} - 216 q^{65} + 696 q^{73} - 24 q^{77} + 528 q^{85} + 240 q^{89} - 444 q^{97}+O(q^{100}) \) 24 * q + 24 * q^13 + 24 * q^17 - 108 * q^25 - 24 * q^29 - 96 * q^37 + 60 * q^41 - 132 * q^49 + 96 * q^53 + 336 * q^61 - 216 * q^65 + 696 * q^73 - 24 * q^77 + 528 * q^85 + 240 * q^89 - 444 * q^97

Introduction

L-functions

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