LMFDB - Embedded newform 1728.3.o.d.1279.1

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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.121550625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 $ x^8 - x^7 - 4x^6 - 9x^5 + 23x^4 + 18x^3 - 16x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1279.1
Root			$2.25820 + 0.369600i$ of defining polynomial
Character	$\chi$	$=$	1728.1279
Dual form			1728.3.o.d.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.31174 + 5.73610i) q^{5} +(-8.46808 + 4.88905i) q^{7} +O(q^{10})$	$q+(-3.31174 + 5.73610i) q^{5} +(-8.46808 + 4.88905i) q^{7} +(10.0623 - 5.80948i) q^{11} +(8.93521 - 15.4762i) q^{13} -2.37652 q^{17} -14.0337i q^{19} +(35.9635 + 20.7636i) q^{23} +(-9.43521 - 16.3423i) q^{25} +(-5.68826 - 9.85236i) q^{29} +(-18.0334 - 10.4116i) q^{31} -64.7650i q^{35} -35.7409 q^{37} +(2.62957 - 4.55456i) q^{41} +(54.4939 - 31.4621i) q^{43} +(-4.28568 + 2.47434i) q^{47} +(23.3056 - 40.3666i) q^{49} +75.7409 q^{53} +76.9578i q^{55} +(50.3115 + 29.0474i) q^{59} +(-5.93521 - 10.2801i) q^{61} +(59.1822 + 102.507i) q^{65} +(20.6216 + 11.9059i) q^{67} +46.4758i q^{71} -104.352 q^{73} +(-56.8056 + 98.3903i) q^{77} +(-18.0334 + 10.4116i) q^{79} +(-24.4103 + 14.0933i) q^{83} +(7.87043 - 13.6320i) q^{85} -73.0122 q^{89} +174.739i q^{91} +(80.4984 + 46.4758i) q^{95} +(71.1113 + 123.168i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{5}+O(q^{10}) $ 8 * q - 6 * q^5	$ 8 q - 6 q^{5} + 10 q^{13} - 60 q^{17} - 14 q^{25} - 66 q^{29} - 40 q^{37} + 144 q^{41} + 2 q^{49} + 360 q^{53} + 14 q^{61} + 330 q^{65} - 220 q^{73} - 270 q^{77} - 60 q^{85} - 912 q^{89} + 200 q^{97}+O(q^{100}) $ 8 * q - 6 * q^5 + 10 * q^13 - 60 * q^17 - 14 * q^25 - 66 * q^29 - 40 * q^37 + 144 * q^41 + 2 * q^49 + 360 * q^53 + 14 * q^61 + 330 * q^65 - 220 * q^73 - 270 * q^77 - 60 * q^85 - 912 * q^89 + 200 * 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
$5$	−3.31174	+	5.73610i	−0.662348	+	1.14722i	0.317650	+	0.948208i	$0.397106\pi$
$5$	−3.31174	+	5.73610i	−0.662348	+	1.14722i	−0.979997	+	0.199011i	$0.936227\pi$
$6$		0			0
$7$	−8.46808	+	4.88905i	−1.20973	+	0.698436i	−0.962700	−	0.270573i	$-0.912787\pi$
$7$	−8.46808	+	4.88905i	−1.20973	+	0.698436i	−0.247027	+	0.969009i	$0.579454\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10.0623	−	5.80948i	0.914755	−	0.528134i	0.0327970	−	0.999462i	$-0.489559\pi$
$11$	10.0623	−	5.80948i	0.914755	−	0.528134i	0.881958	+	0.471328i	$0.156225\pi$
$12$		0			0
$13$	8.93521	−	15.4762i	0.687324	−	1.19048i	−0.285376	−	0.958416i	$-0.592119\pi$
$13$	8.93521	−	15.4762i	0.687324	−	1.19048i	0.972700	−	0.232065i	$-0.0745481\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.37652			−0.139796			−0.0698978	−	0.997554i	$-0.522267\pi$
$17$	−2.37652			−0.139796			−0.0698978	+	0.997554i	$0.522267\pi$
$18$		0			0
$19$		−	14.0337i		−	0.738614i	−0.929308	−	0.369307i	$-0.879595\pi$
$19$		−	14.0337i		−	0.738614i	0.929308	−	0.369307i	$-0.120405\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	35.9635	+	20.7636i	1.56363	+	0.902763i	0.996885	+	0.0788718i	$0.0251318\pi$
$23$	35.9635	+	20.7636i	1.56363	+	0.902763i	0.566747	+	0.823892i	$0.308202\pi$
$24$		0			0
$25$	−9.43521	−	16.3423i	−0.377409	−	0.653691i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.68826	−	9.85236i	−0.196147	−	0.339737i	0.751129	−	0.660156i	$-0.229510\pi$
$29$	−5.68826	−	9.85236i	−0.196147	−	0.339737i	−0.947276	+	0.320419i	$0.896176\pi$
$30$		0			0
$31$	−18.0334	−	10.4116i	−0.581723	−	0.335858i	0.180095	−	0.983649i	$-0.442360\pi$
$31$	−18.0334	−	10.4116i	−0.581723	−	0.335858i	−0.761818	+	0.647791i	$0.775693\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	64.7650i		−	1.85043i
$36$		0			0
$37$	−35.7409			−0.965969			−0.482984	−	0.875629i	$-0.660447\pi$
$37$	−35.7409			−0.965969			−0.482984	+	0.875629i	$0.660447\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.62957	−	4.55456i	0.0641359	−	0.111087i	−0.832174	−	0.554514i	$-0.812904\pi$
$41$	2.62957	−	4.55456i	0.0641359	−	0.111087i	0.896310	+	0.443427i	$0.146238\pi$
$42$		0			0
$43$	54.4939	−	31.4621i	1.26730	−	0.731676i	0.292824	−	0.956166i	$-0.405405\pi$
$43$	54.4939	−	31.4621i	1.26730	−	0.731676i	0.974476	+	0.224490i	$0.0720716\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.28568	+	2.47434i	−0.0911848	+	0.0526456i	−0.544899	−	0.838502i	$-0.683432\pi$
$47$	−4.28568	+	2.47434i	−0.0911848	+	0.0526456i	0.453714	+	0.891147i	$0.350099\pi$
$48$		0			0
$49$	23.3056	−	40.3666i	0.475625	−	0.823807i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	75.7409			1.42907			0.714536	−	0.699598i	$-0.246638\pi$
$53$	75.7409			1.42907			0.714536	+	0.699598i	$0.246638\pi$
$54$		0			0
$55$			76.9578i			1.39923i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	50.3115	+	29.0474i	0.852738	+	0.492328i	0.861574	−	0.507633i	$-0.169479\pi$
$59$	50.3115	+	29.0474i	0.852738	+	0.492328i	−0.00883587	+	0.999961i	$0.502813\pi$
$60$		0			0
$61$	−5.93521	−	10.2801i	−0.0972986	−	0.168526i	0.813267	−	0.581891i	$-0.197687\pi$
$61$	−5.93521	−	10.2801i	−0.0972986	−	0.168526i	−0.910566	+	0.413365i	$0.864354\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	59.1822	+	102.507i	0.910495	+	1.57702i
$66$		0			0
$67$	20.6216	+	11.9059i	0.307785	+	0.177700i	0.645935	−	0.763393i	$-0.276468\pi$
$67$	20.6216	+	11.9059i	0.307785	+	0.177700i	−0.338150	+	0.941092i	$0.609801\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			46.4758i			0.654589i	0.944922	+	0.327294i	$0.106137\pi$
$71$			46.4758i			0.654589i	−0.944922	+	0.327294i	$0.893863\pi$
$72$		0			0
$73$	−104.352			−1.42948			−0.714741	−	0.699390i	$-0.753455\pi$
$73$	−104.352			−1.42948			−0.714741	+	0.699390i	$0.753455\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−56.8056	+	98.3903i	−0.737736	+	1.27780i
$78$		0			0
$79$	−18.0334	+	10.4116i	−0.228271	+	0.131792i	−0.609774	−	0.792575i	$-0.708740\pi$
$79$	−18.0334	+	10.4116i	−0.228271	+	0.131792i	0.381503	+	0.924368i	$0.375407\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−24.4103	+	14.0933i	−0.294100	+	0.169799i	−0.639789	−	0.768550i	$-0.720978\pi$
$83$	−24.4103	+	14.0933i	−0.294100	+	0.169799i	0.345689	+	0.938349i	$0.387645\pi$
$84$		0			0
$85$	7.87043	−	13.6320i	0.0925932	−	0.160376i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−73.0122			−0.820362			−0.410181	−	0.912004i	$-0.634534\pi$
$89$	−73.0122			−0.820362			−0.410181	+	0.912004i	$0.634534\pi$
$90$		0			0
$91$			174.739i			1.92021i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	80.4984	+	46.4758i	0.847352	+	0.489219i
$96$		0			0
$97$	71.1113	+	123.168i	0.733106	+	1.26978i	0.955549	+	0.294831i	$0.0952634\pi$
$97$	71.1113	+	123.168i	0.733106	+	1.26978i	−0.222443	+	0.974946i	$0.571403\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	13.6761	+	23.6876i	0.135407	+	0.234531i	0.925753	−	0.378130i	$-0.123433\pi$
$101$	13.6761	+	23.6876i	0.135407	+	0.234531i	−0.790346	+	0.612661i	$0.790099\pi$
$102$		0			0
$103$	127.021	+	73.3358i	1.23322	+	0.711998i	0.967699	−	0.252108i	$-0.0811240\pi$
$103$	127.021	+	73.3358i	1.23322	+	0.711998i	0.265517	+	0.964106i	$0.414457\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			192.573i			1.79975i	0.436146	+	0.899876i	$0.356343\pi$
$107$			192.573i			1.79975i	−0.436146	+	0.899876i	$0.643657\pi$
$108$		0			0
$109$	−0.259148			−0.00237750			−0.00118875	−	0.999999i	$-0.500378\pi$
$109$	−0.259148			−0.00237750			−0.00118875	+	0.999999i	$0.500378\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	99.4291	−	172.216i	0.879904	−	1.52404i	0.0284580	−	0.999595i	$-0.490940\pi$
$113$	99.4291	−	172.216i	0.879904	−	1.52404i	0.851446	−	0.524443i	$-0.175726\pi$
$114$		0			0
$115$	−238.204	+	137.527i	−2.07134	+	1.19589i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	20.1246	−	11.6190i	0.169114	−	0.0976382i
$120$		0			0
$121$	7.00000	−	12.1244i	0.0578512	−	0.100201i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−40.5991			−0.324793
$126$		0			0
$127$			72.2477i			0.568879i	0.958694	+	0.284440i	$0.0918075\pi$
$127$			72.2477i			0.568879i	−0.958694	+	0.284440i	$0.908192\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	159.693	+	92.1988i	1.21903	+	0.703808i	0.964711	−	0.263312i	$-0.0848149\pi$
$131$	159.693	+	92.1988i	1.21903	+	0.703808i	0.254320	+	0.967120i	$0.418148\pi$
$132$		0			0
$133$	68.6113	+	118.838i	0.515874	+	0.893520i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	69.8765	+	121.030i	0.510048	+	0.883428i	0.999932	+	0.0116411i	$0.00370557\pi$
$137$	69.8765	+	121.030i	0.510048	+	0.883428i	−0.489885	+	0.871787i	$0.662961\pi$
$138$		0			0
$139$	151.722	+	87.5967i	1.09153	+	0.630192i	0.933982	−	0.357320i	$-0.116309\pi$
$139$	151.722	+	87.5967i	1.09153	+	0.630192i	0.157543	+	0.987512i	$0.449643\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	207.636i		−	1.45200i
$144$		0			0
$145$	75.3521			0.519670
$146$		0			0
$147$		0			0
$148$		0			0
$149$	95.5343	−	165.470i	0.641170	−	1.11054i	−0.344002	−	0.938969i	$-0.611783\pi$
$149$	95.5343	−	165.470i	0.641170	−	1.11054i	0.985172	−	0.171570i	$-0.0548839\pi$
$150$		0			0
$151$	6.27360	−	3.62206i	0.0415470	−	0.0239872i	−0.479083	−	0.877770i	$-0.659031\pi$
$151$	6.27360	−	3.62206i	0.0415470	−	0.0239872i	0.520630	+	0.853783i	$0.325697\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	119.444	−	68.9609i	0.770606	−	0.444909i
$156$		0			0
$157$	36.0648	−	62.4660i	0.229712	−	0.397873i	−0.728011	−	0.685566i	$-0.759555\pi$
$157$	36.0648	−	62.4660i	0.229712	−	0.397873i	0.957723	+	0.287693i	$0.0928882\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−406.056			−2.52209
$162$		0			0
$163$			112.269i			0.688769i	0.938829	+	0.344384i	$0.111912\pi$
$163$			112.269i			0.688769i	−0.938829	+	0.344384i	$0.888088\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−15.8389	−	9.14461i	−0.0948439	−	0.0547581i	0.451828	−	0.892105i	$-0.350772\pi$
$167$	−15.8389	−	9.14461i	−0.0948439	−	0.0547581i	−0.546672	+	0.837347i	$0.684105\pi$
$168$		0			0
$169$	−75.1761	−	130.209i	−0.444829	−	0.770466i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−146.652	−	254.008i	−0.847698	−	1.46826i	−0.883258	−	0.468888i	$-0.844655\pi$
$173$	−146.652	−	254.008i	−0.847698	−	1.46826i	0.0355603	−	0.999368i	$-0.488678\pi$
$174$		0			0
$175$	159.796	+	92.2585i	0.913122	+	0.527191i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			92.9516i			0.519283i	0.965705	+	0.259641i	$0.0836043\pi$
$179$			92.9516i			0.519283i	−0.965705	+	0.259641i	$0.916396\pi$
$180$		0			0
$181$	−128.445			−0.709642			−0.354821	−	0.934934i	$-0.615458\pi$
$181$	−128.445			−0.709642			−0.354821	+	0.934934i	$0.615458\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	118.364	−	205.013i	0.639807	−	1.10818i
$186$		0			0
$187$	−23.9133	+	13.8064i	−0.127879	+	0.0738308i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	260.316	−	150.294i	1.36291	−	0.786877i	0.372901	−	0.927871i	$-0.378363\pi$
$191$	260.316	−	150.294i	1.36291	−	0.786877i	0.990011	+	0.140994i	$0.0450297\pi$
$192$		0			0
$193$	19.6296	−	33.9994i	0.101708	−	0.176163i	−0.810681	−	0.585489i	$-0.800903\pi$
$193$	19.6296	−	33.9994i	0.101708	−	0.176163i	0.912388	+	0.409326i	$0.134236\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−94.7530			−0.480980			−0.240490	−	0.970652i	$-0.577308\pi$
$197$	−94.7530			−0.480980			−0.240490	+	0.970652i	$0.577308\pi$
$198$		0			0
$199$		−	111.360i		−	0.559598i	−0.960059	−	0.279799i	$-0.909732\pi$
$199$		−	111.360i		−	0.559598i	0.960059	−	0.279799i	$-0.0902679\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	96.3374	+	55.6204i	0.474568	+	0.273992i
$204$		0			0
$205$	17.4169	+	30.1670i	0.0849606	+	0.147156i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−81.5282	−	141.211i	−0.390087	−	0.675651i
$210$		0			0
$211$	−102.714	−	59.3021i	−0.486797	−	0.281053i	0.236447	−	0.971644i	$-0.424017\pi$
$211$	−102.714	−	59.3021i	−0.486797	−	0.281053i	−0.723245	+	0.690592i	$0.757350\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			416.777i			1.93850i
$216$		0			0
$217$	203.611			0.938301
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−21.2348	+	36.7797i	−0.0960849	+	0.166424i
$222$		0			0
$223$	21.0153	−	12.1332i	0.0942389	−	0.0544089i	−0.452140	−	0.891947i	$-0.649339\pi$
$223$	21.0153	−	12.1332i	0.0942389	−	0.0544089i	0.546379	+	0.837538i	$0.316006\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.0623	−	5.80948i	0.0443273	−	0.0255924i	−0.477673	−	0.878538i	$-0.658519\pi$
$227$	10.0623	−	5.80948i	0.0443273	−	0.0255924i	0.522000	+	0.852946i	$0.325186\pi$
$228$		0			0
$229$	−9.06479	+	15.7007i	−0.0395842	+	0.0685619i	−0.885139	−	0.465327i	$-0.845937\pi$
$229$	−9.06479	+	15.7007i	−0.0395842	+	0.0685619i	0.845555	+	0.533889i	$0.179270\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	219.785			0.943283			0.471642	−	0.881790i	$-0.343662\pi$
$233$	219.785			0.943283			0.471642	+	0.881790i	$0.343662\pi$
$234$		0			0
$235$		−	32.7775i		−	0.139479i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	139.568	+	80.5799i	0.583968	+	0.337154i	0.762709	−	0.646742i	$-0.223869\pi$
$239$	139.568	+	80.5799i	0.583968	+	0.337154i	−0.178740	+	0.983896i	$0.557202\pi$
$240$		0			0
$241$	84.5930	+	146.519i	0.351008	+	0.607964i	0.986426	−	0.164204i	$-0.0525056\pi$
$241$	84.5930	+	146.519i	0.351008	+	0.607964i	−0.635418	+	0.772168i	$0.719172\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	154.364	+	267.367i	0.630058	+	1.09129i
$246$		0			0
$247$	−217.188	−	125.394i	−0.879305	−	0.507667i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	219.255i		−	0.873524i	−0.899577	−	0.436762i	$-0.856125\pi$
$251$		−	219.255i		−	0.873524i	0.899577	−	0.436762i	$-0.143875\pi$
$252$		0			0
$253$	482.502			1.90712
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−88.7348	+	153.693i	−0.345271	+	0.598028i	−0.985403	−	0.170238i	$-0.945546\pi$
$257$	−88.7348	+	153.693i	−0.345271	+	0.598028i	0.640132	+	0.768265i	$0.278880\pi$
$258$		0			0
$259$	302.657	−	174.739i	1.16856	−	0.674667i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	59.0700	−	34.1041i	0.224601	−	0.129673i	−0.383478	−	0.923550i	$-0.625274\pi$
$263$	59.0700	−	34.1041i	0.224601	−	0.129673i	0.608079	+	0.793877i	$0.291940\pi$
$264$		0			0
$265$	−250.834	+	434.457i	−0.946543	+	1.63946i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−77.7652			−0.289090			−0.144545	−	0.989498i	$-0.546172\pi$
$269$	−77.7652			−0.289090			−0.144545	+	0.989498i	$0.546172\pi$
$270$		0			0
$271$			447.259i			1.65040i	0.564840	+	0.825201i	$0.308938\pi$
$271$			447.259i			1.65040i	−0.564840	+	0.825201i	$0.691062\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−189.880	−	109.627i	−0.690473	−	0.398645i
$276$		0			0
$277$	−84.0282	−	145.541i	−0.303351	−	0.525419i	0.673542	−	0.739149i	$-0.264772\pi$
$277$	−84.0282	−	145.541i	−0.303351	−	0.525419i	−0.976893	+	0.213730i	$0.931439\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	103.170	+	178.696i	0.367153	+	0.635927i	0.989119	−	0.147117i	$-0.0469992\pi$
$281$	103.170	+	178.696i	0.367153	+	0.635927i	−0.621966	+	0.783044i	$0.713666\pi$
$282$		0			0
$283$	172.653	+	99.6815i	0.610083	+	0.352231i	0.772998	−	0.634409i	$-0.218756\pi$
$283$	172.653	+	99.6815i	0.610083	+	0.352231i	−0.162915	+	0.986640i	$0.552090\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			51.4245i			0.179179i
$288$		0			0
$289$	−283.352			−0.980457
$290$		0			0
$291$		0			0
$292$		0			0
$293$	28.5953	−	49.5285i	0.0975948	−	0.169039i	−0.813094	−	0.582133i	$-0.802218\pi$
$293$	28.5953	−	49.5285i	0.0975948	−	0.169039i	0.910689	+	0.413094i	$0.135552\pi$
$294$		0			0
$295$	−333.237	+	192.395i	−1.12962	+	0.652185i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	642.684	−	371.054i	2.14944	−	1.24098i
$300$		0			0
$301$	−307.639	+	532.847i	−1.02206	+	1.77026i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	78.6235			0.257782
$306$		0			0
$307$		−	276.775i		−	0.901549i	−0.892638	−	0.450774i	$-0.851148\pi$
$307$		−	276.775i		−	0.901549i	0.892638	−	0.450774i	$-0.148852\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−21.4284	−	12.3717i	−0.0689017	−	0.0397804i	0.465153	−	0.885230i	$-0.345999\pi$
$311$	−21.4284	−	12.3717i	−0.0689017	−	0.0397804i	−0.534055	+	0.845450i	$0.679333\pi$
$312$		0			0
$313$	32.4070	+	56.1306i	0.103537	+	0.179331i	0.913139	−	0.407647i	$-0.133651\pi$
$313$	32.4070	+	56.1306i	0.103537	+	0.179331i	−0.809603	+	0.586978i	$0.800317\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	227.300	+	393.694i	0.717033	+	1.24194i	0.962170	+	0.272450i	$0.0878338\pi$
$317$	227.300	+	393.694i	0.717033	+	1.24194i	−0.245137	+	0.969488i	$0.578833\pi$
$318$		0			0
$319$	−114.474	−	66.0916i	−0.358853	−	0.207184i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			33.3513i			0.103255i
$324$		0			0
$325$	−337.223			−1.03761
$326$		0			0
$327$		0			0
$328$		0			0
$329$	24.1944	−	41.9059i	0.0735391	−	0.127373i
$330$		0			0
$331$	236.009	−	136.260i	0.713019	−	0.411661i	−0.0991592	−	0.995072i	$-0.531615\pi$
$331$	236.009	−	136.260i	0.713019	−	0.411661i	0.812178	+	0.583410i	$0.198282\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−136.587	+	78.8583i	−0.407721	+	0.235398i
$336$		0			0
$337$	246.945	−	427.721i	0.732775	−	1.26920i	−0.222918	−	0.974837i	$-0.571558\pi$
$337$	246.945	−	427.721i	0.732775	−	1.26920i	0.955693	−	0.294366i	$-0.0951085\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−241.944			−0.709512
$342$		0			0
$343$		−	23.3572i		−	0.0680967i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−450.196	−	259.921i	−1.29740	−	0.749051i	−0.317442	−	0.948278i	$-0.602824\pi$
$347$	−450.196	−	259.921i	−1.29740	−	0.749051i	−0.979954	+	0.199226i	$0.936157\pi$
$348$		0			0
$349$	−120.639	−	208.954i	−0.345672	−	0.598721i	0.639804	−	0.768538i	$-0.279016\pi$
$349$	−120.639	−	208.954i	−0.345672	−	0.598721i	−0.985476	+	0.169817i	$0.945682\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	100.463	+	174.008i	0.284599	+	0.492940i	0.972512	−	0.232853i	$-0.0748063\pi$
$353$	100.463	+	174.008i	0.284599	+	0.492940i	−0.687913	+	0.725793i	$0.741473\pi$
$354$		0			0
$355$	−266.590	−	153.916i	−0.750957	−	0.433565i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			159.654i			0.444719i	0.974965	+	0.222360i	$0.0713759\pi$
$359$			159.654i			0.444719i	−0.974965	+	0.222360i	$0.928624\pi$
$360$		0			0
$361$	164.056			0.454450
$362$		0			0
$363$		0			0
$364$		0			0
$365$	345.587	−	598.574i	0.946813	−	1.63993i
$366$		0			0
$367$	320.690	−	185.150i	0.873815	−	0.504497i	0.00520064	−	0.999986i	$-0.498345\pi$
$367$	320.690	−	185.150i	0.873815	−	0.504497i	0.868614	+	0.495489i	$0.165011\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−641.380	+	370.301i	−1.72879	+	0.998116i
$372$		0			0
$373$	163.194	−	282.661i	0.437518	−	0.757804i	−0.559979	−	0.828507i	$-0.689191\pi$
$373$	163.194	−	282.661i	0.437518	−	0.757804i	0.997497	+	0.0707027i	$0.0225242\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−203.303			−0.539266
$378$		0			0
$379$			348.114i			0.918506i	0.888305	+	0.459253i	$0.151883\pi$
$379$			348.114i			0.918506i	−0.888305	+	0.459253i	$0.848117\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	202.550	+	116.942i	0.528851	+	0.305332i	0.740548	−	0.672003i	$-0.234566\pi$
$383$	202.550	+	116.942i	0.528851	+	0.305332i	−0.211697	+	0.977335i	$0.567899\pi$
$384$		0			0
$385$	−376.251	−	651.685i	−0.977275	−	1.69269i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	218.806	+	378.982i	0.562482	+	0.974248i	0.997279	+	0.0737194i	$0.0234869\pi$
$389$	218.806	+	378.982i	0.562482	+	0.974248i	−0.434797	+	0.900529i	$0.643180\pi$
$390$		0			0
$391$	−85.4682	−	49.3451i	−0.218589	−	0.126202i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	137.922i		−	0.349169i
$396$		0			0
$397$	324.259			0.816774			0.408387	−	0.912809i	$-0.366091\pi$
$397$	324.259			0.816774			0.408387	+	0.912809i	$0.366091\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−197.087	+	341.364i	−0.491488	+	0.851283i	−0.999952	−	0.00980052i	$-0.996880\pi$
$401$	−197.087	+	341.364i	−0.491488	+	0.851283i	0.508463	+	0.861084i	$0.330214\pi$
$402$		0			0
$403$	−322.265	+	186.060i	−0.799664	+	0.461686i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−359.635	+	207.636i	−0.883625	+	0.510161i
$408$		0			0
$409$	14.1479	−	24.5048i	0.0345914	−	0.0599140i	−0.848211	−	0.529658i	$-0.822320\pi$
$409$	14.1479	−	24.5048i	0.0345914	−	0.0599140i	0.882803	+	0.469744i	$0.155654\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−568.056			−1.37544
$414$		0			0
$415$		−	186.693i		−	0.449863i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	280.441	+	161.913i	0.669310	+	0.386426i	0.795815	−	0.605540i	$-0.207043\pi$
$419$	280.441	+	161.913i	0.669310	+	0.386426i	−0.126505	+	0.991966i	$0.540376\pi$
$420$		0			0
$421$	−215.676	−	373.562i	−0.512295	−	0.887320i	−0.999898	−	0.0142554i	$-0.995462\pi$
$421$	−215.676	−	373.562i	−0.512295	−	0.887320i	0.487604	−	0.873065i	$-0.337871\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.4230	+	38.8378i	0.0527600	+	0.0913831i
$426$		0			0
$427$	100.520	+	58.0351i	0.235409	+	0.135914i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	345.557i		−	0.801757i	−0.916131	−	0.400879i	$-0.868705\pi$
$431$		−	345.557i		−	0.801757i	0.916131	−	0.400879i	$-0.131295\pi$
$432$		0			0
$433$	369.907			0.854289			0.427144	−	0.904183i	$-0.359520\pi$
$433$	369.907			0.854289			0.427144	+	0.904183i	$0.359520\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	291.389	−	504.700i	0.666793	−	1.15492i
$438$		0			0
$439$	−358.332	+	206.883i	−0.816245	+	0.471259i	−0.849120	−	0.528200i	$-0.822867\pi$
$439$	−358.332	+	206.883i	−0.816245	+	0.471259i	0.0328748	+	0.999459i	$0.489534\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−467.339	+	269.818i	−1.05494	+	0.609070i	−0.924029	−	0.382323i	$-0.875124\pi$
$443$	−467.339	+	269.818i	−1.05494	+	0.609070i	−0.130912	+	0.991394i	$0.541791\pi$
$444$		0			0
$445$	241.797	−	418.805i	0.543365	−	0.941135i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	325.056			0.723956			0.361978	−	0.932187i	$-0.382101\pi$
$449$	325.056			0.723956			0.361978	+	0.932187i	$0.382101\pi$
$450$		0			0
$451$		−	61.1058i		−	0.135490i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1002.32	−	578.689i	−2.20290	−	1.27184i
$456$		0			0
$457$	−309.075	−	535.333i	−0.676312	−	1.17141i	−0.976083	−	0.217396i	$-0.930244\pi$
$457$	−309.075	−	535.333i	−0.676312	−	1.17141i	0.299771	−	0.954011i	$-0.403090\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	424.992	+	736.107i	0.921891	+	1.59676i	0.796487	+	0.604656i	$0.206689\pi$
$461$	424.992	+	736.107i	0.921891	+	1.59676i	0.125404	+	0.992106i	$0.459977\pi$
$462$		0			0
$463$	706.453	+	407.871i	1.52582	+	0.880930i	0.999531	+	0.0306215i	$0.00974865\pi$
$463$	706.453	+	407.871i	1.52582	+	0.880930i	0.526285	+	0.850309i	$0.323585\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			378.909i			0.811368i	0.914013	+	0.405684i	$0.132967\pi$
$467$			378.909i			0.811368i	−0.914013	+	0.405684i	$0.867033\pi$
$468$		0			0
$469$	−232.834			−0.496447
$470$		0			0
$471$		0			0
$472$		0			0
$473$	365.556	−	633.162i	0.772846	−	1.33861i
$474$		0			0
$475$	−229.342	+	132.411i	−0.482825	+	0.278759i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−297.958	+	172.026i	−0.622041	+	0.359136i	−0.777663	−	0.628681i	$-0.783595\pi$
$479$	−297.958	+	172.026i	−0.622041	+	0.359136i	0.155622	+	0.987817i	$0.450262\pi$
$480$		0			0
$481$	−319.352	+	553.134i	−0.663934	+	1.14997i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−942.008			−1.94228
$486$		0			0
$487$		−	587.595i		−	1.20656i	−0.797529	−	0.603281i	$-0.793860\pi$
$487$		−	587.595i		−	1.20656i	0.797529	−	0.603281i	$-0.206140\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	430.072	+	248.302i	0.875909	+	0.505707i	0.869307	−	0.494272i	$-0.164565\pi$
$491$	430.072	+	248.302i	0.875909	+	0.505707i	0.00660193	+	0.999978i	$0.497899\pi$
$492$		0			0
$493$	13.5183	+	23.4144i	0.0274205	+	0.0474937i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−227.223	−	393.561i	−0.457188	−	0.791873i
$498$		0			0
$499$	30.1869	+	17.4284i	0.0604948	+	0.0349267i	0.529942	−	0.848034i	$-0.322214\pi$
$499$	30.1869	+	17.4284i	0.0604948	+	0.0349267i	−0.469448	+	0.882960i	$0.655547\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	630.866i		−	1.25421i	−0.778936	−	0.627104i	$-0.784240\pi$
$503$		−	630.866i		−	1.25421i	0.778936	−	0.627104i	$-0.215760\pi$
$504$		0			0
$505$	−181.166			−0.358745
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−41.0282	+	71.0629i	−0.0806055	+	0.139613i	−0.903510	−	0.428567i	$-0.859019\pi$
$509$	−41.0282	+	71.0629i	−0.0806055	+	0.139613i	0.822905	+	0.568179i	$0.192352\pi$
$510$		0			0
$511$	883.663	−	510.183i	1.72928	−	0.998401i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−841.322	+	485.738i	−1.63364	+	0.943180i
$516$		0			0
$517$	−28.7492	+	49.7952i	−0.0556078	+	0.0963156i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−562.797			−1.08023			−0.540113	−	0.841593i	$-0.681618\pi$
$521$	−562.797			−1.08023			−0.540113	+	0.841593i	$0.681618\pi$
$522$		0			0
$523$		−	909.721i		−	1.73943i	−0.493555	−	0.869714i	$-0.664303\pi$
$523$		−	909.721i		−	1.73943i	0.493555	−	0.869714i	$-0.335697\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	42.8568	+	24.7434i	0.0813223	+	0.0469514i
$528$		0			0
$529$	597.751	+	1035.33i	1.12996	+	1.95715i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−46.9916	−	81.3918i	−0.0881644	−	0.152705i
$534$		0			0
$535$	−1104.62	−	637.753i	−2.06471	−	1.19206i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	541.574i		−	1.00478i
$540$		0			0
$541$	−468.259			−0.865544			−0.432772	−	0.901503i	$-0.642464\pi$
$541$	−468.259			−0.865544			−0.432772	+	0.901503i	$0.642464\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.858229	−	1.48650i	0.00157473	−	0.00272752i
$546$		0			0
$547$	−271.063	+	156.498i	−0.495544	+	0.286102i	−0.726872	−	0.686773i	$-0.759026\pi$
$547$	−271.063	+	156.498i	−0.495544	+	0.286102i	0.231328	+	0.972876i	$0.425693\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−138.265	+	79.8271i	−0.250934	+	0.144877i
$552$		0			0
$553$	101.806	−	176.333i	0.184097	−	0.318865i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	382.841			0.687328			0.343664	−	0.939093i	$-0.388332\pi$
$557$	382.841			0.687328			0.343664	+	0.939093i	$0.388332\pi$
$558$		0			0
$559$		−	1124.48i		−	2.01160i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−518.767	−	299.510i	−0.921434	−	0.531990i	−0.0373414	−	0.999303i	$-0.511889\pi$
$563$	−518.767	−	299.510i	−0.921434	−	0.531990i	−0.884092	+	0.467313i	$0.845222\pi$
$564$		0			0
$565$	658.566	+	1140.67i	1.16560	+	2.01889i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	422.087	+	731.076i	0.741805	+	1.28484i	0.951673	+	0.307114i	$0.0993632\pi$
$569$	422.087	+	731.076i	0.741805	+	1.28484i	−0.209868	+	0.977730i	$0.567303\pi$
$570$		0			0
$571$	−672.354	−	388.184i	−1.17750	−	0.679832i	−0.222067	−	0.975031i	$-0.571281\pi$
$571$	−672.354	−	388.184i	−1.17750	−	0.679832i	−0.955436	+	0.295200i	$0.904614\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	783.634i		−	1.36284i
$576$		0			0
$577$	−487.316			−0.844568			−0.422284	−	0.906464i	$-0.638771\pi$
$577$	−487.316			−0.844568			−0.422284	+	0.906464i	$0.638771\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	137.806	−	238.686i	0.237187	−	0.410820i
$582$		0			0
$583$	762.128	−	440.015i	1.30725	−	0.754742i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	487.463	−	281.437i	0.830432	−	0.479450i	−0.0235687	−	0.999722i	$-0.507503\pi$
$587$	487.463	−	281.437i	0.830432	−	0.479450i	0.854001	+	0.520272i	$0.174170\pi$
$588$		0			0
$589$	−146.113	+	253.075i	−0.248069	+	0.429669i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	316.915			0.534426			0.267213	−	0.963637i	$-0.413897\pi$
$593$	316.915			0.534426			0.267213	+	0.963637i	$0.413897\pi$
$594$		0			0
$595$			153.916i			0.258682i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	76.5870	+	44.2175i	0.127858	+	0.0738189i	0.562565	−	0.826753i	$-0.309815\pi$
$599$	76.5870	+	44.2175i	0.127858	+	0.0738189i	−0.434707	+	0.900572i	$0.643148\pi$
$600$		0			0
$601$	55.9085	+	96.8364i	0.0930258	+	0.161125i	0.908783	−	0.417269i	$-0.137013\pi$
$601$	55.9085	+	96.8364i	0.0930258	+	0.161125i	−0.815757	+	0.578395i	$0.803679\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	46.3643	+	80.3054i	0.0766353	+	0.132736i
$606$		0			0
$607$	596.058	+	344.134i	0.981973	+	0.566942i	0.902865	−	0.429924i	$-0.141460\pi$
$607$	596.058	+	344.134i	0.981973	+	0.566942i	0.0791078	+	0.996866i	$0.474793\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			88.4351i			0.144738i
$612$		0			0
$613$	1021.67			1.66667			0.833334	−	0.552770i	$-0.186429\pi$
$613$	1021.67			1.66667			0.833334	+	0.552770i	$0.186429\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−381.884	+	661.443i	−0.618937	+	1.07203i	0.370743	+	0.928735i	$0.379103\pi$
$617$	−381.884	+	661.443i	−0.618937	+	1.07203i	−0.989680	+	0.143295i	$0.954230\pi$
$618$		0			0
$619$	661.382	−	381.849i	1.06847	−	0.616880i	0.140706	−	0.990051i	$-0.455063\pi$
$619$	661.382	−	381.849i	1.06847	−	0.616880i	0.927763	+	0.373171i	$0.121730\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	618.273	−	356.960i	0.992413	−	0.572970i
$624$		0			0
$625$	370.334	−	641.437i	0.592534	−	1.02630i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	84.9390			0.135038
$630$		0			0
$631$		−	166.586i		−	0.264002i	−0.991250	−	0.132001i	$-0.957860\pi$
$631$		−	166.586i		−	0.264002i	0.991250	−	0.132001i	$-0.0421403\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−414.420	−	239.265i	−0.652629	−	0.376796i
$636$		0			0
$637$	−416.482	−	721.367i	−0.653817	−	1.13245i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	102.136	+	176.904i	0.159338	+	0.275982i	0.934630	−	0.355621i	$-0.115731\pi$
$641$	102.136	+	176.904i	0.159338	+	0.275982i	−0.775292	+	0.631603i	$0.782397\pi$
$642$		0			0
$643$	−826.974	−	477.454i	−1.28612	−	0.742541i	−0.308159	−	0.951335i	$-0.599713\pi$
$643$	−826.974	−	477.454i	−1.28612	−	0.742541i	−0.977960	+	0.208794i	$0.933046\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	424.736i		−	0.656470i	−0.944596	−	0.328235i	$-0.893546\pi$
$647$		−	424.736i		−	0.656470i	0.944596	−	0.328235i	$-0.106454\pi$
$648$		0			0
$649$	675.000			1.04006
$650$		0			0
$651$		0			0
$652$		0			0
$653$	307.178	−	532.047i	0.470410	−	0.814774i	−0.529018	−	0.848611i	$-0.677439\pi$
$653$	307.178	−	532.047i	0.470410	−	0.814774i	0.999427	+	0.0338373i	$0.0107728\pi$
$654$		0			0
$655$	−1057.72	+	610.677i	−1.61484	+	0.932331i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	843.930	−	487.243i	1.28062	−	0.739367i	0.303660	−	0.952780i	$-0.401791\pi$
$659$	843.930	−	487.243i	1.28062	−	0.739367i	0.976962	+	0.213413i	$0.0684580\pi$
$660$		0			0
$661$	−274.380	+	475.241i	−0.415099	+	0.718972i	−0.995439	−	0.0954020i	$-0.969586\pi$
$661$	−274.380	+	475.241i	−0.415099	+	0.718972i	0.580340	+	0.814374i	$0.302920\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−908.890			−1.36675
$666$		0			0
$667$		−	472.434i		−	0.708297i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−119.444	−	68.9609i	−0.178009	−	0.102773i
$672$		0			0
$673$	186.992	+	323.879i	0.277848	+	0.481247i	0.970850	−	0.239689i	$-0.0770456\pi$
$673$	186.992	+	323.879i	0.277848	+	0.481247i	−0.693002	+	0.720936i	$0.743712\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	120.972	+	209.529i	0.178688	+	0.309497i	0.941431	−	0.337205i	$-0.109481\pi$
$677$	120.972	+	209.529i	0.178688	+	0.309497i	−0.762743	+	0.646701i	$0.776148\pi$
$678$		0			0
$679$	−1204.35	−	695.333i	−1.77372	−	1.02405i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	524.358i		−	0.767728i	−0.923390	−	0.383864i	$-0.874593\pi$
$683$		−	524.358i		−	0.767728i	0.923390	−	0.383864i	$-0.125407\pi$
$684$		0			0
$685$	−925.651			−1.35132
$686$		0			0
$687$		0			0
$688$		0			0
$689$	676.761	−	1172.18i	0.982236	−	1.70128i
$690$		0			0
$691$	306.568	−	176.997i	0.443658	−	0.256146i	−0.261490	−	0.965206i	$-0.584214\pi$
$691$	306.568	−	176.997i	0.443658	−	0.256146i	0.705148	+	0.709060i	$0.250880\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1004.93	+	580.195i	−1.44594	+	0.834813i
$696$		0			0
$697$	−6.24925	+	10.8240i	−0.00896592	+	0.0155294i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−621.287			−0.886286			−0.443143	−	0.896451i	$-0.646137\pi$
$701$	−621.287			−0.886286			−0.443143	+	0.896451i	$0.646137\pi$
$702$		0			0
$703$			501.575i			0.713478i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−231.620	−	133.726i	−0.327610	−	0.189146i
$708$		0			0
$709$	393.992	+	682.413i	0.555700	+	0.962501i	0.997849	+	0.0655594i	$0.0208832\pi$
$709$	393.992	+	682.413i	0.555700	+	0.962501i	−0.442148	+	0.896942i	$0.645783\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−432.364	−	748.876i	−0.606400	−	1.05032i
$714$		0			0
$715$	1191.02	+	687.635i	1.66576	+	0.961727i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1248.82i			1.73689i	0.495785	+	0.868445i	$0.334881\pi$
$719$			1248.82i			1.73689i	−0.495785	+	0.868445i	$0.665119\pi$
$720$		0			0
$721$	−1434.17			−1.98914
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−107.340	+	185.918i	−0.148055	+	0.256439i
$726$		0			0
$727$	−634.487	+	366.321i	−0.872746	+	0.503880i	−0.868260	−	0.496110i	$-0.834761\pi$
$727$	−634.487	+	366.321i	−0.872746	+	0.503880i	−0.00448649	+	0.999990i	$0.501428\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−129.506	+	74.7704i	−0.177163	+	0.102285i
$732$		0			0
$733$	−182.361	+	315.858i	−0.248787	+	0.430911i	−0.963189	−	0.268824i	$-0.913365\pi$
$733$	−182.361	+	315.858i	−0.248787	+	0.430911i	0.714403	+	0.699735i	$0.246698\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	276.668			0.375397
$738$		0			0
$739$		−	1162.07i		−	1.57248i	−0.617918	−	0.786242i	$-0.712024\pi$
$739$		−	1162.07i		−	1.57248i	0.617918	−	0.786242i	$-0.287976\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	156.711	+	90.4773i	0.210917	+	0.121773i	0.601737	−	0.798694i	$-0.294475\pi$
$743$	156.711	+	90.4773i	0.210917	+	0.121773i	−0.390821	+	0.920467i	$0.627809\pi$
$744$		0			0
$745$	632.769	+	1095.99i	0.849354	+	1.47113i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−941.502	−	1630.73i	−1.25701	−	2.17721i
$750$		0			0
$751$	152.903	+	88.2786i	0.203599	+	0.117548i	0.598333	−	0.801247i	$-0.295830\pi$
$751$	152.903	+	88.2786i	0.203599	+	0.117548i	−0.394734	+	0.918795i	$0.629163\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			47.9813i			0.0635514i
$756$		0			0
$757$	−146.665			−0.193745			−0.0968723	−	0.995297i	$-0.530884\pi$
$757$	−146.665			−0.193745			−0.0968723	+	0.995297i	$0.530884\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	403.405	−	698.717i	0.530098	−	0.918157i	−0.469285	−	0.883047i	$-0.655488\pi$
$761$	403.405	−	698.717i	0.530098	−	0.918157i	0.999383	−	0.0351104i	$-0.0111783\pi$
$762$		0			0
$763$	2.19448	−	1.26699i	0.00287613	−	0.00166053i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	899.088	−	519.089i	1.17221	−	0.676778i
$768$		0			0
$769$	−275.566	+	477.295i	−0.358344	+	0.620669i	−0.987684	−	0.156460i	$-0.949992\pi$
$769$	−275.566	+	477.295i	−0.358344	+	0.620669i	0.629341	+	0.777130i	$0.283325\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	856.915			1.10856			0.554279	−	0.832331i	$-0.312994\pi$
$773$	856.915			1.10856			0.554279	+	0.832331i	$0.312994\pi$
$774$		0			0
$775$			392.942i			0.507023i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−63.9171	−	36.9025i	−0.0820502	−	0.0473717i
$780$		0			0
$781$	270.000	+	467.654i	0.345711	+	0.598788i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	238.874	+	413.742i	0.304298	+	0.527060i
$786$		0			0
$787$	−466.532	−	269.352i	−0.592798	−	0.342252i	0.173405	−	0.984851i	$-0.444523\pi$
$787$	−466.532	−	269.352i	−0.592798	−	0.342252i	−0.766203	+	0.642598i	$0.777856\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1944.46i			2.45823i
$792$		0			0
$793$	−212.130			−0.267503
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−674.153	+	1167.67i	−0.845863	+	1.46508i	0.0390058	+	0.999239i	$0.487581\pi$
$797$	−674.153	+	1167.67i	−0.845863	+	1.46508i	−0.884869	+	0.465840i	$0.845752\pi$
$798$		0			0
$799$	10.1850	−	5.88033i	0.0127472	−	0.00735962i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1050.02	+	606.231i	−1.30763	+	0.754958i
$804$		0			0
$805$	1344.75	−	2329.18i	1.67050	−	2.89339i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1163.88			−1.43867			−0.719334	−	0.694664i	$-0.755553\pi$
$809$	−1163.88			−1.43867			−0.719334	+	0.694664i	$0.755553\pi$
$810$		0			0
$811$			627.878i			0.774202i	0.922037	+	0.387101i	$0.126524\pi$
$811$			627.878i			0.774202i	−0.922037	+	0.387101i	$0.873476\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−643.988	−	371.806i	−0.790169	−	0.456204i
$816$		0			0
$817$	−441.528	−	764.749i	−0.540426	−	0.936046i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−32.1822	−	55.7411i	−0.0391987	−	0.0678942i	0.845760	−	0.533563i	$-0.179147\pi$
$821$	−32.1822	−	55.7411i	−0.0391987	−	0.0678942i	−0.884959	+	0.465669i	$0.845814\pi$
$822$		0			0
$823$	817.183	+	471.801i	0.992932	+	0.573269i	0.906149	−	0.422958i	$-0.139008\pi$
$823$	817.183	+	471.801i	0.992932	+	0.573269i	0.0867825	+	0.996227i	$0.472341\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	571.482i		−	0.691031i	−0.938413	−	0.345515i	$-0.887704\pi$
$827$		−	571.482i		−	0.691031i	0.938413	−	0.345515i	$-0.112296\pi$
$828$		0			0
$829$	519.555			0.626725			0.313362	−	0.949634i	$-0.398545\pi$
$829$	519.555			0.626725			0.313362	+	0.949634i	$0.398545\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−55.3864	+	95.9321i	−0.0664903	+	0.115165i
$834$		0			0
$835$	104.909	−	60.5691i	0.125639	−	0.0725378i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1039.96	+	600.422i	−1.23952	+	0.715640i	−0.968997	−	0.247073i	$-0.920531\pi$
$839$	−1039.96	+	600.422i	−1.23952	+	0.715640i	−0.270527	+	0.962712i	$0.587198\pi$
$840$		0			0
$841$	355.787	−	616.242i	0.423053	−	0.732749i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	995.854			1.17853
$846$		0			0
$847$			136.893i			0.161622i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1285.37	−	742.107i	−1.51042	−	0.872041i
$852$		0			0
$853$	−299.583	−	518.893i	−0.351211	−	0.608315i	0.635251	−	0.772306i	$-0.280897\pi$
$853$	−299.583	−	518.893i	−0.351211	−	0.608315i	−0.986462	+	0.163990i	$0.947563\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	345.664	+	598.707i	0.403342	+	0.698608i	0.994127	−	0.108221i	$-0.0345153\pi$
$857$	345.664	+	598.707i	0.403342	+	0.698608i	−0.590785	+	0.806829i	$0.701182\pi$
$858$		0			0
$859$	1215.72	+	701.896i	1.41527	+	0.817108i	0.995879	−	0.0906967i	$-0.0289094\pi$
$859$	1215.72	+	701.896i	1.41527	+	0.817108i	0.419394	+	0.907805i	$0.362243\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	253.038i		−	0.293207i	−0.989195	−	0.146604i	$-0.953166\pi$
$863$		−	253.038i		−	0.293207i	0.989195	−	0.146604i	$-0.0468342\pi$
$864$		0			0
$865$	1942.69			2.24588
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−120.972	+	209.529i	−0.139208	+	0.241115i
$870$		0			0
$871$	368.517	−	212.763i	0.423096	−	0.244275i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	343.796	−	198.491i	0.392910	−	0.226847i
$876$		0			0
$877$	−53.9352	+	93.4185i	−0.0614997	+	0.106521i	−0.895136	−	0.445793i	$-0.852922\pi$
$877$	−53.9352	+	93.4185i	−0.0614997	+	0.106521i	0.833636	+	0.552314i	$0.186255\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	119.927			0.136126			0.0680629	−	0.997681i	$-0.478318\pi$
$881$	119.927			0.136126			0.0680629	+	0.997681i	$0.478318\pi$
$882$		0			0
$883$			306.661i			0.347295i	0.984808	+	0.173647i	$0.0555553\pi$
$883$			306.661i			0.347295i	−0.984808	+	0.173647i	$0.944445\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	680.325	+	392.786i	0.766996	+	0.442825i	0.831802	−	0.555073i	$-0.187310\pi$
$887$	680.325	+	392.786i	0.766996	+	0.442825i	−0.0648060	+	0.997898i	$0.520643\pi$
$888$		0			0
$889$	−353.223	−	611.799i	−0.397326	−	0.688188i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	34.7241	+	60.1438i	0.0388847	+	0.0673503i
$894$		0			0
$895$	−533.179	−	307.831i	−0.595731	−	0.343946i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			236.896i			0.263510i
$900$		0			0
$901$	−180.000			−0.199778
$902$		0			0
$903$		0			0
$904$		0			0
$905$	425.377	−	736.774i	0.470029	−	0.814115i
$906$		0			0
$907$	1055.14	−	609.182i	1.16332	−	0.671646i	0.211226	−	0.977437i	$-0.432254\pi$
$907$	1055.14	−	609.182i	1.16332	−	0.671646i	0.952098	+	0.305792i	$0.0989211\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	760.824	−	439.262i	0.835152	−	0.482175i	−0.0204612	−	0.999791i	$-0.506513\pi$
$911$	760.824	−	439.262i	0.835152	−	0.482175i	0.855614	+	0.517615i	$0.173180\pi$
$912$		0			0
$913$	−163.749	+	283.622i	−0.179353	+	0.310648i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1803.06			−1.96626
$918$		0			0
$919$			1036.67i			1.12804i	0.825760	+	0.564022i	$0.190747\pi$
$919$			1036.67i			1.12804i	−0.825760	+	0.564022i	$0.809253\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	719.271	+	415.271i	0.779275	+	0.449915i
$924$		0			0
$925$	337.223	+	584.087i	0.364565	+	0.631445i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	124.911	+	216.352i	0.134457	+	0.232887i	0.925390	−	0.379016i	$-0.123738\pi$
$929$	124.911	+	216.352i	0.134457	+	0.232887i	−0.790933	+	0.611903i	$0.790404\pi$
$930$		0			0
$931$	−566.490	−	327.063i	−0.608475	−	0.351303i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	182.892i		−	0.195607i
$936$		0			0
$937$	190.744			0.203569			0.101784	−	0.994806i	$-0.467545\pi$
$937$	190.744			0.203569			0.101784	+	0.994806i	$0.467545\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−141.150	+	244.479i	−0.150000	+	0.259808i	−0.931227	−	0.364439i	$-0.881261\pi$
$941$	−141.150	+	244.479i	−0.150000	+	0.259808i	0.781227	+	0.624247i	$0.214594\pi$
$942$		0			0
$943$	189.138	−	109.199i	0.200570	−	0.115799i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	573.926	−	331.356i	0.606046	−	0.349901i	−0.165370	−	0.986232i	$-0.552882\pi$
$947$	573.926	−	331.356i	0.606046	−	0.349901i	0.771416	+	0.636331i	$0.219549\pi$
$948$		0			0
$949$	−932.409	+	1614.98i	−0.982517	+	1.70177i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1166.70			−1.22424			−0.612119	−	0.790765i	$-0.709683\pi$
$953$	−1166.70			−1.22424			−0.612119	+	0.790765i	$0.709683\pi$
$954$		0			0
$955$			1990.93i			2.08475i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1183.44	−	683.260i	−1.23404	−	0.712471i
$960$		0			0
$961$	−263.697	−	456.737i	−0.274399	−	0.475273i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	130.016	+	225.194i	0.134732	+	0.233362i
$966$		0			0
$967$	1010.52	+	583.422i	1.04500	+	0.603332i	0.921246	−	0.388981i	$-0.127173\pi$
$967$	1010.52	+	583.422i	1.04500	+	0.603332i	0.123756	+	0.992313i	$0.460506\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1481.20i			1.52544i	0.646728	+	0.762721i	$0.276137\pi$
$971$			1481.20i			1.52544i	−0.646728	+	0.762721i	$0.723863\pi$
$972$		0			0
$973$	−1713.06			−1.76060
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−592.779	+	1026.72i	−0.606734	+	1.05089i	0.385041	+	0.922899i	$0.374187\pi$
$977$	−592.779	+	1026.72i	−0.606734	+	1.05089i	−0.991775	+	0.127994i	$0.959146\pi$
$978$		0			0
$979$	−734.671	+	424.163i	−0.750430	+	0.433261i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−56.0882	+	32.3825i	−0.0570581	+	0.0329425i	−0.528258	−	0.849084i	$-0.677154\pi$
$983$	−56.0882	+	32.3825i	−0.0570581	+	0.0329425i	0.471200	+	0.882027i	$0.343821\pi$
$984$		0			0
$985$	313.797	−	543.513i	0.318576	−	0.551790i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2613.06			2.64212
$990$		0			0
$991$			1394.27i			1.40694i	0.710727	+	0.703468i	$0.248366\pi$
$991$			1394.27i			1.40694i	−0.710727	+	0.703468i	$0.751634\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	638.772	+	368.795i	0.641982	+	0.370649i
$996$		0			0
$997$	−487.271	−	843.977i	−0.488737	−	0.846517i	0.511179	−	0.859474i	$-0.329209\pi$
$997$	−487.271	−	843.977i	−0.488737	−	0.846517i	−0.999916	+	0.0129572i	$0.995875\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.o.d.1279.1		8
3.2	odd	2		576.3.o.e.319.3		8
4.3	odd	2	inner	1728.3.o.d.1279.2		8
8.3	odd	2		432.3.o.c.415.4		8
8.5	even	2		432.3.o.c.415.3		8
9.2	odd	6		576.3.o.e.511.2		8
9.7	even	3	inner	1728.3.o.d.127.2		8
12.11	even	2		576.3.o.e.319.2		8
24.5	odd	2		144.3.o.b.31.2	✓	8
24.11	even	2		144.3.o.b.31.3	yes	8
36.7	odd	6	inner	1728.3.o.d.127.1		8
36.11	even	6		576.3.o.e.511.3		8
72.5	odd	6		1296.3.g.h.1135.3		4
72.11	even	6		144.3.o.b.79.2	yes	8
72.13	even	6		1296.3.g.d.1135.1		4
72.29	odd	6		144.3.o.b.79.3	yes	8
72.43	odd	6		432.3.o.c.127.3		8
72.59	even	6		1296.3.g.h.1135.4		4
72.61	even	6		432.3.o.c.127.4		8
72.67	odd	6		1296.3.g.d.1135.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.o.b.31.2	✓	8	24.5	odd	2
144.3.o.b.31.3	yes	8	24.11	even	2
144.3.o.b.79.2	yes	8	72.11	even	6
144.3.o.b.79.3	yes	8	72.29	odd	6
432.3.o.c.127.3		8	72.43	odd	6
432.3.o.c.127.4		8	72.61	even	6
432.3.o.c.415.3		8	8.5	even	2
432.3.o.c.415.4		8	8.3	odd	2
576.3.o.e.319.2		8	12.11	even	2
576.3.o.e.319.3		8	3.2	odd	2
576.3.o.e.511.2		8	9.2	odd	6
576.3.o.e.511.3		8	36.11	even	6
1296.3.g.d.1135.1		4	72.13	even	6
1296.3.g.d.1135.2		4	72.67	odd	6
1296.3.g.h.1135.3		4	72.5	odd	6
1296.3.g.h.1135.4		4	72.59	even	6
1728.3.o.d.127.1		8	36.7	odd	6	inner
1728.3.o.d.127.2		8	9.7	even	3	inner
1728.3.o.d.1279.1		8	1.1	even	1	trivial
1728.3.o.d.1279.2		8	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-3.31174 + 5.73610i) q^{5} +(-8.46808 + 4.88905i) q^{7} +O(q^{10})\)	\(q+(-3.31174 + 5.73610i) q^{5} +(-8.46808 + 4.88905i) q^{7} +(10.0623 - 5.80948i) q^{11} +(8.93521 - 15.4762i) q^{13} -2.37652 q^{17} -14.0337i q^{19} +(35.9635 + 20.7636i) q^{23} +(-9.43521 - 16.3423i) q^{25} +(-5.68826 - 9.85236i) q^{29} +(-18.0334 - 10.4116i) q^{31} -64.7650i q^{35} -35.7409 q^{37} +(2.62957 - 4.55456i) q^{41} +(54.4939 - 31.4621i) q^{43} +(-4.28568 + 2.47434i) q^{47} +(23.3056 - 40.3666i) q^{49} +75.7409 q^{53} +76.9578i q^{55} +(50.3115 + 29.0474i) q^{59} +(-5.93521 - 10.2801i) q^{61} +(59.1822 + 102.507i) q^{65} +(20.6216 + 11.9059i) q^{67} +46.4758i q^{71} -104.352 q^{73} +(-56.8056 + 98.3903i) q^{77} +(-18.0334 + 10.4116i) q^{79} +(-24.4103 + 14.0933i) q^{83} +(7.87043 - 13.6320i) q^{85} -73.0122 q^{89} +174.739i q^{91} +(80.4984 + 46.4758i) q^{95} +(71.1113 + 123.168i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5}+O(q^{10}) \) 8 * q - 6 * q^5	\( 8 q - 6 q^{5} + 10 q^{13} - 60 q^{17} - 14 q^{25} - 66 q^{29} - 40 q^{37} + 144 q^{41} + 2 q^{49} + 360 q^{53} + 14 q^{61} + 330 q^{65} - 220 q^{73} - 270 q^{77} - 60 q^{85} - 912 q^{89} + 200 q^{97}+O(q^{100}) \) 8 * q - 6 * q^5 + 10 * q^13 - 60 * q^17 - 14 * q^25 - 66 * q^29 - 40 * q^37 + 144 * q^41 + 2 * q^49 + 360 * q^53 + 14 * q^61 + 330 * q^65 - 220 * q^73 - 270 * q^77 - 60 * q^85 - 912 * q^89 + 200 * q^97

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