LMFDB - Embedded newform 1764.4.t.b.521.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(521,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1764.521");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.t (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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + \cdots + 7375227456 $ x^16 - 4x^15 - 290x^14 + 1728x^13 + 29275x^12 - 246984x^11 - 955194x^10 + 14344616x^9 - 18123280x^8 - 273588032x^7 + 1239640536x^6 - 1407381792x^5 - 1961185792x^4 + 4297169408x^3 + 2991779296x^2 - 11217342336*x + 7375227456
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{18} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.4
Root			$8.15703 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.521
Dual form			1764.4.t.b.1097.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.41226 - 5.91021i) q^{5} +O(q^{10})$	$q+(-3.41226 - 5.91021i) q^{5} +(-50.5303 - 29.1737i) q^{11} +38.5535i q^{13} +(-16.1260 + 27.9310i) q^{17} +(-107.846 + 62.2650i) q^{19} +(-174.217 + 100.584i) q^{23} +(39.2130 - 67.9188i) q^{25} -104.357i q^{29} +(240.747 + 138.995i) q^{31} +(23.8286 + 41.2724i) q^{37} +387.272 q^{41} +272.528 q^{43} +(81.5941 + 141.325i) q^{47} +(-313.867 - 181.211i) q^{53} +398.193i q^{55} +(105.853 - 183.342i) q^{59} +(202.919 - 117.155i) q^{61} +(227.859 - 131.555i) q^{65} +(-262.131 + 454.024i) q^{67} +348.689i q^{71} +(-465.143 - 268.550i) q^{73} +(-362.792 - 628.374i) q^{79} +392.121 q^{83} +220.104 q^{85} +(-430.015 - 744.807i) q^{89} +(735.998 + 424.929i) q^{95} -978.030i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 72 q^{19} - 212 q^{25} + 708 q^{31} + 76 q^{37} + 1408 q^{43} + 1632 q^{61} - 1528 q^{67} + 2700 q^{73} - 364 q^{79} + 7392 q^{85}+O(q^{100}) $ 16 * q + 72 * q^19 - 212 * q^25 + 708 * q^31 + 76 * q^37 + 1408 * q^43 + 1632 * q^61 - 1528 * q^67 + 2700 * q^73 - 364 * q^79 + 7392 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	−3.41226	−	5.91021i	−0.305202	−	0.528625i	0.672104	−	0.740456i	$-0.265391\pi$
$5$	−3.41226	−	5.91021i	−0.305202	−	0.528625i	−0.977306	+	0.211831i	$0.932057\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−50.5303	−	29.1737i	−1.38504	−	0.799654i	−0.392290	−	0.919841i	$-0.628317\pi$
$11$	−50.5303	−	29.1737i	−1.38504	−	0.799654i	−0.992751	+	0.120187i	$0.961651\pi$
$12$		0			0
$13$			38.5535i			0.822525i	0.911517	+	0.411262i	$0.134912\pi$
$13$			38.5535i			0.822525i	−0.911517	+	0.411262i	$0.865088\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−16.1260	+	27.9310i	−0.230066	+	0.398486i	−0.957827	−	0.287345i	$-0.907227\pi$
$17$	−16.1260	+	27.9310i	−0.230066	+	0.398486i	0.727761	+	0.685830i	$0.240561\pi$
$18$		0			0
$19$	−107.846	+	62.2650i	−1.30219	+	0.751819i	−0.980779	−	0.195121i	$-0.937490\pi$
$19$	−107.846	+	62.2650i	−1.30219	+	0.751819i	−0.321410	+	0.946940i	$0.604157\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−174.217	+	100.584i	−1.57942	+	0.911880i	−0.584484	+	0.811405i	$0.698703\pi$
$23$	−174.217	+	100.584i	−1.57942	+	0.911880i	−0.994940	+	0.100475i	$0.967964\pi$
$24$		0			0
$25$	39.2130	−	67.9188i	0.313704	−	0.543351i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	104.357i		−	0.668226i	−0.942533	−	0.334113i	$-0.891563\pi$
$29$		−	104.357i		−	0.668226i	0.942533	−	0.334113i	$-0.108437\pi$
$30$		0			0
$31$	240.747	+	138.995i	1.39482	+	0.805299i	0.993844	−	0.110790i	$-0.0353382\pi$
$31$	240.747	+	138.995i	1.39482	+	0.805299i	0.400975	+	0.916089i	$0.368671\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	23.8286	+	41.2724i	0.105876	+	0.183382i	0.914096	−	0.405499i	$-0.132902\pi$
$37$	23.8286	+	41.2724i	0.105876	+	0.183382i	−0.808220	+	0.588881i	$0.799569\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	387.272			1.47516			0.737582	−	0.675258i	$-0.235968\pi$
$41$	387.272			1.47516			0.737582	+	0.675258i	$0.235968\pi$
$42$		0			0
$43$	272.528			0.966515			0.483257	−	0.875478i	$-0.339454\pi$
$43$	272.528			0.966515			0.483257	+	0.875478i	$0.339454\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	81.5941	+	141.325i	0.253228	+	0.438604i	0.964413	−	0.264401i	$-0.0851743\pi$
$47$	81.5941	+	141.325i	0.253228	+	0.438604i	−0.711185	+	0.703005i	$0.751841\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−313.867	−	181.211i	−0.813452	−	0.469646i	0.0347015	−	0.999398i	$-0.488952\pi$
$53$	−313.867	−	181.211i	−0.813452	−	0.469646i	−0.848153	+	0.529751i	$0.822285\pi$
$54$		0			0
$55$			398.193i			0.976224i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	105.853	−	183.342i	0.233573	−	0.404561i	−0.725284	−	0.688450i	$-0.758291\pi$
$59$	105.853	−	183.342i	0.233573	−	0.404561i	0.958857	+	0.283889i	$0.0916248\pi$
$60$		0			0
$61$	202.919	−	117.155i	0.425919	−	0.245905i	−0.271687	−	0.962386i	$-0.587582\pi$
$61$	202.919	−	117.155i	0.425919	−	0.245905i	0.697607	+	0.716481i	$0.254248\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	227.859	−	131.555i	0.434807	−	0.251036i
$66$		0			0
$67$	−262.131	+	454.024i	−0.477976	+	0.827879i	−0.999681	−	0.0252470i	$-0.991963\pi$
$67$	−262.131	+	454.024i	−0.477976	+	0.827879i	0.521705	+	0.853126i	$0.325296\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			348.689i			0.582842i	0.956595	+	0.291421i	$0.0941280\pi$
$71$			348.689i			0.582842i	−0.956595	+	0.291421i	$0.905872\pi$
$72$		0			0
$73$	−465.143	−	268.550i	−0.745765	−	0.430567i	0.0783969	−	0.996922i	$-0.475020\pi$
$73$	−465.143	−	268.550i	−0.745765	−	0.430567i	−0.824162	+	0.566355i	$0.808353\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−362.792	−	628.374i	−0.516675	−	0.894907i	−0.999813	−	0.0193623i	$-0.993836\pi$
$79$	−362.792	−	628.374i	−0.516675	−	0.894907i	0.483138	−	0.875544i	$-0.339497\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	392.121			0.518565			0.259283	−	0.965801i	$-0.416514\pi$
$83$	392.121			0.518565			0.259283	+	0.965801i	$0.416514\pi$
$84$		0			0
$85$	220.104			0.280866
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−430.015	−	744.807i	−0.512151	−	0.887072i	−0.999901	−	0.0140882i	$-0.995515\pi$
$89$	−430.015	−	744.807i	−0.512151	−	0.887072i	0.487750	−	0.872984i	$-0.337818\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	735.998	+	424.929i	0.794861	+	0.458913i
$96$		0			0
$97$		−	978.030i		−	1.02375i	−0.859059	−	0.511876i	$-0.828951\pi$
$97$		−	978.030i		−	1.02375i	0.859059	−	0.511876i	$-0.171049\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−224.150	+	388.240i	−0.220830	+	0.382488i	−0.955060	−	0.296412i	$-0.904210\pi$
$101$	−224.150	+	388.240i	−0.220830	+	0.382488i	0.734230	+	0.678900i	$0.237543\pi$
$102$		0			0
$103$	1137.56	−	656.773i	1.08823	−	0.628289i	0.155124	−	0.987895i	$-0.450422\pi$
$103$	1137.56	−	656.773i	1.08823	−	0.628289i	0.933104	+	0.359606i	$0.117089\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	161.506	−	93.2455i	0.145919	−	0.0842466i	−0.425263	−	0.905070i	$-0.639818\pi$
$107$	161.506	−	93.2455i	0.145919	−	0.0842466i	0.571182	+	0.820823i	$0.306485\pi$
$108$		0			0
$109$	−61.5811	+	106.662i	−0.0541137	+	0.0937277i	−0.891813	−	0.452404i	$-0.850567\pi$
$109$	−61.5811	+	106.662i	−0.0541137	+	0.0937277i	0.837700	+	0.546131i	$0.183900\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2267.03i		−	1.88729i	−0.330956	−	0.943646i	$-0.607371\pi$
$113$		−	2267.03i		−	1.88729i	0.330956	−	0.943646i	$-0.392629\pi$
$114$		0			0
$115$	1188.95	+	686.439i	0.964086	+	0.556615i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1036.71	+	1795.63i	0.778894	+	1.34908i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1388.28			−0.993376
$126$		0			0
$127$	839.285			0.586413			0.293207	−	0.956049i	$-0.405278\pi$
$127$	839.285			0.586413			0.293207	+	0.956049i	$0.405278\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1185.33	−	2053.06i	−0.790559	−	1.36929i	−0.925621	−	0.378451i	$-0.876457\pi$
$131$	−1185.33	−	2053.06i	−0.790559	−	1.36929i	0.135063	−	0.990837i	$-0.456876\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2656.64	+	1533.81i	1.65673	+	0.956513i	0.974209	+	0.225647i	$0.0724495\pi$
$137$	2656.64	+	1533.81i	1.65673	+	0.956513i	0.682520	+	0.730866i	$0.260884\pi$
$138$		0			0
$139$			2191.44i			1.33723i	0.743608	+	0.668616i	$0.233113\pi$
$139$			2191.44i			1.33723i	−0.743608	+	0.668616i	$0.766887\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1124.75	−	1948.12i	0.657735	−	1.13923i
$144$		0			0
$145$	−616.770	+	356.092i	−0.353241	+	0.203944i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−398.867	+	230.286i	−0.219305	+	0.126616i	−0.605628	−	0.795748i	$-0.707078\pi$
$149$	−398.867	+	230.286i	−0.219305	+	0.126616i	0.386323	+	0.922363i	$0.373745\pi$
$150$		0			0
$151$	−317.552	+	550.017i	−0.171139	+	0.296422i	−0.938818	−	0.344412i	$-0.888078\pi$
$151$	−317.552	+	550.017i	−0.171139	+	0.296422i	0.767679	+	0.640834i	$0.221412\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	1897.15i		−	0.983115i
$156$		0			0
$157$	−479.721	−	276.967i	−0.243859	−	0.140792i	0.373090	−	0.927795i	$-0.378298\pi$
$157$	−479.721	−	276.967i	−0.243859	−	0.140792i	−0.616949	+	0.787003i	$0.711632\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−687.621	−	1190.99i	−0.330421	−	0.572306i	0.652173	−	0.758070i	$-0.273857\pi$
$163$	−687.621	−	1190.99i	−0.330421	−	0.572306i	−0.982594	+	0.185764i	$0.940524\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3999.79			1.85337			0.926685	−	0.375839i	$-0.122645\pi$
$167$	3999.79			1.85337			0.926685	+	0.375839i	$0.122645\pi$
$168$		0			0
$169$	710.626			0.323453
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−225.948	−	391.354i	−0.0992979	−	0.171989i	0.812096	−	0.583523i	$-0.198326\pi$
$173$	−225.948	−	391.354i	−0.0992979	−	0.171989i	−0.911394	+	0.411534i	$0.864993\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1959.62	−	1131.39i	−0.818262	−	0.472424i	0.0315548	−	0.999502i	$-0.489954\pi$
$179$	−1959.62	−	1131.39i	−0.818262	−	0.472424i	−0.849817	+	0.527078i	$0.823287\pi$
$180$		0			0
$181$			3035.53i			1.24657i	0.781994	+	0.623286i	$0.214203\pi$
$181$			3035.53i			1.24657i	−0.781994	+	0.623286i	$0.785797\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	162.619	−	281.664i	0.0646270	−	0.111937i
$186$		0			0
$187$	1629.70	−	940.907i	0.637302	−	0.367946i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1676.68	+	968.031i	−0.635184	+	0.366724i	−0.782757	−	0.622327i	$-0.786187\pi$
$191$	−1676.68	+	968.031i	−0.635184	+	0.366724i	0.147573	+	0.989051i	$0.452854\pi$
$192$		0			0
$193$	1062.00	−	1839.43i	0.396084	−	0.686038i	−0.597155	−	0.802126i	$-0.703702\pi$
$193$	1062.00	−	1839.43i	0.396084	−	0.686038i	0.993239	+	0.116088i	$0.0370356\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2122.77i		−	0.767720i	−0.923391	−	0.383860i	$-0.874594\pi$
$197$		−	2122.77i		−	0.767720i	0.923391	−	0.383860i	$-0.125406\pi$
$198$		0			0
$199$	−1035.18	−	597.659i	−0.368752	−	0.212899i	0.304161	−	0.952621i	$-0.401624\pi$
$199$	−1035.18	−	597.659i	−0.368752	−	0.212899i	−0.672913	+	0.739722i	$0.734957\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1321.47	−	2288.86i	−0.450223	−	0.779809i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7265.99			2.40478
$210$		0			0
$211$	188.402			0.0614698			0.0307349	−	0.999528i	$-0.490215\pi$
$211$	188.402			0.0614698			0.0307349	+	0.999528i	$0.490215\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−929.937	−	1610.70i	−0.294982	−	0.510924i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1076.84	−	621.712i	−0.327764	−	0.189235i
$222$		0			0
$223$			2802.67i			0.841618i	0.907149	+	0.420809i	$0.138254\pi$
$223$			2802.67i			0.841618i	−0.907149	+	0.420809i	$0.861746\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	812.590	−	1407.45i	0.237593	−	0.411522i	−0.722430	−	0.691444i	$-0.756975\pi$
$227$	812.590	−	1407.45i	0.237593	−	0.411522i	0.960023	+	0.279921i	$0.0903084\pi$
$228$		0			0
$229$	5733.12	−	3310.02i	1.65439	−	0.955163i	0.679154	−	0.733996i	$-0.262347\pi$
$229$	5733.12	−	3310.02i	1.65439	−	0.955163i	0.975236	−	0.221167i	$-0.0709866\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1356.99	−	783.461i	0.381543	−	0.220284i	−0.296946	−	0.954894i	$-0.595968\pi$
$233$	1356.99	−	783.461i	0.381543	−	0.220284i	0.678490	+	0.734610i	$0.262635\pi$
$234$		0			0
$235$	556.841	−	964.477i	0.154571	−	0.267726i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	5211.23i		−	1.41040i	−0.709006	−	0.705202i	$-0.750856\pi$
$239$		−	5211.23i		−	1.41040i	0.709006	−	0.705202i	$-0.249144\pi$
$240$		0			0
$241$	3449.81	+	1991.75i	0.922082	+	0.532364i	0.884299	−	0.466922i	$-0.154637\pi$
$241$	3449.81	+	1991.75i	0.922082	+	0.532364i	0.0377833	+	0.999286i	$0.487970\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2400.53	−	4157.85i	−0.618390	−	1.07108i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7086.26			1.78199			0.890997	−	0.454009i	$-0.150007\pi$
$251$	7086.26			1.78199			0.890997	+	0.454009i	$0.150007\pi$
$252$		0			0
$253$	11737.6			2.91676
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1221.82	+	2116.25i	0.296556	+	0.513651i	0.975346	−	0.220682i	$-0.0708284\pi$
$257$	1221.82	+	2116.25i	0.296556	+	0.513651i	−0.678789	+	0.734333i	$0.737495\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2167.27	+	1251.27i	0.508135	+	0.293372i	0.732067	−	0.681233i	$-0.238556\pi$
$263$	2167.27	+	1251.27i	0.508135	+	0.293372i	−0.223932	+	0.974605i	$0.571889\pi$
$264$		0			0
$265$			2473.36i			0.573348i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4160.18	−	7205.64i	0.942939	−	1.63322i	0.183114	−	0.983092i	$-0.441382\pi$
$269$	4160.18	−	7205.64i	0.942939	−	1.63322i	0.759825	−	0.650127i	$-0.225284\pi$
$270$		0			0
$271$	−5816.51	+	3358.16i	−1.30379	+	0.752745i	−0.981052	−	0.193742i	$-0.937937\pi$
$271$	−5816.51	+	3358.16i	−1.30379	+	0.752745i	−0.322740	+	0.946488i	$0.604604\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3962.88	+	2287.97i	−0.868985	+	0.501709i
$276$		0			0
$277$	−3335.48	+	5777.22i	−0.723500	+	1.25314i	0.236089	+	0.971732i	$0.424134\pi$
$277$	−3335.48	+	5777.22i	−0.723500	+	1.25314i	−0.959589	+	0.281407i	$0.909199\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	5065.50i		−	1.07538i	−0.843142	−	0.537692i	$-0.819296\pi$
$281$		−	5065.50i		−	1.07538i	0.843142	−	0.537692i	$-0.180704\pi$
$282$		0			0
$283$	2227.91	+	1286.28i	0.467969	+	0.270182i	0.715389	−	0.698726i	$-0.246249\pi$
$283$	2227.91	+	1286.28i	0.467969	+	0.270182i	−0.247420	+	0.968908i	$0.579583\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1936.41	+	3353.96i	0.394139	+	0.682670i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5796.55			−1.15576			−0.577881	−	0.816121i	$-0.696120\pi$
$293$	−5796.55			−1.15576			−0.577881	+	0.816121i	$0.696120\pi$
$294$		0			0
$295$	−1444.79			−0.285148
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3877.87	−	6716.67i	−0.750044	−	1.29911i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1384.82	−	799.528i	−0.259983	−	0.150101i
$306$		0			0
$307$			753.054i			0.139997i	0.997547	+	0.0699985i	$0.0222994\pi$
$307$			753.054i			0.139997i	−0.997547	+	0.0699985i	$0.977701\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2694.45	+	4666.93i	−0.491281	+	0.850924i	−0.999950	−	0.0100384i	$-0.996805\pi$
$311$	−2694.45	+	4666.93i	−0.491281	+	0.850924i	0.508668	+	0.860963i	$0.330138\pi$
$312$		0			0
$313$	6116.48	−	3531.35i	1.10455	−	0.637712i	0.167138	−	0.985934i	$-0.446548\pi$
$313$	6116.48	−	3531.35i	1.10455	−	0.637712i	0.937412	+	0.348221i	$0.113214\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2126.55	+	1227.77i	−0.376780	+	0.217534i	−0.676416	−	0.736520i	$-0.736468\pi$
$317$	−2126.55	+	1227.77i	−0.376780	+	0.217534i	0.299636	+	0.954053i	$0.403135\pi$
$318$		0			0
$319$	−3044.47	+	5273.17i	−0.534350	+	0.925521i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4016.33i		−	0.691872i
$324$		0			0
$325$	2618.51	+	1511.80i	0.446919	+	0.258029i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	4255.12	+	7370.09i	0.706594	+	1.22386i	0.966113	+	0.258119i	$0.0831028\pi$
$331$	4255.12	+	7370.09i	0.706594	+	1.22386i	−0.259519	+	0.965738i	$0.583564\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3577.84			0.583517
$336$		0			0
$337$	1803.01			0.291443			0.145722	−	0.989326i	$-0.453450\pi$
$337$	1803.01			0.291443			0.145722	+	0.989326i	$0.453450\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8110.00	−	14046.9i	−1.28792	−	2.23075i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7237.09	+	4178.34i	1.11962	+	0.646412i	0.941303	−	0.337562i	$-0.109602\pi$
$347$	7237.09	+	4178.34i	1.11962	+	0.646412i	0.178314	+	0.983974i	$0.442936\pi$
$348$		0			0
$349$			4977.74i			0.763474i	0.924271	+	0.381737i	$0.124674\pi$
$349$			4977.74i			0.763474i	−0.924271	+	0.381737i	$0.875326\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3820.07	+	6616.56i	−0.575983	+	0.997631i	0.419952	+	0.907547i	$0.362047\pi$
$353$	−3820.07	+	6616.56i	−0.575983	+	0.997631i	−0.995934	+	0.0900846i	$0.971286\pi$
$354$		0			0
$355$	2060.82	−	1189.82i	0.308105	−	0.177884i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3601.64	−	2079.40i	0.529490	−	0.305701i	−0.211319	−	0.977417i	$-0.567776\pi$
$359$	3601.64	−	2079.40i	0.529490	−	0.305701i	0.740809	+	0.671716i	$0.234442\pi$
$360$		0			0
$361$	4324.35	−	7490.00i	0.630464	−	1.09200i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			3665.45i			0.525640i
$366$		0			0
$367$	1885.57	+	1088.64i	0.268191	+	0.154840i	0.628065	−	0.778161i	$-0.283847\pi$
$367$	1885.57	+	1088.64i	0.268191	+	0.154840i	−0.359874	+	0.933001i	$0.617180\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	190.136	+	329.324i	0.0263937	+	0.0457152i	0.878921	−	0.476968i	$-0.158264\pi$
$373$	190.136	+	329.324i	0.0263937	+	0.0457152i	−0.852527	+	0.522683i	$0.824931\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4023.32			0.549632
$378$		0			0
$379$	6918.48			0.937673			0.468837	−	0.883285i	$-0.344673\pi$
$379$	6918.48			0.937673			0.468837	+	0.883285i	$0.344673\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−63.4117	−	109.832i	−0.00846002	−	0.0146532i	0.861764	−	0.507309i	$-0.169360\pi$
$383$	−63.4117	−	109.832i	−0.00846002	−	0.0146532i	−0.870224	+	0.492656i	$0.836026\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1278.19	+	737.963i	0.166598	+	0.0961856i	0.580981	−	0.813917i	$-0.302669\pi$
$389$	1278.19	+	737.963i	0.166598	+	0.0961856i	−0.414383	+	0.910103i	$0.636002\pi$
$390$		0			0
$391$		−	6488.06i		−	0.839170i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2475.88	+	4288.35i	−0.315380	+	0.546254i
$396$		0			0
$397$	−751.551	+	433.908i	−0.0950107	+	0.0548545i	−0.546753	−	0.837294i	$-0.684136\pi$
$397$	−751.551	+	433.908i	−0.0950107	+	0.0548545i	0.451742	+	0.892149i	$0.350803\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−603.832	+	348.623i	−0.0751968	+	0.0434149i	−0.537127	−	0.843501i	$-0.680490\pi$
$401$	−603.832	+	348.623i	−0.0751968	+	0.0434149i	0.461930	+	0.886916i	$0.347157\pi$
$402$		0			0
$403$	−5358.75	+	9281.63i	−0.662378	+	1.14727i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2780.68i		−	0.338656i
$408$		0			0
$409$	5359.98	+	3094.59i	0.648005	+	0.374126i	0.787692	−	0.616070i	$-0.211276\pi$
$409$	5359.98	+	3094.59i	0.648005	+	0.374126i	−0.139686	+	0.990196i	$0.544609\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1338.02	−	2317.52i	−0.158267	−	0.274127i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1443.53			−0.168308			−0.0841538	−	0.996453i	$-0.526819\pi$
$419$	−1443.53			−0.168308			−0.0841538	+	0.996453i	$0.526819\pi$
$420$		0			0
$421$	−15750.0			−1.82330			−0.911648	−	0.410971i	$-0.865190\pi$
$421$	−15750.0			−1.82330			−0.911648	+	0.410971i	$0.865190\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1264.69	+	2190.51i	0.144345	+	0.250013i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9792.23	+	5653.55i	1.09437	+	0.631837i	0.934738	−	0.355339i	$-0.115634\pi$
$431$	9792.23	+	5653.55i	1.09437	+	0.631837i	0.159637	+	0.987176i	$0.448968\pi$
$432$		0			0
$433$			2318.26i			0.257295i	0.991690	+	0.128647i	$0.0410635\pi$
$433$			2318.26i			0.257295i	−0.991690	+	0.128647i	$0.958936\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12525.7	−	21695.2i	1.37114	−	2.37488i
$438$		0			0
$439$	6678.84	−	3856.03i	0.726113	−	0.419222i	−0.0908855	−	0.995861i	$-0.528970\pi$
$439$	6678.84	−	3856.03i	0.726113	−	0.419222i	0.816998	+	0.576640i	$0.195636\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11158.1	−	6442.15i	1.19670	−	0.690916i	0.236883	−	0.971538i	$-0.423874\pi$
$443$	11158.1	−	6442.15i	1.19670	−	0.690916i	0.959818	+	0.280623i	$0.0905410\pi$
$444$		0			0
$445$	−2934.64	+	5082.95i	−0.312619	+	0.541472i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	319.989i		−	0.0336330i	−0.999859	−	0.0168165i	$-0.994647\pi$
$449$		−	319.989i		−	0.0336330i	0.999859	−	0.0168165i	$-0.00535311\pi$
$450$		0			0
$451$	−19569.0	−	11298.1i	−2.04316	−	1.17962i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3897.48	−	6750.64i	−0.398942	−	0.690988i	0.594653	−	0.803982i	$-0.297289\pi$
$457$	−3897.48	−	6750.64i	−0.398942	−	0.690988i	−0.993596	+	0.112994i	$0.963956\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4610.97			0.465844			0.232922	−	0.972495i	$-0.425171\pi$
$461$	4610.97			0.465844			0.232922	+	0.972495i	$0.425171\pi$
$462$		0			0
$463$	15203.3			1.52604			0.763022	−	0.646373i	$-0.223715\pi$
$463$	15203.3			1.52604			0.763022	+	0.646373i	$0.223715\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8060.07	+	13960.4i	0.798663	+	1.38332i	0.920487	+	0.390773i	$0.127792\pi$
$467$	8060.07	+	13960.4i	0.798663	+	1.38332i	−0.121824	+	0.992552i	$0.538874\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−13770.9	−	7950.65i	−1.33866	−	0.772878i
$474$		0			0
$475$			9766.37i			0.943393i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3361.99	−	5823.14i	0.320696	−	0.555461i	−0.659936	−	0.751322i	$-0.729417\pi$
$479$	3361.99	−	5823.14i	0.320696	−	0.555461i	0.980632	+	0.195861i	$0.0627500\pi$
$480$		0			0
$481$	−1591.20	+	918.678i	−0.150836	+	0.0870854i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5780.36	+	3337.29i	−0.541181	+	0.312451i
$486$		0			0
$487$	5029.52	−	8711.38i	0.467986	−	0.810575i	−0.531345	−	0.847156i	$-0.678313\pi$
$487$	5029.52	−	8711.38i	0.467986	−	0.810575i	0.999331	+	0.0365803i	$0.0116465\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	18097.6i		−	1.66340i	−0.555222	−	0.831702i	$-0.687367\pi$
$491$		−	18097.6i		−	1.66340i	0.555222	−	0.831702i	$-0.312633\pi$
$492$		0			0
$493$	2914.78	+	1682.85i	0.266278	+	0.153736i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−5594.25	−	9689.53i	−0.501870	−	0.869264i	−0.999998	−	0.00216055i	$-0.999312\pi$
$499$	−5594.25	−	9689.53i	−0.501870	−	0.869264i	0.498128	−	0.867104i	$-0.334021\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2912.13			0.258142			0.129071	−	0.991635i	$-0.458800\pi$
$503$	2912.13			0.258142			0.129071	+	0.991635i	$0.458800\pi$
$504$		0			0
$505$	3059.44			0.269591
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2400.48	−	4157.75i	−0.209036	−	0.362061i	0.742375	−	0.669984i	$-0.233699\pi$
$509$	−2400.48	−	4157.75i	−0.209036	−	0.362061i	−0.951411	+	0.307924i	$0.900366\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7763.33	−	4482.16i	−0.664258	−	0.383510i
$516$		0			0
$517$		−	9521.61i		−	0.809980i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3256.63	+	5640.65i	−0.273849	+	0.474321i	−0.969844	−	0.243726i	$-0.921630\pi$
$521$	−3256.63	+	5640.65i	−0.273849	+	0.474321i	0.695995	+	0.718047i	$0.254964\pi$
$522$		0			0
$523$	3856.61	−	2226.62i	0.322443	−	0.186163i	−0.330038	−	0.943968i	$-0.607061\pi$
$523$	3856.61	−	2226.62i	0.322443	−	0.186163i	0.652481	+	0.757805i	$0.273728\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7764.54	+	4482.86i	−0.641800	+	0.370543i
$528$		0			0
$529$	14150.9	−	24510.0i	1.16305	−	2.01447i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			14930.7i			1.21336i
$534$		0			0
$535$	−1102.20	−	636.356i	−0.0890697	−	0.0514244i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−3885.32	−	6729.58i	−0.308767	−	0.534800i	0.669326	−	0.742969i	$-0.266583\pi$
$541$	−3885.32	−	6729.58i	−0.308767	−	0.534800i	−0.978093	+	0.208169i	$0.933250\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	840.523			0.0660624
$546$		0			0
$547$	−4095.62			−0.320139			−0.160069	−	0.987106i	$-0.551172\pi$
$547$	−4095.62			−0.320139			−0.160069	+	0.987106i	$0.551172\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6497.77	+	11254.5i	0.502385	+	0.870156i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−219.268	−	126.594i	−0.0166799	−	0.00963012i	0.491637	−	0.870800i	$-0.336399\pi$
$557$	−219.268	−	126.594i	−0.0166799	−	0.00963012i	−0.508317	+	0.861170i	$0.669732\pi$
$558$		0			0
$559$			10506.9i			0.794982i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10252.3	+	17757.6i	−0.767469	+	1.32929i	0.171463	+	0.985191i	$0.445151\pi$
$563$	−10252.3	+	17757.6i	−0.767469	+	1.32929i	−0.938931	+	0.344104i	$0.888183\pi$
$564$		0			0
$565$	−13398.6	+	7735.69i	−0.997670	+	0.576005i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11746.8	+	6782.03i	−0.865470	+	0.499679i	−0.865840	−	0.500321i	$-0.833215\pi$
$569$	−11746.8	+	6782.03i	−0.865470	+	0.499679i	0.000370481		1.00000i	$0.499882\pi$
$570$		0			0
$571$	2350.96	−	4071.98i	0.172302	−	0.298436i	−0.766922	−	0.641740i	$-0.778213\pi$
$571$	2350.96	−	4071.98i	0.172302	−	0.298436i	0.939224	+	0.343304i	$0.111546\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			15776.8i			1.14424i
$576$		0			0
$577$	3418.50	+	1973.67i	0.246644	+	0.142400i	0.618227	−	0.786000i	$-0.287851\pi$
$577$	3418.50	+	1973.67i	0.246644	+	0.142400i	−0.371582	+	0.928400i	$0.621185\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	10573.2	+	18313.3i	0.751110	+	1.30096i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12531.1			−0.881116			−0.440558	−	0.897724i	$-0.645219\pi$
$587$	−12531.1			−0.881116			−0.440558	+	0.897724i	$0.645219\pi$
$588$		0			0
$589$	−34618.1			−2.42176
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6475.41	+	11215.7i	0.448420	+	0.776686i	0.998283	−	0.0585683i	$-0.0186535\pi$
$593$	6475.41	+	11215.7i	0.448420	+	0.776686i	−0.549863	+	0.835255i	$0.685320\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	121.786	+	70.3130i	0.00830722	+	0.00479618i	0.504148	−	0.863617i	$-0.331807\pi$
$599$	121.786	+	70.3130i	0.00830722	+	0.00479618i	−0.495841	+	0.868414i	$0.665140\pi$
$600$		0			0
$601$		−	19220.3i		−	1.30451i	−0.758000	−	0.652255i	$-0.773823\pi$
$601$		−	19220.3i		−	1.30451i	0.758000	−	0.652255i	$-0.226177\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7075.03	−	12254.3i	0.475440	−	0.823486i
$606$		0			0
$607$	−14750.7	+	8516.30i	−0.986344	+	0.569466i	−0.904180	−	0.427152i	$-0.859517\pi$
$607$	−14750.7	+	8516.30i	−0.986344	+	0.569466i	−0.0821649	+	0.996619i	$0.526183\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5448.58	+	3145.74i	−0.360763	+	0.208287i
$612$		0			0
$613$	−3246.99	+	5623.95i	−0.213939	+	0.370553i	−0.952944	−	0.303147i	$-0.901963\pi$
$613$	−3246.99	+	5623.95i	−0.213939	+	0.370553i	0.739005	+	0.673700i	$0.235296\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	12338.5i		−	0.805074i	−0.915404	−	0.402537i	$-0.868129\pi$
$617$		−	12338.5i		−	0.805074i	0.915404	−	0.402537i	$-0.131871\pi$
$618$		0			0
$619$	8967.52	+	5177.40i	0.582286	+	0.336183i	0.762041	−	0.647528i	$-0.224197\pi$
$619$	8967.52	+	5177.40i	0.582286	+	0.336183i	−0.179755	+	0.983711i	$0.557531\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−164.430	−	284.802i	−0.0105235	−	0.0182273i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1537.04			−0.0974336
$630$		0			0
$631$	−4917.41			−0.310236			−0.155118	−	0.987896i	$-0.549576\pi$
$631$	−4917.41			−0.310236			−0.155118	+	0.987896i	$0.549576\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2863.86	−	4960.35i	−0.178974	−	0.309993i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14808.6	−	8549.73i	−0.912486	−	0.526824i	−0.0312556	−	0.999511i	$-0.509951\pi$
$641$	−14808.6	−	8549.73i	−0.912486	−	0.526824i	−0.881230	+	0.472688i	$0.843284\pi$
$642$		0			0
$643$			14631.4i			0.897365i	0.893691	+	0.448683i	$0.148107\pi$
$643$			14631.4i			0.897365i	−0.893691	+	0.448683i	$0.851893\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14808.8	+	25649.5i	−0.899833	+	1.55856i	−0.0721269	+	0.997395i	$0.522979\pi$
$647$	−14808.8	+	25649.5i	−0.899833	+	1.55856i	−0.827706	+	0.561161i	$0.810355\pi$
$648$		0			0
$649$	−10697.5	+	6176.22i	−0.647018	+	0.373556i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	24397.1	−	14085.7i	1.46207	−	0.844127i	0.462963	−	0.886378i	$-0.346786\pi$
$653$	24397.1	−	14085.7i	1.46207	−	0.844127i	0.999107	+	0.0422510i	$0.0134529\pi$
$654$		0			0
$655$	−8089.34	+	14011.2i	−0.482560	+	0.835818i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			25287.9i			1.49480i	0.664372	+	0.747402i	$0.268699\pi$
$659$			25287.9i			1.49480i	−0.664372	+	0.747402i	$0.731301\pi$
$660$		0			0
$661$	−8541.36	−	4931.35i	−0.502603	−	0.290178i	0.227185	−	0.973852i	$-0.427048\pi$
$661$	−8541.36	−	4931.35i	−0.502603	−	0.290178i	−0.729788	+	0.683674i	$0.760381\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	10496.6	+	18180.7i	0.609342	+	1.05541i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−13671.4			−0.786555
$672$		0			0
$673$	21145.6			1.21115			0.605574	−	0.795789i	$-0.292943\pi$
$673$	21145.6			1.21115			0.605574	+	0.795789i	$0.292943\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12062.4	−	20892.7i	−0.684779	−	1.18607i	−0.973506	−	0.228661i	$-0.926565\pi$
$677$	−12062.4	−	20892.7i	−0.684779	−	1.18607i	0.288727	−	0.957412i	$-0.406768\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	26779.8	+	15461.4i	1.50030	+	0.866197i	1.00000		0.000342338i	$0.000108969\pi$
$683$	26779.8	+	15461.4i	1.50030	+	0.866197i	0.500296	+	0.865854i	$0.333224\pi$
$684$		0			0
$685$		−	20935.1i		−	1.16772i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6986.33	−	12100.7i	0.386296	−	0.669084i
$690$		0			0
$691$	−10946.4	+	6319.92i	−0.602636	+	0.347932i	−0.770078	−	0.637950i	$-0.779783\pi$
$691$	−10946.4	+	6319.92i	−0.602636	+	0.347932i	0.167442	+	0.985882i	$0.446449\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12951.9	−	7477.76i	0.706895	−	0.408126i
$696$		0			0
$697$	−6245.13	+	10816.9i	−0.339385	+	0.587832i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			24051.5i			1.29588i	0.761690	+	0.647942i	$0.224370\pi$
$701$			24051.5i			1.29588i	−0.761690	+	0.647942i	$0.775630\pi$
$702$		0			0
$703$	−5139.65	−	2967.38i	−0.275741	−	0.159199i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5438.14	+	9419.14i	0.288059	+	0.498932i	0.973346	−	0.229340i	$-0.0736568\pi$
$709$	5438.14	+	9419.14i	0.288059	+	0.498932i	−0.685288	+	0.728273i	$0.740323\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−55922.8			−2.93735
$714$		0			0
$715$	−15351.7			−0.802968
$716$		0			0
$717$		0			0
$718$		0			0
$719$	10456.8	+	18111.7i	0.542381	+	0.939431i	0.998767	+	0.0496490i	$0.0158103\pi$
$719$	10456.8	+	18111.7i	0.542381	+	0.939431i	−0.456386	+	0.889782i	$0.650856\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7087.78	−	4092.13i	−0.363081	−	0.209625i
$726$		0			0
$727$		−	22897.9i		−	1.16814i	−0.811704	−	0.584068i	$-0.801460\pi$
$727$		−	22897.9i		−	1.16814i	0.811704	−	0.584068i	$-0.198540\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4394.77	+	7611.97i	−0.222362	+	0.385142i
$732$		0			0
$733$	26899.9	−	15530.7i	1.35549	−	0.782590i	0.366474	−	0.930428i	$-0.380565\pi$
$733$	26899.9	−	15530.7i	1.35549	−	0.782590i	0.989012	+	0.147838i	$0.0472315\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	26491.1	−	15294.7i	1.32403	−	0.764431i
$738$		0			0
$739$	−10170.9	+	17616.5i	−0.506281	+	0.876904i	0.493693	+	0.869636i	$0.335647\pi$
$739$	−10170.9	+	17616.5i	−0.506281	+	0.876904i	−0.999974	+	0.00726747i	$0.997687\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			31458.9i			1.55332i	0.629921	+	0.776660i	$0.283087\pi$
$743$			31458.9i			1.55332i	−0.629921	+	0.776660i	$0.716913\pi$
$744$		0			0
$745$	2722.08	+	1571.59i	0.133865	+	0.0772868i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−9653.81	−	16720.9i	−0.469071	−	0.812456i	0.530303	−	0.847808i	$-0.322078\pi$
$751$	−9653.81	−	16720.9i	−0.469071	−	0.812456i	−0.999375	+	0.0353523i	$0.988745\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4334.29			0.208928
$756$		0			0
$757$	−18791.2			−0.902216			−0.451108	−	0.892469i	$-0.648971\pi$
$757$	−18791.2			−0.902216			−0.451108	+	0.892469i	$0.648971\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	13249.7	+	22949.2i	0.631145	+	1.09318i	0.987318	+	0.158756i	$0.0507482\pi$
$761$	13249.7	+	22949.2i	0.631145	+	1.09318i	−0.356172	+	0.934420i	$0.615918\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7068.48	+	4080.99i	0.332761	+	0.192120i
$768$		0			0
$769$		−	29077.1i		−	1.36352i	−0.731575	−	0.681761i	$-0.761215\pi$
$769$		−	29077.1i		−	1.36352i	0.731575	−	0.681761i	$-0.238785\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14605.3	+	25297.2i	−0.679582	+	1.17707i	0.295525	+	0.955335i	$0.404505\pi$
$773$	−14605.3	+	25297.2i	−0.679582	+	1.17707i	−0.975107	+	0.221735i	$0.928828\pi$
$774$		0			0
$775$	18880.8	−	10900.8i	0.875119	−	0.505250i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−41765.8	+	24113.5i	−1.92094	+	1.10906i
$780$		0			0
$781$	10172.5	−	17619.4i	0.466072	−	0.807260i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			3780.33i			0.171880i
$786$		0			0
$787$	−29887.2	−	17255.4i	−1.35370	−	0.781561i	−0.364937	−	0.931032i	$-0.618910\pi$
$787$	−29887.2	−	17255.4i	−1.35370	−	0.781561i	−0.988766	+	0.149471i	$0.952243\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	4516.74	+	7823.23i	0.202263	+	0.350329i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−40125.5			−1.78333			−0.891667	−	0.452692i	$-0.850464\pi$
$797$	−40125.5			−1.78333			−0.891667	+	0.452692i	$0.850464\pi$
$798$		0			0
$799$	−5263.13			−0.233037
$800$		0			0
$801$		0			0
$802$		0			0
$803$	15669.2	+	27139.8i	0.688610	+	1.19271i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22092.7	−	12755.2i	−0.960121	−	0.554326i	−0.0639107	−	0.997956i	$-0.520357\pi$
$809$	−22092.7	−	12755.2i	−0.960121	−	0.554326i	−0.896210	+	0.443630i	$0.853691\pi$
$810$		0			0
$811$		−	34785.8i		−	1.50616i	−0.657930	−	0.753079i	$-0.728568\pi$
$811$		−	34785.8i		−	1.50616i	0.657930	−	0.753079i	$-0.271432\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4692.68	+	8127.97i	−0.201690	+	0.349338i
$816$		0			0
$817$	−29391.1	+	16968.9i	−1.25858	+	0.726644i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	19574.7	−	11301.4i	0.832108	−	0.480418i	−0.0224658	−	0.999748i	$-0.507152\pi$
$821$	19574.7	−	11301.4i	0.832108	−	0.480418i	0.854574	+	0.519330i	$0.173818\pi$
$822$		0			0
$823$	9554.31	−	16548.6i	0.404669	−	0.700907i	−0.589614	−	0.807685i	$-0.700720\pi$
$823$	9554.31	−	16548.6i	0.404669	−	0.700907i	0.994283	+	0.106778i	$0.0340535\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			7545.77i			0.317282i	0.987336	+	0.158641i	$0.0507112\pi$
$827$			7545.77i			0.317282i	−0.987336	+	0.158641i	$0.949289\pi$
$828$		0			0
$829$	27729.7	+	16009.8i	1.16175	+	0.670738i	0.951723	−	0.306957i	$-0.0993108\pi$
$829$	27729.7	+	16009.8i	1.16175	+	0.670738i	0.210029	+	0.977695i	$0.432644\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−13648.3	−	23639.6i	−0.565652	−	0.979738i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	33582.9			1.38190			0.690948	−	0.722905i	$-0.257194\pi$
$839$	33582.9			1.38190			0.690948	+	0.722905i	$0.257194\pi$
$840$		0			0
$841$	13498.7			0.553474
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2424.84	−	4199.95i	−0.0987185	−	0.170985i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8302.70	−	4793.57i	−0.334445	−	0.193092i
$852$		0			0
$853$		−	119.927i		−	0.00481387i	−0.999997	−	0.00240693i	$-0.999234\pi$
$853$		−	119.927i		−	0.00481387i	0.999997	−	0.00240693i	$-0.000766151\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2606.04	+	4513.80i	−0.103875	+	0.179917i	−0.913278	−	0.407337i	$-0.866457\pi$
$857$	−2606.04	+	4513.80i	−0.103875	+	0.179917i	0.809403	+	0.587253i	$0.199791\pi$
$858$		0			0
$859$	−16503.5	+	9528.29i	−0.655520	+	0.378464i	−0.790568	−	0.612375i	$-0.790214\pi$
$859$	−16503.5	+	9528.29i	−0.655520	+	0.378464i	0.135048	+	0.990839i	$0.456881\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3451.41	−	1992.67i	0.136138	−	0.0785994i	−0.430384	−	0.902646i	$-0.641622\pi$
$863$	3451.41	−	1992.67i	0.136138	−	0.0785994i	0.566522	+	0.824046i	$0.308289\pi$
$864$		0			0
$865$	−1541.99	+	2670.81i	−0.0606118	+	0.104983i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			42335.9i			1.65264i
$870$		0			0
$871$	−17504.2	−	10106.1i	−0.680951	−	0.393147i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	15501.1	+	26848.8i	0.596848	+	1.03377i	0.993283	+	0.115709i	$0.0369140\pi$
$877$	15501.1	+	26848.8i	0.596848	+	1.03377i	−0.396435	+	0.918063i	$0.629753\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−29240.6			−1.11821			−0.559103	−	0.829098i	$-0.688854\pi$
$881$	−29240.6			−1.11821			−0.559103	+	0.829098i	$0.688854\pi$
$882$		0			0
$883$	−42056.9			−1.60286			−0.801431	−	0.598088i	$-0.795927\pi$
$883$	−42056.9			−1.60286			−0.801431	+	0.598088i	$0.795927\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1216.16	+	2106.45i	0.0460367	+	0.0797379i	0.888126	−	0.459601i	$-0.152008\pi$
$887$	1216.16	+	2106.45i	0.0460367	+	0.0797379i	−0.842089	+	0.539339i	$0.818674\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−17599.2	−	10160.9i	−0.659502	−	0.380764i
$894$		0			0
$895$			15442.4i			0.576738i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	14505.1	−	25123.5i	0.538122	−	0.932054i
$900$		0			0
$901$	10122.8	−	5844.41i	0.374295	−	0.216099i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	17940.6	−	10358.0i	0.658969	−	0.380456i
$906$		0			0
$907$	3376.70	−	5848.61i	0.123618	−	0.214112i	−0.797574	−	0.603221i	$-0.793884\pi$
$907$	3376.70	−	5848.61i	0.123618	−	0.214112i	0.921192	+	0.389109i	$0.127217\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	13545.2i		−	0.492615i	−0.969192	−	0.246308i	$-0.920783\pi$
$911$		−	13545.2i		−	0.492615i	0.969192	−	0.246308i	$-0.0792174\pi$
$912$		0			0
$913$	−19814.0	−	11439.6i	−0.718234	−	0.414673i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−19698.8	−	34119.3i	−0.707077	−	1.22469i	−0.965937	−	0.258778i	$-0.916680\pi$
$919$	−19698.8	−	34119.3i	−0.707077	−	1.22469i	0.258860	−	0.965915i	$-0.416653\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−13443.2			−0.479402
$924$		0			0
$925$	3737.56			0.132854
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5574.09	+	9654.60i	0.196857	+	0.340966i	0.947508	−	0.319733i	$-0.103593\pi$
$929$	5574.09	+	9654.60i	0.196857	+	0.340966i	−0.750651	+	0.660699i	$0.770260\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−11121.9	−	6421.24i	−0.389011	−	0.224596i
$936$		0			0
$937$			22882.5i			0.797800i	0.916994	+	0.398900i	$0.130608\pi$
$937$			22882.5i			0.797800i	−0.916994	+	0.398900i	$0.869392\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2825.28	+	4893.52i	−0.0978761	+	0.169526i	−0.910805	−	0.412836i	$-0.864538\pi$
$941$	−2825.28	+	4893.52i	−0.0978761	+	0.169526i	0.812929	+	0.582362i	$0.197872\pi$
$942$		0			0
$943$	−67469.3	+	38953.4i	−2.32991	+	1.34517i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26394.5	+	15238.9i	−0.905709	+	0.522911i	−0.879048	−	0.476733i	$-0.841821\pi$
$947$	−26394.5	+	15238.9i	−0.905709	+	0.522911i	−0.0266611	+	0.999645i	$0.508487\pi$
$948$		0			0
$949$	10353.6	−	17932.9i	0.354152	−	0.613410i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	24139.2i		−	0.820511i	−0.911971	−	0.410255i	$-0.865440\pi$
$953$		−	24139.2i		−	0.820511i	0.911971	−	0.410255i	$-0.134560\pi$
$954$		0			0
$955$	11442.5	+	6606.35i	0.387719	+	0.223850i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	23743.8	+	41125.5i	0.797013	+	1.38047i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−14495.2			−0.483542
$966$		0			0
$967$	11475.1			0.381609			0.190804	−	0.981628i	$-0.438890\pi$
$967$	11475.1			0.381609			0.190804	+	0.981628i	$0.438890\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2519.42	+	4363.76i	0.0832667	+	0.144222i	0.904651	−	0.426153i	$-0.140131\pi$
$971$	2519.42	+	4363.76i	0.0832667	+	0.144222i	−0.821385	+	0.570375i	$0.806798\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11585.2	−	6688.74i	−0.379370	−	0.219029i	0.298174	−	0.954512i	$-0.403622\pi$
$977$	−11585.2	−	6688.74i	−0.379370	−	0.219029i	−0.677544	+	0.735482i	$0.736956\pi$
$978$		0			0
$979$			50180.4i			1.63818i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−5785.08	+	10020.1i	−0.187707	+	0.325117i	−0.944485	−	0.328554i	$-0.893439\pi$
$983$	−5785.08	+	10020.1i	−0.187707	+	0.325117i	0.756779	+	0.653671i	$0.226772\pi$
$984$		0			0
$985$	−12546.0	+	7243.43i	−0.405836	+	0.234310i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−47479.0	+	27412.0i	−1.52654	+	0.881346i
$990$		0			0
$991$	−10029.6	+	17371.7i	−0.321493	+	0.556842i	−0.980796	−	0.195035i	$-0.937518\pi$
$991$	−10029.6	+	17371.7i	−0.321493	+	0.556842i	0.659303	+	0.751877i	$0.270851\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			8157.47i			0.259909i
$996$		0			0
$997$	29515.9	+	17041.0i	0.937592	+	0.541319i	0.889205	−	0.457510i	$-0.151259\pi$
$997$	29515.9	+	17041.0i	0.937592	+	0.541319i	0.0483872	+	0.998829i	$0.484592\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.t.b.521.4		16
3.2	odd	2	inner	1764.4.t.b.521.5		16
7.2	even	3		252.4.t.a.89.4	yes	16
7.3	odd	6		1764.4.f.a.881.8		16
7.4	even	3		1764.4.f.a.881.10		16
7.5	odd	6	inner	1764.4.t.b.1097.5		16
7.6	odd	2		252.4.t.a.17.5	yes	16
21.2	odd	6		252.4.t.a.89.5	yes	16
21.5	even	6	inner	1764.4.t.b.1097.4		16
21.11	odd	6		1764.4.f.a.881.7		16
21.17	even	6		1764.4.f.a.881.9		16
21.20	even	2		252.4.t.a.17.4	✓	16
28.23	odd	6		1008.4.bt.b.593.4		16
28.27	even	2		1008.4.bt.b.17.5		16
84.23	even	6		1008.4.bt.b.593.5		16
84.83	odd	2		1008.4.bt.b.17.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.t.a.17.4	✓	16	21.20	even	2
252.4.t.a.17.5	yes	16	7.6	odd	2
252.4.t.a.89.4	yes	16	7.2	even	3
252.4.t.a.89.5	yes	16	21.2	odd	6
1008.4.bt.b.17.4		16	84.83	odd	2
1008.4.bt.b.17.5		16	28.27	even	2
1008.4.bt.b.593.4		16	28.23	odd	6
1008.4.bt.b.593.5		16	84.23	even	6
1764.4.f.a.881.7		16	21.11	odd	6
1764.4.f.a.881.8		16	7.3	odd	6
1764.4.f.a.881.9		16	21.17	even	6
1764.4.f.a.881.10		16	7.4	even	3
1764.4.t.b.521.4		16	1.1	even	1	trivial
1764.4.t.b.521.5		16	3.2	odd	2	inner
1764.4.t.b.1097.4		16	21.5	even	6	inner
1764.4.t.b.1097.5		16	7.5	odd	6	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.41226 - 5.91021i) q^{5} +O(q^{10})\)	\(q+(-3.41226 - 5.91021i) q^{5} +(-50.5303 - 29.1737i) q^{11} +38.5535i q^{13} +(-16.1260 + 27.9310i) q^{17} +(-107.846 + 62.2650i) q^{19} +(-174.217 + 100.584i) q^{23} +(39.2130 - 67.9188i) q^{25} -104.357i q^{29} +(240.747 + 138.995i) q^{31} +(23.8286 + 41.2724i) q^{37} +387.272 q^{41} +272.528 q^{43} +(81.5941 + 141.325i) q^{47} +(-313.867 - 181.211i) q^{53} +398.193i q^{55} +(105.853 - 183.342i) q^{59} +(202.919 - 117.155i) q^{61} +(227.859 - 131.555i) q^{65} +(-262.131 + 454.024i) q^{67} +348.689i q^{71} +(-465.143 - 268.550i) q^{73} +(-362.792 - 628.374i) q^{79} +392.121 q^{83} +220.104 q^{85} +(-430.015 - 744.807i) q^{89} +(735.998 + 424.929i) q^{95} -978.030i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 72 q^{19} - 212 q^{25} + 708 q^{31} + 76 q^{37} + 1408 q^{43} + 1632 q^{61} - 1528 q^{67} + 2700 q^{73} - 364 q^{79} + 7392 q^{85}+O(q^{100}) \) 16 * q + 72 * q^19 - 212 * q^25 + 708 * q^31 + 76 * q^37 + 1408 * q^43 + 1632 * q^61 - 1528 * q^67 + 2700 * q^73 - 364 * q^79 + 7392 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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