LMFDB - Embedded newform 1764.3.z.n.325.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(325,1764)]

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, 0, 1]))

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

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

chi := DirichletCharacter("1764.325");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.z (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:	$48.0655186332$
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} - 12x^{14} + 114x^{12} - 336x^{10} + 755x^{8} - 336x^{6} + 114x^{4} - 12x^{2} + 1 $ x^16 - 12x^14 + 114x^12 - 336x^10 + 755x^8 - 336x^6 + 114x^4 - 12*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{6}\cdot 7^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			325.4
Root			$-0.293380 - 0.169383i$ of defining polynomial
Character	$\chi$	$=$	1764.325
Dual form			1764.3.z.n.901.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.68448 + 2.12723i) q^{5} +O(q^{10})$	$q+(-3.68448 + 2.12723i) q^{5} +(9.48935 - 16.4360i) q^{11} +4.46088i q^{13} +(-11.0534 - 6.38170i) q^{17} +(-8.11475 + 4.68506i) q^{19} +(8.53562 + 14.7841i) q^{23} +(-3.44975 + 5.97514i) q^{25} +20.8862 q^{29} +(-13.5116 - 7.80092i) q^{31} +(-22.8492 - 39.5760i) q^{37} +63.3894i q^{41} +2.20101 q^{43} +(43.8434 - 25.3130i) q^{47} +(-18.9787 + 32.8721i) q^{53} +80.7443i q^{55} +(58.5813 + 33.8220i) q^{59} +(-54.8814 + 31.6858i) q^{61} +(-9.48935 - 16.4360i) q^{65} +(-13.6985 + 23.7265i) q^{67} +15.1638 q^{71} +(-72.5860 - 41.9075i) q^{73} +(-66.5980 - 115.351i) q^{79} -109.761i q^{83} +54.3015 q^{85} +(-69.2645 + 39.9899i) q^{89} +(19.9324 - 34.5240i) q^{95} +119.547i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 24 q^{25} - 128 q^{37} + 352 q^{43} + 256 q^{67} - 432 q^{79} + 1344 q^{85}+O(q^{100}) $ 16 * q + 24 * q^25 - 128 * q^37 + 352 * q^43 + 256 * q^67 - 432 * q^79 + 1344 * 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
$5$	−3.68448	+	2.12723i	−0.736896	+	0.425447i	−0.820940	−	0.571015i	$-0.806550\pi$
$5$	−3.68448	+	2.12723i	−0.736896	+	0.425447i	0.0840439	+	0.996462i	$0.473216\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	9.48935	−	16.4360i	0.862668	−	1.49418i	−0.00667619	−	0.999978i	$-0.502125\pi$
$11$	9.48935	−	16.4360i	0.862668	−	1.49418i	0.869344	−	0.494207i	$-0.164542\pi$
$12$		0			0
$13$			4.46088i			0.343145i	0.985171	+	0.171572i	$0.0548848\pi$
$13$			4.46088i			0.343145i	−0.985171	+	0.171572i	$0.945115\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−11.0534	−	6.38170i	−0.650202	−	0.375394i	0.138332	−	0.990386i	$-0.455826\pi$
$17$	−11.0534	−	6.38170i	−0.650202	−	0.375394i	−0.788534	+	0.614992i	$0.789159\pi$
$18$		0			0
$19$	−8.11475	+	4.68506i	−0.427092	+	0.246582i	−0.698107	−	0.715993i	$-0.745974\pi$
$19$	−8.11475	+	4.68506i	−0.427092	+	0.246582i	0.271015	+	0.962575i	$0.412641\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	8.53562	+	14.7841i	0.371114	+	0.642788i	0.989737	−	0.142900i	$-0.0456426\pi$
$23$	8.53562	+	14.7841i	0.371114	+	0.642788i	−0.618623	+	0.785688i	$0.712309\pi$
$24$		0			0
$25$	−3.44975	+	5.97514i	−0.137990	+	0.239006i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	20.8862			0.720212			0.360106	−	0.932911i	$-0.382741\pi$
$29$	20.8862			0.720212			0.360106	+	0.932911i	$0.382741\pi$
$30$		0			0
$31$	−13.5116	−	7.80092i	−0.435858	−	0.251642i	0.265981	−	0.963978i	$-0.414304\pi$
$31$	−13.5116	−	7.80092i	−0.435858	−	0.251642i	−0.701839	+	0.712336i	$0.747637\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−22.8492	−	39.5760i	−0.617547	−	1.06962i	−0.989932	−	0.141545i	$-0.954793\pi$
$37$	−22.8492	−	39.5760i	−0.617547	−	1.06962i	0.372385	−	0.928078i	$-0.378540\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			63.3894i			1.54608i	0.634355	+	0.773042i	$0.281266\pi$
$41$			63.3894i			1.54608i	−0.634355	+	0.773042i	$0.718734\pi$
$42$		0			0
$43$	2.20101			0.0511863			0.0255931	−	0.999672i	$-0.491853\pi$
$43$	2.20101			0.0511863			0.0255931	+	0.999672i	$0.491853\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	43.8434	−	25.3130i	0.932839	−	0.538575i	0.0451306	−	0.998981i	$-0.485630\pi$
$47$	43.8434	−	25.3130i	0.932839	−	0.538575i	0.887708	+	0.460406i	$0.152296\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−18.9787	+	32.8721i	−0.358089	+	0.620228i	−0.987641	−	0.156730i	$-0.949905\pi$
$53$	−18.9787	+	32.8721i	−0.358089	+	0.620228i	0.629553	+	0.776958i	$0.283238\pi$
$54$		0			0
$55$			80.7443i			1.46808i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	58.5813	+	33.8220i	0.992904	+	0.573253i	0.906141	−	0.422976i	$-0.139014\pi$
$59$	58.5813	+	33.8220i	0.992904	+	0.573253i	0.0867630	+	0.996229i	$0.472348\pi$
$60$		0			0
$61$	−54.8814	+	31.6858i	−0.899695	+	0.519439i	−0.877101	−	0.480305i	$-0.840526\pi$
$61$	−54.8814	+	31.6858i	−0.899695	+	0.519439i	−0.0225940	+	0.999745i	$0.507193\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−9.48935	−	16.4360i	−0.145990	−	0.252862i
$66$		0			0
$67$	−13.6985	+	23.7265i	−0.204455	+	0.354126i	−0.949959	−	0.312375i	$-0.898876\pi$
$67$	−13.6985	+	23.7265i	−0.204455	+	0.354126i	0.745504	+	0.666501i	$0.232209\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	15.1638			0.213574			0.106787	−	0.994282i	$-0.465944\pi$
$71$	15.1638			0.213574			0.106787	+	0.994282i	$0.465944\pi$
$72$		0			0
$73$	−72.5860	−	41.9075i	−0.994329	−	0.574076i	−0.0877632	−	0.996141i	$-0.527972\pi$
$73$	−72.5860	−	41.9075i	−0.994329	−	0.574076i	−0.906565	+	0.422066i	$0.861305\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−66.5980	−	115.351i	−0.843012	−	1.46014i	−0.887336	−	0.461123i	$-0.847447\pi$
$79$	−66.5980	−	115.351i	−0.843012	−	1.46014i	0.0443238	−	0.999017i	$-0.485887\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	109.761i		−	1.32242i	−0.750200	−	0.661211i	$-0.770043\pi$
$83$		−	109.761i		−	1.32242i	0.750200	−	0.661211i	$-0.229957\pi$
$84$		0			0
$85$	54.3015			0.638841
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−69.2645	+	39.9899i	−0.778252	+	0.449324i	−0.835811	−	0.549018i	$-0.815002\pi$
$89$	−69.2645	+	39.9899i	−0.778252	+	0.449324i	0.0575582	+	0.998342i	$0.481669\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	19.9324	−	34.5240i	0.209815	−	0.363410i
$96$		0			0
$97$			119.547i			1.23245i	0.787572	+	0.616223i	$0.211338\pi$
$97$			119.547i			1.23245i	−0.787572	+	0.616223i	$0.788662\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−127.846	−	73.8118i	−1.26580	−	0.730810i	−0.291610	−	0.956537i	$-0.594191\pi$
$101$	−127.846	−	73.8118i	−1.26580	−	0.730810i	−0.974191	+	0.225727i	$0.927524\pi$
$102$		0			0
$103$	−159.577	+	92.1319i	−1.54929	+	0.894485i	−0.551097	+	0.834441i	$0.685791\pi$
$103$	−159.577	+	92.1319i	−1.54929	+	0.894485i	−0.998196	+	0.0600435i	$0.980876\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−23.6994	−	41.0486i	−0.221490	−	0.383632i	0.733771	−	0.679397i	$-0.237759\pi$
$107$	−23.6994	−	41.0486i	−0.221490	−	0.383632i	−0.955261	+	0.295766i	$0.904425\pi$
$108$		0			0
$109$	−49.4472	+	85.6451i	−0.453644	+	0.785735i	−0.998609	−	0.0527240i	$-0.983210\pi$
$109$	−49.4472	+	85.6451i	−0.453644	+	0.785735i	0.544965	+	0.838459i	$0.316543\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−187.880			−1.66265			−0.831325	−	0.555786i	$-0.812417\pi$
$113$	−187.880			−1.66265			−0.831325	+	0.555786i	$0.812417\pi$
$114$		0			0
$115$	−62.8986	−	36.3145i	−0.546944	−	0.315779i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−119.595	−	207.145i	−0.988392	−	1.71195i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	135.715i		−	1.08572i
$126$		0			0
$127$	−22.9949			−0.181063			−0.0905313	−	0.995894i	$-0.528857\pi$
$127$	−22.9949			−0.181063			−0.0905313	+	0.995894i	$0.528857\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−124.532	+	71.8984i	−0.950623	+	0.548843i	−0.893274	−	0.449512i	$-0.851598\pi$
$131$	−124.532	+	71.8984i	−0.950623	+	0.548843i	−0.0573487	+	0.998354i	$0.518265\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−33.2367	+	57.5676i	−0.242604	+	0.420202i	−0.961455	−	0.274962i	$-0.911335\pi$
$137$	−33.2367	+	57.5676i	−0.242604	+	0.420202i	0.718852	+	0.695164i	$0.244668\pi$
$138$		0			0
$139$			246.268i			1.77171i	0.463961	+	0.885856i	$0.346428\pi$
$139$			246.268i			1.77171i	−0.463961	+	0.885856i	$0.653572\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	73.3193	+	42.3309i	0.512722	+	0.296020i
$144$		0			0
$145$	−76.9546	+	44.4297i	−0.530721	+	0.306412i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	39.8648	+	69.0479i	0.267549	+	0.463409i	0.968228	−	0.250068i	$-0.0804528\pi$
$149$	39.8648	+	69.0479i	0.267549	+	0.463409i	−0.700679	+	0.713477i	$0.747120\pi$
$150$		0			0
$151$	−16.2965	+	28.2263i	−0.107924	+	0.186929i	−0.914929	−	0.403615i	$-0.867754\pi$
$151$	−16.2965	+	28.2263i	−0.107924	+	0.186929i	0.807005	+	0.590544i	$0.201087\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	66.3775			0.428242
$156$		0			0
$157$	−220.515	−	127.314i	−1.40455	−	0.810919i	−0.409698	−	0.912221i	$-0.634366\pi$
$157$	−220.515	−	127.314i	−1.40455	−	0.810919i	−0.994856	+	0.101302i	$0.967699\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−121.497	−	210.440i	−0.745383	−	1.29104i	−0.950015	−	0.312203i	$-0.898933\pi$
$163$	−121.497	−	210.440i	−0.745383	−	1.29104i	0.204632	−	0.978839i	$-0.434400\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			144.224i			0.863619i	0.901965	+	0.431809i	$0.142125\pi$
$167$			144.224i			0.863619i	−0.901965	+	0.431809i	$0.857875\pi$
$168$		0			0
$169$	149.101			0.882252
$170$		0			0
$171$		0			0
$172$		0			0
$173$	289.593	−	167.196i	1.67395	−	0.966453i	0.708550	−	0.705660i	$-0.249349\pi$
$173$	289.593	−	167.196i	1.67395	−	0.966453i	0.965395	−	0.260793i	$-0.0839839\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−58.7956	+	101.837i	−0.328467	+	0.568922i	−0.982208	−	0.187797i	$-0.939865\pi$
$179$	−58.7956	+	101.837i	−0.328467	+	0.568922i	0.653741	+	0.756719i	$0.273199\pi$
$180$		0			0
$181$			173.481i			0.958459i	0.877690	+	0.479230i	$0.159084\pi$
$181$			173.481i			0.958459i	−0.877690	+	0.479230i	$0.840916\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	168.375	+	97.2114i	0.910135	+	0.525467i
$186$		0			0
$187$	−209.780	+	121.116i	−1.12182	+	0.647681i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−130.991	−	226.884i	−0.685819	−	1.18787i	−0.973179	−	0.230050i	$-0.926111\pi$
$191$	−130.991	−	226.884i	−0.685819	−	1.18787i	0.287360	−	0.957823i	$-0.407222\pi$
$192$		0			0
$193$	−14.7990	+	25.6326i	−0.0766787	+	0.132811i	−0.901815	−	0.432122i	$-0.857765\pi$
$193$	−14.7990	+	25.6326i	−0.0766787	+	0.132811i	0.825136	+	0.564934i	$0.191098\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	237.282			1.20448			0.602238	−	0.798317i	$-0.294276\pi$
$197$	237.282			1.20448			0.602238	+	0.798317i	$0.294276\pi$
$198$		0			0
$199$	25.5871	+	14.7727i	0.128579	+	0.0742349i	0.562910	−	0.826518i	$-0.309682\pi$
$199$	25.5871	+	14.7727i	0.128579	+	0.0742349i	−0.434331	+	0.900753i	$0.643015\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−134.844	−	233.557i	−0.657777	−	1.13930i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			177.832i			0.850873i
$210$		0			0
$211$	−157.990			−0.748767			−0.374384	−	0.927274i	$-0.622146\pi$
$211$	−157.990			−0.748767			−0.374384	+	0.927274i	$0.622146\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.10957	+	4.68206i	−0.0377189	+	0.0217770i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	28.4680	−	49.3081i	0.128815	−	0.223114i
$222$		0			0
$223$			225.779i			1.01246i	0.862397	+	0.506232i	$0.168962\pi$
$223$			225.779i			1.01246i	−0.862397	+	0.506232i	$0.831038\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−124.902	−	72.1122i	−0.550229	−	0.317675i	0.198985	−	0.980002i	$-0.436235\pi$
$227$	−124.902	−	72.1122i	−0.550229	−	0.317675i	−0.749214	+	0.662328i	$0.769569\pi$
$228$		0			0
$229$	−107.880	+	62.2845i	−0.471092	+	0.271985i	−0.716697	−	0.697385i	$-0.754347\pi$
$229$	−107.880	+	62.2845i	−0.471092	+	0.271985i	0.245605	+	0.969370i	$0.421013\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	213.486	+	369.769i	0.916251	+	1.58699i	0.805060	+	0.593193i	$0.202133\pi$
$233$	213.486	+	369.769i	0.916251	+	1.58699i	0.111190	+	0.993799i	$0.464534\pi$
$234$		0			0
$235$	−107.693	+	186.530i	−0.458270	+	0.793747i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	28.5160			0.119314			0.0596568	−	0.998219i	$-0.480999\pi$
$239$	28.5160			0.119314			0.0596568	+	0.998219i	$0.480999\pi$
$240$		0			0
$241$	−203.548	−	117.518i	−0.844597	−	0.487628i	0.0142272	−	0.999899i	$-0.495471\pi$
$241$	−203.548	−	117.518i	−0.844597	−	0.487628i	−0.858824	+	0.512271i	$0.828805\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−20.8995	−	36.1990i	−0.0846133	−	0.146555i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			193.568i			0.771186i	0.922669	+	0.385593i	$0.126003\pi$
$251$			193.568i			0.771186i	−0.922669	+	0.385593i	$0.873997\pi$
$252$		0			0
$253$	323.990			1.28059
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−128.586	+	74.2394i	−0.500336	+	0.288869i	−0.728852	−	0.684671i	$-0.759946\pi$
$257$	−128.586	+	74.2394i	−0.500336	+	0.288869i	0.228516	+	0.973540i	$0.426613\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	223.882	−	387.774i	0.851261	−	1.47443i	−0.0288108	−	0.999585i	$-0.509172\pi$
$263$	223.882	−	387.774i	0.851261	−	1.47443i	0.880071	−	0.474842i	$-0.157495\pi$
$264$		0			0
$265$		−	161.489i		−	0.609391i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−158.803	−	91.6849i	−0.590345	−	0.340836i	0.174889	−	0.984588i	$-0.444043\pi$
$269$	−158.803	−	91.6849i	−0.590345	−	0.340836i	−0.765234	+	0.643752i	$0.777377\pi$
$270$		0			0
$271$	−116.984	+	67.5407i	−0.431675	+	0.249228i	−0.700060	−	0.714084i	$-0.746843\pi$
$271$	−116.984	+	67.5407i	−0.431675	+	0.249228i	0.268385	+	0.963312i	$0.413510\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	65.4717	+	113.400i	0.238079	+	0.412365i
$276$		0			0
$277$	−10.3015	+	17.8427i	−0.0371896	+	0.0644143i	−0.884021	−	0.467447i	$-0.845174\pi$
$277$	−10.3015	+	17.8427i	−0.0371896	+	0.0644143i	0.846832	+	0.531861i	$0.178507\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−241.001			−0.857654			−0.428827	−	0.903387i	$-0.641073\pi$
$281$	−241.001			−0.857654			−0.428827	+	0.903387i	$0.641073\pi$
$282$		0			0
$283$	76.8765	+	44.3847i	0.271648	+	0.156836i	0.629637	−	0.776890i	$-0.283204\pi$
$283$	76.8765	+	44.3847i	0.271648	+	0.156836i	−0.357988	+	0.933726i	$0.616537\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−63.0477	−	109.202i	−0.218158	−	0.377861i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	275.258i		−	0.939446i	−0.882814	−	0.469723i	$-0.844354\pi$
$293$		−	275.258i		−	0.939446i	0.882814	−	0.469723i	$-0.155646\pi$
$294$		0			0
$295$	−287.789			−0.975556
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−65.9503	+	38.0764i	−0.220570	+	0.127346i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	134.806	−	233.491i	0.441988	−	0.765545i
$306$		0			0
$307$			366.196i			1.19282i	0.802680	+	0.596410i	$0.203407\pi$
$307$			366.196i			1.19282i	−0.802680	+	0.596410i	$0.796593\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−256.062	−	147.837i	−0.823350	−	0.475361i	0.0282202	−	0.999602i	$-0.491016\pi$
$311$	−256.062	−	147.837i	−0.823350	−	0.475361i	−0.851570	+	0.524240i	$0.824349\pi$
$312$		0			0
$313$	200.791	−	115.927i	0.641504	−	0.370373i	−0.143689	−	0.989623i	$-0.545897\pi$
$313$	200.791	−	115.927i	0.641504	−	0.370373i	0.785194	+	0.619250i	$0.212563\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−154.691	−	267.932i	−0.487983	−	0.845212i	0.511921	−	0.859033i	$-0.328934\pi$
$317$	−154.691	−	267.932i	−0.487983	−	0.845212i	−0.999904	+	0.0138204i	$0.995601\pi$
$318$		0			0
$319$	198.196	−	343.285i	0.621304	−	1.07613i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	119.595			0.370262
$324$		0			0
$325$	−26.6544	−	15.3889i	−0.0820135	−	0.0473505i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	116.593	+	201.945i	0.352244	+	0.610105i	0.986642	−	0.162901i	$-0.0520852\pi$
$331$	116.593	+	201.945i	0.352244	+	0.610105i	−0.634398	+	0.773007i	$0.718752\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	116.560i		−	0.347939i
$336$		0			0
$337$	172.894			0.513040			0.256520	−	0.966539i	$-0.417424\pi$
$337$	172.894			0.513040			0.256520	+	0.966539i	$0.417424\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−256.432	+	148.051i	−0.752001	+	0.434168i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−232.465	+	402.641i	−0.669928	+	1.16035i	0.307996	+	0.951388i	$0.400342\pi$
$347$	−232.465	+	402.641i	−0.669928	+	1.16035i	−0.977924	+	0.208962i	$0.932992\pi$
$348$		0			0
$349$		−	426.071i		−	1.22083i	−0.792080	−	0.610417i	$-0.791002\pi$
$349$		−	426.071i		−	1.22083i	0.792080	−	0.610417i	$-0.208998\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	349.655	+	201.873i	0.990524	+	0.571880i	0.905431	−	0.424493i	$-0.139548\pi$
$353$	349.655	+	201.873i	0.990524	+	0.571880i	0.0850934	+	0.996373i	$0.472881\pi$
$354$		0			0
$355$	−55.8706	+	32.2569i	−0.157382	+	0.0908646i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	37.0516	+	64.1752i	0.103208	+	0.178761i	0.913005	−	0.407949i	$-0.133756\pi$
$359$	37.0516	+	64.1752i	0.103208	+	0.178761i	−0.809797	+	0.586710i	$0.800423\pi$
$360$		0			0
$361$	−136.601	+	236.599i	−0.378395	+	0.655399i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	356.589			0.976955
$366$		0			0
$367$	264.136	+	152.499i	0.719718	+	0.415529i	0.814649	−	0.579955i	$-0.196930\pi$
$367$	264.136	+	152.499i	0.719718	+	0.415529i	−0.0949312	+	0.995484i	$0.530263\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	64.7035	+	112.070i	0.173468	+	0.300455i	0.939630	−	0.342192i	$-0.111169\pi$
$373$	64.7035	+	112.070i	0.173468	+	0.300455i	−0.766162	+	0.642647i	$0.777836\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			93.1707i			0.247137i
$378$		0			0
$379$	−276.804			−0.730354			−0.365177	−	0.930938i	$-0.618991\pi$
$379$	−276.804			−0.730354			−0.365177	+	0.930938i	$0.618991\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	573.297	−	330.993i	1.49686	−	0.864212i	0.496866	−	0.867827i	$-0.334484\pi$
$383$	573.297	−	330.993i	1.49686	−	0.864212i	0.999993	+	0.00361454i	$0.00115055\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−204.045	+	353.416i	−0.524537	+	0.908525i	0.475055	+	0.879956i	$0.342428\pi$
$389$	−204.045	+	353.416i	−0.524537	+	0.908525i	−0.999592	+	0.0285687i	$0.990905\pi$
$390$		0			0
$391$		−	217.887i		−	0.557256i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	490.758	+	283.339i	1.24242	+	0.717314i
$396$		0			0
$397$	614.993	−	355.066i	1.54910	−	0.894373i	0.550889	−	0.834578i	$-0.314289\pi$
$397$	614.993	−	355.066i	1.54910	−	0.894373i	0.998211	−	0.0597948i	$-0.0190446\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−336.800	−	583.355i	−0.839900	−	1.45475i	−0.889978	−	0.456004i	$-0.849280\pi$
$401$	−336.800	−	583.355i	−0.839900	−	1.45475i	0.0500773	−	0.998745i	$-0.484053\pi$
$402$		0			0
$403$	34.7990	−	60.2736i	0.0863499	−	0.149562i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−867.298			−2.13095
$408$		0			0
$409$	−256.703	−	148.207i	−0.627635	−	0.362365i	0.152201	−	0.988350i	$-0.451364\pi$
$409$	−256.703	−	148.207i	−0.627635	−	0.362365i	−0.779835	+	0.625985i	$0.784697\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	233.487	+	404.412i	0.562620	+	0.974487i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	357.803i		−	0.853945i	−0.904265	−	0.426973i	$-0.859580\pi$
$419$		−	357.803i		−	0.853945i	0.904265	−	0.426973i	$-0.140420\pi$
$420$		0			0
$421$	185.186			0.439871			0.219936	−	0.975514i	$-0.429415\pi$
$421$	185.186			0.439871			0.219936	+	0.975514i	$0.429415\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	76.2631	−	44.0305i	0.179443	−	0.103601i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−360.643	+	624.652i	−0.836759	+	1.44931i	0.0558311	+	0.998440i	$0.482219\pi$
$431$	−360.643	+	624.652i	−0.836759	+	1.44931i	−0.892590	+	0.450869i	$0.851114\pi$
$432$		0			0
$433$		−	585.095i		−	1.35126i	−0.737242	−	0.675629i	$-0.763872\pi$
$433$		−	585.095i		−	1.35126i	0.737242	−	0.675629i	$-0.236128\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−138.529	−	79.9797i	−0.317000	−	0.183020i
$438$		0			0
$439$	−131.893	+	76.1482i	−0.300439	+	0.173458i	−0.642640	−	0.766168i	$-0.722161\pi$
$439$	−131.893	+	76.1482i	−0.300439	+	0.173458i	0.342201	+	0.939627i	$0.388828\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	358.736	+	621.348i	0.809787	+	1.40259i	0.913011	+	0.407934i	$0.133751\pi$
$443$	358.736	+	621.348i	0.809787	+	1.40259i	−0.103224	+	0.994658i	$0.532916\pi$
$444$		0			0
$445$	170.136	−	294.684i	0.382327	−	0.662210i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13.3522			−0.0297376			−0.0148688	−	0.999889i	$-0.504733\pi$
$449$	−13.3522			−0.0297376			−0.0148688	+	0.999889i	$0.504733\pi$
$450$		0			0
$451$	1041.87	+	601.524i	2.31013	+	1.33376i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	277.095	+	479.943i	0.606336	+	1.05020i	0.991839	+	0.127498i	$0.0406946\pi$
$457$	277.095	+	479.943i	0.606336	+	1.05020i	−0.385503	+	0.922707i	$0.625972\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			336.103i			0.729074i	0.931189	+	0.364537i	$0.118773\pi$
$461$			336.103i			0.729074i	−0.931189	+	0.364537i	$0.881227\pi$
$462$		0			0
$463$	−481.186			−1.03928			−0.519639	−	0.854386i	$-0.673934\pi$
$463$	−481.186			−1.03928			−0.519639	+	0.854386i	$0.673934\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−334.899	+	193.354i	−0.717128	+	0.414034i	−0.813695	−	0.581293i	$-0.802547\pi$
$467$	−334.899	+	193.354i	−0.717128	+	0.414034i	0.0965670	+	0.995326i	$0.469214\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	20.8862	−	36.1759i	0.0441568	−	0.0764818i
$474$		0			0
$475$		−	64.6490i		−	0.136103i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−232.474	−	134.219i	−0.485332	−	0.280206i	0.237304	−	0.971435i	$-0.423736\pi$
$479$	−232.474	−	134.219i	−0.485332	−	0.280206i	−0.722636	+	0.691229i	$0.757070\pi$
$480$		0			0
$481$	176.544	−	101.928i	0.367036	−	0.211908i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−254.305	−	440.469i	−0.524340	−	0.908184i
$486$		0			0
$487$	42.8995	−	74.3041i	0.0880893	−	0.152575i	−0.818614	−	0.574344i	$-0.805257\pi$
$487$	42.8995	−	74.3041i	0.0880893	−	0.152575i	0.906703	+	0.421769i	$0.138591\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	465.122			0.947295			0.473647	−	0.880715i	$-0.342937\pi$
$491$	465.122			0.947295			0.473647	+	0.880715i	$0.342937\pi$
$492$		0			0
$493$	−230.864	−	133.289i	−0.468283	−	0.270364i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	427.784	+	740.943i	0.857282	+	1.48486i	0.874511	+	0.485005i	$0.161182\pi$
$499$	427.784	+	740.943i	0.857282	+	1.48486i	−0.0172292	+	0.999852i	$0.505485\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	432.245i		−	0.859335i	−0.902987	−	0.429667i	$-0.858631\pi$
$503$		−	432.245i		−	0.859335i	0.902987	−	0.429667i	$-0.141369\pi$
$504$		0			0
$505$	628.060			1.24368
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−231.752	+	133.802i	−0.455308	+	0.262872i	−0.710069	−	0.704132i	$-0.751336\pi$
$509$	−231.752	+	133.802i	−0.455308	+	0.262872i	0.254761	+	0.967004i	$0.418003\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	391.972	−	678.916i	0.761111	−	1.31828i
$516$		0			0
$517$		−	960.816i		−	1.85844i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−290.703	−	167.838i	−0.557972	−	0.322145i	0.194359	−	0.980930i	$-0.437737\pi$
$521$	−290.703	−	167.838i	−0.557972	−	0.322145i	−0.752331	+	0.658785i	$0.771071\pi$
$522$		0			0
$523$	840.091	−	485.027i	1.60629	−	0.927393i	0.616102	−	0.787666i	$-0.288711\pi$
$523$	840.091	−	485.027i	1.60629	−	0.927393i	0.990190	−	0.139727i	$-0.0446224\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	99.5663	+	172.454i	0.188930	+	0.327237i
$528$		0			0
$529$	118.786	−	205.744i	0.224549	−	0.388930i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−282.773			−0.530531
$534$		0			0
$535$	174.640	+	100.828i	0.326430	+	0.188464i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	175.090	+	303.265i	0.323642	+	0.560565i	0.981237	−	0.192807i	$-0.0617593\pi$
$541$	175.090	+	303.265i	0.323642	+	0.560565i	−0.657594	+	0.753372i	$0.728426\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	420.743i		−	0.772006i
$546$		0			0
$547$	−648.794			−1.18609			−0.593047	−	0.805167i	$-0.702075\pi$
$547$	−648.794			−1.18609			−0.593047	+	0.805167i	$0.702075\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−169.486	+	97.8528i	−0.307597	+	0.177591i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	439.371	−	761.013i	0.788817	−	1.36627i	−0.137875	−	0.990450i	$-0.544027\pi$
$557$	439.371	−	761.013i	0.788817	−	1.36627i	0.926692	−	0.375822i	$-0.122640\pi$
$558$		0			0
$559$			9.81845i			0.0175643i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−508.828	−	293.772i	−0.903780	−	0.521798i	−0.0253555	−	0.999678i	$-0.508072\pi$
$563$	−508.828	−	293.772i	−0.903780	−	0.521798i	−0.878425	+	0.477881i	$0.841405\pi$
$564$		0			0
$565$	692.238	−	399.664i	1.22520	−	0.707369i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	52.1195	+	90.2737i	0.0915985	+	0.158653i	0.908184	−	0.418571i	$-0.137469\pi$
$569$	52.1195	+	90.2737i	0.0915985	+	0.158653i	−0.816585	+	0.577225i	$0.804136\pi$
$570$		0			0
$571$	235.693	−	408.233i	0.412773	−	0.714944i	−0.582419	−	0.812889i	$-0.697894\pi$
$571$	235.693	−	408.233i	0.412773	−	0.714944i	0.995192	+	0.0979450i	$0.0312269\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−117.783			−0.204840
$576$		0			0
$577$	411.347	+	237.491i	0.712907	+	0.411597i	0.812136	−	0.583468i	$-0.198304\pi$
$577$	411.347	+	237.491i	0.712907	+	0.411597i	−0.0992296	+	0.995065i	$0.531638\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	360.191	+	623.869i	0.617823	+	1.07010i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			539.013i			0.918251i	0.888371	+	0.459125i	$0.151837\pi$
$587$			539.013i			0.918251i	−0.888371	+	0.459125i	$0.848163\pi$
$588$		0			0
$589$	146.191			0.248202
$590$		0			0
$591$		0			0
$592$		0			0
$593$	89.8901	−	51.8981i	0.151585	−	0.0875178i	−0.422289	−	0.906461i	$-0.638773\pi$
$593$	89.8901	−	51.8981i	0.151585	−	0.0875178i	0.573874	+	0.818944i	$0.305440\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	275.095	−	476.479i	0.459257	−	0.795457i	−0.539664	−	0.841880i	$-0.681449\pi$
$599$	275.095	−	476.479i	0.459257	−	0.795457i	0.998922	+	0.0464229i	$0.0147822\pi$
$600$		0			0
$601$		−	848.487i		−	1.41179i	−0.708315	−	0.705896i	$-0.750544\pi$
$601$		−	848.487i		−	1.41179i	0.708315	−	0.705896i	$-0.249456\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	881.294	+	508.815i	1.45668	+	0.841017i
$606$		0			0
$607$	−515.032	+	297.354i	−0.848488	+	0.489875i	−0.860140	−	0.510057i	$-0.829624\pi$
$607$	−515.032	+	297.354i	−0.848488	+	0.489875i	0.0116523	+	0.999932i	$0.496291\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	112.918	+	195.580i	0.184809	+	0.320099i
$612$		0			0
$613$	−258.754	+	448.175i	−0.422111	+	0.731117i	−0.996146	−	0.0877137i	$-0.972044\pi$
$613$	−258.754	+	448.175i	−0.422111	+	0.731117i	0.574035	+	0.818831i	$0.305377\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	681.421			1.10441			0.552205	−	0.833708i	$-0.313786\pi$
$617$	681.421			1.10441			0.552205	+	0.833708i	$0.313786\pi$
$618$		0			0
$619$	774.084	+	446.918i	1.25054	+	0.721999i	0.971217	−	0.238198i	$-0.0765567\pi$
$619$	774.084	+	446.918i	1.25054	+	0.721999i	0.279323	+	0.960197i	$0.409890\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	202.455	+	350.662i	0.323928	+	0.561059i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			583.268i			0.927295i
$630$		0			0
$631$	−949.588			−1.50489			−0.752447	−	0.658653i	$-0.771127\pi$
$631$	−949.588			−1.50489			−0.752447	+	0.658653i	$0.771127\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	84.7244	−	48.9156i	0.133424	−	0.0770325i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−215.394	+	373.073i	−0.336028	+	0.582017i	−0.983682	−	0.179917i	$-0.942417\pi$
$641$	−215.394	+	373.073i	−0.336028	+	0.582017i	0.647654	+	0.761935i	$0.275750\pi$
$642$		0			0
$643$		−	540.999i		−	0.841368i	−0.907207	−	0.420684i	$-0.861790\pi$
$643$		−	540.999i		−	0.841368i	0.907207	−	0.420684i	$-0.138210\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−85.8353	−	49.5570i	−0.132667	−	0.0765951i	0.432198	−	0.901779i	$-0.357738\pi$
$647$	−85.8353	−	49.5570i	−0.132667	−	0.0765951i	−0.564864	+	0.825184i	$0.691072\pi$
$648$		0			0
$649$	1111.80	−	641.897i	1.71309	−	0.989055i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	414.622	+	718.147i	0.634950	+	1.09977i	0.986526	+	0.163606i	$0.0523126\pi$
$653$	414.622	+	718.147i	0.634950	+	1.09977i	−0.351576	+	0.936159i	$0.614354\pi$
$654$		0			0
$655$	305.889	−	529.816i	0.467007	−	0.808879i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	584.716			0.887278			0.443639	−	0.896206i	$-0.353687\pi$
$659$	584.716			0.887278			0.443639	+	0.896206i	$0.353687\pi$
$660$		0			0
$661$	−333.657	−	192.637i	−0.504776	−	0.291433i	0.225908	−	0.974149i	$-0.427465\pi$
$661$	−333.657	−	192.637i	−0.504776	−	0.291433i	−0.730684	+	0.682716i	$0.760799\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	178.276	+	308.784i	0.267281	+	0.462944i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1202.71i			1.79241i
$672$		0			0
$673$	797.266			1.18465			0.592323	−	0.805701i	$-0.298211\pi$
$673$	797.266			1.18465			0.592323	+	0.805701i	$0.298211\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	415.605	−	239.950i	0.613893	−	0.354431i	−0.160595	−	0.987020i	$-0.551341\pi$
$677$	415.605	−	239.950i	0.613893	−	0.354431i	0.774487	+	0.632589i	$0.218008\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−535.170	+	926.942i	−0.783559	+	1.35716i	0.146298	+	0.989241i	$0.453264\pi$
$683$	−535.170	+	926.942i	−0.783559	+	1.35716i	−0.929856	+	0.367923i	$0.880069\pi$
$684$		0			0
$685$		−	282.809i		−	0.412860i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−146.639	−	84.6618i	−0.212828	−	0.122876i
$690$		0			0
$691$	−25.9344	+	14.9732i	−0.0375317	+	0.0216690i	−0.518648	−	0.854988i	$-0.673565\pi$
$691$	−25.9344	+	14.9732i	−0.0375317	+	0.0216690i	0.481117	+	0.876657i	$0.340231\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−523.870	−	907.369i	−0.753769	−	1.30557i
$696$		0			0
$697$	404.533	−	700.671i	0.580391	−	1.00527i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−396.453			−0.565554			−0.282777	−	0.959186i	$-0.591256\pi$
$701$	−396.453			−0.565554			−0.282777	+	0.959186i	$0.591256\pi$
$702$		0			0
$703$	370.832	+	214.100i	0.527499	+	0.304552i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	173.749	+	300.942i	0.245062	+	0.424459i	0.962149	−	0.272524i	$-0.0878584\pi$
$709$	173.749	+	300.942i	0.245062	+	0.424459i	−0.717087	+	0.696983i	$0.754525\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	266.343i		−	0.373552i
$714$		0			0
$715$	−360.191			−0.503764
$716$		0			0
$717$		0			0
$718$		0			0
$719$	910.047	−	525.416i	1.26571	−	0.730760i	0.291539	−	0.956559i	$-0.405833\pi$
$719$	910.047	−	525.416i	1.26571	−	0.730760i	0.974174	+	0.225799i	$0.0724994\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−72.0519	+	124.798i	−0.0993820	+	0.172135i
$726$		0			0
$727$		−	47.3417i		−	0.0651193i	−0.999470	−	0.0325596i	$-0.989634\pi$
$727$		−	47.3417i		−	0.0651193i	0.999470	−	0.0325596i	$-0.0103659\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−24.3287	−	14.0462i	−0.0332814	−	0.0192150i
$732$		0			0
$733$	−53.9898	+	31.1710i	−0.0736559	+	0.0425252i	−0.536376	−	0.843979i	$-0.680207\pi$
$733$	−53.9898	+	31.1710i	−0.0736559	+	0.0425252i	0.462720	+	0.886505i	$0.346874\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	259.979	+	450.297i	0.352754	+	0.610987i
$738$		0			0
$739$	−669.176	+	1159.05i	−0.905515	+	1.56840i	−0.0852911	+	0.996356i	$0.527182\pi$
$739$	−669.176	+	1159.05i	−0.905515	+	1.56840i	−0.820224	+	0.572042i	$0.806151\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−242.908			−0.326929			−0.163464	−	0.986549i	$-0.552267\pi$
$743$	−242.908			−0.326929			−0.163464	+	0.986549i	$0.552267\pi$
$744$		0			0
$745$	−293.762	−	169.604i	−0.394312	−	0.227656i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	496.291	+	859.602i	0.660841	+	1.14461i	0.980395	+	0.197042i	$0.0631335\pi$
$751$	496.291	+	859.602i	0.660841	+	1.14461i	−0.319554	+	0.947568i	$0.603533\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	138.666i		−	0.183663i
$756$		0			0
$757$	780.060			1.03046			0.515231	−	0.857051i	$-0.327706\pi$
$757$	780.060			1.03046			0.515231	+	0.857051i	$0.327706\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	402.349	−	232.296i	0.528711	−	0.305251i	−0.211781	−	0.977317i	$-0.567926\pi$
$761$	402.349	−	232.296i	0.528711	−	0.305251i	0.740491	+	0.672066i	$0.234593\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−150.876	+	261.325i	−0.196709	+	0.340710i
$768$		0			0
$769$		−	611.409i		−	0.795071i	−0.917587	−	0.397535i	$-0.869866\pi$
$769$		−	611.409i		−	0.795071i	0.917587	−	0.397535i	$-0.130134\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	204.128	+	117.853i	0.264072	+	0.152462i	0.626191	−	0.779670i	$-0.284613\pi$
$773$	204.128	+	117.853i	0.264072	+	0.152462i	−0.362119	+	0.932132i	$0.617947\pi$
$774$		0			0
$775$	93.2231	−	53.8224i	0.120288	−	0.0694482i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−296.983	−	514.390i	−0.381236	−	0.660321i
$780$		0			0
$781$	143.894	−	249.232i	0.184244	−	0.319120i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1083.31			1.38001
$786$		0			0
$787$	515.266	+	297.489i	0.654722	+	0.378004i	0.790263	−	0.612768i	$-0.209944\pi$
$787$	515.266	+	297.489i	0.654722	+	0.378004i	−0.135541	+	0.990772i	$0.543277\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−141.347	−	244.820i	−0.178243	−	0.308726i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	547.544i		−	0.687006i	−0.939152	−	0.343503i	$-0.888386\pi$
$797$		−	547.544i		−	0.687006i	0.939152	−	0.343503i	$-0.111614\pi$
$798$		0			0
$799$	−646.161			−0.808712
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1377.59	+	795.350i	−1.71555	+	0.990474i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	256.069	−	443.524i	0.316525	−	0.548237i	−0.663236	−	0.748411i	$-0.730817\pi$
$809$	256.069	−	443.524i	0.316525	−	0.548237i	0.979760	+	0.200174i	$0.0641506\pi$
$810$		0			0
$811$			1615.33i			1.99178i	0.0905851	+	0.995889i	$0.471126\pi$
$811$			1615.33i			1.99178i	−0.0905851	+	0.995889i	$0.528874\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	895.310	+	516.907i	1.09854	+	0.634242i
$816$		0			0
$817$	−17.8607	+	10.3119i	−0.0218613	+	0.0126216i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	490.777	+	850.050i	0.597779	+	1.03538i	0.993148	+	0.116861i	$0.0372833\pi$
$821$	490.777	+	850.050i	0.597779	+	1.03538i	−0.395369	+	0.918522i	$0.629383\pi$
$822$		0			0
$823$	678.276	−	1174.81i	0.824151	−	1.42747i	−0.0784154	−	0.996921i	$-0.524986\pi$
$823$	678.276	−	1174.81i	0.824151	−	1.42747i	0.902566	−	0.430551i	$-0.141681\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−601.883			−0.727791			−0.363896	−	0.931440i	$-0.618554\pi$
$827$	−601.883			−0.727791			−0.363896	+	0.931440i	$0.618554\pi$
$828$		0			0
$829$	154.783	+	89.3641i	0.186711	+	0.107798i	0.590442	−	0.807080i	$-0.298954\pi$
$829$	154.783	+	89.3641i	0.186711	+	0.107798i	−0.403731	+	0.914878i	$0.632287\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−306.799	−	531.391i	−0.367424	−	0.636397i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	852.989i		−	1.01667i	−0.861158	−	0.508337i	$-0.830261\pi$
$839$		−	852.989i		−	1.01667i	0.861158	−	0.508337i	$-0.169739\pi$
$840$		0			0
$841$	−404.769			−0.481295
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−549.358	+	317.172i	−0.650127	+	0.375351i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	390.065	−	675.612i	0.458361	−	0.793904i
$852$		0			0
$853$		−	855.303i		−	1.00270i	−0.865245	−	0.501350i	$-0.832837\pi$
$853$		−	855.303i		−	1.00270i	0.865245	−	0.501350i	$-0.167163\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1251.22	−	722.394i	−1.46000	−	0.842933i	−0.460992	−	0.887404i	$-0.652506\pi$
$857$	−1251.22	−	722.394i	−1.46000	−	0.842933i	−0.999011	+	0.0444708i	$0.985840\pi$
$858$		0			0
$859$	−367.259	+	212.037i	−0.427543	+	0.246842i	−0.698299	−	0.715806i	$-0.746060\pi$
$859$	−367.259	+	212.037i	−0.427543	+	0.246842i	0.270756	+	0.962648i	$0.412726\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−674.602	−	1168.44i	−0.781694	−	1.35393i	−0.930955	−	0.365135i	$-0.881023\pi$
$863$	−674.602	−	1168.44i	−0.781694	−	1.35393i	0.149261	−	0.988798i	$-0.452311\pi$
$864$		0			0
$865$	−711.332	+	1232.06i	−0.822349	+	1.42435i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2527.89			−2.90896
$870$		0			0
$871$	−105.841	−	61.1074i	−0.121517	−	0.0701577i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−321.829	−	557.424i	−0.366966	−	0.635603i	0.622124	−	0.782919i	$-0.286270\pi$
$877$	−321.829	−	557.424i	−0.366966	−	0.635603i	−0.989090	+	0.147315i	$0.952937\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1206.54i		−	1.36951i	−0.728774	−	0.684754i	$-0.759909\pi$
$881$		−	1206.54i		−	1.36951i	0.728774	−	0.684754i	$-0.240091\pi$
$882$		0			0
$883$	652.754			0.739245			0.369623	−	0.929182i	$-0.379487\pi$
$883$	652.754			0.739245			0.369623	+	0.929182i	$0.379487\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−910.418	+	525.630i	−1.02640	+	0.592593i	−0.915952	−	0.401289i	$-0.868562\pi$
$887$	−910.418	+	525.630i	−1.02640	+	0.592593i	−0.110450	+	0.993882i	$0.535229\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−237.186	+	410.818i	−0.265606	+	0.460042i
$894$		0			0
$895$		−	500.288i		−	0.558981i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−282.205	−	162.931i	−0.313910	−	0.181236i
$900$		0			0
$901$	419.560	−	242.233i	0.465660	−	0.268849i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−369.035	−	639.187i	−0.407773	−	0.706284i
$906$		0			0
$907$	−143.080	+	247.822i	−0.157751	+	0.273233i	−0.934057	−	0.357123i	$-0.883758\pi$
$907$	−143.080	+	247.822i	−0.157751	+	0.273233i	0.776306	+	0.630356i	$0.217091\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	713.465			0.783167			0.391583	−	0.920143i	$-0.371927\pi$
$911$	713.465			0.783167			0.391583	+	0.920143i	$0.371927\pi$
$912$		0			0
$913$	−1804.04	−	1041.56i	−1.97594	−	1.14081i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−170.698	−	295.658i	−0.185744	−	0.321718i	0.758083	−	0.652158i	$-0.226136\pi$
$919$	−170.698	−	295.658i	−0.185744	−	0.321718i	−0.943827	+	0.330440i	$0.892803\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			67.6439i			0.0732870i
$924$		0			0
$925$	315.296			0.340861
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−79.5587	+	45.9332i	−0.0856390	+	0.0494437i	−0.542208	−	0.840244i	$-0.682412\pi$
$929$	−79.5587	+	45.9332i	−0.0856390	+	0.0494437i	0.456569	+	0.889688i	$0.349078\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	515.286	−	892.501i	0.551108	−	0.954547i
$936$		0			0
$937$			906.902i			0.967879i	0.875102	+	0.483939i	$0.160794\pi$
$937$			906.902i			0.967879i	−0.875102	+	0.483939i	$0.839206\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	337.880	+	195.075i	0.359064	+	0.207306i	0.668670	−	0.743559i	$-0.266864\pi$
$941$	337.880	+	195.075i	0.359064	+	0.207306i	−0.309606	+	0.950865i	$0.600197\pi$
$942$		0			0
$943$	−937.158	+	541.068i	−0.993804	+	0.573773i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	304.757	+	527.854i	0.321813	+	0.557396i	0.980862	−	0.194704i	$-0.0623745\pi$
$947$	304.757	+	527.854i	0.321813	+	0.557396i	−0.659049	+	0.752100i	$0.729041\pi$
$948$		0			0
$949$	186.945	−	323.798i	0.196991	−	0.341199i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	110.441			0.115887			0.0579437	−	0.998320i	$-0.481546\pi$
$953$	110.441			0.115887			0.0579437	+	0.998320i	$0.481546\pi$
$954$		0			0
$955$	965.269	+	557.299i	1.01075	+	0.583559i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−358.791	−	621.445i	−0.373352	−	0.646665i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	125.924i		−	0.130491i
$966$		0			0
$967$	−1125.53			−1.16394			−0.581969	−	0.813211i	$-0.697717\pi$
$967$	−1125.53			−1.16394			−0.581969	+	0.813211i	$0.697717\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	977.849	−	564.562i	1.00705	−	0.581423i	0.0967262	−	0.995311i	$-0.469163\pi$
$971$	977.849	−	564.562i	1.00705	−	0.581423i	0.910328	+	0.413888i	$0.135830\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−756.430	+	1310.18i	−0.774238	+	1.34102i	0.160984	+	0.986957i	$0.448533\pi$
$977$	−756.430	+	1310.18i	−0.774238	+	1.34102i	−0.935222	+	0.354062i	$0.884800\pi$
$978$		0			0
$979$			1517.91i			1.55047i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−501.830	−	289.731i	−0.510508	−	0.294742i	0.222534	−	0.974925i	$-0.428567\pi$
$983$	−501.830	−	289.731i	−0.510508	−	0.294742i	−0.733043	+	0.680183i	$0.761900\pi$
$984$		0			0
$985$	−874.259	+	504.754i	−0.887573	+	0.512440i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	18.7870	+	32.5400i	0.0189959	+	0.0329019i
$990$		0			0
$991$	−86.3768	+	149.609i	−0.0871612	+	0.150968i	−0.906310	−	0.422613i	$-0.861113\pi$
$991$	−86.3768	+	149.609i	−0.0871612	+	0.150968i	0.819149	+	0.573581i	$0.194446\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−125.700			−0.126332
$996$		0			0
$997$	−333.267	−	192.412i	−0.334270	−	0.192991i	0.323465	−	0.946240i	$-0.395152\pi$
$997$	−333.267	−	192.412i	−0.334270	−	0.192991i	−0.657735	+	0.753249i	$0.728485\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.z.n.325.4		16
3.2	odd	2	inner	1764.3.z.n.325.5		16
7.2	even	3	inner	1764.3.z.n.901.6		16
7.3	odd	6		1764.3.d.g.685.5	yes	8
7.4	even	3		1764.3.d.g.685.3	✓	8
7.5	odd	6	inner	1764.3.z.n.901.4		16
7.6	odd	2	inner	1764.3.z.n.325.6		16
21.2	odd	6	inner	1764.3.z.n.901.3		16
21.5	even	6	inner	1764.3.z.n.901.5		16
21.11	odd	6		1764.3.d.g.685.6	yes	8
21.17	even	6		1764.3.d.g.685.4	yes	8
21.20	even	2	inner	1764.3.z.n.325.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.3.d.g.685.3	✓	8	7.4	even	3
1764.3.d.g.685.4	yes	8	21.17	even	6
1764.3.d.g.685.5	yes	8	7.3	odd	6
1764.3.d.g.685.6	yes	8	21.11	odd	6
1764.3.z.n.325.3		16	21.20	even	2	inner
1764.3.z.n.325.4		16	1.1	even	1	trivial
1764.3.z.n.325.5		16	3.2	odd	2	inner
1764.3.z.n.325.6		16	7.6	odd	2	inner
1764.3.z.n.901.3		16	21.2	odd	6	inner
1764.3.z.n.901.4		16	7.5	odd	6	inner
1764.3.z.n.901.5		16	21.5	even	6	inner
1764.3.z.n.901.6		16	7.2	even	3	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.68448 + 2.12723i) q^{5} +O(q^{10})\)	\(q+(-3.68448 + 2.12723i) q^{5} +(9.48935 - 16.4360i) q^{11} +4.46088i q^{13} +(-11.0534 - 6.38170i) q^{17} +(-8.11475 + 4.68506i) q^{19} +(8.53562 + 14.7841i) q^{23} +(-3.44975 + 5.97514i) q^{25} +20.8862 q^{29} +(-13.5116 - 7.80092i) q^{31} +(-22.8492 - 39.5760i) q^{37} +63.3894i q^{41} +2.20101 q^{43} +(43.8434 - 25.3130i) q^{47} +(-18.9787 + 32.8721i) q^{53} +80.7443i q^{55} +(58.5813 + 33.8220i) q^{59} +(-54.8814 + 31.6858i) q^{61} +(-9.48935 - 16.4360i) q^{65} +(-13.6985 + 23.7265i) q^{67} +15.1638 q^{71} +(-72.5860 - 41.9075i) q^{73} +(-66.5980 - 115.351i) q^{79} -109.761i q^{83} +54.3015 q^{85} +(-69.2645 + 39.9899i) q^{89} +(19.9324 - 34.5240i) q^{95} +119.547i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 24 q^{25} - 128 q^{37} + 352 q^{43} + 256 q^{67} - 432 q^{79} + 1344 q^{85}+O(q^{100}) \) 16 * q + 24 * q^25 - 128 * q^37 + 352 * q^43 + 256 * q^67 - 432 * q^79 + 1344 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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