LMFDB - Embedded newform 1764.4.k.z.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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, 4]))

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

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

chi := DirichletCharacter("1764.361");

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.k (of order $3$, 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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$-1.63746 - 1.52274i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.z.1549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.91238 - 8.50848i) q^{5} +O(q^{10})$	$q+(-4.91238 - 8.50848i) q^{5} +(-7.08762 + 12.2761i) q^{11} +26.1238 q^{13} +(39.2990 - 68.0679i) q^{17} +(36.5876 + 63.3716i) q^{19} +(-48.0000 - 83.1384i) q^{23} +(14.2371 - 24.6594i) q^{25} -173.021 q^{29} +(33.6238 - 58.2381i) q^{31} +(150.835 + 261.254i) q^{37} +472.042 q^{41} -463.670 q^{43} +(45.5980 + 78.9781i) q^{47} +(-81.6341 + 141.394i) q^{53} +139.268 q^{55} +(300.263 - 520.071i) q^{59} +(285.846 + 495.099i) q^{61} +(-128.330 - 222.274i) q^{65} +(269.711 - 467.154i) q^{67} +1064.39 q^{71} +(-221.340 + 383.372i) q^{73} +(22.8505 + 39.5782i) q^{79} -686.464 q^{83} -772.206 q^{85} +(-330.072 - 571.702i) q^{89} +(359.464 - 622.610i) q^{95} -658.320 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{5}+O(q^{10}) $ 4 * q + 3 * q^5	$ 4 q + 3 q^{5} - 51 q^{11} - 122 q^{13} - 24 q^{17} + 169 q^{19} - 192 q^{23} - 11 q^{25} + 78 q^{29} - 92 q^{31} + 173 q^{37} + 348 q^{41} - 994 q^{43} - 180 q^{47} + 285 q^{53} - 666 q^{55} + 1269 q^{59} + 328 q^{61} - 1374 q^{65} + 875 q^{67} + 2808 q^{71} - 1361 q^{73} + 182 q^{79} - 798 q^{83} - 4176 q^{85} - 822 q^{89} - 510 q^{95} - 1682 q^{97}+O(q^{100}) $ 4 * q + 3 * q^5 - 51 * q^11 - 122 * q^13 - 24 * q^17 + 169 * q^19 - 192 * q^23 - 11 * q^25 + 78 * q^29 - 92 * q^31 + 173 * q^37 + 348 * q^41 - 994 * q^43 - 180 * q^47 + 285 * q^53 - 666 * q^55 + 1269 * q^59 + 328 * q^61 - 1374 * q^65 + 875 * q^67 + 2808 * q^71 - 1361 * q^73 + 182 * q^79 - 798 * q^83 - 4176 * q^85 - 822 * q^89 - 510 * q^95 - 1682 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−4.91238	−	8.50848i	−0.439376	−	0.761022i	0.558265	−	0.829663i	$-0.311467\pi$
$5$	−4.91238	−	8.50848i	−0.439376	−	0.761022i	−0.997641	+	0.0686406i	$0.978134\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.08762	+	12.2761i	−0.194273	+	0.336490i	−0.946662	−	0.322229i	$-0.895568\pi$
$11$	−7.08762	+	12.2761i	−0.194273	+	0.336490i	0.752389	+	0.658719i	$0.228901\pi$
$12$		0			0
$13$	26.1238			0.557341			0.278670	−	0.960387i	$-0.410106\pi$
$13$	26.1238			0.557341			0.278670	+	0.960387i	$0.410106\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	39.2990	−	68.0679i	0.560671	−	0.971111i	−0.436767	−	0.899575i	$-0.643877\pi$
$17$	39.2990	−	68.0679i	0.560671	−	0.971111i	0.997438	−	0.0715361i	$-0.0227901\pi$
$18$		0			0
$19$	36.5876	+	63.3716i	0.441778	+	0.765181i	0.997822	−	0.0659715i	$-0.0210147\pi$
$19$	36.5876	+	63.3716i	0.441778	+	0.765181i	−0.556044	+	0.831153i	$0.687681\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−48.0000	−	83.1384i	−0.435161	−	0.753720i	0.562148	−	0.827037i	$-0.309975\pi$
$23$	−48.0000	−	83.1384i	−0.435161	−	0.753720i	−0.997309	+	0.0733164i	$0.976642\pi$
$24$		0			0
$25$	14.2371	−	24.6594i	0.113897	−	0.197275i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−173.021			−1.10790			−0.553951	−	0.832549i	$-0.686880\pi$
$29$	−173.021			−1.10790			−0.553951	+	0.832549i	$0.686880\pi$
$30$		0			0
$31$	33.6238	−	58.2381i	0.194807	−	0.337415i	−0.752030	−	0.659128i	$-0.770925\pi$
$31$	33.6238	−	58.2381i	0.194807	−	0.337415i	0.946837	+	0.321713i	$0.104259\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	150.835	+	261.254i	0.670193	+	1.16081i	0.977849	+	0.209311i	$0.0671221\pi$
$37$	150.835	+	261.254i	0.670193	+	1.16081i	−0.307656	+	0.951498i	$0.599545\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	472.042			1.79806			0.899031	−	0.437886i	$-0.144273\pi$
$41$	472.042			1.79806			0.899031	+	0.437886i	$0.144273\pi$
$42$		0			0
$43$	−463.670			−1.64440			−0.822198	−	0.569201i	$-0.807253\pi$
$43$	−463.670			−1.64440			−0.822198	+	0.569201i	$0.807253\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	45.5980	+	78.9781i	0.141514	+	0.245109i	0.928067	−	0.372413i	$-0.121470\pi$
$47$	45.5980	+	78.9781i	0.141514	+	0.245109i	−0.786553	+	0.617523i	$0.788136\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−81.6341	+	141.394i	−0.211572	+	0.366453i	−0.952207	−	0.305455i	$-0.901192\pi$
$53$	−81.6341	+	141.394i	−0.211572	+	0.366453i	0.740635	+	0.671908i	$0.234525\pi$
$54$		0			0
$55$	139.268			0.341435
$56$		0			0
$57$		0			0
$58$		0			0
$59$	300.263	−	520.071i	0.662558	−	1.14758i	−0.317384	−	0.948297i	$-0.602804\pi$
$59$	300.263	−	520.071i	0.662558	−	1.14758i	0.979941	−	0.199286i	$-0.0638623\pi$
$60$		0			0
$61$	285.846	+	495.099i	0.599980	+	1.03920i	0.992823	+	0.119590i	$0.0381581\pi$
$61$	285.846	+	495.099i	0.599980	+	1.03920i	−0.392844	+	0.919605i	$0.628509\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−128.330	−	222.274i	−0.244882	−	0.424148i
$66$		0			0
$67$	269.711	−	467.154i	0.491798	−	0.851820i	−0.508157	−	0.861264i	$-0.669673\pi$
$67$	269.711	−	467.154i	0.491798	−	0.851820i	0.999955	+	0.00944469i	$0.00300638\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1064.39			1.77916			0.889578	−	0.456783i	$-0.150998\pi$
$71$	1064.39			1.77916			0.889578	+	0.456783i	$0.150998\pi$
$72$		0			0
$73$	−221.340	+	383.372i	−0.354875	+	0.614662i	−0.987097	−	0.160126i	$-0.948810\pi$
$73$	−221.340	+	383.372i	−0.354875	+	0.614662i	0.632221	+	0.774788i	$0.282143\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	22.8505	+	39.5782i	0.0325428	+	0.0563658i	0.881838	−	0.471552i	$-0.156306\pi$
$79$	22.8505	+	39.5782i	0.0325428	+	0.0563658i	−0.849295	+	0.527918i	$0.822973\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−686.464			−0.907822			−0.453911	−	0.891047i	$-0.649972\pi$
$83$	−686.464			−0.907822			−0.453911	+	0.891047i	$0.649972\pi$
$84$		0			0
$85$	−772.206			−0.985382
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−330.072	−	571.702i	−0.393119	−	0.680902i	0.599740	−	0.800195i	$-0.295271\pi$
$89$	−330.072	−	571.702i	−0.393119	−	0.680902i	−0.992859	+	0.119293i	$0.961937\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	359.464	−	622.610i	0.388213	−	0.672405i
$96$		0			0
$97$	−658.320			−0.689095			−0.344548	−	0.938769i	$-0.611968\pi$
$97$	−658.320			−0.689095			−0.344548	+	0.938769i	$0.611968\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	901.949	−	1562.22i	0.888586	−	1.53908i	0.0470394	−	0.998893i	$-0.485021\pi$
$101$	901.949	−	1562.22i	0.888586	−	1.53908i	0.841547	−	0.540184i	$-0.181645\pi$
$102$		0			0
$103$	−537.454	−	930.898i	−0.514145	−	0.890525i	−0.999865	−	0.0164107i	$-0.994776\pi$
$103$	−537.454	−	930.898i	−0.514145	−	0.890525i	0.485721	−	0.874114i	$-0.338557\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−829.676	−	1437.04i	−0.749605	−	1.29835i	−0.948012	−	0.318235i	$-0.896910\pi$
$107$	−829.676	−	1437.04i	−0.749605	−	1.29835i	0.198407	−	0.980120i	$-0.436423\pi$
$108$		0			0
$109$	−299.464	+	518.687i	−0.263151	+	0.455791i	−0.967077	−	0.254482i	$-0.918095\pi$
$109$	−299.464	+	518.687i	−0.263151	+	0.455791i	0.703927	+	0.710273i	$0.251428\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	781.650			0.650720			0.325360	−	0.945590i	$-0.394514\pi$
$113$	781.650			0.650720			0.325360	+	0.945590i	$0.394514\pi$
$114$		0			0
$115$	−471.588	+	816.815i	−0.382398	+	0.662333i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	565.031	+	978.663i	0.424516	+	0.735284i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1507.85			−1.07893
$126$		0			0
$127$	25.3722			0.0177277			0.00886385	−	0.999961i	$-0.497179\pi$
$127$	25.3722			0.0177277			0.00886385	+	0.999961i	$0.497179\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	237.789	+	411.862i	0.158593	+	0.274691i	0.934362	−	0.356326i	$-0.115971\pi$
$131$	237.789	+	411.862i	0.158593	+	0.274691i	−0.775768	+	0.631018i	$0.782638\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	529.268	−	916.720i	0.330062	−	0.571683i	−0.652462	−	0.757821i	$-0.726264\pi$
$137$	529.268	−	916.720i	0.330062	−	0.571683i	0.982524	+	0.186138i	$0.0595971\pi$
$138$		0			0
$139$	−2580.99			−1.57494			−0.787470	−	0.616352i	$-0.788610\pi$
$139$	−2580.99			−1.57494			−0.787470	+	0.616352i	$0.788610\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−185.155	+	320.699i	−0.108276	+	0.187540i
$144$		0			0
$145$	849.943	+	1472.14i	0.486786	+	0.843138i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−50.9377	−	88.2266i	−0.0280066	−	0.0485088i	0.851682	−	0.524058i	$-0.175583\pi$
$149$	−50.9377	−	88.2266i	−0.0280066	−	0.0485088i	−0.879689	+	0.475549i	$0.842249\pi$
$150$		0			0
$151$	889.211	−	1540.16i	0.479225	−	0.830042i	−0.520491	−	0.853867i	$-0.674251\pi$
$151$	889.211	−	1540.16i	0.479225	−	0.830042i	0.999716	+	0.0238249i	$0.00758441\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−660.690			−0.342374
$156$		0			0
$157$	699.629	−	1211.79i	0.355646	−	0.615997i	−0.631582	−	0.775309i	$-0.717594\pi$
$157$	699.629	−	1211.79i	0.355646	−	0.615997i	0.987228	+	0.159312i	$0.0509275\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1807.71	−	3131.05i	−0.868656	−	1.50456i	−0.863370	−	0.504571i	$-0.831651\pi$
$163$	−1807.71	−	3131.05i	−0.868656	−	1.50456i	−0.00528615	−	0.999986i	$-0.501683\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1076.72			−0.498917			−0.249459	−	0.968385i	$-0.580253\pi$
$167$	−1076.72			−0.498917			−0.249459	+	0.968385i	$0.580253\pi$
$168$		0			0
$169$	−1514.55			−0.689372
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−936.815	−	1622.61i	−0.411704	−	0.713091i	0.583373	−	0.812205i	$-0.301733\pi$
$173$	−936.815	−	1622.61i	−0.411704	−	0.713091i	−0.995076	+	0.0991132i	$0.968399\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1346.15	+	2331.59i	−0.562099	+	0.973583i	0.435215	+	0.900327i	$0.356672\pi$
$179$	−1346.15	+	2331.59i	−0.562099	+	0.973583i	−0.997313	+	0.0732565i	$0.976661\pi$
$180$		0			0
$181$	−461.670			−0.189589			−0.0947947	−	0.995497i	$-0.530219\pi$
$181$	−461.670			−0.189589			−0.0947947	+	0.995497i	$0.530219\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1481.92	−	2566.76i	0.588934	−	1.02006i
$186$		0			0
$187$	557.073	+	964.879i	0.217846	+	0.377321i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1882.41	−	3260.43i	−0.713124	−	1.23517i	−0.963679	−	0.267064i	$-0.913946\pi$
$191$	−1882.41	−	3260.43i	−0.713124	−	1.23517i	0.250555	−	0.968102i	$-0.419387\pi$
$192$		0			0
$193$	−1463.10	+	2534.16i	−0.545680	+	0.945145i	0.452884	+	0.891569i	$0.350395\pi$
$193$	−1463.10	+	2534.16i	−0.545680	+	0.945145i	−0.998564	+	0.0535755i	$0.982938\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5045.55			−1.82477			−0.912387	−	0.409330i	$-0.865763\pi$
$197$	−5045.55			−1.82477			−0.912387	+	0.409330i	$0.865763\pi$
$198$		0			0
$199$	2146.12	−	3717.20i	0.764496	−	1.32415i	−0.176016	−	0.984387i	$-0.556321\pi$
$199$	2146.12	−	3717.20i	0.764496	−	1.32415i	0.940513	−	0.339759i	$-0.110346\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2318.85	−	4016.36i	−0.790025	−	1.36836i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1037.28			−0.343301
$210$		0			0
$211$	3625.76			1.18297			0.591487	−	0.806315i	$-0.298541\pi$
$211$	3625.76			1.18297			0.591487	+	0.806315i	$0.298541\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2277.72	+	3945.13i	0.722509	+	1.25142i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1026.64	−	1778.19i	0.312485	−	0.541239i
$222$		0			0
$223$	3145.00			0.944417			0.472208	−	0.881487i	$-0.343457\pi$
$223$	3145.00			0.944417			0.472208	+	0.881487i	$0.343457\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	772.873	−	1338.65i	0.225980	−	0.391408i	−0.730633	−	0.682770i	$-0.760775\pi$
$227$	772.873	−	1338.65i	0.225980	−	0.391408i	0.956613	+	0.291362i	$0.0941084\pi$
$228$		0			0
$229$	−2036.07	−	3526.58i	−0.587544	−	1.01766i	−0.994553	−	0.104232i	$-0.966762\pi$
$229$	−2036.07	−	3526.58i	−0.587544	−	1.01766i	0.407009	−	0.913424i	$-0.366572\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1507.86	−	2611.68i	−0.423961	−	0.734322i	0.572362	−	0.820001i	$-0.306027\pi$
$233$	−1507.86	−	2611.68i	−0.423961	−	0.734322i	−0.996323	+	0.0856790i	$0.972694\pi$
$234$		0			0
$235$	447.989	−	775.940i	0.124356	−	0.215390i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4647.38			−1.25780			−0.628900	−	0.777486i	$-0.716495\pi$
$239$	−4647.38			−1.25780			−0.628900	+	0.777486i	$0.716495\pi$
$240$		0			0
$241$	559.138	−	968.456i	0.149449	−	0.258854i	−0.781575	−	0.623812i	$-0.785583\pi$
$241$	559.138	−	968.456i	0.149449	−	0.258854i	0.931024	+	0.364958i	$0.118917\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	955.806	+	1655.50i	0.246221	+	0.426467i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	883.518			0.222180			0.111090	−	0.993810i	$-0.464566\pi$
$251$	883.518			0.222180			0.111090	+	0.993810i	$0.464566\pi$
$252$		0			0
$253$	1360.82			0.338159
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2934.44	+	5082.61i	0.712240	+	1.23364i	0.964015	+	0.265849i	$0.0856523\pi$
$257$	2934.44	+	5082.61i	0.712240	+	1.23364i	−0.251775	+	0.967786i	$0.581014\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2437.81	−	4222.41i	0.571565	−	0.989980i	−0.424841	−	0.905268i	$-0.639670\pi$
$263$	2437.81	−	4222.41i	0.571565	−	0.989980i	0.996406	−	0.0847114i	$-0.0269968\pi$
$264$		0			0
$265$	1604.07			0.371839
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2763.55	−	4786.61i	0.626382	−	1.08493i	−0.361890	−	0.932221i	$-0.617868\pi$
$269$	2763.55	−	4786.61i	0.626382	−	1.08493i	0.988272	−	0.152704i	$-0.0487983\pi$
$270$		0			0
$271$	−1612.37	−	2792.70i	−0.361418	−	0.625994i	0.626777	−	0.779199i	$-0.284374\pi$
$271$	−1612.37	−	2792.70i	−0.361418	−	0.625994i	−0.988194	+	0.153205i	$0.951041\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	201.815	+	349.554i	0.0442542	+	0.0766504i
$276$		0			0
$277$	979.022	−	1695.72i	0.212360	−	0.367818i	−0.740093	−	0.672505i	$-0.765218\pi$
$277$	979.022	−	1695.72i	0.212360	−	0.367818i	0.952453	+	0.304687i	$0.0985517\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8531.16			−1.81113			−0.905563	−	0.424212i	$-0.860551\pi$
$281$	−8531.16			−1.81113			−0.905563	+	0.424212i	$0.860551\pi$
$282$		0			0
$283$	3954.93	−	6850.14i	0.830728	−	1.43886i	−0.0667331	−	0.997771i	$-0.521258\pi$
$283$	3954.93	−	6850.14i	0.830728	−	1.43886i	0.897461	−	0.441093i	$-0.145409\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−632.324	−	1095.22i	−0.128704	−	0.222922i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8321.41			−1.65919			−0.829594	−	0.558367i	$-0.811428\pi$
$293$	−8321.41			−1.65919			−0.829594	+	0.558367i	$0.811428\pi$
$294$		0			0
$295$	−5900.02			−1.16445
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1253.94	−	2171.89i	−0.242533	−	0.420079i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2808.36	−	4864.22i	0.527234	−	0.913196i
$306$		0			0
$307$	−2541.79			−0.472533			−0.236267	−	0.971688i	$-0.575924\pi$
$307$	−2541.79			−0.472533			−0.236267	+	0.971688i	$0.575924\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	303.855	−	526.292i	0.0554020	−	0.0959590i	−0.836994	−	0.547212i	$-0.815689\pi$
$311$	303.855	−	526.292i	0.0554020	−	0.0959590i	0.892396	+	0.451253i	$0.149023\pi$
$312$		0			0
$313$	5513.64	+	9549.90i	0.995684	+	1.72458i	0.578213	+	0.815886i	$0.303750\pi$
$313$	5513.64	+	9549.90i	0.995684	+	1.72458i	0.417471	+	0.908690i	$0.362916\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3630.80	+	6288.73i	0.643300	+	1.11423i	0.984691	+	0.174307i	$0.0557685\pi$
$317$	3630.80	+	6288.73i	0.643300	+	1.11423i	−0.341391	+	0.939921i	$0.610898\pi$
$318$		0			0
$319$	1226.31	−	2124.02i	0.215235	−	0.372798i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5751.43			0.990768
$324$		0			0
$325$	371.927	−	644.197i	0.0634794	−	0.109950i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3316.36	−	5744.11i	−0.550706	−	0.953851i	−0.998224	−	0.0595761i	$-0.981025\pi$
$331$	−3316.36	−	5744.11i	−0.550706	−	0.953851i	0.447517	−	0.894275i	$-0.352308\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5299.69			−0.864338
$336$		0			0
$337$	8104.59			1.31005			0.655023	−	0.755609i	$-0.272659\pi$
$337$	8104.59			1.31005			0.655023	+	0.755609i	$0.272659\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	476.625	+	825.539i	0.0756912	+	0.131101i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2968.64	+	5141.83i	−0.459265	+	0.795470i	−0.998922	−	0.0464146i	$-0.985220\pi$
$347$	−2968.64	+	5141.83i	−0.459265	+	0.795470i	0.539657	+	0.841885i	$0.318554\pi$
$348$		0			0
$349$	−268.472			−0.0411775			−0.0205888	−	0.999788i	$-0.506554\pi$
$349$	−268.472			−0.0411775			−0.0205888	+	0.999788i	$0.506554\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1331.40	+	2306.06i	−0.200747	+	0.347703i	−0.948769	−	0.315970i	$-0.897670\pi$
$353$	−1331.40	+	2306.06i	−0.200747	+	0.347703i	0.748023	+	0.663673i	$0.231003\pi$
$354$		0			0
$355$	−5228.69	−	9056.36i	−0.781719	−	1.35398i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−781.637	−	1353.83i	−0.114911	−	0.199032i	0.802833	−	0.596204i	$-0.203325\pi$
$359$	−781.637	−	1353.83i	−0.114911	−	0.199032i	−0.917744	+	0.397172i	$0.869992\pi$
$360$		0			0
$361$	752.192	−	1302.83i	0.109665	−	0.189945i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4349.22			0.623695
$366$		0			0
$367$	605.408	−	1048.60i	0.0861091	−	0.149145i	−0.819754	−	0.572716i	$-0.805890\pi$
$367$	605.408	−	1048.60i	0.0861091	−	0.149145i	0.905863	+	0.423570i	$0.139223\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	2385.93	+	4132.55i	0.331203	+	0.573661i	0.982748	−	0.184949i	$-0.0592121\pi$
$373$	2385.93	+	4132.55i	0.331203	+	0.573661i	−0.651545	+	0.758610i	$0.725879\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4519.95			−0.617479
$378$		0			0
$379$	−4118.66			−0.558209			−0.279104	−	0.960261i	$-0.590038\pi$
$379$	−4118.66			−0.558209			−0.279104	+	0.960261i	$0.590038\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1767.99	−	3062.25i	−0.235875	−	0.408547i	0.723652	−	0.690165i	$-0.242462\pi$
$383$	−1767.99	−	3062.25i	−0.235875	−	0.408547i	−0.959527	+	0.281618i	$0.909129\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4803.23	−	8319.44i	0.626050	−	1.08435i	−0.362287	−	0.932067i	$-0.618004\pi$
$389$	4803.23	−	8319.44i	0.626050	−	1.08435i	0.988337	−	0.152284i	$-0.0486627\pi$
$390$		0			0
$391$	−7545.41			−0.975928
$392$		0			0
$393$		0			0
$394$		0			0
$395$	224.500	−	388.846i	0.0285971	−	0.0495316i
$396$		0			0
$397$	2683.54	+	4648.02i	0.339252	+	0.587601i	0.984292	−	0.176547i	$-0.0564929\pi$
$397$	2683.54	+	4648.02i	0.339252	+	0.587601i	−0.645041	+	0.764148i	$0.723160\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7032.91	−	12181.4i	−0.875828	−	1.51698i	−0.855878	−	0.517177i	$-0.826983\pi$
$401$	−7032.91	−	12181.4i	−0.875828	−	1.51698i	−0.0199492	−	0.999801i	$-0.506350\pi$
$402$		0			0
$403$	878.379	−	1521.40i	0.108574	−	0.188055i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4276.25			−0.520801
$408$		0			0
$409$	1636.22	−	2834.02i	0.197814	−	0.342624i	−0.750005	−	0.661432i	$-0.769949\pi$
$409$	1636.22	−	2834.02i	0.197814	−	0.342624i	0.947819	+	0.318808i	$0.103282\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	3372.17	+	5840.77i	0.398876	+	0.690873i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8077.07			0.941744			0.470872	−	0.882202i	$-0.343939\pi$
$419$	8077.07			0.941744			0.470872	+	0.882202i	$0.343939\pi$
$420$		0			0
$421$	8051.68			0.932101			0.466051	−	0.884758i	$-0.345676\pi$
$421$	8051.68			0.932101			0.466051	+	0.884758i	$0.345676\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1119.01	−	1938.18i	−0.127718	−	0.221213i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4950.11	+	8573.84i	−0.553221	+	0.958207i	0.444818	+	0.895621i	$0.353268\pi$
$431$	−4950.11	+	8573.84i	−0.553221	+	0.958207i	−0.998040	+	0.0625863i	$0.980065\pi$
$432$		0			0
$433$	511.795			0.0568021			0.0284010	−	0.999597i	$-0.490958\pi$
$433$	511.795			0.0568021			0.0284010	+	0.999597i	$0.490958\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3512.41	−	6083.68i	0.384488	−	0.665954i
$438$		0			0
$439$	−651.130	−	1127.79i	−0.0707898	−	0.122612i	0.828458	−	0.560051i	$-0.189219\pi$
$439$	−651.130	−	1127.79i	−0.0707898	−	0.122612i	−0.899248	+	0.437440i	$0.855885\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4100.16	−	7101.69i	−0.439739	−	0.761651i	0.557930	−	0.829888i	$-0.311596\pi$
$443$	−4100.16	−	7101.69i	−0.439739	−	0.761651i	−0.997669	+	0.0682372i	$0.978263\pi$
$444$		0			0
$445$	−3242.88	+	5616.83i	−0.345454	+	0.598344i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5788.12			0.608370			0.304185	−	0.952613i	$-0.401616\pi$
$449$	5788.12			0.608370			0.304185	+	0.952613i	$0.401616\pi$
$450$		0			0
$451$	−3345.65	+	5794.84i	−0.349314	+	0.605030i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3629.59	−	6286.63i	−0.371521	−	0.643493i	0.618279	−	0.785959i	$-0.287830\pi$
$457$	−3629.59	−	6286.63i	−0.371521	−	0.643493i	−0.989800	+	0.142466i	$0.954497\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−5966.92			−0.602835			−0.301418	−	0.953492i	$-0.597460\pi$
$461$	−5966.92			−0.602835			−0.301418	+	0.953492i	$0.597460\pi$
$462$		0			0
$463$	−8884.02			−0.891740			−0.445870	−	0.895098i	$-0.647106\pi$
$463$	−8884.02			−0.891740			−0.445870	+	0.895098i	$0.647106\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2014.68	+	3489.53i	0.199632	+	0.345774i	0.948409	−	0.317049i	$-0.102692\pi$
$467$	2014.68	+	3489.53i	0.199632	+	0.345774i	−0.748777	+	0.662822i	$0.769359\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3286.32	−	5692.07i	0.319461	−	0.553323i
$474$		0			0
$475$	2083.61			0.201269
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8720.33	−	15104.0i	0.831820	−	1.44075i	−0.0647733	−	0.997900i	$-0.520632\pi$
$479$	8720.33	−	15104.0i	0.831820	−	1.44075i	0.896593	−	0.442855i	$-0.146034\pi$
$480$		0			0
$481$	3940.38	+	6824.94i	0.373526	+	0.646966i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3233.91	+	5601.30i	0.302772	+	0.524417i
$486$		0			0
$487$	2136.14	−	3699.90i	0.198763	−	0.344268i	−0.749364	−	0.662158i	$-0.769641\pi$
$487$	2136.14	−	3699.90i	0.198763	−	0.344268i	0.948128	+	0.317890i	$0.102974\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2013.29			−0.185048			−0.0925240	−	0.995710i	$-0.529493\pi$
$491$	−2013.29			−0.185048			−0.0925240	+	0.995710i	$0.529493\pi$
$492$		0			0
$493$	−6799.54	+	11777.2i	−0.621169	+	1.07590i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2440.23	+	4226.60i	0.218917	+	0.379176i	0.954477	−	0.298284i	$-0.0964142\pi$
$499$	2440.23	+	4226.60i	0.218917	+	0.379176i	−0.735560	+	0.677460i	$0.763081\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12961.9			1.14899			0.574496	−	0.818508i	$-0.305198\pi$
$503$	12961.9			1.14899			0.574496	+	0.818508i	$0.305198\pi$
$504$		0			0
$505$	−17722.8			−1.56170
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4077.69	−	7062.77i	−0.355089	−	0.615033i	0.632044	−	0.774932i	$-0.282216\pi$
$509$	−4077.69	−	7062.77i	−0.355089	−	0.615033i	−0.987133	+	0.159900i	$0.948883\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5280.35	+	9145.84i	−0.451806	+	0.782551i
$516$		0			0
$517$	−1292.73			−0.109969
$518$		0			0
$519$		0			0
$520$		0			0
$521$	166.009	−	287.536i	0.0139597	−	0.0241789i	−0.858961	−	0.512041i	$-0.828890\pi$
$521$	166.009	−	287.536i	0.0139597	−	0.0241789i	0.872921	+	0.487862i	$0.162223\pi$
$522$		0			0
$523$	3627.42	+	6282.87i	0.303281	+	0.525298i	0.976877	−	0.213802i	$-0.0685847\pi$
$523$	3627.42	+	6282.87i	0.303281	+	0.525298i	−0.673596	+	0.739099i	$0.735251\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2642.76	−	4577.40i	−0.218445	−	0.378358i
$528$		0			0
$529$	1475.50	−	2555.64i	0.121271	−	0.210047i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12331.5			1.00213
$534$		0			0
$535$	−8151.36	+	14118.6i	−0.658718	+	1.14093i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2705.10	−	4685.36i	−0.214974	−	0.372347i	0.738290	−	0.674483i	$-0.235633\pi$
$541$	−2705.10	−	4685.36i	−0.214974	−	0.372347i	−0.953265	+	0.302137i	$0.902300\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5884.32			0.462489
$546$		0			0
$547$	−18673.9			−1.45967			−0.729834	−	0.683625i	$-0.760403\pi$
$547$	−18673.9			−1.45967			−0.729834	+	0.683625i	$0.760403\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6330.42	−	10964.6i	−0.489446	−	0.847746i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4104.74	+	7109.62i	−0.312250	+	0.540833i	−0.978849	−	0.204583i	$-0.934416\pi$
$557$	−4104.74	+	7109.62i	−0.312250	+	0.540833i	0.666599	+	0.745417i	$0.267749\pi$
$558$		0			0
$559$	−12112.8			−0.916489
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8761.10	+	15174.7i	−0.655837	+	1.13594i	0.325846	+	0.945423i	$0.394351\pi$
$563$	−8761.10	+	15174.7i	−0.655837	+	1.13594i	−0.981683	+	0.190521i	$0.938982\pi$
$564$		0			0
$565$	−3839.76	−	6650.65i	−0.285911	−	0.495212i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4528.41	−	7843.44i	−0.333640	−	0.577881i	0.649583	−	0.760291i	$-0.274944\pi$
$569$	−4528.41	−	7843.44i	−0.333640	−	0.577881i	−0.983223	+	0.182410i	$0.941610\pi$
$570$		0			0
$571$	−5358.27	+	9280.80i	−0.392709	+	0.680191i	−0.992806	−	0.119736i	$-0.961795\pi$
$571$	−5358.27	+	9280.80i	−0.392709	+	0.680191i	0.600097	+	0.799927i	$0.295129\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2733.53			−0.198254
$576$		0			0
$577$	1608.62	−	2786.21i	0.116062	−	0.201025i	−0.802142	−	0.597133i	$-0.796306\pi$
$577$	1608.62	−	2786.21i	0.116062	−	0.201025i	0.918204	+	0.396109i	$0.129640\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1157.18	−	2004.30i	−0.0822053	−	0.142384i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3248.45			0.228412			0.114206	−	0.993457i	$-0.463568\pi$
$587$	3248.45			0.228412			0.114206	+	0.993457i	$0.463568\pi$
$588$		0			0
$589$	4920.85			0.344245
$590$		0			0
$591$		0			0
$592$		0			0
$593$	10555.2	+	18282.1i	0.730945	+	1.26603i	0.956480	+	0.291799i	$0.0942537\pi$
$593$	10555.2	+	18282.1i	0.730945	+	1.26603i	−0.225535	+	0.974235i	$0.572413\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7868.98	+	13629.5i	−0.536758	+	0.929692i	0.462318	+	0.886714i	$0.347018\pi$
$599$	−7868.98	+	13629.5i	−0.536758	+	0.929692i	−0.999076	+	0.0429777i	$0.986316\pi$
$600$		0			0
$601$	4856.96			0.329650			0.164825	−	0.986323i	$-0.447294\pi$
$601$	4856.96			0.329650			0.164825	+	0.986323i	$0.447294\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5551.29	−	9615.12i	0.373045	−	0.646132i
$606$		0			0
$607$	−8629.29	−	14946.4i	−0.577022	−	0.999431i	−0.995819	−	0.0913513i	$-0.970881\pi$
$607$	−8629.29	−	14946.4i	−0.577022	−	0.999431i	0.418797	−	0.908080i	$-0.362452\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1191.19	+	2063.20i	0.0788714	+	0.136609i
$612$		0			0
$613$	−8058.53	+	13957.8i	−0.530964	+	0.919657i	0.468383	+	0.883526i	$0.344837\pi$
$613$	−8058.53	+	13957.8i	−0.530964	+	0.919657i	−0.999347	+	0.0361315i	$0.988496\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17507.3			1.14233			0.571164	−	0.820836i	$-0.306492\pi$
$617$	17507.3			1.14233			0.571164	+	0.820836i	$0.306492\pi$
$618$		0			0
$619$	4859.24	−	8416.46i	0.315524	−	0.546504i	−0.664025	−	0.747711i	$-0.731153\pi$
$619$	4859.24	−	8416.46i	0.315524	−	0.546504i	0.979549	+	0.201207i	$0.0644863\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	5627.47	+	9747.06i	0.360158	+	0.623812i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	23710.7			1.50303
$630$		0			0
$631$	5144.78			0.324580			0.162290	−	0.986743i	$-0.448112\pi$
$631$	5144.78			0.324580			0.162290	+	0.986743i	$0.448112\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−124.638	−	215.879i	−0.00778913	−	0.0134912i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10843.0	−	18780.7i	0.668135	−	1.15724i	−0.310290	−	0.950642i	$-0.600426\pi$
$641$	10843.0	−	18780.7i	0.668135	−	1.15724i	0.978425	−	0.206602i	$-0.0662405\pi$
$642$		0			0
$643$	14171.1			0.869132			0.434566	−	0.900640i	$-0.356902\pi$
$643$	14171.1			0.869132			0.434566	+	0.900640i	$0.356902\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10026.6	+	17366.6i	−0.609254	+	1.05526i	0.382109	+	0.924117i	$0.375198\pi$
$647$	−10026.6	+	17366.6i	−0.609254	+	1.05526i	−0.991364	+	0.131142i	$0.958135\pi$
$648$		0			0
$649$	4256.30	+	7372.13i	0.257434	+	0.445888i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11820.6	+	20473.8i	0.708385	+	1.22696i	0.965456	+	0.260566i	$0.0839091\pi$
$653$	11820.6	+	20473.8i	0.708385	+	1.22696i	−0.257072	+	0.966392i	$0.582758\pi$
$654$		0			0
$655$	2336.21	−	4046.44i	0.139364	−	0.241386i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2367.34			−0.139937			−0.0699685	−	0.997549i	$-0.522290\pi$
$659$	−2367.34			−0.139937			−0.0699685	+	0.997549i	$0.522290\pi$
$660$		0			0
$661$	−194.991	+	337.733i	−0.0114739	+	0.0198734i	−0.871705	−	0.490030i	$-0.836986\pi$
$661$	−194.991	+	337.733i	−0.0114739	+	0.0198734i	0.860231	+	0.509904i	$0.170319\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	8305.00	+	14384.7i	0.482115	+	0.835048i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8103.86			−0.466239
$672$		0			0
$673$	28764.5			1.64753			0.823766	−	0.566930i	$-0.191869\pi$
$673$	28764.5			1.64753			0.823766	+	0.566930i	$0.191869\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	881.061	+	1526.04i	0.0500176	+	0.0866331i	0.889950	−	0.456058i	$-0.150739\pi$
$677$	881.061	+	1526.04i	0.0500176	+	0.0866331i	−0.839933	+	0.542691i	$0.817406\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1114.44	+	1930.26i	−0.0624344	+	0.108140i	−0.895553	−	0.444955i	$-0.853220\pi$
$683$	−1114.44	+	1930.26i	−0.0624344	+	0.108140i	0.833119	+	0.553094i	$0.186553\pi$
$684$		0			0
$685$	−10399.9			−0.580085
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2132.59	+	3693.76i	−0.117918	+	0.204239i
$690$		0			0
$691$	8672.14	+	15020.6i	0.477429	+	0.826932i	0.999665	−	0.0258690i	$-0.00823527\pi$
$691$	8672.14	+	15020.6i	0.477429	+	0.826932i	−0.522236	+	0.852801i	$0.674902\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12678.8	+	21960.3i	0.691992	+	1.19856i
$696$		0			0
$697$	18550.8	−	32130.9i	1.00812	−	1.74612i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	23697.7			1.27682			0.638409	−	0.769697i	$-0.279593\pi$
$701$	23697.7			1.27682			0.638409	+	0.769697i	$0.279593\pi$
$702$		0			0
$703$	−11037.4	+	19117.3i	−0.592153	+	1.02564i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	11395.5	+	19737.6i	0.603620	+	1.04550i	0.992268	+	0.124114i	$0.0396088\pi$
$709$	11395.5	+	19737.6i	0.603620	+	1.04550i	−0.388648	+	0.921386i	$0.627058\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6455.76			−0.339089
$714$		0			0
$715$	3638.21			0.190296
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15922.4	−	27578.4i	−0.825876	−	1.43046i	−0.901248	−	0.433304i	$-0.857348\pi$
$719$	−15922.4	−	27578.4i	−0.825876	−	1.43046i	0.0753721	−	0.997155i	$-0.475986\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2463.32	+	4266.59i	−0.126187	+	0.218562i
$726$		0			0
$727$	22500.0			1.14784			0.573920	−	0.818912i	$-0.305422\pi$
$727$	22500.0			1.14784			0.573920	+	0.818912i	$0.305422\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−18221.8	+	31561.1i	−0.921966	+	1.59689i
$732$		0			0
$733$	−1611.38	−	2790.99i	−0.0811972	−	0.140638i	0.822567	−	0.568668i	$-0.192541\pi$
$733$	−1611.38	−	2790.99i	−0.0811972	−	0.140638i	−0.903764	+	0.428030i	$0.859208\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3823.23	+	6622.02i	0.191086	+	0.330971i
$738$		0			0
$739$	−9179.77	+	15899.8i	−0.456946	+	0.791454i	−0.998798	−	0.0490199i	$-0.984390\pi$
$739$	−9179.77	+	15899.8i	−0.456946	+	0.791454i	0.541851	+	0.840474i	$0.317724\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12890.4			−0.636478			−0.318239	−	0.948010i	$-0.603091\pi$
$743$	−12890.4			−0.636478			−0.318239	+	0.948010i	$0.603091\pi$
$744$		0			0
$745$	−500.450	+	866.805i	−0.0246108	+	0.0426272i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−7695.10	−	13328.3i	−0.373899	−	0.647612i	0.616263	−	0.787541i	$-0.288646\pi$
$751$	−7695.10	−	13328.3i	−0.373899	−	0.647612i	−0.990162	+	0.139929i	$0.955313\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−17472.6			−0.842241
$756$		0			0
$757$	13974.6			0.670957			0.335479	−	0.942048i	$-0.391102\pi$
$757$	13974.6			0.670957			0.335479	+	0.942048i	$0.391102\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1687.57	+	2922.96i	0.0803868	+	0.139234i	0.903416	−	0.428765i	$-0.141051\pi$
$761$	1687.57	+	2922.96i	0.0803868	+	0.139234i	−0.823029	+	0.567999i	$0.807718\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7843.99	−	13586.2i	0.369270	−	0.639595i
$768$		0			0
$769$	31253.9			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$769$	31253.9			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−11581.8	+	20060.2i	−0.538897	+	0.933396i	0.460067	+	0.887884i	$0.347825\pi$
$773$	−11581.8	+	20060.2i	−0.538897	+	0.933396i	−0.998964	+	0.0455122i	$0.985508\pi$
$774$		0			0
$775$	−957.411	−	1658.29i	−0.0443758	−	0.0768611i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	17270.9	+	29914.0i	0.794343	+	1.37584i
$780$		0			0
$781$	−7544.01	+	13066.6i	−0.345641	+	0.598668i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13747.4			−0.625050
$786$		0			0
$787$	−4951.47	+	8576.20i	−0.224271	+	0.388448i	−0.956100	−	0.293039i	$-0.905333\pi$
$787$	−4951.47	+	8576.20i	−0.224271	+	0.388448i	0.731830	+	0.681488i	$0.238667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	7467.36	+	12933.8i	0.334393	+	0.579186i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	16844.5			0.748638			0.374319	−	0.927300i	$-0.377877\pi$
$797$	16844.5			0.748638			0.374319	+	0.927300i	$0.377877\pi$
$798$		0			0
$799$	7167.83			0.317371
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3137.55	−	5434.40i	−0.137885	−	0.238824i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−17215.3	+	29817.8i	−0.748157	+	1.29585i	0.200549	+	0.979684i	$0.435728\pi$
$809$	−17215.3	+	29817.8i	−0.748157	+	1.29585i	−0.948705	+	0.316162i	$0.897606\pi$
$810$		0			0
$811$	62.4498			0.00270396			0.00135198	−	0.999999i	$-0.499570\pi$
$811$	62.4498			0.00270396			0.00135198	+	0.999999i	$0.499570\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−17760.3	+	30761.8i	−0.763334	+	1.32213i
$816$		0			0
$817$	−16964.6	−	29383.5i	−0.726458	−	1.25826i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	19475.8	+	33733.1i	0.827906	+	1.43398i	0.899678	+	0.436555i	$0.143802\pi$
$821$	19475.8	+	33733.1i	0.827906	+	1.43398i	−0.0717711	+	0.997421i	$0.522865\pi$
$822$		0			0
$823$	17741.2	−	30728.6i	0.751420	−	1.30150i	−0.195715	−	0.980661i	$-0.562703\pi$
$823$	17741.2	−	30728.6i	0.751420	−	1.30150i	0.947135	−	0.320836i	$-0.103964\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−7945.87			−0.334105			−0.167053	−	0.985948i	$-0.553425\pi$
$827$	−7945.87			−0.334105			−0.167053	+	0.985948i	$0.553425\pi$
$828$		0			0
$829$	20318.2	−	35192.2i	0.851243	−	1.47440i	−0.0288439	−	0.999584i	$-0.509183\pi$
$829$	20318.2	−	35192.2i	0.851243	−	1.47440i	0.880087	−	0.474812i	$-0.157484\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	5289.26	+	9161.26i	0.219212	+	0.379687i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	27723.5			1.14079			0.570394	−	0.821371i	$-0.306791\pi$
$839$	27723.5			1.14079			0.570394	+	0.821371i	$0.306791\pi$
$840$		0			0
$841$	5547.19			0.227446
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7440.04	+	12886.5i	0.302893	+	0.524627i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	14480.2	−	25080.4i	0.583283	−	1.01028i
$852$		0			0
$853$	−37599.1			−1.50922			−0.754612	−	0.656171i	$-0.772175\pi$
$853$	−37599.1			−1.50922			−0.754612	+	0.656171i	$0.772175\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12805.0	+	22178.9i	−0.510397	+	0.884034i	0.489530	+	0.871986i	$0.337168\pi$
$857$	−12805.0	+	22178.9i	−0.510397	+	0.884034i	−0.999927	+	0.0120476i	$0.996165\pi$
$858$		0			0
$859$	−1989.70	−	3446.26i	−0.0790309	−	0.136886i	0.823801	−	0.566879i	$-0.191849\pi$
$859$	−1989.70	−	3446.26i	−0.0790309	−	0.136886i	−0.902832	+	0.429993i	$0.858516\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12410.2	−	21495.0i	−0.489510	−	0.847856i	0.510417	−	0.859927i	$-0.329491\pi$
$863$	−12410.2	−	21495.0i	−0.489510	−	0.847856i	−0.999927	+	0.0120708i	$0.996158\pi$
$864$		0			0
$865$	−9203.97	+	15941.7i	−0.361786	+	0.626631i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−647.823			−0.0252887
$870$		0			0
$871$	7045.87	−	12203.8i	0.274099	−	0.474754i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−25009.7	−	43318.1i	−0.962963	−	1.66790i	−0.714992	−	0.699133i	$-0.753570\pi$
$877$	−25009.7	−	43318.1i	−0.962963	−	1.66790i	−0.247971	−	0.968767i	$-0.579764\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10386.4			−0.397193			−0.198596	−	0.980081i	$-0.563638\pi$
$881$	−10386.4			−0.397193			−0.198596	+	0.980081i	$0.563638\pi$
$882$		0			0
$883$	36366.4			1.38599			0.692994	−	0.720944i	$-0.256291\pi$
$883$	36366.4			1.38599			0.692994	+	0.720944i	$0.256291\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3803.65	+	6588.12i	0.143984	+	0.249388i	0.928994	−	0.370096i	$-0.120675\pi$
$887$	3803.65	+	6588.12i	0.143984	+	0.249388i	−0.785009	+	0.619484i	$0.787342\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3336.65	+	5779.24i	−0.125035	+	0.216568i
$894$		0			0
$895$	26451.1			0.987891
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5817.61	+	10076.4i	−0.215827	+	0.373823i
$900$		0			0
$901$	6416.28	+	11113.3i	0.237245	+	0.410920i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2267.90	+	3928.11i	0.0833011	+	0.144282i
$906$		0			0
$907$	−12901.5	+	22346.1i	−0.472314	+	0.818072i	−0.999498	−	0.0316794i	$-0.989914\pi$
$907$	−12901.5	+	22346.1i	−0.472314	+	0.818072i	0.527184	+	0.849751i	$0.323248\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8165.21			−0.296954			−0.148477	−	0.988916i	$-0.547437\pi$
$911$	−8165.21			−0.296954			−0.148477	+	0.988916i	$0.547437\pi$
$912$		0			0
$913$	4865.40	−	8427.12i	0.176365	−	0.305473i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−296.804	−	514.079i	−0.0106536	−	0.0184526i	0.860649	−	0.509198i	$-0.170058\pi$
$919$	−296.804	−	514.079i	−0.0106536	−	0.0184526i	−0.871303	+	0.490745i	$0.836725\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	27805.9			0.991596
$924$		0			0
$925$	8589.84			0.305332
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16573.7	+	28706.5i	0.585323	+	1.01381i	0.994835	+	0.101504i	$0.0323655\pi$
$929$	16573.7	+	28706.5i	0.585323	+	1.01381i	−0.409512	+	0.912305i	$0.634301\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5473.11	−	9479.70i	0.191433	−	0.331571i
$936$		0			0
$937$	24883.8			0.867575			0.433788	−	0.901015i	$-0.357177\pi$
$937$	24883.8			0.867575			0.433788	+	0.901015i	$0.357177\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13065.1	−	22629.5i	0.452616	−	0.783954i	−0.545932	−	0.837830i	$-0.683824\pi$
$941$	13065.1	−	22629.5i	0.452616	−	0.783954i	0.998548	+	0.0538758i	$0.0171575\pi$
$942$		0			0
$943$	−22658.0	−	39244.8i	−0.782445	−	1.35523i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	3984.35	+	6901.09i	0.136720	+	0.236806i	0.926253	−	0.376902i	$-0.123011\pi$
$947$	3984.35	+	6901.09i	0.136720	+	0.236806i	−0.789533	+	0.613708i	$0.789677\pi$
$948$		0			0
$949$	−5782.24	+	10015.1i	−0.197786	+	0.342576i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6742.36			0.229178			0.114589	−	0.993413i	$-0.463445\pi$
$953$	6742.36			0.229178			0.114589	+	0.993413i	$0.463445\pi$
$954$		0			0
$955$	−18494.2	+	32033.0i	−0.626659	+	1.08541i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	12634.4	+	21883.4i	0.424101	+	0.734564i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	28749.2			0.959035
$966$		0			0
$967$	−7202.50			−0.239521			−0.119760	−	0.992803i	$-0.538213\pi$
$967$	−7202.50			−0.239521			−0.119760	+	0.992803i	$0.538213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6852.86	−	11869.5i	−0.226487	−	0.392286i	0.730278	−	0.683150i	$-0.239391\pi$
$971$	−6852.86	−	11869.5i	−0.226487	−	0.392286i	−0.956764	+	0.290864i	$0.906057\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−28274.0	+	48972.0i	−0.925860	+	1.60364i	−0.135687	+	0.990752i	$0.543324\pi$
$977$	−28274.0	+	48972.0i	−0.925860	+	1.60364i	−0.790173	+	0.612884i	$0.790009\pi$
$978$		0			0
$979$	9357.71			0.305489
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−8681.87	+	15037.4i	−0.281698	+	0.487914i	−0.971803	−	0.235794i	$-0.924231\pi$
$983$	−8681.87	+	15037.4i	−0.281698	+	0.487914i	0.690105	+	0.723709i	$0.257564\pi$
$984$		0			0
$985$	24785.6	+	42930.0i	0.801762	+	1.38869i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	22256.2	+	38548.8i	0.715576	+	1.23941i
$990$		0			0
$991$	−24519.7	+	42469.4i	−0.785968	+	1.36134i	0.142451	+	0.989802i	$0.454502\pi$
$991$	−24519.7	+	42469.4i	−0.785968	+	1.36134i	−0.928419	+	0.371535i	$0.878832\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−42170.3			−1.34361
$996$		0			0
$997$	−4558.30	+	7895.21i	−0.144797	+	0.250796i	−0.929297	−	0.369333i	$-0.879586\pi$
$997$	−4558.30	+	7895.21i	−0.144797	+	0.250796i	0.784500	+	0.620129i	$0.212920\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.z.361.1		4
3.2	odd	2		588.4.i.i.361.2		4
7.2	even	3	inner	1764.4.k.z.1549.1		4
7.3	odd	6		1764.4.a.x.1.1		2
7.4	even	3		1764.4.a.p.1.2		2
7.5	odd	6		252.4.k.d.37.2		4
7.6	odd	2		252.4.k.d.109.2		4
21.2	odd	6		588.4.i.i.373.2		4
21.5	even	6		84.4.i.b.37.1	yes	4
21.11	odd	6		588.4.a.h.1.1		2
21.17	even	6		588.4.a.g.1.2		2
21.20	even	2		84.4.i.b.25.1	✓	4
84.11	even	6		2352.4.a.bp.1.1		2
84.47	odd	6		336.4.q.h.289.1		4
84.59	odd	6		2352.4.a.cb.1.2		2
84.83	odd	2		336.4.q.h.193.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.i.b.25.1	✓	4	21.20	even	2
84.4.i.b.37.1	yes	4	21.5	even	6
252.4.k.d.37.2		4	7.5	odd	6
252.4.k.d.109.2		4	7.6	odd	2
336.4.q.h.193.1		4	84.83	odd	2
336.4.q.h.289.1		4	84.47	odd	6
588.4.a.g.1.2		2	21.17	even	6
588.4.a.h.1.1		2	21.11	odd	6
588.4.i.i.361.2		4	3.2	odd	2
588.4.i.i.373.2		4	21.2	odd	6
1764.4.a.p.1.2		2	7.4	even	3
1764.4.a.x.1.1		2	7.3	odd	6
1764.4.k.z.361.1		4	1.1	even	1	trivial
1764.4.k.z.1549.1		4	7.2	even	3	inner
2352.4.a.bp.1.1		2	84.11	even	6
2352.4.a.cb.1.2		2	84.59	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.91238 - 8.50848i) q^{5} +O(q^{10})\)	\(q+(-4.91238 - 8.50848i) q^{5} +(-7.08762 + 12.2761i) q^{11} +26.1238 q^{13} +(39.2990 - 68.0679i) q^{17} +(36.5876 + 63.3716i) q^{19} +(-48.0000 - 83.1384i) q^{23} +(14.2371 - 24.6594i) q^{25} -173.021 q^{29} +(33.6238 - 58.2381i) q^{31} +(150.835 + 261.254i) q^{37} +472.042 q^{41} -463.670 q^{43} +(45.5980 + 78.9781i) q^{47} +(-81.6341 + 141.394i) q^{53} +139.268 q^{55} +(300.263 - 520.071i) q^{59} +(285.846 + 495.099i) q^{61} +(-128.330 - 222.274i) q^{65} +(269.711 - 467.154i) q^{67} +1064.39 q^{71} +(-221.340 + 383.372i) q^{73} +(22.8505 + 39.5782i) q^{79} -686.464 q^{83} -772.206 q^{85} +(-330.072 - 571.702i) q^{89} +(359.464 - 622.610i) q^{95} -658.320 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{5}+O(q^{10}) \) 4 * q + 3 * q^5	\( 4 q + 3 q^{5} - 51 q^{11} - 122 q^{13} - 24 q^{17} + 169 q^{19} - 192 q^{23} - 11 q^{25} + 78 q^{29} - 92 q^{31} + 173 q^{37} + 348 q^{41} - 994 q^{43} - 180 q^{47} + 285 q^{53} - 666 q^{55} + 1269 q^{59} + 328 q^{61} - 1374 q^{65} + 875 q^{67} + 2808 q^{71} - 1361 q^{73} + 182 q^{79} - 798 q^{83} - 4176 q^{85} - 822 q^{89} - 510 q^{95} - 1682 q^{97}+O(q^{100}) \) 4 * q + 3 * q^5 - 51 * q^11 - 122 * q^13 - 24 * q^17 + 169 * q^19 - 192 * q^23 - 11 * q^25 + 78 * q^29 - 92 * q^31 + 173 * q^37 + 348 * q^41 - 994 * q^43 - 180 * q^47 + 285 * q^53 - 666 * q^55 + 1269 * q^59 + 328 * q^61 - 1374 * q^65 + 875 * q^67 + 2808 * q^71 - 1361 * q^73 + 182 * q^79 - 798 * q^83 - 4176 * q^85 - 822 * q^89 - 510 * q^95 - 1682 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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