LMFDB - Embedded newform 2160.3.c.m.1889.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(1889,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2160.1889");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.c (of order $2$, degree $1$, 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:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.31744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 14x^{2} + 31 $ x^4 + 14*x^2 + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.4
Root			$1.66053i$ of defining polynomial
Character	$\chi$	$=$	2160.1889
Dual form			2160.3.c.m.1889.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.41421 + 2.34834i) q^{5} +13.6872i q^{7} +O(q^{10})$	$q+(4.41421 + 2.34834i) q^{5} +13.6872i q^{7} +12.3115i q^{11} +17.0082i q^{13} +6.89949 q^{17} +7.24264 q^{19} -34.7990 q^{23} +(13.9706 + 20.7322i) q^{25} +21.1351i q^{29} +38.2132 q^{31} +(-32.1421 + 60.4180i) q^{35} +21.5380i q^{37} -36.3648i q^{41} +6.23921i q^{43} +40.2010 q^{47} -138.338 q^{49} +38.2721 q^{53} +(-28.9117 + 54.3457i) q^{55} -41.6313i q^{59} -15.0589 q^{61} +(-39.9411 + 75.0779i) q^{65} -128.618i q^{67} -104.967i q^{71} -2.11232i q^{73} -168.510 q^{77} +44.0955 q^{79} +55.0244 q^{83} +(30.4558 + 16.2024i) q^{85} -68.1020i q^{89} -232.794 q^{91} +(31.9706 + 17.0082i) q^{95} -101.243i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{5}+O(q^{10}) $ 4 * q + 12 * q^5	$ 4 q + 12 q^{5} - 12 q^{17} + 12 q^{19} - 60 q^{23} - 12 q^{25} + 68 q^{31} - 72 q^{35} + 240 q^{47} - 180 q^{49} + 204 q^{53} + 88 q^{55} - 196 q^{61} - 24 q^{65} - 312 q^{77} - 180 q^{79} - 108 q^{83} + 20 q^{85} - 456 q^{91} + 60 q^{95}+O(q^{100}) $ 4 * q + 12 * q^5 - 12 * q^17 + 12 * q^19 - 60 * q^23 - 12 * q^25 + 68 * q^31 - 72 * q^35 + 240 * q^47 - 180 * q^49 + 204 * q^53 + 88 * q^55 - 196 * q^61 - 24 * q^65 - 312 * q^77 - 180 * q^79 - 108 * q^83 + 20 * q^85 - 456 * q^91 + 60 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	4.41421	+	2.34834i	0.882843	+	0.469669i
$5$	4.41421	+	2.34834i	0.882843	+	0.469669i
$6$		0			0
$7$			13.6872i			1.95531i	0.210222	+	0.977654i	$0.432581\pi$
$7$			13.6872i			1.95531i	−0.210222	+	0.977654i	$0.567419\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			12.3115i			1.11923i	0.828753	+	0.559615i	$0.189051\pi$
$11$			12.3115i			1.11923i	−0.828753	+	0.559615i	$0.810949\pi$
$12$		0			0
$13$			17.0082i			1.30832i	0.756355	+	0.654162i	$0.226979\pi$
$13$			17.0082i			1.30832i	−0.756355	+	0.654162i	$0.773021\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.89949			0.405853			0.202926	−	0.979194i	$-0.434955\pi$
$17$	6.89949			0.405853			0.202926	+	0.979194i	$0.434955\pi$
$18$		0			0
$19$	7.24264			0.381192			0.190596	−	0.981669i	$-0.438958\pi$
$19$	7.24264			0.381192			0.190596	+	0.981669i	$0.438958\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−34.7990			−1.51300			−0.756500	−	0.653994i	$-0.773092\pi$
$23$	−34.7990			−1.51300			−0.756500	+	0.653994i	$0.773092\pi$
$24$		0			0
$25$	13.9706	+	20.7322i	0.558823	+	0.829287i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			21.1351i			0.728796i	0.931243	+	0.364398i	$0.118725\pi$
$29$			21.1351i			0.728796i	−0.931243	+	0.364398i	$0.881275\pi$
$30$		0			0
$31$	38.2132			1.23268			0.616342	−	0.787479i	$-0.288614\pi$
$31$	38.2132			1.23268			0.616342	+	0.787479i	$0.288614\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−32.1421	+	60.4180i	−0.918347	+	1.72623i
$36$		0			0
$37$			21.5380i			0.582108i	0.956707	+	0.291054i	$0.0940060\pi$
$37$			21.5380i			0.582108i	−0.956707	+	0.291054i	$0.905994\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	36.3648i		−	0.886945i	−0.896288	−	0.443473i	$-0.853746\pi$
$41$		−	36.3648i		−	0.886945i	0.896288	−	0.443473i	$-0.146254\pi$
$42$		0			0
$43$			6.23921i			0.145098i	0.997365	+	0.0725489i	$0.0231133\pi$
$43$			6.23921i			0.145098i	−0.997365	+	0.0725489i	$0.976887\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	40.2010			0.855341			0.427670	−	0.903935i	$-0.359334\pi$
$47$	40.2010			0.855341			0.427670	+	0.903935i	$0.359334\pi$
$48$		0			0
$49$	−138.338			−2.82323
$50$		0			0
$51$		0			0
$52$		0			0
$53$	38.2721			0.722115			0.361057	−	0.932544i	$-0.382416\pi$
$53$	38.2721			0.722115			0.361057	+	0.932544i	$0.382416\pi$
$54$		0			0
$55$	−28.9117	+	54.3457i	−0.525667	+	0.988103i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	41.6313i		−	0.705615i	−0.935696	−	0.352807i	$-0.885227\pi$
$59$		−	41.6313i		−	0.705615i	0.935696	−	0.352807i	$-0.114773\pi$
$60$		0			0
$61$	−15.0589			−0.246867			−0.123433	−	0.992353i	$-0.539391\pi$
$61$	−15.0589			−0.246867			−0.123433	+	0.992353i	$0.539391\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−39.9411	+	75.0779i	−0.614479	+	1.15504i
$66$		0			0
$67$		−	128.618i		−	1.91967i	−0.280569	−	0.959834i	$-0.590523\pi$
$67$		−	128.618i		−	1.91967i	0.280569	−	0.959834i	$-0.409477\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	104.967i		−	1.47841i	−0.673478	−	0.739207i	$-0.735200\pi$
$71$		−	104.967i		−	1.47841i	0.673478	−	0.739207i	$-0.264800\pi$
$72$		0			0
$73$		−	2.11232i		−	0.0289359i	−0.999895	−	0.0144680i	$-0.995395\pi$
$73$		−	2.11232i		−	0.0289359i	0.999895	−	0.0144680i	$-0.00460546\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−168.510			−2.18844
$78$		0			0
$79$	44.0955			0.558170			0.279085	−	0.960266i	$-0.409969\pi$
$79$	44.0955			0.558170			0.279085	+	0.960266i	$0.409969\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	55.0244			0.662944			0.331472	−	0.943465i	$-0.392455\pi$
$83$	55.0244			0.662944			0.331472	+	0.943465i	$0.392455\pi$
$84$		0			0
$85$	30.4558	+	16.2024i	0.358304	+	0.190616i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	68.1020i		−	0.765191i	−0.923916	−	0.382595i	$-0.875030\pi$
$89$		−	68.1020i		−	0.765191i	0.923916	−	0.382595i	$-0.124970\pi$
$90$		0			0
$91$	−232.794			−2.55818
$92$		0			0
$93$		0			0
$94$		0			0
$95$	31.9706	+	17.0082i	0.336532	+	0.179034i
$96$		0			0
$97$		−	101.243i		−	1.04375i	−0.853023	−	0.521873i	$-0.825233\pi$
$97$		−	101.243i		−	1.04375i	0.853023	−	0.521873i	$-0.174767\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			15.2297i			0.150789i	0.997154	+	0.0753944i	$0.0240216\pi$
$101$			15.2297i			0.150789i	−0.997154	+	0.0753944i	$0.975978\pi$
$102$		0			0
$103$			77.0924i			0.748470i	0.927334	+	0.374235i	$0.122095\pi$
$103$			77.0924i			0.748470i	−0.927334	+	0.374235i	$0.877905\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	78.4264			0.732957			0.366479	−	0.930427i	$-0.380563\pi$
$107$	78.4264			0.732957			0.366479	+	0.930427i	$0.380563\pi$
$108$		0			0
$109$	−146.279			−1.34201			−0.671006	−	0.741452i	$-0.734137\pi$
$109$	−146.279			−1.34201			−0.671006	+	0.741452i	$0.734137\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	54.4264			0.481650			0.240825	−	0.970569i	$-0.422582\pi$
$113$	54.4264			0.481650			0.240825	+	0.970569i	$0.422582\pi$
$114$		0			0
$115$	−153.610	−	81.7200i	−1.33574	−	0.710609i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			94.4344i			0.793567i
$120$		0			0
$121$	−30.5736			−0.252674
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.9828	+	124.324i	0.103862	+	0.994592i
$126$		0			0
$127$			159.215i			1.25366i	0.779154	+	0.626832i	$0.215649\pi$
$127$			159.215i			1.25366i	−0.779154	+	0.626832i	$0.784351\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	110.303i		−	0.842008i	−0.907059	−	0.421004i	$-0.861678\pi$
$131$		−	110.303i		−	0.842008i	0.907059	−	0.421004i	$-0.138322\pi$
$132$		0			0
$133$			99.1311i			0.745347i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	217.919			1.59065			0.795324	−	0.606184i	$-0.207301\pi$
$137$	217.919			1.59065			0.795324	+	0.606184i	$0.207301\pi$
$138$		0			0
$139$	106.250			0.764387			0.382193	−	0.924082i	$-0.375169\pi$
$139$	106.250			0.764387			0.382193	+	0.924082i	$0.375169\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−209.397			−1.46431
$144$		0			0
$145$	−49.6325	+	93.2948i	−0.342293	+	0.643413i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			14.1592i			0.0950281i	0.998871	+	0.0475141i	$0.0151299\pi$
$149$			14.1592i			0.0950281i	−0.998871	+	0.0475141i	$0.984870\pi$
$150$		0			0
$151$	−73.3381			−0.485683			−0.242841	−	0.970066i	$-0.578079\pi$
$151$	−73.3381			−0.485683			−0.242841	+	0.970066i	$0.578079\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	168.681	+	89.7377i	1.08827	+	0.578953i
$156$		0			0
$157$		−	70.1452i		−	0.446784i	−0.974729	−	0.223392i	$-0.928287\pi$
$157$		−	70.1452i		−	0.446784i	0.974729	−	0.223392i	$-0.0717131\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	476.299i		−	2.95838i
$162$		0			0
$163$			9.05959i			0.0555803i	0.999614	+	0.0277902i	$0.00884702\pi$
$163$			9.05959i			0.0555803i	−0.999614	+	0.0277902i	$0.991153\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	63.8528			0.382352			0.191176	−	0.981556i	$-0.438770\pi$
$167$	63.8528			0.382352			0.191176	+	0.981556i	$0.438770\pi$
$168$		0			0
$169$	−120.279			−0.711711
$170$		0			0
$171$		0			0
$172$		0			0
$173$	27.9878			0.161779			0.0808896	−	0.996723i	$-0.474224\pi$
$173$	27.9878			0.161779			0.0808896	+	0.996723i	$0.474224\pi$
$174$		0			0
$175$	−283.765	+	191.217i	−1.62151	+	1.09267i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			21.0660i			0.117687i	0.998267	+	0.0588435i	$0.0187413\pi$
$179$			21.0660i			0.117687i	−0.998267	+	0.0588435i	$0.981259\pi$
$180$		0			0
$181$	53.6030			0.296149			0.148075	−	0.988976i	$-0.452692\pi$
$181$	53.6030			0.296149			0.148075	+	0.988976i	$0.452692\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−50.5786	+	95.0734i	−0.273398	+	0.513910i
$186$		0			0
$187$			84.9433i			0.454242i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	223.024i		−	1.16766i	−0.811875	−	0.583831i	$-0.801553\pi$
$191$		−	223.024i		−	1.16766i	0.811875	−	0.583831i	$-0.198447\pi$
$192$		0			0
$193$			176.224i			0.913075i	0.889704	+	0.456538i	$0.150911\pi$
$193$			176.224i			0.913075i	−0.889704	+	0.456538i	$0.849089\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	277.586			1.40906			0.704532	−	0.709672i	$-0.251157\pi$
$197$	277.586			1.40906			0.704532	+	0.709672i	$0.251157\pi$
$198$		0			0
$199$	−71.5736			−0.359666			−0.179833	−	0.983697i	$-0.557556\pi$
$199$	−71.5736			−0.359666			−0.179833	+	0.983697i	$0.557556\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−289.279			−1.42502
$204$		0			0
$205$	85.3970	−	160.522i	0.416571	−	0.783033i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			89.1679i			0.426641i
$210$		0			0
$211$	−319.492			−1.51418			−0.757091	−	0.653309i	$-0.773380\pi$
$211$	−319.492			−1.51418			−0.757091	+	0.653309i	$0.773380\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−14.6518	+	27.5412i	−0.0681479	+	0.128099i
$216$		0			0
$217$			523.030i			2.41028i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			117.348i			0.530987i
$222$		0			0
$223$			162.439i			0.728424i	0.931316	+	0.364212i	$0.118662\pi$
$223$			162.439i			0.728424i	−0.931316	+	0.364212i	$0.881338\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−241.098			−1.06210			−0.531052	−	0.847339i	$-0.678203\pi$
$227$	−241.098			−1.06210			−0.531052	+	0.847339i	$0.678203\pi$
$228$		0			0
$229$	102.220			0.446377			0.223189	−	0.974775i	$-0.428353\pi$
$229$	102.220			0.446377			0.223189	+	0.974775i	$0.428353\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	241.966			1.03848			0.519239	−	0.854629i	$-0.326215\pi$
$233$	241.966			1.03848			0.519239	+	0.854629i	$0.326215\pi$
$234$		0			0
$235$	177.456	+	94.4058i	0.755131	+	0.401727i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			375.458i			1.57096i	0.618890	+	0.785478i	$0.287583\pi$
$239$			375.458i			1.57096i	−0.618890	+	0.785478i	$0.712417\pi$
$240$		0			0
$241$	−318.279			−1.32066			−0.660330	−	0.750975i	$-0.729584\pi$
$241$	−318.279			−1.32066			−0.660330	+	0.750975i	$0.729584\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−610.654	−	324.865i	−2.49246	−	1.32598i
$246$		0			0
$247$			123.184i			0.498722i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	85.5417i		−	0.340804i	−0.985375	−	0.170402i	$-0.945493\pi$
$251$		−	85.5417i		−	0.340804i	0.985375	−	0.170402i	$-0.0545066\pi$
$252$		0			0
$253$		−	428.429i		−	1.69339i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−214.227			−0.833570			−0.416785	−	0.909005i	$-0.636843\pi$
$257$	−214.227			−0.833570			−0.416785	+	0.909005i	$0.636843\pi$
$258$		0			0
$259$	−294.794			−1.13820
$260$		0			0
$261$		0			0
$262$		0			0
$263$	106.877			0.406377			0.203189	−	0.979140i	$-0.434870\pi$
$263$	106.877			0.406377			0.203189	+	0.979140i	$0.434870\pi$
$264$		0			0
$265$	168.941	+	89.8760i	0.637514	+	0.339155i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			365.927i			1.36032i	0.733062	+	0.680161i	$0.238090\pi$
$269$			365.927i			1.36032i	−0.733062	+	0.680161i	$0.761910\pi$
$270$		0			0
$271$	92.8894			0.342765			0.171383	−	0.985205i	$-0.445177\pi$
$271$	92.8894			0.342765			0.171383	+	0.985205i	$0.445177\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−255.245	+	171.999i	−0.928163	+	0.625451i
$276$		0			0
$277$			226.137i			0.816380i	0.912897	+	0.408190i	$0.133840\pi$
$277$			226.137i			0.816380i	−0.912897	+	0.408190i	$0.866160\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	63.4053i		−	0.225642i	−0.993615	−	0.112821i	$-0.964011\pi$
$281$		−	63.4053i		−	0.225642i	0.993615	−	0.112821i	$-0.0359886\pi$
$282$		0			0
$283$			363.106i			1.28306i	0.767097	+	0.641531i	$0.221700\pi$
$283$			363.106i			1.28306i	−0.767097	+	0.641531i	$0.778300\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	497.730			1.73425
$288$		0			0
$289$	−241.397			−0.835284
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−227.434			−0.776224			−0.388112	−	0.921612i	$-0.626873\pi$
$293$	−227.434			−0.776224			−0.388112	+	0.921612i	$0.626873\pi$
$294$		0			0
$295$	97.7645	−	183.769i	0.331405	−	0.622947i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	591.869i		−	1.97949i
$300$		0			0
$301$	−85.3970			−0.283711
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−66.4731	−	35.3634i	−0.217945	−	0.115946i
$306$		0			0
$307$			340.164i			1.10803i	0.832508	+	0.554013i	$0.186904\pi$
$307$			340.164i			1.10803i	−0.832508	+	0.554013i	$0.813096\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			386.769i			1.24363i	0.783164	+	0.621815i	$0.213604\pi$
$311$			386.769i			1.24363i	−0.783164	+	0.621815i	$0.786396\pi$
$312$		0			0
$313$			376.403i			1.20256i	0.799037	+	0.601282i	$0.205343\pi$
$313$			376.403i			1.20256i	−0.799037	+	0.601282i	$0.794657\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	355.399			1.12113			0.560566	−	0.828110i	$-0.310583\pi$
$317$	355.399			1.12113			0.560566	+	0.828110i	$0.310583\pi$
$318$		0			0
$319$	−260.205			−0.815690
$320$		0			0
$321$		0			0
$322$		0			0
$323$	49.9706			0.154708
$324$		0			0
$325$	−352.617	+	237.614i	−1.08498	+	0.731121i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			550.237i			1.67245i
$330$		0			0
$331$	444.191			1.34197			0.670983	−	0.741473i	$-0.265872\pi$
$331$	444.191			1.34197			0.670983	+	0.741473i	$0.265872\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	302.039	−	567.746i	0.901608	−	1.69476i
$336$		0			0
$337$		−	407.086i		−	1.20797i	−0.796995	−	0.603985i	$-0.793579\pi$
$337$		−	407.086i		−	1.20797i	0.796995	−	0.603985i	$-0.206421\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			470.463i			1.37966i
$342$		0			0
$343$		−	1222.78i		−	3.56497i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	489.411			1.41041			0.705204	−	0.709005i	$-0.250856\pi$
$347$	489.411			1.41041			0.705204	+	0.709005i	$0.250856\pi$
$348$		0			0
$349$	81.7939			0.234367			0.117183	−	0.993110i	$-0.462613\pi$
$349$	81.7939			0.234367			0.117183	+	0.993110i	$0.462613\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−251.387			−0.712144			−0.356072	−	0.934459i	$-0.615884\pi$
$353$	−251.387			−0.712144			−0.356072	+	0.934459i	$0.615884\pi$
$354$		0			0
$355$	246.500	−	463.349i	0.694365	−	1.30521i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			518.500i			1.44429i	0.691742	+	0.722145i	$0.256844\pi$
$359$			518.500i			1.44429i	−0.691742	+	0.722145i	$0.743156\pi$
$360$		0			0
$361$	−308.544			−0.854693
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.96046	−	9.32425i	0.0135903	−	0.0255459i
$366$		0			0
$367$		−	341.983i		−	0.931834i	−0.884828	−	0.465917i	$-0.845724\pi$
$367$		−	341.983i		−	0.931834i	0.884828	−	0.465917i	$-0.154276\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			523.836i			1.41196i
$372$		0			0
$373$			110.303i			0.295719i	0.989008	+	0.147859i	$0.0472383\pi$
$373$			110.303i			0.295719i	−0.989008	+	0.147859i	$0.952762\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−359.470			−0.953502
$378$		0			0
$379$	−324.345			−0.855792			−0.427896	−	0.903828i	$-0.640745\pi$
$379$	−324.345			−0.855792			−0.427896	+	0.903828i	$0.640745\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−572.480			−1.49473			−0.747363	−	0.664416i	$-0.768680\pi$
$383$	−572.480			−1.49473			−0.747363	+	0.664416i	$0.768680\pi$
$384$		0			0
$385$	−743.838	−	395.719i	−1.93205	−	1.02784i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	721.666i		−	1.85518i	−0.373595	−	0.927592i	$-0.621875\pi$
$389$		−	721.666i		−	1.85518i	0.373595	−	0.927592i	$-0.378125\pi$
$390$		0			0
$391$	−240.095			−0.614055
$392$		0			0
$393$		0			0
$394$		0			0
$395$	194.647	+	103.551i	0.492777	+	0.262155i
$396$		0			0
$397$		−	379.321i		−	0.955468i	−0.878505	−	0.477734i	$-0.841458\pi$
$397$		−	379.321i		−	0.955468i	0.878505	−	0.477734i	$-0.158542\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			350.974i			0.875246i	0.899159	+	0.437623i	$0.144180\pi$
$401$			350.974i			0.875246i	−0.899159	+	0.437623i	$0.855820\pi$
$402$		0			0
$403$			649.938i			1.61275i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−265.166			−0.651513
$408$		0			0
$409$	−533.426			−1.30422			−0.652111	−	0.758124i	$-0.726116\pi$
$409$	−533.426			−1.30422			−0.652111	+	0.758124i	$0.726116\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	569.813			1.37969
$414$		0			0
$415$	242.889	+	129.216i	0.585276	+	0.311364i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	239.755i		−	0.572208i	−0.958199	−	0.286104i	$-0.907640\pi$
$419$		−	239.755i		−	0.572208i	0.958199	−	0.286104i	$-0.0923603\pi$
$420$		0			0
$421$	424.941			1.00936			0.504681	−	0.863306i	$-0.331610\pi$
$421$	424.941			1.00936			0.504681	+	0.863306i	$0.331610\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	96.3898	+	143.042i	0.226800	+	0.336568i
$426$		0			0
$427$		−	206.113i		−	0.482700i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	544.039i		−	1.26227i	−0.775673	−	0.631135i	$-0.782589\pi$
$431$		−	544.039i		−	1.26227i	0.775673	−	0.631135i	$-0.217411\pi$
$432$		0			0
$433$		−	602.735i		−	1.39200i	−0.718043	−	0.695999i	$-0.754962\pi$
$433$		−	602.735i		−	1.39200i	0.718043	−	0.695999i	$-0.245038\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−252.037			−0.576743
$438$		0			0
$439$	−402.286			−0.916370			−0.458185	−	0.888857i	$-0.651500\pi$
$439$	−402.286			−0.916370			−0.458185	+	0.888857i	$0.651500\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−438.515			−0.989875			−0.494938	−	0.868929i	$-0.664809\pi$
$443$	−438.515			−0.989875			−0.494938	+	0.868929i	$0.664809\pi$
$444$		0			0
$445$	159.927	−	300.617i	0.359386	−	0.675543i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	670.866i		−	1.49413i	−0.664749	−	0.747067i	$-0.731462\pi$
$449$		−	670.866i		−	1.49413i	0.664749	−	0.747067i	$-0.268538\pi$
$450$		0			0
$451$	447.706			0.992695
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1027.60	−	546.680i	−2.25847	−	1.20149i
$456$		0			0
$457$			695.029i			1.52085i	0.649425	+	0.760425i	$0.275010\pi$
$457$			695.029i			1.52085i	−0.649425	+	0.760425i	$0.724990\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	13.0887i		−	0.0283920i	−0.999899	−	0.0141960i	$-0.995481\pi$
$461$		−	13.0887i		−	0.0283920i	0.999899	−	0.0141960i	$-0.00451888\pi$
$462$		0			0
$463$		−	234.281i		−	0.506007i	−0.967465	−	0.253004i	$-0.918582\pi$
$463$		−	234.281i		−	0.506007i	0.967465	−	0.253004i	$-0.0814185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	742.882			1.59075			0.795377	−	0.606115i	$-0.207273\pi$
$467$	742.882			1.59075			0.795377	+	0.606115i	$0.207273\pi$
$468$		0			0
$469$	1760.41			3.75354
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−76.8141			−0.162398
$474$		0			0
$475$	101.184	+	150.156i	0.213018	+	0.316117i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			447.411i			0.934052i	0.884244	+	0.467026i	$0.154675\pi$
$479$			447.411i			0.934052i	−0.884244	+	0.467026i	$0.845325\pi$
$480$		0			0
$481$	−366.323			−0.761586
$482$		0			0
$483$		0			0
$484$		0			0
$485$	237.754	−	446.910i	0.490215	−	0.921464i
$486$		0			0
$487$			50.3166i			0.103319i	0.998665	+	0.0516597i	$0.0164511\pi$
$487$			50.3166i			0.103319i	−0.998665	+	0.0516597i	$0.983549\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			790.631i			1.61025i	0.593107	+	0.805124i	$0.297901\pi$
$491$			790.631i			1.61025i	−0.593107	+	0.805124i	$0.702099\pi$
$492$		0			0
$493$			145.821i			0.295784i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1436.70			2.89075
$498$		0			0
$499$	600.390			1.20319			0.601593	−	0.798803i	$-0.294533\pi$
$499$	600.390			1.20319			0.601593	+	0.798803i	$0.294533\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	433.368			0.861566			0.430783	−	0.902456i	$-0.358237\pi$
$503$	433.368			0.861566			0.430783	+	0.902456i	$0.358237\pi$
$504$		0			0
$505$	−35.7645	+	67.2270i	−0.0708208	+	0.133123i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	961.715i		−	1.88942i	−0.327907	−	0.944710i	$-0.606343\pi$
$509$		−	961.715i		−	1.88942i	0.327907	−	0.944710i	$-0.393657\pi$
$510$		0			0
$511$	28.9117			0.0565786
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−181.040	+	340.302i	−0.351533	+	0.660781i
$516$		0			0
$517$			494.936i			0.957322i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	5.90542i		−	0.0113348i	−0.999984	−	0.00566739i	$-0.998196\pi$
$521$		−	5.90542i		−	0.0113348i	0.999984	−	0.00566739i	$-0.00180400\pi$
$522$		0			0
$523$			165.846i			0.317104i	0.987351	+	0.158552i	$0.0506826\pi$
$523$			165.846i			0.317104i	−0.987351	+	0.158552i	$0.949317\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	263.652			0.500288
$528$		0			0
$529$	681.970			1.28917
$530$		0			0
$531$		0			0
$532$		0			0
$533$	618.500			1.16041
$534$		0			0
$535$	346.191	+	184.172i	0.647086	+	0.344247i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1703.15i		−	3.15984i
$540$		0			0
$541$	216.985			0.401081			0.200541	−	0.979685i	$-0.435730\pi$
$541$	216.985			0.401081			0.200541	+	0.979685i	$0.435730\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−645.708	−	343.514i	−1.18478	−	0.630301i
$546$		0			0
$547$			100.645i			0.183995i	0.995759	+	0.0919973i	$0.0293251\pi$
$547$			100.645i			0.183995i	−0.995759	+	0.0919973i	$0.970675\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			153.074i			0.277811i
$552$		0			0
$553$			603.541i			1.09139i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	116.662			0.209447			0.104723	−	0.994501i	$-0.466604\pi$
$557$	116.662			0.209447			0.104723	+	0.994501i	$0.466604\pi$
$558$		0			0
$559$	−106.118			−0.189835
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−533.907			−0.948324			−0.474162	−	0.880438i	$-0.657249\pi$
$563$	−533.907			−0.948324			−0.474162	+	0.880438i	$0.657249\pi$
$564$		0			0
$565$	240.250	+	127.812i	0.425221	+	0.226216i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			112.651i			0.197981i	0.995088	+	0.0989907i	$0.0315614\pi$
$569$			112.651i			0.197981i	−0.995088	+	0.0989907i	$0.968439\pi$
$570$		0			0
$571$	861.535			1.50882			0.754409	−	0.656404i	$-0.227923\pi$
$571$	861.535			1.50882			0.754409	+	0.656404i	$0.227923\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−486.161	−	721.459i	−0.845498	−	1.25471i
$576$		0			0
$577$		−	37.3256i		−	0.0646891i	−0.999477	−	0.0323446i	$-0.989703\pi$
$577$		−	37.3256i		−	0.0646891i	0.999477	−	0.0323446i	$-0.0102974\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			753.127i			1.29626i
$582$		0			0
$583$			471.188i			0.808212i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	538.103			0.916699			0.458350	−	0.888772i	$-0.348441\pi$
$587$	538.103			0.916699			0.458350	+	0.888772i	$0.348441\pi$
$588$		0			0
$589$	276.765			0.469889
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−884.169			−1.49101			−0.745505	−	0.666500i	$-0.767792\pi$
$593$	−884.169			−1.49101			−0.745505	+	0.666500i	$0.767792\pi$
$594$		0			0
$595$	−221.765	+	416.854i	−0.372713	+	0.700595i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	44.7736i		−	0.0747472i	−0.999301	−	0.0373736i	$-0.988101\pi$
$599$		−	44.7736i		−	0.0747472i	0.999301	−	0.0373736i	$-0.0118992\pi$
$600$		0			0
$601$	394.574			0.656528			0.328264	−	0.944586i	$-0.393536\pi$
$601$	394.574			0.656528			0.328264	+	0.944586i	$0.393536\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−134.958	−	71.7973i	−0.223072	−	0.118673i
$606$		0			0
$607$			215.173i			0.354485i	0.984167	+	0.177243i	$0.0567178\pi$
$607$			215.173i			0.354485i	−0.984167	+	0.177243i	$0.943282\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			683.747i			1.11906i
$612$		0			0
$613$		−	333.937i		−	0.544758i	−0.962190	−	0.272379i	$-0.912189\pi$
$613$		−	333.937i		−	0.544758i	0.962190	−	0.272379i	$-0.0878105\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−65.4234			−0.106035			−0.0530174	−	0.998594i	$-0.516884\pi$
$617$	−65.4234			−0.106035			−0.0530174	+	0.998594i	$0.516884\pi$
$618$		0			0
$619$	19.9268			0.0321920			0.0160960	−	0.999870i	$-0.494876\pi$
$619$	19.9268			0.0321920			0.0160960	+	0.999870i	$0.494876\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	932.122			1.49618
$624$		0			0
$625$	−234.647	+	579.281i	−0.375435	+	0.926849i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			148.601i			0.236250i
$630$		0			0
$631$	210.625			0.333796			0.166898	−	0.985974i	$-0.446625\pi$
$631$	210.625			0.333796			0.166898	+	0.985974i	$0.446625\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−373.892	+	702.811i	−0.588807	+	1.10679i
$636$		0			0
$637$		−	2352.88i		−	3.69369i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1177.76i			1.83738i	0.394976	+	0.918692i	$0.370753\pi$
$641$			1177.76i			1.83738i	−0.394976	+	0.918692i	$0.629247\pi$
$642$		0			0
$643$		−	37.6426i		−	0.0585422i	−0.999572	−	0.0292711i	$-0.990681\pi$
$643$		−	37.6426i		−	0.0585422i	0.999572	−	0.0292711i	$-0.00931861\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	695.382			1.07478			0.537389	−	0.843334i	$-0.319411\pi$
$647$	695.382			1.07478			0.537389	+	0.843334i	$0.319411\pi$
$648$		0			0
$649$	512.544			0.789744
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−924.296			−1.41546			−0.707731	−	0.706482i	$-0.750281\pi$
$653$	−924.296			−1.41546			−0.707731	+	0.706482i	$0.750281\pi$
$654$		0			0
$655$	259.029	−	486.901i	0.395465	−	0.743361i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			67.6704i			0.102687i	0.998681	+	0.0513433i	$0.0163503\pi$
$659$			67.6704i			0.102687i	−0.998681	+	0.0513433i	$0.983650\pi$
$660$		0			0
$661$	1032.87			1.56258			0.781291	−	0.624167i	$-0.214561\pi$
$661$	1032.87			1.56258			0.781291	+	0.624167i	$0.214561\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−232.794	+	437.586i	−0.350066	+	0.658024i
$666$		0			0
$667$		−	735.480i		−	1.10267i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	185.398i		−	0.276301i
$672$		0			0
$673$		−	10.1706i		−	0.0151123i	−0.999971	−	0.00755614i	$-0.997595\pi$
$673$		−	10.1706i		−	0.0151123i	0.999971	−	0.00755614i	$-0.00240522\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	576.936			0.852195			0.426098	−	0.904677i	$-0.359888\pi$
$677$	576.936			0.852195			0.426098	+	0.904677i	$0.359888\pi$
$678$		0			0
$679$	1385.73			2.04085
$680$		0			0
$681$		0			0
$682$		0			0
$683$	526.009			0.770145			0.385073	−	0.922886i	$-0.374176\pi$
$683$	526.009			0.770145			0.385073	+	0.922886i	$0.374176\pi$
$684$		0			0
$685$	961.940	+	511.748i	1.40429	+	0.747078i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			650.940i			0.944760i
$690$		0			0
$691$	−422.904			−0.612017			−0.306008	−	0.952029i	$-0.598994\pi$
$691$	−422.904			−0.612017			−0.306008	+	0.952029i	$0.598994\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	469.009	+	249.511i	0.674833	+	0.359009i
$696$		0			0
$697$		−	250.899i		−	0.359969i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	881.146i		−	1.25698i	−0.777816	−	0.628492i	$-0.783672\pi$
$701$		−	881.146i		−	1.25698i	0.777816	−	0.628492i	$-0.216328\pi$
$702$		0			0
$703$			155.992i			0.221895i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−208.451			−0.294838
$708$		0			0
$709$	101.427			0.143057			0.0715284	−	0.997439i	$-0.477212\pi$
$709$	101.427			0.143057			0.0715284	+	0.997439i	$0.477212\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1329.78			−1.86505
$714$		0			0
$715$	−924.323	−	491.736i	−1.29276	−	0.687743i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			36.2956i			0.0504807i	0.999681	+	0.0252404i	$0.00803511\pi$
$719$			36.2956i			0.0504807i	−0.999681	+	0.0252404i	$0.991965\pi$
$720$		0			0
$721$	−1055.18			−1.46349
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−438.177	+	295.269i	−0.604382	+	0.407268i
$726$		0			0
$727$		−	527.755i		−	0.725936i	−0.931802	−	0.362968i	$-0.881763\pi$
$727$		−	527.755i		−	0.725936i	0.931802	−	0.362968i	$-0.118237\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			43.0474i			0.0588883i
$732$		0			0
$733$		−	1305.93i		−	1.78163i	−0.454370	−	0.890813i	$-0.650136\pi$
$733$		−	1305.93i		−	1.78163i	0.454370	−	0.890813i	$-0.349864\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1583.48			2.14855
$738$		0			0
$739$	−754.875			−1.02148			−0.510741	−	0.859735i	$-0.670629\pi$
$739$	−754.875			−1.02148			−0.510741	+	0.859735i	$0.670629\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1312.59			−1.76660			−0.883302	−	0.468804i	$-0.844685\pi$
$743$	−1312.59			−1.76660			−0.883302	+	0.468804i	$0.844685\pi$
$744$		0			0
$745$	−33.2506	+	62.5017i	−0.0446317	+	0.0838949i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1073.43i			1.43316i
$750$		0			0
$751$	−1232.43			−1.64106			−0.820528	−	0.571607i	$-0.806320\pi$
$751$	−1232.43			−1.64106			−0.820528	+	0.571607i	$0.806320\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−323.730	−	172.223i	−0.428781	−	0.228110i
$756$		0			0
$757$		−	401.164i		−	0.529939i	−0.964257	−	0.264970i	$-0.914638\pi$
$757$		−	401.164i		−	0.529939i	0.964257	−	0.264970i	$-0.0853619\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	525.130i		−	0.690053i	−0.938593	−	0.345027i	$-0.887870\pi$
$761$		−	525.130i		−	0.690053i	0.938593	−	0.345027i	$-0.112130\pi$
$762$		0			0
$763$		−	2002.15i		−	2.62404i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	708.073			0.923172
$768$		0			0
$769$	−542.750			−0.705787			−0.352894	−	0.935663i	$-0.614802\pi$
$769$	−542.750			−0.705787			−0.352894	+	0.935663i	$0.614802\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−522.640			−0.676119			−0.338059	−	0.941125i	$-0.609770\pi$
$773$	−522.640			−0.676119			−0.338059	+	0.941125i	$0.609770\pi$
$774$		0			0
$775$	533.860	+	792.243i	0.688852	+	1.02225i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	263.377i		−	0.338096i
$780$		0			0
$781$	1292.31			1.65468
$782$		0			0
$783$		0			0
$784$		0			0
$785$	164.725	−	309.636i	0.209841	−	0.394440i
$786$		0			0
$787$		−	828.884i		−	1.05322i	−0.850107	−	0.526610i	$-0.823463\pi$
$787$		−	828.884i		−	1.05322i	0.850107	−	0.526610i	$-0.176537\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			744.942i			0.941773i
$792$		0			0
$793$		−	256.125i		−	0.322982i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	613.606			0.769895			0.384947	−	0.922939i	$-0.374220\pi$
$797$	613.606			0.769895			0.384947	+	0.922939i	$0.374220\pi$
$798$		0			0
$799$	277.367			0.347142
$800$		0			0
$801$		0			0
$802$		0			0
$803$	26.0059			0.0323859
$804$		0			0
$805$	1118.51	−	2102.49i	1.38946	−	2.61178i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	545.471i		−	0.674254i	−0.941459	−	0.337127i	$-0.890545\pi$
$809$		−	545.471i		−	0.674254i	0.941459	−	0.337127i	$-0.109455\pi$
$810$		0			0
$811$	293.192			0.361519			0.180759	−	0.983527i	$-0.442144\pi$
$811$	293.192			0.361519			0.180759	+	0.983527i	$0.442144\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−21.2750	+	39.9910i	−0.0261043	+	0.0490687i
$816$		0			0
$817$			45.1883i			0.0553101i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	853.691i		−	1.03982i	−0.854221	−	0.519909i	$-0.825966\pi$
$821$		−	853.691i		−	1.03982i	0.854221	−	0.519909i	$-0.174034\pi$
$822$		0			0
$823$			202.475i			0.246021i	0.992405	+	0.123010i	$0.0392548\pi$
$823$			202.475i			0.246021i	−0.992405	+	0.123010i	$0.960745\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	15.3734			0.0185894			0.00929471	−	0.999957i	$-0.497041\pi$
$827$	15.3734			0.0185894			0.00929471	+	0.999957i	$0.497041\pi$
$828$		0			0
$829$	−1135.22			−1.36938			−0.684692	−	0.728833i	$-0.740063\pi$
$829$	−1135.22			−1.36938			−0.684692	+	0.728833i	$0.740063\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−954.463			−1.14581
$834$		0			0
$835$	281.860	+	149.948i	0.337557	+	0.179579i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			248.510i			0.296197i	0.988973	+	0.148099i	$0.0473153\pi$
$839$			248.510i			0.296197i	−0.988973	+	0.148099i	$0.952685\pi$
$840$		0			0
$841$	394.308			0.468856
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−530.938	−	282.457i	−0.628329	−	0.334269i
$846$		0			0
$847$		−	418.465i		−	0.494056i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	749.501i		−	0.880730i
$852$		0			0
$853$			20.1219i			0.0235895i	0.999930	+	0.0117948i	$0.00375448\pi$
$853$			20.1219i			0.0235895i	−0.999930	+	0.0117948i	$0.996246\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	185.567			0.216531			0.108266	−	0.994122i	$-0.465470\pi$
$857$	185.567			0.216531			0.108266	+	0.994122i	$0.465470\pi$
$858$		0			0
$859$	1098.48			1.27879			0.639393	−	0.768880i	$-0.279186\pi$
$859$	1098.48			1.27879			0.639393	+	0.768880i	$0.279186\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−32.5736			−0.0377446			−0.0188723	−	0.999822i	$-0.506008\pi$
$863$	−32.5736			−0.0377446			−0.0188723	+	0.999822i	$0.506008\pi$
$864$		0			0
$865$	123.544	+	65.7250i	0.142826	+	0.0759827i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			542.882i			0.624721i
$870$		0			0
$871$	2187.56			2.51155
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1701.64	+	177.697i	−1.94473	+	0.203082i
$876$		0			0
$877$			1001.29i			1.14172i	0.821048	+	0.570859i	$0.193390\pi$
$877$			1001.29i			1.14172i	−0.821048	+	0.570859i	$0.806610\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1664.18i			1.88897i	0.328559	+	0.944483i	$0.393437\pi$
$881$			1664.18i			1.88897i	−0.328559	+	0.944483i	$0.606563\pi$
$882$		0			0
$883$		−	221.400i		−	0.250736i	−0.992110	−	0.125368i	$-0.959989\pi$
$883$		−	221.400i		−	0.250736i	0.992110	−	0.125368i	$-0.0400112\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1659.54			−1.87096			−0.935481	−	0.353378i	$-0.885033\pi$
$887$	−1659.54			−1.87096			−0.935481	+	0.353378i	$0.885033\pi$
$888$		0			0
$889$	−2179.20			−2.45130
$890$		0			0
$891$		0			0
$892$		0			0
$893$	291.161			0.326049
$894$		0			0
$895$	−49.4701	+	92.9897i	−0.0552739	+	0.103899i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			807.640i			0.898376i
$900$		0			0
$901$	264.058			0.293072
$902$		0			0
$903$		0			0
$904$		0			0
$905$	236.615	+	125.878i	0.261453	+	0.139092i
$906$		0			0
$907$		−	168.494i		−	0.185771i	−0.995677	−	0.0928854i	$-0.970391\pi$
$907$		−	168.494i		−	0.185771i	0.995677	−	0.0928854i	$-0.0296090\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1446.53i		−	1.58785i	−0.608019	−	0.793923i	$-0.708035\pi$
$911$		−	1446.53i		−	1.58785i	0.608019	−	0.793923i	$-0.291965\pi$
$912$		0			0
$913$			677.434i			0.741987i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1509.73			1.64638
$918$		0			0
$919$	1279.60			1.39239			0.696193	−	0.717855i	$-0.254876\pi$
$919$	1279.60			1.39239			0.696193	+	0.717855i	$0.254876\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1785.31			1.93424
$924$		0			0
$925$	−446.530	+	300.898i	−0.482735	+	0.325295i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	983.265i		−	1.05841i	−0.848493	−	0.529206i	$-0.822490\pi$
$929$		−	983.265i		−	1.05841i	0.848493	−	0.529206i	$-0.177510\pi$
$930$		0			0
$931$	−1001.93			−1.07619
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−199.476	+	374.958i	−0.213343	+	0.401024i
$936$		0			0
$937$			964.938i			1.02982i	0.857245	+	0.514908i	$0.172174\pi$
$937$			964.938i			1.02982i	−0.857245	+	0.514908i	$0.827826\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1029.89i		−	1.09446i	−0.836983	−	0.547230i	$-0.815682\pi$
$941$		−	1029.89i		−	1.09446i	0.836983	−	0.547230i	$-0.184318\pi$
$942$		0			0
$943$			1265.46i			1.34195i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−186.868			−0.197326			−0.0986631	−	0.995121i	$-0.531457\pi$
$947$	−186.868			−0.197326			−0.0986631	+	0.995121i	$0.531457\pi$
$948$		0			0
$949$	35.9268			0.0378576
$950$		0			0
$951$		0			0
$952$		0			0
$953$	857.170			0.899444			0.449722	−	0.893169i	$-0.351523\pi$
$953$	857.170			0.899444			0.449722	+	0.893169i	$0.351523\pi$
$954$		0			0
$955$	523.736	−	984.474i	0.548415	−	1.03086i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			2982.69i			3.11021i
$960$		0			0
$961$	499.249			0.519510
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−413.833	+	777.888i	−0.428843	+	0.806102i
$966$		0			0
$967$			1234.04i			1.27615i	0.769972	+	0.638077i	$0.220270\pi$
$967$			1234.04i			1.27615i	−0.769972	+	0.638077i	$0.779730\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1313.38i		−	1.35261i	−0.736624	−	0.676303i	$-0.763581\pi$
$971$		−	1313.38i		−	1.35261i	0.736624	−	0.676303i	$-0.236419\pi$
$972$		0			0
$973$			1454.26i			1.49461i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−109.486			−0.112064			−0.0560318	−	0.998429i	$-0.517845\pi$
$977$	−109.486			−0.112064			−0.0560318	+	0.998429i	$0.517845\pi$
$978$		0			0
$979$	838.439			0.856424
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−341.995			−0.347909			−0.173955	−	0.984754i	$-0.555655\pi$
$983$	−341.995			−0.347909			−0.173955	+	0.984754i	$0.555655\pi$
$984$		0			0
$985$	1225.32	+	651.867i	1.24398	+	0.661794i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	217.118i		−	0.219533i
$990$		0			0
$991$	−140.258			−0.141532			−0.0707658	−	0.997493i	$-0.522544\pi$
$991$	−140.258			−0.141532			−0.0707658	+	0.997493i	$0.522544\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−315.941	−	168.079i	−0.317529	−	0.168924i
$996$		0			0
$997$			619.634i			0.621498i	0.950492	+	0.310749i	$0.100580\pi$
$997$			619.634i			0.621498i	−0.950492	+	0.310749i	$0.899420\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.c.m.1889.4		4
3.2	odd	2		2160.3.c.g.1889.1		4
4.3	odd	2		270.3.b.d.269.4	yes	4
5.4	even	2		2160.3.c.g.1889.2		4
12.11	even	2		270.3.b.a.269.1	✓	4
15.14	odd	2	inner	2160.3.c.m.1889.3		4
20.3	even	4		1350.3.d.o.701.1		8
20.7	even	4		1350.3.d.o.701.8		8
20.19	odd	2		270.3.b.a.269.2	yes	4
36.7	odd	6		810.3.j.a.539.1		8
36.11	even	6		810.3.j.f.539.4		8
36.23	even	6		810.3.j.f.269.3		8
36.31	odd	6		810.3.j.a.269.2		8
60.23	odd	4		1350.3.d.o.701.5		8
60.47	odd	4		1350.3.d.o.701.4		8
60.59	even	2		270.3.b.d.269.3	yes	4
180.59	even	6		810.3.j.a.269.1		8
180.79	odd	6		810.3.j.f.539.3		8
180.119	even	6		810.3.j.a.539.2		8
180.139	odd	6		810.3.j.f.269.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.b.a.269.1	✓	4	12.11	even	2
270.3.b.a.269.2	yes	4	20.19	odd	2
270.3.b.d.269.3	yes	4	60.59	even	2
270.3.b.d.269.4	yes	4	4.3	odd	2
810.3.j.a.269.1		8	180.59	even	6
810.3.j.a.269.2		8	36.31	odd	6
810.3.j.a.539.1		8	36.7	odd	6
810.3.j.a.539.2		8	180.119	even	6
810.3.j.f.269.3		8	36.23	even	6
810.3.j.f.269.4		8	180.139	odd	6
810.3.j.f.539.3		8	180.79	odd	6
810.3.j.f.539.4		8	36.11	even	6
1350.3.d.o.701.1		8	20.3	even	4
1350.3.d.o.701.4		8	60.47	odd	4
1350.3.d.o.701.5		8	60.23	odd	4
1350.3.d.o.701.8		8	20.7	even	4
2160.3.c.g.1889.1		4	3.2	odd	2
2160.3.c.g.1889.2		4	5.4	even	2
2160.3.c.m.1889.3		4	15.14	odd	2	inner
2160.3.c.m.1889.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.41421 + 2.34834i) q^{5} +13.6872i q^{7} +O(q^{10})\)	\(q+(4.41421 + 2.34834i) q^{5} +13.6872i q^{7} +12.3115i q^{11} +17.0082i q^{13} +6.89949 q^{17} +7.24264 q^{19} -34.7990 q^{23} +(13.9706 + 20.7322i) q^{25} +21.1351i q^{29} +38.2132 q^{31} +(-32.1421 + 60.4180i) q^{35} +21.5380i q^{37} -36.3648i q^{41} +6.23921i q^{43} +40.2010 q^{47} -138.338 q^{49} +38.2721 q^{53} +(-28.9117 + 54.3457i) q^{55} -41.6313i q^{59} -15.0589 q^{61} +(-39.9411 + 75.0779i) q^{65} -128.618i q^{67} -104.967i q^{71} -2.11232i q^{73} -168.510 q^{77} +44.0955 q^{79} +55.0244 q^{83} +(30.4558 + 16.2024i) q^{85} -68.1020i q^{89} -232.794 q^{91} +(31.9706 + 17.0082i) q^{95} -101.243i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{5}+O(q^{10}) \) 4 * q + 12 * q^5	\( 4 q + 12 q^{5} - 12 q^{17} + 12 q^{19} - 60 q^{23} - 12 q^{25} + 68 q^{31} - 72 q^{35} + 240 q^{47} - 180 q^{49} + 204 q^{53} + 88 q^{55} - 196 q^{61} - 24 q^{65} - 312 q^{77} - 180 q^{79} - 108 q^{83} + 20 q^{85} - 456 q^{91} + 60 q^{95}+O(q^{100}) \) 4 * q + 12 * q^5 - 12 * q^17 + 12 * q^19 - 60 * q^23 - 12 * q^25 + 68 * q^31 - 72 * q^35 + 240 * q^47 - 180 * q^49 + 204 * q^53 + 88 * q^55 - 196 * q^61 - 24 * q^65 - 312 * q^77 - 180 * q^79 - 108 * q^83 + 20 * q^85 - 456 * q^91 + 60 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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