LMFDB - Embedded newform 1620.3.b.b.809.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(809,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1620.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.b (of order $2$, degree $1$, 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:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.8
Character	$\chi$	$=$	1620.809
Dual form			1620.3.b.b.809.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.98582 + 4.58874i) q^{5} -0.667924i q^{7} +O(q^{10})$	$q+(-1.98582 + 4.58874i) q^{5} -0.667924i q^{7} +19.4601i q^{11} +20.0501i q^{13} +16.3370 q^{17} +2.43805 q^{19} +34.3714 q^{23} +(-17.1130 - 18.2248i) q^{25} +5.73648i q^{29} -26.8521 q^{31} +(3.06493 + 1.32638i) q^{35} +11.3929i q^{37} -29.2502i q^{41} -34.1247i q^{43} -67.9986 q^{47} +48.5539 q^{49} +37.0720 q^{53} +(-89.2972 - 38.6442i) q^{55} +19.0295i q^{59} -113.562 q^{61} +(-92.0047 - 39.8159i) q^{65} +106.718i q^{67} -37.0887i q^{71} +115.846i q^{73} +12.9979 q^{77} +16.7659 q^{79} +90.4200 q^{83} +(-32.4423 + 74.9660i) q^{85} +28.0932i q^{89} +13.3919 q^{91} +(-4.84154 + 11.1876i) q^{95} +9.42958i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 6 q^{25} - 60 q^{31} - 216 q^{49} + 42 q^{55} - 96 q^{61} - 228 q^{79} - 96 q^{85} - 84 q^{91}+O(q^{100}) $ 24 * q - 6 * q^25 - 60 * q^31 - 216 * q^49 + 42 * q^55 - 96 * q^61 - 228 * q^79 - 96 * q^85 - 84 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	−1.98582	+	4.58874i	−0.397164	+	0.917748i
$5$	−1.98582	+	4.58874i	−0.397164	+	0.917748i
$6$		0			0
$7$		−	0.667924i		−	0.0954177i	−0.998861	−	0.0477088i	$-0.984808\pi$
$7$		−	0.667924i		−	0.0954177i	0.998861	−	0.0477088i	$-0.0151920\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			19.4601i			1.76910i	0.466447	+	0.884549i	$0.345534\pi$
$11$			19.4601i			1.76910i	−0.466447	+	0.884549i	$0.654466\pi$
$12$		0			0
$13$			20.0501i			1.54232i	0.636644	+	0.771158i	$0.280322\pi$
$13$			20.0501i			1.54232i	−0.636644	+	0.771158i	$0.719678\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	16.3370			0.960998			0.480499	−	0.876995i	$-0.340456\pi$
$17$	16.3370			0.960998			0.480499	+	0.876995i	$0.340456\pi$
$18$		0			0
$19$	2.43805			0.128319			0.0641593	−	0.997940i	$-0.479563\pi$
$19$	2.43805			0.128319			0.0641593	+	0.997940i	$0.479563\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	34.3714			1.49441			0.747203	−	0.664595i	$-0.231396\pi$
$23$	34.3714			1.49441			0.747203	+	0.664595i	$0.231396\pi$
$24$		0			0
$25$	−17.1130	−	18.2248i	−0.684521	−	0.728993i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			5.73648i			0.197810i	0.995097	+	0.0989048i	$0.0315339\pi$
$29$			5.73648i			0.197810i	−0.995097	+	0.0989048i	$0.968466\pi$
$30$		0			0
$31$	−26.8521			−0.866198			−0.433099	−	0.901346i	$-0.642580\pi$
$31$	−26.8521			−0.866198			−0.433099	+	0.901346i	$0.642580\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.06493	+	1.32638i	0.0875694	+	0.0378965i
$36$		0			0
$37$			11.3929i			0.307916i	0.988077	+	0.153958i	$0.0492021\pi$
$37$			11.3929i			0.307916i	−0.988077	+	0.153958i	$0.950798\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	29.2502i		−	0.713419i	−0.934215	−	0.356709i	$-0.883899\pi$
$41$		−	29.2502i		−	0.713419i	0.934215	−	0.356709i	$-0.116101\pi$
$42$		0			0
$43$		−	34.1247i		−	0.793598i	−0.917906	−	0.396799i	$-0.870121\pi$
$43$		−	34.1247i		−	0.793598i	0.917906	−	0.396799i	$-0.129879\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−67.9986			−1.44678			−0.723389	−	0.690440i	$-0.757417\pi$
$47$	−67.9986			−1.44678			−0.723389	+	0.690440i	$0.757417\pi$
$48$		0			0
$49$	48.5539			0.990895
$50$		0			0
$51$		0			0
$52$		0			0
$53$	37.0720			0.699471			0.349736	−	0.936848i	$-0.386271\pi$
$53$	37.0720			0.699471			0.349736	+	0.936848i	$0.386271\pi$
$54$		0			0
$55$	−89.2972	−	38.6442i	−1.62359	−	0.702622i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			19.0295i			0.322533i	0.986911	+	0.161267i	$0.0515579\pi$
$59$			19.0295i			0.322533i	−0.986911	+	0.161267i	$0.948442\pi$
$60$		0			0
$61$	−113.562			−1.86168			−0.930839	−	0.365430i	$-0.880922\pi$
$61$	−113.562			−1.86168			−0.930839	+	0.365430i	$0.880922\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−92.0047	−	39.8159i	−1.41546	−	0.612553i
$66$		0			0
$67$			106.718i			1.59280i	0.604769	+	0.796401i	$0.293265\pi$
$67$			106.718i			1.59280i	−0.604769	+	0.796401i	$0.706735\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	37.0887i		−	0.522376i	−0.965288	−	0.261188i	$-0.915886\pi$
$71$		−	37.0887i		−	0.522376i	0.965288	−	0.261188i	$-0.0841142\pi$
$72$		0			0
$73$			115.846i			1.58694i	0.608612	+	0.793468i	$0.291727\pi$
$73$			115.846i			1.58694i	−0.608612	+	0.793468i	$0.708273\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.9979			0.168803
$78$		0			0
$79$	16.7659			0.212226			0.106113	−	0.994354i	$-0.466159\pi$
$79$	16.7659			0.212226			0.106113	+	0.994354i	$0.466159\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	90.4200			1.08940			0.544699	−	0.838632i	$-0.316644\pi$
$83$	90.4200			1.08940			0.544699	+	0.838632i	$0.316644\pi$
$84$		0			0
$85$	−32.4423	+	74.9660i	−0.381674	+	0.881954i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			28.0932i			0.315654i	0.987467	+	0.157827i	$0.0504488\pi$
$89$			28.0932i			0.315654i	−0.987467	+	0.157827i	$0.949551\pi$
$90$		0			0
$91$	13.3919			0.147164
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.84154	+	11.1876i	−0.0509636	+	0.117764i
$96$		0			0
$97$			9.42958i			0.0972121i	0.998818	+	0.0486061i	$0.0154779\pi$
$97$			9.42958i			0.0972121i	−0.998818	+	0.0486061i	$0.984522\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			93.7242i			0.927962i	0.885845	+	0.463981i	$0.153579\pi$
$101$			93.7242i			0.927962i	−0.885845	+	0.463981i	$0.846421\pi$
$102$		0			0
$103$		−	101.748i		−	0.987846i	−0.869506	−	0.493923i	$-0.835562\pi$
$103$		−	101.748i		−	0.987846i	0.869506	−	0.493923i	$-0.164438\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−48.7944			−0.456022			−0.228011	−	0.973659i	$-0.573222\pi$
$107$	−48.7944			−0.456022			−0.228011	+	0.973659i	$0.573222\pi$
$108$		0			0
$109$	−78.0486			−0.716042			−0.358021	−	0.933713i	$-0.616548\pi$
$109$	−78.0486			−0.716042			−0.358021	+	0.933713i	$0.616548\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−57.6078			−0.509804			−0.254902	−	0.966967i	$-0.582043\pi$
$113$	−57.6078			−0.509804			−0.254902	+	0.966967i	$0.582043\pi$
$114$		0			0
$115$	−68.2554	+	157.721i	−0.593525	+	1.37149i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	10.9118i		−	0.0916962i
$120$		0			0
$121$	−257.695			−2.12971
$122$		0			0
$123$		0			0
$124$		0			0
$125$	117.612	−	42.3360i	0.940899	−	0.338688i
$126$		0			0
$127$		−	78.7342i		−	0.619954i	−0.950744	−	0.309977i	$-0.899679\pi$
$127$		−	78.7342i		−	0.619954i	0.950744	−	0.309977i	$-0.100321\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	136.725i		−	1.04370i	−0.853036	−	0.521852i	$-0.825241\pi$
$131$		−	136.725i		−	1.04370i	0.853036	−	0.521852i	$-0.174759\pi$
$132$		0			0
$133$		−	1.62843i		−	0.0122439i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	62.2123			0.454105			0.227052	−	0.973883i	$-0.427091\pi$
$137$	62.2123			0.454105			0.227052	+	0.973883i	$0.427091\pi$
$138$		0			0
$139$	−171.778			−1.23581			−0.617907	−	0.786251i	$-0.712019\pi$
$139$	−171.778			−1.23581			−0.617907	+	0.786251i	$0.712019\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−390.177			−2.72851
$144$		0			0
$145$	−26.3232	−	11.3916i	−0.181539	−	0.0785629i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			99.7497i			0.669461i	0.942314	+	0.334731i	$0.108645\pi$
$149$			99.7497i			0.669461i	−0.942314	+	0.334731i	$0.891355\pi$
$150$		0			0
$151$	47.7638			0.316316			0.158158	−	0.987414i	$-0.449444\pi$
$151$	47.7638			0.316316			0.158158	+	0.987414i	$0.449444\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	53.3235	−	123.217i	0.344023	−	0.794951i
$156$		0			0
$157$		−	153.095i		−	0.975127i	−0.873088	−	0.487563i	$-0.837886\pi$
$157$		−	153.095i		−	0.975127i	0.873088	−	0.487563i	$-0.162114\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	22.9575i		−	0.142593i
$162$		0			0
$163$			162.445i			0.996595i	0.867006	+	0.498298i	$0.166041\pi$
$163$			162.445i			0.996595i	−0.867006	+	0.498298i	$0.833959\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−131.441			−0.787074			−0.393537	−	0.919309i	$-0.628749\pi$
$167$	−131.441			−0.787074			−0.393537	+	0.919309i	$0.628749\pi$
$168$		0			0
$169$	−233.007			−1.37874
$170$		0			0
$171$		0			0
$172$		0			0
$173$	221.466			1.28015			0.640075	−	0.768312i	$-0.278903\pi$
$173$	221.466			1.28015			0.640075	+	0.768312i	$0.278903\pi$
$174$		0			0
$175$	−12.1728	+	11.4302i	−0.0695588	+	0.0653154i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			68.5428i			0.382921i	0.981500	+	0.191460i	$0.0613223\pi$
$179$			68.5428i			0.382921i	−0.981500	+	0.191460i	$0.938678\pi$
$180$		0			0
$181$	−197.707			−1.09230			−0.546152	−	0.837686i	$-0.683908\pi$
$181$	−197.707			−1.09230			−0.546152	+	0.837686i	$0.683908\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−52.2790	−	22.6243i	−0.282589	−	0.122293i
$186$		0			0
$187$			317.919i			1.70010i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	269.998i		−	1.41360i	−0.707412	−	0.706802i	$-0.750137\pi$
$191$		−	269.998i		−	1.41360i	0.707412	−	0.706802i	$-0.249863\pi$
$192$		0			0
$193$		−	51.6976i		−	0.267863i	−0.990991	−	0.133932i	$-0.957240\pi$
$193$		−	51.6976i		−	0.267863i	0.990991	−	0.133932i	$-0.0427603\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	293.145			1.48805			0.744024	−	0.668153i	$-0.232915\pi$
$197$	293.145			1.48805			0.744024	+	0.668153i	$0.232915\pi$
$198$		0			0
$199$	−216.611			−1.08850			−0.544248	−	0.838924i	$-0.683185\pi$
$199$	−216.611			−1.08850			−0.544248	+	0.838924i	$0.683185\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.83153			0.0188745
$204$		0			0
$205$	134.221	+	58.0856i	0.654738	+	0.283344i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			47.4447i			0.227008i
$210$		0			0
$211$	278.581			1.32029			0.660144	−	0.751139i	$-0.270495\pi$
$211$	278.581			1.32029			0.660144	+	0.751139i	$0.270495\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	156.589	+	67.7656i	0.728323	+	0.315189i
$216$		0			0
$217$			17.9352i			0.0826506i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			327.558i			1.48216i
$222$		0			0
$223$			84.4757i			0.378815i	0.981899	+	0.189407i	$0.0606567\pi$
$223$			84.4757i			0.378815i	−0.981899	+	0.189407i	$0.939343\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	24.9286			0.109818			0.0549089	−	0.998491i	$-0.482513\pi$
$227$	24.9286			0.109818			0.0549089	+	0.998491i	$0.482513\pi$
$228$		0			0
$229$	−265.181			−1.15800			−0.578998	−	0.815329i	$-0.696556\pi$
$229$	−265.181			−1.15800			−0.578998	+	0.815329i	$0.696556\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−290.507			−1.24681			−0.623405	−	0.781899i	$-0.714251\pi$
$233$	−290.507			−1.24681			−0.623405	+	0.781899i	$0.714251\pi$
$234$		0			0
$235$	135.033	−	312.028i	0.574609	−	1.32778i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			106.849i			0.447069i	0.974696	+	0.223534i	$0.0717595\pi$
$239$			106.849i			0.447069i	−0.974696	+	0.223534i	$0.928241\pi$
$240$		0			0
$241$	−125.991			−0.522782			−0.261391	−	0.965233i	$-0.584181\pi$
$241$	−125.991			−0.522782			−0.261391	+	0.965233i	$0.584181\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−96.4193	+	222.801i	−0.393548	+	0.909392i
$246$		0			0
$247$			48.8832i			0.197908i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			201.781i			0.803908i	0.915660	+	0.401954i	$0.131669\pi$
$251$			201.781i			0.803908i	−0.915660	+	0.401954i	$0.868331\pi$
$252$		0			0
$253$			668.869i			2.64375i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	250.865			0.976128			0.488064	−	0.872808i	$-0.337703\pi$
$257$	250.865			0.976128			0.488064	+	0.872808i	$0.337703\pi$
$258$		0			0
$259$	7.60959			0.0293807
$260$		0			0
$261$		0			0
$262$		0			0
$263$	358.139			1.36175			0.680873	−	0.732401i	$-0.261600\pi$
$263$	358.139			1.36175			0.680873	+	0.732401i	$0.261600\pi$
$264$		0			0
$265$	−73.6183	+	170.114i	−0.277805	+	0.641938i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			366.978i			1.36423i	0.731245	+	0.682115i	$0.238940\pi$
$269$			366.978i			1.36423i	−0.731245	+	0.682115i	$0.761060\pi$
$270$		0			0
$271$	7.74235			0.0285695			0.0142848	−	0.999898i	$-0.495453\pi$
$271$	7.74235			0.0285695			0.0142848	+	0.999898i	$0.495453\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	354.657	−	333.021i	1.28966	−	1.21099i
$276$		0			0
$277$		−	101.934i		−	0.367994i	−0.982927	−	0.183997i	$-0.941096\pi$
$277$		−	101.934i		−	0.367994i	0.982927	−	0.183997i	$-0.0589036\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	181.599i		−	0.646260i	−0.946355	−	0.323130i	$-0.895265\pi$
$281$		−	181.599i		−	0.646260i	0.946355	−	0.323130i	$-0.104735\pi$
$282$		0			0
$283$		−	267.926i		−	0.946736i	−0.880865	−	0.473368i	$-0.843038\pi$
$283$		−	267.926i		−	0.946736i	0.880865	−	0.473368i	$-0.156962\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−19.5369			−0.0680728
$288$		0			0
$289$	−22.1036			−0.0764830
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−269.238			−0.918900			−0.459450	−	0.888204i	$-0.651953\pi$
$293$	−269.238			−0.918900			−0.459450	+	0.888204i	$0.651953\pi$
$294$		0			0
$295$	−87.3212	−	37.7891i	−0.296004	−	0.128099i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			689.149i			2.30485i
$300$		0			0
$301$	−22.7927			−0.0757233
$302$		0			0
$303$		0			0
$304$		0			0
$305$	225.514	−	521.108i	0.739392	−	1.70855i
$306$		0			0
$307$		−	409.604i		−	1.33422i	−0.744961	−	0.667108i	$-0.767532\pi$
$307$		−	409.604i		−	1.33422i	0.744961	−	0.667108i	$-0.232468\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	24.1223i		−	0.0775637i	−0.999248	−	0.0387819i	$-0.987652\pi$
$311$		−	24.1223i		−	0.0775637i	0.999248	−	0.0387819i	$-0.0123477\pi$
$312$		0			0
$313$		−	435.203i		−	1.39042i	−0.718805	−	0.695212i	$-0.755310\pi$
$313$		−	435.203i		−	1.39042i	0.718805	−	0.695212i	$-0.244690\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−392.849			−1.23927			−0.619636	−	0.784890i	$-0.712720\pi$
$317$	−392.849			−1.23927			−0.619636	+	0.784890i	$0.712720\pi$
$318$		0			0
$319$	−111.632			−0.349945
$320$		0			0
$321$		0			0
$322$		0			0
$323$	39.8304			0.123314
$324$		0			0
$325$	365.410	−	343.118i	1.12434	−	1.05575i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			45.4179i			0.138048i
$330$		0			0
$331$	380.480			1.14949			0.574743	−	0.818334i	$-0.305102\pi$
$331$	380.480			1.14949			0.574743	+	0.818334i	$0.305102\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−489.699	−	211.922i	−1.46179	−	0.632604i
$336$		0			0
$337$			636.992i			1.89018i	0.326804	+	0.945092i	$0.394028\pi$
$337$			636.992i			1.89018i	−0.326804	+	0.945092i	$0.605972\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	522.545i		−	1.53239i
$342$		0			0
$343$		−	65.1586i		−	0.189967i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−105.078			−0.302819			−0.151410	−	0.988471i	$-0.548381\pi$
$347$	−105.078			−0.302819			−0.151410	+	0.988471i	$0.548381\pi$
$348$		0			0
$349$	−346.448			−0.992688			−0.496344	−	0.868126i	$-0.665325\pi$
$349$	−346.448			−0.992688			−0.496344	+	0.868126i	$0.665325\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−214.861			−0.608671			−0.304335	−	0.952565i	$-0.598434\pi$
$353$	−214.861			−0.608671			−0.304335	+	0.952565i	$0.598434\pi$
$354$		0			0
$355$	170.190	+	73.6515i	0.479409	+	0.207469i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	221.170i		−	0.616073i	−0.951375	−	0.308037i	$-0.900328\pi$
$359$		−	221.170i		−	0.616073i	0.951375	−	0.308037i	$-0.0996719\pi$
$360$		0			0
$361$	−355.056			−0.983534
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−531.588	−	230.050i	−1.45641	−	0.630274i
$366$		0			0
$367$		−	463.766i		−	1.26367i	−0.775103	−	0.631834i	$-0.782302\pi$
$367$		−	463.766i		−	1.26367i	0.775103	−	0.631834i	$-0.217698\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	24.7613i		−	0.0667419i
$372$		0			0
$373$			61.9988i			0.166217i	0.996541	+	0.0831083i	$0.0264847\pi$
$373$			61.9988i			0.166217i	−0.996541	+	0.0831083i	$0.973515\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−115.017			−0.305085
$378$		0			0
$379$	560.650			1.47929			0.739644	−	0.672998i	$-0.234994\pi$
$379$	560.650			1.47929			0.739644	+	0.672998i	$0.234994\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	499.720			1.30475			0.652377	−	0.757895i	$-0.273772\pi$
$383$	499.720			1.30475			0.652377	+	0.757895i	$0.273772\pi$
$384$		0			0
$385$	−25.8114	+	59.6437i	−0.0670426	+	0.154919i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	100.320i		−	0.257892i	−0.991652	−	0.128946i	$-0.958841\pi$
$389$		−	100.320i		−	0.257892i	0.991652	−	0.128946i	$-0.0411595\pi$
$390$		0			0
$391$	561.524			1.43612
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−33.2940	+	76.9342i	−0.0842886	+	0.194770i
$396$		0			0
$397$			467.421i			1.17738i	0.808357	+	0.588692i	$0.200357\pi$
$397$			467.421i			1.17738i	−0.808357	+	0.588692i	$0.799643\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	735.269i		−	1.83359i	−0.399361	−	0.916794i	$-0.630768\pi$
$401$		−	735.269i		−	1.83359i	0.399361	−	0.916794i	$-0.369232\pi$
$402$		0			0
$403$		−	538.388i		−	1.33595i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−221.707			−0.544734
$408$		0			0
$409$	766.716			1.87461			0.937306	−	0.348509i	$-0.113312\pi$
$409$	766.716			1.87461			0.937306	+	0.348509i	$0.113312\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.7102			0.0307754
$414$		0			0
$415$	−179.558	+	414.914i	−0.432670	+	0.999792i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			388.690i			0.927662i	0.885924	+	0.463831i	$0.153526\pi$
$419$			388.690i			0.927662i	−0.885924	+	0.463831i	$0.846474\pi$
$420$		0			0
$421$	−133.546			−0.317211			−0.158605	−	0.987342i	$-0.550700\pi$
$421$	−133.546			−0.317211			−0.158605	+	0.987342i	$0.550700\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−279.575	−	297.738i	−0.657823	−	0.700561i
$426$		0			0
$427$			75.8510i			0.177637i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	4.42143i		−	0.0102585i	−0.999987	−	0.00512927i	$-0.998367\pi$
$431$		−	4.42143i		−	0.0102585i	0.999987	−	0.00512927i	$-0.00163270\pi$
$432$		0			0
$433$		−	503.803i		−	1.16352i	−0.813361	−	0.581759i	$-0.802365\pi$
$433$		−	503.803i		−	1.16352i	0.813361	−	0.581759i	$-0.197635\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	83.7992			0.191760
$438$		0			0
$439$	797.121			1.81577			0.907883	−	0.419223i	$-0.137698\pi$
$439$	797.121			1.81577			0.907883	+	0.419223i	$0.137698\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−51.1612			−0.115488			−0.0577440	−	0.998331i	$-0.518391\pi$
$443$	−51.1612			−0.115488			−0.0577440	+	0.998331i	$0.518391\pi$
$444$		0			0
$445$	−128.912	−	55.7880i	−0.289691	−	0.125366i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			410.343i			0.913904i	0.889491	+	0.456952i	$0.151059\pi$
$449$			410.343i			0.913904i	−0.889491	+	0.456952i	$0.848941\pi$
$450$		0			0
$451$	569.211			1.26211
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−26.5940	+	61.4521i	−0.0584484	+	0.135060i
$456$		0			0
$457$			590.003i			1.29104i	0.763745	+	0.645518i	$0.223358\pi$
$457$			590.003i			1.29104i	−0.763745	+	0.645518i	$0.776642\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	165.283i		−	0.358530i	−0.983801	−	0.179265i	$-0.942628\pi$
$461$		−	165.283i		−	0.358530i	0.983801	−	0.179265i	$-0.0573720\pi$
$462$		0			0
$463$			3.37297i			0.00728502i	0.999993	+	0.00364251i	$0.00115945\pi$
$463$			3.37297i			0.00728502i	−0.999993	+	0.00364251i	$0.998841\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	553.128			1.18443			0.592214	−	0.805781i	$-0.298254\pi$
$467$	553.128			1.18443			0.592214	+	0.805781i	$0.298254\pi$
$468$		0			0
$469$	71.2793			0.151981
$470$		0			0
$471$		0			0
$472$		0			0
$473$	664.070			1.40395
$474$		0			0
$475$	−41.7225	−	44.4331i	−0.0878368	−	0.0935434i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			305.218i			0.637199i	0.947889	+	0.318600i	$0.103213\pi$
$479$			305.218i			0.637199i	−0.947889	+	0.318600i	$0.896787\pi$
$480$		0			0
$481$	−228.429			−0.474904
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−43.2699	−	18.7254i	−0.0892162	−	0.0386092i
$486$		0			0
$487$		−	263.256i		−	0.540566i	−0.962781	−	0.270283i	$-0.912883\pi$
$487$		−	263.256i		−	0.540566i	0.962781	−	0.270283i	$-0.0871173\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			546.341i			1.11271i	0.830945	+	0.556355i	$0.187801\pi$
$491$			546.341i			1.11271i	−0.830945	+	0.556355i	$0.812199\pi$
$492$		0			0
$493$			93.7166i			0.190095i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−24.7724			−0.0498439
$498$		0			0
$499$	651.657			1.30593			0.652963	−	0.757390i	$-0.273526\pi$
$499$	651.657			1.30593			0.652963	+	0.757390i	$0.273526\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	188.144			0.374045			0.187022	−	0.982356i	$-0.440116\pi$
$503$	188.144			0.374045			0.187022	+	0.982356i	$0.440116\pi$
$504$		0			0
$505$	−430.076	−	186.119i	−0.851635	−	0.368553i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			233.722i			0.459178i	0.973288	+	0.229589i	$0.0737382\pi$
$509$			233.722i			0.459178i	−0.973288	+	0.229589i	$0.926262\pi$
$510$		0			0
$511$	77.3765			0.151422
$512$		0			0
$513$		0			0
$514$		0			0
$515$	466.896	+	202.054i	0.906594	+	0.392337i
$516$		0			0
$517$		−	1323.26i		−	2.55949i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			455.570i			0.874415i	0.899361	+	0.437207i	$0.144032\pi$
$521$			455.570i			0.874415i	−0.899361	+	0.437207i	$0.855968\pi$
$522$		0			0
$523$			176.002i			0.336523i	0.985742	+	0.168261i	$0.0538153\pi$
$523$			176.002i			0.336523i	−0.985742	+	0.168261i	$0.946185\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−438.682			−0.832414
$528$		0			0
$529$	652.390			1.23325
$530$		0			0
$531$		0			0
$532$		0			0
$533$	586.469			1.10032
$534$		0			0
$535$	96.8969	−	223.905i	0.181116	−	0.418513i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			944.862i			1.75299i
$540$		0			0
$541$	409.786			0.757461			0.378730	−	0.925507i	$-0.376361\pi$
$541$	409.786			0.757461			0.378730	+	0.925507i	$0.376361\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	154.991	−	358.145i	0.284386	−	0.657146i
$546$		0			0
$547$		−	959.956i		−	1.75495i	−0.479626	−	0.877473i	$-0.659228\pi$
$547$		−	959.956i		−	1.75495i	0.479626	−	0.877473i	$-0.340772\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			13.9858i			0.0253827i
$552$		0			0
$553$		−	11.1983i		−	0.0202501i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−813.070			−1.45973			−0.729865	−	0.683591i	$-0.760417\pi$
$557$	−813.070			−1.45973			−0.729865	+	0.683591i	$0.760417\pi$
$558$		0			0
$559$	684.204			1.22398
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−720.361			−1.27950			−0.639752	−	0.768581i	$-0.720963\pi$
$563$	−720.361			−1.27950			−0.639752	+	0.768581i	$0.720963\pi$
$564$		0			0
$565$	114.399	−	264.347i	0.202476	−	0.467871i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			741.140i			1.30253i	0.758850	+	0.651265i	$0.225761\pi$
$569$			741.140i			1.30253i	−0.758850	+	0.651265i	$0.774239\pi$
$570$		0			0
$571$	−139.808			−0.244847			−0.122424	−	0.992478i	$-0.539067\pi$
$571$	−139.808			−0.244847			−0.122424	+	0.992478i	$0.539067\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−588.198	−	626.412i	−1.02295	−	1.08941i
$576$		0			0
$577$		−	696.659i		−	1.20738i	−0.797219	−	0.603690i	$-0.793696\pi$
$577$		−	696.659i		−	1.20738i	0.797219	−	0.603690i	$-0.206304\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	60.3937i		−	0.103948i
$582$		0			0
$583$			721.424i			1.23743i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	514.225			0.876022			0.438011	−	0.898970i	$-0.355683\pi$
$587$	514.225			0.876022			0.438011	+	0.898970i	$0.355683\pi$
$588$		0			0
$589$	−65.4670			−0.111149
$590$		0			0
$591$		0			0
$592$		0			0
$593$	319.924			0.539501			0.269751	−	0.962930i	$-0.413059\pi$
$593$	319.924			0.539501			0.269751	+	0.962930i	$0.413059\pi$
$594$		0			0
$595$	50.0716	+	21.6690i	0.0841540	+	0.0364184i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1003.57i			1.67542i	0.546118	+	0.837709i	$0.316105\pi$
$599$			1003.57i			1.67542i	−0.546118	+	0.837709i	$0.683895\pi$
$600$		0			0
$601$	−83.3311			−0.138654			−0.0693271	−	0.997594i	$-0.522085\pi$
$601$	−83.3311			−0.138654			−0.0693271	+	0.997594i	$0.522085\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	511.736	−	1182.49i	0.845844	−	1.95454i
$606$		0			0
$607$			50.6944i			0.0835163i	0.999128	+	0.0417581i	$0.0132959\pi$
$607$			50.6944i			0.0835163i	−0.999128	+	0.0417581i	$0.986704\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1363.38i		−	2.23139i
$612$		0			0
$613$		−	262.022i		−	0.427442i	−0.976895	−	0.213721i	$-0.931442\pi$
$613$		−	262.022i		−	0.427442i	0.976895	−	0.213721i	$-0.0685584\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	342.435			0.555001			0.277500	−	0.960726i	$-0.410494\pi$
$617$	342.435			0.555001			0.277500	+	0.960726i	$0.410494\pi$
$618$		0			0
$619$	545.803			0.881749			0.440875	−	0.897569i	$-0.354668\pi$
$619$	545.803			0.881749			0.440875	+	0.897569i	$0.354668\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.7641			0.0301190
$624$		0			0
$625$	−39.2883	+	623.764i	−0.0628613	+	0.998022i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			186.125i			0.295907i
$630$		0			0
$631$	906.405			1.43646			0.718229	−	0.695807i	$-0.244953\pi$
$631$	906.405			1.43646			0.718229	+	0.695807i	$0.244953\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	361.290	+	156.352i	0.568961	+	0.246224i
$636$		0			0
$637$			973.510i			1.52827i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			809.751i			1.26326i	0.775269	+	0.631631i	$0.217614\pi$
$641$			809.751i			1.26326i	−0.775269	+	0.631631i	$0.782386\pi$
$642$		0			0
$643$		−	425.314i		−	0.661453i	−0.943727	−	0.330726i	$-0.892706\pi$
$643$		−	425.314i		−	0.661453i	0.943727	−	0.330726i	$-0.107294\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	782.663			1.20968			0.604840	−	0.796347i	$-0.293237\pi$
$647$	782.663			1.20968			0.604840	+	0.796347i	$0.293237\pi$
$648$		0			0
$649$	−370.315			−0.570593
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1028.32			−1.57477			−0.787385	−	0.616462i	$-0.788566\pi$
$653$	−1028.32			−1.57477			−0.787385	+	0.616462i	$0.788566\pi$
$654$		0			0
$655$	627.397	+	271.512i	0.957857	+	0.414522i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	266.180i		−	0.403916i	−0.979394	−	0.201958i	$-0.935270\pi$
$659$		−	266.180i		−	0.403916i	0.979394	−	0.201958i	$-0.0647304\pi$
$660$		0			0
$661$	−509.305			−0.770507			−0.385254	−	0.922811i	$-0.625886\pi$
$661$	−509.305			−0.770507			−0.385254	+	0.922811i	$0.625886\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.47246	+	3.23378i	0.0112368	+	0.00486283i
$666$		0			0
$667$			197.171i			0.295608i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	2209.93i		−	3.29349i
$672$		0			0
$673$			796.227i			1.18310i	0.806268	+	0.591551i	$0.201484\pi$
$673$			796.227i			1.18310i	−0.806268	+	0.591551i	$0.798516\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1264.05			1.86713			0.933565	−	0.358408i	$-0.116680\pi$
$677$	1264.05			1.86713			0.933565	+	0.358408i	$0.116680\pi$
$678$		0			0
$679$	6.29824			0.00927576
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−262.103			−0.383753			−0.191876	−	0.981419i	$-0.561457\pi$
$683$	−262.103			−0.383753			−0.191876	+	0.981419i	$0.561457\pi$
$684$		0			0
$685$	−123.543	+	285.476i	−0.180354	+	0.416753i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			743.297i			1.07881i
$690$		0			0
$691$	751.634			1.08775			0.543874	−	0.839167i	$-0.316957\pi$
$691$	751.634			1.08775			0.543874	+	0.839167i	$0.316957\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	341.120	−	788.245i	0.490821	−	1.13416i
$696$		0			0
$697$		−	477.859i		−	0.685594i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			620.953i			0.885810i	0.896569	+	0.442905i	$0.146052\pi$
$701$			620.953i			0.885810i	−0.896569	+	0.442905i	$0.853948\pi$
$702$		0			0
$703$			27.7765i			0.0395114i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	62.6006			0.0885440
$708$		0			0
$709$	−403.710			−0.569408			−0.284704	−	0.958615i	$-0.591895\pi$
$709$	−403.710			−0.569408			−0.284704	+	0.958615i	$0.591895\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−922.945			−1.29445
$714$		0			0
$715$	774.821	−	1790.42i	1.08367	−	2.50408i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			316.833i			0.440658i	0.975426	+	0.220329i	$0.0707132\pi$
$719$			316.833i			0.440658i	−0.975426	+	0.220329i	$0.929287\pi$
$720$		0			0
$721$	−67.9600			−0.0942580
$722$		0			0
$723$		0			0
$724$		0			0
$725$	104.546	−	98.1685i	0.144202	−	0.135405i
$726$		0			0
$727$		−	557.477i		−	0.766818i	−0.923579	−	0.383409i	$-0.874750\pi$
$727$		−	557.477i		−	0.766818i	0.923579	−	0.383409i	$-0.125250\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	557.494i		−	0.762646i
$732$		0			0
$733$			446.752i			0.609485i	0.952435	+	0.304742i	$0.0985704\pi$
$733$			446.752i			0.609485i	−0.952435	+	0.304742i	$0.901430\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2076.73			−2.81782
$738$		0			0
$739$	209.090			0.282937			0.141468	−	0.989943i	$-0.454818\pi$
$739$	209.090			0.282937			0.141468	+	0.989943i	$0.454818\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−45.5445			−0.0612981			−0.0306490	−	0.999530i	$-0.509757\pi$
$743$	−45.5445			−0.0612981			−0.0306490	+	0.999530i	$0.509757\pi$
$744$		0			0
$745$	−457.725	−	198.085i	−0.614396	−	0.265886i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			32.5909i			0.0435126i
$750$		0			0
$751$	17.3154			0.0230564			0.0115282	−	0.999934i	$-0.496330\pi$
$751$	17.3154			0.0230564			0.0115282	+	0.999934i	$0.496330\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−94.8503	+	219.175i	−0.125629	+	0.290298i
$756$		0			0
$757$			1269.99i			1.67767i	0.544388	+	0.838834i	$0.316762\pi$
$757$			1269.99i			1.67767i	−0.544388	+	0.838834i	$0.683238\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1152.17i			1.51402i	0.653403	+	0.757010i	$0.273341\pi$
$761$			1152.17i			1.51402i	−0.653403	+	0.757010i	$0.726659\pi$
$762$		0			0
$763$			52.1305i			0.0683231i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−381.543			−0.497448
$768$		0			0
$769$	−506.118			−0.658151			−0.329076	−	0.944304i	$-0.606737\pi$
$769$	−506.118			−0.658151			−0.329076	+	0.944304i	$0.606737\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	374.490			0.484463			0.242232	−	0.970218i	$-0.422121\pi$
$773$	374.490			0.484463			0.242232	+	0.970218i	$0.422121\pi$
$774$		0			0
$775$	459.521	+	489.375i	0.592931	+	0.631452i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	71.3135i		−	0.0915449i
$780$		0			0
$781$	721.749			0.924134
$782$		0			0
$783$		0			0
$784$		0			0
$785$	702.512	+	304.019i	0.894920	+	0.387285i
$786$		0			0
$787$			856.557i			1.08838i	0.838961	+	0.544191i	$0.183163\pi$
$787$			856.557i			1.08838i	−0.838961	+	0.544191i	$0.816837\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			38.4776i			0.0486443i
$792$		0			0
$793$		−	2276.94i		−	2.87129i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	229.964			0.288536			0.144268	−	0.989539i	$-0.453917\pi$
$797$	229.964			0.288536			0.144268	+	0.989539i	$0.453917\pi$
$798$		0			0
$799$	−1110.89			−1.39035
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2254.38			−2.80745
$804$		0			0
$805$	105.346	+	45.5894i	0.130864	+	0.0566328i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1140.95i			1.41032i	0.709046	+	0.705162i	$0.249126\pi$
$809$			1140.95i			1.41032i	−0.709046	+	0.705162i	$0.750874\pi$
$810$		0			0
$811$	−853.740			−1.05270			−0.526350	−	0.850268i	$-0.676440\pi$
$811$	−853.740			−1.05270			−0.526350	+	0.850268i	$0.676440\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−745.418	−	322.587i	−0.914623	−	0.395812i
$816$		0			0
$817$		−	83.1979i		−	0.101833i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	733.845i		−	0.893843i	−0.894573	−	0.446921i	$-0.852520\pi$
$821$		−	733.845i		−	0.893843i	0.894573	−	0.446921i	$-0.147480\pi$
$822$		0			0
$823$			334.806i			0.406811i	0.979095	+	0.203406i	$0.0652010\pi$
$823$			334.806i			0.406811i	−0.979095	+	0.203406i	$0.934799\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1023.56			1.23768			0.618840	−	0.785517i	$-0.287603\pi$
$827$	1023.56			1.23768			0.618840	+	0.785517i	$0.287603\pi$
$828$		0			0
$829$	−1381.98			−1.66705			−0.833525	−	0.552482i	$-0.813681\pi$
$829$	−1381.98			−1.66705			−0.833525	+	0.552482i	$0.813681\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	793.223			0.952248
$834$		0			0
$835$	261.019	−	603.150i	0.312597	−	0.722335i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1213.98i			1.44693i	0.690358	+	0.723467i	$0.257453\pi$
$839$			1213.98i			1.44693i	−0.690358	+	0.723467i	$0.742547\pi$
$840$		0			0
$841$	808.093			0.960871
$842$		0			0
$843$		0			0
$844$		0			0
$845$	462.710	−	1069.21i	0.547585	−	1.26533i
$846$		0			0
$847$			172.120i			0.203212i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			391.589i			0.460152i
$852$		0			0
$853$		−	100.817i		−	0.118192i	−0.998252	−	0.0590958i	$-0.981178\pi$
$853$		−	100.817i		−	0.118192i	0.998252	−	0.0590958i	$-0.0188217\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−137.768			−0.160756			−0.0803780	−	0.996764i	$-0.525613\pi$
$857$	−137.768			−0.160756			−0.0803780	+	0.996764i	$0.525613\pi$
$858$		0			0
$859$	387.465			0.451065			0.225532	−	0.974236i	$-0.427588\pi$
$859$	387.465			0.451065			0.225532	+	0.974236i	$0.427588\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1033.60			1.19768			0.598841	−	0.800868i	$-0.295628\pi$
$863$	1033.60			1.19768			0.598841	+	0.800868i	$0.295628\pi$
$864$		0			0
$865$	−439.792	+	1016.25i	−0.508430	+	1.17486i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			326.265i			0.375449i
$870$		0			0
$871$	−2139.70			−2.45660
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−28.2772	−	78.5561i	−0.0323168	−	0.0897784i
$876$		0			0
$877$		−	257.248i		−	0.293327i	−0.989186	−	0.146663i	$-0.953147\pi$
$877$		−	257.248i		−	0.293327i	0.989186	−	0.146663i	$-0.0468534\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	861.458i		−	0.977819i	−0.872335	−	0.488909i	$-0.837395\pi$
$881$		−	861.458i		−	0.977819i	0.872335	−	0.488909i	$-0.162605\pi$
$882$		0			0
$883$		−	1059.68i		−	1.20009i	−0.799967	−	0.600045i	$-0.795150\pi$
$883$		−	1059.68i		−	1.20009i	0.799967	−	0.600045i	$-0.204850\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1292.44			−1.45709			−0.728546	−	0.684997i	$-0.759803\pi$
$887$	−1292.44			−1.45709			−0.728546	+	0.684997i	$0.759803\pi$
$888$		0			0
$889$	−52.5884			−0.0591546
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−165.784			−0.185649
$894$		0			0
$895$	−314.525	−	136.114i	−0.351424	−	0.152082i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	154.037i		−	0.171342i
$900$		0			0
$901$	605.643			0.672190
$902$		0			0
$903$		0			0
$904$		0			0
$905$	392.611	−	907.227i	0.433824	−	1.00246i
$906$		0			0
$907$		−	304.430i		−	0.335645i	−0.985817	−	0.167823i	$-0.946326\pi$
$907$		−	304.430i		−	0.335645i	0.985817	−	0.167823i	$-0.0536736\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	253.095i		−	0.277822i	−0.990305	−	0.138911i	$-0.955640\pi$
$911$		−	253.095i		−	0.277822i	0.990305	−	0.138911i	$-0.0443601\pi$
$912$		0			0
$913$			1759.58i			1.92725i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−91.3221			−0.0995879
$918$		0			0
$919$	−782.852			−0.851852			−0.425926	−	0.904758i	$-0.640052\pi$
$919$	−782.852			−0.851852			−0.425926	+	0.904758i	$0.640052\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	743.632			0.805668
$924$		0			0
$925$	207.634	−	194.967i	0.224469	−	0.210775i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			253.182i			0.272532i	0.990672	+	0.136266i	$0.0435102\pi$
$929$			253.182i			0.272532i	−0.990672	+	0.136266i	$0.956490\pi$
$930$		0			0
$931$	118.377			0.127150
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1458.85	−	631.329i	−1.56026	−	0.675219i
$936$		0			0
$937$			578.052i			0.616918i	0.951238	+	0.308459i	$0.0998133\pi$
$937$			578.052i			0.616918i	−0.951238	+	0.308459i	$0.900187\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	768.963i		−	0.817176i	−0.912719	−	0.408588i	$-0.866021\pi$
$941$		−	768.963i		−	0.817176i	0.912719	−	0.408588i	$-0.133979\pi$
$942$		0			0
$943$		−	1005.37i		−	1.06614i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−23.3925			−0.0247016			−0.0123508	−	0.999924i	$-0.503931\pi$
$947$	−23.3925			−0.0247016			−0.0123508	+	0.999924i	$0.503931\pi$
$948$		0			0
$949$	−2322.73			−2.44756
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1230.20			1.29087			0.645437	−	0.763814i	$-0.276675\pi$
$953$	1230.20			1.29087			0.645437	+	0.763814i	$0.276675\pi$
$954$		0			0
$955$	1238.95	+	536.168i	1.29733	+	0.561433i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	41.5531i		−	0.0433296i
$960$		0			0
$961$	−239.963			−0.249701
$962$		0			0
$963$		0			0
$964$		0			0
$965$	237.227	+	102.662i	0.245831	+	0.106386i
$966$		0			0
$967$			1023.85i			1.05879i	0.848375	+	0.529396i	$0.177581\pi$
$967$			1023.85i			1.05879i	−0.848375	+	0.529396i	$0.822419\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	180.852i		−	0.186253i	−0.995654	−	0.0931267i	$-0.970314\pi$
$971$		−	180.852i		−	0.186253i	0.995654	−	0.0931267i	$-0.0296862\pi$
$972$		0			0
$973$			114.735i			0.117918i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1738.78			1.77971			0.889856	−	0.456242i	$-0.150805\pi$
$977$	1738.78			1.77971			0.889856	+	0.456242i	$0.150805\pi$
$978$		0			0
$979$	−546.696			−0.558423
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−693.798			−0.705796			−0.352898	−	0.935662i	$-0.614804\pi$
$983$	−693.798			−0.705796			−0.352898	+	0.935662i	$0.614804\pi$
$984$		0			0
$985$	−582.134	+	1345.17i	−0.590999	+	1.36565i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1172.91i		−	1.18596i
$990$		0			0
$991$	−587.142			−0.592474			−0.296237	−	0.955114i	$-0.595732\pi$
$991$	−587.142			−0.592474			−0.296237	+	0.955114i	$0.595732\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	430.150	−	993.970i	0.432312	−	0.998965i
$996$		0			0
$997$		−	4.24185i		−	0.00425462i	−0.999998	−	0.00212731i	$-0.999323\pi$
$997$		−	4.24185i		−	0.00425462i	0.999998	−	0.00212731i	$-0.000677144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.b.b.809.8		24
3.2	odd	2	inner	1620.3.b.b.809.17		24
5.4	even	2	inner	1620.3.b.b.809.18		24
9.2	odd	6		540.3.t.a.449.8		24
9.4	even	3		540.3.t.a.89.12		24
9.5	odd	6		180.3.t.a.29.2	✓	24
9.7	even	3		180.3.t.a.149.11	yes	24
15.14	odd	2	inner	1620.3.b.b.809.7		24
45.2	even	12		2700.3.p.f.2501.6		24
45.4	even	6		540.3.t.a.89.8		24
45.7	odd	12		900.3.p.f.401.5		24
45.13	odd	12		2700.3.p.f.1601.7		24
45.14	odd	6		180.3.t.a.29.11	yes	24
45.22	odd	12		2700.3.p.f.1601.6		24
45.23	even	12		900.3.p.f.101.8		24
45.29	odd	6		540.3.t.a.449.12		24
45.32	even	12		900.3.p.f.101.5		24
45.34	even	6		180.3.t.a.149.2	yes	24
45.38	even	12		2700.3.p.f.2501.7		24
45.43	odd	12		900.3.p.f.401.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.t.a.29.2	✓	24	9.5	odd	6
180.3.t.a.29.11	yes	24	45.14	odd	6
180.3.t.a.149.2	yes	24	45.34	even	6
180.3.t.a.149.11	yes	24	9.7	even	3
540.3.t.a.89.8		24	45.4	even	6
540.3.t.a.89.12		24	9.4	even	3
540.3.t.a.449.8		24	9.2	odd	6
540.3.t.a.449.12		24	45.29	odd	6
900.3.p.f.101.5		24	45.32	even	12
900.3.p.f.101.8		24	45.23	even	12
900.3.p.f.401.5		24	45.7	odd	12
900.3.p.f.401.8		24	45.43	odd	12
1620.3.b.b.809.7		24	15.14	odd	2	inner
1620.3.b.b.809.8		24	1.1	even	1	trivial
1620.3.b.b.809.17		24	3.2	odd	2	inner
1620.3.b.b.809.18		24	5.4	even	2	inner
2700.3.p.f.1601.6		24	45.22	odd	12
2700.3.p.f.1601.7		24	45.13	odd	12
2700.3.p.f.2501.6		24	45.2	even	12
2700.3.p.f.2501.7		24	45.38	even	12

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.98582 + 4.58874i) q^{5} -0.667924i q^{7} +O(q^{10})\)	\(q+(-1.98582 + 4.58874i) q^{5} -0.667924i q^{7} +19.4601i q^{11} +20.0501i q^{13} +16.3370 q^{17} +2.43805 q^{19} +34.3714 q^{23} +(-17.1130 - 18.2248i) q^{25} +5.73648i q^{29} -26.8521 q^{31} +(3.06493 + 1.32638i) q^{35} +11.3929i q^{37} -29.2502i q^{41} -34.1247i q^{43} -67.9986 q^{47} +48.5539 q^{49} +37.0720 q^{53} +(-89.2972 - 38.6442i) q^{55} +19.0295i q^{59} -113.562 q^{61} +(-92.0047 - 39.8159i) q^{65} +106.718i q^{67} -37.0887i q^{71} +115.846i q^{73} +12.9979 q^{77} +16.7659 q^{79} +90.4200 q^{83} +(-32.4423 + 74.9660i) q^{85} +28.0932i q^{89} +13.3919 q^{91} +(-4.84154 + 11.1876i) q^{95} +9.42958i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 6 q^{25} - 60 q^{31} - 216 q^{49} + 42 q^{55} - 96 q^{61} - 228 q^{79} - 96 q^{85} - 84 q^{91}+O(q^{100}) \) 24 * q - 6 * q^25 - 60 * q^31 - 216 * q^49 + 42 * q^55 - 96 * q^61 - 228 * q^79 - 96 * q^85 - 84 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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