LMFDB - Embedded newform 804.2.ba.a.785.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(41,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([0, 33, 53]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("804.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.ba (of order $66$, degree $20$, 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:	$6.41997232251$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{66}]$

Embedding invariants

Embedding label			785.1
Root			$0.235759 - 0.971812i$ of defining polynomial
Character	$\chi$	$=$	804.785
Dual form			804.2.ba.a.677.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.487975 + 1.66189i) q^{3} +(-5.07182 + 0.241600i) q^{7} +(-2.52376 + 1.62192i) q^{9} +O(q^{10})$	$q+(0.487975 + 1.66189i) q^{3} +(-5.07182 + 0.241600i) q^{7} +(-2.52376 + 1.62192i) q^{9} +(-0.619962 - 1.54859i) q^{13} +(0.411674 - 8.64209i) q^{19} +(-2.87643 - 8.31091i) q^{21} +(0.711574 - 4.94911i) q^{25} +(-3.92699 - 3.40276i) q^{27} +(-3.51128 + 8.77075i) q^{31} +(-2.71727 - 4.70645i) q^{37} +(2.27106 - 1.78598i) q^{39} +(-11.9291 + 5.44783i) q^{43} +(18.6966 - 1.78531i) q^{49} +(14.5631 - 3.53297i) q^{57} +(-2.88395 + 14.9634i) q^{61} +(12.4082 - 8.83583i) q^{63} +(-5.21774 - 6.30676i) q^{67} +(-16.0693 - 3.09710i) q^{73} +(8.57211 - 1.23248i) q^{75} +(-9.02089 + 11.4710i) q^{79} +(3.73874 - 8.18669i) q^{81} +(3.51847 + 7.70438i) q^{91} +(-16.2895 - 1.55545i) q^{93} +(17.0516 - 9.84473i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 20 * q + 6 * q^7 + 6 * q^9	$ 20 q + 6 q^{7} + 6 q^{9} + 9 q^{13} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{31} + 10 q^{37} - 9 q^{39} - 5 q^{49} + 141 q^{57} + 27 q^{61} + 147 q^{63} + 11 q^{67} - 180 q^{73} - 166 q^{79} - 18 q^{81} - 36 q^{91} - 3 q^{93} + 33 q^{97}+O(q^{100}) $ 20 * q + 6 * q^7 + 6 * q^9 + 9 * q^13 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 3 * q^31 + 10 * q^37 - 9 * q^39 - 5 * q^49 + 141 * q^57 + 27 * q^61 + 147 * q^63 + 11 * q^67 - 180 * q^73 - 166 * q^79 - 18 * q^81 - 36 * q^91 - 3 * q^93 + 33 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{43}{66}\right)$	$1$

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
$3$	0.487975	+	1.66189i	0.281733	+	0.959493i
$3$	0.487975	+	1.66189i	0.281733	+	0.959493i
$4$		0			0
$5$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$5$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$6$		0			0
$7$	−5.07182	+	0.241600i	−1.91697	+	0.0913164i	−0.971876	−	0.235493i	$-0.924330\pi$
$7$	−5.07182	+	0.241600i	−1.91697	+	0.0913164i	−0.945090	+	0.326809i	$0.894027\pi$
$8$		0			0
$9$	−2.52376	+	1.62192i	−0.841254	+	0.540641i
$10$		0			0
$11$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$11$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$12$		0			0
$13$	−0.619962	−	1.54859i	−0.171946	−	0.429501i	0.817215	−	0.576333i	$-0.195517\pi$
$13$	−0.619962	−	1.54859i	−0.171946	−	0.429501i	−0.989162	+	0.146831i	$0.953093\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$17$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$18$		0			0
$19$	0.411674	−	8.64209i	0.0944444	−	1.98263i	−0.0753671	−	0.997156i	$-0.524013\pi$
$19$	0.411674	−	8.64209i	0.0944444	−	1.98263i	0.169812	−	0.985477i	$-0.445684\pi$
$20$		0			0
$21$	−2.87643	−	8.31091i	−0.627689	−	1.81359i
$22$		0			0
$23$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$23$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$24$		0			0
$25$	0.711574	−	4.94911i	0.142315	−	0.989821i
$26$		0			0
$27$	−3.92699	−	3.40276i	−0.755750	−	0.654861i
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	−3.51128	+	8.77075i	−0.630645	+	1.57527i	0.176099	+	0.984372i	$0.443652\pi$
$31$	−3.51128	+	8.77075i	−0.630645	+	1.57527i	−0.806744	+	0.590901i	$0.798772\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.71727	−	4.70645i	−0.446716	−	0.773735i	0.551454	−	0.834205i	$-0.314073\pi$
$37$	−2.71727	−	4.70645i	−0.446716	−	0.773735i	−0.998170	+	0.0604704i	$0.980740\pi$
$38$		0			0
$39$	2.27106	−	1.78598i	0.363661	−	0.285986i
$40$		0			0
$41$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$41$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$42$		0			0
$43$	−11.9291	+	5.44783i	−1.81917	+	0.830786i	−0.905286	+	0.424803i	$0.860343\pi$
$43$	−11.9291	+	5.44783i	−1.81917	+	0.830786i	−0.913881	+	0.405983i	$0.866929\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$47$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$48$		0			0
$49$	18.6966	−	1.78531i	2.67095	−	0.255045i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$53$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	14.5631	−	3.53297i	1.92893	−	0.467953i
$58$		0			0
$59$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$59$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$60$		0			0
$61$	−2.88395	+	14.9634i	−0.369252	+	1.91586i	0.0311325	+	0.999515i	$0.490089\pi$
$61$	−2.88395	+	14.9634i	−0.369252	+	1.91586i	−0.400385	+	0.916347i	$0.631124\pi$
$62$		0			0
$63$	12.4082	−	8.83583i	1.56329	−	1.11321i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.21774	−	6.30676i	−0.637449	−	0.770493i
$67$	−5.21774	−	6.30676i	−0.637449	−	0.770493i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$71$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$72$		0			0
$73$	−16.0693	−	3.09710i	−1.88076	−	0.362488i	−0.885915	−	0.463848i	$-0.846468\pi$
$73$	−16.0693	−	3.09710i	−1.88076	−	0.362488i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$	8.57211	−	1.23248i	0.989821	−	0.142315i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−9.02089	+	11.4710i	−1.01493	+	1.29059i	−0.0586047	+	0.998281i	$0.518665\pi$
$79$	−9.02089	+	11.4710i	−1.01493	+	1.29059i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$	3.73874	−	8.18669i	0.415415	−	0.909632i
$82$		0			0
$83$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$83$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$89$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$90$		0			0
$91$	3.51847	+	7.70438i	0.368836	+	0.807638i
$92$		0			0
$93$	−16.2895	−	1.55545i	−1.68914	−	0.161293i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	17.0516	−	9.84473i	1.73132	−	0.999581i	0.851187	−	0.524862i	$-0.175883\pi$
$97$	17.0516	−	9.84473i	1.73132	−	0.999581i	0.880138	−	0.474718i	$-0.157450\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$101$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$102$		0			0
$103$	−17.3585	−	6.94931i	−1.71039	−	0.684736i	−0.710566	−	0.703631i	$-0.751561\pi$
$103$	−17.3585	−	6.94931i	−1.71039	−	0.684736i	−0.999821	+	0.0188949i	$0.993985\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$107$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$108$		0			0
$109$	8.59955	+	1.23643i	0.823688	+	0.118428i	0.541255	−	0.840859i	$-0.317949\pi$
$109$	8.59955	+	1.23643i	0.823688	+	0.118428i	0.282433	+	0.959287i	$0.408859\pi$
$110$		0			0
$111$	6.49564	−	6.81243i	0.616539	−	0.646607i
$112$		0			0
$113$		0			0		0.945001	−	0.327068i	$-0.106061\pi$
$113$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	4.07633	+	2.90274i	0.376857	+	0.268358i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.2120	+	4.08829i	−0.928368	+	0.371662i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	0.442367	+	9.28643i	0.0392538	+	0.824038i	0.929559	+	0.368674i	$0.120188\pi$
$127$	0.442367	+	9.28643i	0.0392538	+	0.824038i	−0.890305	+	0.455364i	$0.849509\pi$
$128$		0			0
$129$	−14.8748	−	17.1664i	−1.30965	−	1.51142i
$130$		0			0
$131$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$131$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$132$		0			0
$133$			43.9306i			3.80926i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$137$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$138$		0			0
$139$	−1.12922	+	0.978478i	−0.0957795	+	0.0829934i	−0.701434	−	0.712734i	$-0.747457\pi$
$139$	−1.12922	+	0.978478i	−0.0957795	+	0.0829934i	0.605655	+	0.795727i	$0.292911\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	12.0905	+	30.2006i	0.997207	+	2.49090i
$148$		0			0
$149$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$149$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$150$		0			0
$151$	−4.94024	+	6.93759i	−0.402031	+	0.564573i	−0.965115	−	0.261826i	$-0.915675\pi$
$151$	−4.94024	+	6.93759i	−0.402031	+	0.564573i	0.563084	+	0.826399i	$0.309615\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.391208	+	0.373016i	0.0312218	+	0.0297699i	0.705529	−	0.708681i	$-0.250709\pi$
$157$	0.391208	+	0.373016i	0.0312218	+	0.0297699i	−0.674308	+	0.738451i	$0.735558\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	7.55598	−	13.0873i	0.591830	−	1.02508i	−0.402156	−	0.915571i	$-0.631739\pi$
$163$	7.55598	−	13.0873i	0.591830	−	1.02508i	0.993986	−	0.109509i	$-0.0349277\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$167$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$168$		0			0
$169$	7.39477	−	7.05089i	0.568828	−	0.542376i
$170$		0			0
$171$	12.9778	+	22.4783i	0.992440	+	1.71896i
$172$		0			0
$173$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$173$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$174$		0			0
$175$	−2.41327	+	25.2729i	−0.182426	+	1.91045i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$179$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$180$		0			0
$181$	5.56346	−	22.9329i	0.413529	−	1.70459i	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	5.56346	−	22.9329i	0.413529	−	1.70459i	0.673682	−	0.739022i	$-0.264712\pi$
$182$		0			0
$183$	−26.2748	+	2.50894i	−1.94229	+	0.185466i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	20.7391	+	16.3094i	1.50855	+	1.18633i
$190$		0			0
$191$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$191$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$192$		0			0
$193$	2.06021	+	14.3291i	0.148297	+	1.03143i	0.919007	+	0.394242i	$0.128993\pi$
$193$	2.06021	+	14.3291i	0.148297	+	1.03143i	−0.770710	+	0.637187i	$0.780098\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$197$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$198$		0			0
$199$	24.4820	+	12.6214i	1.73548	+	0.894704i	0.966977	+	0.254864i	$0.0820306\pi$
$199$	24.4820	+	12.6214i	1.73548	+	0.894704i	0.768507	+	0.639841i	$0.221000\pi$
$200$		0			0
$201$	7.93501	−	11.7489i	0.559692	−	0.828700i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	4.16868	+	17.1835i	0.286984	+	1.18296i	0.914574	+	0.404419i	$0.132526\pi$
$211$	4.16868	+	17.1835i	0.286984	+	1.18296i	−0.627590	+	0.778544i	$0.715958\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	15.6896	−	45.3320i	1.06508	−	3.07734i
$218$		0			0
$219$	−2.69436	−	28.2167i	−0.182068	−	1.90670i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	12.5627	+	3.68874i	0.841259	+	0.247016i	0.673847	−	0.738871i	$-0.264641\pi$
$223$	12.5627	+	3.68874i	0.841259	+	0.247016i	0.167412	+	0.985887i	$0.446459\pi$
$224$		0			0
$225$	6.23123	+	13.6445i	0.415415	+	0.909632i
$226$		0			0
$227$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$227$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$228$		0			0
$229$	−16.9823	−	21.5947i	−1.12222	−	1.42702i	−0.890934	−	0.454133i	$-0.849949\pi$
$229$	−16.9823	−	21.5947i	−1.12222	−	1.42702i	−0.231287	−	0.972886i	$-0.574293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$233$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−23.4655	−	9.39417i	−1.52425	−	0.610217i
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−15.2563	+	17.6067i	−0.982743	+	1.13415i	0.00821459	+	0.999966i	$0.497385\pi$
$241$	−15.2563	+	17.6067i	−0.982743	+	1.13415i	−0.990957	+	0.134179i	$0.957160\pi$
$242$		0			0
$243$	15.4298	+	2.21847i	0.989821	+	0.142315i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−13.6383	+	4.72025i	−0.867783	+	0.300343i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$251$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$257$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$258$		0			0
$259$	14.9186	+	23.2137i	0.926994	+	1.44243i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$263$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	0.722570	+	2.46085i	0.0438930	+	0.149486i	0.978524	−	0.206133i	$-0.0660881\pi$
$271$	0.722570	+	2.46085i	0.0438930	+	0.149486i	−0.934631	+	0.355619i	$0.884270\pi$
$272$		0			0
$273$	−11.0869	+	9.60686i	−0.671010	+	0.581434i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.55562	−	0.999733i	0.0934679	−	0.0600682i	−0.493072	−	0.869989i	$-0.664126\pi$
$277$	1.55562	−	0.999733i	0.0934679	−	0.0600682i	0.586540	+	0.809920i	$0.300490\pi$
$278$		0			0
$279$	−5.36385	−	27.8303i	−0.321125	−	1.66616i
$280$		0			0
$281$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$281$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$282$		0			0
$283$	−11.0217	−	7.08324i	−0.655174	−	0.421055i	0.170379	−	0.985379i	$-0.445501\pi$
$283$	−11.0217	−	7.08324i	−0.655174	−	0.421055i	−0.825554	+	0.564324i	$0.809137\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5.56016	−	16.0650i	−0.327068	−	0.945001i
$290$		0			0
$291$	24.6816	+	23.5339i	1.44686	+	1.37958i
$292$		0			0
$293$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$293$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	59.1859	−	30.5124i	3.41142	−	1.75871i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−14.8494	+	11.6777i	−0.847501	+	0.666482i	−0.944395	−	0.328813i	$-0.893351\pi$
$307$	−14.8494	+	11.6777i	−0.847501	+	0.666482i	0.0968945	+	0.995295i	$0.469109\pi$
$308$		0			0
$309$	3.07846	−	32.2391i	0.175127	−	1.83402i
$310$		0			0
$311$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$311$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$312$		0			0
$313$	−3.21722	+	10.9568i	−0.181848	+	0.619317i	0.817227	+	0.576316i	$0.195510\pi$
$313$	−3.21722	+	10.9568i	−0.181848	+	0.619317i	−0.999075	+	0.0430013i	$0.986308\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$317$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−8.10529	+	1.96632i	−0.449600	+	0.109072i
$326$		0			0
$327$	2.14156	+	14.8949i	0.118428	+	0.823688i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	22.0054	−	15.6700i	1.20952	−	0.861298i	0.216166	−	0.976357i	$-0.430645\pi$
$331$	22.0054	−	15.6700i	1.20952	−	0.861298i	0.993359	+	0.115058i	$0.0367055\pi$
$332$		0			0
$333$	14.4912	+	7.47075i	0.794114	+	0.409394i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	4.63979	−	8.99993i	0.252745	−	0.490258i	−0.727993	−	0.685585i	$-0.759547\pi$
$337$	4.63979	−	8.99993i	0.252745	−	0.490258i	0.980738	+	0.195327i	$0.0625769\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−59.2134	+	8.51361i	−3.19723	+	0.459692i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$347$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$348$		0			0
$349$	−14.6508	+	32.0808i	−0.784241	+	1.71725i	−0.0917948	+	0.995778i	$0.529260\pi$
$349$	−14.6508	+	32.0808i	−0.784241	+	1.71725i	−0.692446	+	0.721470i	$0.743467\pi$
$350$		0			0
$351$	−2.83489	+	8.19087i	−0.151315	+	0.437197i
$352$		0			0
$353$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$353$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$359$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$360$		0			0
$361$	−55.6024	−	5.30938i	−2.92644	−	0.279441i
$362$		0			0
$363$	−11.7775	−	14.9763i	−0.618159	−	0.786053i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−15.1338	−	15.8719i	−0.789978	−	0.828506i	0.198463	−	0.980108i	$-0.436405\pi$
$367$	−15.1338	−	15.8719i	−0.789978	−	0.828506i	−0.988441	+	0.151603i	$0.951557\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−9.74675	−	5.62729i	−0.504667	−	0.291370i	0.225971	−	0.974134i	$-0.427444\pi$
$373$	−9.74675	−	5.62729i	−0.504667	−	0.291370i	−0.730639	+	0.682764i	$0.760778\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	20.8427	−	21.8592i	1.07062	−	1.12283i	0.0785782	−	0.996908i	$-0.474962\pi$
$379$	20.8427	−	21.8592i	1.07062	−	1.12283i	0.992040	−	0.125924i	$-0.0401895\pi$
$380$		0			0
$381$	−15.2172	+	5.26672i	−0.779599	+	0.269822i
$382$		0			0
$383$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$383$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	21.2702	−	33.0970i	1.08122	−	1.68242i
$388$		0			0
$389$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$389$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	20.8706	+	24.0860i	1.04747	+	1.20884i	0.977422	+	0.211298i	$0.0677690\pi$
$397$	20.8706	+	24.0860i	1.04747	+	1.20884i	0.0700455	+	0.997544i	$0.477686\pi$
$398$		0			0
$399$	−73.0078	+	21.4370i	−3.65496	+	1.07319i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	15.7592			0.785020
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	19.6222	−	0.934721i	0.970256	−	0.0462190i	0.443535	−	0.896257i	$-0.353724\pi$
$409$	19.6222	−	0.934721i	0.970256	−	0.0462190i	0.526721	+	0.850038i	$0.323421\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.17716	−	1.39917i	−0.106616	−	0.0685178i
$418$		0			0
$419$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$419$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$420$		0			0
$421$	0.196061	−	4.11582i	0.00955541	−	0.200593i	−0.989041	−	0.147638i	$-0.952833\pi$
$421$	0.196061	−	4.11582i	0.00955541	−	0.200593i	0.998597	−	0.0529550i	$-0.0168640\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	11.0117	−	76.5882i	0.532895	−	3.70636i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	−1.46297	+	3.65431i	−0.0703056	+	0.175615i	−0.959359	−	0.282189i	$-0.908940\pi$
$433$	−1.46297	+	3.65431i	−0.0703056	+	0.175615i	0.889053	+	0.457804i	$0.151364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−20.8188	−	36.0592i	−0.993626	−	1.72101i	−0.594437	−	0.804142i	$-0.702625\pi$
$439$	−20.8188	−	36.0592i	−0.993626	−	1.72101i	−0.399190	−	0.916868i	$-0.630708\pi$
$440$		0			0
$441$	−44.2902	+	34.8302i	−2.10906	+	1.65858i
$442$		0			0
$443$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$443$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$449$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−13.9402	−	4.82476i	−0.654969	−	0.226687i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−32.2646	−	25.3731i	−1.50927	−	1.18691i	−0.927652	−	0.373447i	$-0.878176\pi$
$457$	−32.2646	−	25.3731i	−1.50927	−	1.18691i	−0.581622	−	0.813459i	$-0.697582\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$461$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$462$		0			0
$463$	8.11299	−	42.0942i	0.377043	−	1.95628i	0.102362	−	0.994747i	$-0.467360\pi$
$463$	8.11299	−	42.0942i	0.377043	−	1.95628i	0.274681	−	0.961535i	$-0.411428\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$467$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$468$		0			0
$469$	27.9871	+	30.7261i	1.29233	+	1.41880i
$470$		0			0
$471$	−0.429012	+	0.832167i	−0.0197678	+	0.0383443i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−42.4777	−	8.18691i	−1.94901	−	0.375641i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$479$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$480$		0			0
$481$	−5.60375	+	7.12575i	−0.255509	+	0.324906i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−3.21593	−	33.6788i	−0.145728	−	1.52613i	−0.712157	−	0.702020i	$-0.752282\pi$
$487$	−3.21593	−	33.6788i	−0.145728	−	1.52613i	0.566429	−	0.824110i	$-0.308325\pi$
$488$		0			0
$489$	25.4369	+	6.17092i	1.15029	+	0.279058i
$490$		0			0
$491$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$491$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−31.2732	+	18.0556i	−1.39998	+	0.808280i	−0.994390	−	0.105774i	$-0.966268\pi$
$499$	−31.2732	+	18.0556i	−1.39998	+	0.808280i	−0.405592	+	0.914054i	$0.632935\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$503$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	15.3263	+	8.84863i	0.680664	+	0.392981i
$508$		0			0
$509$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$509$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$510$		0			0
$511$	82.2486	+	11.8256i	3.63846	+	0.523132i
$512$		0			0
$513$	−31.0236	+	32.5366i	−1.36972	+	1.43653i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$521$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$522$		0			0
$523$	16.3549	−	6.54751i	0.715150	−	0.286303i	0.0145869	−	0.999894i	$-0.495357\pi$
$523$	16.3549	−	6.54751i	0.715150	−	0.286303i	0.700563	+	0.713591i	$0.252932\pi$
$524$		0			0
$525$	−43.1784	+	8.32195i	−1.88446	+	0.363200i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1.09438	+	22.9739i	0.0475819	+	0.998867i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	31.3555	−	27.1697i	1.34808	−	1.16812i	0.377865	−	0.925861i	$-0.376658\pi$
$541$	31.3555	−	27.1697i	1.34808	−	1.16812i	0.970213	−	0.242255i	$-0.0778870\pi$
$542$		0			0
$543$	40.8268	−	1.94482i	1.75205	−	0.0834602i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−3.50444	−	18.1828i	−0.149839	−	0.777439i	−0.976630	−	0.214929i	$-0.931048\pi$
$547$	−3.50444	−	18.1828i	−0.149839	−	0.777439i	0.826791	−	0.562510i	$-0.190164\pi$
$548$		0			0
$549$	−16.9910	−	42.4415i	−0.725159	−	1.81136i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	42.9809	−	60.3582i	1.82773	−	2.56669i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$557$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$558$		0			0
$559$	15.8320	+	15.0958i	0.669623	+	0.638484i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$563$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−16.9843	+	42.4247i	−0.713272	+	1.78167i
$568$		0			0
$569$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$569$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$570$		0			0
$571$	28.9434	−	27.5975i	1.21124	−	1.15492i	0.227799	−	0.973708i	$-0.426847\pi$
$571$	28.9434	−	27.5975i	1.21124	−	1.15492i	0.983444	−	0.181210i	$-0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−4.56288	+	47.7847i	−0.189955	+	1.98930i	−0.0545549	+	0.998511i	$0.517374\pi$
$577$	−4.56288	+	47.7847i	−0.189955	+	1.98930i	−0.135400	+	0.990791i	$0.543232\pi$
$578$		0			0
$579$	−22.8080	+	10.4161i	−0.947869	+	0.432877i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$587$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$588$		0			0
$589$	74.3522	+	33.9555i	3.06363	+	1.39911i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$593$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−9.02870	+	46.8453i	−0.369520	+	1.91725i
$598$		0			0
$599$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$599$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$600$		0			0
$601$	29.6846	+	15.3035i	1.21086	+	0.624242i	0.940918	−	0.338635i	$-0.109965\pi$
$601$	29.6846	+	15.3035i	1.21086	+	0.624242i	0.269942	+	0.962877i	$0.412995\pi$
$602$		0			0
$603$	23.3974	+	7.45397i	0.952816	+	0.303549i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−22.2812	−	31.2896i	−0.904368	−	1.27001i	−0.962384	−	0.271691i	$-0.912417\pi$
$607$	−22.2812	−	31.2896i	−0.904368	−	1.27001i	0.0580166	−	0.998316i	$-0.481522\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	3.43376	+	14.1541i	0.138688	+	0.571680i	0.998168	+	0.0605099i	$0.0192727\pi$
$613$	3.43376	+	14.1541i	0.138688	+	0.571680i	−0.859479	+	0.511170i	$0.829212\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$617$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$618$		0			0
$619$	−7.91841	+	22.8787i	−0.318268	+	0.919574i	0.666471	+	0.745531i	$0.267804\pi$
$619$	−7.91841	+	22.8787i	−0.318268	+	0.919574i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−23.9873	−	7.04331i	−0.959493	−	0.281733i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−22.0452	−	28.0328i	−0.877606	−	1.11597i	−0.992458	−	0.122585i	$-0.960882\pi$
$631$	−22.0452	−	28.0328i	−0.877606	−	1.11597i	0.114852	−	0.993383i	$-0.463361\pi$
$632$		0			0
$633$	−26.5229	+	15.3130i	−1.05419	+	0.608638i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−14.3559	−	27.8466i	−0.568803	−	1.10332i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	7.68998	−	8.87471i	0.303263	−	0.349984i	−0.583579	−	0.812056i	$-0.698348\pi$
$643$	7.68998	−	8.87471i	0.303263	−	0.349984i	0.886843	+	0.462072i	$0.152894\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$647$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	82.9929	+	3.95344i	3.25275	+	0.154947i
$652$		0			0
$653$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$653$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	45.5782	−	18.2468i	1.77818	−	0.711874i
$658$		0			0
$659$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$659$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$660$		0			0
$661$	5.87143	+	9.13612i	0.228372	+	0.355354i	0.936462	−	0.350768i	$-0.114079\pi$
$661$	5.87143	+	9.13612i	0.228372	+	0.355354i	−0.708090	+	0.706122i	$0.750443\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			22.6778i			0.876775i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.16734	+	3.97560i	0.0449977	+	0.153248i	0.978927	−	0.204209i	$-0.0654621\pi$
$673$	1.16734	+	3.97560i	0.0449977	+	0.153248i	−0.933930	+	0.357457i	$0.883644\pi$
$674$		0			0
$675$	−19.6349	+	17.0138i	−0.755750	+	0.654861i
$676$		0			0
$677$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$677$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$678$		0			0
$679$	−84.1040	+	54.0503i	−3.22761	+	2.07426i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$683$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	27.6011	−	38.7604i	1.05305	−	1.47880i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	16.4496	+	47.5280i	0.625772	+	1.80805i	0.588142	+	0.808758i	$0.299860\pi$
$691$	16.4496	+	47.5280i	0.625772	+	1.80805i	0.0376301	+	0.999292i	$0.488019\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$701$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$702$		0			0
$703$	−41.7922	+	21.5454i	−1.57622	+	0.812599i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5.85259	−	4.60253i	0.219799	−	0.172852i	−0.502188	−	0.864758i	$-0.667472\pi$
$709$	5.85259	−	4.60253i	0.219799	−	0.172852i	0.721987	+	0.691907i	$0.243229\pi$
$710$		0			0
$711$	4.16150	−	43.5812i	0.156069	−	1.63442i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$719$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$720$		0			0
$721$	89.7183	+	31.0518i	3.34128	+	1.15643i
$722$		0			0
$723$	−36.7050	−	16.7626i	−1.36508	−	0.623409i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−34.2255	+	8.30301i	−1.26935	+	0.307942i	−0.813213	−	0.581967i	$-0.802283\pi$
$727$	−34.2255	+	8.30301i	−1.26935	+	0.307942i	−0.456140	+	0.889908i	$0.650768\pi$
$728$		0			0
$729$	3.84250	+	26.7252i	0.142315	+	0.989821i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	7.77669	−	5.53775i	0.287238	−	0.204542i	−0.427351	−	0.904086i	$-0.640553\pi$
$733$	7.77669	−	5.53775i	0.287238	−	0.204542i	0.714589	+	0.699544i	$0.246614\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	23.0054	−	44.6243i	0.846268	−	1.64153i	0.0819999	−	0.996632i	$-0.473869\pi$
$739$	23.0054	−	44.6243i	0.846268	−	1.64153i	0.764268	−	0.644899i	$-0.223100\pi$
$740$		0			0
$741$	−14.4997	−	20.3620i	−0.532659	−	0.748015i
$742$		0			0
$743$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$743$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−2.42864	+	5.31797i	−0.0886222	+	0.194056i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−2.42864	+	5.31797i	−0.0886222	+	0.194056i	0.860131	+	0.510073i	$0.170382\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−51.0605	−	12.3871i	−1.85582	−	0.450218i	−0.857854	−	0.513893i	$-0.828203\pi$
$757$	−51.0605	−	12.3871i	−1.85582	−	0.450218i	−0.997971	+	0.0636752i	$0.979718\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$761$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$762$		0			0
$763$	−43.9141	−	4.19328i	−1.58980	−	0.151807i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−29.2118	−	30.6365i	−1.05341	−	1.10478i	−0.994310	−	0.106528i	$-0.966026\pi$
$769$	−29.2118	−	30.6365i	−1.05341	−	1.10478i	−0.0590966	−	0.998252i	$-0.518822\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$773$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$774$		0			0
$775$	40.9089	+	23.6187i	1.46949	+	0.848410i
$776$		0			0
$777$	−31.2988	+	36.1207i	−1.12284	+	1.29582i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	44.7504	+	31.8666i	1.59518	+	1.13592i	0.915431	+	0.402475i	$0.131850\pi$
$787$	44.7504	+	31.8666i	1.59518	+	1.13592i	0.679747	+	0.733447i	$0.262090\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	24.9601	−	4.81066i	0.886357	−	0.170831i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$797$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$809$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$810$		0			0
$811$	−11.1526	+	0.531262i	−0.391620	+	0.0186551i	−0.242469	−	0.970159i	$-0.577957\pi$
$811$	−11.1526	+	0.531262i	−0.391620	+	0.0186551i	−0.149151	+	0.988814i	$0.547654\pi$
$812$		0			0
$813$	−3.73707	+	2.40167i	−0.131065	+	0.0842301i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	42.1697	+	105.335i	1.47533	+	3.68520i
$818$		0			0
$819$	−21.3757	−	13.7373i	−0.746927	−	0.480021i
$820$		0			0
$821$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$821$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$822$		0			0
$823$	−2.72791	+	57.2659i	−0.0950890	+	1.99616i	−0.00794450	+	0.999968i	$0.502529\pi$
$823$	−2.72791	+	57.2659i	−0.0950890	+	1.99616i	−0.0871445	+	0.996196i	$0.527774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$827$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$828$		0			0
$829$	−7.81314	+	54.3416i	−0.271361	+	1.88736i	0.163007	+	0.986625i	$0.447881\pi$
$829$	−7.81314	+	54.3416i	−0.271361	+	1.88736i	−0.434369	+	0.900735i	$0.643028\pi$
$830$		0			0
$831$	2.42055	+	2.09742i	0.0839679	+	0.0727586i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	43.6335	−	22.4946i	1.50819	−	0.777528i
$838$		0			0
$839$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$839$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	50.8059	−	23.2023i	1.74571	−	0.797240i
$848$		0			0
$849$	6.39324	−	21.7734i	0.219415	−	0.747260i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−41.1213	+	3.92661i	−1.40797	+	0.134445i	−0.771191	−	0.636604i	$-0.780338\pi$
$853$	−41.1213	+	3.92661i	−1.40797	+	0.134445i	−0.636776	+	0.771049i	$0.719732\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$857$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$858$		0			0
$859$	43.0174	+	33.8292i	1.46773	+	1.15424i	0.955348	+	0.295484i	$0.0954809\pi$
$859$	43.0174	+	33.8292i	1.46773	+	1.15424i	0.512387	+	0.858755i	$0.328762\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$863$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	23.9851	−	17.0797i	0.814576	−	0.580057i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−6.53178	+	11.9901i	−0.221321	+	0.406269i
$872$		0			0
$873$	−27.0667	+	52.5021i	−0.916069	+	1.77693i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−45.9785	−	8.86163i	−1.55258	−	0.299236i	−0.660402	−	0.750912i	$-0.729614\pi$
$877$	−45.9785	−	8.86163i	−1.55258	−	0.299236i	−0.892182	+	0.451676i	$0.850826\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$881$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$882$		0			0
$883$	−33.5354	+	42.6437i	−1.12856	+	1.43508i	−0.243312	+	0.969948i	$0.578234\pi$
$883$	−33.5354	+	42.6437i	−1.12856	+	1.43508i	−0.885244	+	0.465127i	$0.846009\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$887$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$888$		0			0
$889$	−4.48721	−	46.9922i	−0.150496	−	1.57607i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	79.5896	+	83.4711i	2.64858	+	2.77775i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	55.9141	+	22.3846i	1.85660	+	0.743269i	0.932861	+	0.360238i	$0.117304\pi$
$907$	55.9141	+	22.3846i	1.85660	+	0.743269i	0.923735	+	0.383031i	$0.125120\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$911$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−18.8611	−	0.898463i	−0.622169	−	0.0296375i	−0.265869	−	0.964009i	$-0.585659\pi$
$919$	−18.8611	−	0.898463i	−0.622169	−	0.0296375i	−0.356300	+	0.934372i	$0.615962\pi$
$920$		0			0
$921$	−26.6532	−	18.9797i	−0.878253	−	0.625401i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−25.2262	+	10.0991i	−0.829434	+	0.332055i
$926$		0			0
$927$	55.0800	−	10.6158i	1.80907	−	0.348669i
$928$		0			0
$929$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$929$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$930$		0			0
$931$	−7.73193	−	162.313i	−0.253404	−	5.31960i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			52.7032i			1.72174i	0.508826	+	0.860869i	$0.330080\pi$
$937$			52.7032i			1.72174i	−0.508826	+	0.860869i	$0.669920\pi$
$938$		0			0
$939$	−19.7790			−0.645463
$940$		0			0
$941$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$941$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$947$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$948$		0			0
$949$	5.16620	+	26.8048i	0.167702	+	0.870120i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$953$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−42.1613	−	40.2007i	−1.36004	−	1.29680i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	27.5566	−	47.7295i	0.886161	−	1.53488i	0.0417844	−	0.999127i	$-0.486696\pi$
$967$	27.5566	−	47.7295i	0.886161	−	1.53488i	0.844377	−	0.535750i	$-0.179971\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$971$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$972$		0			0
$973$	5.49081	−	5.23548i	0.176027	−	0.167842i
$974$		0			0
$975$	−7.22299	−	12.5106i	−0.231321	−	0.400659i
$976$		0			0
$977$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$977$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−23.7086	+	10.8274i	−0.756957	+	0.345691i
$982$		0			0
$983$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$983$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−57.0735	−	26.0646i	−1.81300	−	0.827970i	−0.940942	−	0.338567i	$-0.890058\pi$
$991$	−57.0735	−	26.0646i	−1.81300	−	0.827970i	−0.872058	−	0.489402i	$-0.837215\pi$
$992$		0			0
$993$	36.7798	+	28.9240i	1.16717	+	0.917874i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.86556	+	26.8856i	0.122424	+	0.851474i	0.954797	+	0.297259i	$0.0960725\pi$
$997$	3.86556	+	26.8856i	0.122424	+	0.851474i	−0.832373	+	0.554215i	$0.813018\pi$
$998$		0			0
$999$	−5.34420	+	27.7284i	−0.169083	+	0.877287i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.ba.a.785.1	yes	20
3.2	odd	2	CM	804.2.ba.a.785.1	yes	20
67.7	odd	66	inner	804.2.ba.a.677.1	✓	20
201.74	even	66	inner	804.2.ba.a.677.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.ba.a.677.1	✓	20	67.7	odd	66	inner
804.2.ba.a.677.1	✓	20	201.74	even	66	inner
804.2.ba.a.785.1	yes	20	1.1	even	1	trivial
804.2.ba.a.785.1	yes	20	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.ba (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(0.487975 + 1.66189i) q^{3} +(-5.07182 + 0.241600i) q^{7} +(-2.52376 + 1.62192i) q^{9} +O(q^{10})\)	\(q+(0.487975 + 1.66189i) q^{3} +(-5.07182 + 0.241600i) q^{7} +(-2.52376 + 1.62192i) q^{9} +(-0.619962 - 1.54859i) q^{13} +(0.411674 - 8.64209i) q^{19} +(-2.87643 - 8.31091i) q^{21} +(0.711574 - 4.94911i) q^{25} +(-3.92699 - 3.40276i) q^{27} +(-3.51128 + 8.77075i) q^{31} +(-2.71727 - 4.70645i) q^{37} +(2.27106 - 1.78598i) q^{39} +(-11.9291 + 5.44783i) q^{43} +(18.6966 - 1.78531i) q^{49} +(14.5631 - 3.53297i) q^{57} +(-2.88395 + 14.9634i) q^{61} +(12.4082 - 8.83583i) q^{63} +(-5.21774 - 6.30676i) q^{67} +(-16.0693 - 3.09710i) q^{73} +(8.57211 - 1.23248i) q^{75} +(-9.02089 + 11.4710i) q^{79} +(3.73874 - 8.18669i) q^{81} +(3.51847 + 7.70438i) q^{91} +(-16.2895 - 1.55545i) q^{93} +(17.0516 - 9.84473i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 20 * q + 6 * q^7 + 6 * q^9	\( 20 q + 6 q^{7} + 6 q^{9} + 9 q^{13} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{31} + 10 q^{37} - 9 q^{39} - 5 q^{49} + 141 q^{57} + 27 q^{61} + 147 q^{63} + 11 q^{67} - 180 q^{73} - 166 q^{79} - 18 q^{81} - 36 q^{91} - 3 q^{93} + 33 q^{97}+O(q^{100}) \) 20 * q + 6 * q^7 + 6 * q^9 + 9 * q^13 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 3 * q^31 + 10 * q^37 - 9 * q^39 - 5 * q^49 + 141 * q^57 + 27 * q^61 + 147 * q^63 + 11 * q^67 - 180 * q^73 - 166 * q^79 - 18 * q^81 - 36 * q^91 - 3 * q^93 + 33 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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