LMFDB - Embedded newform 804.2.ba.a.605.1

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			605.1
Root			$0.723734 - 0.690079i$ of defining polynomial
Character	$\chi$	$=$	804.605
Dual form			804.2.ba.a.101.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.487975 + 1.66189i) q^{3} +(1.00786 + 1.95498i) q^{7} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})$	$q+(-0.487975 + 1.66189i) q^{3} +(1.00786 + 1.95498i) q^{7} +(-2.52376 - 1.62192i) q^{9} +(4.37316 + 5.56092i) q^{13} +(-5.95519 - 3.07011i) q^{19} +(-3.74078 + 0.720976i) q^{21} +(0.711574 + 4.94911i) q^{25} +(3.92699 - 3.40276i) q^{27} +(-4.43776 + 5.64307i) q^{31} +(4.14042 + 7.17141i) q^{37} +(-11.3756 + 4.55412i) q^{39} +(-11.4806 - 5.24303i) q^{43} +(1.25423 - 1.76132i) q^{49} +(8.00818 - 8.39874i) q^{57} +(-13.8682 - 4.79981i) q^{61} +(0.627224 - 6.56859i) q^{63} +(6.78321 + 4.58128i) q^{67} +(-2.35389 + 6.80112i) q^{73} +(-8.57211 - 1.23248i) q^{75} +(-4.14114 + 10.3441i) q^{79} +(3.73874 + 8.18669i) q^{81} +(-6.46396 + 14.1541i) q^{91} +(-7.21265 - 10.1287i) q^{93} +(14.6117 - 8.43607i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

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

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	$-0.772727\pi$
$5$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$6$		0			0
$7$	1.00786	+	1.95498i	0.380937	+	0.738914i	0.998880	−	0.0473083i	$-0.0150643\pi$
$7$	1.00786	+	1.95498i	0.380937	+	0.738914i	−0.617944	+	0.786222i	$0.712034\pi$
$8$		0			0
$9$	−2.52376	−	1.62192i	−0.841254	−	0.540641i
$10$		0			0
$11$		0			0		0.945001	−	0.327068i	$-0.106061\pi$
$11$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$12$		0			0
$13$	4.37316	+	5.56092i	1.21290	+	1.54232i	0.758731	+	0.651404i	$0.225820\pi$
$13$	4.37316	+	5.56092i	1.21290	+	1.54232i	0.454165	+	0.890918i	$0.349938\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$17$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$18$		0			0
$19$	−5.95519	−	3.07011i	−1.36621	−	0.704333i	−0.389934	−	0.920843i	$-0.627502\pi$
$19$	−5.95519	−	3.07011i	−1.36621	−	0.704333i	−0.976280	+	0.216510i	$0.930533\pi$
$20$		0			0
$21$	−3.74078	+	0.720976i	−0.816305	+	0.157330i
$22$		0			0
$23$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$23$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$24$		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
$31$	−4.43776	+	5.64307i	−0.797045	+	1.01353i	0.202369	+	0.979309i	$0.435136\pi$
$31$	−4.43776	+	5.64307i	−0.797045	+	1.01353i	−0.999414	+	0.0342160i	$0.989107\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.14042	+	7.17141i	0.680680	+	1.17897i	0.974774	+	0.223196i	$0.0716490\pi$
$37$	4.14042	+	7.17141i	0.680680	+	1.17897i	−0.294093	+	0.955777i	$0.595018\pi$
$38$		0			0
$39$	−11.3756	+	4.55412i	−1.82156	+	0.729242i
$40$		0			0
$41$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$41$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$42$		0			0
$43$	−11.4806	−	5.24303i	−1.75078	−	0.799554i	−0.988296	−	0.152547i	$-0.951253\pi$
$43$	−11.4806	−	5.24303i	−1.75078	−	0.799554i	−0.762484	−	0.647008i	$-0.776020\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$47$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$48$		0			0
$49$	1.25423	−	1.76132i	0.179176	−	0.251617i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$53$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	8.00818	−	8.39874i	1.06071	−	1.11244i
$58$		0			0
$59$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$59$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$60$		0			0
$61$	−13.8682	−	4.79981i	−1.77564	−	0.614553i	−0.775771	−	0.631015i	$-0.782639\pi$
$61$	−13.8682	−	4.79981i	−1.77564	−	0.614553i	−0.999864	+	0.0164616i	$0.994760\pi$
$62$		0			0
$63$	0.627224	−	6.56859i	0.0790228	−	0.827564i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.78321	+	4.58128i	0.828700	+	0.559692i
$67$	6.78321	+	4.58128i	0.828700	+	0.559692i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$71$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$72$		0			0
$73$	−2.35389	+	6.80112i	−0.275502	+	0.796011i	0.719348	+	0.694650i	$0.244441\pi$
$73$	−2.35389	+	6.80112i	−0.275502	+	0.796011i	−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
$79$	−4.14114	+	10.3441i	−0.465914	+	1.16380i	0.490410	+	0.871492i	$0.336847\pi$
$79$	−4.14114	+	10.3441i	−0.465914	+	1.16380i	−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		−0.981929	−	0.189251i	$-0.939394\pi$
$83$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$89$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$90$		0			0
$91$	−6.46396	+	14.1541i	−0.677607	+	1.48375i
$92$		0			0
$93$	−7.21265	−	10.1287i	−0.747917	−	1.05030i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	14.6117	−	8.43607i	1.48359	−	0.856553i	0.483767	−	0.875197i	$-0.339268\pi$
$97$	14.6117	−	8.43607i	1.48359	−	0.856553i	0.999826	+	0.0186441i	$0.00593496\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$101$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$102$		0			0
$103$	11.7533	+	9.24286i	1.15808	+	0.910726i	0.997090	−	0.0762298i	$-0.0242882\pi$
$103$	11.7533	+	9.24286i	1.15808	+	0.910726i	0.160992	+	0.986956i	$0.448531\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$107$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$108$		0			0
$109$	19.8266	−	2.85064i	1.89905	−	0.273042i	0.909329	−	0.416079i	$-0.136596\pi$
$109$	19.8266	−	2.85064i	1.89905	−	0.273042i	0.989717	+	0.143037i	$0.0456869\pi$
$110$		0			0
$111$	−13.9385	+	3.38145i	−1.32299	+	0.320953i
$112$		0			0
$113$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$113$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.01742	−	21.1274i	−0.186510	−	1.95322i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.64658	−	6.79975i	0.786053	−	0.618159i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	13.0703	−	6.73819i	1.15980	−	0.597918i	0.232631	−	0.972565i	$-0.425267\pi$
$127$	13.0703	−	6.73819i	1.15980	−	0.597918i	0.927167	+	0.374648i	$0.122236\pi$
$128$		0			0
$129$	14.3156	−	16.5211i	1.26042	−	1.45460i
$130$		0			0
$131$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$131$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$132$		0			0
$133$		−	14.7366i		−	1.27782i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$137$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$138$		0			0
$139$	−8.66506	−	7.50832i	−0.734961	−	0.636847i	0.204751	−	0.978814i	$-0.434362\pi$
$139$	−8.66506	−	7.50832i	−0.734961	−	0.636847i	−0.939712	+	0.341967i	$0.888907\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2.31509	+	2.94387i	0.190945	+	0.242806i
$148$		0			0
$149$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$149$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$150$		0			0
$151$	21.5089	−	2.05386i	1.75037	−	0.167140i	0.829891	−	0.557926i	$-0.188403\pi$
$151$	21.5089	−	2.05386i	1.75037	−	0.167140i	0.920482	+	0.390785i	$0.127796\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.42589	+	22.3658i	0.433033	+	1.78499i	0.601060	+	0.799204i	$0.294745\pi$
$157$	5.42589	+	22.3658i	0.433033	+	1.78499i	−0.168027	+	0.985782i	$0.553740\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.82939	−	4.90065i	0.221615	−	0.383849i	−0.733683	−	0.679492i	$-0.762200\pi$
$163$	2.82939	−	4.90065i	0.221615	−	0.383849i	0.955299	+	0.295643i	$0.0955338\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$167$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$168$		0			0
$169$	−8.73448	+	36.0040i	−0.671883	+	2.76954i
$170$		0			0
$171$	10.0500	+	17.4071i	0.768542	+	1.33115i
$172$		0			0
$173$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$173$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$174$		0			0
$175$	−8.95825	+	6.37914i	−0.677180	+	0.482218i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$179$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$180$		0			0
$181$	9.30913	−	8.87624i	0.691942	−	0.659766i	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	9.30913	−	8.87624i	0.691942	−	0.659766i	0.952095	+	0.305802i	$0.0989245\pi$
$182$		0			0
$183$	14.7441	−	20.7052i	1.08991	−	1.53057i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	10.6102	+	4.24768i	0.771779	+	0.308974i
$190$		0			0
$191$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$191$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$192$		0			0
$193$	3.92593	−	27.3055i	0.282595	−	1.96549i	0.0231538	−	0.999732i	$-0.492629\pi$
$193$	3.92593	−	27.3055i	0.282595	−	1.96549i	0.259441	−	0.965759i	$-0.416462\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$197$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$198$		0			0
$199$	0.638541	−	13.4046i	0.0452650	−	0.950228i	−0.856004	−	0.516969i	$-0.827060\pi$
$199$	0.638541	−	13.4046i	0.0452650	−	0.950228i	0.901269	−	0.433260i	$-0.142637\pi$
$200$		0			0
$201$	−10.9236	+	9.03739i	−0.770493	+	0.637449i
$202$		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
$211$	14.6918	+	14.0086i	1.01142	+	0.964390i	0.999368	−	0.0355373i	$-0.0113142\pi$
$211$	14.6918	+	14.0086i	1.01142	+	0.964390i	0.0120548	+	0.999927i	$0.496163\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−15.5048	−	2.98830i	−1.05253	−	0.202859i
$218$		0			0
$219$	−10.1541	−	7.23068i	−0.686149	−	0.488604i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−3.35641	+	0.985532i	−0.224762	+	0.0659961i	−0.392174	−	0.919891i	$-0.628277\pi$
$223$	−3.35641	+	0.985532i	−0.224762	+	0.0659961i	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		−0.580057	−	0.814576i	$-0.696970\pi$
$227$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$228$		0			0
$229$	−11.1266	−	27.7928i	−0.735265	−	1.83660i	−0.503978	−	0.863716i	$-0.668131\pi$
$229$	−11.1266	−	27.7928i	−0.735265	−	1.83660i	−0.231287	−	0.972886i	$-0.574293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$233$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−15.1699	−	11.9298i	−0.985393	−	0.774921i
$238$		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
$241$	18.4249	+	21.2635i	1.18685	+	1.36970i	0.913014	+	0.407929i	$0.133749\pi$
$241$	18.4249	+	21.2635i	1.18685	+	1.36970i	0.273841	+	0.961775i	$0.411706\pi$
$242$		0			0
$243$	−15.4298	+	2.21847i	−0.989821	+	0.142315i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−8.97032	−	46.5424i	−0.570768	−	2.96142i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$251$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$257$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$258$		0			0
$259$	−9.84701	+	15.3222i	−0.611864	+	0.952078i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$263$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$264$		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
$271$	8.11176	−	27.6261i	0.492755	−	1.67817i	−0.218988	−	0.975728i	$-0.570276\pi$
$271$	8.11176	−	27.6261i	0.492755	−	1.67817i	0.711742	−	0.702440i	$-0.247906\pi$
$272$		0			0
$273$	−20.3683	−	17.6492i	−1.23275	−	1.06818i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	20.1115	+	12.9249i	1.20838	+	0.776580i	0.980387	−	0.197082i	$-0.0631466\pi$
$277$	20.1115	+	12.9249i	1.20838	+	0.776580i	0.227995	+	0.973662i	$0.426783\pi$
$278$		0			0
$279$	20.3525	−	7.04406i	1.21847	−	0.421717i
$280$		0			0
$281$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$281$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$282$		0			0
$283$	−24.2361	+	15.5756i	−1.44069	+	0.925874i	−0.441091	+	0.897462i	$0.645408\pi$
$283$	−24.2361	+	15.5756i	−1.44069	+	0.925874i	−0.999596	+	0.0284112i	$0.990955\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.6928	−	3.21727i	0.981929	−	0.189251i
$290$		0			0
$291$	6.88968	+	28.3996i	0.403880	+	1.66482i
$292$		0			0
$293$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$293$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1.32088	−	27.7287i	−0.0761343	−	1.59826i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−32.1929	+	12.8881i	−1.83735	+	0.735562i	−0.865100	+	0.501599i	$0.832745\pi$
$307$	−32.1929	+	12.8881i	−1.83735	+	0.735562i	−0.972245	+	0.233964i	$0.924830\pi$
$308$		0			0
$309$	−21.0959	+	15.0223i	−1.20010	+	0.854591i
$310$		0			0
$311$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$311$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$312$		0			0
$313$	9.91913	+	33.7814i	0.560662	+	1.90944i	0.375915	+	0.926654i	$0.377328\pi$
$313$	9.91913	+	33.7814i	0.560662	+	1.90944i	0.184747	+	0.982786i	$0.440853\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$317$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−24.4098	+	25.6002i	−1.35401	+	1.42005i
$326$		0			0
$327$	−4.93745	+	34.3407i	−0.273042	+	1.89905i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.907508	+	9.50386i	−0.0498812	+	0.522379i	0.935655	+	0.352915i	$0.114810\pi$
$331$	−0.907508	+	9.50386i	−0.0498812	+	0.522379i	−0.985536	+	0.169464i	$0.945796\pi$
$332$		0			0
$333$	1.18205	−	24.8144i	0.0647761	−	1.35982i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.227062	+	0.0108163i	0.0123689	+	0.000589202i	0.0537654	−	0.998554i	$-0.482878\pi$
$337$	0.227062	+	0.0108163i	0.0123689	+	0.000589202i	−0.0413966	+	0.999143i	$0.513181\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	19.9471	+	2.86797i	1.07704	+	0.154856i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$347$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$348$		0			0
$349$	−1.42476	−	3.11980i	−0.0762659	−	0.166999i	0.867659	−	0.497161i	$-0.165624\pi$
$349$	−1.42476	−	3.11980i	−0.0762659	−	0.166999i	−0.943924	+	0.330162i	$0.892897\pi$
$350$		0			0
$351$	36.0958	+	6.95690i	1.92665	+	0.371332i
$352$		0			0
$353$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$353$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$359$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$360$		0			0
$361$	15.0176	+	21.0893i	0.790401	+	1.10996i
$362$		0			0
$363$	7.08112	+	17.6878i	0.371662	+	0.928368i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−27.1879	−	6.59572i	−1.41920	−	0.344294i	−0.548516	−	0.836140i	$-0.684807\pi$
$367$	−27.1879	−	6.59572i	−1.41920	−	0.344294i	−0.870682	+	0.491846i	$0.836322\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−33.1572	−	19.1433i	−1.71681	−	0.991203i	−0.924582	−	0.380982i	$-0.875586\pi$
$373$	−33.1572	−	19.1433i	−1.71681	−	0.991203i	−0.792231	−	0.610221i	$-0.791081\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−9.52769	+	2.31139i	−0.489405	+	0.118728i	−0.472865	−	0.881135i	$-0.656780\pi$
$379$	−9.52769	+	2.31139i	−0.489405	+	0.118728i	−0.0165397	+	0.999863i	$0.505265\pi$
$380$		0			0
$381$	4.82017	+	25.0094i	0.246945	+	1.28127i
$382$		0			0
$383$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$383$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	20.4706	+	31.8528i	1.04058	+	1.61917i
$388$		0			0
$389$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$389$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.82791	+	2.10952i	−0.0917400	+	0.105874i	−0.799763	−	0.600315i	$-0.795042\pi$
$397$	−1.82791	+	2.10952i	−0.0917400	+	0.105874i	0.708023	+	0.706189i	$0.249587\pi$
$398$		0			0
$399$	24.4905	+	7.19107i	1.22606	+	0.360004i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−50.7877			−2.52991
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−17.3175	−	33.5913i	−0.856297	−	1.66098i	−0.744869	−	0.667211i	$-0.767488\pi$
$409$	−17.3175	−	33.5913i	−0.856297	−	1.66098i	−0.111428	−	0.993773i	$-0.535542\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	16.7063	−	10.7365i	0.818113	−	0.525769i
$418$		0			0
$419$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$419$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$420$		0			0
$421$	−6.71011	−	3.45930i	−0.327031	−	0.168596i	0.286890	−	0.957963i	$-0.407378\pi$
$421$	−6.71011	−	3.45930i	−0.327031	−	0.168596i	−0.613921	+	0.789367i	$0.710409\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−4.59365	−	31.9496i	−0.222303	−	1.54615i
$428$		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
$433$	4.88071	−	6.20633i	0.234552	−	0.298257i	−0.654501	−	0.756061i	$-0.727121\pi$
$433$	4.88071	−	6.20633i	0.234552	−	0.298257i	0.889053	+	0.457804i	$0.151364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	15.4184	+	26.7054i	0.735879	+	1.27458i	0.954336	+	0.298734i	$0.0965643\pi$
$439$	15.4184	+	26.7054i	0.735879	+	1.27458i	−0.218457	+	0.975847i	$0.570102\pi$
$440$		0			0
$441$	−6.02210	+	2.41088i	−0.286766	+	0.114804i
$442$		0			0
$443$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$443$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$449$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−7.08255	+	36.7477i	−0.332767	+	1.72656i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	25.9353	+	10.3829i	1.21320	+	0.485693i	0.887953	−	0.459935i	$-0.152127\pi$
$457$	25.9353	+	10.3829i	1.21320	+	0.485693i	0.325251	+	0.945628i	$0.394551\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$461$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$462$		0			0
$463$	20.0749	+	6.94799i	0.932959	+	0.322900i	0.750922	−	0.660391i	$-0.229610\pi$
$463$	20.0749	+	6.94799i	0.932959	+	0.322900i	0.182037	+	0.983292i	$0.441731\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$467$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$468$		0			0
$469$	−2.11978	+	17.8784i	−0.0978822	+	0.825546i
$470$		0			0
$471$	−39.8172	−	1.89673i	−1.83468	−	0.0873966i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	10.9568	−	31.6575i	0.502731	−	1.45255i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$479$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$480$		0			0
$481$	−21.7730	+	54.3862i	−0.992762	+	2.47980i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−0.597939	−	0.425791i	−0.0270952	−	0.0192944i	0.566429	−	0.824110i	$-0.308325\pi$
$487$	−0.597939	−	0.425791i	−0.0270952	−	0.0192944i	−0.593524	+	0.804816i	$0.702264\pi$
$488$		0			0
$489$	6.76368	+	7.09354i	0.305864	+	0.320781i
$490$		0			0
$491$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$491$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	23.5882	−	13.6187i	1.05595	−	0.609656i	0.131644	−	0.991297i	$-0.457974\pi$
$499$	23.5882	−	13.6187i	1.05595	−	0.609656i	0.924310	+	0.381641i	$0.124641\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$503$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−55.5725	−	32.0848i	−2.46806	−	1.42494i
$508$		0			0
$509$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$509$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$510$		0			0
$511$	−15.6685	+	2.25279i	−0.693132	+	0.0996574i
$512$		0			0
$513$	−33.8328	+	8.20776i	−1.49376	+	0.362381i
$514$		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	$-0.681818\pi$
$521$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$522$		0			0
$523$	24.4281	−	19.2105i	1.06817	−	0.840014i	0.0805251	−	0.996753i	$-0.474340\pi$
$523$	24.4281	−	19.2105i	1.06817	−	0.840014i	0.987640	+	0.156738i	$0.0500979\pi$
$524$		0			0
$525$	−6.23003	−	18.0005i	−0.271901	−	0.785606i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−20.4432	+	10.5392i	−0.888835	+	0.458227i
$530$		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
$541$	8.51685	+	7.37989i	0.366168	+	0.317286i	0.818438	−	0.574595i	$-0.194840\pi$
$541$	8.51685	+	7.37989i	0.366168	+	0.317286i	−0.452270	+	0.891881i	$0.649386\pi$
$542$		0			0
$543$	10.2087	+	19.8021i	0.438098	+	0.849791i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−19.2182	+	6.65147i	−0.821710	+	0.284396i	−0.705394	−	0.708816i	$-0.749230\pi$
$547$	−19.2182	+	6.65147i	−0.821710	+	0.284396i	−0.116316	+	0.993212i	$0.537109\pi$
$548$		0			0
$549$	27.2150	+	34.6067i	1.16151	+	1.47698i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−24.3962	+	2.32955i	−1.03743	+	0.0990626i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$557$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$558$		0			0
$559$	−21.0505	−	86.7715i	−0.890343	−	3.67004i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$563$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−12.2367	+	15.5602i	−0.513893	+	0.653468i
$568$		0			0
$569$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$569$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$570$		0			0
$571$	4.59655	−	18.9472i	0.192359	−	0.792917i	−0.791085	−	0.611706i	$-0.790483\pi$
$571$	4.59655	−	18.9472i	0.192359	−	0.792917i	0.983444	−	0.181210i	$-0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−3.99179	+	2.84254i	−0.166180	+	0.118336i	−0.660121	−	0.751159i	$-0.729495\pi$
$577$	−3.99179	+	2.84254i	−0.166180	+	0.118336i	0.493941	+	0.869495i	$0.335556\pi$
$578$		0			0
$579$	43.4629	+	19.8489i	1.80626	+	0.824891i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$587$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$588$		0			0
$589$	43.7526	−	19.9811i	1.80279	−	0.823308i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$593$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	21.9654	+	7.60231i	0.898985	+	0.311142i
$598$		0			0
$599$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$599$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$600$		0			0
$601$	−1.46453	+	30.7443i	−0.0597395	+	1.25409i	0.747660	+	0.664082i	$0.231177\pi$
$601$	−1.46453	+	30.7443i	−0.0597395	+	1.25409i	−0.807400	+	0.590005i	$0.799126\pi$
$602$		0			0
$603$	−9.68871	−	22.5639i	−0.394555	−	0.918873i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	43.8313	+	4.18538i	1.77906	+	0.169879i	0.932201	−	0.361942i	$-0.117886\pi$
$607$	43.8313	+	4.18538i	1.77906	+	0.169879i	0.846857	+	0.531821i	$0.178492\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−32.7891	−	31.2643i	−1.32434	−	1.26275i	−0.940687	−	0.339275i	$-0.889818\pi$
$613$	−32.7891	−	31.2643i	−1.32434	−	1.26275i	−0.383651	−	0.923478i	$-0.625333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$617$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$618$		0			0
$619$	−45.6357	−	8.79555i	−1.83425	−	0.353523i	−0.849514	−	0.527566i	$-0.823105\pi$
$619$	−45.6357	−	8.79555i	−1.83425	−	0.353523i	−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
$631$	4.04006	+	10.0916i	0.160832	+	0.401739i	0.986793	−	0.161989i	$-0.0517908\pi$
$631$	4.04006	+	10.0916i	0.160832	+	0.401739i	−0.825960	+	0.563728i	$0.809367\pi$
$632$		0			0
$633$	−30.4499	+	17.5803i	−1.21028	+	0.698753i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	15.2795	−	0.727852i	0.605396	−	0.0288385i
$638$		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
$643$	−29.4531	−	33.9907i	−1.16152	−	1.34046i	−0.929969	−	0.367638i	$-0.880167\pi$
$643$	−29.4531	−	33.9907i	−1.16152	−	1.34046i	−0.231548	−	0.972824i	$-0.574379\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$647$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	12.5322	−	24.3090i	0.491174	−	0.952745i
$652$		0			0
$653$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$653$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	16.9715	−	13.3466i	0.662123	−	0.520699i
$658$		0			0
$659$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$659$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$660$		0			0
$661$	−16.6907	+	25.9712i	−0.649192	+	1.01016i	0.348160	+	0.937435i	$0.386807\pi$
$661$	−16.6907	+	25.9712i	−0.649192	+	1.01016i	−0.997352	+	0.0727275i	$0.976830\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	6.05891i		−	0.234251i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	14.5887	−	49.6845i	0.562352	−	1.91520i	0.220906	−	0.975295i	$-0.429098\pi$
$673$	14.5887	−	49.6845i	0.562352	−	1.91520i	0.341446	−	0.939901i	$-0.389083\pi$
$674$		0			0
$675$	19.6349	+	17.0138i	0.755750	+	0.654861i
$676$		0			0
$677$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$677$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$678$		0			0
$679$	31.2190	+	20.0632i	1.19807	+	0.769955i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$683$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	51.6181	−	4.92893i	1.96935	−	0.188051i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	51.6211	−	9.94916i	1.96376	−	0.378484i	0.979995	−	0.199023i	$-0.0637767\pi$
$691$	51.6211	−	9.94916i	1.96376	−	0.378484i	0.983765	−	0.179461i	$-0.0574354\pi$
$692$		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		0.618159	−	0.786053i	$-0.287879\pi$
$701$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$702$		0			0
$703$	−2.63992	−	55.4187i	−0.0995664	−	2.09016i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−32.2813	+	12.9235i	−1.21235	+	0.485352i	−0.887674	−	0.460472i	$-0.847680\pi$
$709$	−32.2813	+	12.9235i	−1.21235	+	0.485352i	−0.324677	+	0.945825i	$0.605256\pi$
$710$		0			0
$711$	27.2285	−	19.3893i	1.02115	−	0.727157i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$719$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$720$		0			0
$721$	−6.22396	+	32.2930i	−0.231792	+	1.20265i
$722$		0			0
$723$	−44.3286	+	20.2442i	−1.64860	+	0.752889i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	37.2130	−	39.0279i	1.38015	−	1.44746i	0.688379	−	0.725351i	$-0.258323\pi$
$727$	37.2130	−	39.0279i	1.38015	−	1.44746i	0.691775	−	0.722113i	$-0.256829\pi$
$728$		0			0
$729$	3.84250	−	26.7252i	0.142315	−	0.989821i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	4.23163	−	44.3156i	0.156299	−	1.63683i	−0.488559	−	0.872531i	$-0.662477\pi$
$733$	4.23163	−	44.3156i	0.156299	−	1.63683i	0.644858	−	0.764303i	$-0.276917\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−17.0880	−	0.814004i	−0.628594	−	0.0299436i	−0.269132	−	0.963103i	$-0.586737\pi$
$739$	−17.0880	−	0.814004i	−0.628594	−	0.0299436i	−0.359462	+	0.933160i	$0.617040\pi$
$740$		0			0
$741$	81.7257	+	7.80386i	3.00227	+	0.286682i
$742$		0			0
$743$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$743$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−15.5186	−	33.9810i	−0.566282	−	1.23998i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−15.5186	−	33.9810i	−0.566282	−	1.23998i	0.382472	−	0.923967i	$-0.375073\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−6.00925	−	6.30233i	−0.218410	−	0.229062i	0.605409	−	0.795914i	$-0.293009\pi$
$757$	−6.00925	−	6.30233i	−0.218410	−	0.229062i	−0.823819	+	0.566852i	$0.808161\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$761$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$762$		0			0
$763$	25.5555	+	35.8877i	0.925171	+	1.29922i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	49.2826	+	11.9558i	1.77717	+	0.431138i	0.984046	−	0.177916i	$-0.0569356\pi$
$769$	49.2826	+	11.9558i	1.77717	+	0.431138i	0.793129	+	0.609054i	$0.208451\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$773$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$774$		0			0
$775$	−31.0860	−	17.9475i	−1.11664	−	0.644693i
$776$		0			0
$777$	−20.6588	−	23.8415i	−0.741131	−	0.855310i
$778$		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
$787$	5.31582	+	55.6698i	0.189488	+	1.98441i	0.175309	+	0.984514i	$0.443908\pi$
$787$	5.31582	+	55.6698i	0.189488	+	1.98441i	0.0141799	+	0.999899i	$0.495486\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−33.9562	−	98.1101i	−1.20582	−	3.48399i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$797$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$798$		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	$-0.772727\pi$
$809$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$810$		0			0
$811$	12.1192	+	23.5079i	0.425562	+	0.825475i	0.999976	+	0.00691001i	$0.00219954\pi$
$811$	12.1192	+	23.5079i	0.425562	+	0.825475i	−0.574414	+	0.818565i	$0.694770\pi$
$812$		0			0
$813$	41.9533	+	26.9617i	1.47137	+	0.945589i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	52.2727	+	66.4701i	1.82879	+	2.32549i
$818$		0			0
$819$	39.2703	−	25.2375i	1.37222	−	0.881870i
$820$		0			0
$821$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$821$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$822$		0			0
$823$	19.3295	+	9.96505i	0.673784	+	0.347360i	0.760928	−	0.648836i	$-0.224744\pi$
$823$	19.3295	+	9.96505i	0.673784	+	0.347360i	−0.0871445	+	0.996196i	$0.527774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$827$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$828$		0			0
$829$	6.98546	+	48.5849i	0.242615	+	1.68742i	0.638894	+	0.769295i	$0.279392\pi$
$829$	6.98546	+	48.5849i	0.242615	+	1.68742i	−0.396279	+	0.918130i	$0.629699\pi$
$830$		0			0
$831$	−31.2936	+	27.1161i	−1.08556	+	0.940646i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.77496	+	37.2609i	0.0613514	+	1.28792i
$838$		0			0
$839$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$839$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$840$		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
$847$	22.0080	+	10.0507i	0.756203	+	0.345346i
$848$		0			0
$849$	−14.0583	−	47.8783i	−0.482481	−	1.64318i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−29.7397	+	41.7636i	−1.01827	+	1.42996i	−0.118830	+	0.992915i	$0.537914\pi$
$853$	−29.7397	+	41.7636i	−1.01827	+	1.42996i	−0.899440	+	0.437045i	$0.856025\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$857$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$858$		0			0
$859$	54.2408	+	21.7147i	1.85067	+	0.740898i	0.955348	+	0.295484i	$0.0954809\pi$
$859$	54.2408	+	21.7147i	1.85067	+	0.740898i	0.895325	+	0.445414i	$0.146943\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$863$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−2.79891	+	29.3115i	−0.0950560	+	0.995472i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	4.18788	+	57.7555i	0.141901	+	1.95697i
$872$		0			0
$873$	−50.5591	−	2.40843i	−1.71117	−	0.0815128i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	17.1525	−	49.5589i	0.579199	−	1.67348i	−0.149592	−	0.988748i	$-0.547796\pi$
$877$	17.1525	−	49.5589i	0.579199	−	1.67348i	0.728791	−	0.684737i	$-0.240083\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$881$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$882$		0			0
$883$	21.8382	−	54.5492i	0.734914	−	1.83573i	0.228192	−	0.973616i	$-0.426719\pi$
$883$	21.8382	−	54.5492i	0.734914	−	1.83573i	0.506722	−	0.862110i	$-0.330857\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$887$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$888$		0			0
$889$	26.3461	+	18.7610i	0.883619	+	0.629223i
$890$		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
$903$	46.7266	+	11.3358i	1.55496	+	0.377230i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	6.16225	+	4.84605i	0.204614	+	0.160910i	0.715208	−	0.698912i	$-0.246332\pi$
$907$	6.16225	+	4.84605i	0.204614	+	0.160910i	−0.510594	+	0.859822i	$0.670574\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$911$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	27.3632	−	53.0773i	0.902630	−	1.75086i	0.297520	−	0.954716i	$-0.403841\pi$
$919$	27.3632	−	53.0773i	0.902630	−	1.75086i	0.605110	−	0.796142i	$-0.293129\pi$
$920$		0			0
$921$	−5.70927	−	59.7901i	−0.188127	−	1.97015i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−32.5459	+	25.5944i	−1.07010	+	0.841537i
$926$		0			0
$927$	−14.6712	−	42.3896i	−0.481865	−	1.39226i
$928$		0			0
$929$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$929$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$930$		0			0
$931$	−12.8766	+	6.63836i	−0.422014	+	0.217564i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	10.9710i		−	0.358408i	−0.983812	−	0.179204i	$-0.942648\pi$
$937$		−	10.9710i		−	0.358408i	0.983812	−	0.179204i	$-0.0573523\pi$
$938$		0			0
$939$	−60.9814			−1.99005
$940$		0			0
$941$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$941$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$947$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$948$		0			0
$949$	−48.1144	+	16.6526i	−1.56186	+	0.540565i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$953$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−4.84200	−	19.9590i	−0.156194	−	0.643839i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	30.9724	−	53.6459i	0.996007	−	1.72513i	0.420687	−	0.907206i	$-0.361789\pi$
$967$	30.9724	−	53.6459i	0.996007	−	1.72513i	0.575320	−	0.817928i	$-0.304878\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$971$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$972$		0			0
$973$	5.94543	−	24.5074i	0.190602	−	0.785672i
$974$		0			0
$975$	−30.6334	−	53.0586i	−0.981055	−	1.69924i
$976$		0			0
$977$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$977$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−54.6612	−	24.9629i	−1.74520	−	0.797005i
$982$		0			0
$983$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$983$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−3.42242	+	1.56297i	−0.108717	+	0.0496493i	−0.469031	−	0.883182i	$-0.655397\pi$
$991$	−3.42242	+	1.56297i	−0.108717	+	0.0496493i	0.360314	+	0.932831i	$0.382669\pi$
$992$		0			0
$993$	−15.3515	−	6.14583i	−0.487166	−	0.195032i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.13456	+	7.89103i	−0.0359318	+	0.249911i	−0.999868	−	0.0162206i	$-0.994837\pi$
$997$	−1.13456	+	7.89103i	−0.0359318	+	0.249911i	0.963937	+	0.266132i	$0.0857457\pi$
$998$		0			0
$999$	40.6619	+	14.0732i	1.28649	+	0.445257i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.ba.a.605.1	yes	20
3.2	odd	2	CM	804.2.ba.a.605.1	yes	20
67.34	odd	66	inner	804.2.ba.a.101.1	✓	20
201.101	even	66	inner	804.2.ba.a.101.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.ba.a.101.1	✓	20	67.34	odd	66	inner
804.2.ba.a.101.1	✓	20	201.101	even	66	inner
804.2.ba.a.605.1	yes	20	1.1	even	1	trivial
804.2.ba.a.605.1	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)

\(f(q)\)	\(=\)	\(q+(-0.487975 + 1.66189i) q^{3} +(1.00786 + 1.95498i) q^{7} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\)	\(q+(-0.487975 + 1.66189i) q^{3} +(1.00786 + 1.95498i) q^{7} +(-2.52376 - 1.62192i) q^{9} +(4.37316 + 5.56092i) q^{13} +(-5.95519 - 3.07011i) q^{19} +(-3.74078 + 0.720976i) q^{21} +(0.711574 + 4.94911i) q^{25} +(3.92699 - 3.40276i) q^{27} +(-4.43776 + 5.64307i) q^{31} +(4.14042 + 7.17141i) q^{37} +(-11.3756 + 4.55412i) q^{39} +(-11.4806 - 5.24303i) q^{43} +(1.25423 - 1.76132i) q^{49} +(8.00818 - 8.39874i) q^{57} +(-13.8682 - 4.79981i) q^{61} +(0.627224 - 6.56859i) q^{63} +(6.78321 + 4.58128i) q^{67} +(-2.35389 + 6.80112i) q^{73} +(-8.57211 - 1.23248i) q^{75} +(-4.14114 + 10.3441i) q^{79} +(3.73874 + 8.18669i) q^{81} +(-6.46396 + 14.1541i) q^{91} +(-7.21265 - 10.1287i) q^{93} +(14.6117 - 8.43607i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 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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