LMFDB - Embedded newform 804.2.ba.a.497.1

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			497.1
Root			$0.981929 - 0.189251i$ of defining polynomial
Character	$\chi$	$=$	804.497
Dual form			804.2.ba.a.749.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.30900 + 1.13425i) q^{3} +(-1.35800 + 1.72684i) q^{7} +(0.426945 - 2.96946i) q^{9} +O(q^{10})$	$q+(-1.30900 + 1.13425i) q^{3} +(-1.35800 + 1.72684i) q^{7} +(0.426945 - 2.96946i) q^{9} +(-3.42905 - 2.44182i) q^{13} +(-2.06016 + 1.62013i) q^{19} +(-0.181052 - 3.80075i) q^{21} +(-2.07708 - 4.54816i) q^{25} +(2.80925 + 4.37128i) q^{27} +(6.03124 - 4.29482i) q^{31} +(-5.22813 - 9.05539i) q^{37} +(7.25826 - 0.693079i) q^{39} +(-3.12884 - 10.6558i) q^{43} +(0.512504 + 2.11257i) q^{49} +(0.859110 - 4.45748i) q^{57} +(4.33139 - 8.40173i) q^{61} +(4.54800 + 4.76981i) q^{63} +(-7.79913 - 2.48466i) q^{67} +(-12.7250 - 6.56021i) q^{73} +(7.87764 + 3.59760i) q^{75} +(-0.645295 + 6.75784i) q^{79} +(-8.63544 - 2.53559i) q^{81} +(8.87330 - 2.60544i) q^{91} +(-3.02346 + 12.4629i) q^{93} +(7.53271 - 4.34901i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{25}{66}\right)$	$1$

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
$3$	−1.30900	+	1.13425i	−0.755750	+	0.654861i
$3$	−1.30900	+	1.13425i	−0.755750	+	0.654861i
$4$		0			0
$5$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$5$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$6$		0			0
$7$	−1.35800	+	1.72684i	−0.513277	+	0.652685i	−0.971747	−	0.236025i	$-0.924155\pi$
$7$	−1.35800	+	1.72684i	−0.513277	+	0.652685i	0.458470	+	0.888710i	$0.348398\pi$
$8$		0			0
$9$	0.426945	−	2.96946i	0.142315	−	0.989821i
$10$		0			0
$11$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$11$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$12$		0			0
$13$	−3.42905	−	2.44182i	−0.951048	−	0.677238i	−0.00456441	−	0.999990i	$-0.501453\pi$
$13$	−3.42905	−	2.44182i	−0.951048	−	0.677238i	−0.946484	+	0.322751i	$0.895392\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$17$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$18$		0			0
$19$	−2.06016	+	1.62013i	−0.472633	+	0.371683i	−0.825879	−	0.563848i	$-0.809320\pi$
$19$	−2.06016	+	1.62013i	−0.472633	+	0.371683i	0.353245	+	0.935531i	$0.385078\pi$
$20$		0			0
$21$	−0.181052	−	3.80075i	−0.0395088	−	0.829391i
$22$		0			0
$23$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$23$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$24$		0			0
$25$	−2.07708	−	4.54816i	−0.415415	−	0.909632i
$26$		0			0
$27$	2.80925	+	4.37128i	0.540641	+	0.841254i
$28$		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
$31$	6.03124	−	4.29482i	1.08324	−	0.771373i	0.108364	−	0.994111i	$-0.465439\pi$
$31$	6.03124	−	4.29482i	1.08324	−	0.771373i	0.974878	+	0.222738i	$0.0714995\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.22813	−	9.05539i	−0.859500	−	1.48870i	−0.872407	−	0.488780i	$-0.837442\pi$
$37$	−5.22813	−	9.05539i	−0.859500	−	1.48870i	0.0129071	−	0.999917i	$-0.495891\pi$
$38$		0			0
$39$	7.25826	−	0.693079i	1.16225	−	0.110982i
$40$		0			0
$41$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$41$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$42$		0			0
$43$	−3.12884	−	10.6558i	−0.477143	−	1.62500i	−0.748935	−	0.662644i	$-0.769434\pi$
$43$	−3.12884	−	10.6558i	−0.477143	−	1.62500i	0.271792	−	0.962356i	$-0.412384\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$47$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$48$		0			0
$49$	0.512504	+	2.11257i	0.0732148	+	0.301796i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$53$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.859110	−	4.45748i	0.113792	−	0.590408i
$58$		0			0
$59$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$59$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$60$		0			0
$61$	4.33139	−	8.40173i	0.554578	−	1.07573i	−0.430333	−	0.902670i	$-0.641604\pi$
$61$	4.33139	−	8.40173i	0.554578	−	1.07573i	0.984911	−	0.173061i	$-0.0553659\pi$
$62$		0			0
$63$	4.54800	+	4.76981i	0.572994	+	0.600939i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−7.79913	−	2.48466i	−0.952816	−	0.303549i
$67$	−7.79913	−	2.48466i	−0.952816	−	0.303549i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$71$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$72$		0			0
$73$	−12.7250	−	6.56021i	−1.48935	−	0.767815i	−0.494504	−	0.869176i	$-0.664650\pi$
$73$	−12.7250	−	6.56021i	−1.48935	−	0.767815i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$	7.87764	+	3.59760i	0.909632	+	0.415415i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.645295	+	6.75784i	−0.0726014	+	0.760316i	0.883723	+	0.468010i	$0.155029\pi$
$79$	−0.645295	+	6.75784i	−0.0726014	+	0.760316i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$	−8.63544	−	2.53559i	−0.959493	−	0.281733i
$82$		0			0
$83$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$83$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$89$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$90$		0			0
$91$	8.87330	−	2.60544i	0.930174	−	0.273124i
$92$		0			0
$93$	−3.02346	+	12.4629i	−0.313518	+	1.29234i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	7.53271	−	4.34901i	0.764831	−	0.441575i	−0.0661967	−	0.997807i	$-0.521086\pi$
$97$	7.53271	−	4.34901i	0.764831	−	0.441575i	0.831028	+	0.556231i	$0.187753\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$101$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$102$		0			0
$103$	3.63994	+	5.11158i	0.358654	+	0.503659i	0.953998	−	0.299812i	$-0.0969239\pi$
$103$	3.63994	+	5.11158i	0.358654	+	0.503659i	−0.595345	+	0.803470i	$0.702984\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$107$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$108$		0			0
$109$	−2.71680	+	1.24072i	−0.260223	+	0.118840i	−0.541255	−	0.840859i	$-0.682051\pi$
$109$	−2.71680	+	1.24072i	−0.260223	+	0.118840i	0.281032	+	0.959698i	$0.409323\pi$
$110$		0			0
$111$	17.1147	+	5.92346i	1.62446	+	0.562230i
$112$		0			0
$113$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$113$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−8.71491	+	9.13993i	−0.805693	+	0.844987i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−6.38063	+	8.96034i	−0.580057	+	0.814576i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−14.8734	−	11.6965i	−1.31980	−	1.03790i	−0.995773	−	0.0918526i	$-0.970721\pi$
$127$	−14.8734	−	11.6965i	−1.31980	−	1.03790i	−0.324026	−	0.946048i	$-0.605036\pi$
$128$		0			0
$129$	16.1820	+	10.3996i	1.42475	+	0.915631i
$130$		0			0
$131$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$131$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$132$		0			0
$133$		−	5.75771i		−	0.499257i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$137$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$138$		0			0
$139$	−11.2372	+	17.4855i	−0.953128	+	1.48310i	−0.0793066	+	0.996850i	$0.525271\pi$
$139$	−11.2372	+	17.4855i	−0.953128	+	1.48310i	−0.873822	+	0.486246i	$0.838366\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−3.06705	−	2.18404i	−0.252966	−	0.180136i
$148$		0			0
$149$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$149$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$150$		0			0
$151$	10.6148	+	10.1212i	0.863822	+	0.823653i	0.985633	−	0.168901i	$-0.0540217\pi$
$151$	10.6148	+	10.1212i	0.863822	+	0.823653i	−0.121811	+	0.992553i	$0.538870\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.07713	+	17.5587i	−0.485008	+	1.40134i	0.391601	+	0.920135i	$0.371921\pi$
$157$	−6.07713	+	17.5587i	−0.485008	+	1.40134i	−0.876609	+	0.481204i	$0.840200\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−11.2617	+	19.5058i	−0.882084	+	1.52781i	−0.0330650	+	0.999453i	$0.510527\pi$
$163$	−11.2617	+	19.5058i	−0.882084	+	1.52781i	−0.849019	+	0.528362i	$0.822806\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$167$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$168$		0			0
$169$	1.54405	+	4.46124i	0.118773	+	0.343172i
$170$		0			0
$171$	3.93134	+	6.80928i	0.300637	+	0.520719i
$172$		0			0
$173$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$173$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$174$		0			0
$175$	10.6746	+	2.58964i	0.806926	+	0.195758i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$179$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$180$		0			0
$181$	−1.92125	+	0.370291i	−0.142805	+	0.0275235i	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−1.92125	+	0.370291i	−0.142805	+	0.0275235i	0.117348	+	0.993091i	$0.462561\pi$
$182$		0			0
$183$	3.85990	+	15.9107i	0.285332	+	1.17616i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−11.3635	−	1.08508i	−0.826572	−	0.0789280i
$190$		0			0
$191$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$191$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$192$		0			0
$193$	−3.50738	+	7.68009i	−0.252467	+	0.552825i	−0.992851	−	0.119359i	$-0.961916\pi$
$193$	−3.50738	+	7.68009i	−0.252467	+	0.552825i	0.740384	+	0.672184i	$0.234643\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$197$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$198$		0			0
$199$	9.25370	−	3.70462i	0.655977	−	0.262614i	−0.0197060	−	0.999806i	$-0.506273\pi$
$199$	9.25370	−	3.70462i	0.655977	−	0.262614i	0.675683	+	0.737192i	$0.263849\pi$
$200$		0			0
$201$	13.0273	−	5.59378i	0.918873	−	0.394555i
$202$		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
$211$	−23.0360	−	4.43982i	−1.58586	−	0.305650i	−0.681304	−	0.732000i	$-0.738587\pi$
$211$	−23.0360	−	4.43982i	−1.58586	−	0.305650i	−0.904558	+	0.426350i	$0.859799\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.773958	+	16.2474i	−0.0525397	+	1.10294i
$218$		0			0
$219$	24.0980	−	5.84610i	1.62839	−	0.395043i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−12.4284	+	14.3431i	−0.832265	+	0.960485i	−0.999677	−	0.0254020i	$-0.991913\pi$
$223$	−12.4284	+	14.3431i	−0.832265	+	0.960485i	0.167412	+	0.985887i	$0.446459\pi$
$224$		0			0
$225$	−14.3924	+	4.22599i	−0.959493	+	0.281733i
$226$		0			0
$227$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$227$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$228$		0			0
$229$	−2.84949	−	29.8412i	−0.188299	−	1.97196i	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−2.84949	−	29.8412i	−0.188299	−	1.97196i	0.0429870	−	0.999076i	$-0.486313\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.945001	−	0.327068i	$-0.106061\pi$
$233$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−6.82040	−	9.57791i	−0.443033	−	0.622152i
$238$		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
$241$	25.8223	−	16.5950i	1.66336	−	1.06898i	0.750345	−	0.661047i	$-0.229888\pi$
$241$	25.8223	−	16.5950i	1.66336	−	1.06898i	0.913014	−	0.407929i	$-0.133749\pi$
$242$		0			0
$243$	14.1798	−	6.47568i	0.909632	−	0.415415i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	11.0205	−	0.524969i	0.701215	−	0.0334030i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$251$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$257$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$258$		0			0
$259$	22.7370	+	3.26909i	1.41281	+	0.203132i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$263$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$264$		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
$271$	20.0171	−	17.3449i	1.21595	−	1.05363i	0.218988	−	0.975728i	$-0.429724\pi$
$271$	20.0171	−	17.3449i	1.21595	−	1.05363i	0.996961	−	0.0778984i	$-0.0248210\pi$
$272$		0			0
$273$	−8.65990	+	13.4751i	−0.524121	+	0.815548i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	4.71918	−	32.8226i	0.283548	−	1.97212i	0.0555527	−	0.998456i	$-0.482308\pi$
$277$	4.71918	−	32.8226i	0.283548	−	1.97212i	0.227995	−	0.973662i	$-0.426783\pi$
$278$		0			0
$279$	−10.1783	−	19.7432i	−0.609360	−	1.18199i
$280$		0			0
$281$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$281$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$282$		0			0
$283$	−3.03154	−	21.0848i	−0.180206	−	1.25336i	−0.856274	−	0.516522i	$-0.827227\pi$
$283$	−3.03154	−	21.0848i	−0.180206	−	1.25336i	0.676068	−	0.736839i	$-0.263682\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.808893	+	16.9807i	0.0475819	+	0.998867i
$290$		0			0
$291$	−4.92742	+	14.2368i	−0.288850	+	0.834578i
$292$		0			0
$293$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$293$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	22.6499	+	9.06766i	1.30552	+	0.522651i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−26.4058	+	2.52145i	−1.50706	+	0.143907i	−0.815646	−	0.578551i	$-0.803618\pi$
$307$	−26.4058	+	2.52145i	−1.50706	+	0.143907i	−0.691415	+	0.722458i	$0.743012\pi$
$308$		0			0
$309$	−10.5625	−	2.56243i	−0.600879	−	0.145772i
$310$		0			0
$311$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$311$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$312$		0			0
$313$	13.4766	+	11.6776i	0.761744	+	0.660055i	0.946491	−	0.322731i	$-0.104601\pi$
$313$	13.4766	+	11.6776i	0.761744	+	0.660055i	−0.184747	+	0.982786i	$0.559147\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$317$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−3.98338	+	20.6677i	−0.220958	+	1.14644i
$326$		0			0
$327$	2.14899	−	4.70564i	0.118840	−	0.260223i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	7.55738	+	7.92595i	0.415391	+	0.435649i	0.897872	−	0.440257i	$-0.145113\pi$
$331$	7.55738	+	7.92595i	0.415391	+	0.435649i	−0.482481	+	0.875907i	$0.660264\pi$
$332$		0			0
$333$	−29.1218	+	11.6586i	−1.59586	+	0.638888i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3.92541	−	9.80520i	−0.213831	−	0.534123i	0.782211	−	0.623014i	$-0.214092\pi$
$337$	−3.92541	−	9.80520i	−0.213831	−	0.534123i	−0.996041	+	0.0888908i	$0.971668\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.3323	−	8.37210i	−0.989853	−	0.452051i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$347$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$348$		0			0
$349$	−31.1052	−	9.13332i	−1.66502	−	0.488895i	−0.692446	−	0.721470i	$-0.743467\pi$
$349$	−31.1052	−	9.13332i	−1.66502	−	0.488895i	−0.972579	+	0.232574i	$0.925285\pi$
$350$		0			0
$351$	1.04080	−	21.8490i	0.0555537	−	1.16622i
$352$		0			0
$353$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$353$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$359$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$360$		0			0
$361$	−2.85997	+	11.7890i	−0.150525	+	0.620472i
$362$		0			0
$363$	−1.81106	−	18.9663i	−0.0950560	−	0.995472i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	28.2494	−	9.77721i	1.47461	−	0.510366i	0.532718	−	0.846293i	$-0.321171\pi$
$367$	28.2494	−	9.77721i	1.47461	−	0.510366i	0.941888	+	0.335927i	$0.109050\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−17.8012	−	10.2775i	−0.921713	−	0.532151i	−0.0375318	−	0.999295i	$-0.511950\pi$
$373$	−17.8012	−	10.2775i	−0.921713	−	0.532151i	−0.884181	+	0.467144i	$0.845283\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−36.2397	−	12.5427i	−1.86151	−	0.644275i	−0.987332	−	0.158667i	$-0.949280\pi$
$379$	−36.2397	−	12.5427i	−1.86151	−	0.644275i	−0.874177	−	0.485608i	$-0.838598\pi$
$380$		0			0
$381$	32.7360	−	1.55941i	1.67712	−	0.0798910i
$382$		0			0
$383$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$383$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−32.9780	+	4.74152i	−1.67636	+	0.241025i
$388$		0			0
$389$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$389$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	23.7356	+	15.2539i	1.19125	+	0.765573i	0.977422	−	0.211298i	$-0.0677690\pi$
$397$	23.7356	+	15.2539i	1.19125	+	0.765573i	0.213832	+	0.976870i	$0.431405\pi$
$398$		0			0
$399$	6.53069	+	7.53682i	0.326944	+	0.377313i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−31.1686			−1.55262
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	5.49395	−	6.98613i	0.271658	−	0.345442i	−0.631013	−	0.775772i	$-0.717361\pi$
$409$	5.49395	−	6.98613i	0.271658	−	0.345442i	0.902671	+	0.430331i	$0.141603\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−5.12343	−	35.6342i	−0.250895	−	1.74502i
$418$		0			0
$419$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$419$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$420$		0			0
$421$	14.6265	−	11.5024i	0.712853	−	0.560594i	−0.194531	−	0.980896i	$-0.562319\pi$
$421$	14.6265	−	11.5024i	0.712853	−	0.560594i	0.907384	+	0.420303i	$0.138076\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	8.62641	+	18.8892i	0.417461	+	0.914113i
$428$		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
$433$	15.5484	−	11.0720i	0.747209	−	0.532085i	−0.141844	−	0.989889i	$-0.545303\pi$
$433$	15.5484	−	11.0720i	0.747209	−	0.532085i	0.889053	+	0.457804i	$0.151364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	0.624991	+	1.08252i	0.0298292	+	0.0516656i	0.880555	−	0.473945i	$-0.157170\pi$
$439$	0.624991	+	1.08252i	0.0298292	+	0.0516656i	−0.850725	+	0.525610i	$0.823837\pi$
$440$		0			0
$441$	6.49201	−	0.619912i	0.309143	−	0.0295196i
$442$		0			0
$443$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$443$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$449$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−25.3748	−	1.20875i	−1.19221	−	0.0567920i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	17.7551	+	1.69541i	0.830548	+	0.0793077i	0.501664	−	0.865063i	$-0.332721\pi$
$457$	17.7551	+	1.69541i	0.830548	+	0.0793077i	0.328884	+	0.944370i	$0.393327\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$461$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$462$		0			0
$463$	11.5562	−	22.4159i	0.537061	−	1.04175i	−0.451689	−	0.892175i	$-0.649178\pi$
$463$	11.5562	−	22.4159i	0.537061	−	1.04175i	0.988750	−	0.149578i	$-0.0477914\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$467$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$468$		0			0
$469$	14.8819	−	10.0937i	0.687180	−	0.466084i
$470$		0			0
$471$	−11.9611	−	29.8773i	−0.551137	−	1.37667i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	11.6477	+	6.00481i	0.534434	+	0.275520i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$479$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$480$		0			0
$481$	−4.18407	+	43.8176i	−0.190777	+	1.99791i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	31.9440	−	7.74953i	1.44752	−	0.351165i	0.566429	−	0.824110i	$-0.308325\pi$
$487$	31.9440	−	7.74953i	1.44752	−	0.351165i	0.881092	+	0.472946i	$0.156809\pi$
$488$		0			0
$489$	−7.38301	−	38.3067i	−0.333871	−	1.73229i
$490$		0			0
$491$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$491$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−13.9923	+	8.07843i	−0.626379	+	0.361640i	−0.779349	−	0.626591i	$-0.784450\pi$
$499$	−13.9923	+	8.07843i	−0.626379	+	0.361640i	0.152969	+	0.988231i	$0.451116\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$503$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−7.08132	−	4.08840i	−0.314492	−	0.181572i
$508$		0			0
$509$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$509$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$510$		0			0
$511$	28.6091	−	13.0653i	1.26559	−	0.577976i
$512$		0			0
$513$	−12.8695	−	4.45419i	−0.568204	−	0.196657i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$521$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$522$		0			0
$523$	−25.1956	+	35.3823i	−1.10173	+	1.54716i	−0.295699	+	0.955281i	$0.595552\pi$
$523$	−25.1956	+	35.3823i	−1.10173	+	1.54716i	−0.806028	+	0.591877i	$0.798387\pi$
$524$		0			0
$525$	−16.9103	+	8.71789i	−0.738028	+	0.380480i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−18.0792	−	14.2177i	−0.786053	−	0.618159i
$530$		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
$541$	14.4510	−	22.4863i	0.621299	−	0.966759i	−0.377865	−	0.925861i	$-0.623342\pi$
$541$	14.4510	−	22.4863i	0.621299	−	0.966759i	0.999163	−	0.0408986i	$-0.0130221\pi$
$542$		0			0
$543$	2.09491	−	2.66389i	0.0899011	−	0.114319i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−20.6902	−	40.1334i	−0.884649	−	1.71598i	−0.674449	−	0.738321i	$-0.735619\pi$
$547$	−20.6902	−	40.1334i	−0.884649	−	1.71598i	−0.210200	−	0.977658i	$-0.567412\pi$
$548$		0			0
$549$	−23.0994	−	16.4490i	−0.985858	−	0.702026i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−10.7934	−	10.2915i	−0.458982	−	0.437639i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$557$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$558$		0			0
$559$	−15.2907	+	44.1795i	−0.646726	+	1.86859i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$563$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	16.1055	−	11.4687i	0.676368	−	0.481640i
$568$		0			0
$569$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$569$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$570$		0			0
$571$	2.35105	+	6.79290i	0.0983882	+	0.284274i	0.983444	−	0.181210i	$-0.0580014\pi$
$571$	2.35105	+	6.79290i	0.0983882	+	0.284274i	−0.885056	+	0.465484i	$0.845880\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	45.2934	+	10.9881i	1.88559	+	0.457439i	0.999976	−	0.00698200i	$-0.00222246\pi$
$577$	45.2934	+	10.9881i	1.88559	+	0.457439i	0.885614	+	0.464421i	$0.153738\pi$
$578$		0			0
$579$	−4.12001	−	14.0315i	−0.171222	−	0.583128i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$587$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$588$		0			0
$589$	−5.46715	+	18.6194i	−0.225270	+	0.767199i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$593$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.91108	+	15.3454i	−0.323779	+	0.628044i
$598$		0			0
$599$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$599$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$600$		0			0
$601$	39.1380	−	15.6685i	1.59647	−	0.639131i	0.608471	−	0.793576i	$-0.291783\pi$
$601$	39.1380	−	15.6685i	1.59647	−	0.639131i	0.988001	+	0.154446i	$0.0493591\pi$
$602$		0			0
$603$	−10.7079	+	22.0984i	−0.436059	+	0.899918i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1.32455	−	1.26296i	0.0537619	−	0.0512619i	−0.662722	−	0.748866i	$-0.730599\pi$
$607$	1.32455	−	1.26296i	0.0537619	−	0.0512619i	0.716484	+	0.697604i	$0.245750\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	43.9637	+	8.47330i	1.77568	+	0.342233i	0.968678	−	0.248321i	$-0.0798787\pi$
$613$	43.9637	+	8.47330i	1.77568	+	0.342233i	0.806998	+	0.590554i	$0.201091\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$617$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$618$		0			0
$619$	−0.522543	+	10.9695i	−0.0210028	+	0.440902i	0.963735	+	0.266860i	$0.0859860\pi$
$619$	−0.522543	+	10.9695i	−0.0210028	+	0.440902i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−16.3715	+	18.8937i	−0.654861	+	0.755750i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.65145	+	17.2948i	0.0657433	+	0.688494i	0.966926	+	0.255056i	$0.0820939\pi$
$631$	1.65145	+	17.2948i	0.0657433	+	0.688494i	−0.901183	+	0.433439i	$0.857300\pi$
$632$		0			0
$633$	35.1899	−	20.3169i	1.39867	−	0.807524i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.40111	−	8.49555i	0.134757	−	0.336606i
$638$		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
$643$	−39.6763	+	25.4984i	−1.56468	+	1.00556i	−0.583579	+	0.812056i	$0.698348\pi$
$643$	−39.6763	+	25.4984i	−1.56468	+	1.00556i	−0.981100	+	0.193502i	$0.938015\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$647$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−17.4155	−	22.1456i	−0.682568	−	0.867955i
$652$		0			0
$653$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$653$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−24.9132	+	34.9857i	−0.971957	+	1.36492i
$658$		0			0
$659$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$659$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$660$		0			0
$661$	50.7718	+	7.29988i	1.97479	+	0.283932i	0.997352	+	0.0727275i	$0.0231703\pi$
$661$	50.7718	+	7.29988i	1.97479	+	0.283932i	0.977442	+	0.211205i	$0.0677388\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	32.8720i		−	1.27090i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−8.00736	+	6.93842i	−0.308661	+	0.267456i	−0.795392	−	0.606096i	$-0.792735\pi$
$673$	−8.00736	+	6.93842i	−0.308661	+	0.267456i	0.486731	+	0.873552i	$0.338189\pi$
$674$		0			0
$675$	14.0463	−	21.8564i	0.540641	−	0.841254i
$676$		0			0
$677$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$677$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$678$		0			0
$679$	−2.71939	+	18.9138i	−0.104361	+	0.725844i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$683$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	37.5774	+	35.8300i	1.43367	+	1.36700i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−2.40713	−	50.5319i	−0.0915716	−	1.92232i	−0.317639	−	0.948212i	$-0.602890\pi$
$691$	−2.40713	−	50.5319i	−0.0915716	−	1.92232i	0.226067	−	0.974112i	$-0.427413\pi$
$692$		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		0.814576	−	0.580057i	$-0.196970\pi$
$701$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$702$		0			0
$703$	25.4417	+	10.1853i	0.959551	+	0.384146i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	50.8276	−	4.85345i	1.90887	−	0.182275i	0.927425	−	0.374009i	$-0.122017\pi$
$709$	50.8276	−	4.85345i	1.90887	−	0.182275i	0.981447	+	0.191733i	$0.0614109\pi$
$710$		0			0
$711$	19.7917	+	4.80140i	0.742245	+	0.180067i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$719$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$720$		0			0
$721$	−13.7699	−	0.655943i	−0.512819	−	0.0244286i
$722$		0			0
$723$	−14.9784	+	51.0117i	−0.557052	+	1.89715i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−6.70129	+	34.7696i	−0.248537	+	1.28953i	0.616313	+	0.787501i	$0.288626\pi$
$727$	−6.70129	+	34.7696i	−0.248537	+	1.28953i	−0.864850	+	0.502031i	$0.832586\pi$
$728$		0			0
$729$	−11.2162	+	24.5601i	−0.415415	+	0.909632i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	32.1115	+	33.6776i	1.18607	+	1.24391i	0.963118	+	0.269080i	$0.0867197\pi$
$733$	32.1115	+	33.6776i	1.18607	+	1.24391i	0.222948	+	0.974830i	$0.428432\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−10.3320	−	25.8081i	−0.380068	−	0.949365i	−0.987871	−	0.155277i	$-0.950373\pi$
$739$	−10.3320	−	25.8081i	−0.380068	−	0.949365i	0.607803	−	0.794088i	$-0.292051\pi$
$740$		0			0
$741$	−13.8303	+	13.1872i	−0.508069	+	0.484442i
$742$		0			0
$743$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$743$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−52.5547	−	15.4314i	−1.91775	−	0.563101i	−0.968994	−	0.247084i	$-0.920528\pi$
$751$	−52.5547	−	15.4314i	−1.91775	−	0.563101i	−0.948753	−	0.316017i	$-0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	10.4126	+	54.0258i	0.378453	+	1.96360i	0.205053	+	0.978751i	$0.434263\pi$
$757$	10.4126	+	54.0258i	0.378453	+	1.96360i	0.173400	+	0.984851i	$0.444525\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$761$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$762$		0			0
$763$	1.54690	−	6.37639i	0.0560014	−	0.230841i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−51.9604	+	17.9837i	−1.87374	+	0.648508i	−0.894060	+	0.447947i	$0.852155\pi$
$769$	−51.9604	+	17.9837i	−1.87374	+	0.648508i	−0.979681	+	0.200561i	$0.935723\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$773$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$774$		0			0
$775$	−32.0609	−	18.5104i	−1.15166	−	0.664912i
$776$		0			0
$777$	−33.4707	+	21.5103i	−1.20075	+	0.771678i
$778$		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
$787$	−36.1453	+	37.9081i	−1.28844	+	1.35128i	−0.384789	+	0.923004i	$0.625726\pi$
$787$	−36.1453	+	37.9081i	−1.28844	+	1.35128i	−0.903650	+	0.428272i	$0.859123\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−35.3681	+	18.2335i	−1.25596	+	0.647491i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$797$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$798$		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		0.540641	−	0.841254i	$-0.318182\pi$
$809$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$810$		0			0
$811$	24.4718	−	31.1184i	0.859320	−	1.09271i	−0.135472	−	0.990781i	$-0.543255\pi$
$811$	24.4718	−	31.1184i	0.859320	−	1.09271i	0.994791	−	0.101932i	$-0.0325026\pi$
$812$		0			0
$813$	−6.52879	+	45.4088i	−0.228975	+	1.59255i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	23.7097	+	16.8836i	0.829498	+	0.590683i
$818$		0			0
$819$	−3.94834	−	27.4613i	−0.137966	−	0.959576i
$820$		0			0
$821$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$821$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$822$		0			0
$823$	−30.8626	+	24.2707i	−1.07580	+	0.846022i	−0.988660	−	0.150174i	$-0.952017\pi$
$823$	−30.8626	+	24.2707i	−1.07580	+	0.846022i	−0.0871445	+	0.996196i	$0.527774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$827$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$828$		0			0
$829$	3.89939	+	8.53847i	0.135431	+	0.296553i	0.965181	−	0.261582i	$-0.0842442\pi$
$829$	3.89939	+	8.53847i	0.135431	+	0.296553i	−0.829750	+	0.558135i	$0.811517\pi$
$830$		0			0
$831$	31.0517	+	48.3174i	1.07717	+	1.67611i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	35.7172	+	14.2990i	1.23457	+	0.494245i
$838$		0			0
$839$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$839$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$840$		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
$847$	−6.80817	−	23.1865i	−0.233931	−	0.796697i
$848$		0			0
$849$	27.8837	+	24.1614i	0.956967	+	0.829217i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	13.4568	+	55.4698i	0.460753	+	1.89925i	0.436844	+	0.899537i	$0.356096\pi$
$853$	13.4568	+	55.4698i	0.460753	+	1.89925i	0.0239089	+	0.999714i	$0.492389\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$857$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$858$		0			0
$859$	57.1330	+	5.45554i	1.94935	+	0.186141i	0.994004	−	0.109344i	$-0.0348749\pi$
$859$	57.1330	+	5.45554i	1.94935	+	0.186141i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$863$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−20.3193	−	21.3103i	−0.690079	−	0.723734i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	20.6766	+	27.5641i	0.700599	+	0.933973i
$872$		0			0
$873$	−9.69819	−	24.2249i	−0.328234	−	0.819889i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−2.89166	−	1.49075i	−0.0976444	−	0.0503392i	0.408713	−	0.912663i	$-0.365978\pi$
$877$	−2.89166	−	1.49075i	−0.0976444	−	0.0503392i	−0.506357	+	0.862324i	$0.669008\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$881$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$882$		0			0
$883$	−4.24050	+	44.4086i	−0.142704	+	1.49447i	0.586376	+	0.810039i	$0.300554\pi$
$883$	−4.24050	+	44.4086i	−0.142704	+	1.49447i	−0.729080	+	0.684428i	$0.760052\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$887$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$888$		0			0
$889$	40.3962	−	9.80001i	1.35484	−	0.328682i
$890$		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
$903$	−39.9337	+	13.8212i	−1.32891	+	0.459940i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−33.6416	−	47.2430i	−1.11705	−	1.56868i	−0.778405	−	0.627762i	$-0.783971\pi$
$907$	−33.6416	−	47.2430i	−1.11705	−	1.56868i	−0.338644	−	0.940914i	$-0.609968\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$911$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.16562	−	1.48221i	−0.0384503	−	0.0488935i	0.766448	−	0.642307i	$-0.222022\pi$
$919$	−1.16562	−	1.48221i	−0.0384503	−	0.0488935i	−0.804898	+	0.593413i	$0.797780\pi$
$920$		0			0
$921$	31.7052	−	33.2514i	1.04472	−	1.09567i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−30.3261	+	42.5871i	−0.997118	+	1.40026i
$926$		0			0
$927$	16.7327	−	8.62631i	0.549574	−	0.283325i
$928$		0			0
$929$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$929$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$930$		0			0
$931$	−4.47847	−	3.52191i	−0.146776	−	0.115426i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	38.3342i		−	1.25232i	−0.779693	−	0.626162i	$-0.784625\pi$
$937$		−	38.3342i		−	1.25232i	0.779693	−	0.626162i	$-0.215375\pi$
$938$		0			0
$939$	−30.8862			−1.00793
$940$		0			0
$941$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$941$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$947$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$948$		0			0
$949$	27.6160	+	53.5676i	0.896453	+	1.73888i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$953$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.79119	−	22.5112i	0.251329	−	0.726167i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	24.5547	−	42.5301i	0.789627	−	1.36767i	−0.136568	−	0.990631i	$-0.543607\pi$
$967$	24.5547	−	42.5301i	0.789627	−	1.36767i	0.926195	−	0.377044i	$-0.123059\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$971$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$972$		0			0
$973$	−14.9344	−	43.1502i	−0.478776	−	1.38333i
$974$		0			0
$975$	−18.2282	−	31.5721i	−0.583769	−	1.01112i
$976$		0			0
$977$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$977$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	2.52436	+	8.59717i	0.0805965	+	0.274487i
$982$		0			0
$983$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$983$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	17.3752	−	59.1745i	0.551941	−	1.87974i	0.0829100	−	0.996557i	$-0.473579\pi$
$991$	17.3752	−	59.1745i	0.551941	−	1.87974i	0.469031	−	0.883182i	$-0.344603\pi$
$992$		0			0
$993$	−18.8826	−	1.80307i	−0.599221	−	0.0572187i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	16.8556	−	36.9086i	0.533821	−	1.16891i	−0.430115	−	0.902774i	$-0.641527\pi$
$997$	16.8556	−	36.9086i	0.533821	−	1.16891i	0.963937	−	0.266132i	$-0.0857457\pi$
$998$		0			0
$999$	24.8965	−	48.2925i	0.787691	−	1.52791i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.ba.a.497.1	✓	20
3.2	odd	2	CM	804.2.ba.a.497.1	✓	20
67.12	odd	66	inner	804.2.ba.a.749.1	yes	20
201.146	even	66	inner	804.2.ba.a.749.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.ba.a.497.1	✓	20	1.1	even	1	trivial
804.2.ba.a.497.1	✓	20	3.2	odd	2	CM
804.2.ba.a.749.1	yes	20	67.12	odd	66	inner
804.2.ba.a.749.1	yes	20	201.146	even	66	inner

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)

\(f(q)\)	\(=\)	\(q+(-1.30900 + 1.13425i) q^{3} +(-1.35800 + 1.72684i) q^{7} +(0.426945 - 2.96946i) q^{9} +O(q^{10})\)	\(q+(-1.30900 + 1.13425i) q^{3} +(-1.35800 + 1.72684i) q^{7} +(0.426945 - 2.96946i) q^{9} +(-3.42905 - 2.44182i) q^{13} +(-2.06016 + 1.62013i) q^{19} +(-0.181052 - 3.80075i) q^{21} +(-2.07708 - 4.54816i) q^{25} +(2.80925 + 4.37128i) q^{27} +(6.03124 - 4.29482i) q^{31} +(-5.22813 - 9.05539i) q^{37} +(7.25826 - 0.693079i) q^{39} +(-3.12884 - 10.6558i) q^{43} +(0.512504 + 2.11257i) q^{49} +(0.859110 - 4.45748i) q^{57} +(4.33139 - 8.40173i) q^{61} +(4.54800 + 4.76981i) q^{63} +(-7.79913 - 2.48466i) q^{67} +(-12.7250 - 6.56021i) q^{73} +(7.87764 + 3.59760i) q^{75} +(-0.645295 + 6.75784i) q^{79} +(-8.63544 - 2.53559i) q^{81} +(8.87330 - 2.60544i) q^{91} +(-3.02346 + 12.4629i) q^{93} +(7.53271 - 4.34901i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 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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