LMFDB - Embedded newform 2664.2.r.n.433.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2664,2,Mod(433,2664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2664, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("2664.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2664 = 2^{3} \cdot 3^{2} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2664.r (of order $3$ , degree $2$ , 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:	$21.2721470985$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - x^{9} + 9x^{8} + 4x^{7} + 54x^{6} + 6x^{5} + 98x^{4} - 8x^{3} + 148x^{2} - 24x + 4$ x^10 - x^9 + 9x^8 + 4x^7 + 54x^6 + 6x^5 + 98x^4 - 8x^3 + 148x^2 - 24x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	no (minimal twist has level 296)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.5
Root			$-0.917643 + 1.58940i$ of defining polynomial
Character	$\chi$	$=$	2664.433
Dual form			2664.2.r.n.1009.5

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73584 - 3.00655i) q^{5} +(1.47954 - 2.56263i) q^{7} -5.67057 q^{11} +(3.28255 - 5.68554i) q^{13} +(-0.500000 - 0.866025i) q^{17} +(-1.44726 + 2.50673i) q^{19} +5.85359 q^{23} +(-3.52625 - 6.10764i) q^{25} +1.19523 q^{29} -5.67057 q^{31} +(-5.13646 - 8.89661i) q^{35} +(-4.18841 + 4.41103i) q^{37} +(5.49792 - 9.52267i) q^{41} +4.47167 q^{43} -7.40187 q^{47} +(-0.878051 - 1.52083i) q^{49} +(2.65082 + 4.59135i) q^{53} +(-9.84318 + 17.0489i) q^{55} +(1.38803 + 2.40413i) q^{59} +(1.72586 - 2.98928i) q^{61} +(-11.3959 - 19.7383i) q^{65} +(3.02441 - 5.23843i) q^{67} +(0.585015 - 1.01328i) q^{71} -15.3611 q^{73} +(-8.38981 + 14.5316i) q^{77} +(-2.66611 + 4.61784i) q^{79} +(5.94287 + 10.2934i) q^{83} -3.47167 q^{85} +(3.49792 + 6.05857i) q^{89} +(-9.71329 - 16.8239i) q^{91} +(5.02441 + 8.70253i) q^{95} -11.0655 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 3 q^{5} + 3 q^{7} - 16 q^{11} + q^{13} - 5 q^{17} - 3 q^{19} + 12 q^{23} - 12 q^{25} - 16 q^{31} - 5 q^{35} - 12 q^{37} - 9 q^{41} + 4 q^{43} - 32 q^{47} - 18 q^{49} - 5 q^{53} + 4 q^{55} + 5 q^{59}+ \cdots - 36 q^{97}+O(q^{100})$ 10 * q - 3 * q^5 + 3 * q^7 - 16 * q^11 + q^13 - 5 * q^17 - 3 * q^19 + 12 * q^23 - 12 * q^25 - 16 * q^31 - 5 * q^35 - 12 * q^37 - 9 * q^41 + 4 * q^43 - 32 * q^47 - 18 * q^49 - 5 * q^53 + 4 * q^55 + 5 * q^59 + 15 * q^61 - 23 * q^65 + q^67 + 17 * q^71 - 36 * q^73 + 4 * q^77 + 21 * q^79 + 15 * q^83 + 6 * q^85 - 29 * q^89 - q^91 + 21 * q^95 - 36 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times$ .

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.73584	−	3.00655i	0.776289	−	1.34457i	−0.157778	−	0.987475i	$-0.550433\pi$
$5$	1.73584	−	3.00655i	0.776289	−	1.34457i	0.934067	−	0.357097i	$-0.116234\pi$
$6$		0			0
$7$	1.47954	−	2.56263i	0.559212	−	0.968583i	−0.438351	−	0.898804i	$-0.644437\pi$
$7$	1.47954	−	2.56263i	0.559212	−	0.968583i	0.997562	−	0.0697793i	$-0.0222295\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.67057			−1.70974			−0.854871	−	0.518841i	$-0.826364\pi$
$11$	−5.67057			−1.70974			−0.854871	+	0.518841i	$0.826364\pi$
$12$		0			0
$13$	3.28255	−	5.68554i	0.910414	−	1.57688i	0.0969348	−	0.995291i	$-0.469096\pi$
$13$	3.28255	−	5.68554i	0.910414	−	1.57688i	0.813480	−	0.581593i	$-0.197570\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.500000	−	0.866025i	−0.121268	−	0.210042i	0.799000	−	0.601331i	$-0.205363\pi$
$17$	−0.500000	−	0.866025i	−0.121268	−	0.210042i	−0.920268	+	0.391289i	$0.872029\pi$
$18$		0			0
$19$	−1.44726	+	2.50673i	−0.332024	+	0.575083i	−0.982909	−	0.184094i	$-0.941065\pi$
$19$	−1.44726	+	2.50673i	−0.332024	+	0.575083i	0.650884	+	0.759177i	$0.274398\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.85359			1.22056			0.610279	−	0.792186i	$-0.291057\pi$
$23$	5.85359			1.22056			0.610279	+	0.792186i	$0.291057\pi$
$24$		0			0
$25$	−3.52625	−	6.10764i	−0.705249	−	1.22153i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.19523			0.221948			0.110974	−	0.993823i	$-0.464603\pi$
$29$	1.19523			0.221948			0.110974	+	0.993823i	$0.464603\pi$
$30$		0			0
$31$	−5.67057			−1.01846			−0.509232	−	0.860629i	$-0.670071\pi$
$31$	−5.67057			−1.01846			−0.509232	+	0.860629i	$0.670071\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.13646	−	8.89661i	−0.868220	−	1.50380i
$36$		0			0
$37$	−4.18841	+	4.41103i	−0.688571	+	0.725169i
$37$	−4.18841	+	4.41103i	−0.688571	+	0.725169i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.49792	−	9.52267i	0.858630	−	1.48719i	−0.0146054	−	0.999893i	$-0.504649\pi$
$41$	5.49792	−	9.52267i	0.858630	−	1.48719i	0.873236	−	0.487298i	$-0.162017\pi$
$42$		0			0
$43$	4.47167			0.681923			0.340962	−	0.940077i	$-0.389247\pi$
$43$	4.47167			0.681923			0.340962	+	0.940077i	$0.389247\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.40187			−1.07967			−0.539837	−	0.841770i	$-0.681514\pi$
$47$	−7.40187			−1.07967			−0.539837	+	0.841770i	$0.681514\pi$
$48$		0			0
$49$	−0.878051	−	1.52083i	−0.125436	−	0.217261i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.65082	+	4.59135i	0.364118	+	0.630671i	0.988634	−	0.150341i	$-0.0480371\pi$
$53$	2.65082	+	4.59135i	0.364118	+	0.630671i	−0.624516	+	0.781012i	$0.714704\pi$
$54$		0			0
$55$	−9.84318	+	17.0489i	−1.32725	+	2.29887i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.38803	+	2.40413i	0.180706	+	0.312991i	0.942121	−	0.335273i	$-0.108829\pi$
$59$	1.38803	+	2.40413i	0.180706	+	0.312991i	−0.761415	+	0.648264i	$0.775495\pi$
$60$		0			0
$61$	1.72586	−	2.98928i	0.220974	−	0.382738i	−0.734130	−	0.679009i	$-0.762410\pi$
$61$	1.72586	−	2.98928i	0.220974	−	0.382738i	0.955104	+	0.296271i	$0.0957431\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−11.3959	−	19.7383i	−1.41349	−	2.44824i
$66$		0			0
$67$	3.02441	−	5.23843i	0.369490	−	0.639976i	−0.619995	−	0.784605i	$-0.712866\pi$
$67$	3.02441	−	5.23843i	0.369490	−	0.639976i	0.989486	+	0.144629i	$0.0461989\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.585015	−	1.01328i	0.0694285	−	0.120254i	−0.829221	−	0.558920i	$-0.811216\pi$
$71$	0.585015	−	1.01328i	0.0694285	−	0.120254i	0.898650	+	0.438667i	$0.144549\pi$
$72$		0			0
$73$	−15.3611			−1.79788			−0.898940	−	0.438072i	$-0.855662\pi$
$73$	−15.3611			−1.79788			−0.898940	+	0.438072i	$0.855662\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.38981	+	14.5316i	−0.956108	+	1.65603i
$78$		0			0
$79$	−2.66611	+	4.61784i	−0.299961	+	0.519547i	−0.976127	−	0.217202i	$-0.930307\pi$
$79$	−2.66611	+	4.61784i	−0.299961	+	0.519547i	0.676166	+	0.736749i	$0.263640\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.94287	+	10.2934i	0.652315	+	1.12984i	0.982560	+	0.185948i	$0.0595355\pi$
$83$	5.94287	+	10.2934i	0.652315	+	1.12984i	−0.330244	+	0.943895i	$0.607131\pi$
$84$		0			0
$85$	−3.47167			−0.376555
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.49792	+	6.05857i	0.370778	+	0.642207i	0.989685	−	0.143257i	$-0.0457576\pi$
$89$	3.49792	+	6.05857i	0.370778	+	0.642207i	−0.618907	+	0.785464i	$0.712424\pi$
$90$		0			0
$91$	−9.71329	−	16.8239i	−1.01823	−	1.76362i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.02441	+	8.70253i	0.515494	+	0.892861i
$96$		0			0
$97$	−11.0655			−1.12353			−0.561765	−	0.827297i	$-0.689877\pi$
$97$	−11.0655			−1.12353			−0.561765	+	0.827297i	$0.689877\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.1186			1.20585			0.602924	−	0.797799i	$-0.294002\pi$
$101$	12.1186			1.20585			0.602924	+	0.797799i	$0.294002\pi$
$102$		0			0
$103$	−3.94608			−0.388819			−0.194409	−	0.980920i	$-0.562279\pi$
$103$	−3.94608			−0.388819			−0.194409	+	0.980920i	$0.562279\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.847809	−	1.46845i	0.0819608	−	0.141960i	−0.822131	−	0.569298i	$-0.807215\pi$
$107$	0.847809	−	1.46845i	0.0819608	−	0.141960i	0.904092	+	0.427338i	$0.140548\pi$
$108$		0			0
$109$	7.06318	+	12.2338i	0.676530	+	1.17178i	0.976019	+	0.217685i	$0.0698504\pi$
$109$	7.06318	+	12.2338i	0.676530	+	1.17178i	−0.299489	+	0.954100i	$0.596816\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.33712	−	4.04801i	−0.219858	−	0.380805i	0.734906	−	0.678169i	$-0.237226\pi$
$113$	−2.33712	−	4.04801i	−0.219858	−	0.380805i	−0.954764	+	0.297363i	$0.903893\pi$
$114$		0			0
$115$	10.1609	−	17.5991i	0.947506	−	1.64113i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.95907			−0.271258
$120$		0			0
$121$	21.1554			1.92322
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.12558			−0.637331
$126$		0			0
$127$	1.20934	+	2.09464i	0.107312	+	0.185869i	0.914680	−	0.404178i	$-0.132442\pi$
$127$	1.20934	+	2.09464i	0.107312	+	0.185869i	−0.807369	+	0.590047i	$0.799109\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.00047	−	8.66106i	−0.436893	−	0.756720i	0.560555	−	0.828117i	$-0.310588\pi$
$131$	−5.00047	−	8.66106i	−0.436893	−	0.756720i	−0.997448	+	0.0713966i	$0.977254\pi$
$132$		0			0
$133$	4.28255	+	7.41759i	0.371344	+	0.643186i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.29561			−0.196127			−0.0980636	−	0.995180i	$-0.531265\pi$
$137$	−2.29561			−0.196127			−0.0980636	+	0.995180i	$0.531265\pi$
$138$		0			0
$139$	−0.686769	−	1.18952i	−0.0582509	−	0.100894i	0.835429	−	0.549598i	$-0.185219\pi$
$139$	−0.686769	−	1.18952i	−0.0582509	−	0.100894i	−0.893680	+	0.448704i	$0.851886\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−18.6139	+	32.2402i	−1.55657	+	2.69606i
$144$		0			0
$145$	2.07472	−	3.59352i	0.172296	−	0.298425i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.22043			−0.509597			−0.254799	−	0.966994i	$-0.582009\pi$
$149$	−6.22043			−0.509597			−0.254799	+	0.966994i	$0.582009\pi$
$150$		0			0
$151$	−0.221400	+	0.383475i	−0.0180172	+	0.0312068i	−0.874893	−	0.484315i	$-0.839069\pi$
$151$	−0.221400	+	0.383475i	−0.0180172	+	0.0312068i	0.856876	+	0.515522i	$0.172402\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.84318	+	17.0489i	−0.790623	+	1.36940i
$156$		0			0
$157$	4.19753	+	7.27034i	0.334999	+	0.580236i	0.983485	−	0.180991i	$-0.0579305\pi$
$157$	4.19753	+	7.27034i	0.334999	+	0.580236i	−0.648485	+	0.761227i	$0.724597\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.66060	−	15.0006i	0.682551	−	1.18221i
$162$		0			0
$163$	−4.77816	−	8.27602i	−0.374254	−	0.648228i	0.615961	−	0.787777i	$-0.288768\pi$
$163$	−4.77816	−	8.27602i	−0.374254	−	0.648228i	−0.990215	+	0.139549i	$0.955435\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.54742	−	4.41227i	0.197126	−	0.341431i	−0.750470	−	0.660905i	$-0.770173\pi$
$167$	2.54742	−	4.41227i	0.197126	−	0.341431i	0.947595	+	0.319473i	$0.103506\pi$
$168$		0			0
$169$	−15.0502	−	26.0677i	−1.15771	−	2.00521i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.64400	−	9.77570i	−0.429106	−	0.743233i	0.567688	−	0.823244i	$-0.307838\pi$
$173$	−5.64400	−	9.77570i	−0.429106	−	0.743233i	−0.996794	+	0.0800108i	$0.974505\pi$
$174$		0			0
$175$	−20.8688			−1.57754
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.89294			0.216229			0.108114	−	0.994138i	$-0.465519\pi$
$179$	2.89294			0.216229			0.108114	+	0.994138i	$0.465519\pi$
$180$		0			0
$181$	4.91886	−	8.51971i	0.365616	−	0.633265i	−0.623259	−	0.782015i	$-0.714192\pi$
$181$	4.91886	−	8.51971i	0.365616	−	0.633265i	0.988875	+	0.148751i	$0.0475251\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.99162	+	20.2495i	0.440513	+	1.48877i
$186$		0			0
$187$	2.83529	+	4.91086i	0.207337	+	0.359118i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.03309			0.291824			0.145912	−	0.989298i	$-0.453388\pi$
$191$	4.03309			0.291824			0.145912	+	0.989298i	$0.453388\pi$
$192$		0			0
$193$	−1.54196			−0.110993			−0.0554965	−	0.998459i	$-0.517674\pi$
$193$	−1.54196			−0.110993			−0.0554965	+	0.998459i	$0.517674\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.5135	−	18.2099i	−0.749054	−	1.29740i	−0.948277	−	0.317446i	$-0.897175\pi$
$197$	−10.5135	−	18.2099i	−0.749054	−	1.29740i	0.199222	−	0.979954i	$-0.436158\pi$
$198$		0			0
$199$	2.94686			0.208898			0.104449	−	0.994530i	$-0.466692\pi$
$199$	2.94686			0.208898			0.104449	+	0.994530i	$0.466692\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.76838	−	3.06293i	0.124116	−	0.214975i
$204$		0			0
$205$	−19.0870	−	33.0596i	−1.33309	−	2.30898i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.20679	−	14.2146i	0.567676	−	0.983243i
$210$		0			0
$211$	13.8286			0.952000			0.476000	−	0.879445i	$-0.342086\pi$
$211$	13.8286			0.952000			0.476000	+	0.879445i	$0.342086\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.76208	−	13.4443i	0.529370	−	0.916895i
$216$		0			0
$217$	−8.38981	+	14.5316i	−0.569538	+	0.986468i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.56509			−0.441616
$222$		0			0
$223$	−1.03935			−0.0696002			−0.0348001	−	0.999394i	$-0.511079\pi$
$223$	−1.03935			−0.0696002			−0.0348001	+	0.999394i	$0.511079\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.18748	+	5.52088i	−0.211561	+	0.366434i	−0.952203	−	0.305466i	$-0.901188\pi$
$227$	−3.18748	+	5.52088i	−0.211561	+	0.366434i	0.740643	+	0.671899i	$0.234521\pi$
$228$		0			0
$229$	8.48618	−	14.6985i	0.560783	−	0.971304i	−0.436646	−	0.899634i	$-0.643834\pi$
$229$	8.48618	−	14.6985i	0.560783	−	0.971304i	0.997428	−	0.0716704i	$-0.0228330\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.01221			0.0663120			0.0331560	−	0.999450i	$-0.489444\pi$
$233$	1.01221			0.0663120			0.0331560	+	0.999450i	$0.489444\pi$
$234$		0			0
$235$	−12.8484	+	22.2541i	−0.838139	+	1.45170i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.04463	−	13.9337i	−0.520364	−	0.901296i	−0.999720	−	0.0236758i	$-0.992463\pi$
$239$	−8.04463	−	13.9337i	−0.520364	−	0.901296i	0.479356	−	0.877621i	$-0.340870\pi$
$240$		0			0
$241$	−2.08063	+	3.60376i	−0.134025	+	0.232138i	−0.925225	−	0.379420i	$-0.876124\pi$
$241$	−2.08063	+	3.60376i	−0.134025	+	0.232138i	0.791200	+	0.611558i	$0.209457\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.09661			−0.389498
$246$		0			0
$247$	9.50140	+	16.4569i	0.604559	+	1.04713i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.560874			−0.0354021			−0.0177010	−	0.999843i	$-0.505635\pi$
$251$	−0.560874			−0.0354021			−0.0177010	+	0.999843i	$0.505635\pi$
$252$		0			0
$253$	−33.1932			−2.08684
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.12805	−	10.6141i	−0.382257	−	0.662089i	0.609127	−	0.793073i	$-0.291520\pi$
$257$	−6.12805	−	10.6141i	−0.382257	−	0.662089i	−0.991385	+	0.130983i	$0.958187\pi$
$258$		0			0
$259$	5.10694	+	17.2596i	0.317330	+	1.07246i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−13.1970	+	22.8579i	−0.813763	+	1.40948i	0.0964501	+	0.995338i	$0.469251\pi$
$263$	−13.1970	+	22.8579i	−0.813763	+	1.40948i	−0.910213	+	0.414141i	$0.864082\pi$
$264$		0			0
$265$	18.4055			1.13064
$266$		0			0
$267$		0			0
$268$		0			0
$269$	30.8272			1.87957			0.939783	−	0.341772i	$-0.111027\pi$
$269$	30.8272			1.87957			0.939783	+	0.341772i	$0.111027\pi$
$270$		0			0
$271$	−14.3376	−	24.8335i	−0.870948	−	1.50853i	−0.861017	−	0.508576i	$-0.830172\pi$
$271$	−14.3376	−	24.8335i	−0.870948	−	1.50853i	−0.00993062	−	0.999951i	$-0.503161\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	19.9958	+	34.6338i	1.20579	+	2.08850i
$276$		0			0
$277$	9.22745	−	15.9824i	0.554424	−	0.960290i	−0.443524	−	0.896262i	$-0.646272\pi$
$277$	9.22745	−	15.9824i	0.554424	−	0.960290i	0.997948	−	0.0640281i	$-0.0203947\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.12597	−	10.6105i	−0.365445	−	0.632969i	0.623403	−	0.781901i	$-0.285750\pi$
$281$	−6.12597	−	10.6105i	−0.365445	−	0.632969i	−0.988847	+	0.148932i	$0.952416\pi$
$282$		0			0
$283$	16.5026	−	28.5834i	0.980981	−	1.69911i	0.322390	−	0.946607i	$-0.395514\pi$
$283$	16.5026	−	28.5834i	0.980981	−	1.69911i	0.658590	−	0.752502i	$-0.271153\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−16.2687	−	28.1783i	−0.960313	−	1.66331i
$288$		0			0
$289$	8.00000	−	13.8564i	0.470588	−	0.815083i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.0870	−	24.3993i	0.822969	−	1.42542i	−0.0804934	−	0.996755i	$-0.525650\pi$
$293$	14.0870	−	24.3993i	0.822969	−	1.42542i	0.903462	−	0.428668i	$-0.141017\pi$
$294$		0			0
$295$	9.63753			0.561119
$296$		0			0
$297$		0			0
$298$		0			0
$299$	19.2147	−	33.2808i	1.11121	−	1.92468i
$300$		0			0
$301$	6.61600	−	11.4592i	0.381340	−	0.660500i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.99162	−	10.3778i	−0.343079	−	0.594230i
$306$		0			0
$307$	5.96603			0.340499			0.170250	−	0.985401i	$-0.445543\pi$
$307$	5.96603			0.340499			0.170250	+	0.985401i	$0.445543\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.42562	+	5.93334i	0.194249	+	0.336449i	0.946654	−	0.322252i	$-0.104440\pi$
$311$	3.42562	+	5.93334i	0.194249	+	0.336449i	−0.752405	+	0.658701i	$0.771106\pi$
$312$		0			0
$313$	−0.741420	−	1.28418i	−0.0419075	−	0.0725860i	0.844311	−	0.535854i	$-0.180010\pi$
$313$	−0.741420	−	1.28418i	−0.0419075	−	0.0725860i	−0.886218	+	0.463268i	$0.846677\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.10570	−	8.84333i	−0.286765	−	0.496691i	0.686271	−	0.727346i	$-0.259246\pi$
$317$	−5.10570	−	8.84333i	−0.286765	−	0.496691i	−0.973036	+	0.230655i	$0.925913\pi$
$318$		0			0
$319$	−6.77763			−0.379474
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.89452			0.161055
$324$		0			0
$325$	−46.3003			−2.56828
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−10.9513	+	18.9683i	−0.603766	+	1.04575i
$330$		0			0
$331$	13.2127	+	22.8851i	0.726238	+	1.25788i	0.958462	+	0.285219i	$0.0920664\pi$
$331$	13.2127	+	22.8851i	0.726238	+	1.25788i	−0.232224	+	0.972662i	$0.574600\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.4998	−	18.1861i	−0.573663	−	0.993613i
$336$		0			0
$337$	−6.07874	+	10.5287i	−0.331130	+	0.573534i	−0.982734	−	0.185026i	$-0.940763\pi$
$337$	−6.07874	+	10.5287i	−0.331130	+	0.573534i	0.651604	+	0.758560i	$0.274097\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	32.1554			1.74131
$342$		0			0
$343$	15.5171			0.837843
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.9890			−0.858336			−0.429168	−	0.903225i	$-0.641193\pi$
$347$	−15.9890			−0.858336			−0.429168	+	0.903225i	$0.641193\pi$
$348$		0			0
$349$	9.96783	+	17.2648i	0.533565	+	0.924162i	0.999231	+	0.0392018i	$0.0124815\pi$
$349$	9.96783	+	17.2648i	0.533565	+	0.924162i	−0.465666	+	0.884961i	$0.654185\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.08016	+	12.2632i	0.376839	+	0.652705i	0.990600	−	0.136787i	$-0.0436776\pi$
$353$	7.08016	+	12.2632i	0.376839	+	0.652705i	−0.613761	+	0.789492i	$0.710344\pi$
$354$		0			0
$355$	−2.03098	−	3.51776i	−0.107793	−	0.186703i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	31.9371			1.68557			0.842787	−	0.538247i	$-0.180913\pi$
$359$	31.9371			1.68557			0.842787	+	0.538247i	$0.180913\pi$
$360$		0			0
$361$	5.31088	+	9.19871i	0.279520	+	0.484142i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−26.6643	+	46.1840i	−1.39567	+	2.41738i
$366$		0			0
$367$	−8.62511	+	14.9391i	−0.450227	+	0.779817i	−0.998400	−	0.0565487i	$-0.981990\pi$
$367$	−8.62511	+	14.9391i	−0.450227	+	0.779817i	0.548173	+	0.836365i	$0.315324\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	15.6879			0.814477
$372$		0			0
$373$	−10.9713	+	19.0029i	−0.568075	+	0.983935i	0.428681	+	0.903456i	$0.358978\pi$
$373$	−10.9713	+	19.0029i	−0.568075	+	0.983935i	−0.996756	+	0.0804788i	$0.974355\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.92339	−	6.79551i	0.202065	−	0.349987i
$378$		0			0
$379$	−2.85619	−	4.94707i	−0.146713	−	0.254114i	0.783298	−	0.621647i	$-0.213536\pi$
$379$	−2.85619	−	4.94707i	−0.146713	−	0.254114i	−0.930011	+	0.367533i	$0.880203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.80862	+	16.9890i	−0.501197	+	0.868099i	0.498802	+	0.866716i	$0.333774\pi$
$383$	−9.80862	+	16.9890i	−0.501197	+	0.868099i	−0.999999	+	0.00138286i	$0.999560\pi$
$384$		0			0
$385$	29.1267	+	50.4489i	1.48443	+	2.57111i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.67918	+	13.3007i	−0.389350	+	0.674373i	−0.992362	−	0.123358i	$-0.960634\pi$
$389$	−7.67918	+	13.3007i	−0.389350	+	0.674373i	0.603013	+	0.797732i	$0.293967\pi$
$390$		0			0
$391$	−2.92680	−	5.06936i	−0.148014	−	0.256369i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.25586	+	16.0316i	0.465713	+	0.806638i
$396$		0			0
$397$	14.7190			0.738728			0.369364	−	0.929285i	$-0.379576\pi$
$397$	14.7190			0.738728			0.369364	+	0.929285i	$0.379576\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.94829			−0.0972929			−0.0486464	−	0.998816i	$-0.515491\pi$
$401$	−1.94829			−0.0972929			−0.0486464	+	0.998816i	$0.515491\pi$
$402$		0			0
$403$	−18.6139	+	32.2402i	−0.927225	+	1.60600i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	23.7507	−	25.0131i	1.17728	−	1.23985i
$408$		0			0
$409$	−11.1518	−	19.3155i	−0.551422	−	0.955091i	−0.998172	−	0.0604326i	$-0.980752\pi$
$409$	−11.1518	−	19.3155i	−0.551422	−	0.955091i	0.446750	−	0.894659i	$-0.352581\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.21453			0.404211
$414$		0			0
$415$	41.2634			2.02554
$416$		0			0
$417$		0			0
$418$		0			0
$419$	19.5824	+	33.9178i	0.956665	+	1.65699i	0.730512	+	0.682900i	$0.239282\pi$
$419$	19.5824	+	33.9178i	0.956665	+	1.65699i	0.226153	+	0.974092i	$0.427385\pi$
$420$		0			0
$421$	13.6602			0.665759			0.332879	−	0.942969i	$-0.391980\pi$
$421$	13.6602			0.665759			0.332879	+	0.942969i	$0.391980\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.52625	+	6.10764i	−0.171048	+	0.296264i
$426$		0			0
$427$	−5.10694	−	8.84549i	−0.247142	−	0.428063i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3.41932	−	5.92244i	0.164703	−	0.285274i	−0.771847	−	0.635809i	$-0.780667\pi$
$431$	3.41932	−	5.92244i	0.164703	−	0.285274i	0.936550	+	0.350535i	$0.114000\pi$
$432$		0			0
$433$	0.552496			0.0265513			0.0132756	−	0.999912i	$-0.495774\pi$
$433$	0.552496			0.0265513			0.0132756	+	0.999912i	$0.495774\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.47167	+	14.6734i	−0.405255	+	0.701922i
$438$		0			0
$439$	11.2789	−	19.5356i	0.538311	−	0.932382i	−0.460684	−	0.887564i	$-0.652396\pi$
$439$	11.2789	−	19.5356i	0.538311	−	0.932382i	0.998995	−	0.0448181i	$-0.0142708\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	27.5511			1.30899			0.654496	−	0.756065i	$-0.272881\pi$
$443$	27.5511			1.30899			0.654496	+	0.756065i	$0.272881\pi$
$444$		0			0
$445$	24.2872			1.15132
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3.22272	+	5.58192i	−0.152090	+	0.263427i	−0.931995	−	0.362470i	$-0.881934\pi$
$449$	−3.22272	+	5.58192i	−0.152090	+	0.263427i	0.779906	+	0.625897i	$0.215267\pi$
$450$		0			0
$451$	−31.1763	+	53.9990i	−1.46804	+	2.54271i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−67.4427			−3.16176
$456$		0			0
$457$	−3.37090	+	5.83857i	−0.157684	+	0.273117i	−0.934033	−	0.357186i	$-0.883736\pi$
$457$	−3.37090	+	5.83857i	−0.157684	+	0.273117i	0.776349	+	0.630303i	$0.217069\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.51870	−	14.7548i	−0.396756	−	0.687201i	0.596568	−	0.802563i	$-0.296531\pi$
$461$	−8.51870	−	14.7548i	−0.396756	−	0.687201i	−0.993324	+	0.115362i	$0.963197\pi$
$462$		0			0
$463$	−12.6720	+	21.9486i	−0.588919	+	1.02004i	0.405456	+	0.914115i	$0.367113\pi$
$463$	−12.6720	+	21.9486i	−0.588919	+	1.02004i	−0.994374	+	0.105923i	$0.966220\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.0670			−0.882314			−0.441157	−	0.897430i	$-0.645432\pi$
$467$	−19.0670			−0.882314			−0.441157	+	0.897430i	$0.645432\pi$
$468$		0			0
$469$	−8.94944	−	15.5009i	−0.413247	−	0.715765i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−25.3569			−1.16591
$474$		0			0
$475$	20.4136			0.936640
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14.6566	+	25.3860i	0.669678	+	1.15992i	0.977994	+	0.208633i	$0.0669014\pi$
$479$	14.6566	+	25.3860i	0.669678	+	1.15992i	−0.308316	+	0.951284i	$0.599765\pi$
$480$		0			0
$481$	11.3304	+	38.2928i	0.516623	+	1.74600i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−19.2079	+	33.2690i	−0.872184	+	1.51067i
$486$		0			0
$487$	30.4309			1.37896			0.689478	−	0.724307i	$-0.257840\pi$
$487$	30.4309			1.37896			0.689478	+	0.724307i	$0.257840\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.08416			0.0940567			0.0470283	−	0.998894i	$-0.485025\pi$
$491$	2.08416			0.0940567			0.0470283	+	0.998894i	$0.485025\pi$
$492$		0			0
$493$	−0.597614	−	1.03510i	−0.0269152	−	0.0466185i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.73110	−	2.99836i	−0.0776505	−	0.134495i
$498$		0			0
$499$	1.43505	−	2.48558i	0.0642418	−	0.111270i	−0.832116	−	0.554602i	$-0.812870\pi$
$499$	1.43505	−	2.48558i	0.0642418	−	0.111270i	0.896357	+	0.443332i	$0.146204\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.09054	−	5.35297i	−0.137800	−	0.238677i	0.788863	−	0.614569i	$-0.210670\pi$
$503$	−3.09054	−	5.35297i	−0.137800	−	0.238677i	−0.926664	+	0.375892i	$0.877337\pi$
$504$		0			0
$505$	21.0359	−	36.4353i	0.936086	−	1.62135i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.92410	−	10.2608i	−0.262581	−	0.454804i	0.704346	−	0.709857i	$-0.251240\pi$
$509$	−5.92410	−	10.2608i	−0.262581	−	0.454804i	−0.966927	+	0.255053i	$0.917907\pi$
$510$		0			0
$511$	−22.7273	+	39.3648i	−1.00540	+	1.74140i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.84975	+	11.8641i	−0.301836	+	0.522795i
$516$		0			0
$517$	41.9728			1.84596
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.60622	−	11.4423i	0.289424	−	0.501296i	−0.684249	−	0.729249i	$-0.739870\pi$
$521$	6.60622	−	11.4423i	0.289424	−	0.501296i	0.973672	+	0.227952i	$0.0732030\pi$
$522$		0			0
$523$	12.8531	−	22.2623i	0.562028	−	0.973461i	−0.435291	−	0.900290i	$-0.643355\pi$
$523$	12.8531	−	22.2623i	0.562028	−	0.973461i	0.997319	−	0.0731714i	$-0.0233120\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.83529	+	4.91086i	0.123507	+	0.213920i
$528$		0			0
$529$	11.2645			0.489762
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−36.0943	−	62.5172i	−1.56342	−	2.70792i
$534$		0			0
$535$	−2.94331	−	5.09797i	−0.127251	−	0.220404i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.97905	+	8.62397i	0.214463	+	0.371461i
$540$		0			0
$541$	7.42197			0.319095			0.159548	−	0.987190i	$-0.448996\pi$
$541$	7.42197			0.319095			0.159548	+	0.987190i	$0.448996\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	49.0421			2.10073
$546$		0			0
$547$	−29.2749			−1.25171			−0.625853	−	0.779941i	$-0.715249\pi$
$547$	−29.2749			−1.25171			−0.625853	+	0.779941i	$0.715249\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.72981	+	2.99611i	−0.0736922	+	0.127639i
$552$		0			0
$553$	7.88921	+	13.6645i	0.335483	+	0.581074i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.23270	+	15.9915i	0.391202	+	0.677582i	0.992608	−	0.121361i	$-0.0387260\pi$
$557$	9.23270	+	15.9915i	0.391202	+	0.677582i	−0.601406	+	0.798943i	$0.705393\pi$
$558$		0			0
$559$	14.6785	−	25.4238i	0.620833	−	1.07531i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	41.0334			1.72935			0.864676	−	0.502329i	$-0.167524\pi$
$563$	41.0334			1.72935			0.864676	+	0.502329i	$0.167524\pi$
$564$		0			0
$565$	−16.2274			−0.682693
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−37.1894			−1.55906			−0.779530	−	0.626364i	$-0.784542\pi$
$569$	−37.1894			−1.55906			−0.779530	+	0.626364i	$0.784542\pi$
$570$		0			0
$571$	−22.1390	−	38.3459i	−0.926490	−	1.60473i	−0.789148	−	0.614204i	$-0.789477\pi$
$571$	−22.1390	−	38.3459i	−0.926490	−	1.60473i	−0.137342	−	0.990524i	$-0.543856\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−20.6412	−	35.7516i	−0.860798	−	1.49095i
$576$		0			0
$577$	20.9558	+	36.2964i	0.872400	+	1.51104i	0.859507	+	0.511124i	$0.170771\pi$
$577$	20.9558	+	36.2964i	0.872400	+	1.51104i	0.0128925	+	0.999917i	$0.495896\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	35.1708			1.45913
$582$		0			0
$583$	−15.0317	−	26.0356i	−0.622548	−	1.07828i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.82583	+	8.35858i	−0.199183	+	0.344996i	−0.948264	−	0.317483i	$-0.897162\pi$
$587$	−4.82583	+	8.35858i	−0.199183	+	0.344996i	0.749081	+	0.662479i	$0.230496\pi$
$588$		0			0
$589$	8.20679	−	14.2146i	0.338155	−	0.585702i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	35.2684			1.44830			0.724150	−	0.689643i	$-0.242232\pi$
$593$	35.2684			1.44830			0.724150	+	0.689643i	$0.242232\pi$
$594$		0			0
$595$	−5.13646	+	8.89661i	−0.210574	+	0.364725i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.2599	+	21.2348i	−0.500927	+	0.867631i	0.499073	+	0.866560i	$0.333674\pi$
$599$	−12.2599	+	21.2348i	−0.500927	+	0.867631i	−0.999999	+	0.00107059i	$0.999659\pi$
$600$		0			0
$601$	−5.63161	−	9.75423i	−0.229718	−	0.397883i	0.728006	−	0.685570i	$-0.240447\pi$
$601$	−5.63161	−	9.75423i	−0.229718	−	0.397883i	−0.957725	+	0.287687i	$0.907114\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	36.7223	−	63.6048i	1.49297	−	2.58590i
$606$		0			0
$607$	−5.61455	−	9.72468i	−0.227888	−	0.394713i	0.729294	−	0.684200i	$-0.239849\pi$
$607$	−5.61455	−	9.72468i	−0.227888	−	0.394713i	−0.957182	+	0.289487i	$0.906515\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.2970	+	42.0836i	−0.982951	+	1.70252i
$612$		0			0
$613$	18.5965	+	32.2101i	0.751107	+	1.30096i	0.947287	+	0.320388i	$0.103813\pi$
$613$	18.5965	+	32.2101i	0.751107	+	1.30096i	−0.196180	+	0.980568i	$0.562854\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.29094	+	14.3603i	0.333781	+	0.578125i	0.983250	−	0.182263i	$-0.0583421\pi$
$617$	8.29094	+	14.3603i	0.333781	+	0.578125i	−0.649469	+	0.760388i	$0.725009\pi$
$618$		0			0
$619$	9.98466			0.401317			0.200659	−	0.979661i	$-0.435692\pi$
$619$	9.98466			0.401317			0.200659	+	0.979661i	$0.435692\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	20.7012			0.829375
$624$		0			0
$625$	5.26240	−	9.11475i	0.210496	−	0.364590i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.91427	+	1.42175i	0.235817	+	0.0566891i
$630$		0			0
$631$	−0.851042	−	1.47405i	−0.0338795	−	0.0586809i	0.848589	−	0.529053i	$-0.177453\pi$
$631$	−0.851042	−	1.47405i	−0.0338795	−	0.0586809i	−0.882468	+	0.470372i	$0.844120\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.39687			0.333220
$636$		0			0
$637$	−11.5290			−0.456795
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.31984	+	4.01809i	0.0916284	+	0.158705i	0.908196	−	0.418544i	$-0.137459\pi$
$641$	2.31984	+	4.01809i	0.0916284	+	0.158705i	−0.816568	+	0.577249i	$0.804126\pi$
$642$		0			0
$643$	−42.5723			−1.67889			−0.839443	−	0.543448i	$-0.817119\pi$
$643$	−42.5723			−1.67889			−0.839443	+	0.543448i	$0.817119\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.662706	−	1.14784i	0.0260537	−	0.0451263i	−0.852705	−	0.522393i	$-0.825039\pi$
$647$	0.662706	−	1.14784i	0.0260537	−	0.0451263i	0.878758	+	0.477267i	$0.158373\pi$
$648$		0			0
$649$	−7.87090	−	13.6328i	−0.308960	−	0.535134i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−21.9139	+	37.9560i	−0.857557	+	1.48533i	0.0166945	+	0.999861i	$0.494686\pi$
$653$	−21.9139	+	37.9560i	−0.857557	+	1.48533i	−0.874252	+	0.485472i	$0.838648\pi$
$654$		0			0
$655$	−34.7199			−1.35662
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11.2811	−	19.5395i	0.439450	−	0.761150i	−0.558197	−	0.829708i	$-0.688507\pi$
$659$	11.2811	−	19.5395i	0.439450	−	0.761150i	0.997647	+	0.0685586i	$0.0218400\pi$
$660$		0			0
$661$	−8.36705	+	14.4922i	−0.325441	+	0.563680i	−0.981601	−	0.190941i	$-0.938846\pi$
$661$	−8.36705	+	14.4922i	−0.325441	+	0.563680i	0.656161	+	0.754621i	$0.272179\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	29.7352			1.15308
$666$		0			0
$667$	6.99638			0.270901
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.78662	+	16.9509i	−0.377808	+	0.654383i
$672$		0			0
$673$	7.51137	−	13.0101i	0.289542	−	0.501502i	−0.684158	−	0.729333i	$-0.739830\pi$
$673$	7.51137	−	13.0101i	0.289542	−	0.501502i	0.973700	+	0.227832i	$0.0731637\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	39.6180			1.52264			0.761322	−	0.648374i	$-0.224551\pi$
$677$	39.6180			1.52264			0.761322	+	0.648374i	$0.224551\pi$
$678$		0			0
$679$	−16.3718	+	28.3567i	−0.628291	+	1.08823i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19.4817	−	33.7432i	−0.745445	−	1.29115i	−0.949986	−	0.312291i	$-0.898904\pi$
$683$	−19.4817	−	33.7432i	−0.745445	−	1.29115i	0.204541	−	0.978858i	$-0.434430\pi$
$684$		0			0
$685$	−3.98480	+	6.90187i	−0.152251	+	0.263707i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	34.8058			1.32599
$690$		0			0
$691$	18.3572	+	31.7955i	0.698340	+	1.20956i	0.969042	+	0.246897i	$0.0794108\pi$
$691$	18.3572	+	31.7955i	0.698340	+	1.20956i	−0.270702	+	0.962663i	$0.587256\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.76847			−0.180878
$696$		0			0
$697$	−10.9958			−0.416497
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10.4484	+	18.0972i	0.394631	+	0.683521i	0.993054	−	0.117659i	$-0.0375391\pi$
$701$	10.4484	+	18.0972i	0.394631	+	0.683521i	−0.598423	+	0.801180i	$0.704206\pi$
$702$		0			0
$703$	−4.99554	−	16.8831i	−0.188410	−	0.636759i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	17.9299	−	31.0555i	0.674324	−	1.16796i
$708$		0			0
$709$	25.0407			0.940422			0.470211	−	0.882554i	$-0.344178\pi$
$709$	25.0407			0.940422			0.470211	+	0.882554i	$0.344178\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−33.1932			−1.24310
$714$		0			0
$715$	64.6214	+	111.927i	2.41670	+	4.18585i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10.6596	−	18.4630i	−0.397537	−	0.688555i	0.595884	−	0.803070i	$-0.296802\pi$
$719$	−10.6596	−	18.4630i	−0.397537	−	0.688555i	−0.993421	+	0.114515i	$0.963468\pi$
$720$		0			0
$721$	−5.83837	+	10.1124i	−0.217432	+	0.376604i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.21467	−	7.30002i	−0.156529	−	0.271116i
$726$		0			0
$727$	25.7943	−	44.6770i	0.956656	−	1.65698i	0.226125	−	0.974098i	$-0.427394\pi$
$727$	25.7943	−	44.6770i	0.956656	−	1.65698i	0.730532	−	0.682879i	$-0.239272\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.23584	−	3.87258i	−0.0826953	−	0.143233i
$732$		0			0
$733$	20.9698	−	36.3207i	0.774537	−	1.34154i	−0.160518	−	0.987033i	$-0.551317\pi$
$733$	20.9698	−	36.3207i	0.774537	−	1.34154i	0.935055	−	0.354504i	$-0.115350\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−17.1501	+	29.7049i	−0.631733	+	1.09419i
$738$		0			0
$739$	8.72502			0.320955			0.160478	−	0.987039i	$-0.448697\pi$
$739$	8.72502			0.320955			0.160478	+	0.987039i	$0.448697\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.37379	+	5.84357i	−0.123772	+	0.214380i	−0.921252	−	0.388965i	$-0.872833\pi$
$743$	−3.37379	+	5.84357i	−0.123772	+	0.214380i	0.797480	+	0.603345i	$0.206166\pi$
$744$		0			0
$745$	−10.7976	+	18.7020i	−0.395595	+	0.685190i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.50873	−	4.34525i	−0.0916669	−	0.158772i
$750$		0			0
$751$	−28.7334			−1.04849			−0.524247	−	0.851566i	$-0.675653\pi$
$751$	−28.7334			−1.04849			−0.524247	+	0.851566i	$0.675653\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.768626	+	1.33130i	0.0279732	+	0.0484509i
$756$		0			0
$757$	−3.98132	−	6.89585i	−0.144704	−	0.250634i	0.784559	−	0.620055i	$-0.212890\pi$
$757$	−3.98132	−	6.89585i	−0.144704	−	0.250634i	−0.929262	+	0.369421i	$0.879556\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7.62636	−	13.2092i	−0.276455	−	0.478835i	0.694046	−	0.719931i	$-0.255826\pi$
$761$	−7.62636	−	13.2092i	−0.276455	−	0.478835i	−0.970501	+	0.241096i	$0.922493\pi$
$762$		0			0
$763$	41.8009			1.51329
$764$		0			0
$765$		0			0
$766$		0			0
$767$	18.2250			0.658068
$768$		0			0
$769$	28.1883			1.01649			0.508247	−	0.861211i	$-0.330294\pi$
$769$	28.1883			1.01649			0.508247	+	0.861211i	$0.330294\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.364769	+	0.631798i	−0.0131198	+	0.0227242i	−0.872511	−	0.488595i	$-0.837510\pi$
$773$	−0.364769	+	0.631798i	−0.0131198	+	0.0227242i	0.859391	+	0.511319i	$0.170843\pi$
$774$		0			0
$775$	19.9958	+	34.6338i	0.718272	+	1.24408i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	15.9138	+	27.5636i	0.570172	+	0.987567i
$780$		0			0
$781$	−3.31737	+	5.74585i	−0.118705	+	0.205603i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	29.1449			1.04023
$786$		0			0
$787$	31.8972			1.13701			0.568506	−	0.822679i	$-0.307522\pi$
$787$	31.8972			1.13701			0.568506	+	0.822679i	$0.307522\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−13.8314			−0.491789
$792$		0			0
$793$	−11.3304	−	19.6249i	−0.402356	−	0.696900i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	25.7045	+	44.5215i	0.910501	+	1.57703i	0.813358	+	0.581763i	$0.197637\pi$
$797$	25.7045	+	44.5215i	0.910501	+	1.57703i	0.0971422	+	0.995271i	$0.469030\pi$
$798$		0			0
$799$	3.70094	+	6.41021i	0.130930	+	0.226777i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	87.1062			3.07391
$804$		0			0
$805$	−30.0667	−	52.0771i	−1.05971	−	1.83548i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10.9064	+	18.8905i	−0.383450	+	0.664154i	−0.991553	−	0.129704i	$-0.958597\pi$
$809$	−10.9064	+	18.8905i	−0.383450	+	0.664154i	0.608103	+	0.793858i	$0.291931\pi$
$810$		0			0
$811$	−3.37318	+	5.84251i	−0.118448	+	0.205158i	−0.919153	−	0.393901i	$-0.871125\pi$
$811$	−3.37318	+	5.84251i	−0.118448	+	0.205158i	0.800705	+	0.599059i	$0.204459\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−33.1764			−1.16212
$816$		0			0
$817$	−6.47167	+	11.2093i	−0.226415	+	0.392162i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.20736	+	2.09121i	−0.0421372	+	0.0729839i	−0.886325	−	0.463064i	$-0.846750\pi$
$821$	−1.20736	+	2.09121i	−0.0421372	+	0.0729839i	0.844188	+	0.536048i	$0.180083\pi$
$822$		0			0
$823$	10.7708	+	18.6555i	0.375445	+	0.650290i	0.990394	−	0.138277i	$-0.0441565\pi$
$823$	10.7708	+	18.6555i	0.375445	+	0.650290i	−0.614948	+	0.788567i	$0.710823\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.59099	+	6.21978i	−0.124871	+	0.216283i	−0.921683	−	0.387945i	$-0.873185\pi$
$827$	−3.59099	+	6.21978i	−0.124871	+	0.216283i	0.796811	+	0.604228i	$0.206518\pi$
$828$		0			0
$829$	7.00650	+	12.1356i	0.243346	+	0.421487i	0.961665	−	0.274226i	$-0.0884217\pi$
$829$	7.00650	+	12.1356i	0.243346	+	0.421487i	−0.718319	+	0.695713i	$0.755088\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.878051	+	1.52083i	−0.0304227	+	0.0526936i
$834$		0			0
$835$	−8.84381	−	15.3179i	−0.306053	−	0.530099i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−21.1686	−	36.6650i	−0.730820	−	1.26582i	−0.956533	−	0.291624i	$-0.905805\pi$
$839$	−21.1686	−	36.6650i	−0.730820	−	1.26582i	0.225713	−	0.974194i	$-0.427529\pi$
$840$		0			0
$841$	−27.5714			−0.950739
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−104.499			−3.59487
$846$		0			0
$847$	31.3001	−	54.2134i	1.07549	−	1.86280i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.5172	+	25.8204i	−0.840441	+	0.885111i
$852$		0			0
$853$	18.2990	+	31.6948i	0.626546	+	1.08521i	0.988240	+	0.152912i	$0.0488651\pi$
$853$	18.2990	+	31.6948i	0.626546	+	1.08521i	−0.361694	+	0.932297i	$0.617802\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	32.4623			1.10889			0.554446	−	0.832220i	$-0.312930\pi$
$857$	32.4623			1.10889			0.554446	+	0.832220i	$0.312930\pi$
$858$		0			0
$859$	−14.3584			−0.489901			−0.244950	−	0.969536i	$-0.578772\pi$
$859$	−14.3584			−0.489901			−0.244950	+	0.969536i	$0.578772\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−16.1233	−	27.9264i	−0.548843	−	0.950625i	−0.998354	−	0.0573498i	$-0.981735\pi$
$863$	−16.1233	−	27.9264i	−0.548843	−	0.950625i	0.449511	−	0.893275i	$-0.351598\pi$
$864$		0			0
$865$	−39.1882			−1.33244
$866$		0			0
$867$		0			0
$868$		0			0
$869$	15.1184	−	26.1858i	0.512856	−	0.888292i
$870$		0			0
$871$	−19.8555	−	34.3908i	−0.672779	−	1.16529i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10.5425	+	18.2602i	−0.356403	+	0.617308i
$876$		0			0
$877$	7.94227			0.268192			0.134096	−	0.990968i	$-0.457187\pi$
$877$	7.94227			0.268192			0.134096	+	0.990968i	$0.457187\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−8.05813	+	13.9571i	−0.271485	+	0.470226i	−0.969242	−	0.246108i	$-0.920848\pi$
$881$	−8.05813	+	13.9571i	−0.271485	+	0.470226i	0.697757	+	0.716335i	$0.254182\pi$
$882$		0			0
$883$	18.5606	−	32.1479i	0.624614	−	1.08186i	−0.364001	−	0.931399i	$-0.618590\pi$
$883$	18.5606	−	32.1479i	0.624614	−	1.08186i	0.988615	−	0.150465i	$-0.0480771\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−52.7711			−1.77188			−0.885939	−	0.463801i	$-0.846485\pi$
$887$	−52.7711			−1.77188			−0.885939	+	0.463801i	$0.846485\pi$
$888$		0			0
$889$	7.15706			0.240040
$890$		0			0
$891$		0			0
$892$		0			0
$893$	10.7124	−	18.5545i	0.358478	−	0.620902i
$894$		0			0
$895$	5.02167	−	8.69780i	0.167856	−	0.290735i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−6.77763			−0.226047
$900$		0			0
$901$	2.65082	−	4.59135i	0.0883116	−	0.152960i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−17.0766	−	29.5776i	−0.567647	−	0.983193i
$906$		0			0
$907$	−17.2800	+	29.9299i	−0.573774	+	0.993805i	0.422400	+	0.906410i	$0.361188\pi$
$907$	−17.2800	+	29.9299i	−0.573774	+	0.993805i	−0.996174	+	0.0873957i	$0.972146\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	27.0339			0.895672			0.447836	−	0.894116i	$-0.352195\pi$
$911$	27.0339			0.895672			0.447836	+	0.894116i	$0.352195\pi$
$912$		0			0
$913$	−33.6995	−	58.3692i	−1.11529	−	1.93174i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−29.5935			−0.977262
$918$		0			0
$919$	0.901918			0.0297515			0.0148758	−	0.999889i	$-0.495265\pi$
$919$	0.901918			0.0297515			0.0148758	+	0.999889i	$0.495265\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.84068	−	6.65225i	−0.126417	−	0.218961i
$924$		0			0
$925$	41.7104	+	10.0269i	1.37143	+	0.329683i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.0987	+	33.0799i	−0.626608	+	1.08532i	0.361620	+	0.932326i	$0.382224\pi$
$929$	−19.0987	+	33.0799i	−0.626608	+	1.08532i	−0.988228	+	0.152991i	$0.951110\pi$
$930$		0			0
$931$	5.08307			0.166591
$932$		0			0
$933$		0			0
$934$		0			0
$935$	19.6864			0.643813
$936$		0			0
$937$	12.7488	+	22.0815i	0.416484	+	0.721372i	0.995583	−	0.0938855i	$-0.0299288\pi$
$937$	12.7488	+	22.0815i	0.416484	+	0.721372i	−0.579099	+	0.815257i	$0.696595\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.38085	+	5.85581i	0.110213	+	0.190894i	0.915856	−	0.401507i	$-0.131513\pi$
$941$	3.38085	+	5.85581i	0.110213	+	0.190894i	−0.805643	+	0.592401i	$0.798180\pi$
$942$		0			0
$943$	32.1826	−	55.7418i	1.04801	−	1.81520i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24.6236	−	42.6493i	−0.800159	−	1.38592i	−0.919511	−	0.393064i	$-0.871415\pi$
$947$	−24.6236	−	42.6493i	−0.800159	−	1.38592i	0.119352	−	0.992852i	$-0.461918\pi$
$948$		0			0
$949$	−50.4235	+	87.3360i	−1.63682	+	2.83505i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−17.2798	−	29.9295i	−0.559748	−	0.969512i	−0.997517	−	0.0704248i	$-0.977565\pi$
$953$	−17.2798	−	29.9295i	−0.559748	−	0.969512i	0.437769	−	0.899088i	$-0.355769\pi$
$954$		0			0
$955$	7.00078	−	12.1257i	0.226540	−	0.392378i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.39644	+	5.88280i	−0.109677	+	0.189965i
$960$		0			0
$961$	1.15538			0.0372704
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.67659	+	4.63600i	−0.0861626	+	0.149238i
$966$		0			0
$967$	1.28304	−	2.22229i	0.0412598	−	0.0714640i	−0.844658	−	0.535306i	$-0.820196\pi$
$967$	1.28304	−	2.22229i	0.0412598	−	0.0714640i	0.885918	+	0.463842i	$0.153530\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.28387	−	3.95577i	−0.0732928	−	0.126947i	0.827050	−	0.562129i	$-0.190017\pi$
$971$	−2.28387	−	3.95577i	−0.0732928	−	0.126947i	−0.900343	+	0.435182i	$0.856684\pi$
$972$		0			0
$973$	−4.06439			−0.130298
$974$		0			0
$975$		0			0
$976$		0			0
$977$	8.18562	+	14.1779i	0.261881	+	0.453591i	0.966742	−	0.255754i	$-0.0823239\pi$
$977$	8.18562	+	14.1779i	0.261881	+	0.453591i	−0.704861	+	0.709346i	$0.748991\pi$
$978$		0			0
$979$	−19.8352	−	34.3556i	−0.633935	−	1.09801i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−22.2959	−	38.6176i	−0.711129	−	1.23171i	−0.964434	−	0.264325i	$-0.914851\pi$
$983$	−22.2959	−	38.6176i	−0.711129	−	1.23171i	0.253305	−	0.967387i	$-0.418482\pi$
$984$		0			0
$985$	−72.9986			−2.32593
$986$		0			0
$987$		0			0
$988$		0			0
$989$	26.1753			0.832327
$990$		0			0
$991$	42.0614			1.33612			0.668062	−	0.744106i	$-0.267124\pi$
$991$	42.0614			1.33612			0.668062	+	0.744106i	$0.267124\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.11527	−	8.85990i	0.162165	−	0.280878i
$996$		0			0
$997$	−14.2958	−	24.7611i	−0.452754	−	0.784192i	0.545802	−	0.837914i	$-0.316225\pi$
$997$	−14.2958	−	24.7611i	−0.452754	−	0.784192i	−0.998556	+	0.0537216i	$0.982892\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2664.2.r.n.433.5		10
3.2	odd	2		296.2.i.c.137.4	yes	10
12.11	even	2		592.2.i.h.433.2		10
37.10	even	3	inner	2664.2.r.n.1009.5		10
111.47	odd	6		296.2.i.c.121.4	✓	10
444.47	even	6		592.2.i.h.417.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.i.c.121.4	✓	10	111.47	odd	6
296.2.i.c.137.4	yes	10	3.2	odd	2
592.2.i.h.417.2		10	444.47	even	6
592.2.i.h.433.2		10	12.11	even	2
2664.2.r.n.433.5		10	1.1	even	1	trivial
2664.2.r.n.1009.5		10	37.10	even	3	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2664.2.r.n.433.5

Level	$2664$
Weight	$2$
Character	2664.433
Analytic conductor	$21.272$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$