LMFDB - Embedded newform 588.2.be.a.341.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [588,2,Mod(5,588)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 21, 29])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("588.5"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	588.be (of order $42$, degree $12$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.69520363885$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + 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:	$\mathrm{U}(1)[D_{42}]$

Embedding invariants

Embedding label			341.1
Root			$-0.988831 + 0.149042i$ of defining polynomial
Character	$\chi$	$=$	588.341
Dual form			588.2.be.a.269.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.61232 + 0.632789i) q^{3} +(1.40061 + 2.24461i) q^{7} +(2.19916 + 2.04052i) q^{9} +(-0.652626 + 0.148958i) q^{13} +(0.824595 + 0.476080i) q^{19} +(0.837874 + 4.50533i) q^{21} +(-4.77786 + 1.47378i) q^{25} +(2.25453 + 4.68157i) q^{27} +(5.48475 - 3.16662i) q^{31} +(6.32579 - 4.31285i) q^{37} +(-1.14650 - 0.172807i) q^{39} +(-0.750774 + 0.941440i) q^{43} +(-3.07656 + 6.28767i) q^{49} +(1.02825 + 1.28939i) q^{57} +(-6.93811 - 10.1763i) q^{61} +(-1.50000 + 7.79423i) q^{63} +(-7.65250 - 13.2545i) q^{67} +(2.63409 + 8.53952i) q^{73} +(-8.63604 - 0.647182i) q^{75} +(6.53097 - 11.3120i) q^{79} +(0.672571 + 8.97483i) q^{81} +(-1.24843 - 1.25626i) q^{91} +(10.8470 - 1.63492i) q^{93} +10.5657i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 5 q^{7} - 3 q^{9} + 15 q^{19} + 6 q^{21} - 5 q^{25} + 3 q^{31} - 4 q^{37} - 30 q^{39} - 26 q^{43} - 11 q^{49} + 30 q^{57} + 19 q^{61} - 18 q^{63} + 5 q^{67} + 3 q^{73} - 15 q^{75} + 17 q^{79}+ \cdots + 3 q^{93}+O(q^{100}) $ 12 * q - 3 * q^3 + 5 * q^7 - 3 * q^9 + 15 * q^19 + 6 * q^21 - 5 * q^25 + 3 * q^31 - 4 * q^37 - 30 * q^39 - 26 * q^43 - 11 * q^49 + 30 * q^57 + 19 * q^61 - 18 * q^63 + 5 * q^67 + 3 * q^73 - 15 * q^75 + 17 * q^79 + 9 * q^81 + 9 * q^91 + 3 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{42}\right)$

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
$3$	1.61232	+	0.632789i	0.930874	+	0.365341i
$3$	1.61232	+	0.632789i	0.930874	+	0.365341i
$4$		0			0
$5$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$5$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$6$		0			0
$7$	1.40061	+	2.24461i	0.529383	+	0.848383i
$7$	1.40061	+	2.24461i	0.529383	+	0.848383i
$8$		0			0
$9$	2.19916	+	2.04052i	0.733052	+	0.680173i
$10$		0			0
$11$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$11$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$12$		0			0
$13$	−0.652626	+	0.148958i	−0.181006	+	0.0413134i	−0.312063	−	0.950061i	$-0.601020\pi$
$13$	−0.652626	+	0.148958i	−0.181006	+	0.0413134i	0.131057	+	0.991375i	$0.458163\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$17$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$18$		0			0
$19$	0.824595	+	0.476080i	0.189175	+	0.109220i	0.591596	−	0.806234i	$-0.298498\pi$
$19$	0.824595	+	0.476080i	0.189175	+	0.109220i	−0.402421	+	0.915455i	$0.631831\pi$
$20$		0			0
$21$	0.837874	+	4.50533i	0.182839	+	0.983143i
$22$		0			0
$23$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$23$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$24$		0			0
$25$	−4.77786	+	1.47378i	−0.955573	+	0.294755i
$26$		0			0
$27$	2.25453	+	4.68157i	0.433884	+	0.900969i
$28$		0			0
$29$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$29$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$30$		0			0
$31$	5.48475	−	3.16662i	0.985090	−	0.568742i	0.0812873	−	0.996691i	$-0.474097\pi$
$31$	5.48475	−	3.16662i	0.985090	−	0.568742i	0.903803	+	0.427949i	$0.140764\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.32579	−	4.31285i	1.03995	−	0.709029i	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	6.32579	−	4.31285i	1.03995	−	0.709029i	0.957754	+	0.287587i	$0.0928532\pi$
$38$		0			0
$39$	−1.14650	−	0.172807i	−0.183587	−	0.0276713i
$40$		0			0
$41$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$41$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$42$		0			0
$43$	−0.750774	+	0.941440i	−0.114492	+	0.143568i	−0.835775	−	0.549072i	$-0.814981\pi$
$43$	−0.750774	+	0.941440i	−0.114492	+	0.143568i	0.721283	+	0.692640i	$0.243553\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$47$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$48$		0			0
$49$	−3.07656	+	6.28767i	−0.439508	+	0.898239i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$53$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.02825	+	1.28939i	0.136195	+	0.170784i
$58$		0			0
$59$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$59$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$60$		0			0
$61$	−6.93811	−	10.1763i	−0.888334	−	1.30295i	−0.952353	−	0.304999i	$-0.901344\pi$
$61$	−6.93811	−	10.1763i	−0.888334	−	1.30295i	0.0640184	−	0.997949i	$-0.479608\pi$
$62$		0			0
$63$	−1.50000	+	7.79423i	−0.188982	+	0.981981i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−7.65250	−	13.2545i	−0.934901	−	1.61930i	−0.774810	−	0.632195i	$-0.782154\pi$
$67$	−7.65250	−	13.2545i	−0.934901	−	1.61930i	−0.160092	−	0.987102i	$-0.551179\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$71$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$72$		0			0
$73$	2.63409	+	8.53952i	0.308297	+	0.999475i	0.968631	+	0.248503i	$0.0799386\pi$
$73$	2.63409	+	8.53952i	0.308297	+	0.999475i	−0.660334	+	0.750972i	$0.729585\pi$
$74$		0			0
$75$	−8.63604	−	0.647182i	−0.997204	−	0.0747301i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	6.53097	−	11.3120i	0.734792	−	1.27270i	−0.220023	−	0.975495i	$-0.570613\pi$
$79$	6.53097	−	11.3120i	0.734792	−	1.27270i	0.954815	−	0.297202i	$-0.0960535\pi$
$80$		0			0
$81$	0.672571	+	8.97483i	0.0747301	+	0.997204i
$82$		0			0
$83$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$83$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$89$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$90$		0			0
$91$	−1.24843	−	1.25626i	−0.130871	−	0.131692i
$92$		0			0
$93$	10.8470	−	1.63492i	1.12478	−	0.169533i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			10.5657i			1.07278i	0.843970	+	0.536390i	$0.180212\pi$
$97$			10.5657i			1.07278i	−0.843970	+	0.536390i	$0.819788\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$101$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$102$		0			0
$103$	−2.40596	−	15.9625i	−0.237067	−	1.57283i	−0.717979	−	0.696065i	$-0.754933\pi$
$103$	−2.40596	−	15.9625i	−0.237067	−	1.57283i	0.480912	−	0.876769i	$-0.340306\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$107$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$108$		0			0
$109$	−3.72057	+	3.45218i	−0.356366	+	0.330659i	−0.837956	−	0.545737i	$-0.816250\pi$
$109$	−3.72057	+	3.45218i	−0.356366	+	0.330659i	0.481591	+	0.876396i	$0.340059\pi$
$110$		0			0
$111$	12.9283	−	2.95081i	1.22710	−	0.280078i
$112$		0			0
$113$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$113$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.73918	−	1.00411i	−0.160787	−	0.0928304i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.822031	−	10.9692i	0.0747301	−	0.997204i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−19.9787	−	9.62122i	−1.77282	−	0.853745i	−0.964095	−	0.265557i	$-0.914444\pi$
$127$	−19.9787	−	9.62122i	−1.77282	−	0.853745i	−0.808724	−	0.588188i	$-0.799842\pi$
$128$		0			0
$129$	−1.80622	+	1.04282i	−0.159029	+	0.0918154i
$130$		0			0
$131$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$131$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$132$		0			0
$133$	0.0863251	+	2.51770i	0.00748533	+	0.218312i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$137$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$138$		0			0
$139$	11.5261	−	9.19172i	0.977627	−	0.779632i	0.00220985	−	0.999998i	$-0.499297\pi$
$139$	11.5261	−	9.19172i	0.977627	−	0.779632i	0.975417	+	0.220366i	$0.0707252\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−8.93917	+	8.19093i	−0.737290	+	0.675576i
$148$		0			0
$149$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$149$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$150$		0			0
$151$	−20.2683	−	13.8187i	−1.64941	−	1.12455i	−0.858999	−	0.511976i	$-0.828913\pi$
$151$	−20.2683	−	13.8187i	−1.64941	−	1.12455i	−0.790413	−	0.612574i	$-0.790134\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.300074	+	1.99086i	−0.0239485	+	0.158888i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−0.300074	+	1.99086i	−0.0239485	+	0.158888i	0.973660	+	0.228005i	$0.0732201\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−5.10399	−	13.0048i	−0.399776	−	1.01861i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−5.10399	−	13.0048i	−0.399776	−	1.01861i	0.579300	−	0.815114i	$-0.303326\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$167$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$168$		0			0
$169$	−11.3089	+	5.44606i	−0.869913	+	0.418928i
$170$		0			0
$171$	0.841962	+	2.72957i	0.0643865	+	0.208736i
$172$		0			0
$173$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$173$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$174$		0			0
$175$	−10.0000	−	8.66025i	−0.755929	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$179$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$180$		0			0
$181$	−2.90111	−	0.662160i	−0.215638	−	0.0492180i	0.113337	−	0.993557i	$-0.463846\pi$
$181$	−2.90111	−	0.662160i	−0.215638	−	0.0492180i	−0.328975	+	0.944339i	$0.606703\pi$
$182$		0			0
$183$	−4.74698	−	20.7979i	−0.350907	−	1.53742i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−7.35058	+	11.6176i	−0.534676	+	0.845057i
$190$		0			0
$191$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$191$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$192$		0			0
$193$	−9.69170	+	24.6940i	−0.697624	+	1.77752i	−0.0719816	+	0.997406i	$0.522932\pi$
$193$	−9.69170	+	24.6940i	−0.697624	+	1.77752i	−0.625642	+	0.780110i	$0.715163\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	24.6934	+	9.69145i	1.75047	+	0.687009i	0.999649	+	0.0265041i	$0.00843751\pi$
$199$	24.6934	+	9.69145i	1.75047	+	0.687009i	0.750822	+	0.660505i	$0.229658\pi$
$200$		0			0
$201$	−3.95097	−	26.2129i	−0.278680	−	1.84892i
$202$		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
$211$	0.866581	−	3.79674i	0.0596579	−	0.261378i	−0.936300	−	0.351202i	$-0.885773\pi$
$211$	0.866581	−	3.79674i	0.0596579	−	0.261378i	0.995958	+	0.0898234i	$0.0286303\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	14.7899	+	7.87591i	1.00400	+	0.534652i
$218$		0			0
$219$	−1.15671	+	15.4353i	−0.0781635	+	1.04302i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.20863	+	2.50974i	0.0809357	+	0.168065i	0.937509	−	0.347960i	$-0.113126\pi$
$223$	1.20863	+	2.50974i	0.0809357	+	0.168065i	−0.856574	+	0.516025i	$0.827411\pi$
$224$		0			0
$225$	−13.5145	−	6.50826i	−0.900969	−	0.433884i
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	28.1124	−	11.0333i	1.85772	−	0.729101i	0.904837	−	0.425757i	$-0.139992\pi$
$229$	28.1124	−	11.0333i	1.85772	−	0.729101i	0.952881	−	0.303344i	$-0.0981030\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$233$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	17.6881	−	14.1058i	1.14897	−	0.916270i
$238$		0			0
$239$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$239$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$240$		0			0
$241$	−15.8993	+	23.3200i	−1.02416	+	1.50217i	−0.167787	+	0.985823i	$0.553662\pi$
$241$	−15.8993	+	23.3200i	−1.02416	+	1.50217i	−0.856377	+	0.516351i	$0.827290\pi$
$242$		0			0
$243$	−4.59478	+	14.8959i	−0.294755	+	0.955573i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.609068	−	0.187872i	−0.0387540	−	0.0119540i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$251$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$257$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$258$		0			0
$259$	18.5407	+	8.15830i	1.15206	+	0.506932i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$269$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$270$		0			0
$271$	11.0611	+	0.828915i	0.671914	+	0.0503530i	0.406319	−	0.913731i	$-0.366812\pi$
$271$	11.0611	+	0.828915i	0.671914	+	0.0503530i	0.265596	+	0.964084i	$0.414431\pi$
$272$		0			0
$273$	−1.21792	−	2.81548i	−0.0737119	−	0.170401i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.15572	+	28.7661i	0.129525	+	1.72838i	0.562343	+	0.826904i	$0.309900\pi$
$277$	2.15572	+	28.7661i	0.129525	+	1.72838i	−0.432819	+	0.901481i	$0.642481\pi$
$278$		0			0
$279$	18.5234	+	4.22784i	1.10897	+	0.253114i
$280$		0			0
$281$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$281$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$282$		0			0
$283$	21.1764	+	22.8228i	1.25881	+	1.35667i	0.907298	+	0.420489i	$0.138141\pi$
$283$	21.1764	+	22.8228i	1.25881	+	1.35667i	0.351510	+	0.936184i	$0.385668\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.8101	−	2.53372i	0.988831	−	0.149042i
$290$		0			0
$291$	−6.68583	+	17.0352i	−0.391930	+	0.998622i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−3.16471	−	0.366600i	−0.182411	−	0.0211304i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	11.6714	−	2.66392i	0.666121	−	0.152038i	0.123927	−	0.992291i	$-0.460451\pi$
$307$	11.6714	−	2.66392i	0.666121	−	0.152038i	0.542194	+	0.840254i	$0.317594\pi$
$308$		0			0
$309$	6.22173	−	27.2592i	0.353942	−	1.55072i
$310$		0			0
$311$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$311$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$312$		0			0
$313$	30.5624	+	17.6452i	1.72749	+	0.997367i	0.900006	+	0.435878i	$0.143562\pi$
$313$	30.5624	+	17.6452i	1.72749	+	0.997367i	0.827484	+	0.561489i	$0.189771\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$317$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	2.89863	−	1.67352i	0.160787	−	0.0928304i
$326$		0			0
$327$	−8.18325	+	3.21169i	−0.452535	+	0.177607i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−21.0468	+	14.3495i	−1.15684	+	0.788718i	−0.980675	−	0.195643i	$-0.937320\pi$
$331$	−21.0468	+	14.3495i	−1.15684	+	0.788718i	−0.176161	+	0.984361i	$0.556368\pi$
$332$		0			0
$333$	22.7119	+	3.42326i	1.24460	+	0.187594i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−22.2495	+	27.9000i	−1.21201	+	1.51981i	−0.422143	+	0.906529i	$0.638722\pi$
$337$	−22.2495	+	27.9000i	−1.21201	+	1.51981i	−0.789865	+	0.613280i	$0.789850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.4224	+	1.90093i	−0.994719	+	0.102641i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$347$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$348$		0			0
$349$	16.6978	+	13.3161i	0.893815	+	0.712794i	0.958494	−	0.285112i	$-0.0920309\pi$
$349$	16.6978	+	13.3161i	0.893815	+	0.712794i	−0.0646791	+	0.997906i	$0.520602\pi$
$350$		0			0
$351$	−2.16872	−	2.71949i	−0.115758	−	0.145155i
$352$		0			0
$353$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$353$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$359$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$360$		0			0
$361$	−9.04670	−	15.6693i	−0.476142	−	0.824702i
$362$		0			0
$363$	8.26660	−	17.1658i	0.433884	−	0.900969i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	11.2820	+	36.5752i	0.588913	+	1.90921i	0.337452	+	0.941343i	$0.390435\pi$
$367$	11.2820	+	36.5752i	0.588913	+	1.90921i	0.251461	+	0.967867i	$0.419089\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.5718	−	20.0429i	0.599165	−	1.03778i	−0.393780	−	0.919205i	$-0.628833\pi$
$373$	11.5718	−	20.0429i	0.599165	−	1.03778i	0.992945	−	0.118579i	$-0.0378340\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	4.20036	+	18.4030i	0.215758	+	0.945298i	0.960573	+	0.278027i	$0.0896806\pi$
$379$	4.20036	+	18.4030i	0.215758	+	0.945298i	−0.744815	+	0.667271i	$0.767462\pi$
$380$		0			0
$381$	−26.1238	−	28.1548i	−1.33836	−	1.44241i
$382$		0			0
$383$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$383$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.57209	+	0.538407i	−0.181580	+	0.0273687i
$388$		0			0
$389$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$389$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−5.28477	−	35.0621i	−0.265235	−	1.75972i	−0.583877	−	0.811842i	$-0.698465\pi$
$397$	−5.28477	−	35.0621i	−0.265235	−	1.75972i	0.318642	−	0.947875i	$-0.396773\pi$
$398$		0			0
$399$	−1.45399	+	4.11396i	−0.0727905	+	0.205956i
$400$		0			0
$401$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$401$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$402$		0			0
$403$	−3.10780	+	2.88362i	−0.154810	+	0.143643i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−9.92962	+	0.744122i	−0.490988	+	0.0367945i	−0.317924	−	0.948116i	$-0.602986\pi$
$409$	−9.92962	+	0.744122i	−0.490988	+	0.0367945i	−0.173064	+	0.984911i	$0.555367\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	24.4001	−	7.52644i	1.19488	−	0.368571i
$418$		0			0
$419$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$419$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$420$		0			0
$421$	−21.8115	−	10.5039i	−1.06303	−	0.511926i	−0.181173	−	0.983451i	$-0.557990\pi$
$421$	−21.8115	−	10.5039i	−1.06303	−	0.511926i	−0.881853	+	0.471525i	$0.843704\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	13.1243	−	29.8265i	0.635130	−	1.44341i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$431$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$432$		0			0
$433$	−31.5171	+	25.1341i	−1.51462	+	1.20787i	−0.602376	+	0.798213i	$0.705779\pi$
$433$	−31.5171	+	25.1341i	−1.51462	+	1.20787i	−0.912241	+	0.409654i	$0.865649\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−12.1822	+	39.4936i	−0.581423	+	1.88493i	−0.132971	+	0.991120i	$0.542452\pi$
$439$	−12.1822	+	39.4936i	−0.581423	+	1.88493i	−0.448452	+	0.893807i	$0.648024\pi$
$440$		0			0
$441$	−19.5959	+	7.54979i	−0.933140	+	0.359514i
$442$		0			0
$443$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$443$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$449$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−23.9347	−	35.1058i	−1.12455	−	1.64941i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.53652	−	14.1068i	−0.258987	−	0.659889i	0.740944	−	0.671566i	$-0.234378\pi$
$457$	−5.53652	−	14.1068i	−0.258987	−	0.659889i	−0.999932	+	0.0116771i	$0.996283\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$461$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$462$		0			0
$463$	22.9350	−	11.0449i	1.06588	−	0.513300i	0.183102	−	0.983094i	$-0.441386\pi$
$463$	22.9350	−	11.0449i	1.06588	−	0.513300i	0.882776	+	0.469794i	$0.155672\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$467$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$468$		0			0
$469$	19.0330	−	35.7413i	0.878864	−	1.65038i
$470$		0			0
$471$	−1.74361	+	3.02003i	−0.0803415	+	0.139156i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−4.64144	−	1.05938i	−0.212964	−	0.0486076i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$479$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$480$		0			0
$481$	−3.48594	+	3.75695i	−0.158945	+	0.171302i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.0607530	−	0.154796i	0.00275298	−	0.00701448i	−0.929491	−	0.368846i	$-0.879753\pi$
$487$	0.0607530	−	0.154796i	0.00275298	−	0.00701448i	0.932244	+	0.361831i	$0.117848\pi$
$488$		0			0
$489$		−	24.1976i		−	1.09425i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6.50735	+	6.03793i	0.291309	+	0.270295i	0.812265	−	0.583288i	$-0.198234\pi$
$499$	6.50735	+	6.03793i	0.291309	+	0.270295i	−0.520957	+	0.853583i	$0.674425\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$503$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−21.6797	+	1.62467i	−0.962830	+	0.0721542i
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−15.4786	+	17.8731i	−0.684731	+	0.790659i
$512$		0			0
$513$	−0.369732	+	4.93373i	−0.0163241	+	0.217830i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	4.29346	−	1.68506i	0.187740	−	0.0736824i	−0.269609	−	0.962970i	$-0.586894\pi$
$523$	4.29346	−	1.68506i	0.187740	−	0.0736824i	0.457349	+	0.889288i	$0.348799\pi$
$524$		0			0
$525$	−10.6431	−	20.2910i	−0.464503	−	0.885572i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−22.7431	−	3.42797i	−0.988831	−	0.149042i
$530$		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
$541$	−43.9549	−	13.5583i	−1.88977	−	0.582916i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−43.9549	−	13.5583i	−1.88977	−	0.582916i	−0.900921	−	0.433983i	$-0.857108\pi$
$542$		0			0
$543$	−4.25852	−	2.90341i	−0.182751	−	0.124597i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	29.0296	+	36.4020i	1.24122	+	1.55644i	0.695689	+	0.718343i	$0.255099\pi$
$547$	29.0296	+	36.4020i	1.24122	+	1.55644i	0.545528	+	0.838093i	$0.316329\pi$
$548$		0			0
$549$	5.50702	−	36.5367i	0.235034	−	1.55935i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	34.5384	−	1.18423i	1.46872	−	0.0503585i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$	0.349740	−	0.726242i	0.0147924	−	0.0307168i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$563$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−19.2030	+	14.0799i	−0.806450	+	0.591302i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	−3.44529	−	45.9742i	−0.144181	−	1.92396i	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−3.44529	−	45.9742i	−0.144181	−	1.92396i	0.190609	−	0.981666i	$-0.438954\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−13.9827	−	15.0698i	−0.582108	−	0.627363i	0.371493	−	0.928436i	$-0.378846\pi$
$577$	−13.9827	−	15.0698i	−0.582108	−	0.627363i	−0.953601	+	0.301073i	$0.902655\pi$
$578$		0			0
$579$	−31.2522	+	33.6819i	−1.29880	+	1.39977i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	6.03026			0.248473
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$593$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	33.6811	+	31.2515i	1.37847	+	1.27904i
$598$		0			0
$599$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$599$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$600$		0			0
$601$	−42.2156	+	9.63544i	−1.72201	+	0.393038i	−0.965389	−	0.260815i	$-0.916009\pi$
$601$	−42.2156	+	9.63544i	−1.72201	+	0.393038i	−0.756622	+	0.653852i	$0.773152\pi$
$602$		0			0
$603$	10.2170	−	44.7638i	0.416070	−	1.82292i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	34.5000	+	19.9186i	1.40031	+	0.808470i	0.994424	−	0.105453i	$-0.0336291\pi$
$607$	34.5000	+	19.9186i	1.40031	+	0.808470i	0.405887	+	0.913923i	$0.366962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	28.7168	−	8.85796i	1.15986	−	0.357770i	0.345664	−	0.938358i	$-0.387654\pi$
$613$	28.7168	−	8.85796i	1.15986	−	0.357770i	0.814197	+	0.580589i	$0.197178\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$617$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$618$		0			0
$619$	40.5000	−	23.3827i	1.62783	−	0.939829i	0.643094	−	0.765787i	$-0.277650\pi$
$619$	40.5000	−	23.3827i	1.62783	−	0.939829i	0.984738	−	0.174042i	$-0.0556830\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	20.6560	−	14.0830i	0.826239	−	0.563320i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−18.1570	+	22.7681i	−0.722817	+	0.906384i	−0.998494	−	0.0548663i	$-0.982527\pi$
$631$	−18.1570	+	22.7681i	−0.722817	+	0.906384i	0.275676	+	0.961250i	$0.411098\pi$
$632$		0			0
$633$	3.79974	−	5.57320i	0.151026	−	0.221515i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.07124	−	4.56177i	0.0424443	−	0.180744i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$641$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$642$		0			0
$643$	−38.0167	−	30.3173i	−1.49923	−	1.19560i	−0.926750	−	0.375680i	$-0.877409\pi$
$643$	−38.0167	−	30.3173i	−1.49923	−	1.19560i	−0.572482	−	0.819917i	$-0.694019\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$647$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	18.8622	+	22.0574i	0.739268	+	0.864496i
$652$		0			0
$653$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$653$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−11.6323	+	24.1547i	−0.453818	+	0.942363i
$658$		0			0
$659$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$659$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$660$		0			0
$661$	−14.8054	−	47.9979i	−0.575863	−	1.86690i	−0.492353	−	0.870396i	$-0.663863\pi$
$661$	−14.8054	−	47.9979i	−0.575863	−	1.86690i	−0.0835105	−	0.996507i	$-0.526613\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.360558	+	4.81131i	0.0139400	+	0.186016i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	6.93500	+	30.3842i	0.267325	+	1.17123i	0.913112	+	0.407709i	$0.133672\pi$
$673$	6.93500	+	30.3842i	0.267325	+	1.17123i	−0.645787	+	0.763517i	$0.723471\pi$
$674$		0			0
$675$	−17.6714	−	19.0452i	−0.680173	−	0.733052i
$676$		0			0
$677$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$677$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$678$		0			0
$679$	−23.7158	+	14.7984i	−0.910128	+	0.567911i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$683$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	52.3079			1.99567
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−4.90483	−	32.5414i	−0.186588	−	1.23793i	−0.864441	−	0.502735i	$-0.832327\pi$
$691$	−4.90483	−	32.5414i	−0.186588	−	1.23793i	0.677852	−	0.735198i	$-0.262911\pi$
$692$		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		0.222521	−	0.974928i	$-0.428571\pi$
$701$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$702$		0			0
$703$	7.26948	−	0.544772i	0.274174	−	0.0205465i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3.89919	+	52.0311i	−0.146437	+	1.95407i	0.126333	+	0.991988i	$0.459679\pi$
$709$	−3.89919	+	52.0311i	−0.146437	+	1.95407i	−0.272770	+	0.962079i	$0.587940\pi$
$710$		0			0
$711$	37.4449	−	11.5502i	1.40429	−	0.433167i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$719$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$720$		0			0
$721$	32.4598	−	27.7578i	1.20887	−	1.03375i
$722$		0			0
$723$	−40.3914	+	27.5384i	−1.50217	+	1.02416i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−41.9793	+	33.4774i	−1.55693	+	1.24161i	−0.720654	+	0.693295i	$0.756158\pi$
$727$	−41.9793	+	33.4774i	−1.55693	+	1.24161i	−0.836273	+	0.548313i	$0.815270\pi$
$728$		0			0
$729$	−16.8342	+	21.1095i	−0.623490	+	0.781831i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−10.1982	+	33.0618i	−0.376680	+	1.22117i	0.546716	+	0.837318i	$0.315878\pi$
$733$	−10.1982	+	33.0618i	−0.376680	+	1.22117i	−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−44.7946	−	30.5404i	−1.64780	−	1.12345i	−0.866271	−	0.499574i	$-0.833490\pi$
$739$	−44.7946	−	30.5404i	−1.64780	−	1.12345i	−0.781524	−	0.623875i	$-0.785558\pi$
$740$		0			0
$741$	−0.863128	−	0.688322i	−0.0317078	−	0.0252861i
$742$		0			0
$743$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$743$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−18.2304	−	46.4503i	−0.665236	−	1.69499i	−0.717683	−	0.696370i	$-0.754797\pi$
$751$	−18.2304	−	46.4503i	−0.665236	−	1.69499i	0.0524465	−	0.998624i	$-0.483298\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	48.7672	−	23.4851i	1.77248	−	0.853579i	0.808098	−	0.589048i	$-0.200497\pi$
$757$	48.7672	−	23.4851i	1.77248	−	0.853579i	0.964377	−	0.264531i	$-0.0852172\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$761$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$762$		0			0
$763$	−12.9599	−	3.51605i	−0.469179	−	0.127289i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−28.7066	−	6.55210i	−1.03519	−	0.236275i	−0.329022	−	0.944322i	$-0.606719\pi$
$769$	−28.7066	−	6.55210i	−1.03519	−	0.236275i	−0.706165	+	0.708047i	$0.749576\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$773$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$774$		0			0
$775$	−21.5385	+	23.2130i	−0.773686	+	0.833835i
$776$		0			0
$777$	24.7310	+	24.8861i	0.887221	+	0.892785i
$778$		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
$787$	−25.5232	−	10.0171i	−0.909804	−	0.357072i	−0.136156	−	0.990687i	$-0.543475\pi$
$787$	−25.5232	−	10.0171i	−0.909804	−	0.357072i	−0.773648	+	0.633616i	$0.781570\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.04384	+	5.60786i	0.214623	+	0.199141i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$797$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$798$		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		0.955573	−	0.294755i	$-0.0952381\pi$
$809$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$810$		0			0
$811$	−16.1852	−	33.6089i	−0.568339	−	1.18017i	−0.965011	−	0.262208i	$-0.915549\pi$
$811$	−16.1852	−	33.6089i	−0.568339	−	1.18017i	0.396672	−	0.917960i	$-0.370165\pi$
$812$		0			0
$813$	17.3095	+	8.33583i	0.607071	+	0.292350i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.06728	+	0.418878i	−0.0373396	+	0.0146547i
$818$		0			0
$819$	−0.182071	−	5.31015i	−0.00636207	−	0.185552i
$820$		0			0
$821$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$821$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$822$		0			0
$823$	11.9348	+	1.79889i	0.416023	+	0.0627053i	0.353719	−	0.935352i	$-0.384917\pi$
$823$	11.9348	+	1.79889i	0.416023	+	0.0627053i	0.0623041	+	0.998057i	$0.480155\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$827$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$828$		0			0
$829$	−1.52605	+	2.23830i	−0.0530019	+	0.0777395i	−0.851825	−	0.523827i	$-0.824504\pi$
$829$	−1.52605	+	2.23830i	−0.0530019	+	0.0777395i	0.798823	+	0.601566i	$0.205456\pi$
$830$		0			0
$831$	−14.7271	+	47.7442i	−0.510879	+	1.65623i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	27.1903	+	18.5380i	0.939834	+	0.640768i
$838$		0			0
$839$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$839$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$840$		0			0
$841$	18.0812	+	22.6731i	0.623490	+	0.781831i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	25.7730	−	13.5185i	0.885572	−	0.464503i
$848$		0			0
$849$	19.7012	+	50.1978i	0.676143	+	1.72279i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−14.9131	+	30.9675i	−0.510616	+	1.06031i	0.473172	+	0.880970i	$0.343109\pi$
$853$	−14.9131	+	30.9675i	−0.510616	+	1.06031i	−0.983788	+	0.179335i	$0.942605\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$857$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$858$		0			0
$859$	58.2867	+	4.36798i	1.98872	+	0.149034i	1.00000		0.000762858i	$-0.000242825\pi$
$859$	58.2867	+	4.36798i	1.98872	+	0.149034i	0.988717	+	0.149797i	$0.0478619\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	28.7066	+	6.55210i	0.974928	+	0.222521i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	6.96858	+	7.51034i	0.236121	+	0.254478i
$872$		0			0
$873$	−21.5594	+	23.2355i	−0.729675	+	0.786403i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	39.2722	−	5.91933i	1.32613	−	0.199882i	0.552495	−	0.833516i	$-0.313676\pi$
$877$	39.2722	−	5.91933i	1.32613	−	0.199882i	0.773632	+	0.633635i	$0.218438\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	51.8962			1.74645			0.873223	−	0.487321i	$-0.162026\pi$
$883$	51.8962			1.74645			0.873223	+	0.487321i	$0.162026\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$887$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$888$		0			0
$889$	−6.38652	−	58.3199i	−0.214197	−	1.95599i
$890$		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
$903$	−4.87055	−	2.59367i	−0.162082	−	0.0863120i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	54.0223	−	16.6637i	1.79378	−	0.553308i	0.795190	−	0.606360i	$-0.207371\pi$
$907$	54.0223	−	16.6637i	1.79378	−	0.553308i	0.998592	+	0.0530515i	$0.0168947\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$911$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−46.3238	+	31.5830i	−1.52808	+	1.04183i	−0.549616	+	0.835417i	$0.685226\pi$
$919$	−46.3238	+	31.5830i	−1.52808	+	1.04183i	−0.978465	+	0.206411i	$0.933822\pi$
$920$		0			0
$921$	20.5037	+	3.09044i	0.675620	+	0.101833i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−23.8676	+	29.9290i	−0.784762	+	0.984060i
$926$		0			0
$927$	27.2807	−	40.0135i	0.896017	−	1.31421i
$928$		0			0
$929$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$929$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$930$		0			0
$931$	−5.53035	+	3.72009i	−0.181250	+	0.121921i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	45.5920	+	36.3584i	1.48943	+	1.18778i	0.934544	+	0.355848i	$0.115808\pi$
$937$	45.5920	+	36.3584i	1.48943	+	1.18778i	0.554882	+	0.831929i	$0.312764\pi$
$938$		0			0
$939$	38.1107	+	47.7893i	1.24370	+	1.55955i
$940$		0			0
$941$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$941$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$947$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$948$		0			0
$949$	−2.99110	−	5.18074i	−0.0970954	−	0.168174i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$953$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	4.55500	−	7.88948i	0.146935	−	0.254500i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	12.1755	+	53.3444i	0.391538	+	1.71544i	0.659236	+	0.751936i	$0.270880\pi$
$967$	12.1755	+	53.3444i	0.391538	+	1.71544i	−0.267698	+	0.963503i	$0.586263\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$971$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$972$		0			0
$973$	36.7754	+	12.9974i	1.17897	+	0.416679i
$974$		0			0
$975$	5.73250	−	0.864036i	0.183587	−	0.0276713i
$976$		0			0
$977$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$977$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−15.2263			−0.486140
$982$		0			0
$983$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$983$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	21.0909	−	19.5695i	0.669975	−	0.621646i	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	21.0909	−	19.5695i	0.669975	−	0.621646i	0.939987	+	0.341211i	$0.110837\pi$
$992$		0			0
$993$	−43.0144	+	9.81775i	−1.36502	+	0.311557i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−22.4537	+	1.68267i	−0.711116	+	0.0532908i	−0.425376	−	0.905017i	$-0.639858\pi$
$997$	−22.4537	+	1.68267i	−0.711116	+	0.0532908i	−0.285739	+	0.958307i	$0.592239\pi$
$998$		0			0
$999$	34.4526	+	19.8912i	1.09003	+	0.629330i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.be.a.341.1	yes	12
3.2	odd	2	CM	588.2.be.a.341.1	yes	12
49.24	odd	42	inner	588.2.be.a.269.1	✓	12
147.122	even	42	inner	588.2.be.a.269.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.be.a.269.1	✓	12	49.24	odd	42	inner
588.2.be.a.269.1	✓	12	147.122	even	42	inner
588.2.be.a.341.1	yes	12	1.1	even	1	trivial
588.2.be.a.341.1	yes	12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	588.be (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(1.61232 + 0.632789i) q^{3} +(1.40061 + 2.24461i) q^{7} +(2.19916 + 2.04052i) q^{9} +(-0.652626 + 0.148958i) q^{13} +(0.824595 + 0.476080i) q^{19} +(0.837874 + 4.50533i) q^{21} +(-4.77786 + 1.47378i) q^{25} +(2.25453 + 4.68157i) q^{27} +(5.48475 - 3.16662i) q^{31} +(6.32579 - 4.31285i) q^{37} +(-1.14650 - 0.172807i) q^{39} +(-0.750774 + 0.941440i) q^{43} +(-3.07656 + 6.28767i) q^{49} +(1.02825 + 1.28939i) q^{57} +(-6.93811 - 10.1763i) q^{61} +(-1.50000 + 7.79423i) q^{63} +(-7.65250 - 13.2545i) q^{67} +(2.63409 + 8.53952i) q^{73} +(-8.63604 - 0.647182i) q^{75} +(6.53097 - 11.3120i) q^{79} +(0.672571 + 8.97483i) q^{81} +(-1.24843 - 1.25626i) q^{91} +(10.8470 - 1.63492i) q^{93} +10.5657i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 5 q^{7} - 3 q^{9} + 15 q^{19} + 6 q^{21} - 5 q^{25} + 3 q^{31} - 4 q^{37} - 30 q^{39} - 26 q^{43} - 11 q^{49} + 30 q^{57} + 19 q^{61} - 18 q^{63} + 5 q^{67} + 3 q^{73} - 15 q^{75} + 17 q^{79}+ \cdots + 3 q^{93}+O(q^{100}) \) 12 * q - 3 * q^3 + 5 * q^7 - 3 * q^9 + 15 * q^19 + 6 * q^21 - 5 * q^25 + 3 * q^31 - 4 * q^37 - 30 * q^39 - 26 * q^43 - 11 * q^49 + 30 * q^57 + 19 * q^61 - 18 * q^63 + 5 * q^67 + 3 * q^73 - 15 * q^75 + 17 * q^79 + 9 * q^81 + 9 * q^91 + 3 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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