LMFDB - Embedded newform 448.2.m.d.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(113,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("448.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.m (of order $4$, degree $2$, not 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:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.20138089353117696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 $ x^12 - 3x^10 - 2x^9 + 2x^8 + 4x^7 + 2x^6 + 8x^5 + 8x^4 - 16x^3 - 48*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			337.2
Root			$-0.605558 + 1.27801i$ of defining polynomial
Character	$\chi$	$=$	448.337
Dual form			448.2.m.d.113.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.39123 - 1.39123i) q^{3} +(2.16478 - 2.16478i) q^{5} +1.00000i q^{7} +0.871066i q^{9} +O(q^{10})$	\(q+(-1.39123 - 1.39123i) q^{3} +(2.16478 - 2.16478i) q^{5} +1.00000i q^{7} +0.871066i q^{9} +(3.09563 - 3.09563i) q^{11} +(1.75410 + 1.75410i) q^{13} -6.02343 q^{15} -5.20470 q^{17} +(0.851620 + 0.851620i) q^{19} +(1.39123 - 1.39123i) q^{21} -6.15500i q^{23} -4.37253i q^{25} +(-2.96185 + 2.96185i) q^{27} +(-6.24096 - 6.24096i) q^{29} +2.78247 q^{31} -8.61348 q^{33} +(2.16478 + 2.16478i) q^{35} +(4.11202 - 4.11202i) q^{37} -4.88072i q^{39} +6.32956i q^{41} +(-3.05937 + 3.05937i) q^{43} +(1.88567 + 1.88567i) q^{45} -3.60383 q^{47} -1.00000 q^{49} +(7.24096 + 7.24096i) q^{51} +(5.28393 - 5.28393i) q^{53} -13.4027i q^{55} -2.36961i q^{57} +(7.13555 - 7.13555i) q^{59} +(1.03992 + 1.03992i) q^{61} -0.871066 q^{63} +7.59446 q^{65} +(0.966693 + 0.966693i) q^{67} +(-8.56304 + 8.56304i) q^{69} +10.0597i q^{71} +15.1717i q^{73} +(-6.08321 + 6.08321i) q^{75} +(3.09563 + 3.09563i) q^{77} +6.61348 q^{79} +10.8544 q^{81} +(7.41730 + 7.41730i) q^{83} +(-11.2670 + 11.2670i) q^{85} +17.3653i q^{87} +3.26144i q^{89} +(-1.75410 + 1.75410i) q^{91} +(-3.87107 - 3.87107i) q^{93} +3.68714 q^{95} -7.66352 q^{97} +(2.69650 + 2.69650i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 4 q^{5}+O(q^{10}) $ 12 * q - 4 * q^3 + 4 * q^5	$ 12 q - 4 q^{3} + 4 q^{5} + 24 q^{15} - 8 q^{17} + 4 q^{21} - 4 q^{27} - 4 q^{29} + 8 q^{31} + 4 q^{35} - 20 q^{37} - 16 q^{43} + 40 q^{45} - 16 q^{47} - 12 q^{49} + 16 q^{51} + 4 q^{53} + 16 q^{59} - 20 q^{61} - 12 q^{63} + 32 q^{65} - 24 q^{67} - 4 q^{69} + 40 q^{75} - 24 q^{79} - 44 q^{81} + 20 q^{83} - 8 q^{85} - 48 q^{93} + 48 q^{97} - 32 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 4 * q^5 + 24 * q^15 - 8 * q^17 + 4 * q^21 - 4 * q^27 - 4 * q^29 + 8 * q^31 + 4 * q^35 - 20 * q^37 - 16 * q^43 + 40 * q^45 - 16 * q^47 - 12 * q^49 + 16 * q^51 + 4 * q^53 + 16 * q^59 - 20 * q^61 - 12 * q^63 + 32 * q^65 - 24 * q^67 - 4 * q^69 + 40 * q^75 - 24 * q^79 - 44 * q^81 + 20 * q^83 - 8 * q^85 - 48 * q^93 + 48 * q^97 - 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$	−1.39123	−	1.39123i	−0.803230	−	0.803230i	0.180369	−	0.983599i	$-0.442271\pi$
$3$	−1.39123	−	1.39123i	−0.803230	−	0.803230i	−0.983599	+	0.180369i	$0.942271\pi$
$4$		0			0
$5$	2.16478	−	2.16478i	0.968118	−	0.968118i	−0.0313891	−	0.999507i	$-0.509993\pi$
$5$	2.16478	−	2.16478i	0.968118	−	0.968118i	0.999507	+	0.0313891i	$0.00999312\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$			0.871066i			0.290355i
$10$		0			0
$11$	3.09563	−	3.09563i	0.933367	−	0.933367i	−0.0645481	−	0.997915i	$-0.520561\pi$
$11$	3.09563	−	3.09563i	0.933367	−	0.933367i	0.997915	+	0.0645481i	$0.0205606\pi$
$12$		0			0
$13$	1.75410	+	1.75410i	0.486499	+	0.486499i	0.907200	−	0.420701i	$-0.138216\pi$
$13$	1.75410	+	1.75410i	0.486499	+	0.486499i	−0.420701	+	0.907200i	$0.638216\pi$
$14$		0			0
$15$	−6.02343			−1.55524
$16$		0			0
$17$	−5.20470			−1.26233			−0.631163	−	0.775651i	$-0.717422\pi$
$17$	−5.20470			−1.26233			−0.631163	+	0.775651i	$0.717422\pi$
$18$		0			0
$19$	0.851620	+	0.851620i	0.195375	+	0.195375i	0.798014	−	0.602639i	$-0.205884\pi$
$19$	0.851620	+	0.851620i	0.195375	+	0.195375i	−0.602639	+	0.798014i	$0.705884\pi$
$20$		0			0
$21$	1.39123	−	1.39123i	0.303592	−	0.303592i
$22$		0			0
$23$		−	6.15500i		−	1.28341i	−0.766954	−	0.641703i	$-0.778228\pi$
$23$		−	6.15500i		−	1.28341i	0.766954	−	0.641703i	$-0.221772\pi$
$24$		0			0
$25$		−	4.37253i		−	0.874505i
$26$		0			0
$27$	−2.96185	+	2.96185i	−0.570007	+	0.570007i
$28$		0			0
$29$	−6.24096	−	6.24096i	−1.15892	−	1.15892i	−0.984709	−	0.174208i	$-0.944264\pi$
$29$	−6.24096	−	6.24096i	−1.15892	−	1.15892i	−0.174208	−	0.984709i	$-0.555736\pi$
$30$		0			0
$31$	2.78247			0.499746			0.249873	−	0.968279i	$-0.419611\pi$
$31$	2.78247			0.499746			0.249873	+	0.968279i	$0.419611\pi$
$32$		0			0
$33$	−8.61348			−1.49942
$34$		0			0
$35$	2.16478	+	2.16478i	0.365914	+	0.365914i
$36$		0			0
$37$	4.11202	−	4.11202i	0.676013	−	0.676013i	−0.283083	−	0.959095i	$-0.591357\pi$
$37$	4.11202	−	4.11202i	0.676013	−	0.676013i	0.959095	+	0.283083i	$0.0913571\pi$
$38$		0			0
$39$		−	4.88072i		−	0.781541i
$40$		0			0
$41$			6.32956i			0.988511i	0.869317	+	0.494255i	$0.164559\pi$
$41$			6.32956i			0.988511i	−0.869317	+	0.494255i	$0.835441\pi$
$42$		0			0
$43$	−3.05937	+	3.05937i	−0.466549	+	0.466549i	−0.900795	−	0.434245i	$-0.857015\pi$
$43$	−3.05937	+	3.05937i	−0.466549	+	0.466549i	0.434245	+	0.900795i	$0.357015\pi$
$44$		0			0
$45$	1.88567	+	1.88567i	0.281098	+	0.281098i
$46$		0			0
$47$	−3.60383			−0.525673			−0.262836	−	0.964840i	$-0.584658\pi$
$47$	−3.60383			−0.525673			−0.262836	+	0.964840i	$0.584658\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	7.24096	+	7.24096i	1.01394	+	1.01394i
$52$		0			0
$53$	5.28393	−	5.28393i	0.725803	−	0.725803i	−0.243977	−	0.969781i	$-0.578452\pi$
$53$	5.28393	−	5.28393i	0.725803	−	0.725803i	0.969781	+	0.243977i	$0.0784523\pi$
$54$		0			0
$55$		−	13.4027i		−	1.80722i
$56$		0			0
$57$		−	2.36961i		−	0.313862i
$58$		0			0
$59$	7.13555	−	7.13555i	0.928969	−	0.928969i	−0.0686701	−	0.997639i	$-0.521876\pi$
$59$	7.13555	−	7.13555i	0.928969	−	0.928969i	0.997639	+	0.0686701i	$0.0218756\pi$
$60$		0			0
$61$	1.03992	+	1.03992i	0.133148	+	0.133148i	0.770540	−	0.637392i	$-0.219987\pi$
$61$	1.03992	+	1.03992i	0.133148	+	0.133148i	−0.637392	+	0.770540i	$0.719987\pi$
$62$		0			0
$63$	−0.871066			−0.109744
$64$		0			0
$65$	7.59446			0.941977
$66$		0			0
$67$	0.966693	+	0.966693i	0.118100	+	0.118100i	0.763687	−	0.645587i	$-0.223387\pi$
$67$	0.966693	+	0.966693i	0.118100	+	0.118100i	−0.645587	+	0.763687i	$0.723387\pi$
$68$		0			0
$69$	−8.56304	+	8.56304i	−1.03087	+	1.03087i
$70$		0			0
$71$			10.0597i			1.19386i	0.802291	+	0.596932i	$0.203614\pi$
$71$			10.0597i			1.19386i	−0.802291	+	0.596932i	$0.796386\pi$
$72$		0			0
$73$			15.1717i			1.77571i	0.460119	+	0.887857i	$0.347807\pi$
$73$			15.1717i			1.77571i	−0.460119	+	0.887857i	$0.652193\pi$
$74$		0			0
$75$	−6.08321	+	6.08321i	−0.702428	+	0.702428i
$76$		0			0
$77$	3.09563	+	3.09563i	0.352779	+	0.352779i
$78$		0			0
$79$	6.61348			0.744075			0.372038	−	0.928218i	$-0.378659\pi$
$79$	6.61348			0.744075			0.372038	+	0.928218i	$0.378659\pi$
$80$		0			0
$81$	10.8544			1.20605
$82$		0			0
$83$	7.41730	+	7.41730i	0.814154	+	0.814154i	0.985254	−	0.171100i	$-0.0547320\pi$
$83$	7.41730	+	7.41730i	0.814154	+	0.814154i	−0.171100	+	0.985254i	$0.554732\pi$
$84$		0			0
$85$	−11.2670	+	11.2670i	−1.22208	+	1.22208i
$86$		0			0
$87$			17.3653i			1.86175i
$88$		0			0
$89$			3.26144i			0.345712i	0.984947	+	0.172856i	$0.0552996\pi$
$89$			3.26144i			0.345712i	−0.984947	+	0.172856i	$0.944700\pi$
$90$		0			0
$91$	−1.75410	+	1.75410i	−0.183879	+	0.183879i
$92$		0			0
$93$	−3.87107	−	3.87107i	−0.401411	−	0.401411i
$94$		0			0
$95$	3.68714			0.378292
$96$		0			0
$97$	−7.66352			−0.778112			−0.389056	−	0.921214i	$-0.627199\pi$
$97$	−7.66352			−0.778112			−0.389056	+	0.921214i	$0.627199\pi$
$98$		0			0
$99$	2.69650	+	2.69650i	0.271008	+	0.271008i
$100$		0			0
$101$	9.93252	−	9.93252i	0.988323	−	0.988323i	−0.0116100	−	0.999933i	$-0.503696\pi$
$101$	9.93252	−	9.93252i	0.988323	−	0.988323i	0.999933	+	0.0116100i	$0.00369566\pi$
$102$		0			0
$103$			1.61199i			0.158834i	0.996841	+	0.0794169i	$0.0253058\pi$
$103$			1.61199i			0.158834i	−0.996841	+	0.0794169i	$0.974694\pi$
$104$		0			0
$105$		−	6.02343i		−	0.587826i
$106$		0			0
$107$	−7.92372	+	7.92372i	−0.766015	+	0.766015i	−0.977402	−	0.211387i	$-0.932202\pi$
$107$	−7.92372	+	7.92372i	−0.766015	+	0.766015i	0.211387	+	0.977402i	$0.432202\pi$
$108$		0			0
$109$	5.68979	+	5.68979i	0.544983	+	0.544983i	0.924986	−	0.380002i	$-0.124077\pi$
$109$	5.68979	+	5.68979i	0.544983	+	0.544983i	−0.380002	+	0.924986i	$0.624077\pi$
$110$		0			0
$111$	−11.4416			−1.08599
$112$		0			0
$113$	15.2609			1.43563			0.717813	−	0.696235i	$-0.245143\pi$
$113$	15.2609			1.43563			0.717813	+	0.696235i	$0.245143\pi$
$114$		0			0
$115$	−13.3242	−	13.3242i	−1.24249	−	1.24249i
$116$		0			0
$117$	−1.52794	+	1.52794i	−0.141258	+	0.141258i
$118$		0			0
$119$		−	5.20470i		−	0.477114i
$120$		0			0
$121$		−	8.16581i		−	0.742346i
$122$		0			0
$123$	8.80590	−	8.80590i	0.794001	−	0.794001i
$124$		0			0
$125$	1.35834	+	1.35834i	0.121494	+	0.121494i
$126$		0			0
$127$	1.80529			0.160193			0.0800966	−	0.996787i	$-0.474477\pi$
$127$	1.80529			0.160193			0.0800966	+	0.996787i	$0.474477\pi$
$128$		0			0
$129$	8.51260			0.749492
$130$		0			0
$131$	−9.28573	−	9.28573i	−0.811298	−	0.811298i	0.173531	−	0.984828i	$-0.444482\pi$
$131$	−9.28573	−	9.28573i	−0.811298	−	0.811298i	−0.984828	+	0.173531i	$0.944482\pi$
$132$		0			0
$133$	−0.851620	+	0.851620i	−0.0738448	+	0.0738448i
$134$		0			0
$135$			12.8235i			1.10367i
$136$		0			0
$137$			7.93652i			0.678063i	0.940775	+	0.339031i	$0.110099\pi$
$137$			7.93652i			0.678063i	−0.940775	+	0.339031i	$0.889901\pi$
$138$		0			0
$139$	2.06915	−	2.06915i	0.175503	−	0.175503i	−0.613889	−	0.789392i	$-0.710396\pi$
$139$	2.06915	−	2.06915i	0.175503	−	0.175503i	0.789392	+	0.613889i	$0.210396\pi$
$140$		0			0
$141$	5.01377	+	5.01377i	0.422236	+	0.422236i
$142$		0			0
$143$	10.8601			0.908164
$144$		0			0
$145$	−27.0206			−2.24394
$146$		0			0
$147$	1.39123	+	1.39123i	0.114747	+	0.114747i
$148$		0			0
$149$	9.15500	−	9.15500i	0.750006	−	0.750006i	−0.224474	−	0.974480i	$-0.572066\pi$
$149$	9.15500	−	9.15500i	0.750006	−	0.750006i	0.974480	+	0.224474i	$0.0720663\pi$
$150$		0			0
$151$		−	2.80295i		−	0.228101i	−0.993475	−	0.114051i	$-0.963617\pi$
$151$		−	2.80295i		−	0.228101i	0.993475	−	0.114051i	$-0.0363826\pi$
$152$		0			0
$153$		−	4.53364i		−	0.366523i
$154$		0			0
$155$	6.02343	−	6.02343i	0.483813	−	0.483813i
$156$		0			0
$157$	0.958797	+	0.958797i	0.0765204	+	0.0765204i	0.744331	−	0.667811i	$-0.232768\pi$
$157$	0.958797	+	0.958797i	0.0765204	+	0.0765204i	−0.667811	+	0.744331i	$0.732768\pi$
$158$		0			0
$159$	−14.7024			−1.16597
$160$		0			0
$161$	6.15500			0.485082
$162$		0			0
$163$	−12.9205	−	12.9205i	−1.01202	−	1.01202i	−0.999927	−	0.0120883i	$-0.996152\pi$
$163$	−12.9205	−	12.9205i	−1.01202	−	1.01202i	−0.0120883	−	0.999927i	$-0.503848\pi$
$164$		0			0
$165$	−18.6463	+	18.6463i	−1.45161	+	1.45161i
$166$		0			0
$167$			1.96111i			0.151755i	0.997117	+	0.0758775i	$0.0241758\pi$
$167$			1.96111i			0.151755i	−0.997117	+	0.0758775i	$0.975824\pi$
$168$		0			0
$169$		−	6.84629i		−	0.526637i
$170$		0			0
$171$	−0.741818	+	0.741818i	−0.0567282	+	0.0567282i
$172$		0			0
$173$	−4.14040	−	4.14040i	−0.314789	−	0.314789i	0.531973	−	0.846761i	$-0.321451\pi$
$173$	−4.14040	−	4.14040i	−0.314789	−	0.314789i	−0.846761	+	0.531973i	$0.821451\pi$
$174$		0			0
$175$	4.37253			0.330532
$176$		0			0
$177$	−19.8544			−1.49235
$178$		0			0
$179$	9.51522	+	9.51522i	0.711201	+	0.711201i	0.966787	−	0.255585i	$-0.0822682\pi$
$179$	9.51522	+	9.51522i	0.711201	+	0.711201i	−0.255585	+	0.966787i	$0.582268\pi$
$180$		0			0
$181$	7.60424	−	7.60424i	0.565219	−	0.565219i	−0.365566	−	0.930785i	$-0.619125\pi$
$181$	7.60424	−	7.60424i	0.565219	−	0.565219i	0.930785	+	0.365566i	$0.119125\pi$
$182$		0			0
$183$		−	2.89355i		−	0.213897i
$184$		0			0
$185$		−	17.8032i		−	1.30892i
$186$		0			0
$187$	−16.1118	+	16.1118i	−1.17821	+	1.17821i
$188$		0			0
$189$	−2.96185	−	2.96185i	−0.215443	−	0.215443i
$190$		0			0
$191$	20.4878			1.48245			0.741223	−	0.671259i	$-0.234246\pi$
$191$	20.4878			1.48245			0.741223	+	0.671259i	$0.234246\pi$
$192$		0			0
$193$	12.7155			0.915284			0.457642	−	0.889137i	$-0.348694\pi$
$193$	12.7155			0.915284			0.457642	+	0.889137i	$0.348694\pi$
$194$		0			0
$195$	−10.5657	−	10.5657i	−0.756624	−	0.756624i
$196$		0			0
$197$	−12.8638	+	12.8638i	−0.916509	+	0.916509i	−0.996774	−	0.0802649i	$-0.974423\pi$
$197$	−12.8638	+	12.8638i	−0.916509	+	0.916509i	0.0802649	+	0.996774i	$0.474423\pi$
$198$		0			0
$199$			1.46847i			0.104097i	0.998645	+	0.0520487i	$0.0165751\pi$
$199$			1.46847i			0.104097i	−0.998645	+	0.0520487i	$0.983425\pi$
$200$		0			0
$201$		−	2.68979i		−	0.189723i
$202$		0			0
$203$	6.24096	−	6.24096i	0.438029	−	0.438029i
$204$		0			0
$205$	13.7021	+	13.7021i	0.956995	+	0.956995i
$206$		0			0
$207$	5.36141			0.372644
$208$		0			0
$209$	5.27260			0.364713
$210$		0			0
$211$	−0.534767	−	0.534767i	−0.0368149	−	0.0368149i	0.688460	−	0.725275i	$-0.258287\pi$
$211$	−0.534767	−	0.534767i	−0.0368149	−	0.0368149i	−0.725275	+	0.688460i	$0.758287\pi$
$212$		0			0
$213$	13.9954	−	13.9954i	0.958948	−	0.958948i
$214$		0			0
$215$			13.2457i			0.903350i
$216$		0			0
$217$			2.78247i			0.188886i
$218$		0			0
$219$	21.1074	−	21.1074i	1.42631	−	1.42631i
$220$		0			0
$221$	−9.12955	−	9.12955i	−0.614120	−	0.614120i
$222$		0			0
$223$	7.83775			0.524855			0.262427	−	0.964952i	$-0.415477\pi$
$223$	7.83775			0.524855			0.262427	+	0.964952i	$0.415477\pi$
$224$		0			0
$225$	3.80876			0.253917
$226$		0			0
$227$	−0.247573	−	0.247573i	−0.0164320	−	0.0164320i	0.698843	−	0.715275i	$-0.253699\pi$
$227$	−0.247573	−	0.247573i	−0.0164320	−	0.0164320i	−0.715275	+	0.698843i	$0.753699\pi$
$228$		0			0
$229$	−15.6030	+	15.6030i	−1.03107	+	1.03107i	−0.0315714	+	0.999502i	$0.510051\pi$
$229$	−15.6030	+	15.6030i	−1.03107	+	1.03107i	−0.999502	+	0.0315714i	$0.989949\pi$
$230$		0			0
$231$		−	8.61348i		−	0.566726i
$232$		0			0
$233$			9.51493i			0.623344i	0.950190	+	0.311672i	$0.100889\pi$
$233$			9.51493i			0.623344i	−0.950190	+	0.311672i	$0.899111\pi$
$234$		0			0
$235$	−7.80149	+	7.80149i	−0.508913	+	0.508913i
$236$		0			0
$237$	−9.20091	−	9.20091i	−0.597663	−	0.597663i
$238$		0			0
$239$	−18.8469			−1.21910			−0.609552	−	0.792746i	$-0.708651\pi$
$239$	−18.8469			−1.21910			−0.609552	+	0.792746i	$0.708651\pi$
$240$		0			0
$241$	−6.39828			−0.412150			−0.206075	−	0.978536i	$-0.566069\pi$
$241$	−6.39828			−0.412150			−0.206075	+	0.978536i	$0.566069\pi$
$242$		0			0
$243$	−6.21554	−	6.21554i	−0.398727	−	0.398727i
$244$		0			0
$245$	−2.16478	+	2.16478i	−0.138303	+	0.138303i
$246$		0			0
$247$			2.98765i			0.190100i
$248$		0			0
$249$		−	20.6384i		−	1.30791i
$250$		0			0
$251$	2.93159	−	2.93159i	0.185040	−	0.185040i	−0.608508	−	0.793548i	$-0.708232\pi$
$251$	2.93159	−	2.93159i	0.185040	−	0.185040i	0.793548	+	0.608508i	$0.208232\pi$
$252$		0			0
$253$	−19.0536	−	19.0536i	−1.19789	−	1.19789i
$254$		0			0
$255$	31.3501			1.96322
$256$		0			0
$257$	28.9676			1.80695			0.903475	−	0.428640i	$-0.141007\pi$
$257$	28.9676			1.80695			0.903475	+	0.428640i	$0.141007\pi$
$258$		0			0
$259$	4.11202	+	4.11202i	0.255509	+	0.255509i
$260$		0			0
$261$	5.43629	−	5.43629i	0.336498	−	0.336498i
$262$		0			0
$263$		−	0.344446i		−	0.0212395i	−0.999944	−	0.0106197i	$-0.996620\pi$
$263$		−	0.344446i		−	0.0212395i	0.999944	−	0.0106197i	$-0.00338043\pi$
$264$		0			0
$265$		−	22.8771i		−	1.40533i
$266$		0			0
$267$	4.53743	−	4.53743i	0.277686	−	0.277686i
$268$		0			0
$269$	−9.87874	−	9.87874i	−0.602317	−	0.602317i	0.338610	−	0.940927i	$-0.390043\pi$
$269$	−9.87874	−	9.87874i	−0.602317	−	0.602317i	−0.940927	+	0.338610i	$0.890043\pi$
$270$		0			0
$271$	12.4969			0.759130			0.379565	−	0.925165i	$-0.376074\pi$
$271$	12.4969			0.759130			0.379565	+	0.925165i	$0.376074\pi$
$272$		0			0
$273$	4.88072			0.295395
$274$		0			0
$275$	−13.5357	−	13.5357i	−0.816234	−	0.816234i
$276$		0			0
$277$	−5.99946	+	5.99946i	−0.360472	+	0.360472i	−0.863987	−	0.503514i	$-0.832040\pi$
$277$	−5.99946	+	5.99946i	−0.360472	+	0.360472i	0.503514	+	0.863987i	$0.332040\pi$
$278$		0			0
$279$			2.42372i			0.145104i
$280$		0			0
$281$		−	9.56494i		−	0.570596i	−0.958439	−	0.285298i	$-0.907907\pi$
$281$		−	9.56494i		−	0.570596i	0.958439	−	0.285298i	$-0.0920926\pi$
$282$		0			0
$283$	−11.6878	+	11.6878i	−0.694768	+	0.694768i	−0.963277	−	0.268509i	$-0.913469\pi$
$283$	−11.6878	+	11.6878i	−0.694768	+	0.694768i	0.268509	+	0.963277i	$0.413469\pi$
$284$		0			0
$285$	−5.12967	−	5.12967i	−0.303856	−	0.303856i
$286$		0			0
$287$	−6.32956			−0.373622
$288$		0			0
$289$	10.0889			0.593465
$290$		0			0
$291$	10.6617	+	10.6617i	0.625003	+	0.625003i
$292$		0			0
$293$	−15.2913	+	15.2913i	−0.893328	+	0.893328i	−0.994835	−	0.101507i	$-0.967634\pi$
$293$	−15.2913	+	15.2913i	−0.893328	+	0.893328i	0.101507	+	0.994835i	$0.467634\pi$
$294$		0			0
$295$		−	30.8938i		−	1.79870i
$296$		0			0
$297$			18.3375i			1.06405i
$298$		0			0
$299$	10.7965	−	10.7965i	0.624375	−	0.624375i
$300$		0			0
$301$	−3.05937	−	3.05937i	−0.176339	−	0.176339i
$302$		0			0
$303$	−27.6369			−1.58770
$304$		0			0
$305$	4.50240			0.257807
$306$		0			0
$307$	4.16259	+	4.16259i	0.237571	+	0.237571i	0.815844	−	0.578272i	$-0.196273\pi$
$307$	4.16259	+	4.16259i	0.237571	+	0.237571i	−0.578272	+	0.815844i	$0.696273\pi$
$308$		0			0
$309$	2.24265	−	2.24265i	0.127580	−	0.127580i
$310$		0			0
$311$		−	0.802623i		−	0.0455126i	−0.999741	−	0.0227563i	$-0.992756\pi$
$311$		−	0.802623i		−	0.0455126i	0.999741	−	0.0227563i	$-0.00724418\pi$
$312$		0			0
$313$			17.7285i			1.00207i	0.865427	+	0.501036i	$0.167047\pi$
$313$			17.7285i			1.00207i	−0.865427	+	0.501036i	$0.832953\pi$
$314$		0			0
$315$	−1.88567	+	1.88567i	−0.106245	+	0.106245i
$316$		0			0
$317$	10.3691	+	10.3691i	0.582385	+	0.582385i	0.935558	−	0.353173i	$-0.114897\pi$
$317$	10.3691	+	10.3691i	0.582385	+	0.582385i	−0.353173	+	0.935558i	$0.614897\pi$
$318$		0			0
$319$	−38.6394			−2.16339
$320$		0			0
$321$	22.0475			1.23057
$322$		0			0
$323$	−4.43243	−	4.43243i	−0.246627	−	0.246627i
$324$		0			0
$325$	7.66984	−	7.66984i	0.425446	−	0.425446i
$326$		0			0
$327$		−	15.8317i		−	0.875493i
$328$		0			0
$329$		−	3.60383i		−	0.198686i
$330$		0			0
$331$	0.223062	−	0.223062i	0.0122606	−	0.0122606i	−0.700950	−	0.713211i	$-0.747240\pi$
$331$	0.223062	−	0.223062i	0.0122606	−	0.0122606i	0.713211	+	0.700950i	$0.247240\pi$
$332$		0			0
$333$	3.58185	+	3.58185i	0.196284	+	0.196284i
$334$		0			0
$335$	4.18535			0.228670
$336$		0			0
$337$	−26.7633			−1.45789			−0.728944	−	0.684573i	$-0.759989\pi$
$337$	−26.7633			−1.45789			−0.728944	+	0.684573i	$0.759989\pi$
$338$		0			0
$339$	−21.2315	−	21.2315i	−1.15314	−	1.15314i
$340$		0			0
$341$	8.61348	−	8.61348i	0.466446	−	0.466446i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			37.0742i			1.99601i
$346$		0			0
$347$	−24.6077	+	24.6077i	−1.32101	+	1.32101i	−0.408050	+	0.912960i	$0.633791\pi$
$347$	−24.6077	+	24.6077i	−1.32101	+	1.32101i	−0.912960	+	0.408050i	$0.866209\pi$
$348$		0			0
$349$	13.7818	+	13.7818i	0.737725	+	0.737725i	0.972137	−	0.234412i	$-0.0753166\pi$
$349$	13.7818	+	13.7818i	0.737725	+	0.737725i	−0.234412	+	0.972137i	$0.575317\pi$
$350$		0			0
$351$	−10.3907			−0.554616
$352$		0			0
$353$	−26.0443			−1.38620			−0.693099	−	0.720842i	$-0.743755\pi$
$353$	−26.0443			−1.38620			−0.693099	+	0.720842i	$0.743755\pi$
$354$		0			0
$355$	21.7770	+	21.7770i	1.15580	+	1.15580i
$356$		0			0
$357$	−7.24096	+	7.24096i	−0.383232	+	0.383232i
$358$		0			0
$359$			21.0382i			1.11035i	0.831733	+	0.555176i	$0.187349\pi$
$359$			21.0382i			1.11035i	−0.831733	+	0.555176i	$0.812651\pi$
$360$		0			0
$361$		−	17.5495i		−	0.923657i
$362$		0			0
$363$	−11.3606	+	11.3606i	−0.596274	+	0.596274i
$364$		0			0
$365$	32.8434	+	32.8434i	1.71910	+	1.71910i
$366$		0			0
$367$	−7.41340			−0.386977			−0.193488	−	0.981103i	$-0.561980\pi$
$367$	−7.41340			−0.386977			−0.193488	+	0.981103i	$0.561980\pi$
$368$		0			0
$369$	−5.51346			−0.287019
$370$		0			0
$371$	5.28393	+	5.28393i	0.274328	+	0.274328i
$372$		0			0
$373$	3.75295	−	3.75295i	0.194320	−	0.194320i	−0.603240	−	0.797560i	$-0.706124\pi$
$373$	3.75295	−	3.75295i	0.194320	−	0.194320i	0.797560	+	0.603240i	$0.206124\pi$
$374$		0			0
$375$		−	3.77954i		−	0.195175i
$376$		0			0
$377$		−	21.8945i		−	1.12762i
$378$		0			0
$379$	−18.8387	+	18.8387i	−0.967677	+	0.967677i	−0.999494	−	0.0318167i	$-0.989871\pi$
$379$	−18.8387	+	18.8387i	−0.967677	+	0.967677i	0.0318167	+	0.999494i	$0.489871\pi$
$380$		0			0
$381$	−2.51157	−	2.51157i	−0.128672	−	0.128672i
$382$		0			0
$383$	4.94620			0.252739			0.126369	−	0.991983i	$-0.459668\pi$
$383$	4.94620			0.252739			0.126369	+	0.991983i	$0.459668\pi$
$384$		0			0
$385$	13.4027			0.683064
$386$		0			0
$387$	−2.66491	−	2.66491i	−0.135465	−	0.135465i
$388$		0			0
$389$	14.0267	−	14.0267i	0.711181	−	0.711181i	−0.255601	−	0.966782i	$-0.582273\pi$
$389$	14.0267	−	14.0267i	0.711181	−	0.711181i	0.966782	+	0.255601i	$0.0822735\pi$
$390$		0			0
$391$			32.0349i			1.62007i
$392$		0			0
$393$			25.8372i			1.30332i
$394$		0			0
$395$	14.3167	−	14.3167i	0.720353	−	0.720353i
$396$		0			0
$397$	−0.0638914	−	0.0638914i	−0.00320662	−	0.00320662i	0.705502	−	0.708708i	$-0.250722\pi$
$397$	−0.0638914	−	0.0638914i	−0.00320662	−	0.00320662i	−0.708708	+	0.705502i	$0.750722\pi$
$398$		0			0
$399$	2.36961			0.118629
$400$		0			0
$401$	−25.4103			−1.26893			−0.634466	−	0.772951i	$-0.718780\pi$
$401$	−25.4103			−1.26893			−0.634466	+	0.772951i	$0.718780\pi$
$402$		0			0
$403$	4.88072	+	4.88072i	0.243126	+	0.243126i
$404$		0			0
$405$	23.4975	−	23.4975i	1.16760	−	1.16760i
$406$		0			0
$407$		−	25.4586i		−	1.26194i
$408$		0			0
$409$			22.4054i			1.10788i	0.832557	+	0.553939i	$0.186876\pi$
$409$			22.4054i			1.10788i	−0.832557	+	0.553939i	$0.813124\pi$
$410$		0			0
$411$	11.0416	−	11.0416i	0.544640	−	0.544640i
$412$		0			0
$413$	7.13555	+	7.13555i	0.351117	+	0.351117i
$414$		0			0
$415$	32.1136			1.57639
$416$		0			0
$417$	−5.75735			−0.281939
$418$		0			0
$419$	24.2758	+	24.2758i	1.18595	+	1.18595i	0.978177	+	0.207775i	$0.0666220\pi$
$419$	24.2758	+	24.2758i	1.18595	+	1.18595i	0.207775	+	0.978177i	$0.433378\pi$
$420$		0			0
$421$	16.0270	−	16.0270i	0.781108	−	0.781108i	−0.198910	−	0.980018i	$-0.563740\pi$
$421$	16.0270	−	16.0270i	0.781108	−	0.781108i	0.980018	+	0.198910i	$0.0637400\pi$
$422$		0			0
$423$		−	3.13918i		−	0.152632i
$424$		0			0
$425$			22.7577i			1.10391i
$426$		0			0
$427$	−1.03992	+	1.03992i	−0.0503254	+	0.0503254i
$428$		0			0
$429$	−15.1089	−	15.1089i	−0.729464	−	0.729464i
$430$		0			0
$431$	31.6615			1.52508			0.762540	−	0.646941i	$-0.223952\pi$
$431$	31.6615			1.52508			0.762540	+	0.646941i	$0.223952\pi$
$432$		0			0
$433$	−11.6823			−0.561413			−0.280707	−	0.959794i	$-0.590569\pi$
$433$	−11.6823			−0.561413			−0.280707	+	0.959794i	$0.590569\pi$
$434$		0			0
$435$	37.5920	+	37.5920i	1.80240	+	1.80240i
$436$		0			0
$437$	5.24172	−	5.24172i	0.250745	−	0.250745i
$438$		0			0
$439$			13.1217i			0.626265i	0.949709	+	0.313133i	$0.101378\pi$
$439$			13.1217i			0.626265i	−0.949709	+	0.313133i	$0.898622\pi$
$440$		0			0
$441$		−	0.871066i		−	0.0414794i
$442$		0			0
$443$	16.2200	−	16.2200i	0.770634	−	0.770634i	−0.207584	−	0.978217i	$-0.566560\pi$
$443$	16.2200	−	16.2200i	0.770634	−	0.770634i	0.978217	+	0.207584i	$0.0665599\pi$
$444$		0			0
$445$	7.06030	+	7.06030i	0.334690	+	0.334690i
$446$		0			0
$447$	−25.4735			−1.20485
$448$		0			0
$449$	−15.6396			−0.738077			−0.369038	−	0.929414i	$-0.620313\pi$
$449$	−15.6396			−0.738077			−0.369038	+	0.929414i	$0.620313\pi$
$450$		0			0
$451$	19.5939	+	19.5939i	0.922643	+	0.922643i
$452$		0			0
$453$	−3.89957	+	3.89957i	−0.183218	+	0.183218i
$454$		0			0
$455$			7.59446i			0.356034i
$456$		0			0
$457$		−	12.2305i		−	0.572117i	−0.958212	−	0.286058i	$-0.907655\pi$
$457$		−	12.2305i		−	0.572117i	0.958212	−	0.286058i	$-0.0923451\pi$
$458$		0			0
$459$	15.4155	−	15.4155i	0.719535	−	0.719535i
$460$		0			0
$461$	−13.4658	−	13.4658i	−0.627163	−	0.627163i	0.320191	−	0.947353i	$-0.396253\pi$
$461$	−13.4658	−	13.4658i	−0.627163	−	0.627163i	−0.947353	+	0.320191i	$0.896253\pi$
$462$		0			0
$463$	24.7807			1.15166			0.575829	−	0.817570i	$-0.304679\pi$
$463$	24.7807			1.15166			0.575829	+	0.817570i	$0.304679\pi$
$464$		0			0
$465$	−16.7600			−0.777226
$466$		0			0
$467$	−23.1683	−	23.1683i	−1.07210	−	1.07210i	−0.997190	−	0.0749131i	$-0.976132\pi$
$467$	−23.1683	−	23.1683i	−1.07210	−	1.07210i	−0.0749131	−	0.997190i	$-0.523868\pi$
$468$		0			0
$469$	−0.966693	+	0.966693i	−0.0446377	+	0.0446377i
$470$		0			0
$471$		−	2.66782i		−	0.122927i
$472$		0			0
$473$			18.9413i			0.870923i
$474$		0			0
$475$	3.72373	−	3.72373i	0.170857	−	0.170857i
$476$		0			0
$477$	4.60265	+	4.60265i	0.210741	+	0.210741i
$478$		0			0
$479$	−35.3648			−1.61586			−0.807930	−	0.589278i	$-0.799412\pi$
$479$	−35.3648			−1.61586			−0.807930	+	0.589278i	$0.799412\pi$
$480$		0			0
$481$	14.4258			0.657759
$482$		0			0
$483$	−8.56304	−	8.56304i	−0.389632	−	0.389632i
$484$		0			0
$485$	−16.5898	+	16.5898i	−0.753304	+	0.753304i
$486$		0			0
$487$		−	16.7258i		−	0.757921i	−0.925413	−	0.378960i	$-0.876282\pi$
$487$		−	16.7258i		−	0.757921i	0.925413	−	0.378960i	$-0.123718\pi$
$488$		0			0
$489$			35.9510i			1.62576i
$490$		0			0
$491$	−5.93243	+	5.93243i	−0.267727	+	0.267727i	−0.828184	−	0.560457i	$-0.810626\pi$
$491$	−5.93243	+	5.93243i	−0.267727	+	0.267727i	0.560457	+	0.828184i	$0.310626\pi$
$492$		0			0
$493$	32.4823	+	32.4823i	1.46293	+	1.46293i
$494$		0			0
$495$	11.6746			0.524736
$496$		0			0
$497$	−10.0597			−0.451239
$498$		0			0
$499$	5.55675	+	5.55675i	0.248754	+	0.248754i	0.820459	−	0.571705i	$-0.193718\pi$
$499$	5.55675	+	5.55675i	0.248754	+	0.248754i	−0.571705	+	0.820459i	$0.693718\pi$
$500$		0			0
$501$	2.72836	−	2.72836i	0.121894	−	0.121894i
$502$		0			0
$503$			6.99765i			0.312010i	0.987756	+	0.156005i	$0.0498616\pi$
$503$			6.99765i			0.312010i	−0.987756	+	0.156005i	$0.950138\pi$
$504$		0			0
$505$		−	43.0034i		−	1.91363i
$506$		0			0
$507$	−9.52479	+	9.52479i	−0.423011	+	0.423011i
$508$		0			0
$509$	3.57736	+	3.57736i	0.158563	+	0.158563i	0.781930	−	0.623366i	$-0.214235\pi$
$509$	3.57736	+	3.57736i	0.158563	+	0.158563i	−0.623366	+	0.781930i	$0.714235\pi$
$510$		0			0
$511$	−15.1717			−0.671157
$512$		0			0
$513$	−5.04473			−0.222730
$514$		0			0
$515$	3.48959	+	3.48959i	0.153770	+	0.153770i
$516$		0			0
$517$	−11.1561	+	11.1561i	−0.490645	+	0.490645i
$518$		0			0
$519$			11.5205i			0.505695i
$520$		0			0
$521$			17.3647i			0.760761i	0.924830	+	0.380380i	$0.124207\pi$
$521$			17.3647i			0.760761i	−0.924830	+	0.380380i	$0.875793\pi$
$522$		0			0
$523$	25.9531	−	25.9531i	1.13485	−	1.13485i	0.145491	−	0.989360i	$-0.453524\pi$
$523$	25.9531	−	25.9531i	1.13485	−	1.13485i	0.989360	−	0.145491i	$-0.0464762\pi$
$524$		0			0
$525$	−6.08321	−	6.08321i	−0.265493	−	0.265493i
$526$		0			0
$527$	−14.4819			−0.630842
$528$		0			0
$529$	−14.8840			−0.647129
$530$		0			0
$531$	6.21554	+	6.21554i	0.269731	+	0.269731i
$532$		0			0
$533$	−11.1027	+	11.1027i	−0.480909	+	0.480909i
$534$		0			0
$535$			34.3062i			1.48319i
$536$		0			0
$537$		−	26.4758i		−	1.14252i
$538$		0			0
$539$	−3.09563	+	3.09563i	−0.133338	+	0.133338i
$540$		0			0
$541$	−11.3284	−	11.3284i	−0.487046	−	0.487046i	0.420327	−	0.907373i	$-0.361915\pi$
$541$	−11.3284	−	11.3284i	−0.487046	−	0.487046i	−0.907373	+	0.420327i	$0.861915\pi$
$542$		0			0
$543$	−21.1586			−0.908001
$544$		0			0
$545$	24.6343			1.05522
$546$		0			0
$547$	10.8422	+	10.8422i	0.463578	+	0.463578i	0.899826	−	0.436248i	$-0.143693\pi$
$547$	10.8422	+	10.8422i	0.463578	+	0.463578i	−0.436248	+	0.899826i	$0.643693\pi$
$548$		0			0
$549$	−0.905841	+	0.905841i	−0.0386604	+	0.0386604i
$550$		0			0
$551$		−	10.6299i		−	0.452847i
$552$		0			0
$553$			6.61348i			0.281234i
$554$		0			0
$555$	−24.7685	+	24.7685i	−1.05136	+	1.05136i
$556$		0			0
$557$	−15.7787	−	15.7787i	−0.668564	−	0.668564i	0.288820	−	0.957384i	$-0.406737\pi$
$557$	−15.7787	−	15.7787i	−0.668564	−	0.668564i	−0.957384	+	0.288820i	$0.906737\pi$
$558$		0			0
$559$	−10.7329			−0.453952
$560$		0			0
$561$	44.8306			1.89275
$562$		0			0
$563$	−14.7663	−	14.7663i	−0.622324	−	0.622324i	0.323801	−	0.946125i	$-0.395039\pi$
$563$	−14.7663	−	14.7663i	−0.622324	−	0.622324i	−0.946125	+	0.323801i	$0.895039\pi$
$564$		0			0
$565$	33.0365	−	33.0365i	1.38986	−	1.38986i
$566$		0			0
$567$			10.8544i			0.455844i
$568$		0			0
$569$		−	17.6170i		−	0.738542i	−0.929322	−	0.369271i	$-0.879607\pi$
$569$		−	17.6170i		−	0.738542i	0.929322	−	0.369271i	$-0.120393\pi$
$570$		0			0
$571$	6.07198	−	6.07198i	0.254105	−	0.254105i	−0.568547	−	0.822651i	$-0.692494\pi$
$571$	6.07198	−	6.07198i	0.254105	−	0.254105i	0.822651	+	0.568547i	$0.192494\pi$
$572$		0			0
$573$	−28.5034	−	28.5034i	−1.19074	−	1.19074i
$574$		0			0
$575$	−26.9129			−1.12234
$576$		0			0
$577$	−35.3028			−1.46968			−0.734838	−	0.678242i	$-0.762742\pi$
$577$	−35.3028			−1.46968			−0.734838	+	0.678242i	$0.762742\pi$
$578$		0			0
$579$	−17.6903	−	17.6903i	−0.735183	−	0.735183i
$580$		0			0
$581$	−7.41730	+	7.41730i	−0.307721	+	0.307721i
$582$		0			0
$583$		−	32.7141i		−	1.35488i
$584$		0			0
$585$			6.61528i			0.273508i
$586$		0			0
$587$	2.38838	−	2.38838i	0.0985791	−	0.0985791i	−0.656097	−	0.754676i	$-0.727794\pi$
$587$	2.38838	−	2.38838i	0.0985791	−	0.0985791i	0.754676	+	0.656097i	$0.227794\pi$
$588$		0			0
$589$	2.36961	+	2.36961i	0.0976379	+	0.0976379i
$590$		0			0
$591$	35.7932			1.47233
$592$		0			0
$593$	11.5641			0.474882			0.237441	−	0.971402i	$-0.423691\pi$
$593$	11.5641			0.474882			0.237441	+	0.971402i	$0.423691\pi$
$594$		0			0
$595$	−11.2670	−	11.2670i	−0.461903	−	0.461903i
$596$		0			0
$597$	2.04299	−	2.04299i	0.0836141	−	0.0836141i
$598$		0			0
$599$		−	12.2713i		−	0.501393i	−0.968066	−	0.250697i	$-0.919340\pi$
$599$		−	12.2713i		−	0.501393i	0.968066	−	0.250697i	$-0.0806596\pi$
$600$		0			0
$601$			14.1538i			0.577347i	0.957428	+	0.288674i	$0.0932143\pi$
$601$			14.1538i			0.577347i	−0.957428	+	0.288674i	$0.906786\pi$
$602$		0			0
$603$	−0.842054	+	0.842054i	−0.0342911	+	0.0342911i
$604$		0			0
$605$	−17.6772	−	17.6772i	−0.718679	−	0.718679i
$606$		0			0
$607$	25.6349			1.04049			0.520244	−	0.854018i	$-0.325841\pi$
$607$	25.6349			1.04049			0.520244	+	0.854018i	$0.325841\pi$
$608$		0			0
$609$	−17.3653			−0.703676
$610$		0			0
$611$	−6.32147	−	6.32147i	−0.255739	−	0.255739i
$612$		0			0
$613$	6.80822	−	6.80822i	0.274981	−	0.274981i	−0.556120	−	0.831102i	$-0.687711\pi$
$613$	6.80822	−	6.80822i	0.274981	−	0.274981i	0.831102	+	0.556120i	$0.187711\pi$
$614$		0			0
$615$		−	38.1256i		−	1.53737i
$616$		0			0
$617$			23.1951i			0.933800i	0.884310	+	0.466900i	$0.154629\pi$
$617$			23.1951i			0.933800i	−0.884310	+	0.466900i	$0.845371\pi$
$618$		0			0
$619$	−23.7380	+	23.7380i	−0.954110	+	0.954110i	−0.998992	−	0.0448821i	$-0.985709\pi$
$619$	−23.7380	+	23.7380i	−0.954110	+	0.954110i	0.0448821	+	0.998992i	$0.485709\pi$
$620$		0			0
$621$	18.2301	+	18.2301i	0.731551	+	0.731551i
$622$		0			0
$623$	−3.26144			−0.130667
$624$		0			0
$625$	27.7436			1.10975
$626$		0			0
$627$	−7.33542	−	7.33542i	−0.292948	−	0.292948i
$628$		0			0
$629$	−21.4019	+	21.4019i	−0.853348	+	0.853348i
$630$		0			0
$631$		−	30.2574i		−	1.20453i	−0.798296	−	0.602265i	$-0.794265\pi$
$631$		−	30.2574i		−	1.20453i	0.798296	−	0.602265i	$-0.205735\pi$
$632$		0			0
$633$			1.48797i			0.0591416i
$634$		0			0
$635$	3.90804	−	3.90804i	0.155086	−	0.155086i
$636$		0			0
$637$	−1.75410	−	1.75410i	−0.0694999	−	0.0694999i
$638$		0			0
$639$	−8.76265			−0.346645
$640$		0			0
$641$	−6.69390			−0.264393			−0.132197	−	0.991224i	$-0.542203\pi$
$641$	−6.69390			−0.264393			−0.132197	+	0.991224i	$0.542203\pi$
$642$		0			0
$643$	−24.7491	−	24.7491i	−0.976007	−	0.976007i	0.0237114	−	0.999719i	$-0.492452\pi$
$643$	−24.7491	−	24.7491i	−0.976007	−	0.976007i	−0.999719	+	0.0237114i	$0.992452\pi$
$644$		0			0
$645$	18.4279	−	18.4279i	0.725597	−	0.725597i
$646$		0			0
$647$		−	43.6311i		−	1.71532i	−0.514219	−	0.857659i	$-0.671918\pi$
$647$		−	43.6311i		−	1.71532i	0.514219	−	0.857659i	$-0.328082\pi$
$648$		0			0
$649$		−	44.1780i		−	1.73414i
$650$		0			0
$651$	3.87107	−	3.87107i	0.151719	−	0.151719i
$652$		0			0
$653$	−17.8880	−	17.8880i	−0.700013	−	0.700013i	0.264400	−	0.964413i	$-0.414826\pi$
$653$	−17.8880	−	17.8880i	−0.700013	−	0.700013i	−0.964413	+	0.264400i	$0.914826\pi$
$654$		0			0
$655$	−40.2031			−1.57086
$656$		0			0
$657$	−13.2156			−0.515588
$658$		0			0
$659$	16.0357	+	16.0357i	0.624662	+	0.624662i	0.946720	−	0.322058i	$-0.104374\pi$
$659$	16.0357	+	16.0357i	0.624662	+	0.624662i	−0.322058	+	0.946720i	$0.604374\pi$
$660$		0			0
$661$	7.70847	−	7.70847i	0.299825	−	0.299825i	−0.541120	−	0.840945i	$-0.682000\pi$
$661$	7.70847	−	7.70847i	0.299825	−	0.299825i	0.840945	+	0.541120i	$0.182000\pi$
$662$		0			0
$663$			25.4027i			0.986559i
$664$		0			0
$665$			3.68714i			0.142981i
$666$		0			0
$667$	−38.4131	+	38.4131i	−1.48736	+	1.48736i
$668$		0			0
$669$	−10.9041	−	10.9041i	−0.421579	−	0.421579i
$670$		0			0
$671$	6.43842			0.248553
$672$		0			0
$673$	1.82580			0.0703795			0.0351897	−	0.999381i	$-0.488796\pi$
$673$	1.82580			0.0703795			0.0351897	+	0.999381i	$0.488796\pi$
$674$		0			0
$675$	12.9507	+	12.9507i	0.498475	+	0.498475i
$676$		0			0
$677$	19.1385	−	19.1385i	0.735552	−	0.735552i	−0.236162	−	0.971714i	$-0.575889\pi$
$677$	19.1385	−	19.1385i	0.735552	−	0.735552i	0.971714	+	0.236162i	$0.0758894\pi$
$678$		0			0
$679$		−	7.66352i		−	0.294099i
$680$		0			0
$681$			0.688866i			0.0263974i
$682$		0			0
$683$	14.1568	−	14.1568i	0.541695	−	0.541695i	−0.382330	−	0.924026i	$-0.624878\pi$
$683$	14.1568	−	14.1568i	0.541695	−	0.541695i	0.924026	+	0.382330i	$0.124878\pi$
$684$		0			0
$685$	17.1808	+	17.1808i	0.656445	+	0.656445i
$686$		0			0
$687$	43.4148			1.65638
$688$		0			0
$689$	18.5370			0.706205
$690$		0			0
$691$	10.8188	+	10.8188i	0.411565	+	0.411565i	0.882283	−	0.470719i	$-0.156005\pi$
$691$	10.8188	+	10.8188i	0.411565	+	0.411565i	−0.470719	+	0.882283i	$0.656005\pi$
$692$		0			0
$693$	−2.69650	+	2.69650i	−0.102431	+	0.102431i
$694$		0			0
$695$		−	8.95851i		−	0.339815i
$696$		0			0
$697$		−	32.9434i		−	1.24782i
$698$		0			0
$699$	13.2375	−	13.2375i	0.500688	−	0.500688i
$700$		0			0
$701$	2.64009	+	2.64009i	0.0997148	+	0.0997148i	0.755204	−	0.655490i	$-0.227538\pi$
$701$	2.64009	+	2.64009i	0.0997148	+	0.0997148i	−0.655490	+	0.755204i	$0.727538\pi$
$702$		0			0
$703$	7.00377			0.264152
$704$		0			0
$705$	21.7074			0.817548
$706$		0			0
$707$	9.93252	+	9.93252i	0.373551	+	0.373551i
$708$		0			0
$709$	−20.2724	+	20.2724i	−0.761345	+	0.761345i	−0.976565	−	0.215221i	$-0.930953\pi$
$709$	−20.2724	+	20.2724i	−0.761345	+	0.761345i	0.215221	+	0.976565i	$0.430953\pi$
$710$		0			0
$711$			5.76078i			0.216046i
$712$		0			0
$713$		−	17.1261i		−	0.641377i
$714$		0			0
$715$	23.5096	−	23.5096i	0.879210	−	0.879210i
$716$		0			0
$717$	26.2205	+	26.2205i	0.979221	+	0.979221i
$718$		0			0
$719$	−20.1357			−0.750936			−0.375468	−	0.926835i	$-0.622518\pi$
$719$	−20.1357			−0.750936			−0.375468	+	0.926835i	$0.622518\pi$
$720$		0			0
$721$	−1.61199			−0.0600335
$722$		0			0
$723$	8.90151	+	8.90151i	0.331051	+	0.331051i
$724$		0			0
$725$	−27.2888	+	27.2888i	−1.01348	+	1.01348i
$726$		0			0
$727$		−	30.0313i		−	1.11380i	−0.830580	−	0.556900i	$-0.811991\pi$
$727$		−	30.0313i		−	1.11380i	0.830580	−	0.556900i	$-0.188009\pi$
$728$		0			0
$729$		−	15.2688i		−	0.565511i
$730$		0			0
$731$	15.9231	−	15.9231i	0.588937	−	0.588937i
$732$		0			0
$733$	−9.05256	−	9.05256i	−0.334364	−	0.334364i	0.519877	−	0.854241i	$-0.325978\pi$
$733$	−9.05256	−	9.05256i	−0.334364	−	0.334364i	−0.854241	+	0.519877i	$0.825978\pi$
$734$		0			0
$735$	6.02343			0.222177
$736$		0			0
$737$	5.98504			0.220462
$738$		0			0
$739$	−20.2007	−	20.2007i	−0.743096	−	0.743096i	0.230077	−	0.973172i	$-0.426102\pi$
$739$	−20.2007	−	20.2007i	−0.743096	−	0.743096i	−0.973172	+	0.230077i	$0.926102\pi$
$740$		0			0
$741$	4.15652	−	4.15652i	0.152694	−	0.152694i
$742$		0			0
$743$			32.8469i			1.20504i	0.798106	+	0.602518i	$0.205836\pi$
$743$			32.8469i			1.20504i	−0.798106	+	0.602518i	$0.794164\pi$
$744$		0			0
$745$		−	39.6371i		−	1.45219i
$746$		0			0
$747$	−6.46096	+	6.46096i	−0.236394	+	0.236394i
$748$		0			0
$749$	−7.92372	−	7.92372i	−0.289527	−	0.289527i
$750$		0			0
$751$	−17.6012			−0.642277			−0.321138	−	0.947032i	$-0.604065\pi$
$751$	−17.6012			−0.642277			−0.321138	+	0.947032i	$0.604065\pi$
$752$		0			0
$753$	−8.15705			−0.297259
$754$		0			0
$755$	−6.06777	−	6.06777i	−0.220829	−	0.220829i
$756$		0			0
$757$	−10.8261	+	10.8261i	−0.393480	+	0.393480i	−0.875926	−	0.482446i	$-0.839749\pi$
$757$	−10.8261	+	10.8261i	−0.393480	+	0.393480i	0.482446	+	0.875926i	$0.339749\pi$
$758$		0			0
$759$			53.0160i			1.92436i
$760$		0			0
$761$		−	1.30937i		−	0.0474648i	−0.999718	−	0.0237324i	$-0.992445\pi$
$761$		−	1.30937i		−	0.0474648i	0.999718	−	0.0237324i	$-0.00755496\pi$
$762$		0			0
$763$	−5.68979	+	5.68979i	−0.205984	+	0.205984i
$764$		0			0
$765$	−9.81432	−	9.81432i	−0.354838	−	0.354838i
$766$		0			0
$767$	25.0329			0.903885
$768$		0			0
$769$	22.6761			0.817720			0.408860	−	0.912597i	$-0.365926\pi$
$769$	22.6761			0.817720			0.408860	+	0.912597i	$0.365926\pi$
$770$		0			0
$771$	−40.3008	−	40.3008i	−1.45140	−	1.45140i
$772$		0			0
$773$	−2.77217	+	2.77217i	−0.0997079	+	0.0997079i	−0.755201	−	0.655493i	$-0.772461\pi$
$773$	−2.77217	+	2.77217i	−0.0997079	+	0.0997079i	0.655493	+	0.755201i	$0.272461\pi$
$774$		0			0
$775$		−	12.1664i		−	0.437031i
$776$		0			0
$777$		−	11.4416i		−	0.410464i
$778$		0			0
$779$	−5.39038	+	5.39038i	−0.193130	+	0.193130i
$780$		0			0
$781$	31.1410	+	31.1410i	1.11431	+	1.11431i
$782$		0			0
$783$	36.9695			1.32118
$784$		0			0
$785$	4.15117			0.148161
$786$		0			0
$787$	−21.9765	−	21.9765i	−0.783377	−	0.783377i	0.197022	−	0.980399i	$-0.436873\pi$
$787$	−21.9765	−	21.9765i	−0.783377	−	0.783377i	−0.980399	+	0.197022i	$0.936873\pi$
$788$		0			0
$789$	−0.479205	+	0.479205i	−0.0170602	+	0.0170602i
$790$		0			0
$791$			15.2609i			0.542616i
$792$		0			0
$793$			3.64825i			0.129553i
$794$		0			0
$795$	−31.8274	+	31.8274i	−1.12880	+	1.12880i
$796$		0			0
$797$	27.7986	+	27.7986i	0.984679	+	0.984679i	0.999884	−	0.0152058i	$-0.00484033\pi$
$797$	27.7986	+	27.7986i	0.984679	+	0.984679i	−0.0152058	+	0.999884i	$0.504840\pi$
$798$		0			0
$799$	18.7569			0.663570
$800$		0			0
$801$	−2.84093			−0.100379
$802$		0			0
$803$	46.9659	+	46.9659i	1.65739	+	1.65739i
$804$		0			0
$805$	13.3242	−	13.3242i	0.469616	−	0.469616i
$806$		0			0
$807$			27.4873i			0.967598i
$808$		0			0
$809$			49.3996i			1.73680i	0.495868	+	0.868398i	$0.334850\pi$
$809$			49.3996i			1.73680i	−0.495868	+	0.868398i	$0.665150\pi$
$810$		0			0
$811$	13.1428	−	13.1428i	0.461506	−	0.461506i	−0.437643	−	0.899149i	$-0.644187\pi$
$811$	13.1428	−	13.1428i	0.461506	−	0.461506i	0.899149	+	0.437643i	$0.144187\pi$
$812$		0			0
$813$	−17.3861	−	17.3861i	−0.609756	−	0.609756i
$814$		0			0
$815$	−55.9402			−1.95950
$816$		0			0
$817$	−5.21084			−0.182304
$818$		0			0
$819$	−1.52794	−	1.52794i	−0.0533904	−	0.0533904i
$820$		0			0
$821$	0.760309	−	0.760309i	0.0265350	−	0.0265350i	−0.693715	−	0.720250i	$-0.744027\pi$
$821$	0.760309	−	0.760309i	0.0265350	−	0.0265350i	0.720250	+	0.693715i	$0.244027\pi$
$822$		0			0
$823$			20.0833i			0.700058i	0.936739	+	0.350029i	$0.113828\pi$
$823$			20.0833i			0.700058i	−0.936739	+	0.350029i	$0.886172\pi$
$824$		0			0
$825$			37.6627i			1.31125i
$826$		0			0
$827$	−6.99770	+	6.99770i	−0.243334	+	0.243334i	−0.818228	−	0.574894i	$-0.805043\pi$
$827$	−6.99770	+	6.99770i	−0.243334	+	0.243334i	0.574894	+	0.818228i	$0.305043\pi$
$828$		0			0
$829$	−32.7219	−	32.7219i	−1.13648	−	1.13648i	−0.989077	−	0.147403i	$-0.952909\pi$
$829$	−32.7219	−	32.7219i	−1.13648	−	1.13648i	−0.147403	−	0.989077i	$-0.547091\pi$
$830$		0			0
$831$	16.6933			0.579084
$832$		0			0
$833$	5.20470			0.180332
$834$		0			0
$835$	4.24536	+	4.24536i	0.146917	+	0.146917i
$836$		0			0
$837$	−8.24124	+	8.24124i	−0.284859	+	0.284859i
$838$		0			0
$839$			11.1872i			0.386225i	0.981177	+	0.193113i	$0.0618583\pi$
$839$			11.1872i			0.386225i	−0.981177	+	0.193113i	$0.938142\pi$
$840$		0			0
$841$			48.8991i			1.68618i
$842$		0			0
$843$	−13.3071	+	13.3071i	−0.458320	+	0.458320i
$844$		0			0
$845$	−14.8207	−	14.8207i	−0.509847	−	0.509847i
$846$		0			0
$847$	8.16581			0.280580
$848$		0			0
$849$	32.5209			1.11612
$850$		0			0
$851$	−25.3095	−	25.3095i	−0.867598	−	0.867598i
$852$		0			0
$853$	−25.4369	+	25.4369i	−0.870944	+	0.870944i	−0.992575	−	0.121632i	$-0.961187\pi$
$853$	−25.4369	+	25.4369i	−0.870944	+	0.870944i	0.121632	+	0.992575i	$0.461187\pi$
$854$		0			0
$855$			3.21174i			0.109839i
$856$		0			0
$857$		−	32.9414i		−	1.12526i	−0.826710	−	0.562628i	$-0.809790\pi$
$857$		−	32.9414i		−	1.12526i	0.826710	−	0.562628i	$-0.190210\pi$
$858$		0			0
$859$	29.9192	−	29.9192i	1.02083	−	1.02083i	0.0210508	−	0.999778i	$-0.493299\pi$
$859$	29.9192	−	29.9192i	1.02083	−	1.02083i	0.999778	−	0.0210508i	$-0.00670116\pi$
$860$		0			0
$861$	8.80590	+	8.80590i	0.300104	+	0.300104i
$862$		0			0
$863$	−21.8142			−0.742562			−0.371281	−	0.928520i	$-0.621081\pi$
$863$	−21.8142			−0.742562			−0.371281	+	0.928520i	$0.621081\pi$
$864$		0			0
$865$	−17.9261			−0.609505
$866$		0			0
$867$	−14.0360	−	14.0360i	−0.476689	−	0.476689i
$868$		0			0
$869$	20.4729	−	20.4729i	0.694495	−	0.694495i
$870$		0			0
$871$			3.39135i			0.114911i
$872$		0			0
$873$		−	6.67543i		−	0.225929i
$874$		0			0
$875$	−1.35834	+	1.35834i	−0.0459203	+	0.0459203i
$876$		0			0
$877$	9.89510	+	9.89510i	0.334134	+	0.334134i	0.854154	−	0.520020i	$-0.174076\pi$
$877$	9.89510	+	9.89510i	0.334134	+	0.334134i	−0.520020	+	0.854154i	$0.674076\pi$
$878$		0			0
$879$	42.5476			1.43510
$880$		0			0
$881$	−12.3319			−0.415471			−0.207735	−	0.978185i	$-0.566609\pi$
$881$	−12.3319			−0.415471			−0.207735	+	0.978185i	$0.566609\pi$
$882$		0			0
$883$	−24.3508	−	24.3508i	−0.819469	−	0.819469i	0.166562	−	0.986031i	$-0.446733\pi$
$883$	−24.3508	−	24.3508i	−0.819469	−	0.819469i	−0.986031	+	0.166562i	$0.946733\pi$
$884$		0			0
$885$	−42.9805	+	42.9805i	−1.44477	+	1.44477i
$886$		0			0
$887$			4.26921i			0.143346i	0.997428	+	0.0716731i	$0.0228338\pi$
$887$			4.26921i			0.143346i	−0.997428	+	0.0716731i	$0.977166\pi$
$888$		0			0
$889$			1.80529i			0.0605473i
$890$		0			0
$891$	33.6013	−	33.6013i	1.12569	−	1.12569i
$892$		0			0
$893$	−3.06909	−	3.06909i	−0.102703	−	0.102703i
$894$		0			0
$895$	41.1967			1.37705
$896$		0			0
$897$	−30.0408			−1.00303
$898$		0			0
$899$	−17.3653	−	17.3653i	−0.579164	−	0.579164i
$900$		0			0
$901$	−27.5013	+	27.5013i	−0.916200	+	0.916200i
$902$		0			0
$903$			8.51260i			0.283282i
$904$		0			0
$905$		−	32.9230i		−	1.09440i
$906$		0			0
$907$	23.8097	−	23.8097i	0.790587	−	0.790587i	−0.191002	−	0.981590i	$-0.561174\pi$
$907$	23.8097	−	23.8097i	0.790587	−	0.790587i	0.981590	+	0.191002i	$0.0611738\pi$
$908$		0			0
$909$	8.65188	+	8.65188i	0.286965	+	0.286965i
$910$		0			0
$911$	−4.22749			−0.140063			−0.0700315	−	0.997545i	$-0.522310\pi$
$911$	−4.22749			−0.140063			−0.0700315	+	0.997545i	$0.522310\pi$
$912$		0			0
$913$	45.9224			1.51981
$914$		0			0
$915$	−6.26390	−	6.26390i	−0.207078	−	0.207078i
$916$		0			0
$917$	9.28573	−	9.28573i	0.306642	−	0.306642i
$918$		0			0
$919$			21.7824i			0.718536i	0.933234	+	0.359268i	$0.116974\pi$
$919$			21.7824i			0.718536i	−0.933234	+	0.359268i	$0.883026\pi$
$920$		0			0
$921$		−	11.5823i		−	0.381649i
$922$		0			0
$923$	−17.6457	+	17.6457i	−0.580814	+	0.580814i
$924$		0			0
$925$	−17.9799	−	17.9799i	−0.591177	−	0.591177i
$926$		0			0
$927$	−1.40415			−0.0461182
$928$		0			0
$929$	−31.3367			−1.02812			−0.514062	−	0.857753i	$-0.671860\pi$
$929$	−31.3367			−1.02812			−0.514062	+	0.857753i	$0.671860\pi$
$930$		0			0
$931$	−0.851620	−	0.851620i	−0.0279107	−	0.0279107i
$932$		0			0
$933$	−1.11664	+	1.11664i	−0.0365571	+	0.0365571i
$934$		0			0
$935$			69.7570i			2.28130i
$936$		0			0
$937$		−	53.9341i		−	1.76195i	−0.473162	−	0.880975i	$-0.656887\pi$
$937$		−	53.9341i		−	1.76195i	0.473162	−	0.880975i	$-0.343113\pi$
$938$		0			0
$939$	24.6644	−	24.6644i	0.804893	−	0.804893i
$940$		0			0
$941$	−35.0609	−	35.0609i	−1.14295	−	1.14295i	−0.987907	−	0.155046i	$-0.950448\pi$
$941$	−35.0609	−	35.0609i	−1.14295	−	1.14295i	−0.155046	−	0.987907i	$-0.549552\pi$
$942$		0			0
$943$	38.9584			1.26866
$944$		0			0
$945$	−12.8235			−0.417148
$946$		0			0
$947$	−0.0642739	−	0.0642739i	−0.00208862	−	0.00208862i	0.706062	−	0.708150i	$-0.250470\pi$
$947$	−0.0642739	−	0.0642739i	−0.00208862	−	0.00208862i	−0.708150	+	0.706062i	$0.750470\pi$
$948$		0			0
$949$	−26.6127	+	26.6127i	−0.863883	+	0.863883i
$950$		0			0
$951$		−	28.8516i		−	0.935577i
$952$		0			0
$953$		−	45.4195i		−	1.47128i	−0.677371	−	0.735641i	$-0.736881\pi$
$953$		−	45.4195i		−	1.47128i	0.677371	−	0.735641i	$-0.263119\pi$
$954$		0			0
$955$	44.3516	−	44.3516i	1.43518	−	1.43518i
$956$		0			0
$957$	53.7564	+	53.7564i	1.73770	+	1.73770i
$958$		0			0
$959$	−7.93652			−0.256284
$960$		0			0
$961$	−23.2579			−0.750254
$962$		0			0
$963$	−6.90209	−	6.90209i	−0.222417	−	0.222417i
$964$		0			0
$965$	27.5263	−	27.5263i	0.886103	−	0.886103i
$966$		0			0
$967$			60.1289i			1.93362i	0.255506	+	0.966808i	$0.417758\pi$
$967$			60.1289i			1.93362i	−0.255506	+	0.966808i	$0.582242\pi$
$968$		0			0
$969$			12.3331i			0.396196i
$970$		0			0
$971$	−26.6561	+	26.6561i	−0.855436	+	0.855436i	−0.990796	−	0.135360i	$-0.956781\pi$
$971$	−26.6561	+	26.6561i	−0.855436	+	0.855436i	0.135360	+	0.990796i	$0.456781\pi$
$972$		0			0
$973$	2.06915	+	2.06915i	0.0663339	+	0.0663339i
$974$		0			0
$975$	−21.3411			−0.683462
$976$		0			0
$977$	−24.6888			−0.789864			−0.394932	−	0.918710i	$-0.629232\pi$
$977$	−24.6888			−0.789864			−0.394932	+	0.918710i	$0.629232\pi$
$978$		0			0
$979$	10.0962	+	10.0962i	0.322676	+	0.322676i
$980$		0			0
$981$	−4.95619	+	4.95619i	−0.158239	+	0.158239i
$982$		0			0
$983$		−	5.11704i		−	0.163208i	−0.996665	−	0.0816041i	$-0.973996\pi$
$983$		−	5.11704i		−	0.163208i	0.996665	−	0.0816041i	$-0.0260043\pi$
$984$		0			0
$985$			55.6946i			1.77458i
$986$		0			0
$987$	−5.01377	+	5.01377i	−0.159590	+	0.159590i
$988$		0			0
$989$	18.8304	+	18.8304i	0.598772	+	0.598772i
$990$		0			0
$991$	5.43929			0.172785			0.0863924	−	0.996261i	$-0.472466\pi$
$991$	5.43929			0.172785			0.0863924	+	0.996261i	$0.472466\pi$
$992$		0			0
$993$	−0.620663			−0.0196961
$994$		0			0
$995$	3.17892	+	3.17892i	0.100779	+	0.100779i
$996$		0			0
$997$	−8.55025	+	8.55025i	−0.270789	+	0.270789i	−0.829418	−	0.558629i	$-0.811328\pi$
$997$	−8.55025	+	8.55025i	−0.270789	+	0.270789i	0.558629	+	0.829418i	$0.311328\pi$
$998$		0			0
$999$			24.3584i			0.770665i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.m.d.337.2		12
4.3	odd	2		112.2.m.d.29.1	✓	12
8.3	odd	2		896.2.m.g.673.2		12
8.5	even	2		896.2.m.h.673.5		12
16.3	odd	4		896.2.m.g.225.2		12
16.5	even	4	inner	448.2.m.d.113.2		12
16.11	odd	4		112.2.m.d.85.1	yes	12
16.13	even	4		896.2.m.h.225.5		12
28.3	even	6		784.2.x.m.765.4		24
28.11	odd	6		784.2.x.l.765.4		24
28.19	even	6		784.2.x.m.557.6		24
28.23	odd	6		784.2.x.l.557.6		24
28.27	even	2		784.2.m.h.589.1		12
32.5	even	8		7168.2.a.bi.1.10		12
32.11	odd	8		7168.2.a.bj.1.10		12
32.21	even	8		7168.2.a.bi.1.3		12
32.27	odd	8		7168.2.a.bj.1.3		12
112.11	odd	12		784.2.x.l.373.6		24
112.27	even	4		784.2.m.h.197.1		12
112.59	even	12		784.2.x.m.373.6		24
112.75	even	12		784.2.x.m.165.4		24
112.107	odd	12		784.2.x.l.165.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.d.29.1	✓	12	4.3	odd	2
112.2.m.d.85.1	yes	12	16.11	odd	4
448.2.m.d.113.2		12	16.5	even	4	inner
448.2.m.d.337.2		12	1.1	even	1	trivial
784.2.m.h.197.1		12	112.27	even	4
784.2.m.h.589.1		12	28.27	even	2
784.2.x.l.165.4		24	112.107	odd	12
784.2.x.l.373.6		24	112.11	odd	12
784.2.x.l.557.6		24	28.23	odd	6
784.2.x.l.765.4		24	28.11	odd	6
784.2.x.m.165.4		24	112.75	even	12
784.2.x.m.373.6		24	112.59	even	12
784.2.x.m.557.6		24	28.19	even	6
784.2.x.m.765.4		24	28.3	even	6
896.2.m.g.225.2		12	16.3	odd	4
896.2.m.g.673.2		12	8.3	odd	2
896.2.m.h.225.5		12	16.13	even	4
896.2.m.h.673.5		12	8.5	even	2
7168.2.a.bi.1.3		12	32.21	even	8
7168.2.a.bi.1.10		12	32.5	even	8
7168.2.a.bj.1.3		12	32.27	odd	8
7168.2.a.bj.1.10		12	32.11	odd	8

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.39123 - 1.39123i) q^{3} +(2.16478 - 2.16478i) q^{5} +1.00000i q^{7} +0.871066i q^{9} +O(q^{10})\)	\(q+(-1.39123 - 1.39123i) q^{3} +(2.16478 - 2.16478i) q^{5} +1.00000i q^{7} +0.871066i q^{9} +(3.09563 - 3.09563i) q^{11} +(1.75410 + 1.75410i) q^{13} -6.02343 q^{15} -5.20470 q^{17} +(0.851620 + 0.851620i) q^{19} +(1.39123 - 1.39123i) q^{21} -6.15500i q^{23} -4.37253i q^{25} +(-2.96185 + 2.96185i) q^{27} +(-6.24096 - 6.24096i) q^{29} +2.78247 q^{31} -8.61348 q^{33} +(2.16478 + 2.16478i) q^{35} +(4.11202 - 4.11202i) q^{37} -4.88072i q^{39} +6.32956i q^{41} +(-3.05937 + 3.05937i) q^{43} +(1.88567 + 1.88567i) q^{45} -3.60383 q^{47} -1.00000 q^{49} +(7.24096 + 7.24096i) q^{51} +(5.28393 - 5.28393i) q^{53} -13.4027i q^{55} -2.36961i q^{57} +(7.13555 - 7.13555i) q^{59} +(1.03992 + 1.03992i) q^{61} -0.871066 q^{63} +7.59446 q^{65} +(0.966693 + 0.966693i) q^{67} +(-8.56304 + 8.56304i) q^{69} +10.0597i q^{71} +15.1717i q^{73} +(-6.08321 + 6.08321i) q^{75} +(3.09563 + 3.09563i) q^{77} +6.61348 q^{79} +10.8544 q^{81} +(7.41730 + 7.41730i) q^{83} +(-11.2670 + 11.2670i) q^{85} +17.3653i q^{87} +3.26144i q^{89} +(-1.75410 + 1.75410i) q^{91} +(-3.87107 - 3.87107i) q^{93} +3.68714 q^{95} -7.66352 q^{97} +(2.69650 + 2.69650i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 4 q^{5}+O(q^{10}) \) 12 * q - 4 * q^3 + 4 * q^5	\( 12 q - 4 q^{3} + 4 q^{5} + 24 q^{15} - 8 q^{17} + 4 q^{21} - 4 q^{27} - 4 q^{29} + 8 q^{31} + 4 q^{35} - 20 q^{37} - 16 q^{43} + 40 q^{45} - 16 q^{47} - 12 q^{49} + 16 q^{51} + 4 q^{53} + 16 q^{59} - 20 q^{61} - 12 q^{63} + 32 q^{65} - 24 q^{67} - 4 q^{69} + 40 q^{75} - 24 q^{79} - 44 q^{81} + 20 q^{83} - 8 q^{85} - 48 q^{93} + 48 q^{97} - 32 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 4 * q^5 + 24 * q^15 - 8 * q^17 + 4 * q^21 - 4 * q^27 - 4 * q^29 + 8 * q^31 + 4 * q^35 - 20 * q^37 - 16 * q^43 + 40 * q^45 - 16 * q^47 - 12 * q^49 + 16 * q^51 + 4 * q^53 + 16 * q^59 - 20 * q^61 - 12 * q^63 + 32 * q^65 - 24 * q^67 - 4 * q^69 + 40 * q^75 - 24 * q^79 - 44 * q^81 + 20 * q^83 - 8 * q^85 - 48 * q^93 + 48 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more