LMFDB - Embedded newform 2400.2.w.h.2143.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(607,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 0, 1]))

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

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

chi := DirichletCharacter("2400.607");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.w (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2143.1
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2400.2143
Dual form			2400.2.w.h.607.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{3} +(-3.43916 - 3.43916i) q^{7} -1.00000i q^{9} -3.03528i q^{11} +(-2.63896 - 2.63896i) q^{13} +(0.682163 - 0.682163i) q^{17} -2.69677 q^{19} +4.86370 q^{21} +(-5.18154 + 5.18154i) q^{23} +(0.707107 + 0.707107i) q^{27} +0.399602i q^{29} +4.96008i q^{31} +(2.14626 + 2.14626i) q^{33} +(-4.29253 + 4.29253i) q^{37} +3.73205 q^{39} +10.9847 q^{41} +(5.83876 - 5.83876i) q^{43} +(-4.04989 - 4.04989i) q^{47} +16.6556i q^{49} +0.964724i q^{51} +(10.0599 + 10.0599i) q^{53} +(1.90691 - 1.90691i) q^{57} +3.46410 q^{59} -2.87780 q^{61} +(-3.43916 + 3.43916i) q^{63} +(-7.35275 - 7.35275i) q^{67} -7.32780i q^{69} +14.0857i q^{71} +(9.46410 + 9.46410i) q^{73} +(-10.4388 + 10.4388i) q^{77} +2.60645 q^{79} -1.00000 q^{81} +(-1.97934 + 1.97934i) q^{83} +(-0.282561 - 0.282561i) q^{87} -0.799203i q^{89} +18.1516i q^{91} +(-3.50731 - 3.50731i) q^{93} +(-8.94558 + 8.94558i) q^{97} -3.03528 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} + 8 q^{17} - 16 q^{19} + 8 q^{21} - 8 q^{23} - 8 q^{33} + 16 q^{37} + 16 q^{39} - 16 q^{41} + 24 q^{43} - 16 q^{47} + 8 q^{53} + 8 q^{57} - 24 q^{61} - 8 q^{63} + 8 q^{67} + 48 q^{73} - 8 q^{77}+ \cdots - 16 q^{99}+O(q^{100}) $ 8 * q - 8 * q^7 + 8 * q^17 - 16 * q^19 + 8 * q^21 - 8 * q^23 - 8 * q^33 + 16 * q^37 + 16 * q^39 - 16 * q^41 + 24 * q^43 - 16 * q^47 + 8 * q^53 + 8 * q^57 - 24 * q^61 - 8 * q^63 + 8 * q^67 + 48 * q^73 - 8 * q^77 + 32 * q^79 - 8 * q^81 - 16 * q^83 - 8 * q^87 - 24 * q^93 - 32 * q^97 - 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$	$-1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0
$2$		0			0
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.43916	−	3.43916i	−1.29988	−	1.29988i	−0.928469	−	0.371411i	$-0.878874\pi$
$7$	−3.43916	−	3.43916i	−1.29988	−	1.29988i	−0.371411	−	0.928469i	$-0.621126\pi$
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$		−	3.03528i		−	0.915170i	−0.889166	−	0.457585i	$-0.848715\pi$
$11$		−	3.03528i		−	0.915170i	0.889166	−	0.457585i	$-0.151285\pi$
$12$		0			0
$13$	−2.63896	−	2.63896i	−0.731915	−	0.731915i	0.239084	−	0.970999i	$-0.423153\pi$
$13$	−2.63896	−	2.63896i	−0.731915	−	0.731915i	−0.970999	+	0.239084i	$0.923153\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.682163	−	0.682163i	0.165449	−	0.165449i	−0.619527	−	0.784976i	$-0.712675\pi$
$17$	0.682163	−	0.682163i	0.165449	−	0.165449i	0.784976	+	0.619527i	$0.212675\pi$
$18$		0			0
$19$	−2.69677			−0.618683			−0.309341	−	0.950951i	$-0.600109\pi$
$19$	−2.69677			−0.618683			−0.309341	+	0.950951i	$0.600109\pi$
$20$		0			0
$21$	4.86370			1.06135
$22$		0			0
$23$	−5.18154	+	5.18154i	−1.08043	+	1.08043i	−0.0839565	+	0.996469i	$0.526756\pi$
$23$	−5.18154	+	5.18154i	−1.08043	+	1.08043i	−0.996469	+	0.0839565i	$0.973244\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.707107	+	0.707107i	0.136083	+	0.136083i
$28$		0			0
$29$			0.399602i			0.0742042i	0.999311	+	0.0371021i	$0.0118127\pi$
$29$			0.399602i			0.0742042i	−0.999311	+	0.0371021i	$0.988187\pi$
$30$		0			0
$31$			4.96008i			0.890857i	0.895318	+	0.445428i	$0.146949\pi$
$31$			4.96008i			0.890857i	−0.895318	+	0.445428i	$0.853051\pi$
$32$		0			0
$33$	2.14626	+	2.14626i	0.373617	+	0.373617i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.29253	+	4.29253i	−0.705687	+	0.705687i	−0.965625	−	0.259938i	$-0.916298\pi$
$37$	−4.29253	+	4.29253i	−0.705687	+	0.705687i	0.259938	+	0.965625i	$0.416298\pi$
$38$		0			0
$39$	3.73205			0.597606
$40$		0			0
$41$	10.9847			1.71552			0.857758	−	0.514054i	$-0.171857\pi$
$41$	10.9847			1.71552			0.857758	+	0.514054i	$0.171857\pi$
$42$		0			0
$43$	5.83876	−	5.83876i	0.890402	−	0.890402i	−0.104158	−	0.994561i	$-0.533215\pi$
$43$	5.83876	−	5.83876i	0.890402	−	0.890402i	0.994561	+	0.104158i	$0.0332149\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.04989	−	4.04989i	−0.590737	−	0.590737i	0.347094	−	0.937830i	$-0.387169\pi$
$47$	−4.04989	−	4.04989i	−0.590737	−	0.590737i	−0.937830	+	0.347094i	$0.887169\pi$
$48$		0			0
$49$			16.6556i			2.37937i
$50$		0			0
$51$			0.964724i			0.135088i
$52$		0			0
$53$	10.0599	+	10.0599i	1.38183	+	1.38183i	0.841374	+	0.540454i	$0.181747\pi$
$53$	10.0599	+	10.0599i	1.38183	+	1.38183i	0.540454	+	0.841374i	$0.318253\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.90691	−	1.90691i	0.252576	−	0.252576i
$58$		0			0
$59$	3.46410			0.450988			0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410			0.450988			0.225494	+	0.974245i	$0.427600\pi$
$60$		0			0
$61$	−2.87780			−0.368464			−0.184232	−	0.982883i	$-0.558980\pi$
$61$	−2.87780			−0.368464			−0.184232	+	0.982883i	$0.558980\pi$
$62$		0			0
$63$	−3.43916	+	3.43916i	−0.433293	+	0.433293i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−7.35275	−	7.35275i	−0.898281	−	0.898281i	0.0970029	−	0.995284i	$-0.469074\pi$
$67$	−7.35275	−	7.35275i	−0.898281	−	0.898281i	−0.995284	+	0.0970029i	$0.969074\pi$
$68$		0			0
$69$		−	7.32780i		−	0.882164i
$70$		0			0
$71$			14.0857i			1.67166i	0.548986	+	0.835831i	$0.315014\pi$
$71$			14.0857i			1.67166i	−0.548986	+	0.835831i	$0.684986\pi$
$72$		0			0
$73$	9.46410	+	9.46410i	1.10769	+	1.10769i	0.993454	+	0.114236i	$0.0364419\pi$
$73$	9.46410	+	9.46410i	1.10769	+	1.10769i	0.114236	+	0.993454i	$0.463558\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.4388	+	10.4388i	−1.18961	+	1.18961i
$78$		0			0
$79$	2.60645			0.293249			0.146624	−	0.989192i	$-0.453159\pi$
$79$	2.60645			0.293249			0.146624	+	0.989192i	$0.453159\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	−1.97934	+	1.97934i	−0.217260	+	0.217260i	−0.807343	−	0.590083i	$-0.799095\pi$
$83$	−1.97934	+	1.97934i	−0.217260	+	0.217260i	0.590083	+	0.807343i	$0.299095\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.282561	−	0.282561i	−0.0302937	−	0.0302937i
$88$		0			0
$89$		−	0.799203i		−	0.0847154i	−0.999103	−	0.0423577i	$-0.986513\pi$
$89$		−	0.799203i		−	0.0847154i	0.999103	−	0.0423577i	$-0.0134869\pi$
$90$		0			0
$91$			18.1516i			1.90280i
$92$		0			0
$93$	−3.50731	−	3.50731i	−0.363691	−	0.363691i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.94558	+	8.94558i	−0.908286	+	0.908286i	−0.996134	−	0.0878475i	$-0.972001\pi$
$97$	−8.94558	+	8.94558i	−0.908286	+	0.908286i	0.0878475	+	0.996134i	$0.472001\pi$
$98$		0			0
$99$	−3.03528			−0.305057
$100$		0			0
$101$	−4.52860			−0.450613			−0.225306	−	0.974288i	$-0.572338\pi$
$101$	−4.52860			−0.450613			−0.225306	+	0.974288i	$0.572338\pi$
$102$		0			0
$103$	5.36433	−	5.36433i	0.528563	−	0.528563i	−0.391581	−	0.920144i	$-0.628072\pi$
$103$	5.36433	−	5.36433i	0.528563	−	0.528563i	0.920144	+	0.391581i	$0.128072\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.62498	−	8.62498i	−0.833808	−	0.833808i	0.154227	−	0.988035i	$-0.450711\pi$
$107$	−8.62498	−	8.62498i	−0.833808	−	0.833808i	−0.988035	+	0.154227i	$0.950711\pi$
$108$		0			0
$109$		−	1.66490i		−	0.159468i	−0.996816	−	0.0797342i	$-0.974593\pi$
$109$		−	1.66490i		−	0.159468i	0.996816	−	0.0797342i	$-0.0254071\pi$
$110$		0			0
$111$		−	6.07055i		−	0.576191i
$112$		0			0
$113$	−8.00000	−	8.00000i	−0.752577	−	0.752577i	0.222383	−	0.974959i	$-0.428617\pi$
$113$	−8.00000	−	8.00000i	−0.752577	−	0.752577i	−0.974959	+	0.222383i	$0.928617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.63896	+	2.63896i	−0.243972	+	0.243972i
$118$		0			0
$119$	−4.69213			−0.430127
$120$		0			0
$121$	1.78710			0.162464
$122$		0			0
$123$	−7.76733	+	7.76733i	−0.700356	+	0.700356i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	7.43488	+	7.43488i	0.659739	+	0.659739i	0.955318	−	0.295579i	$-0.0955127\pi$
$127$	7.43488	+	7.43488i	0.659739	+	0.659739i	−0.295579	+	0.955318i	$0.595513\pi$
$128$		0			0
$129$			8.25725i			0.727011i
$130$		0			0
$131$			2.88488i			0.252053i	0.992027	+	0.126027i	$0.0402225\pi$
$131$			2.88488i			0.252053i	−0.992027	+	0.126027i	$0.959777\pi$
$132$		0			0
$133$	9.27463	+	9.27463i	0.804213	+	0.804213i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.47067	−	9.47067i	0.809134	−	0.809134i	−0.175369	−	0.984503i	$-0.556112\pi$
$137$	9.47067	−	9.47067i	0.809134	−	0.809134i	0.984503	+	0.175369i	$0.0561118\pi$
$138$		0			0
$139$	−0.535898			−0.0454543			−0.0227272	−	0.999742i	$-0.507235\pi$
$139$	−0.535898			−0.0454543			−0.0227272	+	0.999742i	$0.507235\pi$
$140$		0			0
$141$	5.72741			0.482335
$142$		0			0
$143$	−8.00997	+	8.00997i	−0.669827	+	0.669827i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−11.7773	−	11.7773i	−0.971375	−	0.971375i
$148$		0			0
$149$		−	13.0693i		−	1.07068i	−0.844637	−	0.535340i	$-0.820184\pi$
$149$		−	13.0693i		−	1.07068i	0.844637	−	0.535340i	$-0.179816\pi$
$150$		0			0
$151$			5.95884i			0.484923i	0.970161	+	0.242462i	$0.0779548\pi$
$151$			5.95884i			0.484923i	−0.970161	+	0.242462i	$0.922045\pi$
$152$		0			0
$153$	−0.682163	−	0.682163i	−0.0551496	−	0.0551496i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−12.8609	+	12.8609i	−1.02641	+	1.02641i	−0.0267728	+	0.999642i	$0.508523\pi$
$157$	−12.8609	+	12.8609i	−1.02641	+	1.02641i	−0.999642	+	0.0267728i	$0.991477\pi$
$158$		0			0
$159$	−14.2268			−1.12826
$160$		0			0
$161$	35.6403			2.80885
$162$		0			0
$163$	−12.4445	+	12.4445i	−0.974727	+	0.974727i	−0.999688	−	0.0249614i	$-0.992054\pi$
$163$	−12.4445	+	12.4445i	−0.974727	+	0.974727i	0.0249614	+	0.999688i	$0.492054\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.49473	−	3.49473i	−0.270431	−	0.270431i	0.558843	−	0.829274i	$-0.311245\pi$
$167$	−3.49473	−	3.49473i	−0.270431	−	0.270431i	−0.829274	+	0.558843i	$0.811245\pi$
$168$		0			0
$169$			0.928203i			0.0714002i
$170$		0			0
$171$			2.69677i			0.206228i
$172$		0			0
$173$	−17.3085	−	17.3085i	−1.31594	−	1.31594i	−0.916956	−	0.398989i	$-0.869361\pi$
$173$	−17.3085	−	17.3085i	−1.31594	−	1.31594i	−0.398989	−	0.916956i	$-0.630639\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.44949	+	2.44949i	−0.184115	+	0.184115i
$178$		0			0
$179$	−25.8471			−1.93190			−0.965952	−	0.258721i	$-0.916699\pi$
$179$	−25.8471			−1.93190			−0.965952	+	0.258721i	$0.916699\pi$
$180$		0			0
$181$	−0.605206			−0.0449846			−0.0224923	−	0.999747i	$-0.507160\pi$
$181$	−0.605206			−0.0449846			−0.0224923	+	0.999747i	$0.507160\pi$
$182$		0			0
$183$	2.03491	−	2.03491i	0.150425	−	0.150425i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.07055	−	2.07055i	−0.151414	−	0.151414i
$188$		0			0
$189$		−	4.86370i		−	0.353782i
$190$		0			0
$191$		−	16.6191i		−	1.20252i	−0.799055	−	0.601258i	$-0.794667\pi$
$191$		−	16.6191i		−	1.20252i	0.799055	−	0.601258i	$-0.205333\pi$
$192$		0			0
$193$	16.3457	+	16.3457i	1.17659	+	1.17659i	0.980608	+	0.195982i	$0.0627893\pi$
$193$	16.3457	+	16.3457i	1.17659	+	1.17659i	0.195982	+	0.980608i	$0.437211\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.2273	+	15.2273i	−1.08490	+	1.08490i	−0.0888557	+	0.996045i	$0.528321\pi$
$197$	−15.2273	+	15.2273i	−1.08490	+	1.08490i	−0.996045	+	0.0888557i	$0.971679\pi$
$198$		0			0
$199$	14.8165			1.05031			0.525156	−	0.851006i	$-0.324007\pi$
$199$	14.8165			1.05031			0.525156	+	0.851006i	$0.324007\pi$
$200$		0			0
$201$	10.3984			0.733444
$202$		0			0
$203$	1.37429	−	1.37429i	0.0964565	−	0.0964565i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5.18154	+	5.18154i	0.360142	+	0.360142i
$208$		0			0
$209$			8.18546i			0.566200i
$210$		0			0
$211$			13.8298i			0.952085i	0.879422	+	0.476043i	$0.157929\pi$
$211$			13.8298i			0.952085i	−0.879422	+	0.476043i	$0.842071\pi$
$212$		0			0
$213$	−9.96008	−	9.96008i	−0.682453	−	0.682453i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	17.0585	−	17.0585i	1.15801	−	1.15801i
$218$		0			0
$219$	−13.3843			−0.904425
$220$		0			0
$221$	−3.60040			−0.242189
$222$		0			0
$223$	−3.92444	+	3.92444i	−0.262800	+	0.262800i	−0.826191	−	0.563391i	$-0.809497\pi$
$223$	−3.92444	+	3.92444i	−0.262800	+	0.262800i	0.563391	+	0.826191i	$0.309497\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	20.6862	+	20.6862i	1.37299	+	1.37299i	0.855972	+	0.517022i	$0.172959\pi$
$227$	20.6862	+	20.6862i	1.37299	+	1.37299i	0.517022	+	0.855972i	$0.327041\pi$
$228$		0			0
$229$		−	21.9068i		−	1.44764i	−0.689987	−	0.723821i	$-0.742384\pi$
$229$		−	21.9068i		−	1.44764i	0.689987	−	0.723821i	$-0.257616\pi$
$230$		0			0
$231$		−	14.7627i		−	0.971313i
$232$		0			0
$233$	2.52520	+	2.52520i	0.165431	+	0.165431i	0.784968	−	0.619536i	$-0.212679\pi$
$233$	2.52520	+	2.52520i	0.165431	+	0.165431i	−0.619536	+	0.784968i	$0.712679\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.84304	+	1.84304i	−0.119718	+	0.119718i
$238$		0			0
$239$	−6.91308			−0.447170			−0.223585	−	0.974684i	$-0.571776\pi$
$239$	−6.91308			−0.447170			−0.223585	+	0.974684i	$0.571776\pi$
$240$		0			0
$241$	−13.6677			−0.880415			−0.440207	−	0.897896i	$-0.645095\pi$
$241$	−13.6677			−0.880415			−0.440207	+	0.897896i	$0.645095\pi$
$242$		0			0
$243$	0.707107	−	0.707107i	0.0453609	−	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	7.11668	+	7.11668i	0.452823	+	0.452823i
$248$		0			0
$249$		−	2.79920i		−	0.177392i
$250$		0			0
$251$		−	9.59841i		−	0.605846i	−0.953015	−	0.302923i	$-0.902037\pi$
$251$		−	9.59841i		−	0.605846i	0.953015	−	0.302923i	$-0.0979625\pi$
$252$		0			0
$253$	15.7274	+	15.7274i	0.988774	+	0.988774i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.24213	+	7.24213i	−0.451751	+	0.451751i	−0.895936	−	0.444184i	$-0.853494\pi$
$257$	−7.24213	+	7.24213i	−0.451751	+	0.451751i	0.444184	+	0.895936i	$0.353494\pi$
$258$		0			0
$259$	29.5254			1.83462
$260$		0			0
$261$	0.399602			0.0247347
$262$		0			0
$263$	2.18206	−	2.18206i	0.134551	−	0.134551i	−0.636623	−	0.771175i	$-0.719669\pi$
$263$	2.18206	−	2.18206i	0.134551	−	0.134551i	0.771175	+	0.636623i	$0.219669\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.565122	+	0.565122i	0.0345849	+	0.0345849i
$268$		0			0
$269$		−	6.25601i		−	0.381436i	−0.981645	−	0.190718i	$-0.938918\pi$
$269$		−	6.25601i		−	0.381436i	0.981645	−	0.190718i	$-0.0610815\pi$
$270$		0			0
$271$			16.1318i			0.979938i	0.871740	+	0.489969i	$0.162992\pi$
$271$			16.1318i			0.979938i	−0.871740	+	0.489969i	$0.837008\pi$
$272$		0			0
$273$	−12.8351	−	12.8351i	−0.776816	−	0.776816i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−7.05922	+	7.05922i	−0.424148	+	0.424148i	−0.886629	−	0.462481i	$-0.846959\pi$
$277$	−7.05922	+	7.05922i	−0.424148	+	0.424148i	0.462481	+	0.886629i	$0.346959\pi$
$278$		0			0
$279$	4.96008			0.296952
$280$		0			0
$281$	−20.8436			−1.24342			−0.621711	−	0.783247i	$-0.713562\pi$
$281$	−20.8436			−1.24342			−0.621711	+	0.783247i	$0.713562\pi$
$282$		0			0
$283$	10.3540	−	10.3540i	0.615481	−	0.615481i	−0.328888	−	0.944369i	$-0.606674\pi$
$283$	10.3540	−	10.3540i	0.615481	−	0.615481i	0.944369	+	0.328888i	$0.106674\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−37.7780	−	37.7780i	−2.22996	−	2.22996i
$288$		0			0
$289$			16.0693i			0.945253i
$290$		0			0
$291$		−	12.6510i		−	0.741613i
$292$		0			0
$293$	−1.50918	−	1.50918i	−0.0881673	−	0.0881673i	0.661648	−	0.749815i	$-0.269857\pi$
$293$	−1.50918	−	1.50918i	−0.0881673	−	0.0881673i	−0.749815	+	0.661648i	$0.769857\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	2.14626	−	2.14626i	0.124539	−	0.124539i
$298$		0			0
$299$	27.3477			1.58156
$300$		0			0
$301$	−40.1608			−2.31483
$302$		0			0
$303$	3.20220	−	3.20220i	0.183962	−	0.183962i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	5.64601	+	5.64601i	0.322235	+	0.322235i	0.849624	−	0.527389i	$-0.176829\pi$
$307$	5.64601	+	5.64601i	0.322235	+	0.322235i	−0.527389	+	0.849624i	$0.676829\pi$
$308$		0			0
$309$			7.58630i			0.431570i
$310$		0			0
$311$			29.3205i			1.66261i	0.555814	+	0.831307i	$0.312407\pi$
$311$			29.3205i			1.66261i	−0.555814	+	0.831307i	$0.687593\pi$
$312$		0			0
$313$	0.139581	+	0.139581i	0.00788956	+	0.00788956i	0.711041	−	0.703151i	$-0.248224\pi$
$313$	0.139581	+	0.139581i	0.00788956	+	0.00788956i	−0.703151	+	0.711041i	$0.748224\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.8272	−	15.8272i	0.888943	−	0.888943i	−0.105478	−	0.994422i	$-0.533637\pi$
$317$	15.8272	−	15.8272i	0.888943	−	0.888943i	0.994422	+	0.105478i	$0.0336374\pi$
$318$		0			0
$319$	1.21290			0.0679094
$320$		0			0
$321$	12.1976			0.680801
$322$		0			0
$323$	−1.83964	+	1.83964i	−0.102360	+	0.102360i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1.17726	+	1.17726i	0.0651027	+	0.0651027i
$328$		0			0
$329$			27.8564i			1.53577i
$330$		0			0
$331$			29.4616i			1.61936i	0.586874	+	0.809678i	$0.300358\pi$
$331$			29.4616i			1.61936i	−0.586874	+	0.809678i	$0.699642\pi$
$332$		0			0
$333$	4.29253	+	4.29253i	0.235229	+	0.235229i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	24.8244	−	24.8244i	1.35227	−	1.35227i	0.469157	−	0.883115i	$-0.344558\pi$
$337$	24.8244	−	24.8244i	1.35227	−	1.35227i	0.883115	−	0.469157i	$-0.155442\pi$
$338$		0			0
$339$	11.3137			0.614476
$340$		0			0
$341$	15.0552			0.815285
$342$		0			0
$343$	33.2072	−	33.2072i	1.79302	−	1.79302i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.7468	+	18.7468i	1.00638	+	1.00638i	0.999980	+	0.00640288i	$0.00203811\pi$
$347$	18.7468	+	18.7468i	1.00638	+	1.00638i	0.00640288	+	0.999980i	$0.497962\pi$
$348$		0			0
$349$			5.45481i			0.291989i	0.989285	+	0.145995i	$0.0466383\pi$
$349$			5.45481i			0.291989i	−0.989285	+	0.145995i	$0.953362\pi$
$350$		0			0
$351$		−	3.73205i		−	0.199202i
$352$		0			0
$353$	12.8233	+	12.8233i	0.682514	+	0.682514i	0.960566	−	0.278052i	$-0.0896889\pi$
$353$	12.8233	+	12.8233i	0.682514	+	0.682514i	−0.278052	+	0.960566i	$0.589689\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	3.31784	−	3.31784i	0.175599	−	0.175599i
$358$		0			0
$359$	−17.6135			−0.929607			−0.464803	−	0.885414i	$-0.653875\pi$
$359$	−17.6135			−0.929607			−0.464803	+	0.885414i	$0.653875\pi$
$360$		0			0
$361$	−11.7274			−0.617232
$362$		0			0
$363$	−1.26367	+	1.26367i	−0.0663254	+	0.0663254i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	4.95315	+	4.95315i	0.258552	+	0.258552i	0.824465	−	0.565913i	$-0.191476\pi$
$367$	4.95315	+	4.95315i	0.258552	+	0.258552i	−0.565913	+	0.824465i	$0.691476\pi$
$368$		0			0
$369$		−	10.9847i		−	0.571839i
$370$		0			0
$371$		−	69.1949i		−	3.59242i
$372$		0			0
$373$	−0.0166514	−	0.0166514i	−0.000862177	−	0.000862177i	0.706676	−	0.707538i	$-0.250194\pi$
$373$	−0.0166514	−	0.0166514i	−0.000862177	−	0.000862177i	−0.707538	+	0.706676i	$0.750194\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.05453	−	1.05453i	0.0543112	−	0.0543112i
$378$		0			0
$379$	8.23143			0.422820			0.211410	−	0.977397i	$-0.432194\pi$
$379$	8.23143			0.422820			0.211410	+	0.977397i	$0.432194\pi$
$380$		0			0
$381$	−10.5145			−0.538674
$382$		0			0
$383$	−15.9202	+	15.9202i	−0.813482	+	0.813482i	−0.985154	−	0.171672i	$-0.945083\pi$
$383$	−15.9202	+	15.9202i	−0.813482	+	0.813482i	0.171672	+	0.985154i	$0.445083\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−5.83876	−	5.83876i	−0.296801	−	0.296801i
$388$		0			0
$389$		−	19.7012i		−	0.998891i	−0.866345	−	0.499445i	$-0.833537\pi$
$389$		−	19.7012i		−	0.998891i	0.866345	−	0.499445i	$-0.166463\pi$
$390$		0			0
$391$			7.06931i			0.357510i
$392$		0			0
$393$	−2.03992	−	2.03992i	−0.102900	−	0.102900i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	5.08188	−	5.08188i	0.255052	−	0.255052i	−0.567986	−	0.823038i	$-0.692277\pi$
$397$	5.08188	−	5.08188i	0.255052	−	0.255052i	0.823038	+	0.567986i	$0.192277\pi$
$398$		0			0
$399$	−13.1163			−0.656637
$400$		0			0
$401$	−19.0976			−0.953687			−0.476844	−	0.878988i	$-0.658219\pi$
$401$	−19.0976			−0.953687			−0.476844	+	0.878988i	$0.658219\pi$
$402$		0			0
$403$	13.0894	−	13.0894i	0.652032	−	0.652032i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.0290	+	13.0290i	0.645824	+	0.645824i
$408$		0			0
$409$			27.1104i			1.34052i	0.742124	+	0.670262i	$0.233818\pi$
$409$			27.1104i			1.34052i	−0.742124	+	0.670262i	$0.766182\pi$
$410$		0			0
$411$			13.3935i			0.660655i
$412$		0			0
$413$	−11.9136	−	11.9136i	−0.586230	−	0.586230i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.378937	−	0.378937i	0.0185566	−	0.0185566i
$418$		0			0
$419$	−13.9542			−0.681707			−0.340853	−	0.940116i	$-0.610716\pi$
$419$	−13.9542			−0.681707			−0.340853	+	0.940116i	$0.610716\pi$
$420$		0			0
$421$	−17.0411			−0.830533			−0.415267	−	0.909700i	$-0.636312\pi$
$421$	−17.0411			−0.830533			−0.415267	+	0.909700i	$0.636312\pi$
$422$		0			0
$423$	−4.04989	+	4.04989i	−0.196912	+	0.196912i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	9.89721	+	9.89721i	0.478959	+	0.478959i
$428$		0			0
$429$		−	11.3278i		−	0.546912i
$430$		0			0
$431$		−	11.0751i		−	0.533471i	−0.963770	−	0.266736i	$-0.914055\pi$
$431$		−	11.0751i		−	0.533471i	0.963770	−	0.266736i	$-0.0859450\pi$
$432$		0			0
$433$	−13.7234	−	13.7234i	−0.659504	−	0.659504i	0.295759	−	0.955263i	$-0.404428\pi$
$433$	−13.7234	−	13.7234i	−0.659504	−	0.659504i	−0.955263	+	0.295759i	$0.904428\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	13.9734	−	13.9734i	0.668441	−	0.668441i
$438$		0			0
$439$	9.48669			0.452775			0.226387	−	0.974037i	$-0.427308\pi$
$439$	9.48669			0.452775			0.226387	+	0.974037i	$0.427308\pi$
$440$		0			0
$441$	16.6556			0.793124
$442$		0			0
$443$	24.6050	−	24.6050i	1.16902	−	1.16902i	0.186578	−	0.982440i	$-0.440260\pi$
$443$	24.6050	−	24.6050i	1.16902	−	1.16902i	0.982440	−	0.186578i	$-0.0597398\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	9.24140	+	9.24140i	0.437103	+	0.437103i
$448$		0			0
$449$		−	12.0000i		−	0.566315i	−0.959073	−	0.283158i	$-0.908618\pi$
$449$		−	12.0000i		−	0.566315i	0.959073	−	0.283158i	$-0.0913819\pi$
$450$		0			0
$451$		−	33.3415i		−	1.56999i
$452$		0			0
$453$	−4.21353	−	4.21353i	−0.197969	−	0.197969i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−10.8284	+	10.8284i	−0.506532	+	0.506532i	−0.913460	−	0.406928i	$-0.866600\pi$
$457$	−10.8284	+	10.8284i	−0.506532	+	0.506532i	0.406928	+	0.913460i	$0.366600\pi$
$458$		0			0
$459$	0.964724			0.0450295
$460$		0			0
$461$	6.26811			0.291935			0.145968	−	0.989289i	$-0.453370\pi$
$461$	6.26811			0.291935			0.145968	+	0.989289i	$0.453370\pi$
$462$		0			0
$463$	2.73545	−	2.73545i	0.127127	−	0.127127i	−0.640680	−	0.767808i	$-0.721348\pi$
$463$	2.73545	−	2.73545i	0.127127	−	0.127127i	0.767808	+	0.640680i	$0.221348\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.3519	+	11.3519i	0.525302	+	0.525302i	0.919168	−	0.393866i	$-0.128863\pi$
$467$	11.3519	+	11.3519i	0.525302	+	0.525302i	−0.393866	+	0.919168i	$0.628863\pi$
$468$		0			0
$469$			50.5745i			2.33531i
$470$		0			0
$471$		−	18.1881i		−	0.838064i
$472$		0			0
$473$	−17.7222	−	17.7222i	−0.814870	−	0.814870i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	10.0599	−	10.0599i	0.460609	−	0.460609i
$478$		0			0
$479$	17.3750			0.793883			0.396941	−	0.917844i	$-0.370072\pi$
$479$	17.3750			0.793883			0.396941	+	0.917844i	$0.370072\pi$
$480$		0			0
$481$	22.6556			1.03301
$482$		0			0
$483$	−25.2015	+	25.2015i	−1.14671	+	1.14671i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−2.11792	−	2.11792i	−0.0959721	−	0.0959721i	0.657491	−	0.753463i	$-0.271618\pi$
$487$	−2.11792	−	2.11792i	−0.0959721	−	0.0959721i	−0.753463	+	0.657491i	$0.771618\pi$
$488$		0			0
$489$		−	17.5992i		−	0.795861i
$490$		0			0
$491$		−	20.3855i		−	0.919985i	−0.887923	−	0.459992i	$-0.847852\pi$
$491$		−	20.3855i		−	0.919985i	0.887923	−	0.459992i	$-0.152148\pi$
$492$		0			0
$493$	0.272593	+	0.272593i	0.0122770	+	0.0122770i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	48.4429	−	48.4429i	2.17296	−	2.17296i
$498$		0			0
$499$	−9.63427			−0.431289			−0.215644	−	0.976472i	$-0.569185\pi$
$499$	−9.63427			−0.431289			−0.215644	+	0.976472i	$0.569185\pi$
$500$		0			0
$501$	4.94230			0.220806
$502$		0			0
$503$	−8.10118	+	8.10118i	−0.361214	+	0.361214i	−0.864260	−	0.503046i	$-0.832213\pi$
$503$	−8.10118	+	8.10118i	−0.361214	+	0.361214i	0.503046	+	0.864260i	$0.332213\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.656339	−	0.656339i	−0.0291490	−	0.0291490i
$508$		0			0
$509$		−	33.3233i		−	1.47703i	−0.674237	−	0.738515i	$-0.735527\pi$
$509$		−	33.3233i		−	1.47703i	0.674237	−	0.738515i	$-0.264473\pi$
$510$		0			0
$511$		−	65.0971i		−	2.87973i
$512$		0			0
$513$	−1.90691	−	1.90691i	−0.0841920	−	0.0841920i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−12.2925	+	12.2925i	−0.540625	+	0.540625i
$518$		0			0
$519$	24.4780			1.07446
$520$		0			0
$521$	−5.64276			−0.247214			−0.123607	−	0.992331i	$-0.539446\pi$
$521$	−5.64276			−0.247214			−0.123607	+	0.992331i	$0.539446\pi$
$522$		0			0
$523$	13.3837	−	13.3837i	0.585230	−	0.585230i	−0.351106	−	0.936336i	$-0.614194\pi$
$523$	13.3837	−	13.3837i	0.585230	−	0.585230i	0.936336	+	0.351106i	$0.114194\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.38358	+	3.38358i	0.147391	+	0.147391i
$528$		0			0
$529$		−	30.6967i		−	1.33464i
$530$		0			0
$531$		−	3.46410i		−	0.150329i
$532$		0			0
$533$	−28.9881	−	28.9881i	−1.25561	−	1.25561i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	18.2767	−	18.2767i	0.788697	−	0.788697i
$538$		0			0
$539$	50.5544			2.17753
$540$		0			0
$541$	−10.9907			−0.472528			−0.236264	−	0.971689i	$-0.575923\pi$
$541$	−10.9907			−0.472528			−0.236264	+	0.971689i	$0.575923\pi$
$542$		0			0
$543$	0.427946	−	0.427946i	0.0183649	−	0.0183649i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−15.1623	−	15.1623i	−0.648292	−	0.648292i	0.304288	−	0.952580i	$-0.401582\pi$
$547$	−15.1623	−	15.1623i	−0.648292	−	0.648292i	−0.952580	+	0.304288i	$0.901582\pi$
$548$		0			0
$549$			2.87780i			0.122821i
$550$		0			0
$551$		−	1.07764i		−	0.0459088i
$552$		0			0
$553$	−8.96399	−	8.96399i	−0.381188	−	0.381188i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.144856	−	0.144856i	0.00613775	−	0.00613775i	−0.704031	−	0.710169i	$-0.748619\pi$
$557$	0.144856	−	0.144856i	0.00613775	−	0.00613775i	0.710169	+	0.704031i	$0.248619\pi$
$558$		0			0
$559$	−30.8165			−1.30340
$560$		0			0
$561$	2.92820			0.123629
$562$		0			0
$563$	−21.8182	+	21.8182i	−0.919529	+	0.919529i	−0.996995	−	0.0774656i	$-0.975317\pi$
$563$	−21.8182	+	21.8182i	−0.919529	+	0.919529i	0.0774656	+	0.996995i	$0.475317\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	3.43916	+	3.43916i	0.144431	+	0.144431i
$568$		0			0
$569$		−	0.0725439i		−	0.00304120i	−0.999999	−	0.00152060i	$-0.999516\pi$
$569$		−	0.0725439i		−	0.00304120i	0.999999	−	0.00152060i	$-0.000484022\pi$
$570$		0			0
$571$			10.7741i			0.450883i	0.974257	+	0.225442i	$0.0723825\pi$
$571$			10.7741i			0.450883i	−0.974257	+	0.225442i	$0.927618\pi$
$572$		0			0
$573$	11.7515	+	11.7515i	0.490925	+	0.490925i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−24.9587	+	24.9587i	−1.03905	+	1.03905i	−0.0398390	+	0.999206i	$0.512684\pi$
$577$	−24.9587	+	24.9587i	−1.03905	+	1.03905i	−0.999206	+	0.0398390i	$0.987316\pi$
$578$		0			0
$579$	−23.1163			−0.960681
$580$		0			0
$581$	13.6145			0.564824
$582$		0			0
$583$	30.5344	−	30.5344i	1.26461	−	1.26461i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.51467	+	7.51467i	0.310164	+	0.310164i	0.844973	−	0.534809i	$-0.179617\pi$
$587$	7.51467	+	7.51467i	0.310164	+	0.310164i	−0.534809	+	0.844973i	$0.679617\pi$
$588$		0			0
$589$		−	13.3762i		−	0.551157i
$590$		0			0
$591$		−	21.5347i		−	0.885817i
$592$		0			0
$593$	22.8790	+	22.8790i	0.939528	+	0.939528i	0.998273	−	0.0587453i	$-0.0187100\pi$
$593$	22.8790	+	22.8790i	0.939528	+	0.939528i	−0.0587453	+	0.998273i	$0.518710\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−10.4768	+	10.4768i	−0.428788	+	0.428788i
$598$		0			0
$599$	−24.5947			−1.00491			−0.502456	−	0.864603i	$-0.667570\pi$
$599$	−24.5947			−1.00491			−0.502456	+	0.864603i	$0.667570\pi$
$600$		0			0
$601$	−40.2830			−1.64318			−0.821589	−	0.570080i	$-0.806912\pi$
$601$	−40.2830			−1.64318			−0.821589	+	0.570080i	$0.806912\pi$
$602$		0			0
$603$	−7.35275	+	7.35275i	−0.299427	+	0.299427i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	25.1399	+	25.1399i	1.02040	+	1.02040i	0.999788	+	0.0206078i	$0.00656014\pi$
$607$	25.1399	+	25.1399i	1.02040	+	1.02040i	0.0206078	+	0.999788i	$0.493440\pi$
$608$		0			0
$609$			1.94354i			0.0787564i
$610$		0			0
$611$			21.3750i			0.864739i
$612$		0			0
$613$	−18.5558	−	18.5558i	−0.749463	−	0.749463i	0.224915	−	0.974378i	$-0.427789\pi$
$613$	−18.5558	−	18.5558i	−0.749463	−	0.749463i	−0.974378	+	0.224915i	$0.927789\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−20.4960	+	20.4960i	−0.825137	+	0.825137i	−0.986840	−	0.161703i	$-0.948301\pi$
$617$	−20.4960	+	20.4960i	−0.825137	+	0.825137i	0.161703	+	0.986840i	$0.448301\pi$
$618$		0			0
$619$	−24.6169			−0.989438			−0.494719	−	0.869053i	$-0.664729\pi$
$619$	−24.6169			−0.989438			−0.494719	+	0.869053i	$0.664729\pi$
$620$		0			0
$621$	−7.32780			−0.294055
$622$		0			0
$623$	−2.74859	+	2.74859i	−0.110120	+	0.110120i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−5.78799	−	5.78799i	−0.231150	−	0.231150i
$628$		0			0
$629$			5.85641i			0.233510i
$630$		0			0
$631$			5.36847i			0.213716i	0.994274	+	0.106858i	$0.0340789\pi$
$631$			5.36847i			0.213716i	−0.994274	+	0.106858i	$0.965921\pi$
$632$		0			0
$633$	−9.77917	−	9.77917i	−0.388687	−	0.388687i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	43.9535	−	43.9535i	1.74150	−	1.74150i
$638$		0			0
$639$	14.0857			0.557221
$640$		0			0
$641$	−8.91610			−0.352165			−0.176082	−	0.984375i	$-0.556343\pi$
$641$	−8.91610			−0.352165			−0.176082	+	0.984375i	$0.556343\pi$
$642$		0			0
$643$	−4.05165	+	4.05165i	−0.159781	+	0.159781i	−0.782470	−	0.622688i	$-0.786040\pi$
$643$	−4.05165	+	4.05165i	−0.159781	+	0.159781i	0.622688	+	0.782470i	$0.286040\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.00804	+	3.00804i	0.118258	+	0.118258i	0.763759	−	0.645501i	$-0.223351\pi$
$647$	3.00804	+	3.00804i	0.118258	+	0.118258i	−0.645501	+	0.763759i	$0.723351\pi$
$648$		0			0
$649$		−	10.5145i		−	0.412730i
$650$		0			0
$651$			24.1244i			0.945508i
$652$		0			0
$653$	6.20861	+	6.20861i	0.242962	+	0.242962i	0.818074	−	0.575113i	$-0.195042\pi$
$653$	6.20861	+	6.20861i	0.242962	+	0.242962i	−0.575113	+	0.818074i	$0.695042\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	9.46410	−	9.46410i	0.369230	−	0.369230i
$658$		0			0
$659$	−12.6002			−0.490834			−0.245417	−	0.969418i	$-0.578925\pi$
$659$	−12.6002			−0.490834			−0.245417	+	0.969418i	$0.578925\pi$
$660$		0			0
$661$	−36.5241			−1.42062			−0.710312	−	0.703887i	$-0.751446\pi$
$661$	−36.5241			−1.42062			−0.710312	+	0.703887i	$0.751446\pi$
$662$		0			0
$663$	2.54587	−	2.54587i	0.0988732	−	0.0988732i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.07055	−	2.07055i	−0.0801721	−	0.0801721i
$668$		0			0
$669$		−	5.54999i		−	0.214575i
$670$		0			0
$671$			8.73492i			0.337208i
$672$		0			0
$673$	0.879044	+	0.879044i	0.0338847	+	0.0338847i	0.723846	−	0.689961i	$-0.242373\pi$
$673$	0.879044	+	0.879044i	0.0338847	+	0.0338847i	−0.689961	+	0.723846i	$0.742373\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.7525	−	21.7525i	0.836017	−	0.836017i	−0.152315	−	0.988332i	$-0.548673\pi$
$677$	21.7525	−	21.7525i	0.836017	−	0.836017i	0.988332	+	0.152315i	$0.0486729\pi$
$678$		0			0
$679$	61.5305			2.36133
$680$		0			0
$681$	−29.2548			−1.12104
$682$		0			0
$683$	17.7181	−	17.7181i	0.677965	−	0.677965i	−0.281574	−	0.959539i	$-0.590857\pi$
$683$	17.7181	−	17.7181i	0.677965	−	0.677965i	0.959539	+	0.281574i	$0.0908566\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	15.4905	+	15.4905i	0.590998	+	0.590998i
$688$		0			0
$689$		−	53.0951i		−	2.02276i
$690$		0			0
$691$			32.3762i			1.23165i	0.787883	+	0.615825i	$0.211177\pi$
$691$			32.3762i			1.23165i	−0.787883	+	0.615825i	$0.788823\pi$
$692$		0			0
$693$	10.4388	+	10.4388i	0.396537	+	0.396537i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	7.49333	−	7.49333i	0.283830	−	0.283830i
$698$		0			0
$699$	−3.57117			−0.135074
$700$		0			0
$701$	2.94230			0.111129			0.0555646	−	0.998455i	$-0.482304\pi$
$701$	2.94230			0.111129			0.0555646	+	0.998455i	$0.482304\pi$
$702$		0			0
$703$	11.5760	−	11.5760i	0.436596	−	0.436596i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	15.5746	+	15.5746i	0.585742	+	0.585742i
$708$		0			0
$709$			43.7471i			1.64296i	0.570239	+	0.821479i	$0.306851\pi$
$709$			43.7471i			1.64296i	−0.570239	+	0.821479i	$0.693149\pi$
$710$		0			0
$711$		−	2.60645i		−	0.0977495i
$712$		0			0
$713$	−25.7009	−	25.7009i	−0.962505	−	0.962505i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	4.88828	−	4.88828i	0.182556	−	0.182556i
$718$		0			0
$719$	4.90628			0.182973			0.0914866	−	0.995806i	$-0.470838\pi$
$719$	4.90628			0.182973			0.0914866	+	0.995806i	$0.470838\pi$
$720$		0			0
$721$	−36.8975			−1.37414
$722$		0			0
$723$	9.66453	−	9.66453i	0.359428	−	0.359428i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	29.6226	+	29.6226i	1.09864	+	1.09864i	0.994570	+	0.104073i	$0.0331874\pi$
$727$	29.6226	+	29.6226i	1.09864	+	1.09864i	0.104073	+	0.994570i	$0.466813\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$		−	7.96597i		−	0.294632i
$732$		0			0
$733$	17.5254	+	17.5254i	0.647314	+	0.647314i	0.952343	−	0.305029i	$-0.0986661\pi$
$733$	17.5254	+	17.5254i	0.647314	+	0.647314i	−0.305029	+	0.952343i	$0.598666\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−22.3176	+	22.3176i	−0.822080	+	0.822080i
$738$		0			0
$739$	−19.5322			−0.718502			−0.359251	−	0.933241i	$-0.616968\pi$
$739$	−19.5322			−0.718502			−0.359251	+	0.933241i	$0.616968\pi$
$740$		0			0
$741$	−10.0645			−0.369729
$742$		0			0
$743$	9.53568	−	9.53568i	0.349830	−	0.349830i	−0.510216	−	0.860046i	$-0.670435\pi$
$743$	9.53568	−	9.53568i	0.349830	−	0.349830i	0.860046	+	0.510216i	$0.170435\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.97934	+	1.97934i	0.0724201	+	0.0724201i
$748$		0			0
$749$			59.3253i			2.16770i
$750$		0			0
$751$		−	21.5649i		−	0.786915i	−0.919343	−	0.393457i	$-0.871279\pi$
$751$		−	21.5649i		−	0.786915i	0.919343	−	0.393457i	$-0.128721\pi$
$752$		0			0
$753$	6.78710	+	6.78710i	0.247336	+	0.247336i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	4.86093	−	4.86093i	0.176674	−	0.176674i	−0.613230	−	0.789904i	$-0.710130\pi$
$757$	4.86093	−	4.86093i	0.176674	−	0.176674i	0.789904	+	0.613230i	$0.210130\pi$
$758$		0			0
$759$	−22.2419			−0.807330
$760$		0			0
$761$	−15.6241			−0.566374			−0.283187	−	0.959065i	$-0.591392\pi$
$761$	−15.6241			−0.566374			−0.283187	+	0.959065i	$0.591392\pi$
$762$		0			0
$763$	−5.72585	+	5.72585i	−0.207290	+	0.207290i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.14162	−	9.14162i	−0.330085	−	0.330085i
$768$		0			0
$769$			26.2855i			0.947880i	0.880557	+	0.473940i	$0.157169\pi$
$769$			26.2855i			0.947880i	−0.880557	+	0.473940i	$0.842831\pi$
$770$		0			0
$771$		−	10.2419i		−	0.368853i
$772$		0			0
$773$	21.2221	+	21.2221i	0.763307	+	0.763307i	0.976919	−	0.213611i	$-0.0685226\pi$
$773$	21.2221	+	21.2221i	0.763307	+	0.763307i	−0.213611	+	0.976919i	$0.568523\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−20.8776	+	20.8776i	−0.748979	+	0.748979i
$778$		0			0
$779$	−29.6232			−1.06136
$780$		0			0
$781$	42.7539			1.52986
$782$		0			0
$783$	−0.282561	+	0.282561i	−0.0100979	+	0.0100979i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−15.6367	−	15.6367i	−0.557389	−	0.557389i	0.371174	−	0.928563i	$-0.378955\pi$
$787$	−15.6367	−	15.6367i	−0.557389	−	0.557389i	−0.928563	+	0.371174i	$0.878955\pi$
$788$		0			0
$789$			3.08589i			0.109861i
$790$		0			0
$791$			55.0265i			1.95652i
$792$		0			0
$793$	7.59439	+	7.59439i	0.269685	+	0.269685i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.9282	+	18.9282i	−0.670471	+	0.670471i	−0.957825	−	0.287353i	$-0.907225\pi$
$797$	−18.9282	+	18.9282i	−0.670471	+	0.670471i	0.287353	+	0.957825i	$0.407225\pi$
$798$		0			0
$799$	−5.52537			−0.195473
$800$		0			0
$801$	−0.799203			−0.0282385
$802$		0			0
$803$	28.7262	−	28.7262i	1.01372	−	1.01372i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	4.42367	+	4.42367i	0.155720	+	0.155720i
$808$		0			0
$809$			21.1283i			0.742830i	0.928467	+	0.371415i	$0.121127\pi$
$809$			21.1283i			0.742830i	−0.928467	+	0.371415i	$0.878873\pi$
$810$		0			0
$811$			38.3229i			1.34570i	0.739779	+	0.672850i	$0.234930\pi$
$811$			38.3229i			1.34570i	−0.739779	+	0.672850i	$0.765070\pi$
$812$		0			0
$813$	−11.4069	−	11.4069i	−0.400058	−	0.400058i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−15.7458	+	15.7458i	−0.550876	+	0.550876i
$818$		0			0
$819$	18.1516			0.634268
$820$		0			0
$821$	−18.3855			−0.641659			−0.320829	−	0.947137i	$-0.603962\pi$
$821$	−18.3855			−0.641659			−0.320829	+	0.947137i	$0.603962\pi$
$822$		0			0
$823$	−19.3576	+	19.3576i	−0.674762	+	0.674762i	−0.958810	−	0.284048i	$-0.908322\pi$
$823$	−19.3576	+	19.3576i	−0.674762	+	0.674762i	0.284048	+	0.958810i	$0.408322\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.39496	+	7.39496i	0.257148	+	0.257148i	0.823893	−	0.566745i	$-0.191798\pi$
$827$	7.39496	+	7.39496i	0.257148	+	0.257148i	−0.566745	+	0.823893i	$0.691798\pi$
$828$		0			0
$829$		−	49.8403i		−	1.73103i	−0.500887	−	0.865513i	$-0.666993\pi$
$829$		−	49.8403i		−	1.73103i	0.500887	−	0.865513i	$-0.333007\pi$
$830$		0			0
$831$		−	9.98325i		−	0.346315i
$832$		0			0
$833$	11.3618	+	11.3618i	0.393664	+	0.393664i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−3.50731	+	3.50731i	−0.121230	+	0.121230i
$838$		0			0
$839$	45.9303			1.58569			0.792845	−	0.609423i	$-0.208599\pi$
$839$	45.9303			1.58569			0.792845	+	0.609423i	$0.208599\pi$
$840$		0			0
$841$	28.8403			0.994494
$842$		0			0
$843$	14.7386	−	14.7386i	0.507625	−	0.507625i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−6.14611	−	6.14611i	−0.211183	−	0.211183i
$848$		0			0
$849$			14.6428i			0.502538i
$850$		0			0
$851$		−	44.4838i		−	1.52489i
$852$		0			0
$853$	−5.20305	−	5.20305i	−0.178149	−	0.178149i	0.612399	−	0.790548i	$-0.290204\pi$
$853$	−5.20305	−	5.20305i	−0.178149	−	0.178149i	−0.790548	+	0.612399i	$0.790204\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−11.7473	+	11.7473i	−0.401282	+	0.401282i	−0.878684	−	0.477403i	$-0.841578\pi$
$857$	−11.7473	+	11.7473i	−0.401282	+	0.401282i	0.477403	+	0.878684i	$0.341578\pi$
$858$		0			0
$859$	17.3205			0.590968			0.295484	−	0.955348i	$-0.404519\pi$
$859$	17.3205			0.590968			0.295484	+	0.955348i	$0.404519\pi$
$860$		0			0
$861$	53.4261			1.82076
$862$		0			0
$863$	−25.7181	+	25.7181i	−0.875455	+	0.875455i	−0.993060	−	0.117605i	$-0.962478\pi$
$863$	−25.7181	+	25.7181i	−0.875455	+	0.875455i	0.117605	+	0.993060i	$0.462478\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−11.3627	−	11.3627i	−0.385898	−	0.385898i
$868$		0			0
$869$		−	7.91130i		−	0.268372i
$870$		0			0
$871$			38.8072i			1.31493i
$872$		0			0
$873$	8.94558	+	8.94558i	0.302762	+	0.302762i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−5.78154	+	5.78154i	−0.195229	+	0.195229i	−0.797951	−	0.602722i	$-0.794083\pi$
$877$	−5.78154	+	5.78154i	−0.195229	+	0.195229i	0.602722	+	0.797951i	$0.294083\pi$
$878$		0			0
$879$	2.13431			0.0719883
$880$		0			0
$881$	−15.3419			−0.516882			−0.258441	−	0.966027i	$-0.583209\pi$
$881$	−15.3419			−0.516882			−0.258441	+	0.966027i	$0.583209\pi$
$882$		0			0
$883$	−31.4856	+	31.4856i	−1.05957	+	1.05957i	−0.0614655	+	0.998109i	$0.519577\pi$
$883$	−31.4856	+	31.4856i	−1.05957	+	1.05957i	−0.998109	+	0.0614655i	$0.980423\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4.89722	−	4.89722i	−0.164433	−	0.164433i	0.620095	−	0.784527i	$-0.287094\pi$
$887$	−4.89722	−	4.89722i	−0.164433	−	0.164433i	−0.784527	+	0.620095i	$0.787094\pi$
$888$		0			0
$889$		−	51.1394i		−	1.71516i
$890$		0			0
$891$			3.03528i			0.101686i
$892$		0			0
$893$	10.9216	+	10.9216i	0.365479	+	0.365479i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−19.3378	+	19.3378i	−0.645669	+	0.645669i
$898$		0			0
$899$	−1.98206			−0.0661053
$900$		0			0
$901$	13.7249			0.457243
$902$		0			0
$903$	28.3980	−	28.3980i	0.945026	−	0.945026i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	7.90022	+	7.90022i	0.262323	+	0.262323i	0.825997	−	0.563674i	$-0.190613\pi$
$907$	7.90022	+	7.90022i	0.262323	+	0.262323i	−0.563674	+	0.825997i	$0.690613\pi$
$908$		0			0
$909$			4.52860i			0.150204i
$910$		0			0
$911$		−	11.3608i		−	0.376400i	−0.982131	−	0.188200i	$-0.939735\pi$
$911$		−	11.3608i		−	0.376400i	0.982131	−	0.188200i	$-0.0602654\pi$
$912$		0			0
$913$	6.00783	+	6.00783i	0.198830	+	0.198830i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.92157	−	9.92157i	0.327639	−	0.327639i
$918$		0			0
$919$	−25.9891			−0.857301			−0.428650	−	0.903470i	$-0.641011\pi$
$919$	−25.9891			−0.857301			−0.428650	+	0.903470i	$0.641011\pi$
$920$		0			0
$921$	−7.98466			−0.263103
$922$		0			0
$923$	37.1715	−	37.1715i	1.22352	−	1.22352i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−5.36433	−	5.36433i	−0.176188	−	0.176188i
$928$		0			0
$929$		−	28.0444i		−	0.920105i	−0.887892	−	0.460053i	$-0.847831\pi$
$929$		−	28.0444i		−	0.920105i	0.887892	−	0.460053i	$-0.152169\pi$
$930$		0			0
$931$		−	44.9164i		−	1.47208i
$932$		0			0
$933$	−20.7327	−	20.7327i	−0.678759	−	0.678759i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−9.20335	+	9.20335i	−0.300660	+	0.300660i	−0.841272	−	0.540612i	$-0.818193\pi$
$937$	−9.20335	+	9.20335i	−0.300660	+	0.300660i	0.540612	+	0.841272i	$0.318193\pi$
$938$		0			0
$939$	−0.197397			−0.00644180
$940$		0			0
$941$	−22.1507			−0.722093			−0.361046	−	0.932548i	$-0.617580\pi$
$941$	−22.1507			−0.722093			−0.361046	+	0.932548i	$0.617580\pi$
$942$		0			0
$943$	−56.9175	+	56.9175i	−1.85349	+	1.85349i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−8.34101	−	8.34101i	−0.271046	−	0.271046i	0.558475	−	0.829521i	$-0.311387\pi$
$947$	−8.34101	−	8.34101i	−0.271046	−	0.271046i	−0.829521	+	0.558475i	$0.811387\pi$
$948$		0			0
$949$		−	49.9507i		−	1.62147i
$950$		0			0
$951$			22.3830i			0.725819i
$952$		0			0
$953$	−39.6461	−	39.6461i	−1.28426	−	1.28426i	−0.938218	−	0.346046i	$-0.887524\pi$
$953$	−39.6461	−	39.6461i	−1.28426	−	1.28426i	−0.346046	−	0.938218i	$-0.612476\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−0.857651	+	0.857651i	−0.0277239	+	0.0277239i
$958$		0			0
$959$	−65.1422			−2.10355
$960$		0			0
$961$	6.39761			0.206375
$962$		0			0
$963$	−8.62498	+	8.62498i	−0.277936	+	0.277936i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−12.7165	−	12.7165i	−0.408937	−	0.408937i	0.472431	−	0.881368i	$-0.343377\pi$
$967$	−12.7165	−	12.7165i	−0.408937	−	0.408937i	−0.881368	+	0.472431i	$0.843377\pi$
$968$		0			0
$969$		−	2.60164i		−	0.0835768i
$970$		0			0
$971$		−	28.2049i		−	0.905137i	−0.891730	−	0.452568i	$-0.850508\pi$
$971$		−	28.2049i		−	0.905137i	0.891730	−	0.452568i	$-0.149492\pi$
$972$		0			0
$973$	1.84304	+	1.84304i	0.0590851	+	0.0590851i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17.6221	+	17.6221i	−0.563781	+	0.563781i	−0.930379	−	0.366599i	$-0.880522\pi$
$977$	−17.6221	+	17.6221i	−0.563781	+	0.563781i	0.366599	+	0.930379i	$0.380522\pi$
$978$		0			0
$979$	−2.42580			−0.0775290
$980$		0			0
$981$	−1.66490			−0.0531561
$982$		0			0
$983$	−4.24088	+	4.24088i	−0.135263	+	0.135263i	−0.771497	−	0.636233i	$-0.780492\pi$
$983$	−4.24088	+	4.24088i	−0.135263	+	0.135263i	0.636233	+	0.771497i	$0.280492\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−19.6975	−	19.6975i	−0.626977	−	0.626977i
$988$		0			0
$989$			60.5075i			1.92403i
$990$		0			0
$991$		−	3.08702i		−	0.0980623i	−0.998797	−	0.0490312i	$-0.984387\pi$
$991$		−	3.08702i		−	0.0980623i	0.998797	−	0.0490312i	$-0.0156134\pi$
$992$		0			0
$993$	−20.8325	−	20.8325i	−0.661100	−	0.661100i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	32.3125	−	32.3125i	1.02335	−	1.02335i	0.0236251	−	0.999721i	$-0.492479\pi$
$997$	32.3125	−	32.3125i	1.02335	−	1.02335i	0.999721	−	0.0236251i	$-0.00752081\pi$
$998$		0			0
$999$	−6.07055			−0.192064

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.w.h.2143.1	yes	8
4.3	odd	2		2400.2.w.l.2143.4	yes	8
5.2	odd	4		2400.2.w.l.607.4	yes	8
5.3	odd	4		2400.2.w.g.607.1	✓	8
5.4	even	2		2400.2.w.k.2143.4	yes	8
20.3	even	4		2400.2.w.k.607.4	yes	8
20.7	even	4	inner	2400.2.w.h.607.1	yes	8
20.19	odd	2		2400.2.w.g.2143.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.2.w.g.607.1	✓	8	5.3	odd	4
2400.2.w.g.2143.1	yes	8	20.19	odd	2
2400.2.w.h.607.1	yes	8	20.7	even	4	inner
2400.2.w.h.2143.1	yes	8	1.1	even	1	trivial
2400.2.w.k.607.4	yes	8	20.3	even	4
2400.2.w.k.2143.4	yes	8	5.4	even	2
2400.2.w.l.607.4	yes	8	5.2	odd	4
2400.2.w.l.2143.4	yes	8	4.3	odd	2

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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{3} +(-3.43916 - 3.43916i) q^{7} -1.00000i q^{9} -3.03528i q^{11} +(-2.63896 - 2.63896i) q^{13} +(0.682163 - 0.682163i) q^{17} -2.69677 q^{19} +4.86370 q^{21} +(-5.18154 + 5.18154i) q^{23} +(0.707107 + 0.707107i) q^{27} +0.399602i q^{29} +4.96008i q^{31} +(2.14626 + 2.14626i) q^{33} +(-4.29253 + 4.29253i) q^{37} +3.73205 q^{39} +10.9847 q^{41} +(5.83876 - 5.83876i) q^{43} +(-4.04989 - 4.04989i) q^{47} +16.6556i q^{49} +0.964724i q^{51} +(10.0599 + 10.0599i) q^{53} +(1.90691 - 1.90691i) q^{57} +3.46410 q^{59} -2.87780 q^{61} +(-3.43916 + 3.43916i) q^{63} +(-7.35275 - 7.35275i) q^{67} -7.32780i q^{69} +14.0857i q^{71} +(9.46410 + 9.46410i) q^{73} +(-10.4388 + 10.4388i) q^{77} +2.60645 q^{79} -1.00000 q^{81} +(-1.97934 + 1.97934i) q^{83} +(-0.282561 - 0.282561i) q^{87} -0.799203i q^{89} +18.1516i q^{91} +(-3.50731 - 3.50731i) q^{93} +(-8.94558 + 8.94558i) q^{97} -3.03528 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} + 8 q^{17} - 16 q^{19} + 8 q^{21} - 8 q^{23} - 8 q^{33} + 16 q^{37} + 16 q^{39} - 16 q^{41} + 24 q^{43} - 16 q^{47} + 8 q^{53} + 8 q^{57} - 24 q^{61} - 8 q^{63} + 8 q^{67} + 48 q^{73} - 8 q^{77}+ \cdots - 16 q^{99}+O(q^{100}) \) 8 * q - 8 * q^7 + 8 * q^17 - 16 * q^19 + 8 * q^21 - 8 * q^23 - 8 * q^33 + 16 * q^37 + 16 * q^39 - 16 * q^41 + 24 * q^43 - 16 * q^47 + 8 * q^53 + 8 * q^57 - 24 * q^61 - 8 * q^63 + 8 * q^67 + 48 * q^73 - 8 * q^77 + 32 * q^79 - 8 * q^81 - 16 * q^83 - 8 * q^87 - 24 * q^93 - 32 * q^97 - 16 * q^99

Introduction

L-functions

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