LMFDB - Embedded newform 2400.2.o.k.2399.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(2399,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2400.2399");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.o (of order $2$, degree $1$, 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:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 8 x^{14} - 16 x^{13} + 41 x^{12} - 64 x^{11} + 56 x^{10} - 100 x^{9} + 256 x^{8} + \cdots + 6561 $ x^16 - 4x^15 + 8x^14 - 16x^13 + 41x^12 - 64x^11 + 56x^10 - 100x^9 + 256x^8 - 300x^7 + 504x^6 - 1728x^5 + 3321x^4 - 3888x^3 + 5832x^2 - 8748*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2399.3
Root			$-0.901513 - 1.47894i$ of defining polynomial
Character	$\chi$	$=$	2400.2399
Dual form			2400.2.o.k.2399.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.68324 + 0.408305i) q^{3} +1.40591 q^{7} +(2.66657 - 1.37455i) q^{9} +O(q^{10})$	$q+(-1.68324 + 0.408305i) q^{3} +1.40591 q^{7} +(2.66657 - 1.37455i) q^{9} -6.15156 q^{11} +3.52133i q^{13} +1.39065 q^{17} +2.77238i q^{19} +(-2.36647 + 0.574039i) q^{21} -4.18687i q^{23} +(-3.92724 + 3.40247i) q^{27} -8.16338i q^{29} -6.44858i q^{31} +(10.3545 - 2.51171i) q^{33} +4.73295i q^{37} +(-1.43778 - 5.92724i) q^{39} +3.02387i q^{41} +5.71563 q^{43} +5.49819i q^{47} -5.02343 q^{49} +(-2.34079 + 0.567809i) q^{51} +12.0284 q^{53} +(-1.13198 - 4.66657i) q^{57} -3.86497 q^{59} -7.94456 q^{61} +(3.74895 - 1.93249i) q^{63} -7.89392 q^{67} +(1.70952 + 7.04750i) q^{69} -10.9918 q^{71} +0.909922i q^{73} -8.64852 q^{77} -1.12153i q^{79} +(5.22124 - 7.33067i) q^{81} -14.3621i q^{83} +(3.33315 + 13.7409i) q^{87} -14.0628i q^{89} +4.95067i q^{91} +(2.63299 + 10.8545i) q^{93} -9.44247i q^{97} +(-16.4036 + 8.45562i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{3} - 4 q^{9}+O(q^{10}) $ 16 * q - 4 * q^3 - 4 * q^9	$ 16 q - 4 q^{3} - 4 q^{9} + 8 q^{21} + 8 q^{27} + 64 q^{43} + 8 q^{49} + 8 q^{61} + 80 q^{63} - 8 q^{67} + 24 q^{69} + 36 q^{81} - 40 q^{87}+O(q^{100}) $ 16 * q - 4 * q^3 - 4 * q^9 + 8 * q^21 + 8 * q^27 + 64 * q^43 + 8 * q^49 + 8 * q^61 + 80 * q^63 - 8 * q^67 + 24 * q^69 + 36 * q^81 - 40 * q^87

Display 100 coefficients 10 coefficients

Character values

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

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	−1.68324	+	0.408305i	−0.971817	+	0.235735i
$3$	−1.68324	+	0.408305i	−0.971817	+	0.235735i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.40591			0.531383			0.265691	−	0.964058i	$-0.414400\pi$
$7$	1.40591			0.531383			0.265691	+	0.964058i	$0.414400\pi$
$8$		0			0
$9$	2.66657	−	1.37455i	0.888858	−	0.458183i
$10$		0			0
$11$	−6.15156			−1.85477			−0.927383	−	0.374114i	$-0.877947\pi$
$11$	−6.15156			−1.85477			−0.927383	+	0.374114i	$0.877947\pi$
$12$		0			0
$13$			3.52133i			0.976643i	0.872664	+	0.488321i	$0.162391\pi$
$13$			3.52133i			0.976643i	−0.872664	+	0.488321i	$0.837609\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.39065			0.337282			0.168641	−	0.985678i	$-0.446062\pi$
$17$	1.39065			0.337282			0.168641	+	0.985678i	$0.446062\pi$
$18$		0			0
$19$			2.77238i			0.636028i	0.948086	+	0.318014i	$0.103016\pi$
$19$			2.77238i			0.636028i	−0.948086	+	0.318014i	$0.896984\pi$
$20$		0			0
$21$	−2.36647	+	0.574039i	−0.516407	+	0.125266i
$22$		0			0
$23$		−	4.18687i		−	0.873024i	−0.899698	−	0.436512i	$-0.856214\pi$
$23$		−	4.18687i		−	0.873024i	0.899698	−	0.436512i	$-0.143786\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.92724	+	3.40247i	−0.755798	+	0.654805i
$28$		0			0
$29$		−	8.16338i		−	1.51590i	−0.652312	−	0.757951i	$-0.726201\pi$
$29$		−	8.16338i		−	1.51590i	0.652312	−	0.757951i	$-0.273799\pi$
$30$		0			0
$31$		−	6.44858i		−	1.15820i	−0.815257	−	0.579099i	$-0.803404\pi$
$31$		−	6.44858i		−	1.15820i	0.815257	−	0.579099i	$-0.196596\pi$
$32$		0			0
$33$	10.3545	−	2.51171i	1.80249	−	0.437233i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			4.73295i			0.778092i	0.921218	+	0.389046i	$0.127195\pi$
$37$			4.73295i			0.778092i	−0.921218	+	0.389046i	$0.872805\pi$
$38$		0			0
$39$	−1.43778	−	5.92724i	−0.230229	−	0.949118i
$40$		0			0
$41$			3.02387i			0.472249i	0.971723	+	0.236124i	$0.0758773\pi$
$41$			3.02387i			0.472249i	−0.971723	+	0.236124i	$0.924123\pi$
$42$		0			0
$43$	5.71563			0.871625			0.435813	−	0.900037i	$-0.356461\pi$
$43$	5.71563			0.871625			0.435813	+	0.900037i	$0.356461\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			5.49819i			0.801994i	0.916079	+	0.400997i	$0.131336\pi$
$47$			5.49819i			0.801994i	−0.916079	+	0.400997i	$0.868664\pi$
$48$		0			0
$49$	−5.02343			−0.717632
$50$		0			0
$51$	−2.34079	+	0.567809i	−0.327776	+	0.0795091i
$52$		0			0
$53$	12.0284			1.65222			0.826111	−	0.563508i	$-0.190549\pi$
$53$	12.0284			1.65222			0.826111	+	0.563508i	$0.190549\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.13198	−	4.66657i	−0.149934	−	0.618103i
$58$		0			0
$59$	−3.86497			−0.503177			−0.251588	−	0.967834i	$-0.580953\pi$
$59$	−3.86497			−0.503177			−0.251588	+	0.967834i	$0.580953\pi$
$60$		0			0
$61$	−7.94456			−1.01720			−0.508598	−	0.861004i	$-0.669836\pi$
$61$	−7.94456			−1.01720			−0.508598	+	0.861004i	$0.669836\pi$
$62$		0			0
$63$	3.74895	−	1.93249i	0.472324	−	0.243470i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−7.89392			−0.964395			−0.482198	−	0.876062i	$-0.660161\pi$
$67$	−7.89392			−0.964395			−0.482198	+	0.876062i	$0.660161\pi$
$68$		0			0
$69$	1.70952	+	7.04750i	0.205802	+	0.848420i
$70$		0			0
$71$	−10.9918			−1.30449			−0.652244	−	0.758009i	$-0.726172\pi$
$71$	−10.9918			−1.30449			−0.652244	+	0.758009i	$0.726172\pi$
$72$		0			0
$73$			0.909922i			0.106498i	0.998581	+	0.0532491i	$0.0169577\pi$
$73$			0.909922i			0.106498i	−0.998581	+	0.0532491i	$0.983042\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.64852			−0.985591
$78$		0			0
$79$		−	1.12153i		−	0.126183i	−0.998008	−	0.0630913i	$-0.979904\pi$
$79$		−	1.12153i		−	0.126183i	0.998008	−	0.0630913i	$-0.0200959\pi$
$80$		0			0
$81$	5.22124	−	7.33067i	0.580137	−	0.814519i
$82$		0			0
$83$		−	14.3621i		−	1.57644i	−0.615392	−	0.788221i	$-0.711002\pi$
$83$		−	14.3621i		−	1.57644i	0.615392	−	0.788221i	$-0.288998\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	3.33315	+	13.7409i	0.357351	+	1.47318i
$88$		0			0
$89$		−	14.0628i		−	1.49065i	−0.666699	−	0.745327i	$-0.732293\pi$
$89$		−	14.0628i		−	1.49065i	0.666699	−	0.745327i	$-0.267707\pi$
$90$		0			0
$91$			4.95067i			0.518971i
$92$		0			0
$93$	2.63299	+	10.8545i	0.273028	+	1.12556i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	9.44247i		−	0.958738i	−0.877614	−	0.479369i	$-0.840866\pi$
$97$		−	9.44247i		−	0.958738i	0.877614	−	0.479369i	$-0.159134\pi$
$98$		0			0
$99$	−16.4036	+	8.45562i	−1.64862	+	0.849822i
$100$		0			0
$101$		−	16.9280i		−	1.68440i	−0.539165	−	0.842200i	$-0.681260\pi$
$101$		−	16.9280i		−	1.68440i	0.539165	−	0.842200i	$-0.318740\pi$
$102$		0			0
$103$	−11.2350			−1.10702			−0.553511	−	0.832842i	$-0.686712\pi$
$103$	−11.2350			−1.10702			−0.553511	+	0.832842i	$0.686712\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	7.78478i		−	0.752583i	−0.926501	−	0.376292i	$-0.877199\pi$
$107$		−	7.78478i		−	0.752583i	0.926501	−	0.376292i	$-0.122801\pi$
$108$		0			0
$109$	18.1876			1.74206			0.871030	−	0.491231i	$-0.163453\pi$
$109$	18.1876			1.74206			0.871030	+	0.491231i	$0.163453\pi$
$110$		0			0
$111$	−1.93249	−	7.96667i	−0.183423	−	0.756163i
$112$		0			0
$113$	7.09921			0.667838			0.333919	−	0.942602i	$-0.391629\pi$
$113$	7.09921			0.667838			0.333919	+	0.942602i	$0.391629\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	4.84024	+	9.38990i	0.447481	+	0.868097i
$118$		0			0
$119$	1.95512			0.179226
$120$		0			0
$121$	26.8417			2.44016
$122$		0			0
$123$	−1.23466	−	5.08989i	−0.111326	−	0.458940i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	9.54476			0.846961			0.423480	−	0.905905i	$-0.360808\pi$
$127$	9.54476			0.846961			0.423480	+	0.905905i	$0.360808\pi$
$128$		0			0
$129$	−9.62076	+	2.33372i	−0.847060	+	0.205473i
$130$		0			0
$131$	−1.40558			−0.122806			−0.0614029	−	0.998113i	$-0.519557\pi$
$131$	−1.40558			−0.122806			−0.0614029	+	0.998113i	$0.519557\pi$
$132$		0			0
$133$			3.89771i			0.337974i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.9444			−1.02048			−0.510241	−	0.860031i	$-0.670444\pi$
$137$	−11.9444			−1.02048			−0.510241	+	0.860031i	$0.670444\pi$
$138$		0			0
$139$		−	16.4091i		−	1.39181i	−0.718136	−	0.695903i	$-0.755004\pi$
$139$		−	16.4091i		−	1.39181i	0.718136	−	0.695903i	$-0.244996\pi$
$140$		0			0
$141$	−2.24494	−	9.25476i	−0.189058	−	0.779392i
$142$		0			0
$143$		−	21.6617i		−	1.81144i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	8.45562	−	2.05109i	0.697408	−	0.169171i
$148$		0			0
$149$		−	12.3974i		−	1.01563i	−0.861465	−	0.507817i	$-0.830453\pi$
$149$		−	12.3974i		−	1.01563i	0.861465	−	0.507817i	$-0.169547\pi$
$150$		0			0
$151$			12.0600i			0.981428i	0.871321	+	0.490714i	$0.163264\pi$
$151$			12.0600i			0.981428i	−0.871321	+	0.490714i	$0.836736\pi$
$152$		0			0
$153$	3.70827	−	1.91151i	0.299796	−	0.154537i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	3.45468i		−	0.275714i	−0.990452	−	0.137857i	$-0.955979\pi$
$157$		−	3.45468i		−	0.275714i	0.990452	−	0.137857i	$-0.0440214\pi$
$158$		0			0
$159$	−20.2466	+	4.91124i	−1.60566	+	0.389486i
$160$		0			0
$161$		−	5.88635i		−	0.463910i
$162$		0			0
$163$	17.0032			1.33180			0.665898	−	0.746043i	$-0.268049\pi$
$163$	17.0032			1.33180			0.665898	+	0.746043i	$0.268049\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	14.2582i		−	1.10334i	−0.834064	−	0.551668i	$-0.813992\pi$
$167$		−	14.2582i		−	1.10334i	0.834064	−	0.551668i	$-0.186008\pi$
$168$		0			0
$169$	0.600200			0.0461692
$170$		0			0
$171$	3.81077	+	7.39276i	0.291417	+	0.565338i
$172$		0			0
$173$	−13.0630			−0.993164			−0.496582	−	0.867990i	$-0.665412\pi$
$173$	−13.0630			−0.993164			−0.496582	+	0.867990i	$0.665412\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	6.50567	−	1.57809i	0.488996	−	0.118616i
$178$		0			0
$179$	8.86388			0.662517			0.331259	−	0.943540i	$-0.392527\pi$
$179$	8.86388			0.662517			0.331259	+	0.943540i	$0.392527\pi$
$180$		0			0
$181$	−24.3011			−1.80629			−0.903145	−	0.429336i	$-0.858747\pi$
$181$	−24.3011			−1.80629			−0.903145	+	0.429336i	$0.858747\pi$
$182$		0			0
$183$	13.3726	−	3.24380i	0.988530	−	0.239789i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−8.55466			−0.625579
$188$		0			0
$189$	−5.52133	+	4.78355i	−0.401618	+	0.347952i
$190$		0			0
$191$	9.74947			0.705447			0.352723	−	0.935728i	$-0.385256\pi$
$191$	9.74947			0.705447			0.352723	+	0.935728i	$0.385256\pi$
$192$		0			0
$193$		−	25.8651i		−	1.86181i	−0.365257	−	0.930907i	$-0.619019\pi$
$193$		−	25.8651i		−	1.86181i	0.365257	−	0.930907i	$-0.380981\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−22.3745			−1.59412			−0.797058	−	0.603902i	$-0.793612\pi$
$197$	−22.3745			−1.59412			−0.797058	+	0.603902i	$0.793612\pi$
$198$		0			0
$199$		−	6.36971i		−	0.451537i	−0.974181	−	0.225768i	$-0.927511\pi$
$199$		−	6.36971i		−	0.451537i	0.974181	−	0.225768i	$-0.0724893\pi$
$200$		0			0
$201$	13.2873	−	3.22312i	0.937216	−	0.227342i
$202$		0			0
$203$		−	11.4769i		−	0.805524i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−5.75506	−	11.1646i	−0.400004	−	0.775994i
$208$		0			0
$209$		−	17.0545i		−	1.17968i
$210$		0			0
$211$		−	25.4387i		−	1.75127i	−0.482973	−	0.875635i	$-0.660443\pi$
$211$		−	25.4387i		−	1.75127i	0.482973	−	0.875635i	$-0.339557\pi$
$212$		0			0
$213$	18.5018	−	4.48801i	1.26772	−	0.307513i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	9.06610i		−	0.615447i
$218$		0			0
$219$	−0.371525	−	1.53161i	−0.0251054	−	0.103497i
$220$		0			0
$221$			4.89694i			0.329404i
$222$		0			0
$223$	3.99334			0.267414			0.133707	−	0.991021i	$-0.457312\pi$
$223$	3.99334			0.267414			0.133707	+	0.991021i	$0.457312\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			9.59081i			0.636564i	0.947996	+	0.318282i	$0.103106\pi$
$227$			9.59081i			0.636564i	−0.947996	+	0.318282i	$0.896894\pi$
$228$		0			0
$229$	−9.45468			−0.624783			−0.312392	−	0.949953i	$-0.601130\pi$
$229$	−9.45468			−0.624783			−0.312392	+	0.949953i	$0.601130\pi$
$230$		0			0
$231$	14.5575	−	3.53123i	0.957814	−	0.232338i
$232$		0			0
$233$	1.42285			0.0932140			0.0466070	−	0.998913i	$-0.485159\pi$
$233$	1.42285			0.0932140			0.0466070	+	0.998913i	$0.485159\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.457928	+	1.88781i	0.0297456	+	0.122626i
$238$		0			0
$239$	10.6653			0.689883			0.344941	−	0.938624i	$-0.387899\pi$
$239$	10.6653			0.689883			0.344941	+	0.938624i	$0.387899\pi$
$240$		0			0
$241$	4.53255			0.291967			0.145984	−	0.989287i	$-0.453365\pi$
$241$	4.53255			0.291967			0.145984	+	0.989287i	$0.453365\pi$
$242$		0			0
$243$	−5.79543	+	14.4711i	−0.371777	+	0.928322i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−9.76248			−0.621172
$248$		0			0
$249$	5.86410	+	24.1748i	0.371623	+	1.53201i
$250$		0			0
$251$	−12.1303			−0.765659			−0.382829	−	0.923819i	$-0.625050\pi$
$251$	−12.1303			−0.765659			−0.382829	+	0.923819i	$0.625050\pi$
$252$		0			0
$253$			25.7558i			1.61925i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.3268			−1.01844			−0.509218	−	0.860638i	$-0.670065\pi$
$257$	−16.3268			−1.01844			−0.509218	+	0.860638i	$0.670065\pi$
$258$		0			0
$259$			6.65408i			0.413465i
$260$		0			0
$261$	−11.2210	−	21.7683i	−0.694560	−	1.34742i
$262$		0			0
$263$		−	10.3933i		−	0.640877i	−0.947269	−	0.320438i	$-0.896170\pi$
$263$		−	10.3933i		−	0.640877i	0.947269	−	0.320438i	$-0.103830\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	5.74191	+	23.6710i	0.351399	+	1.44864i
$268$		0			0
$269$			9.47928i			0.577962i	0.957335	+	0.288981i	$0.0933164\pi$
$269$			9.47928i			0.577962i	−0.957335	+	0.288981i	$0.906684\pi$
$270$		0			0
$271$			23.7878i			1.44501i	0.691367	+	0.722504i	$0.257009\pi$
$271$			23.7878i			1.44501i	−0.691367	+	0.722504i	$0.742991\pi$
$272$		0			0
$273$	−2.02138	−	8.33315i	−0.122340	−	0.504345i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			21.6535i			1.30103i	0.759492	+	0.650517i	$0.225448\pi$
$277$			21.6535i			1.30103i	−0.759492	+	0.650517i	$0.774552\pi$
$278$		0			0
$279$	−8.86388	−	17.1956i	−0.530667	−	1.02947i
$280$		0			0
$281$			7.40618i			0.441816i	0.975295	+	0.220908i	$0.0709020\pi$
$281$			7.40618i			0.441816i	−0.975295	+	0.220908i	$0.929098\pi$
$282$		0			0
$283$	−11.5907			−0.688994			−0.344497	−	0.938787i	$-0.611950\pi$
$283$	−11.5907			−0.688994			−0.344497	+	0.938787i	$0.611950\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			4.25128i			0.250945i
$288$		0			0
$289$	−15.0661			−0.886241
$290$		0			0
$291$	3.85541	+	15.8939i	0.226008	+	0.931718i
$292$		0			0
$293$	−32.0550			−1.87267			−0.936336	−	0.351105i	$-0.885806\pi$
$293$	−32.0550			−1.87267			−0.936336	+	0.351105i	$0.885806\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	24.1587	−	20.9305i	1.40183	−	1.21451i
$298$		0			0
$299$	14.7434			0.852632
$300$		0			0
$301$	8.03564			0.463167
$302$		0			0
$303$	6.91179	+	28.4939i	0.397072	+	1.63693i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	25.2143			1.43906			0.719528	−	0.694464i	$-0.244358\pi$
$307$	25.2143			1.43906			0.719528	+	0.694464i	$0.244358\pi$
$308$		0			0
$309$	18.9112	−	4.58732i	1.07582	−	0.260964i
$310$		0			0
$311$	−10.3979			−0.589608			−0.294804	−	0.955558i	$-0.595254\pi$
$311$	−10.3979			−0.589608			−0.294804	+	0.955558i	$0.595254\pi$
$312$		0			0
$313$			8.68972i			0.491172i	0.969375	+	0.245586i	$0.0789804\pi$
$313$			8.68972i			0.491172i	−0.969375	+	0.245586i	$0.921020\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.0094			0.899179			0.449590	−	0.893235i	$-0.351570\pi$
$317$	16.0094			0.899179			0.449590	+	0.893235i	$0.351570\pi$
$318$		0			0
$319$			50.2175i			2.81164i
$320$		0			0
$321$	3.17857	+	13.1036i	0.177410	+	0.731373i
$322$		0			0
$323$			3.85541i			0.214521i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−30.6141	+	7.42610i	−1.69296	+	0.410664i
$328$		0			0
$329$			7.72995i			0.426166i
$330$		0			0
$331$		−	5.67620i		−	0.311992i	−0.987758	−	0.155996i	$-0.950141\pi$
$331$		−	5.67620i		−	0.311992i	0.987758	−	0.155996i	$-0.0498587\pi$
$332$		0			0
$333$	6.50567	+	12.6208i	0.356508	+	0.691613i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	13.4232i		−	0.731210i	−0.930770	−	0.365605i	$-0.880862\pi$
$337$		−	13.4232i		−	0.731210i	0.930770	−	0.365605i	$-0.119138\pi$
$338$		0			0
$339$	−11.9497	+	2.89864i	−0.649016	+	0.157433i
$340$		0			0
$341$			39.6688i			2.14819i
$342$		0			0
$343$	−16.9038			−0.912720
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	20.5685i		−	1.10417i	−0.833787	−	0.552087i	$-0.813832\pi$
$347$		−	20.5685i		−	1.10417i	0.833787	−	0.552087i	$-0.186168\pi$
$348$		0			0
$349$	3.10932			0.166438			0.0832191	−	0.996531i	$-0.473480\pi$
$349$	3.10932			0.166438			0.0832191	+	0.996531i	$0.473480\pi$
$350$		0			0
$351$	−11.9812	−	13.8291i	−0.639510	−	0.738145i
$352$		0			0
$353$	−1.61137			−0.0857645			−0.0428823	−	0.999080i	$-0.513654\pi$
$353$	−1.61137			−0.0857645			−0.0428823	+	0.999080i	$0.513654\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−3.29093	+	0.798286i	−0.174175	+	0.0422498i
$358$		0			0
$359$	13.2688			0.700302			0.350151	−	0.936693i	$-0.386130\pi$
$359$	13.2688			0.700302			0.350151	+	0.936693i	$0.386130\pi$
$360$		0			0
$361$	11.3139			0.595469
$362$		0			0
$363$	−45.1810	+	10.9596i	−2.37139	+	0.575230i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	13.4726			0.703262			0.351631	−	0.936139i	$-0.385627\pi$
$367$	13.4726			0.703262			0.351631	+	0.936139i	$0.385627\pi$
$368$		0			0
$369$	4.15645	+	8.06337i	0.216376	+	0.419762i
$370$		0			0
$371$	16.9107			0.877962
$372$		0			0
$373$		−	5.03146i		−	0.260519i	−0.991480	−	0.130259i	$-0.958419\pi$
$373$		−	5.03146i		−	0.260519i	0.991480	−	0.130259i	$-0.0415810\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	28.7460			1.48049
$378$		0			0
$379$		−	23.6977i		−	1.21727i	−0.793451	−	0.608635i	$-0.791718\pi$
$379$		−	23.6977i		−	1.21727i	0.793451	−	0.608635i	$-0.208282\pi$
$380$		0			0
$381$	−16.0661	+	3.89717i	−0.823091	+	0.199658i
$382$		0			0
$383$			34.0338i			1.73905i	0.493892	+	0.869523i	$0.335574\pi$
$383$			34.0338i			1.73905i	−0.493892	+	0.869523i	$0.664426\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	15.2411	−	7.85641i	0.774751	−	0.399364i
$388$		0			0
$389$		−	12.5779i		−	0.637725i	−0.947801	−	0.318862i	$-0.896699\pi$
$389$		−	12.5779i		−	0.637725i	0.947801	−	0.318862i	$-0.103301\pi$
$390$		0			0
$391$		−	5.82247i		−	0.294455i
$392$		0			0
$393$	2.36592	−	0.573904i	0.119345	−	0.0289496i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			32.5442i			1.63335i	0.577100	+	0.816673i	$0.304184\pi$
$397$			32.5442i			1.63335i	−0.577100	+	0.816673i	$0.695816\pi$
$398$		0			0
$399$	−1.59145	−	6.56077i	−0.0796723	−	0.328449i
$400$		0			0
$401$			20.4987i			1.02366i	0.859088	+	0.511828i	$0.171032\pi$
$401$			20.4987i			1.02366i	−0.859088	+	0.511828i	$0.828968\pi$
$402$		0			0
$403$	22.7076			1.13115
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	29.1150i		−	1.44318i
$408$		0			0
$409$	1.58096			0.0781733			0.0390866	−	0.999236i	$-0.487555\pi$
$409$	1.58096			0.0781733			0.0390866	+	0.999236i	$0.487555\pi$
$410$		0			0
$411$	20.1053	−	4.87698i	0.991723	−	0.240563i
$412$		0			0
$413$	−5.43379			−0.267379
$414$		0			0
$415$		0			0
$416$		0			0
$417$	6.69993	+	27.6205i	0.328097	+	1.35258i
$418$		0			0
$419$	28.7337			1.40373			0.701867	−	0.712308i	$-0.252350\pi$
$419$	28.7337			1.40373			0.701867	+	0.712308i	$0.252350\pi$
$420$		0			0
$421$	−21.1280			−1.02972			−0.514858	−	0.857275i	$-0.672155\pi$
$421$	−21.1280			−1.02972			−0.514858	+	0.857275i	$0.672155\pi$
$422$		0			0
$423$	7.55753	+	14.6613i	0.367460	+	0.712859i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−11.1693			−0.540521
$428$		0			0
$429$	8.84458	+	36.4618i	0.427021	+	1.76039i
$430$		0			0
$431$	−26.2349			−1.26369			−0.631845	−	0.775095i	$-0.717702\pi$
$431$	−26.2349			−1.26369			−0.631845	+	0.775095i	$0.717702\pi$
$432$		0			0
$433$		−	8.13375i		−	0.390883i	−0.980715	−	0.195441i	$-0.937386\pi$
$433$		−	8.13375i		−	0.390883i	0.980715	−	0.195441i	$-0.0626140\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	11.6076			0.555267
$438$		0			0
$439$		−	8.54987i		−	0.408063i	−0.978964	−	0.204031i	$-0.934596\pi$
$439$		−	8.54987i		−	0.408063i	0.978964	−	0.204031i	$-0.0654045\pi$
$440$		0			0
$441$	−13.3953	+	6.90494i	−0.637873	+	0.328807i
$442$		0			0
$443$			5.28012i			0.250866i	0.992102	+	0.125433i	$0.0400320\pi$
$443$			5.28012i			0.250866i	−0.992102	+	0.125433i	$0.959968\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	5.06191	+	20.8677i	0.239420	+	0.987010i
$448$		0			0
$449$		−	35.9522i		−	1.69669i	−0.529446	−	0.848344i	$-0.677600\pi$
$449$		−	35.9522i		−	1.69669i	0.529446	−	0.848344i	$-0.322400\pi$
$450$		0			0
$451$		−	18.6015i		−	0.875911i
$452$		0			0
$453$	−4.92415	−	20.2998i	−0.231357	−	0.953769i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			14.0192i			0.655793i	0.944714	+	0.327896i	$0.106340\pi$
$457$			14.0192i			0.655793i	−0.944714	+	0.327896i	$0.893660\pi$
$458$		0			0
$459$	−5.46141	+	4.73163i	−0.254917	+	0.220854i
$460$		0			0
$461$			1.19979i			0.0558796i	0.999610	+	0.0279398i	$0.00889467\pi$
$461$			1.19979i			0.0558796i	−0.999610	+	0.0279398i	$0.991105\pi$
$462$		0			0
$463$	−0.341733			−0.0158817			−0.00794085	−	0.999968i	$-0.502528\pi$
$463$	−0.341733			−0.0158817			−0.00794085	+	0.999968i	$0.502528\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			10.2346i			0.473601i	0.971558	+	0.236801i	$0.0760988\pi$
$467$			10.2346i			0.473601i	−0.971558	+	0.236801i	$0.923901\pi$
$468$		0			0
$469$	−11.0981			−0.512463
$470$		0			0
$471$	1.41056	+	5.81505i	0.0649954	+	0.267943i
$472$		0			0
$473$	−35.1600			−1.61666
$474$		0			0
$475$		0			0
$476$		0			0
$477$	32.0745	−	16.5335i	1.46859	−	0.757019i
$478$		0			0
$479$	22.5177			1.02886			0.514431	−	0.857532i	$-0.328003\pi$
$479$	22.5177			1.02886			0.514431	+	0.857532i	$0.328003\pi$
$480$		0			0
$481$	−16.6663			−0.759918
$482$		0			0
$483$	2.40343	+	9.90813i	0.109360	+	0.450836i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−32.1922			−1.45877			−0.729384	−	0.684105i	$-0.760193\pi$
$487$	−32.1922			−1.45877			−0.729384	+	0.684105i	$0.760193\pi$
$488$		0			0
$489$	−28.6205	+	6.94251i	−1.29426	+	0.313951i
$490$		0			0
$491$	11.9122			0.537592			0.268796	−	0.963197i	$-0.413374\pi$
$491$	11.9122			0.537592			0.268796	+	0.963197i	$0.413374\pi$
$492$		0			0
$493$		−	11.3524i		−	0.511286i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−15.4535			−0.693182
$498$		0			0
$499$			5.37389i			0.240569i	0.992739	+	0.120284i	$0.0383806\pi$
$499$			5.37389i			0.240569i	−0.992739	+	0.120284i	$0.961619\pi$
$500$		0			0
$501$	5.82171	+	24.0000i	0.260095	+	1.07224i
$502$		0			0
$503$			35.5337i			1.58437i	0.610283	+	0.792184i	$0.291056\pi$
$503$			35.5337i			1.58437i	−0.610283	+	0.792184i	$0.708944\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.01028	+	0.245065i	−0.0448681	+	0.0108837i
$508$		0			0
$509$		−	0.485143i		−	0.0215036i	−0.999942	−	0.0107518i	$-0.996578\pi$
$509$		−	0.485143i		−	0.0215036i	0.999942	−	0.0107518i	$-0.00342246\pi$
$510$		0			0
$511$			1.27926i			0.0565913i
$512$		0			0
$513$	−9.43293	−	10.8878i	−0.416474	−	0.480709i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	33.8225i		−	1.48751i
$518$		0			0
$519$	21.9882	−	5.33370i	0.965174	−	0.234124i
$520$		0			0
$521$			25.0373i			1.09691i	0.836181	+	0.548453i	$0.184783\pi$
$521$			25.0373i			1.09691i	−0.836181	+	0.548453i	$0.815217\pi$
$522$		0			0
$523$	8.54152			0.373495			0.186747	−	0.982408i	$-0.440205\pi$
$523$	8.54152			0.373495			0.186747	+	0.982408i	$0.440205\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	8.96770i		−	0.390639i
$528$		0			0
$529$	5.47008			0.237830
$530$		0			0
$531$	−10.3062	+	5.31259i	−0.447253	+	0.230547i
$532$		0			0
$533$	−10.6481			−0.461218
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−14.9200	+	3.61917i	−0.643846	+	0.156179i
$538$		0			0
$539$	30.9019			1.33104
$540$		0			0
$541$	−4.65508			−0.200138			−0.100069	−	0.994981i	$-0.531906\pi$
$541$	−4.65508			−0.200138			−0.100069	+	0.994981i	$0.531906\pi$
$542$		0			0
$543$	40.9046	−	9.92227i	1.75538	−	0.425806i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	2.56114			0.109506			0.0547531	−	0.998500i	$-0.482563\pi$
$547$	2.56114			0.109506			0.0547531	+	0.998500i	$0.482563\pi$
$548$		0			0
$549$	−21.1848	+	10.9202i	−0.904144	+	0.466062i
$550$		0			0
$551$	22.6320			0.964155
$552$		0			0
$553$		−	1.57677i		−	0.0670512i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.9388			−0.844834			−0.422417	−	0.906402i	$-0.638818\pi$
$557$	−19.9388			−0.844834			−0.422417	+	0.906402i	$0.638818\pi$
$558$		0			0
$559$			20.1266i			0.851266i
$560$		0			0
$561$	14.3995	−	3.49291i	0.607948	−	0.147471i
$562$		0			0
$563$			31.5744i			1.33070i	0.746530	+	0.665352i	$0.231718\pi$
$563$			31.5744i			1.33070i	−0.746530	+	0.665352i	$0.768282\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	7.34057	−	10.3062i	0.308275	−	0.432821i
$568$		0			0
$569$			27.5657i			1.15561i	0.816173	+	0.577807i	$0.196092\pi$
$569$			27.5657i			1.15561i	−0.816173	+	0.577807i	$0.803908\pi$
$570$		0			0
$571$			7.80097i			0.326460i	0.986588	+	0.163230i	$0.0521913\pi$
$571$			7.80097i			0.326460i	−0.986588	+	0.163230i	$0.947809\pi$
$572$		0			0
$573$	−16.4107	+	3.98076i	−0.685565	+	0.166298i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	0.115060i		−	0.00479001i	−0.999997	−	0.00239500i	$-0.999238\pi$
$577$		−	0.115060i		−	0.00479001i	0.999997	−	0.00239500i	$-0.000762354\pi$
$578$		0			0
$579$	10.5609	+	43.5372i	0.438895	+	1.80934i
$580$		0			0
$581$		−	20.1917i		−	0.837694i
$582$		0			0
$583$	−73.9932			−3.06448
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	24.5967i		−	1.01521i	−0.861589	−	0.507607i	$-0.830530\pi$
$587$		−	24.5967i		−	1.01521i	0.861589	−	0.507607i	$-0.169470\pi$
$588$		0			0
$589$	17.8779			0.736646
$590$		0			0
$591$	37.6616	−	9.13562i	1.54919	−	0.375789i
$592$		0			0
$593$	−29.8619			−1.22628			−0.613140	−	0.789974i	$-0.710094\pi$
$593$	−29.8619			−1.22628			−0.613140	+	0.789974i	$0.710094\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.60078	+	10.7217i	0.106443	+	0.438811i
$598$		0			0
$599$	13.6643			0.558308			0.279154	−	0.960246i	$-0.409946\pi$
$599$	13.6643			0.558308			0.279154	+	0.960246i	$0.409946\pi$
$600$		0			0
$601$	−9.35657			−0.381663			−0.190831	−	0.981623i	$-0.561118\pi$
$601$	−9.35657			−0.381663			−0.190831	+	0.981623i	$0.561118\pi$
$602$		0			0
$603$	−21.0497	+	10.8506i	−0.857211	+	0.441869i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	27.9867			1.13594			0.567972	−	0.823048i	$-0.307728\pi$
$607$	27.9867			1.13594			0.567972	+	0.823048i	$0.307728\pi$
$608$		0			0
$609$	4.68610	+	19.3184i	0.189890	+	0.782822i
$610$		0			0
$611$	−19.3610			−0.783261
$612$		0			0
$613$		−	2.46171i		−	0.0994277i	−0.998764	−	0.0497138i	$-0.984169\pi$
$613$		−	2.46171i		−	0.0994277i	0.998764	−	0.0497138i	$-0.0158309\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.39313			−0.0963440			−0.0481720	−	0.998839i	$-0.515340\pi$
$617$	−2.39313			−0.0963440			−0.0481720	+	0.998839i	$0.515340\pi$
$618$		0			0
$619$		−	37.4939i		−	1.50701i	−0.657444	−	0.753503i	$-0.728363\pi$
$619$		−	37.4939i		−	1.50701i	0.657444	−	0.753503i	$-0.271637\pi$
$620$		0			0
$621$	14.2457	+	16.4429i	0.571660	+	0.659830i
$622$		0			0
$623$		−	19.7710i		−	0.792108i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	6.96343	+	28.7067i	0.278092	+	1.14644i
$628$		0			0
$629$			6.58187i			0.262436i
$630$		0			0
$631$			36.3255i			1.44609i	0.690798	+	0.723047i	$0.257259\pi$
$631$			36.3255i			1.44609i	−0.690798	+	0.723047i	$0.742741\pi$
$632$		0			0
$633$	10.3867	+	42.8193i	0.412836	+	1.70192i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	17.6892i		−	0.700870i
$638$		0			0
$639$	−29.3105	+	15.1088i	−1.15950	+	0.597694i
$640$		0			0
$641$			4.46351i			0.176298i	0.996107	+	0.0881489i	$0.0280951\pi$
$641$			4.46351i			0.176298i	−0.996107	+	0.0881489i	$0.971905\pi$
$642$		0			0
$643$	17.4537			0.688306			0.344153	−	0.938914i	$-0.388166\pi$
$643$	17.4537			0.688306			0.344153	+	0.938914i	$0.388166\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.84087i			0.0723721i	0.999345	+	0.0361860i	$0.0115209\pi$
$647$			1.84087i			0.0723721i	−0.999345	+	0.0361860i	$0.988479\pi$
$648$		0			0
$649$	23.7756			0.933275
$650$		0			0
$651$	3.70173	+	15.2604i	0.145082	+	0.598102i
$652$		0			0
$653$	−9.19535			−0.359842			−0.179921	−	0.983681i	$-0.557584\pi$
$653$	−9.19535			−0.359842			−0.179921	+	0.983681i	$0.557584\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.25073	+	2.42637i	0.0487957	+	0.0946618i
$658$		0			0
$659$	−22.6914			−0.883932			−0.441966	−	0.897032i	$-0.645719\pi$
$659$	−22.6914			−0.883932			−0.441966	+	0.897032i	$0.645719\pi$
$660$		0			0
$661$	16.8651			0.655978			0.327989	−	0.944682i	$-0.393629\pi$
$661$	16.8651			0.655978			0.327989	+	0.944682i	$0.393629\pi$
$662$		0			0
$663$	−1.99944	−	8.24271i	−0.0776520	−	0.320120i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−34.1790			−1.32342
$668$		0			0
$669$	−6.72173	+	1.63050i	−0.259877	+	0.0630388i
$670$		0			0
$671$	48.8715			1.88666
$672$		0			0
$673$			1.53829i			0.0592966i	0.999560	+	0.0296483i	$0.00943873\pi$
$673$			1.53829i			0.0592966i	−0.999560	+	0.0296483i	$0.990561\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.76238			0.259899			0.129950	−	0.991521i	$-0.458518\pi$
$677$	6.76238			0.259899			0.129950	+	0.991521i	$0.458518\pi$
$678$		0			0
$679$		−	13.2752i		−	0.509457i
$680$		0			0
$681$	−3.91597	−	16.1436i	−0.150060	−	0.618624i
$682$		0			0
$683$			11.0956i			0.424563i	0.977209	+	0.212281i	$0.0680893\pi$
$683$			11.0956i			0.424563i	−0.977209	+	0.212281i	$0.931911\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	15.9145	−	3.86039i	0.607175	−	0.147283i
$688$		0			0
$689$			42.3559i			1.61363i
$690$		0			0
$691$			15.6949i			0.597062i	0.954400	+	0.298531i	$0.0964965\pi$
$691$			15.6949i			0.597062i	−0.954400	+	0.298531i	$0.903503\pi$
$692$		0			0
$693$	−23.0619	+	11.8878i	−0.876050	+	0.451581i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			4.20514i			0.159281i
$698$		0			0
$699$	−2.39499	+	0.580957i	−0.0905870	+	0.0219738i
$700$		0			0
$701$			9.45099i			0.356959i	0.983944	+	0.178479i	$0.0571178\pi$
$701$			9.45099i			0.356959i	−0.983944	+	0.178479i	$0.942882\pi$
$702$		0			0
$703$	−13.1215			−0.494888
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	23.7992i		−	0.895061i
$708$		0			0
$709$	−18.3011			−0.687314			−0.343657	−	0.939095i	$-0.611666\pi$
$709$	−18.3011			−0.687314			−0.343657	+	0.939095i	$0.611666\pi$
$710$		0			0
$711$	−1.54160	−	2.99066i	−0.0578147	−	0.112158i
$712$		0			0
$713$	−26.9994			−1.01113
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−17.9523	+	4.35471i	−0.670440	+	0.162629i
$718$		0			0
$719$	−10.1404			−0.378171			−0.189086	−	0.981961i	$-0.560552\pi$
$719$	−10.1404			−0.378171			−0.189086	+	0.981961i	$0.560552\pi$
$720$		0			0
$721$	−15.7954			−0.588252
$722$		0			0
$723$	−7.62935	+	1.85066i	−0.283739	+	0.0688269i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	34.3819			1.27516			0.637578	−	0.770386i	$-0.279937\pi$
$727$	34.3819			1.27516			0.637578	+	0.770386i	$0.279937\pi$
$728$		0			0
$729$	3.84645	−	26.7246i	0.142461	−	0.989800i
$730$		0			0
$731$	7.94843			0.293983
$732$		0			0
$733$		−	25.1562i		−	0.929165i	−0.885530	−	0.464582i	$-0.846204\pi$
$733$		−	25.1562i		−	0.929165i	0.885530	−	0.464582i	$-0.153796\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	48.5599			1.78873
$738$		0			0
$739$			21.1626i			0.778481i	0.921136	+	0.389240i	$0.127262\pi$
$739$			21.1626i			0.778481i	−0.921136	+	0.389240i	$0.872738\pi$
$740$		0			0
$741$	16.4326	−	3.98607i	0.603666	−	0.146432i
$742$		0			0
$743$		−	46.7087i		−	1.71358i	−0.515669	−	0.856788i	$-0.672457\pi$
$743$		−	46.7087i		−	1.71358i	0.515669	−	0.856788i	$-0.327543\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−19.7414	−	38.2975i	−0.722299	−	1.40123i
$748$		0			0
$749$		−	10.9447i		−	0.399910i
$750$		0			0
$751$			9.61141i			0.350725i	0.984504	+	0.175363i	$0.0561098\pi$
$751$			9.61141i			0.350725i	−0.984504	+	0.175363i	$0.943890\pi$
$752$		0			0
$753$	20.4182	−	4.95287i	0.744081	−	0.180493i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	29.9388i		−	1.08815i	−0.839038	−	0.544073i	$-0.816882\pi$
$757$		−	29.9388i		−	1.08815i	0.839038	−	0.544073i	$-0.183118\pi$
$758$		0			0
$759$	−10.5162	−	43.3532i	−0.381715	−	1.57362i
$760$		0			0
$761$		−	21.7438i		−	0.788211i	−0.919065	−	0.394105i	$-0.871055\pi$
$761$		−	21.7438i		−	0.788211i	0.919065	−	0.394105i	$-0.128945\pi$
$762$		0			0
$763$	25.5701			0.925700
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	13.6099i		−	0.491424i
$768$		0			0
$769$	−9.30972			−0.335717			−0.167859	−	0.985811i	$-0.553685\pi$
$769$	−9.30972			−0.335717			−0.167859	+	0.985811i	$0.553685\pi$
$770$		0			0
$771$	27.4818	−	6.66630i	0.989733	−	0.240081i
$772$		0			0
$773$	30.7908			1.10747			0.553734	−	0.832694i	$-0.313202\pi$
$773$	30.7908			1.10747			0.553734	+	0.832694i	$0.313202\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−2.71690	−	11.2004i	−0.0974681	−	0.401812i
$778$		0			0
$779$	−8.38331			−0.300363
$780$		0			0
$781$	67.6168			2.41952
$782$		0			0
$783$	27.7756	+	32.0596i	0.992620	+	1.14572i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	0.0413019			0.00147225			0.000736127	−	1.00000i	$-0.499766\pi$
$787$	0.0413019			0.00147225			0.000736127		1.00000i	$0.499766\pi$
$788$		0			0
$789$	4.24363	+	17.4943i	0.151077	+	0.622815i
$790$		0			0
$791$	9.98083			0.354877
$792$		0			0
$793$		−	27.9755i		−	0.993438i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.12834			−0.0753895			−0.0376948	−	0.999289i	$-0.512001\pi$
$797$	−2.12834			−0.0753895			−0.0376948	+	0.999289i	$0.512001\pi$
$798$		0			0
$799$			7.64605i			0.270498i
$800$		0			0
$801$	−19.3300	−	37.4995i	−0.682992	−	1.32498i
$802$		0			0
$803$		−	5.59744i		−	0.197529i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−3.87044	−	15.9559i	−0.136246	−	0.561673i
$808$		0			0
$809$		−	19.9170i		−	0.700243i	−0.936704	−	0.350121i	$-0.886140\pi$
$809$		−	19.9170i		−	0.700243i	0.936704	−	0.350121i	$-0.113860\pi$
$810$		0			0
$811$		−	5.43792i		−	0.190951i	−0.995432	−	0.0954756i	$-0.969563\pi$
$811$		−	5.43792i		−	0.190951i	0.995432	−	0.0954756i	$-0.0304372\pi$
$812$		0			0
$813$	−9.71269	−	40.0406i	−0.340639	−	1.40428i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			15.8459i			0.554378i
$818$		0			0
$819$	6.80493	+	13.2013i	0.237784	+	0.461292i
$820$		0			0
$821$		−	21.1720i		−	0.738908i	−0.929249	−	0.369454i	$-0.879545\pi$
$821$		−	21.1720i		−	0.738908i	0.929249	−	0.369454i	$-0.120455\pi$
$822$		0			0
$823$	12.4551			0.434156			0.217078	−	0.976154i	$-0.430347\pi$
$823$	12.4551			0.434156			0.217078	+	0.976154i	$0.430347\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			22.3105i			0.775812i	0.921699	+	0.387906i	$0.126802\pi$
$827$			22.3105i			0.775812i	−0.921699	+	0.387906i	$0.873198\pi$
$828$		0			0
$829$	−31.5699			−1.09647			−0.548234	−	0.836325i	$-0.684700\pi$
$829$	−31.5699			−1.09647			−0.548234	+	0.836325i	$0.684700\pi$
$830$		0			0
$831$	−8.84124	−	36.4480i	−0.306699	−	1.26437i
$832$		0			0
$833$	−6.98582			−0.242044
$834$		0			0
$835$		0			0
$836$		0			0
$837$	21.9411	+	25.3251i	0.758394	+	0.875364i
$838$		0			0
$839$	−8.30477			−0.286712			−0.143356	−	0.989671i	$-0.545789\pi$
$839$	−8.30477			−0.286712			−0.143356	+	0.989671i	$0.545789\pi$
$840$		0			0
$841$	−37.6408			−1.29796
$842$		0			0
$843$	−3.02398	−	12.4664i	−0.104151	−	0.429364i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	37.7369			1.29666
$848$		0			0
$849$	19.5098	−	4.73253i	0.669576	−	0.162420i
$850$		0			0
$851$	19.8163			0.679293
$852$		0			0
$853$		−	11.6348i		−	0.398369i	−0.979962	−	0.199185i	$-0.936171\pi$
$853$		−	11.6348i		−	0.398369i	0.979962	−	0.199185i	$-0.0638294\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−51.8482			−1.77110			−0.885550	−	0.464543i	$-0.846219\pi$
$857$	−51.8482			−1.77110			−0.885550	+	0.464543i	$0.846219\pi$
$858$		0			0
$859$		−	42.9844i		−	1.46661i	−0.679901	−	0.733304i	$-0.737977\pi$
$859$		−	42.9844i		−	1.46661i	0.679901	−	0.733304i	$-0.262023\pi$
$860$		0			0
$861$	−1.73582	−	7.15591i	−0.0591565	−	0.243873i
$862$		0			0
$863$			31.0656i			1.05748i	0.848783	+	0.528742i	$0.177336\pi$
$863$			31.0656i			1.05748i	−0.848783	+	0.528742i	$0.822664\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	25.3598	−	6.15156i	0.861264	−	0.208918i
$868$		0			0
$869$			6.89919i			0.234039i
$870$		0			0
$871$		−	27.7971i		−	0.941870i
$872$		0			0
$873$	−12.9791	−	25.1790i	−0.439277	−	0.852182i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			52.8726i			1.78538i	0.450670	+	0.892691i	$0.351185\pi$
$877$			52.8726i			1.78538i	−0.450670	+	0.892691i	$0.648815\pi$
$878$		0			0
$879$	53.9561	−	13.0882i	1.81990	−	0.441454i
$880$		0			0
$881$		−	16.9797i		−	0.572061i	−0.958220	−	0.286031i	$-0.907664\pi$
$881$		−	16.9797i		−	0.572061i	0.958220	−	0.286031i	$-0.0923359\pi$
$882$		0			0
$883$	34.5602			1.16304			0.581522	−	0.813531i	$-0.302457\pi$
$883$	34.5602			1.16304			0.581522	+	0.813531i	$0.302457\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			46.9163i			1.57530i	0.616125	+	0.787648i	$0.288702\pi$
$887$			46.9163i			1.57530i	−0.616125	+	0.787648i	$0.711298\pi$
$888$		0			0
$889$	13.4190			0.450060
$890$		0			0
$891$	−32.1188	+	45.0951i	−1.07602	+	1.51074i
$892$		0			0
$893$	−15.2431			−0.510090
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−24.8166	+	6.01980i	−0.828603	+	0.200995i
$898$		0			0
$899$	−52.6422			−1.75571
$900$		0			0
$901$	16.7272			0.557264
$902$		0			0
$903$	−13.5259	+	3.28099i	−0.450113	+	0.109185i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−39.8545			−1.32335			−0.661673	−	0.749792i	$-0.730153\pi$
$907$	−39.8545			−1.32335			−0.661673	+	0.749792i	$0.730153\pi$
$908$		0			0
$909$	−23.2684	−	45.1398i	−0.771763	−	1.49719i
$910$		0			0
$911$	−32.6627			−1.08216			−0.541081	−	0.840970i	$-0.681985\pi$
$911$	−32.6627			−1.08216			−0.541081	+	0.840970i	$0.681985\pi$
$912$		0			0
$913$			88.3492i			2.92393i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.97611			−0.0652569
$918$		0			0
$919$		−	16.9449i		−	0.558962i	−0.960151	−	0.279481i	$-0.909838\pi$
$919$		−	16.9449i		−	0.558962i	0.960151	−	0.279481i	$-0.0901623\pi$
$920$		0			0
$921$	−42.4416	+	10.2951i	−1.39850	+	0.339236i
$922$		0			0
$923$		−	38.7058i		−	1.27402i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−29.9591	+	15.4431i	−0.983985	+	0.507218i
$928$		0			0
$929$			57.3567i			1.88181i	0.338665	+	0.940907i	$0.390025\pi$
$929$			57.3567i			1.88181i	−0.338665	+	0.940907i	$0.609975\pi$
$930$		0			0
$931$		−	13.9269i		−	0.456434i
$932$		0			0
$933$	17.5020	−	4.24549i	0.572991	−	0.138991i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			26.7756i			0.874721i	0.899286	+	0.437361i	$0.144087\pi$
$937$			26.7756i			0.874721i	−0.899286	+	0.437361i	$0.855913\pi$
$938$		0			0
$939$	−3.54806	−	14.6269i	−0.115786	−	0.477330i
$940$		0			0
$941$			18.1922i			0.593049i	0.955025	+	0.296524i	$0.0958276\pi$
$941$			18.1922i			0.593049i	−0.955025	+	0.296524i	$0.904172\pi$
$942$		0			0
$943$	12.6606			0.412285
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	34.3856i		−	1.11738i	−0.829376	−	0.558691i	$-0.811304\pi$
$947$		−	34.3856i		−	1.11738i	0.829376	−	0.558691i	$-0.188696\pi$
$948$		0			0
$949$	−3.20414			−0.104011
$950$		0			0
$951$	−26.9477	+	6.53673i	−0.873838	+	0.211968i
$952$		0			0
$953$	42.3018			1.37029			0.685145	−	0.728407i	$-0.259739\pi$
$953$	42.3018			1.37029			0.685145	+	0.728407i	$0.259739\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−20.5041	−	84.5280i	−0.662802	−	2.73240i
$958$		0			0
$959$	−16.7928			−0.542267
$960$		0			0
$961$	−10.5841			−0.341424
$962$		0			0
$963$	−10.7006	−	20.7587i	−0.344821	−	0.668940i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−20.4110			−0.656374			−0.328187	−	0.944613i	$-0.606438\pi$
$967$	−20.4110			−0.656374			−0.328187	+	0.944613i	$0.606438\pi$
$968$		0			0
$969$	−1.57418	−	6.48956i	−0.0505700	−	0.208475i
$970$		0			0
$971$	−9.57208			−0.307183			−0.153591	−	0.988134i	$-0.549084\pi$
$971$	−9.57208			−0.307183			−0.153591	+	0.988134i	$0.549084\pi$
$972$		0			0
$973$		−	23.0697i		−	0.739581i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11.2815			−0.360927			−0.180464	−	0.983582i	$-0.557760\pi$
$977$	−11.2815			−0.360927			−0.180464	+	0.983582i	$0.557760\pi$
$978$		0			0
$979$			86.5082i			2.76481i
$980$		0			0
$981$	48.4987	−	24.9998i	1.54844	−	0.798181i
$982$		0			0
$983$		−	4.05349i		−	0.129286i	−0.997908	−	0.0646432i	$-0.979409\pi$
$983$		−	4.05349i		−	0.129286i	0.997908	−	0.0646432i	$-0.0205909\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−3.15617	−	13.0113i	−0.100462	−	0.414155i
$988$		0			0
$989$		−	23.9306i		−	0.760949i
$990$		0			0
$991$		−	48.7598i		−	1.54891i	−0.632631	−	0.774453i	$-0.718025\pi$
$991$		−	48.7598i		−	1.54891i	0.632631	−	0.774453i	$-0.281975\pi$
$992$		0			0
$993$	2.31762	+	9.55438i	0.0735474	+	0.303199i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	61.9989i		−	1.96352i	−0.190113	−	0.981762i	$-0.560885\pi$
$997$		−	61.9989i		−	1.96352i	0.190113	−	0.981762i	$-0.439115\pi$
$998$		0			0
$999$	−16.1037	−	18.5874i	−0.509498	−	0.588080i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.o.k.2399.3		16
3.2	odd	2	inner	2400.2.o.k.2399.2		16
4.3	odd	2		2400.2.o.l.2399.14		16
5.2	odd	4		2400.2.h.g.1151.10	yes	16
5.3	odd	4		2400.2.h.f.1151.7	✓	16
5.4	even	2		2400.2.o.l.2399.13		16
12.11	even	2		2400.2.o.l.2399.15		16
15.2	even	4		2400.2.h.g.1151.8	yes	16
15.8	even	4		2400.2.h.f.1151.9	yes	16
15.14	odd	2		2400.2.o.l.2399.16		16
20.3	even	4		2400.2.h.f.1151.10	yes	16
20.7	even	4		2400.2.h.g.1151.7	yes	16
20.19	odd	2	inner	2400.2.o.k.2399.4		16
60.23	odd	4		2400.2.h.f.1151.8	yes	16
60.47	odd	4		2400.2.h.g.1151.9	yes	16
60.59	even	2	inner	2400.2.o.k.2399.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.2.h.f.1151.7	✓	16	5.3	odd	4
2400.2.h.f.1151.8	yes	16	60.23	odd	4
2400.2.h.f.1151.9	yes	16	15.8	even	4
2400.2.h.f.1151.10	yes	16	20.3	even	4
2400.2.h.g.1151.7	yes	16	20.7	even	4
2400.2.h.g.1151.8	yes	16	15.2	even	4
2400.2.h.g.1151.9	yes	16	60.47	odd	4
2400.2.h.g.1151.10	yes	16	5.2	odd	4
2400.2.o.k.2399.1		16	60.59	even	2	inner
2400.2.o.k.2399.2		16	3.2	odd	2	inner
2400.2.o.k.2399.3		16	1.1	even	1	trivial
2400.2.o.k.2399.4		16	20.19	odd	2	inner
2400.2.o.l.2399.13		16	5.4	even	2
2400.2.o.l.2399.14		16	4.3	odd	2
2400.2.o.l.2399.15		16	12.11	even	2
2400.2.o.l.2399.16		16	15.14	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.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.68324 + 0.408305i) q^{3} +1.40591 q^{7} +(2.66657 - 1.37455i) q^{9} +O(q^{10})\)	\(q+(-1.68324 + 0.408305i) q^{3} +1.40591 q^{7} +(2.66657 - 1.37455i) q^{9} -6.15156 q^{11} +3.52133i q^{13} +1.39065 q^{17} +2.77238i q^{19} +(-2.36647 + 0.574039i) q^{21} -4.18687i q^{23} +(-3.92724 + 3.40247i) q^{27} -8.16338i q^{29} -6.44858i q^{31} +(10.3545 - 2.51171i) q^{33} +4.73295i q^{37} +(-1.43778 - 5.92724i) q^{39} +3.02387i q^{41} +5.71563 q^{43} +5.49819i q^{47} -5.02343 q^{49} +(-2.34079 + 0.567809i) q^{51} +12.0284 q^{53} +(-1.13198 - 4.66657i) q^{57} -3.86497 q^{59} -7.94456 q^{61} +(3.74895 - 1.93249i) q^{63} -7.89392 q^{67} +(1.70952 + 7.04750i) q^{69} -10.9918 q^{71} +0.909922i q^{73} -8.64852 q^{77} -1.12153i q^{79} +(5.22124 - 7.33067i) q^{81} -14.3621i q^{83} +(3.33315 + 13.7409i) q^{87} -14.0628i q^{89} +4.95067i q^{91} +(2.63299 + 10.8545i) q^{93} -9.44247i q^{97} +(-16.4036 + 8.45562i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) 16 * q - 4 * q^3 - 4 * q^9	\( 16 q - 4 q^{3} - 4 q^{9} + 8 q^{21} + 8 q^{27} + 64 q^{43} + 8 q^{49} + 8 q^{61} + 80 q^{63} - 8 q^{67} + 24 q^{69} + 36 q^{81} - 40 q^{87}+O(q^{100}) \) 16 * q - 4 * q^3 - 4 * q^9 + 8 * q^21 + 8 * q^27 + 64 * q^43 + 8 * q^49 + 8 * q^61 + 80 * q^63 - 8 * q^67 + 24 * q^69 + 36 * q^81 - 40 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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