LMFDB - Embedded newform 2400.2.h.f.1151.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(1151,2400)]

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, 0]))

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

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

chi := DirichletCharacter("2400.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.h (of order $2$, degree $1$, 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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.7
Root			$1.47894 + 0.901513i$ of defining polynomial
Character	$\chi$	$=$	2400.1151
Dual form			2400.2.h.f.1151.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.408305 - 1.68324i) q^{3} -1.40591i q^{7} +(-2.66657 + 1.37455i) q^{9} +O(q^{10})$	$q+(-0.408305 - 1.68324i) q^{3} -1.40591i q^{7} +(-2.66657 + 1.37455i) q^{9} -6.15156 q^{11} -3.52133 q^{13} -1.39065i q^{17} -2.77238i q^{19} +(-2.36647 + 0.574039i) q^{21} +4.18687 q^{23} +(3.40247 + 3.92724i) q^{27} +8.16338i q^{29} -6.44858i q^{31} +(2.51171 + 10.3545i) q^{33} +4.73295 q^{37} +(1.43778 + 5.92724i) q^{39} +3.02387i q^{41} +5.71563i q^{43} +5.49819 q^{47} +5.02343 q^{49} +(-2.34079 + 0.567809i) q^{51} +12.0284i q^{53} +(-4.66657 + 1.13198i) q^{57} +3.86497 q^{59} -7.94456 q^{61} +(1.93249 + 3.74895i) q^{63} +7.89392i q^{67} +(-1.70952 - 7.04750i) q^{69} -10.9918 q^{71} -0.909922 q^{73} +8.64852i q^{77} +1.12153i q^{79} +(5.22124 - 7.33067i) q^{81} +14.3621 q^{83} +(13.7409 - 3.33315i) q^{87} +14.0628i q^{89} +4.95067i q^{91} +(-10.8545 + 2.63299i) q^{93} -9.44247 q^{97} +(16.4036 - 8.45562i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{9}+O(q^{10}) $ 16 * q + 4 * q^9	$ 16 q + 4 q^{9} - 8 q^{13} + 8 q^{21} - 4 q^{33} - 16 q^{37} - 8 q^{49} - 28 q^{57} + 8 q^{61} - 24 q^{69} - 8 q^{73} + 36 q^{81} - 32 q^{93} - 56 q^{97}+O(q^{100}) $ 16 * q + 4 * q^9 - 8 * q^13 + 8 * q^21 - 4 * q^33 - 16 * q^37 - 8 * q^49 - 28 * q^57 + 8 * q^61 - 24 * q^69 - 8 * q^73 + 36 * q^81 - 32 * q^93 - 56 * q^97

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
$3$	−0.408305	−	1.68324i	−0.235735	−	0.971817i
$3$	−0.408305	−	1.68324i	−0.235735	−	0.971817i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	1.40591i		−	0.531383i	−0.964058	−	0.265691i	$-0.914400\pi$
$7$		−	1.40591i		−	0.531383i	0.964058	−	0.265691i	$-0.0856002\pi$
$8$		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
$13$	−3.52133			−0.976643			−0.488321	−	0.872664i	$-0.662391\pi$
$13$	−3.52133			−0.976643			−0.488321	+	0.872664i	$0.662391\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.39065i		−	0.337282i	−0.985678	−	0.168641i	$-0.946062\pi$
$17$		−	1.39065i		−	0.337282i	0.985678	−	0.168641i	$-0.0539378\pi$
$18$		0			0
$19$		−	2.77238i		−	0.636028i	−0.948086	−	0.318014i	$-0.896984\pi$
$19$		−	2.77238i		−	0.636028i	0.948086	−	0.318014i	$-0.103016\pi$
$20$		0			0
$21$	−2.36647	+	0.574039i	−0.516407	+	0.125266i
$22$		0			0
$23$	4.18687			0.873024			0.436512	−	0.899698i	$-0.356214\pi$
$23$	4.18687			0.873024			0.436512	+	0.899698i	$0.356214\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.40247	+	3.92724i	0.654805	+	0.755798i
$28$		0			0
$29$			8.16338i			1.51590i	0.652312	+	0.757951i	$0.273799\pi$
$29$			8.16338i			1.51590i	−0.652312	+	0.757951i	$0.726201\pi$
$30$		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
$33$	2.51171	+	10.3545i	0.437233	+	1.80249i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.73295			0.778092			0.389046	−	0.921218i	$-0.372805\pi$
$37$	4.73295			0.778092			0.389046	+	0.921218i	$0.372805\pi$
$38$		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
$43$			5.71563i			0.871625i	0.900037	+	0.435813i	$0.143539\pi$
$43$			5.71563i			0.871625i	−0.900037	+	0.435813i	$0.856461\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.49819			0.801994			0.400997	−	0.916079i	$-0.368664\pi$
$47$	5.49819			0.801994			0.400997	+	0.916079i	$0.368664\pi$
$48$		0			0
$49$	5.02343			0.717632
$50$		0			0
$51$	−2.34079	+	0.567809i	−0.327776	+	0.0795091i
$52$		0			0
$53$			12.0284i			1.65222i	0.563508	+	0.826111i	$0.309451\pi$
$53$			12.0284i			1.65222i	−0.563508	+	0.826111i	$0.690549\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.66657	+	1.13198i	−0.618103	+	0.149934i
$58$		0			0
$59$	3.86497			0.503177			0.251588	−	0.967834i	$-0.419047\pi$
$59$	3.86497			0.503177			0.251588	+	0.967834i	$0.419047\pi$
$60$		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
$63$	1.93249	+	3.74895i	0.243470	+	0.472324i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			7.89392i			0.964395i	0.876062	+	0.482198i	$0.160161\pi$
$67$			7.89392i			0.964395i	−0.876062	+	0.482198i	$0.839839\pi$
$68$		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
$73$	−0.909922			−0.106498			−0.0532491	−	0.998581i	$-0.516958\pi$
$73$	−0.909922			−0.106498			−0.0532491	+	0.998581i	$0.516958\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			8.64852i			0.985591i
$78$		0			0
$79$			1.12153i			0.126183i	0.998008	+	0.0630913i	$0.0200959\pi$
$79$			1.12153i			0.126183i	−0.998008	+	0.0630913i	$0.979904\pi$
$80$		0			0
$81$	5.22124	−	7.33067i	0.580137	−	0.814519i
$82$		0			0
$83$	14.3621			1.57644			0.788221	−	0.615392i	$-0.211002\pi$
$83$	14.3621			1.57644			0.788221	+	0.615392i	$0.211002\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	13.7409	−	3.33315i	1.47318	−	0.357351i
$88$		0			0
$89$			14.0628i			1.49065i	0.666699	+	0.745327i	$0.267707\pi$
$89$			14.0628i			1.49065i	−0.666699	+	0.745327i	$0.732293\pi$
$90$		0			0
$91$			4.95067i			0.518971i
$92$		0			0
$93$	−10.8545	+	2.63299i	−1.12556	+	0.273028i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−9.44247			−0.958738			−0.479369	−	0.877614i	$-0.659134\pi$
$97$	−9.44247			−0.958738			−0.479369	+	0.877614i	$0.659134\pi$
$98$		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
$103$		−	11.2350i		−	1.10702i	−0.832842	−	0.553511i	$-0.813288\pi$
$103$		−	11.2350i		−	1.10702i	0.832842	−	0.553511i	$-0.186712\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.78478			−0.752583			−0.376292	−	0.926501i	$-0.622801\pi$
$107$	−7.78478			−0.752583			−0.376292	+	0.926501i	$0.622801\pi$
$108$		0			0
$109$	−18.1876			−1.74206			−0.871030	−	0.491231i	$-0.836547\pi$
$109$	−18.1876			−1.74206			−0.871030	+	0.491231i	$0.836547\pi$
$110$		0			0
$111$	−1.93249	−	7.96667i	−0.183423	−	0.756163i
$112$		0			0
$113$			7.09921i			0.667838i	0.942602	+	0.333919i	$0.108371\pi$
$113$			7.09921i			0.667838i	−0.942602	+	0.333919i	$0.891629\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	9.38990	−	4.84024i	0.868097	−	0.447481i
$118$		0			0
$119$	−1.95512			−0.179226
$120$		0			0
$121$	26.8417			2.44016
$122$		0			0
$123$	5.08989	−	1.23466i	0.458940	−	0.111326i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	9.54476i		−	0.846961i	−0.905905	−	0.423480i	$-0.860808\pi$
$127$		−	9.54476i		−	0.846961i	0.905905	−	0.423480i	$-0.139192\pi$
$128$		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
$133$	−3.89771			−0.337974
$134$		0			0
$135$		0			0
$136$		0			0
$137$			11.9444i			1.02048i	0.860031	+	0.510241i	$0.170444\pi$
$137$			11.9444i			1.02048i	−0.860031	+	0.510241i	$0.829556\pi$
$138$		0			0
$139$			16.4091i			1.39181i	0.718136	+	0.695903i	$0.244996\pi$
$139$			16.4091i			1.39181i	−0.718136	+	0.695903i	$0.755004\pi$
$140$		0			0
$141$	−2.24494	−	9.25476i	−0.189058	−	0.779392i
$142$		0			0
$143$	21.6617			1.81144
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.05109	−	8.45562i	−0.169171	−	0.697408i
$148$		0			0
$149$			12.3974i			1.01563i	0.861465	+	0.507817i	$0.169547\pi$
$149$			12.3974i			1.01563i	−0.861465	+	0.507817i	$0.830453\pi$
$150$		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
$153$	1.91151	+	3.70827i	0.154537	+	0.299796i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−3.45468			−0.275714			−0.137857	−	0.990452i	$-0.544021\pi$
$157$	−3.45468			−0.275714			−0.137857	+	0.990452i	$0.544021\pi$
$158$		0			0
$159$	20.2466	−	4.91124i	1.60566	−	0.389486i
$160$		0			0
$161$		−	5.88635i		−	0.463910i
$162$		0			0
$163$			17.0032i			1.33180i	0.746043	+	0.665898i	$0.231951\pi$
$163$			17.0032i			1.33180i	−0.746043	+	0.665898i	$0.768049\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.2582			−1.10334			−0.551668	−	0.834064i	$-0.686008\pi$
$167$	−14.2582			−1.10334			−0.551668	+	0.834064i	$0.686008\pi$
$168$		0			0
$169$	−0.600200			−0.0461692
$170$		0			0
$171$	3.81077	+	7.39276i	0.291417	+	0.565338i
$172$		0			0
$173$		−	13.0630i		−	0.993164i	−0.867990	−	0.496582i	$-0.834588\pi$
$173$		−	13.0630i		−	0.993164i	0.867990	−	0.496582i	$-0.165412\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.57809	−	6.50567i	−0.118616	−	0.488996i
$178$		0			0
$179$	−8.86388			−0.662517			−0.331259	−	0.943540i	$-0.607473\pi$
$179$	−8.86388			−0.662517			−0.331259	+	0.943540i	$0.607473\pi$
$180$		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
$183$	3.24380	+	13.3726i	0.239789	+	0.988530i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			8.55466i			0.625579i
$188$		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
$193$	25.8651			1.86181			0.930907	−	0.365257i	$-0.119019\pi$
$193$	25.8651			1.86181			0.930907	+	0.365257i	$0.119019\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			22.3745i			1.59412i	0.603902	+	0.797058i	$0.293612\pi$
$197$			22.3745i			1.59412i	−0.603902	+	0.797058i	$0.706388\pi$
$198$		0			0
$199$			6.36971i			0.451537i	0.974181	+	0.225768i	$0.0724893\pi$
$199$			6.36971i			0.451537i	−0.974181	+	0.225768i	$0.927511\pi$
$200$		0			0
$201$	13.2873	−	3.22312i	0.937216	−	0.227342i
$202$		0			0
$203$	11.4769			0.805524
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−11.1646	+	5.75506i	−0.775994	+	0.400004i
$208$		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
$213$	4.48801	+	18.5018i	0.307513	+	1.26772i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−9.06610			−0.615447
$218$		0			0
$219$	0.371525	+	1.53161i	0.0251054	+	0.103497i
$220$		0			0
$221$			4.89694i			0.329404i
$222$		0			0
$223$			3.99334i			0.267414i	0.991021	+	0.133707i	$0.0426881\pi$
$223$			3.99334i			0.267414i	−0.991021	+	0.133707i	$0.957312\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.59081			0.636564			0.318282	−	0.947996i	$-0.396894\pi$
$227$	9.59081			0.636564			0.318282	+	0.947996i	$0.396894\pi$
$228$		0			0
$229$	9.45468			0.624783			0.312392	−	0.949953i	$-0.398870\pi$
$229$	9.45468			0.624783			0.312392	+	0.949953i	$0.398870\pi$
$230$		0			0
$231$	14.5575	−	3.53123i	0.957814	−	0.232338i
$232$		0			0
$233$			1.42285i			0.0932140i	0.998913	+	0.0466070i	$0.0148408\pi$
$233$			1.42285i			0.0932140i	−0.998913	+	0.0466070i	$0.985159\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.88781	−	0.457928i	0.122626	−	0.0297456i
$238$		0			0
$239$	−10.6653			−0.689883			−0.344941	−	0.938624i	$-0.612101\pi$
$239$	−10.6653			−0.689883			−0.344941	+	0.938624i	$0.612101\pi$
$240$		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
$243$	−14.4711	−	5.79543i	−0.928322	−	0.371777i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			9.76248i			0.621172i
$248$		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
$253$	−25.7558			−1.61925
$254$		0			0
$255$		0			0
$256$		0			0
$257$			16.3268i			1.01844i	0.860638	+	0.509218i	$0.170065\pi$
$257$			16.3268i			1.01844i	−0.860638	+	0.509218i	$0.829935\pi$
$258$		0			0
$259$		−	6.65408i		−	0.413465i
$260$		0			0
$261$	−11.2210	−	21.7683i	−0.694560	−	1.34742i
$262$		0			0
$263$	10.3933			0.640877			0.320438	−	0.947269i	$-0.396170\pi$
$263$	10.3933			0.640877			0.320438	+	0.947269i	$0.396170\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	23.6710	−	5.74191i	1.44864	−	0.351399i
$268$		0			0
$269$		−	9.47928i		−	0.577962i	−0.957335	−	0.288981i	$-0.906684\pi$
$269$		−	9.47928i		−	0.577962i	0.957335	−	0.288981i	$-0.0933164\pi$
$270$		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
$273$	8.33315	−	2.02138i	0.504345	−	0.122340i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	21.6535			1.30103			0.650517	−	0.759492i	$-0.274552\pi$
$277$	21.6535			1.30103			0.650517	+	0.759492i	$0.274552\pi$
$278$		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
$283$		−	11.5907i		−	0.688994i	−0.938787	−	0.344497i	$-0.888050\pi$
$283$		−	11.5907i		−	0.688994i	0.938787	−	0.344497i	$-0.111950\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.25128			0.250945
$288$		0			0
$289$	15.0661			0.886241
$290$		0			0
$291$	3.85541	+	15.8939i	0.226008	+	0.931718i
$292$		0			0
$293$		−	32.0550i		−	1.87267i	−0.351105	−	0.936336i	$-0.614194\pi$
$293$		−	32.0550i		−	1.87267i	0.351105	−	0.936336i	$-0.385806\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−20.9305	−	24.1587i	−1.21451	−	1.40183i
$298$		0			0
$299$	−14.7434			−0.852632
$300$		0			0
$301$	8.03564			0.463167
$302$		0			0
$303$	−28.4939	+	6.91179i	−1.63693	+	0.397072i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	25.2143i		−	1.43906i	−0.694464	−	0.719528i	$-0.744358\pi$
$307$		−	25.2143i		−	1.43906i	0.694464	−	0.719528i	$-0.255642\pi$
$308$		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
$313$	−8.68972			−0.491172			−0.245586	−	0.969375i	$-0.578980\pi$
$313$	−8.68972			−0.491172			−0.245586	+	0.969375i	$0.578980\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	16.0094i		−	0.899179i	−0.893235	−	0.449590i	$-0.851570\pi$
$317$		−	16.0094i		−	0.899179i	0.893235	−	0.449590i	$-0.148430\pi$
$318$		0			0
$319$		−	50.2175i		−	2.81164i
$320$		0			0
$321$	3.17857	+	13.1036i	0.177410	+	0.731373i
$322$		0			0
$323$	−3.85541			−0.214521
$324$		0			0
$325$		0			0
$326$		0			0
$327$	7.42610	+	30.6141i	0.410664	+	1.69296i
$328$		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
$333$	−12.6208	+	6.50567i	−0.691613	+	0.356508i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−13.4232			−0.731210			−0.365605	−	0.930770i	$-0.619138\pi$
$337$	−13.4232			−0.731210			−0.365605	+	0.930770i	$0.619138\pi$
$338$		0			0
$339$	11.9497	−	2.89864i	0.649016	−	0.157433i
$340$		0			0
$341$			39.6688i			2.14819i
$342$		0			0
$343$		−	16.9038i		−	0.912720i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−20.5685			−1.10417			−0.552087	−	0.833787i	$-0.686168\pi$
$347$	−20.5685			−1.10417			−0.552087	+	0.833787i	$0.686168\pi$
$348$		0			0
$349$	−3.10932			−0.166438			−0.0832191	−	0.996531i	$-0.526520\pi$
$349$	−3.10932			−0.166438			−0.0832191	+	0.996531i	$0.526520\pi$
$350$		0			0
$351$	−11.9812	−	13.8291i	−0.639510	−	0.738145i
$352$		0			0
$353$		−	1.61137i		−	0.0857645i	−0.999080	−	0.0428823i	$-0.986346\pi$
$353$		−	1.61137i		−	0.0857645i	0.999080	−	0.0428823i	$-0.0136540\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	0.798286	+	3.29093i	0.0422498	+	0.174175i
$358$		0			0
$359$	−13.2688			−0.700302			−0.350151	−	0.936693i	$-0.613870\pi$
$359$	−13.2688			−0.700302			−0.350151	+	0.936693i	$0.613870\pi$
$360$		0			0
$361$	11.3139			0.595469
$362$		0			0
$363$	−10.9596	−	45.1810i	−0.575230	−	2.37139i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	13.4726i		−	0.703262i	−0.936139	−	0.351631i	$-0.885627\pi$
$367$		−	13.4726i		−	0.703262i	0.936139	−	0.351631i	$-0.114373\pi$
$368$		0			0
$369$	−4.15645	−	8.06337i	−0.216376	−	0.419762i
$370$		0			0
$371$	16.9107			0.877962
$372$		0			0
$373$	5.03146			0.260519			0.130259	−	0.991480i	$-0.458419\pi$
$373$	5.03146			0.260519			0.130259	+	0.991480i	$0.458419\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	28.7460i		−	1.48049i
$378$		0			0
$379$			23.6977i			1.21727i	0.793451	+	0.608635i	$0.208282\pi$
$379$			23.6977i			1.21727i	−0.793451	+	0.608635i	$0.791718\pi$
$380$		0			0
$381$	−16.0661	+	3.89717i	−0.823091	+	0.199658i
$382$		0			0
$383$	−34.0338			−1.73905			−0.869523	−	0.493892i	$-0.835574\pi$
$383$	−34.0338			−1.73905			−0.869523	+	0.493892i	$0.835574\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−7.85641	−	15.2411i	−0.399364	−	0.774751i
$388$		0			0
$389$			12.5779i			0.637725i	0.947801	+	0.318862i	$0.103301\pi$
$389$			12.5779i			0.637725i	−0.947801	+	0.318862i	$0.896699\pi$
$390$		0			0
$391$		−	5.82247i		−	0.294455i
$392$		0			0
$393$	0.573904	+	2.36592i	0.0289496	+	0.119345i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	32.5442			1.63335			0.816673	−	0.577100i	$-0.195816\pi$
$397$	32.5442			1.63335			0.816673	+	0.577100i	$0.195816\pi$
$398$		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
$403$			22.7076i			1.13115i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−29.1150			−1.44318
$408$		0			0
$409$	−1.58096			−0.0781733			−0.0390866	−	0.999236i	$-0.512445\pi$
$409$	−1.58096			−0.0781733			−0.0390866	+	0.999236i	$0.512445\pi$
$410$		0			0
$411$	20.1053	−	4.87698i	0.991723	−	0.240563i
$412$		0			0
$413$		−	5.43379i		−	0.267379i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	27.6205	−	6.69993i	1.35258	−	0.328097i
$418$		0			0
$419$	−28.7337			−1.40373			−0.701867	−	0.712308i	$-0.747650\pi$
$419$	−28.7337			−1.40373			−0.701867	+	0.712308i	$0.747650\pi$
$420$		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
$423$	−14.6613	+	7.55753i	−0.712859	+	0.367460i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			11.1693i			0.540521i
$428$		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
$433$	8.13375			0.390883			0.195441	−	0.980715i	$-0.437386\pi$
$433$	8.13375			0.390883			0.195441	+	0.980715i	$0.437386\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	11.6076i		−	0.555267i
$438$		0			0
$439$			8.54987i			0.408063i	0.978964	+	0.204031i	$0.0654045\pi$
$439$			8.54987i			0.408063i	−0.978964	+	0.204031i	$0.934596\pi$
$440$		0			0
$441$	−13.3953	+	6.90494i	−0.637873	+	0.328807i
$442$		0			0
$443$	−5.28012			−0.250866			−0.125433	−	0.992102i	$-0.540032\pi$
$443$	−5.28012			−0.250866			−0.125433	+	0.992102i	$0.540032\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	20.8677	−	5.06191i	0.987010	−	0.239420i
$448$		0			0
$449$			35.9522i			1.69669i	0.529446	+	0.848344i	$0.322400\pi$
$449$			35.9522i			1.69669i	−0.529446	+	0.848344i	$0.677600\pi$
$450$		0			0
$451$		−	18.6015i		−	0.875911i
$452$		0			0
$453$	20.2998	−	4.92415i	0.953769	−	0.231357i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.0192			0.655793			0.327896	−	0.944714i	$-0.393660\pi$
$457$	14.0192			0.655793			0.327896	+	0.944714i	$0.393660\pi$
$458$		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
$463$		−	0.341733i		−	0.0158817i	−0.999968	−	0.00794085i	$-0.997472\pi$
$463$		−	0.341733i		−	0.0158817i	0.999968	−	0.00794085i	$-0.00252768\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10.2346			0.473601			0.236801	−	0.971558i	$-0.423901\pi$
$467$	10.2346			0.473601			0.236801	+	0.971558i	$0.423901\pi$
$468$		0			0
$469$	11.0981			0.512463
$470$		0			0
$471$	1.41056	+	5.81505i	0.0649954	+	0.267943i
$472$		0			0
$473$		−	35.1600i		−	1.61666i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−16.5335	−	32.0745i	−0.757019	−	1.46859i
$478$		0			0
$479$	−22.5177			−1.02886			−0.514431	−	0.857532i	$-0.671997\pi$
$479$	−22.5177			−1.02886			−0.514431	+	0.857532i	$0.671997\pi$
$480$		0			0
$481$	−16.6663			−0.759918
$482$		0			0
$483$	−9.90813	+	2.40343i	−0.450836	+	0.109360i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			32.1922i			1.45877i	0.684105	+	0.729384i	$0.260193\pi$
$487$			32.1922i			1.45877i	−0.684105	+	0.729384i	$0.739807\pi$
$488$		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
$493$	11.3524			0.511286
$494$		0			0
$495$		0			0
$496$		0			0
$497$			15.4535i			0.693182i
$498$		0			0
$499$		−	5.37389i		−	0.240569i	−0.992739	−	0.120284i	$-0.961619\pi$
$499$		−	5.37389i		−	0.240569i	0.992739	−	0.120284i	$-0.0383806\pi$
$500$		0			0
$501$	5.82171	+	24.0000i	0.260095	+	1.07224i
$502$		0			0
$503$	−35.5337			−1.58437			−0.792184	−	0.610283i	$-0.791056\pi$
$503$	−35.5337			−1.58437			−0.792184	+	0.610283i	$0.791056\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.245065	+	1.01028i	0.0108837	+	0.0448681i
$508$		0			0
$509$			0.485143i			0.0215036i	0.999942	+	0.0107518i	$0.00342246\pi$
$509$			0.485143i			0.0215036i	−0.999942	+	0.0107518i	$0.996578\pi$
$510$		0			0
$511$			1.27926i			0.0565913i
$512$		0			0
$513$	10.8878	−	9.43293i	0.480709	−	0.416474i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−33.8225			−1.48751
$518$		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
$523$			8.54152i			0.373495i	0.982408	+	0.186747i	$0.0597946\pi$
$523$			8.54152i			0.373495i	−0.982408	+	0.186747i	$0.940205\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.96770			−0.390639
$528$		0			0
$529$	−5.47008			−0.237830
$530$		0			0
$531$	−10.3062	+	5.31259i	−0.447253	+	0.230547i
$532$		0			0
$533$		−	10.6481i		−	0.461218i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.61917	+	14.9200i	0.156179	+	0.643846i
$538$		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
$543$	9.92227	+	40.9046i	0.425806	+	1.75538i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	2.56114i		−	0.109506i	−0.998500	−	0.0547531i	$-0.982563\pi$
$547$		−	2.56114i		−	0.109506i	0.998500	−	0.0547531i	$-0.0174372\pi$
$548$		0			0
$549$	21.1848	−	10.9202i	0.904144	−	0.466062i
$550$		0			0
$551$	22.6320			0.964155
$552$		0			0
$553$	1.57677			0.0670512
$554$		0			0
$555$		0			0
$556$		0			0
$557$			19.9388i			0.844834i	0.906402	+	0.422417i	$0.138818\pi$
$557$			19.9388i			0.844834i	−0.906402	+	0.422417i	$0.861182\pi$
$558$		0			0
$559$		−	20.1266i		−	0.851266i
$560$		0			0
$561$	14.3995	−	3.49291i	0.607948	−	0.147471i
$562$		0			0
$563$	−31.5744			−1.33070			−0.665352	−	0.746530i	$-0.731718\pi$
$563$	−31.5744			−1.33070			−0.665352	+	0.746530i	$0.731718\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−10.3062	−	7.34057i	−0.432821	−	0.308275i
$568$		0			0
$569$		−	27.5657i		−	1.15561i	−0.816173	−	0.577807i	$-0.803908\pi$
$569$		−	27.5657i		−	1.15561i	0.816173	−	0.577807i	$-0.196092\pi$
$570$		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
$573$	−3.98076	−	16.4107i	−0.166298	−	0.685565i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.115060			−0.00479001			−0.00239500	−	0.999997i	$-0.500762\pi$
$577$	−0.115060			−0.00479001			−0.00239500	+	0.999997i	$0.500762\pi$
$578$		0			0
$579$	−10.5609	−	43.5372i	−0.438895	−	1.80934i
$580$		0			0
$581$		−	20.1917i		−	0.837694i
$582$		0			0
$583$		−	73.9932i		−	3.06448i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−24.5967			−1.01521			−0.507607	−	0.861589i	$-0.669470\pi$
$587$	−24.5967			−1.01521			−0.507607	+	0.861589i	$0.669470\pi$
$588$		0			0
$589$	−17.8779			−0.736646
$590$		0			0
$591$	37.6616	−	9.13562i	1.54919	−	0.375789i
$592$		0			0
$593$		−	29.8619i		−	1.22628i	−0.789974	−	0.613140i	$-0.789906\pi$
$593$		−	29.8619i		−	1.22628i	0.789974	−	0.613140i	$-0.210094\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	10.7217	−	2.60078i	0.438811	−	0.106443i
$598$		0			0
$599$	−13.6643			−0.558308			−0.279154	−	0.960246i	$-0.590054\pi$
$599$	−13.6643			−0.558308			−0.279154	+	0.960246i	$0.590054\pi$
$600$		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
$603$	−10.8506	−	21.0497i	−0.441869	−	0.857211i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	27.9867i		−	1.13594i	−0.823048	−	0.567972i	$-0.807728\pi$
$607$		−	27.9867i		−	1.13594i	0.823048	−	0.567972i	$-0.192272\pi$
$608$		0			0
$609$	−4.68610	−	19.3184i	−0.189890	−	0.782822i
$610$		0			0
$611$	−19.3610			−0.783261
$612$		0			0
$613$	2.46171			0.0994277			0.0497138	−	0.998764i	$-0.484169\pi$
$613$	2.46171			0.0994277			0.0497138	+	0.998764i	$0.484169\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2.39313i			0.0963440i	0.998839	+	0.0481720i	$0.0153396\pi$
$617$			2.39313i			0.0963440i	−0.998839	+	0.0481720i	$0.984660\pi$
$618$		0			0
$619$			37.4939i			1.50701i	0.657444	+	0.753503i	$0.271637\pi$
$619$			37.4939i			1.50701i	−0.657444	+	0.753503i	$0.728363\pi$
$620$		0			0
$621$	14.2457	+	16.4429i	0.571660	+	0.659830i
$622$		0			0
$623$	19.7710			0.792108
$624$		0			0
$625$		0			0
$626$		0			0
$627$	28.7067	−	6.96343i	1.14644	−	0.278092i
$628$		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
$633$	−42.8193	+	10.3867i	−1.70192	+	0.412836i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−17.6892			−0.700870
$638$		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
$643$			17.4537i			0.688306i	0.938914	+	0.344153i	$0.111834\pi$
$643$			17.4537i			0.688306i	−0.938914	+	0.344153i	$0.888166\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.84087			0.0723721			0.0361860	−	0.999345i	$-0.488479\pi$
$647$	1.84087			0.0723721			0.0361860	+	0.999345i	$0.488479\pi$
$648$		0			0
$649$	−23.7756			−0.933275
$650$		0			0
$651$	3.70173	+	15.2604i	0.145082	+	0.598102i
$652$		0			0
$653$		−	9.19535i		−	0.359842i	−0.983681	−	0.179921i	$-0.942416\pi$
$653$		−	9.19535i		−	0.359842i	0.983681	−	0.179921i	$-0.0575842\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	2.42637	−	1.25073i	0.0946618	−	0.0487957i
$658$		0			0
$659$	22.6914			0.883932			0.441966	−	0.897032i	$-0.354281\pi$
$659$	22.6914			0.883932			0.441966	+	0.897032i	$0.354281\pi$
$660$		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
$663$	8.24271	−	1.99944i	0.320120	−	0.0776520i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			34.1790i			1.32342i
$668$		0			0
$669$	6.72173	−	1.63050i	0.259877	−	0.0630388i
$670$		0			0
$671$	48.8715			1.88666
$672$		0			0
$673$	−1.53829			−0.0592966			−0.0296483	−	0.999560i	$-0.509439\pi$
$673$	−1.53829			−0.0592966			−0.0296483	+	0.999560i	$0.509439\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	6.76238i		−	0.259899i	−0.991521	−	0.129950i	$-0.958518\pi$
$677$		−	6.76238i		−	0.259899i	0.991521	−	0.129950i	$-0.0414816\pi$
$678$		0			0
$679$			13.2752i			0.509457i
$680$		0			0
$681$	−3.91597	−	16.1436i	−0.150060	−	0.618624i
$682$		0			0
$683$	−11.0956			−0.424563			−0.212281	−	0.977209i	$-0.568089\pi$
$683$	−11.0956			−0.424563			−0.212281	+	0.977209i	$0.568089\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−3.86039	−	15.9145i	−0.147283	−	0.607175i
$688$		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
$693$	−11.8878	−	23.0619i	−0.451581	−	0.876050i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	4.20514			0.159281
$698$		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
$703$		−	13.1215i		−	0.494888i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−23.7992			−0.895061
$708$		0			0
$709$	18.3011			0.687314			0.343657	−	0.939095i	$-0.388334\pi$
$709$	18.3011			0.687314			0.343657	+	0.939095i	$0.388334\pi$
$710$		0			0
$711$	−1.54160	−	2.99066i	−0.0578147	−	0.112158i
$712$		0			0
$713$		−	26.9994i		−	1.01113i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	4.35471	+	17.9523i	0.162629	+	0.670440i
$718$		0			0
$719$	10.1404			0.378171			0.189086	−	0.981961i	$-0.439448\pi$
$719$	10.1404			0.378171			0.189086	+	0.981961i	$0.439448\pi$
$720$		0			0
$721$	−15.7954			−0.588252
$722$		0			0
$723$	−1.85066	−	7.62935i	−0.0688269	−	0.283739i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	34.3819i		−	1.27516i	−0.770386	−	0.637578i	$-0.779937\pi$
$727$		−	34.3819i		−	1.27516i	0.770386	−	0.637578i	$-0.220063\pi$
$728$		0			0
$729$	−3.84645	+	26.7246i	−0.142461	+	0.989800i
$730$		0			0
$731$	7.94843			0.293983
$732$		0			0
$733$	25.1562			0.929165			0.464582	−	0.885530i	$-0.346204\pi$
$733$	25.1562			0.929165			0.464582	+	0.885530i	$0.346204\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	48.5599i		−	1.78873i
$738$		0			0
$739$		−	21.1626i		−	0.778481i	−0.921136	−	0.389240i	$-0.872738\pi$
$739$		−	21.1626i		−	0.778481i	0.921136	−	0.389240i	$-0.127262\pi$
$740$		0			0
$741$	16.4326	−	3.98607i	0.603666	−	0.146432i
$742$		0			0
$743$	46.7087			1.71358			0.856788	−	0.515669i	$-0.172457\pi$
$743$	46.7087			1.71358			0.856788	+	0.515669i	$0.172457\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−38.2975	+	19.7414i	−1.40123	+	0.722299i
$748$		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
$753$	4.95287	+	20.4182i	0.180493	+	0.744081i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−29.9388			−1.08815			−0.544073	−	0.839038i	$-0.683118\pi$
$757$	−29.9388			−1.08815			−0.544073	+	0.839038i	$0.683118\pi$
$758$		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
$763$			25.5701i			0.925700i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−13.6099			−0.491424
$768$		0			0
$769$	9.30972			0.335717			0.167859	−	0.985811i	$-0.446315\pi$
$769$	9.30972			0.335717			0.167859	+	0.985811i	$0.446315\pi$
$770$		0			0
$771$	27.4818	−	6.66630i	0.989733	−	0.240081i
$772$		0			0
$773$			30.7908i			1.10747i	0.832694	+	0.553734i	$0.186798\pi$
$773$			30.7908i			1.10747i	−0.832694	+	0.553734i	$0.813202\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−11.2004	+	2.71690i	−0.401812	+	0.0974681i
$778$		0			0
$779$	8.38331			0.300363
$780$		0			0
$781$	67.6168			2.41952
$782$		0			0
$783$	−32.0596	+	27.7756i	−1.14572	+	0.992620i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	0.0413019i		−	0.00147225i	−1.00000		0.000736127i	$-0.999766\pi$
$787$		−	0.0413019i		−	0.00147225i	1.00000		0.000736127i	$-0.000234316\pi$
$788$		0			0
$789$	−4.24363	−	17.4943i	−0.151077	−	0.622815i
$790$		0			0
$791$	9.98083			0.354877
$792$		0			0
$793$	27.9755			0.993438
$794$		0			0
$795$		0			0
$796$		0			0
$797$			2.12834i			0.0753895i	0.999289	+	0.0376948i	$0.0120015\pi$
$797$			2.12834i			0.0753895i	−0.999289	+	0.0376948i	$0.987999\pi$
$798$		0			0
$799$		−	7.64605i		−	0.270498i
$800$		0			0
$801$	−19.3300	−	37.4995i	−0.682992	−	1.32498i
$802$		0			0
$803$	5.59744			0.197529
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−15.9559	+	3.87044i	−0.561673	+	0.136246i
$808$		0			0
$809$			19.9170i			0.700243i	0.936704	+	0.350121i	$0.113860\pi$
$809$			19.9170i			0.700243i	−0.936704	+	0.350121i	$0.886140\pi$
$810$		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
$813$	40.0406	−	9.71269i	1.40428	−	0.340639i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	15.8459			0.554378
$818$		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
$823$			12.4551i			0.434156i	0.976154	+	0.217078i	$0.0696526\pi$
$823$			12.4551i			0.434156i	−0.976154	+	0.217078i	$0.930347\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22.3105			0.775812			0.387906	−	0.921699i	$-0.373198\pi$
$827$	22.3105			0.775812			0.387906	+	0.921699i	$0.373198\pi$
$828$		0			0
$829$	31.5699			1.09647			0.548234	−	0.836325i	$-0.315300\pi$
$829$	31.5699			1.09647			0.548234	+	0.836325i	$0.315300\pi$
$830$		0			0
$831$	−8.84124	−	36.4480i	−0.306699	−	1.26437i
$832$		0			0
$833$		−	6.98582i		−	0.242044i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	25.3251	−	21.9411i	0.875364	−	0.758394i
$838$		0			0
$839$	8.30477			0.286712			0.143356	−	0.989671i	$-0.454211\pi$
$839$	8.30477			0.286712			0.143356	+	0.989671i	$0.454211\pi$
$840$		0			0
$841$	−37.6408			−1.29796
$842$		0			0
$843$	12.4664	−	3.02398i	0.429364	−	0.104151i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	37.7369i		−	1.29666i
$848$		0			0
$849$	−19.5098	+	4.73253i	−0.669576	+	0.162420i
$850$		0			0
$851$	19.8163			0.679293
$852$		0			0
$853$	11.6348			0.398369			0.199185	−	0.979962i	$-0.436171\pi$
$853$	11.6348			0.398369			0.199185	+	0.979962i	$0.436171\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			51.8482i			1.77110i	0.464543	+	0.885550i	$0.346219\pi$
$857$			51.8482i			1.77110i	−0.464543	+	0.885550i	$0.653781\pi$
$858$		0			0
$859$			42.9844i			1.46661i	0.679901	+	0.733304i	$0.262023\pi$
$859$			42.9844i			1.46661i	−0.679901	+	0.733304i	$0.737977\pi$
$860$		0			0
$861$	−1.73582	−	7.15591i	−0.0591565	−	0.243873i
$862$		0			0
$863$	−31.0656			−1.05748			−0.528742	−	0.848783i	$-0.677336\pi$
$863$	−31.0656			−1.05748			−0.528742	+	0.848783i	$0.677336\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−6.15156	−	25.3598i	−0.208918	−	0.861264i
$868$		0			0
$869$		−	6.89919i		−	0.234039i
$870$		0			0
$871$		−	27.7971i		−	0.941870i
$872$		0			0
$873$	25.1790	−	12.9791i	0.852182	−	0.439277i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	52.8726			1.78538			0.892691	−	0.450670i	$-0.148815\pi$
$877$	52.8726			1.78538			0.892691	+	0.450670i	$0.148815\pi$
$878$		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
$883$			34.5602i			1.16304i	0.813531	+	0.581522i	$0.197543\pi$
$883$			34.5602i			1.16304i	−0.813531	+	0.581522i	$0.802457\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	46.9163			1.57530			0.787648	−	0.616125i	$-0.211298\pi$
$887$	46.9163			1.57530			0.787648	+	0.616125i	$0.211298\pi$
$888$		0			0
$889$	−13.4190			−0.450060
$890$		0			0
$891$	−32.1188	+	45.0951i	−1.07602	+	1.51074i
$892$		0			0
$893$		−	15.2431i		−	0.510090i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	6.01980	+	24.8166i	0.200995	+	0.828603i
$898$		0			0
$899$	52.6422			1.75571
$900$		0			0
$901$	16.7272			0.557264
$902$		0			0
$903$	−3.28099	−	13.5259i	−0.109185	−	0.450113i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			39.8545i			1.32335i	0.749792	+	0.661673i	$0.230153\pi$
$907$			39.8545i			1.32335i	−0.749792	+	0.661673i	$0.769847\pi$
$908$		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
$913$	−88.3492			−2.92393
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1.97611i			0.0652569i
$918$		0			0
$919$			16.9449i			0.558962i	0.960151	+	0.279481i	$0.0901623\pi$
$919$			16.9449i			0.558962i	−0.960151	+	0.279481i	$0.909838\pi$
$920$		0			0
$921$	−42.4416	+	10.2951i	−1.39850	+	0.339236i
$922$		0			0
$923$	38.7058			1.27402
$924$		0			0
$925$		0			0
$926$		0			0
$927$	15.4431	+	29.9591i	0.507218	+	0.983985i
$928$		0			0
$929$		−	57.3567i		−	1.88181i	−0.338665	−	0.940907i	$-0.609975\pi$
$929$		−	57.3567i		−	1.88181i	0.338665	−	0.940907i	$-0.390025\pi$
$930$		0			0
$931$		−	13.9269i		−	0.456434i
$932$		0			0
$933$	4.24549	+	17.5020i	0.138991	+	0.572991i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	26.7756			0.874721			0.437361	−	0.899286i	$-0.355913\pi$
$937$	26.7756			0.874721			0.437361	+	0.899286i	$0.355913\pi$
$938$		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
$943$			12.6606i			0.412285i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−34.3856			−1.11738			−0.558691	−	0.829376i	$-0.688696\pi$
$947$	−34.3856			−1.11738			−0.558691	+	0.829376i	$0.688696\pi$
$948$		0			0
$949$	3.20414			0.104011
$950$		0			0
$951$	−26.9477	+	6.53673i	−0.873838	+	0.211968i
$952$		0			0
$953$			42.3018i			1.37029i	0.728407	+	0.685145i	$0.240261\pi$
$953$			42.3018i			1.37029i	−0.728407	+	0.685145i	$0.759739\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−84.5280	+	20.5041i	−2.73240	+	0.662802i
$958$		0			0
$959$	16.7928			0.542267
$960$		0			0
$961$	−10.5841			−0.341424
$962$		0			0
$963$	20.7587	−	10.7006i	0.668940	−	0.344821i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			20.4110i			0.656374i	0.944613	+	0.328187i	$0.106438\pi$
$967$			20.4110i			0.656374i	−0.944613	+	0.328187i	$0.893562\pi$
$968$		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
$973$	23.0697			0.739581
$974$		0			0
$975$		0			0
$976$		0			0
$977$			11.2815i			0.360927i	0.983582	+	0.180464i	$0.0577598\pi$
$977$			11.2815i			0.360927i	−0.983582	+	0.180464i	$0.942240\pi$
$978$		0			0
$979$		−	86.5082i		−	2.76481i
$980$		0			0
$981$	48.4987	−	24.9998i	1.54844	−	0.798181i
$982$		0			0
$983$	4.05349			0.129286			0.0646432	−	0.997908i	$-0.479409\pi$
$983$	4.05349			0.129286			0.0646432	+	0.997908i	$0.479409\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−13.0113	+	3.15617i	−0.414155	+	0.100462i
$988$		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
$993$	−9.55438	+	2.31762i	−0.303199	+	0.0735474i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−61.9989			−1.96352			−0.981762	−	0.190113i	$-0.939115\pi$
$997$	−61.9989			−1.96352			−0.981762	+	0.190113i	$0.939115\pi$
$998$		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.h.f.1151.7	✓	16
3.2	odd	2	inner	2400.2.h.f.1151.9	yes	16
4.3	odd	2	inner	2400.2.h.f.1151.10	yes	16
5.2	odd	4		2400.2.o.k.2399.3		16
5.3	odd	4		2400.2.o.l.2399.13		16
5.4	even	2		2400.2.h.g.1151.10	yes	16
12.11	even	2	inner	2400.2.h.f.1151.8	yes	16
15.2	even	4		2400.2.o.k.2399.2		16
15.8	even	4		2400.2.o.l.2399.16		16
15.14	odd	2		2400.2.h.g.1151.8	yes	16
20.3	even	4		2400.2.o.k.2399.4		16
20.7	even	4		2400.2.o.l.2399.14		16
20.19	odd	2		2400.2.h.g.1151.7	yes	16
60.23	odd	4		2400.2.o.k.2399.1		16
60.47	odd	4		2400.2.o.l.2399.15		16
60.59	even	2		2400.2.h.g.1151.9	yes	16

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

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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.408305 - 1.68324i) q^{3} -1.40591i q^{7} +(-2.66657 + 1.37455i) q^{9} +O(q^{10})\)	\(q+(-0.408305 - 1.68324i) q^{3} -1.40591i q^{7} +(-2.66657 + 1.37455i) q^{9} -6.15156 q^{11} -3.52133 q^{13} -1.39065i q^{17} -2.77238i q^{19} +(-2.36647 + 0.574039i) q^{21} +4.18687 q^{23} +(3.40247 + 3.92724i) q^{27} +8.16338i q^{29} -6.44858i q^{31} +(2.51171 + 10.3545i) q^{33} +4.73295 q^{37} +(1.43778 + 5.92724i) q^{39} +3.02387i q^{41} +5.71563i q^{43} +5.49819 q^{47} +5.02343 q^{49} +(-2.34079 + 0.567809i) q^{51} +12.0284i q^{53} +(-4.66657 + 1.13198i) q^{57} +3.86497 q^{59} -7.94456 q^{61} +(1.93249 + 3.74895i) q^{63} +7.89392i q^{67} +(-1.70952 - 7.04750i) q^{69} -10.9918 q^{71} -0.909922 q^{73} +8.64852i q^{77} +1.12153i q^{79} +(5.22124 - 7.33067i) q^{81} +14.3621 q^{83} +(13.7409 - 3.33315i) q^{87} +14.0628i q^{89} +4.95067i q^{91} +(-10.8545 + 2.63299i) q^{93} -9.44247 q^{97} +(16.4036 - 8.45562i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{9}+O(q^{10}) \) 16 * q + 4 * q^9	\( 16 q + 4 q^{9} - 8 q^{13} + 8 q^{21} - 4 q^{33} - 16 q^{37} - 8 q^{49} - 28 q^{57} + 8 q^{61} - 24 q^{69} - 8 q^{73} + 36 q^{81} - 32 q^{93} - 56 q^{97}+O(q^{100}) \) 16 * q + 4 * q^9 - 8 * q^13 + 8 * q^21 - 4 * q^33 - 16 * q^37 - 8 * q^49 - 28 * q^57 + 8 * q^61 - 24 * q^69 - 8 * q^73 + 36 * q^81 - 32 * q^93 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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