LMFDB - Embedded newform 4800.2.d.o.1249.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4800,2,Mod(1249,4800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("4800.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4800 = 2^{6} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4800.d (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:	$38.3281929702$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	4800.1249
Dual form			4800.2.d.o.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.46410i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.46410i q^{7} +1.00000 q^{9} -6.00000i q^{17} +4.00000i q^{19} +3.46410i q^{21} +6.92820i q^{23} +1.00000 q^{27} +3.46410i q^{29} +3.46410 q^{31} +6.92820 q^{37} -6.00000 q^{41} -4.00000 q^{43} -6.92820i q^{47} -5.00000 q^{49} -6.00000i q^{51} -3.46410 q^{53} +4.00000i q^{57} +12.0000i q^{59} +6.92820i q^{61} +3.46410i q^{63} +4.00000 q^{67} +6.92820i q^{69} -6.92820 q^{71} +2.00000i q^{73} +10.3923 q^{79} +1.00000 q^{81} +3.46410i q^{87} -6.00000 q^{89} +3.46410 q^{93} +2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} + 4 q^{27} - 24 q^{41} - 16 q^{43} - 20 q^{49} + 16 q^{67} + 4 q^{81} - 24 q^{89}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 4 * q^27 - 24 * q^41 - 16 * q^43 - 20 * q^49 + 16 * q^67 + 4 * q^81 - 24 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$4351$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.46410i	1.30931i	0.755929	+	0.654654i	$0.227186\pi$
$7$	3.46410i	1.30931i	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	3.46410i	0.755929i
$22$	0	0
$23$	6.92820i	1.44463i	0.691564	+	0.722315i	$0.256922\pi$
$23$	6.92820i	1.44463i	−0.691564	+	0.722315i	$0.743078\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.46410i	0.643268i	0.946864	+	0.321634i	$0.104232\pi$
$29$	3.46410i	0.643268i	−0.946864	+	0.321634i	$0.895768\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.92820	1.13899	0.569495	−	0.821995i	$-0.307139\pi$
$37$	6.92820	1.13899	0.569495	+	0.821995i	$0.307139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.92820i	− 1.01058i	−0.862949	−	0.505291i	$-0.831385\pi$
$47$	− 6.92820i	− 1.01058i	0.862949	−	0.505291i	$-0.168615\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	− 6.00000i	− 0.840168i
$52$	0	0
$53$	−3.46410	−0.475831	−0.237915	−	0.971286i	$-0.576464\pi$
$53$	−3.46410	−0.475831	−0.237915	+	0.971286i	$0.576464\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	12.0000i	1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$	12.0000i	1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$	0	0
$61$	6.92820i	0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$	6.92820i	0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	3.46410i	0.436436i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	6.92820i	0.834058i
$70$	0	0
$71$	−6.92820	−0.822226	−0.411113	−	0.911584i	$-0.634860\pi$
$71$	−6.92820	−0.822226	−0.411113	+	0.911584i	$0.634860\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.3923	1.16923	0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923	1.16923	0.584613	+	0.811312i	$0.301246\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.46410i	0.371391i
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.46410	0.359211
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.46410i	0.344691i	0.985037	+	0.172345i	$0.0551346\pi$
$101$	3.46410i	0.344691i	−0.985037	+	0.172345i	$0.944865\pi$
$102$	0	0
$103$	17.3205i	1.70664i	0.521387	+	0.853320i	$0.325415\pi$
$103$	17.3205i	1.70664i	−0.521387	+	0.853320i	$0.674585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	13.8564i	1.32720i	0.748086	+	0.663602i	$0.230973\pi$
$109$	13.8564i	1.32720i	−0.748086	+	0.663602i	$0.769027\pi$
$110$	0	0
$111$	6.92820	0.657596
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20.7846	1.90532
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.46410i	0.307389i	0.988118	+	0.153695i	$0.0491172\pi$
$127$	3.46410i	0.307389i	−0.988118	+	0.153695i	$0.950883\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	12.0000i	1.04844i	0.851581	+	0.524222i	$0.175644\pi$
$131$	12.0000i	1.04844i	−0.851581	+	0.524222i	$0.824356\pi$
$132$	0	0
$133$	−13.8564	−1.20150
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	− 20.0000i	− 1.69638i	−0.529694	−	0.848189i	$-0.677693\pi$
$139$	− 20.0000i	− 1.69638i	0.529694	−	0.848189i	$-0.322307\pi$
$140$	0	0
$141$	− 6.92820i	− 0.583460i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5.00000	−0.412393
$148$	0	0
$149$	− 10.3923i	− 0.851371i	−0.904871	−	0.425685i	$-0.860033\pi$
$149$	− 10.3923i	− 0.851371i	0.904871	−	0.425685i	$-0.139967\pi$
$150$	0	0
$151$	−3.46410	−0.281905	−0.140952	−	0.990016i	$-0.545016\pi$
$151$	−3.46410	−0.281905	−0.140952	+	0.990016i	$0.545016\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.92820	−0.552931	−0.276465	−	0.961024i	$-0.589163\pi$
$157$	−6.92820	−0.552931	−0.276465	+	0.961024i	$0.589163\pi$
$158$	0	0
$159$	−3.46410	−0.274721
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 13.8564i	− 1.07224i	−0.844141	−	0.536120i	$-0.819889\pi$
$167$	− 13.8564i	− 1.07224i	0.844141	−	0.536120i	$-0.180111\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	−17.3205	−1.31685	−0.658427	−	0.752645i	$-0.728778\pi$
$173$	−17.3205	−1.31685	−0.658427	+	0.752645i	$0.728778\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	0	0
$179$	12.0000i	0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$	12.0000i	0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$	0	0
$181$	13.8564i	1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$	13.8564i	1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$	0	0
$183$	6.92820i	0.512148i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.46410i	0.251976i
$190$	0	0
$191$	13.8564	1.00261	0.501307	−	0.865269i	$-0.332853\pi$
$191$	13.8564	1.00261	0.501307	+	0.865269i	$0.332853\pi$
$192$	0	0
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.3923	−0.740421	−0.370211	−	0.928948i	$-0.620714\pi$
$197$	−10.3923	−0.740421	−0.370211	+	0.928948i	$0.620714\pi$
$198$	0	0
$199$	10.3923	0.736691	0.368345	−	0.929689i	$-0.379924\pi$
$199$	10.3923	0.736691	0.368345	+	0.929689i	$0.379924\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.92820i	0.481543i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.00000i	0.275371i	0.990476	+	0.137686i	$0.0439664\pi$
$211$	4.00000i	0.275371i	−0.990476	+	0.137686i	$0.956034\pi$
$212$	0	0
$213$	−6.92820	−0.474713
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0000i	0.814613i
$218$	0	0
$219$	2.00000i	0.135147i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 3.46410i	− 0.231973i	−0.993251	−	0.115987i	$-0.962997\pi$
$223$	− 3.46410i	− 0.231973i	0.993251	−	0.115987i	$-0.0370030\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	27.7128i	1.83131i	0.401960	+	0.915657i	$0.368329\pi$
$229$	27.7128i	1.83131i	−0.401960	+	0.915657i	$0.631671\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.3923	0.675053
$238$	0	0
$239$	27.7128	1.79259	0.896296	−	0.443455i	$-0.146248\pi$
$239$	27.7128	1.79259	0.896296	+	0.443455i	$0.146248\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	24.0000i	1.49129i
$260$	0	0
$261$	3.46410i	0.214423i
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	31.1769i	1.90089i	0.310893	+	0.950445i	$0.399372\pi$
$269$	31.1769i	1.90089i	−0.310893	+	0.950445i	$0.600628\pi$
$270$	0	0
$271$	−3.46410	−0.210429	−0.105215	−	0.994450i	$-0.533553\pi$
$271$	−3.46410	−0.210429	−0.105215	+	0.994450i	$0.533553\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.8564	0.832551	0.416275	−	0.909239i	$-0.363335\pi$
$277$	13.8564	0.832551	0.416275	+	0.909239i	$0.363335\pi$
$278$	0	0
$279$	3.46410	0.207390
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 20.7846i	− 1.22688i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	2.00000i	0.117242i
$292$	0	0
$293$	−3.46410	−0.202375	−0.101187	−	0.994867i	$-0.532264\pi$
$293$	−3.46410	−0.202375	−0.101187	+	0.994867i	$0.532264\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 13.8564i	− 0.798670i
$302$	0	0
$303$	3.46410i	0.199007i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	17.3205i	0.985329i
$310$	0	0
$311$	13.8564	0.785725	0.392862	−	0.919597i	$-0.371485\pi$
$311$	13.8564	0.785725	0.392862	+	0.919597i	$0.371485\pi$
$312$	0	0
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.1769	−1.75107	−0.875535	−	0.483155i	$-0.839491\pi$
$317$	−31.1769	−1.75107	−0.875535	+	0.483155i	$0.839491\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	24.0000	1.33540
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13.8564i	0.766261i
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	20.0000i	1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$	20.0000i	1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	0	0
$333$	6.92820	0.379663
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6.92820i	0.374088i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	− 6.92820i	− 0.370858i	−0.982658	−	0.185429i	$-0.940632\pi$
$349$	− 6.92820i	− 0.370858i	0.982658	−	0.185429i	$-0.0593675\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	20.7846	1.10004
$358$	0	0
$359$	6.92820	0.365657	0.182828	−	0.983145i	$-0.441475\pi$
$359$	6.92820	0.365657	0.182828	+	0.983145i	$0.441475\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	0	0
$366$	0	0
$367$	24.2487i	1.26577i	0.774245	+	0.632886i	$0.218130\pi$
$367$	24.2487i	1.26577i	−0.774245	+	0.632886i	$0.781870\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	− 12.0000i	− 0.623009i
$372$	0	0
$373$	−20.7846	−1.07619	−0.538093	−	0.842885i	$-0.680855\pi$
$373$	−20.7846	−1.07619	−0.538093	+	0.842885i	$0.680855\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	20.0000i	1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$	20.0000i	1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$	0	0
$381$	3.46410i	0.177471i
$382$	0	0
$383$	13.8564i	0.708029i	0.935240	+	0.354015i	$0.115184\pi$
$383$	13.8564i	0.708029i	−0.935240	+	0.354015i	$0.884816\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	− 24.2487i	− 1.22946i	−0.788738	−	0.614729i	$-0.789265\pi$
$389$	− 24.2487i	− 1.22946i	0.788738	−	0.614729i	$-0.210735\pi$
$390$	0	0
$391$	41.5692	2.10225
$392$	0	0
$393$	12.0000i	0.605320i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	34.6410	1.73858	0.869291	−	0.494300i	$-0.164576\pi$
$397$	34.6410	1.73858	0.869291	+	0.494300i	$0.164576\pi$
$398$	0	0
$399$	−13.8564	−0.693688
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	6.00000i	0.295958i
$412$	0	0
$413$	−41.5692	−2.04549
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.0000i	− 0.979404i
$418$	0	0
$419$	24.0000i	1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$	24.0000i	1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$	0	0
$421$	− 13.8564i	− 0.675320i	−0.941268	−	0.337660i	$-0.890365\pi$
$421$	− 13.8564i	− 0.675320i	0.941268	−	0.337660i	$-0.109635\pi$
$422$	0	0
$423$	− 6.92820i	− 0.336861i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−24.0000	−1.16144
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.92820	−0.333720	−0.166860	−	0.985981i	$-0.553363\pi$
$431$	−6.92820	−0.333720	−0.166860	+	0.985981i	$0.553363\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−27.7128	−1.32568
$438$	0	0
$439$	−31.1769	−1.48799	−0.743996	−	0.668184i	$-0.767072\pi$
$439$	−31.1769	−1.48799	−0.743996	+	0.668184i	$0.767072\pi$
$440$	0	0
$441$	−5.00000	−0.238095
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 10.3923i	− 0.491539i
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−3.46410	−0.162758
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	0	0
$459$	− 6.00000i	− 0.280056i
$460$	0	0
$461$	31.1769i	1.45205i	0.687666	+	0.726027i	$0.258635\pi$
$461$	31.1769i	1.45205i	−0.687666	+	0.726027i	$0.741365\pi$
$462$	0	0
$463$	10.3923i	0.482971i	0.970404	+	0.241486i	$0.0776347\pi$
$463$	10.3923i	0.482971i	−0.970404	+	0.241486i	$0.922365\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	0	0
$469$	13.8564i	0.639829i
$470$	0	0
$471$	−6.92820	−0.319235
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.46410	−0.158610
$478$	0	0
$479$	20.7846	0.949673	0.474837	−	0.880074i	$-0.342507\pi$
$479$	20.7846	0.949673	0.474837	+	0.880074i	$0.342507\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−24.0000	−1.09204
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 10.3923i	− 0.470920i	−0.971884	−	0.235460i	$-0.924340\pi$
$487$	− 10.3923i	− 0.470920i	0.971884	−	0.235460i	$-0.0756597\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	− 36.0000i	− 1.62466i	−0.583200	−	0.812329i	$-0.698200\pi$
$491$	− 36.0000i	− 1.62466i	0.583200	−	0.812329i	$-0.301800\pi$
$492$	0	0
$493$	20.7846	0.936092
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 24.0000i	− 1.07655i
$498$	0	0
$499$	− 28.0000i	− 1.25345i	−0.779240	−	0.626726i	$-0.784395\pi$
$499$	− 28.0000i	− 1.25345i	0.779240	−	0.626726i	$-0.215605\pi$
$500$	0	0
$501$	− 13.8564i	− 0.619059i
$502$	0	0
$503$	− 34.6410i	− 1.54457i	−0.635278	−	0.772283i	$-0.719115\pi$
$503$	− 34.6410i	− 1.54457i	0.635278	−	0.772283i	$-0.280885\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−13.0000	−0.577350
$508$	0	0
$509$	− 24.2487i	− 1.07481i	−0.843326	−	0.537403i	$-0.819406\pi$
$509$	− 24.2487i	− 1.07481i	0.843326	−	0.537403i	$-0.180594\pi$
$510$	0	0
$511$	−6.92820	−0.306486
$512$	0	0
$513$	4.00000i	0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−17.3205	−0.760286
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 20.7846i	− 0.905392i
$528$	0	0
$529$	−25.0000	−1.08696
$530$	0	0
$531$	12.0000i	0.520756i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.0000i	0.517838i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 13.8564i	− 0.595733i	−0.954607	−	0.297867i	$-0.903725\pi$
$541$	− 13.8564i	− 0.595733i	0.954607	−	0.297867i	$-0.0962751\pi$
$542$	0	0
$543$	13.8564i	0.594635i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	6.92820i	0.295689i
$550$	0	0
$551$	−13.8564	−0.590303
$552$	0	0
$553$	36.0000i	1.53088i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.3923	−0.440336	−0.220168	−	0.975462i	$-0.570661\pi$
$557$	−10.3923	−0.440336	−0.220168	+	0.975462i	$0.570661\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.46410i	0.145479i
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	− 4.00000i	− 0.167395i	−0.996491	−	0.0836974i	$-0.973327\pi$
$571$	− 4.00000i	− 0.167395i	0.996491	−	0.0836974i	$-0.0266729\pi$
$572$	0	0
$573$	13.8564	0.578860
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	0	0
$579$	2.00000i	0.0831172i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	13.8564i	0.570943i
$590$	0	0
$591$	−10.3923	−0.427482
$592$	0	0
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.3923	0.425329
$598$	0	0
$599$	34.6410	1.41539	0.707697	−	0.706516i	$-0.249734\pi$
$599$	34.6410	1.41539	0.707697	+	0.706516i	$0.249734\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.46410i	0.140604i	0.997526	+	0.0703018i	$0.0223962\pi$
$607$	3.46410i	0.140604i	−0.997526	+	0.0703018i	$0.977604\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	34.6410	1.39914	0.699569	−	0.714565i	$-0.253375\pi$
$613$	34.6410	1.39914	0.699569	+	0.714565i	$0.253375\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	4.00000i	0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$	4.00000i	0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	0	0
$621$	6.92820i	0.278019i
$622$	0	0
$623$	− 20.7846i	− 0.832718i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 41.5692i	− 1.65747i
$630$	0	0
$631$	−17.3205	−0.689519	−0.344759	−	0.938691i	$-0.612039\pi$
$631$	−17.3205	−0.689519	−0.344759	+	0.938691i	$0.612039\pi$
$632$	0	0
$633$	4.00000i	0.158986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.92820	−0.274075
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 34.6410i	− 1.36188i	−0.732340	−	0.680939i	$-0.761572\pi$
$647$	− 34.6410i	− 1.36188i	0.732340	−	0.680939i	$-0.238428\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	12.0000i	0.470317i
$652$	0	0
$653$	10.3923	0.406682	0.203341	−	0.979108i	$-0.434820\pi$
$653$	10.3923	0.406682	0.203341	+	0.979108i	$0.434820\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	− 12.0000i	− 0.467454i	−0.972302	−	0.233727i	$-0.924908\pi$
$659$	− 12.0000i	− 0.467454i	0.972302	−	0.233727i	$-0.0750921\pi$
$660$	0	0
$661$	6.92820i	0.269476i	0.990881	+	0.134738i	$0.0430193\pi$
$661$	6.92820i	0.269476i	−0.990881	+	0.134738i	$0.956981\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	− 3.46410i	− 0.133930i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.46410	−0.133136	−0.0665681	−	0.997782i	$-0.521205\pi$
$677$	−3.46410	−0.133136	−0.0665681	+	0.997782i	$0.521205\pi$
$678$	0	0
$679$	−6.92820	−0.265880
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	27.7128i	1.05731i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 28.0000i	− 1.06517i	−0.846376	−	0.532585i	$-0.821221\pi$
$691$	− 28.0000i	− 1.06517i	0.846376	−	0.532585i	$-0.178779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	36.0000i	1.36360i
$698$	0	0
$699$	− 18.0000i	− 0.680823i
$700$	0	0
$701$	10.3923i	0.392512i	0.980553	+	0.196256i	$0.0628784\pi$
$701$	10.3923i	0.392512i	−0.980553	+	0.196256i	$0.937122\pi$
$702$	0	0
$703$	27.7128i	1.04521i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.0000	−0.451306
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	10.3923	0.389742
$712$	0	0
$713$	24.0000i	0.898807i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	27.7128	1.03495
$718$	0	0
$719$	−20.7846	−0.775135	−0.387568	−	0.921841i	$-0.626685\pi$
$719$	−20.7846	−0.775135	−0.387568	+	0.921841i	$0.626685\pi$
$720$	0	0
$721$	−60.0000	−2.23452
$722$	0	0
$723$	−22.0000	−0.818189
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 10.3923i	− 0.385429i	−0.981255	−	0.192715i	$-0.938271\pi$
$727$	− 10.3923i	− 0.385429i	0.981255	−	0.192715i	$-0.0617292\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.0000i	0.887672i
$732$	0	0
$733$	−27.7128	−1.02360	−0.511798	−	0.859106i	$-0.671020\pi$
$733$	−27.7128	−1.02360	−0.511798	+	0.859106i	$0.671020\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 28.0000i	− 1.03000i	−0.857191	−	0.514998i	$-0.827793\pi$
$739$	− 28.0000i	− 1.03000i	0.857191	−	0.514998i	$-0.172207\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 27.7128i	− 1.01668i	−0.861155	−	0.508342i	$-0.830258\pi$
$743$	− 27.7128i	− 1.01668i	0.861155	−	0.508342i	$-0.169742\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 41.5692i	− 1.51891i
$750$	0	0
$751$	−24.2487	−0.884848	−0.442424	−	0.896806i	$-0.645881\pi$
$751$	−24.2487	−0.884848	−0.442424	+	0.896806i	$0.645881\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−27.7128	−1.00724	−0.503620	−	0.863925i	$-0.667999\pi$
$757$	−27.7128	−1.00724	−0.503620	+	0.863925i	$0.667999\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	−48.0000	−1.73772
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	− 18.0000i	− 0.648254i
$772$	0	0
$773$	38.1051	1.37055	0.685273	−	0.728286i	$-0.259683\pi$
$773$	38.1051	1.37055	0.685273	+	0.728286i	$0.259683\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	24.0000i	0.860995i
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	3.46410i	0.123797i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	20.7846	0.739016
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.2487	−0.858933	−0.429467	−	0.903083i	$-0.641298\pi$
$797$	−24.2487	−0.858933	−0.429467	+	0.903083i	$0.641298\pi$
$798$	0	0
$799$	−41.5692	−1.47061
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	31.1769i	1.09748i
$808$	0	0
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	− 20.0000i	− 0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$	− 20.0000i	− 0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	0	0
$813$	−3.46410	−0.121491
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 24.2487i	− 0.846286i	−0.906063	−	0.423143i	$-0.860927\pi$
$821$	− 24.2487i	− 0.846286i	0.906063	−	0.423143i	$-0.139073\pi$
$822$	0	0
$823$	− 31.1769i	− 1.08676i	−0.839487	−	0.543379i	$-0.817144\pi$
$823$	− 31.1769i	− 1.08676i	0.839487	−	0.543379i	$-0.182856\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	− 41.5692i	− 1.44376i	−0.692019	−	0.721879i	$-0.743279\pi$
$829$	− 41.5692i	− 1.44376i	0.692019	−	0.721879i	$-0.256721\pi$
$830$	0	0
$831$	13.8564	0.480673
$832$	0	0
$833$	30.0000i	1.03944i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.46410	0.119737
$838$	0	0
$839$	−6.92820	−0.239188	−0.119594	−	0.992823i	$-0.538159\pi$
$839$	−6.92820	−0.239188	−0.119594	+	0.992823i	$0.538159\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	30.0000	1.03325
$844$	0	0
$845$	0	0
$846$	0	0
$847$	38.1051i	1.30931i
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	48.0000i	1.64542i
$852$	0	0
$853$	48.4974	1.66052	0.830260	−	0.557376i	$-0.188192\pi$
$853$	48.4974	1.66052	0.830260	+	0.557376i	$0.188192\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	0	0
$859$	− 4.00000i	− 0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$	− 4.00000i	− 0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$	0	0
$861$	− 20.7846i	− 0.708338i
$862$	0	0
$863$	13.8564i	0.471678i	0.971792	+	0.235839i	$0.0757837\pi$
$863$	13.8564i	0.471678i	−0.971792	+	0.235839i	$0.924216\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−19.0000	−0.645274
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.92820	0.233949	0.116974	−	0.993135i	$-0.462680\pi$
$877$	6.92820	0.233949	0.116974	+	0.993135i	$0.462680\pi$
$878$	0	0
$879$	−3.46410	−0.116841
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	27.7128	0.927374
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.0000i	0.400222i
$900$	0	0
$901$	20.7846i	0.692436i
$902$	0	0
$903$	− 13.8564i	− 0.461112i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	0	0
$909$	3.46410i	0.114897i
$910$	0	0
$911$	−55.4256	−1.83633	−0.918166	−	0.396195i	$-0.870330\pi$
$911$	−55.4256	−1.83633	−0.918166	+	0.396195i	$0.870330\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−41.5692	−1.37274
$918$	0	0
$919$	−51.9615	−1.71405	−0.857026	−	0.515273i	$-0.827691\pi$
$919$	−51.9615	−1.71405	−0.857026	+	0.515273i	$0.827691\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	17.3205i	0.568880i
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	− 20.0000i	− 0.655474i
$932$	0	0
$933$	13.8564	0.453638
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.0000i	0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$	26.0000i	0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$	0	0
$939$	− 26.0000i	− 0.848478i
$940$	0	0
$941$	− 24.2487i	− 0.790485i	−0.918577	−	0.395243i	$-0.870660\pi$
$941$	− 24.2487i	− 0.790485i	0.918577	−	0.395243i	$-0.129340\pi$
$942$	0	0
$943$	− 41.5692i	− 1.35368i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−31.1769	−1.01098
$952$	0	0
$953$	− 30.0000i	− 0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$	− 30.0000i	− 0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.7846	−0.671170
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38.1051i	1.22538i	0.790324	+	0.612689i	$0.209912\pi$
$967$	38.1051i	1.22538i	−0.790324	+	0.612689i	$0.790088\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	− 48.0000i	− 1.54039i	−0.637806	−	0.770197i	$-0.720158\pi$
$971$	− 48.0000i	− 1.54039i	0.637806	−	0.770197i	$-0.279842\pi$
$972$	0	0
$973$	69.2820	2.22108
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	13.8564i	0.442401i
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.0000	0.763928
$988$	0	0
$989$	− 27.7128i	− 0.881216i
$990$	0	0
$991$	−24.2487	−0.770286	−0.385143	−	0.922857i	$-0.625848\pi$
$991$	−24.2487	−0.770286	−0.385143	+	0.922857i	$0.625848\pi$
$992$	0	0
$993$	20.0000i	0.634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	20.7846	0.658255	0.329128	−	0.944285i	$-0.393245\pi$
$997$	20.7846	0.658255	0.329128	+	0.944285i	$0.393245\pi$
$998$	0	0
$999$	6.92820	0.219199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4800.2.d.o.1249.3		4
4.3	odd	2		4800.2.d.j.1249.1		4
5.2	odd	4		192.2.d.a.97.1	✓	4
5.3	odd	4		4800.2.k.j.2401.4		4
5.4	even	2		4800.2.d.j.1249.2		4
8.3	odd	2	inner	4800.2.d.o.1249.1		4
8.5	even	2		4800.2.d.j.1249.3		4
15.2	even	4		576.2.d.b.289.3		4
20.3	even	4		4800.2.k.j.2401.1		4
20.7	even	4		192.2.d.a.97.3	yes	4
20.19	odd	2	inner	4800.2.d.o.1249.4		4
40.3	even	4		4800.2.k.j.2401.3		4
40.13	odd	4		4800.2.k.j.2401.2		4
40.19	odd	2		4800.2.d.j.1249.4		4
40.27	even	4		192.2.d.a.97.2	yes	4
40.29	even	2	inner	4800.2.d.o.1249.2		4
40.37	odd	4		192.2.d.a.97.4	yes	4
60.47	odd	4		576.2.d.b.289.4		4
80.27	even	4		768.2.a.k.1.2		2
80.37	odd	4		768.2.a.j.1.2		2
80.67	even	4		768.2.a.j.1.1		2
80.77	odd	4		768.2.a.k.1.1		2
120.77	even	4		576.2.d.b.289.1		4
120.107	odd	4		576.2.d.b.289.2		4
240.77	even	4		2304.2.a.u.1.2		2
240.107	odd	4		2304.2.a.u.1.1		2
240.197	even	4		2304.2.a.s.1.1		2
240.227	odd	4		2304.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.2.d.a.97.1	✓	4	5.2	odd	4
192.2.d.a.97.2	yes	4	40.27	even	4
192.2.d.a.97.3	yes	4	20.7	even	4
192.2.d.a.97.4	yes	4	40.37	odd	4
576.2.d.b.289.1		4	120.77	even	4
576.2.d.b.289.2		4	120.107	odd	4
576.2.d.b.289.3		4	15.2	even	4
576.2.d.b.289.4		4	60.47	odd	4
768.2.a.j.1.1		2	80.67	even	4
768.2.a.j.1.2		2	80.37	odd	4
768.2.a.k.1.1		2	80.77	odd	4
768.2.a.k.1.2		2	80.27	even	4
2304.2.a.s.1.1		2	240.197	even	4
2304.2.a.s.1.2		2	240.227	odd	4
2304.2.a.u.1.1		2	240.107	odd	4
2304.2.a.u.1.2		2	240.77	even	4
4800.2.d.j.1249.1		4	4.3	odd	2
4800.2.d.j.1249.2		4	5.4	even	2
4800.2.d.j.1249.3		4	8.5	even	2
4800.2.d.j.1249.4		4	40.19	odd	2
4800.2.d.o.1249.1		4	8.3	odd	2	inner
4800.2.d.o.1249.2		4	40.29	even	2	inner
4800.2.d.o.1249.3		4	1.1	even	1	trivial
4800.2.d.o.1249.4		4	20.19	odd	2	inner
4800.2.k.j.2401.1		4	20.3	even	4
4800.2.k.j.2401.2		4	40.13	odd	4
4800.2.k.j.2401.3		4	40.3	even	4
4800.2.k.j.2401.4		4	5.3	odd	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:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.46410i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.46410i q^{7} +1.00000 q^{9} -6.00000i q^{17} +4.00000i q^{19} +3.46410i q^{21} +6.92820i q^{23} +1.00000 q^{27} +3.46410i q^{29} +3.46410 q^{31} +6.92820 q^{37} -6.00000 q^{41} -4.00000 q^{43} -6.92820i q^{47} -5.00000 q^{49} -6.00000i q^{51} -3.46410 q^{53} +4.00000i q^{57} +12.0000i q^{59} +6.92820i q^{61} +3.46410i q^{63} +4.00000 q^{67} +6.92820i q^{69} -6.92820 q^{71} +2.00000i q^{73} +10.3923 q^{79} +1.00000 q^{81} +3.46410i q^{87} -6.00000 q^{89} +3.46410 q^{93} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} + 4 q^{27} - 24 q^{41} - 16 q^{43} - 20 q^{49} + 16 q^{67} + 4 q^{81} - 24 q^{89}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 4 * q^27 - 24 * q^41 - 16 * q^43 - 20 * q^49 + 16 * q^67 + 4 * q^81 - 24 * q^89

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