LMFDB - Embedded newform 4800.2.f.a.3649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4800,2,Mod(3649,4800)]

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, 0, 0, 1]))

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

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

chi := DirichletCharacter("4800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4800 = 2^{6} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4800.f (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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4800.3649
Dual form			4800.2.f.a.3649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -5.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -5.00000i q^{7} -1.00000 q^{9} -6.00000 q^{11} +3.00000i q^{13} +2.00000i q^{17} -1.00000 q^{19} +5.00000 q^{21} +2.00000i q^{23} -1.00000i q^{27} +6.00000 q^{29} -3.00000 q^{31} -6.00000i q^{33} -6.00000i q^{37} -3.00000 q^{39} +4.00000 q^{41} +11.0000i q^{43} -10.0000i q^{47} -18.0000 q^{49} -2.00000 q^{51} +8.00000i q^{53} -1.00000i q^{57} +6.00000 q^{59} -3.00000 q^{61} +5.00000i q^{63} +1.00000i q^{67} -2.00000 q^{69} +12.0000 q^{71} +10.0000i q^{73} +30.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +6.00000i q^{87} +16.0000 q^{89} +15.0000 q^{91} -3.00000i q^{93} +7.00000i q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 12 q^{11} - 2 q^{19} + 10 q^{21} + 12 q^{29} - 6 q^{31} - 6 q^{39} + 8 q^{41} - 36 q^{49} - 4 q^{51} + 12 q^{59} - 6 q^{61} - 4 q^{69} + 24 q^{71} - 16 q^{79} + 2 q^{81} + 32 q^{89} + 30 q^{91} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 12 * q^11 - 2 * q^19 + 10 * q^21 + 12 * q^29 - 6 * q^31 - 6 * q^39 + 8 * q^41 - 36 * q^49 - 4 * q^51 + 12 * q^59 - 6 * q^61 - 4 * q^69 + 24 * q^71 - 16 * q^79 + 2 * q^81 + 32 * q^89 + 30 * q^91 + 12 * q^99

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 5.00000i	− 1.88982i	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	− 5.00000i	− 1.88982i	0.327327	−	0.944911i	$-0.393852\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	0	0
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	− 6.00000i	− 1.04447i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	11.0000i	1.67748i	0.544529	+	0.838742i	$0.316708\pi$
$43$	11.0000i	1.67748i	−0.544529	+	0.838742i	$0.683292\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.0000i	− 1.45865i	−0.684167	−	0.729325i	$-0.739834\pi$
$47$	− 10.0000i	− 1.45865i	0.684167	−	0.729325i	$-0.260166\pi$
$48$	0	0
$49$	−18.0000	−2.57143
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	8.00000i	1.09888i	0.835532	+	0.549442i	$0.185160\pi$
$53$	8.00000i	1.09888i	−0.835532	+	0.549442i	$0.814840\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.00000i	− 0.132453i
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	0	0
$63$	5.00000i	0.629941i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.00000i	0.122169i	0.998133	+	0.0610847i	$0.0194560\pi$
$67$	1.00000i	0.122169i	−0.998133	+	0.0610847i	$0.980544\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	30.0000i	3.41882i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000i	0.643268i
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	15.0000	1.57243
$92$	0	0
$93$	− 3.00000i	− 0.311086i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.00000i	0.710742i	0.934725	+	0.355371i	$0.115646\pi$
$97$	7.00000i	0.710742i	−0.934725	+	0.355371i	$0.884354\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.00000i	− 0.277350i
$118$	0	0
$119$	10.0000	0.916698
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	4.00000i	0.360668i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	−11.0000	−0.968496
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	5.00000i	0.433555i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	0	0
$143$	− 18.0000i	− 1.50524i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 18.0000i	− 1.48461i
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.00000i	0.558661i	0.960195	+	0.279330i	$0.0901125\pi$
$157$	7.00000i	0.558661i	−0.960195	+	0.279330i	$0.909888\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	10.0000	0.788110
$162$	0	0
$163$	− 7.00000i	− 0.548282i	−0.961689	−	0.274141i	$-0.911606\pi$
$163$	− 7.00000i	− 0.548282i	0.961689	−	0.274141i	$-0.0883936\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.00000i	0.450988i
$178$	0	0
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	19.0000	1.41226	0.706129	−	0.708083i	$-0.250440\pi$
$181$	19.0000	1.41226	0.706129	+	0.708083i	$0.250440\pi$
$182$	0	0
$183$	− 3.00000i	− 0.221766i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	3.00000i	0.215945i	0.994154	+	0.107972i	$0.0344358\pi$
$193$	3.00000i	0.215945i	−0.994154	+	0.107972i	$0.965564\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	0	0
$199$	21.0000	1.48865	0.744325	−	0.667817i	$-0.232771\pi$
$199$	21.0000	1.48865	0.744325	+	0.667817i	$0.232771\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	− 30.0000i	− 2.10559i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.00000i	− 0.139010i
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	0	0
$213$	12.0000i	0.822226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	15.0000i	1.01827i
$218$	0	0
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	− 3.00000i	− 0.200895i	−0.994942	−	0.100447i	$-0.967973\pi$
$223$	− 3.00000i	− 0.200895i	0.994942	−	0.100447i	$-0.0320274\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	−3.00000	−0.198246	−0.0991228	−	0.995075i	$-0.531604\pi$
$229$	−3.00000	−0.198246	−0.0991228	+	0.995075i	$0.531604\pi$
$230$	0	0
$231$	−30.0000	−1.97386
$232$	0	0
$233$	20.0000i	1.31024i	0.755523	+	0.655122i	$0.227383\pi$
$233$	20.0000i	1.31024i	−0.755523	+	0.655122i	$0.772617\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.00000i	− 0.190885i
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	− 12.0000i	− 0.754434i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	0	0
$259$	−30.0000	−1.86411
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.0000i	0.979184i
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	15.0000i	0.907841i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.00000i	0.0600842i	0.999549	+	0.0300421i	$0.00956413\pi$
$277$	1.00000i	0.0600842i	−0.999549	+	0.0300421i	$0.990436\pi$
$278$	0	0
$279$	3.00000	0.179605
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	13.0000i	0.772770i	0.922338	+	0.386385i	$0.126276\pi$
$283$	13.0000i	0.772770i	−0.922338	+	0.386385i	$0.873724\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 20.0000i	− 1.18056i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	−7.00000	−0.410347
$292$	0	0
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.00000i	0.348155i
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	55.0000	3.17015
$302$	0	0
$303$	8.00000i	0.459588i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 13.0000i	− 0.741949i	−0.928643	−	0.370975i	$-0.879024\pi$
$307$	− 13.0000i	− 0.741949i	0.928643	−	0.370975i	$-0.120976\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	0	0
$313$	− 29.0000i	− 1.63918i	−0.572953	−	0.819588i	$-0.694202\pi$
$313$	− 29.0000i	− 1.63918i	0.572953	−	0.819588i	$-0.305798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 16.0000i	− 0.898650i	−0.893368	−	0.449325i	$-0.851665\pi$
$317$	− 16.0000i	− 0.898650i	0.893368	−	0.449325i	$-0.148335\pi$
$318$	0	0
$319$	−36.0000	−2.01561
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	− 2.00000i	− 0.111283i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 7.00000i	− 0.387101i
$328$	0	0
$329$	−50.0000	−2.75659
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.0000i	1.25289i	0.779466	+	0.626445i	$0.215491\pi$
$337$	23.0000i	1.25289i	−0.779466	+	0.626445i	$0.784509\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	0	0
$341$	18.0000	0.974755
$342$	0	0
$343$	55.0000i	2.96972i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	3.00000	0.160128
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.0000i	0.529256i
$358$	0	0
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	25.0000i	1.31216i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13.0000i	0.678594i	0.940679	+	0.339297i	$0.110189\pi$
$367$	13.0000i	0.678594i	−0.940679	+	0.339297i	$0.889811\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	40.0000	2.07670
$372$	0	0
$373$	25.0000i	1.29445i	0.762299	+	0.647225i	$0.224071\pi$
$373$	25.0000i	1.29445i	−0.762299	+	0.647225i	$0.775929\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.0000i	0.927047i
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	− 20.0000i	− 1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$	− 20.0000i	− 1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 11.0000i	− 0.559161i
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	16.0000i	0.807093i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.0000i	0.552074i	0.961147	+	0.276037i	$0.0890213\pi$
$397$	11.0000i	0.552074i	−0.961147	+	0.276037i	$0.910979\pi$
$398$	0	0
$399$	−5.00000	−0.250313
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	− 9.00000i	− 0.448322i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	36.0000i	1.78445i
$408$	0	0
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	− 30.0000i	− 1.47620i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.0000i	0.587643i
$418$	0	0
$419$	−32.0000	−1.56330	−0.781651	−	0.623716i	$-0.785622\pi$
$419$	−32.0000	−1.56330	−0.781651	+	0.623716i	$0.785622\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	0	0
$423$	10.0000i	0.486217i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	15.0000i	0.725901i
$428$	0	0
$429$	18.0000	0.869048
$430$	0	0
$431$	−22.0000	−1.05970	−0.529851	−	0.848091i	$-0.677752\pi$
$431$	−22.0000	−1.05970	−0.529851	+	0.848091i	$0.677752\pi$
$432$	0	0
$433$	19.0000i	0.913082i	0.889702	+	0.456541i	$0.150912\pi$
$433$	19.0000i	0.913082i	−0.889702	+	0.456541i	$0.849088\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.00000i	− 0.0956730i
$438$	0	0
$439$	−5.00000	−0.238637	−0.119318	−	0.992856i	$-0.538071\pi$
$439$	−5.00000	−0.238637	−0.119318	+	0.992856i	$0.538071\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 6.00000i	− 0.283790i
$448$	0	0
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	− 9.00000i	− 0.422857i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	16.0000	0.745194	0.372597	−	0.927993i	$-0.378467\pi$
$461$	16.0000	0.745194	0.372597	+	0.927993i	$0.378467\pi$
$462$	0	0
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.0000i	0.647843i	0.946084	+	0.323921i	$0.105001\pi$
$467$	14.0000i	0.647843i	−0.946084	+	0.323921i	$0.894999\pi$
$468$	0	0
$469$	5.00000	0.230879
$470$	0	0
$471$	−7.00000	−0.322543
$472$	0	0
$473$	− 66.0000i	− 3.03468i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 8.00000i	− 0.366295i
$478$	0	0
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	0	0
$483$	10.0000i	0.455016i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.00000i	− 0.135943i	−0.997687	−	0.0679715i	$-0.978347\pi$
$487$	− 3.00000i	− 0.135943i	0.997687	−	0.0679715i	$-0.0216527\pi$
$488$	0	0
$489$	7.00000	0.316551
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 60.0000i	− 2.69137i
$498$	0	0
$499$	−17.0000	−0.761025	−0.380512	−	0.924776i	$-0.624252\pi$
$499$	−17.0000	−0.761025	−0.380512	+	0.924776i	$0.624252\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	− 36.0000i	− 1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$	− 36.0000i	− 1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.00000i	0.177646i
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	50.0000	2.21187
$512$	0	0
$513$	1.00000i	0.0441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	60.0000i	2.63880i
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	0	0
$523$	21.0000i	0.918266i	0.888368	+	0.459133i	$0.151840\pi$
$523$	21.0000i	0.918266i	−0.888368	+	0.459133i	$0.848160\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6.00000i	− 0.261364i
$528$	0	0
$529$	19.0000	0.826087
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.00000i	− 0.0863064i
$538$	0	0
$539$	108.000	4.65189
$540$	0	0
$541$	−17.0000	−0.730887	−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000	−0.730887	−0.365444	+	0.930834i	$0.619083\pi$
$542$	0	0
$543$	19.0000i	0.815368i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 16.0000i	− 0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$	− 16.0000i	− 0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$	0	0
$549$	3.00000	0.128037
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	40.0000i	1.70097i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.0000i	− 0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$	− 14.0000i	− 0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$	0	0
$559$	−33.0000	−1.39575
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	− 18.0000i	− 0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$	− 18.0000i	− 0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 5.00000i	− 0.209980i
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	1.00000	0.0418487	0.0209243	−	0.999781i	$-0.493339\pi$
$571$	1.00000	0.0418487	0.0209243	+	0.999781i	$0.493339\pi$
$572$	0	0
$573$	10.0000i	0.417756i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 43.0000i	− 1.79011i	−0.445952	−	0.895057i	$-0.647135\pi$
$577$	− 43.0000i	− 1.79011i	0.445952	−	0.895057i	$-0.352865\pi$
$578$	0	0
$579$	−3.00000	−0.124676
$580$	0	0
$581$	−30.0000	−1.24461
$582$	0	0
$583$	− 48.0000i	− 1.98796i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 24.0000i	− 0.990586i	−0.868726	−	0.495293i	$-0.835061\pi$
$587$	− 24.0000i	− 0.990586i	0.868726	−	0.495293i	$-0.164939\pi$
$588$	0	0
$589$	3.00000	0.123613
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	4.00000i	0.164260i	0.996622	+	0.0821302i	$0.0261723\pi$
$593$	4.00000i	0.164260i	−0.996622	+	0.0821302i	$0.973828\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.0000i	0.859473i
$598$	0	0
$599$	44.0000	1.79779	0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000	1.79779	0.898896	+	0.438163i	$0.144371\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	0	0
$603$	− 1.00000i	− 0.0407231i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	30.0000	1.21566
$610$	0	0
$611$	30.0000	1.21367
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.0000i	1.28827i	0.764911	+	0.644136i	$0.222783\pi$
$617$	32.0000i	1.28827i	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	19.0000	0.763674	0.381837	−	0.924230i	$-0.375291\pi$
$619$	19.0000	0.763674	0.381837	+	0.924230i	$0.375291\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	− 80.0000i	− 3.20513i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.00000i	0.239617i
$628$	0	0
$629$	12.0000	0.478471
$630$	0	0
$631$	−31.0000	−1.23409	−0.617045	−	0.786928i	$-0.711670\pi$
$631$	−31.0000	−1.23409	−0.617045	+	0.786928i	$0.711670\pi$
$632$	0	0
$633$	− 15.0000i	− 0.596196i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 54.0000i	− 2.13956i
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	−15.0000	−0.587896
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	− 6.00000i	− 0.233021i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0000i	0.464642i
$668$	0	0
$669$	3.00000	0.115987
$670$	0	0
$671$	18.0000	0.694882
$672$	0	0
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	35.0000	1.34318
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.00000i	− 0.114457i
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	0	0
$693$	− 30.0000i	− 1.13961i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.00000i	0.303022i
$698$	0	0
$699$	−20.0000	−0.756469
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	6.00000i	0.226294i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 40.0000i	− 1.50435i
$708$	0	0
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	− 6.00000i	− 0.224702i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.0000i	0.896296i
$718$	0	0
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	0	0
$723$	− 7.00000i	− 0.260333i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 13.0000i	− 0.482143i	−0.970507	−	0.241072i	$-0.922501\pi$
$727$	− 13.0000i	− 0.482143i	0.970507	−	0.241072i	$-0.0774989\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−22.0000	−0.813699
$732$	0	0
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6.00000i	− 0.221013i
$738$	0	0
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	3.00000	0.110208
$742$	0	0
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.00000i	0.219529i
$748$	0	0
$749$	40.0000	1.46157
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	− 20.0000i	− 0.728841i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3.00000i	− 0.109037i	−0.998513	−	0.0545184i	$-0.982638\pi$
$757$	− 3.00000i	− 0.109037i	0.998513	−	0.0545184i	$-0.0173624\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	−52.0000	−1.88500	−0.942499	−	0.334208i	$-0.891531\pi$
$761$	−52.0000	−1.88500	−0.942499	+	0.334208i	$0.891531\pi$
$762$	0	0
$763$	35.0000i	1.26709i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.0000i	0.649942i
$768$	0	0
$769$	−27.0000	−0.973645	−0.486822	−	0.873501i	$-0.661844\pi$
$769$	−27.0000	−0.973645	−0.486822	+	0.873501i	$0.661844\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	0	0
$773$	− 28.0000i	− 1.00709i	−0.863969	−	0.503545i	$-0.832029\pi$
$773$	− 28.0000i	− 1.00709i	0.863969	−	0.503545i	$-0.167971\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 30.0000i	− 1.07624i
$778$	0	0
$779$	−4.00000	−0.143315
$780$	0	0
$781$	−72.0000	−2.57636
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.00000i	0.106938i	0.998569	+	0.0534692i	$0.0170279\pi$
$787$	3.00000i	0.106938i	−0.998569	+	0.0534692i	$0.982972\pi$
$788$	0	0
$789$	−12.0000	−0.427211
$790$	0	0
$791$	60.0000	2.13335
$792$	0	0
$793$	− 9.00000i	− 0.319599i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.0000i	0.991811i	0.868377	+	0.495905i	$0.165164\pi$
$797$	28.0000i	0.991811i	−0.868377	+	0.495905i	$0.834836\pi$
$798$	0	0
$799$	20.0000	0.707549
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	− 60.0000i	− 2.11735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 30.0000i	− 1.05605i
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	9.00000	0.316033	0.158016	−	0.987436i	$-0.449490\pi$
$811$	9.00000	0.316033	0.158016	+	0.987436i	$0.449490\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 11.0000i	− 0.384841i
$818$	0	0
$819$	−15.0000	−0.524142
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	49.0000i	1.70803i	0.520246	+	0.854016i	$0.325840\pi$
$823$	49.0000i	1.70803i	−0.520246	+	0.854016i	$0.674160\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.0000i	1.11275i	0.830932	+	0.556375i	$0.187808\pi$
$827$	32.0000i	1.11275i	−0.830932	+	0.556375i	$0.812192\pi$
$828$	0	0
$829$	50.0000	1.73657	0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	0	0
$831$	−1.00000	−0.0346896
$832$	0	0
$833$	− 36.0000i	− 1.24733i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.00000i	0.103695i
$838$	0	0
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	26.0000i	0.895488i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 125.000i	− 4.29505i
$848$	0	0
$849$	−13.0000	−0.446159
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	− 55.0000i	− 1.88316i	−0.336784	−	0.941582i	$-0.609339\pi$
$853$	− 55.0000i	− 1.88316i	0.336784	−	0.941582i	$-0.390661\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	52.0000i	1.77629i	0.459567	+	0.888143i	$0.348005\pi$
$857$	52.0000i	1.77629i	−0.459567	+	0.888143i	$0.651995\pi$
$858$	0	0
$859$	−48.0000	−1.63774	−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000	−1.63774	−0.818869	+	0.573980i	$0.805399\pi$
$860$	0	0
$861$	20.0000	0.681598
$862$	0	0
$863$	8.00000i	0.272323i	0.990687	+	0.136162i	$0.0434766\pi$
$863$	8.00000i	0.272323i	−0.990687	+	0.136162i	$0.956523\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	48.0000	1.62829
$870$	0	0
$871$	−3.00000	−0.101651
$872$	0	0
$873$	− 7.00000i	− 0.236914i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 17.0000i	− 0.574049i	−0.957923	−	0.287025i	$-0.907334\pi$
$877$	− 17.0000i	− 0.574049i	0.957923	−	0.287025i	$-0.0926662\pi$
$878$	0	0
$879$	−2.00000	−0.0674583
$880$	0	0
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	0	0
$883$	19.0000i	0.639401i	0.947519	+	0.319700i	$0.103582\pi$
$883$	19.0000i	0.639401i	−0.947519	+	0.319700i	$0.896418\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.0000i	1.41022i	0.709097	+	0.705111i	$0.249103\pi$
$887$	42.0000i	1.41022i	−0.709097	+	0.705111i	$0.750897\pi$
$888$	0	0
$889$	40.0000	1.34156
$890$	0	0
$891$	−6.00000	−0.201008
$892$	0	0
$893$	10.0000i	0.334637i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 6.00000i	− 0.200334i
$898$	0	0
$899$	−18.0000	−0.600334
$900$	0	0
$901$	−16.0000	−0.533037
$902$	0	0
$903$	55.0000i	1.83029i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	44.0000i	1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$	44.0000i	1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$	0	0
$909$	−8.00000	−0.265343
$910$	0	0
$911$	14.0000	0.463841	0.231920	−	0.972735i	$-0.425499\pi$
$911$	14.0000	0.463841	0.231920	+	0.972735i	$0.425499\pi$
$912$	0	0
$913$	36.0000i	1.19143i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 80.0000i	− 2.64183i
$918$	0	0
$919$	1.00000	0.0329870	0.0164935	−	0.999864i	$-0.494750\pi$
$919$	1.00000	0.0329870	0.0164935	+	0.999864i	$0.494750\pi$
$920$	0	0
$921$	13.0000	0.428365
$922$	0	0
$923$	36.0000i	1.18495i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.00000i	0.131377i
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	0	0
$933$	14.0000i	0.458339i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 23.0000i	− 0.751377i	−0.926746	−	0.375689i	$-0.877406\pi$
$937$	− 23.0000i	− 0.751377i	0.926746	−	0.375689i	$-0.122594\pi$
$938$	0	0
$939$	29.0000	0.946379
$940$	0	0
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	0	0
$943$	8.00000i	0.260516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.00000i	0.0649913i	0.999472	+	0.0324956i	$0.0103455\pi$
$947$	2.00000i	0.0649913i	−0.999472	+	0.0324956i	$0.989654\pi$
$948$	0	0
$949$	−30.0000	−0.973841
$950$	0	0
$951$	16.0000	0.518836
$952$	0	0
$953$	24.0000i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$	24.0000i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 36.0000i	− 1.16371i
$958$	0	0
$959$	−30.0000	−0.968751
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	− 8.00000i	− 0.257796i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	56.0000i	1.80084i	0.435023	+	0.900419i	$0.356740\pi$
$967$	56.0000i	1.80084i	−0.435023	+	0.900419i	$0.643260\pi$
$968$	0	0
$969$	2.00000	0.0642493
$970$	0	0
$971$	46.0000	1.47621	0.738105	−	0.674686i	$-0.235721\pi$
$971$	46.0000	1.47621	0.738105	+	0.674686i	$0.235721\pi$
$972$	0	0
$973$	− 60.0000i	− 1.92351i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 10.0000i	− 0.319928i	−0.987123	−	0.159964i	$-0.948862\pi$
$977$	− 10.0000i	− 0.319928i	0.987123	−	0.159964i	$-0.0511379\pi$
$978$	0	0
$979$	−96.0000	−3.06817
$980$	0	0
$981$	7.00000	0.223493
$982$	0	0
$983$	36.0000i	1.14822i	0.818778	+	0.574111i	$0.194652\pi$
$983$	36.0000i	1.14822i	−0.818778	+	0.574111i	$0.805348\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 50.0000i	− 1.59152i
$988$	0	0
$989$	−22.0000	−0.699559
$990$	0	0
$991$	25.0000	0.794151	0.397076	−	0.917786i	$-0.370025\pi$
$991$	25.0000	0.794151	0.397076	+	0.917786i	$0.370025\pi$
$992$	0	0
$993$	− 20.0000i	− 0.634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 54.0000i	− 1.71020i	−0.518465	−	0.855099i	$-0.673497\pi$
$997$	− 54.0000i	− 1.71020i	0.518465	−	0.855099i	$-0.326503\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4800.2.f.a.3649.2		2
4.3	odd	2		4800.2.f.bj.3649.1		2
5.2	odd	4		4800.2.a.cs.1.1		1
5.3	odd	4		4800.2.a.a.1.1		1
5.4	even	2	inner	4800.2.f.a.3649.1		2
8.3	odd	2		600.2.f.a.49.2		2
8.5	even	2		1200.2.f.i.49.1		2
20.3	even	4		4800.2.a.ct.1.1		1
20.7	even	4		4800.2.a.b.1.1		1
20.19	odd	2		4800.2.f.bj.3649.2		2
24.5	odd	2		3600.2.f.b.2449.1		2
24.11	even	2		1800.2.f.k.649.2		2
40.3	even	4		600.2.a.e.1.1	✓	1
40.13	odd	4		1200.2.a.j.1.1		1
40.19	odd	2		600.2.f.a.49.1		2
40.27	even	4		600.2.a.f.1.1	yes	1
40.29	even	2		1200.2.f.i.49.2		2
40.37	odd	4		1200.2.a.i.1.1		1
120.29	odd	2		3600.2.f.b.2449.2		2
120.53	even	4		3600.2.a.a.1.1		1
120.59	even	2		1800.2.f.k.649.1		2
120.77	even	4		3600.2.a.bq.1.1		1
120.83	odd	4		1800.2.a.x.1.1		1
120.107	odd	4		1800.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.a.e.1.1	✓	1	40.3	even	4
600.2.a.f.1.1	yes	1	40.27	even	4
600.2.f.a.49.1		2	40.19	odd	2
600.2.f.a.49.2		2	8.3	odd	2
1200.2.a.i.1.1		1	40.37	odd	4
1200.2.a.j.1.1		1	40.13	odd	4
1200.2.f.i.49.1		2	8.5	even	2
1200.2.f.i.49.2		2	40.29	even	2
1800.2.a.a.1.1		1	120.107	odd	4
1800.2.a.x.1.1		1	120.83	odd	4
1800.2.f.k.649.1		2	120.59	even	2
1800.2.f.k.649.2		2	24.11	even	2
3600.2.a.a.1.1		1	120.53	even	4
3600.2.a.bq.1.1		1	120.77	even	4
3600.2.f.b.2449.1		2	24.5	odd	2
3600.2.f.b.2449.2		2	120.29	odd	2
4800.2.a.a.1.1		1	5.3	odd	4
4800.2.a.b.1.1		1	20.7	even	4
4800.2.a.cs.1.1		1	5.2	odd	4
4800.2.a.ct.1.1		1	20.3	even	4
4800.2.f.a.3649.1		2	5.4	even	2	inner
4800.2.f.a.3649.2		2	1.1	even	1	trivial
4800.2.f.bj.3649.1		2	4.3	odd	2
4800.2.f.bj.3649.2		2	20.19	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:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -5.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -5.00000i q^{7} -1.00000 q^{9} -6.00000 q^{11} +3.00000i q^{13} +2.00000i q^{17} -1.00000 q^{19} +5.00000 q^{21} +2.00000i q^{23} -1.00000i q^{27} +6.00000 q^{29} -3.00000 q^{31} -6.00000i q^{33} -6.00000i q^{37} -3.00000 q^{39} +4.00000 q^{41} +11.0000i q^{43} -10.0000i q^{47} -18.0000 q^{49} -2.00000 q^{51} +8.00000i q^{53} -1.00000i q^{57} +6.00000 q^{59} -3.00000 q^{61} +5.00000i q^{63} +1.00000i q^{67} -2.00000 q^{69} +12.0000 q^{71} +10.0000i q^{73} +30.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +6.00000i q^{87} +16.0000 q^{89} +15.0000 q^{91} -3.00000i q^{93} +7.00000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 12 q^{11} - 2 q^{19} + 10 q^{21} + 12 q^{29} - 6 q^{31} - 6 q^{39} + 8 q^{41} - 36 q^{49} - 4 q^{51} + 12 q^{59} - 6 q^{61} - 4 q^{69} + 24 q^{71} - 16 q^{79} + 2 q^{81} + 32 q^{89} + 30 q^{91} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 12 * q^11 - 2 * q^19 + 10 * q^21 + 12 * q^29 - 6 * q^31 - 6 * q^39 + 8 * q^41 - 36 * q^49 - 4 * q^51 + 12 * q^59 - 6 * q^61 - 4 * q^69 + 24 * q^71 - 16 * q^79 + 2 * q^81 + 32 * q^89 + 30 * q^91 + 12 * q^99

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