LMFDB - Embedded newform 4600.2.e.d.4049.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4600,2,Mod(4049,4600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4600.4049"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,10,0,0,0,0,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$36.7311849298$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.d.4049.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} -5.00000i q^{13} +4.00000i q^{17} -1.00000 q^{19} -6.00000 q^{21} -1.00000i q^{23} +4.00000i q^{27} -9.00000 q^{29} -2.00000 q^{31} +10.0000i q^{33} +2.00000i q^{37} +10.0000 q^{39} +3.00000 q^{41} -7.00000i q^{43} +12.0000i q^{47} -2.00000 q^{49} -8.00000 q^{51} +12.0000i q^{53} -2.00000i q^{57} +6.00000 q^{59} -10.0000 q^{61} -3.00000i q^{63} +8.00000i q^{67} +2.00000 q^{69} +2.00000 q^{71} +1.00000i q^{73} +15.0000i q^{77} +11.0000 q^{79} -11.0000 q^{81} +9.00000i q^{83} -18.0000i q^{87} +14.0000 q^{89} +15.0000 q^{91} -4.00000i q^{93} +16.0000i q^{97} -5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} + 10 q^{11} - 2 q^{19} - 12 q^{21} - 18 q^{29} - 4 q^{31} + 20 q^{39} + 6 q^{41} - 4 q^{49} - 16 q^{51} + 12 q^{59} - 20 q^{61} + 4 q^{69} + 4 q^{71} + 22 q^{79} - 22 q^{81} + 28 q^{89} + 30 q^{91}+ \cdots - 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 10 * q^11 - 2 * q^19 - 12 * q^21 - 18 * q^29 - 4 * q^31 + 20 * q^39 + 6 * q^41 - 4 * q^49 - 16 * q^51 + 12 * q^59 - 20 * q^61 + 4 * q^69 + 4 * q^71 + 22 * q^79 - 22 * q^81 + 28 * q^89 + 30 * q^91 - 10 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	− 5.00000i	− 1.38675i	−0.720577	−	0.693375i	$-0.756123\pi$
$13$	− 5.00000i	− 1.38675i	0.720577	−	0.693375i	$-0.243877\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	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
$21$	−6.00000	−1.30931
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	10.0000i	1.74078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	10.0000	1.60128
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	− 7.00000i	− 1.06749i	−0.845645	−	0.533745i	$-0.820784\pi$
$43$	− 7.00000i	− 1.06749i	0.845645	−	0.533745i	$-0.179216\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.00000i	− 0.264906i
$58$	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
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	− 3.00000i	− 0.377964i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	1.00000i	0.117041i	0.998286	+	0.0585206i	$0.0186383\pi$
$73$	1.00000i	0.117041i	−0.998286	+	0.0585206i	$0.981362\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.0000i	1.70941i
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	9.00000i	0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$	9.00000i	0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 18.0000i	− 1.92980i
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	15.0000	1.57243
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.0000i	1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$	16.0000i	1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	− 9.00000i	− 0.886796i	−0.896325	−	0.443398i	$-0.853773\pi$
$103$	− 9.00000i	− 0.886796i	0.896325	−	0.443398i	$-0.146227\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.0000i	− 1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$	− 16.0000i	− 1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	8.00000i	0.752577i	0.926503	+	0.376288i	$0.122800\pi$
$113$	8.00000i	0.752577i	−0.926503	+	0.376288i	$0.877200\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.00000i	0.462250i
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	6.00000i	0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.0000i	0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$127$	10.0000i	0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$128$	0	0
$129$	14.0000	1.23263
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	− 3.00000i	− 0.260133i
$134$	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	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	−24.0000	−2.02116
$142$	0	0
$143$	− 25.0000i	− 2.09061i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.00000i	− 0.329914i
$148$	0	0
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	− 4.00000i	− 0.323381i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.00000i	− 0.638470i	−0.947676	−	0.319235i	$-0.896574\pi$
$157$	− 8.00000i	− 0.638470i	0.947676	−	0.319235i	$-0.103426\pi$
$158$	0	0
$159$	−24.0000	−1.90332
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.0000i	1.08335i	0.840587	+	0.541676i	$0.182210\pi$
$167$	14.0000i	1.08335i	−0.840587	+	0.541676i	$0.817790\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	5.00000i	0.380143i	0.981770	+	0.190071i	$0.0608720\pi$
$173$	5.00000i	0.380143i	−0.981770	+	0.190071i	$0.939128\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	− 20.0000i	− 1.47844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.0000i	1.46254i
$188$	0	0
$189$	−12.0000	−0.872872
$190$	0	0
$191$	−1.00000	−0.0723575	−0.0361787	−	0.999345i	$-0.511519\pi$
$191$	−1.00000	−0.0723575	−0.0361787	+	0.999345i	$0.511519\pi$
$192$	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
$197$	23.0000i	1.63868i	0.573306	+	0.819341i	$0.305660\pi$
$197$	23.0000i	1.63868i	−0.573306	+	0.819341i	$0.694340\pi$
$198$	0	0
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	− 27.0000i	− 1.89503i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	0	0
$209$	−5.00000	−0.345857
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	0	0
$213$	4.00000i	0.274075i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 6.00000i	− 0.407307i
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	20.0000	1.34535
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	0	0
$229$	30.0000	1.98246	0.991228	−	0.132164i	$-0.0421925\pi$
$229$	30.0000	1.98246	0.991228	+	0.132164i	$0.0421925\pi$
$230$	0	0
$231$	−30.0000	−1.97386
$232$	0	0
$233$	− 25.0000i	− 1.63780i	−0.573933	−	0.818902i	$-0.694583\pi$
$233$	− 25.0000i	− 1.63780i	0.573933	−	0.818902i	$-0.305417\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	22.0000i	1.42905i
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	− 10.0000i	− 0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.00000i	0.318142i
$248$	0	0
$249$	−18.0000	−1.14070
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	− 5.00000i	− 0.314347i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	9.00000	0.557086
$262$	0	0
$263$	− 16.0000i	− 0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$	− 16.0000i	− 0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	28.0000i	1.71357i
$268$	0	0
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	30.0000i	1.81568i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 13.0000i	− 0.781094i	−0.920583	−	0.390547i	$-0.872286\pi$
$277$	− 13.0000i	− 0.781094i	0.920583	−	0.390547i	$-0.127714\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.00000i	0.531253i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−32.0000	−1.87587
$292$	0	0
$293$	28.0000i	1.63578i	0.575376	+	0.817889i	$0.304856\pi$
$293$	28.0000i	1.63578i	−0.575376	+	0.817889i	$0.695144\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	20.0000i	1.16052i
$298$	0	0
$299$	−5.00000	−0.289157
$300$	0	0
$301$	21.0000	1.21042
$302$	0	0
$303$	− 28.0000i	− 1.60856i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	0	0
$309$	18.0000	1.02398
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 17.0000i	− 0.954815i	−0.878682	−	0.477408i	$-0.841577\pi$
$317$	− 17.0000i	− 0.954815i	0.878682	−	0.477408i	$-0.158423\pi$
$318$	0	0
$319$	−45.0000	−2.51952
$320$	0	0
$321$	32.0000	1.78607
$322$	0	0
$323$	− 4.00000i	− 0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	20.0000i	1.10600i
$328$	0	0
$329$	−36.0000	−1.98474
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 32.0000i	− 1.74315i	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	− 32.0000i	− 1.74315i	0.490261	−	0.871576i	$-0.336901\pi$
$338$	0	0
$339$	−16.0000	−0.869001
$340$	0	0
$341$	−10.0000	−0.541530
$342$	0	0
$343$	15.0000i	0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 32.0000i	− 1.71785i	−0.512101	−	0.858925i	$-0.671133\pi$
$347$	− 32.0000i	− 1.71785i	0.512101	−	0.858925i	$-0.328867\pi$
$348$	0	0
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	0	0
$353$	21.0000i	1.11772i	0.829263	+	0.558859i	$0.188761\pi$
$353$	21.0000i	1.11772i	−0.829263	+	0.558859i	$0.811239\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 24.0000i	− 1.27021i
$358$	0	0
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	28.0000i	1.46962i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	15.0000i	0.782994i	0.920179	+	0.391497i	$0.128043\pi$
$367$	15.0000i	0.782994i	−0.920179	+	0.391497i	$0.871957\pi$
$368$	0	0
$369$	−3.00000	−0.156174
$370$	0	0
$371$	−36.0000	−1.86903
$372$	0	0
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	45.0000i	2.31762i
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	−20.0000	−1.02463
$382$	0	0
$383$	− 23.0000i	− 1.17525i	−0.809135	−	0.587623i	$-0.800064\pi$
$383$	− 23.0000i	− 1.17525i	0.809135	−	0.587623i	$-0.199936\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.00000i	0.355830i
$388$	0	0
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	12.0000i	0.605320i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	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
$403$	10.0000i	0.498135i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.0000i	0.495682i
$408$	0	0
$409$	35.0000	1.73064	0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000	1.73064	0.865319	+	0.501221i	$0.167116\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	18.0000i	0.885722i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 24.0000i	− 1.17529i
$418$	0	0
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	− 12.0000i	− 0.583460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 30.0000i	− 1.45180i
$428$	0	0
$429$	50.0000	2.41402
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	6.00000i	0.288342i	0.989553	+	0.144171i	$0.0460515\pi$
$433$	6.00000i	0.288342i	−0.989553	+	0.144171i	$0.953949\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.00000i	0.0478365i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	14.0000i	0.665160i	0.943075	+	0.332580i	$0.107919\pi$
$443$	14.0000i	0.665160i	−0.943075	+	0.332580i	$0.892081\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 28.0000i	− 1.32435i
$448$	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
$451$	15.0000	0.706322
$452$	0	0
$453$	− 36.0000i	− 1.69143i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.0000i	1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$	28.0000i	1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	0	0
$459$	−16.0000	−0.746816
$460$	0	0
$461$	41.0000	1.90956	0.954780	−	0.297313i	$-0.0960904\pi$
$461$	41.0000	1.90956	0.954780	+	0.297313i	$0.0960904\pi$
$462$	0	0
$463$	24.0000i	1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$	24.0000i	1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.0000i	0.971764i	0.874024	+	0.485882i	$0.161502\pi$
$467$	21.0000i	0.971764i	−0.874024	+	0.485882i	$0.838498\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	16.0000	0.737241
$472$	0	0
$473$	− 35.0000i	− 1.60930i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 12.0000i	− 0.549442i
$478$	0	0
$479$	17.0000	0.776750	0.388375	−	0.921501i	$-0.373037\pi$
$479$	17.0000	0.776750	0.388375	+	0.921501i	$0.373037\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	0	0
$483$	6.00000i	0.273009i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	32.0000i	1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$	32.0000i	1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$	0	0
$489$	−32.0000	−1.44709
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	− 36.0000i	− 1.62136i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.00000i	0.269137i
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	−28.0000	−1.25095
$502$	0	0
$503$	− 5.00000i	− 0.222939i	−0.993768	−	0.111469i	$-0.964444\pi$
$503$	− 5.00000i	− 0.222939i	0.993768	−	0.111469i	$-0.0355557\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 24.0000i	− 1.06588i
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−3.00000	−0.132712
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	60.0000i	2.63880i
$518$	0	0
$519$	−10.0000	−0.438951
$520$	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
$523$	7.00000i	0.306089i	0.988219	+	0.153044i	$0.0489077\pi$
$523$	7.00000i	0.306089i	−0.988219	+	0.153044i	$0.951092\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.00000i	− 0.348485i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	− 15.0000i	− 0.649722i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 24.0000i	− 1.03568i
$538$	0	0
$539$	−10.0000	−0.430730
$540$	0	0
$541$	−23.0000	−0.988847	−0.494424	−	0.869221i	$-0.664621\pi$
$541$	−23.0000	−0.988847	−0.494424	+	0.869221i	$0.664621\pi$
$542$	0	0
$543$	− 40.0000i	− 1.71656i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 44.0000i	− 1.88130i	−0.339372	−	0.940652i	$-0.610215\pi$
$547$	− 44.0000i	− 1.88130i	0.339372	−	0.940652i	$-0.389785\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	9.00000	0.383413
$552$	0	0
$553$	33.0000i	1.40330i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	0	0
$559$	−35.0000	−1.48034
$560$	0	0
$561$	−40.0000	−1.68880
$562$	0	0
$563$	− 39.0000i	− 1.64365i	−0.569737	−	0.821827i	$-0.692955\pi$
$563$	− 39.0000i	− 1.64365i	0.569737	−	0.821827i	$-0.307045\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 33.0000i	− 1.38587i
$568$	0	0
$569$	4.00000	0.167689	0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000	0.167689	0.0838444	+	0.996479i	$0.473280\pi$
$570$	0	0
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	0	0
$573$	− 2.00000i	− 0.0835512i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.0000i	0.957503i	0.877951	+	0.478751i	$0.158910\pi$
$577$	23.0000i	0.957503i	−0.877951	+	0.478751i	$0.841090\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	−27.0000	−1.12015
$582$	0	0
$583$	60.0000i	2.48495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 30.0000i	− 1.23823i	−0.785299	−	0.619116i	$-0.787491\pi$
$587$	− 30.0000i	− 1.23823i	0.785299	−	0.619116i	$-0.212509\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	0	0
$591$	−46.0000	−1.89219
$592$	0	0
$593$	− 17.0000i	− 0.698106i	−0.937103	−	0.349053i	$-0.886503\pi$
$593$	− 17.0000i	− 0.698106i	0.937103	−	0.349053i	$-0.113497\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 10.0000i	− 0.409273i
$598$	0	0
$599$	14.0000	0.572024	0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000	0.572024	0.286012	+	0.958226i	$0.407670\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	− 8.00000i	− 0.325785i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	0	0
$609$	54.0000	2.18819
$610$	0	0
$611$	60.0000	2.42734
$612$	0	0
$613$	18.0000i	0.727013i	0.931592	+	0.363507i	$0.118421\pi$
$613$	18.0000i	0.727013i	−0.931592	+	0.363507i	$0.881579\pi$
$614$	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
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	42.0000i	1.68269i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 10.0000i	− 0.399362i
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	0	0
$633$	− 4.00000i	− 0.158986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.0000i	0.396214i
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	29.0000i	1.14365i	0.820376	+	0.571824i	$0.193764\pi$
$643$	29.0000i	1.14365i	−0.820376	+	0.571824i	$0.806236\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.0000i	1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$	32.0000i	1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$	0	0
$649$	30.0000	1.17760
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	− 29.0000i	− 1.13486i	−0.823422	−	0.567429i	$-0.807938\pi$
$653$	− 29.0000i	− 1.13486i	0.823422	−	0.567429i	$-0.192062\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.00000i	− 0.0390137i
$658$	0	0
$659$	45.0000	1.75295	0.876476	−	0.481446i	$-0.159888\pi$
$659$	45.0000	1.75295	0.876476	+	0.481446i	$0.159888\pi$
$660$	0	0
$661$	8.00000	0.311164	0.155582	−	0.987823i	$-0.450275\pi$
$661$	8.00000	0.311164	0.155582	+	0.987823i	$0.450275\pi$
$662$	0	0
$663$	40.0000i	1.55347i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.00000i	0.348481i
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−50.0000	−1.93023
$672$	0	0
$673$	15.0000i	0.578208i	0.957298	+	0.289104i	$0.0933573\pi$
$673$	15.0000i	0.578208i	−0.957298	+	0.289104i	$0.906643\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	−48.0000	−1.84207
$680$	0	0
$681$	40.0000	1.53280
$682$	0	0
$683$	− 34.0000i	− 1.30097i	−0.759517	−	0.650487i	$-0.774565\pi$
$683$	− 34.0000i	− 1.30097i	0.759517	−	0.650487i	$-0.225435\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	60.0000i	2.28914i
$688$	0	0
$689$	60.0000	2.28582
$690$	0	0
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	0	0
$693$	− 15.0000i	− 0.569803i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.0000i	0.454532i
$698$	0	0
$699$	50.0000	1.89117
$700$	0	0
$701$	8.00000	0.302156	0.151078	−	0.988522i	$-0.451726\pi$
$701$	8.00000	0.302156	0.151078	+	0.988522i	$0.451726\pi$
$702$	0	0
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 42.0000i	− 1.57957i
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	−11.0000	−0.412532
$712$	0	0
$713$	2.00000i	0.0749006i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 24.0000i	− 0.896296i
$718$	0	0
$719$	34.0000	1.26799	0.633993	−	0.773339i	$-0.281415\pi$
$719$	34.0000	1.26799	0.633993	+	0.773339i	$0.281415\pi$
$720$	0	0
$721$	27.0000	1.00553
$722$	0	0
$723$	− 8.00000i	− 0.297523i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 16.0000i	− 0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$	− 16.0000i	− 0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	28.0000	1.03562
$732$	0	0
$733$	− 26.0000i	− 0.960332i	−0.877178	−	0.480166i	$-0.840576\pi$
$733$	− 26.0000i	− 0.960332i	0.877178	−	0.480166i	$-0.159424\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	40.0000i	1.47342i
$738$	0	0
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	0	0
$741$	−10.0000	−0.367359
$742$	0	0
$743$	21.0000i	0.770415i	0.922830	+	0.385208i	$0.125870\pi$
$743$	21.0000i	0.770415i	−0.922830	+	0.385208i	$0.874130\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 9.00000i	− 0.329293i
$748$	0	0
$749$	48.0000	1.75388
$750$	0	0
$751$	19.0000	0.693320	0.346660	−	0.937991i	$-0.387316\pi$
$751$	19.0000	0.693320	0.346660	+	0.937991i	$0.387316\pi$
$752$	0	0
$753$	40.0000i	1.45768i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	0	0
$759$	10.0000	0.362977
$760$	0	0
$761$	1.00000	0.0362500	0.0181250	−	0.999836i	$-0.494230\pi$
$761$	1.00000	0.0362500	0.0181250	+	0.999836i	$0.494230\pi$
$762$	0	0
$763$	30.0000i	1.08607i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 30.0000i	− 1.08324i
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	−4.00000	−0.144056
$772$	0	0
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 12.0000i	− 0.430498i
$778$	0	0
$779$	−3.00000	−0.107486
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	− 36.0000i	− 1.28654i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	25.0000i	0.891154i	0.895244	+	0.445577i	$0.147001\pi$
$787$	25.0000i	0.891154i	−0.895244	+	0.445577i	$0.852999\pi$
$788$	0	0
$789$	32.0000	1.13923
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	50.0000i	1.77555i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	5.00000i	0.176446i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 10.0000i	− 0.352017i
$808$	0	0
$809$	23.0000	0.808637	0.404318	−	0.914618i	$-0.367509\pi$
$809$	23.0000	0.808637	0.404318	+	0.914618i	$0.367509\pi$
$810$	0	0
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	0	0
$813$	24.0000i	0.841717i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.00000i	0.244899i
$818$	0	0
$819$	−15.0000	−0.524142
$820$	0	0
$821$	51.0000	1.77991	0.889956	−	0.456046i	$-0.150735\pi$
$821$	51.0000	1.77991	0.889956	+	0.456046i	$0.150735\pi$
$822$	0	0
$823$	12.0000i	0.418294i	0.977884	+	0.209147i	$0.0670687\pi$
$823$	12.0000i	0.418294i	−0.977884	+	0.209147i	$0.932931\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 7.00000i	− 0.243414i	−0.992566	−	0.121707i	$-0.961163\pi$
$827$	− 7.00000i	− 0.243414i	0.992566	−	0.121707i	$-0.0388368\pi$
$828$	0	0
$829$	31.0000	1.07667	0.538337	−	0.842729i	$-0.319053\pi$
$829$	31.0000	1.07667	0.538337	+	0.842729i	$0.319053\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	0	0
$833$	− 8.00000i	− 0.277184i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 8.00000i	− 0.276520i
$838$	0	0
$839$	−15.0000	−0.517858	−0.258929	−	0.965896i	$-0.583369\pi$
$839$	−15.0000	−0.517858	−0.258929	+	0.965896i	$0.583369\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	− 16.0000i	− 0.551069i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	42.0000i	1.44314i
$848$	0	0
$849$	8.00000	0.274559
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	19.0000i	0.650548i	0.945620	+	0.325274i	$0.105456\pi$
$853$	19.0000i	0.650548i	−0.945620	+	0.325274i	$0.894544\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.0000i	1.29806i	0.760765	+	0.649028i	$0.224824\pi$
$857$	38.0000i	1.29806i	−0.760765	+	0.649028i	$0.775176\pi$
$858$	0	0
$859$	2.00000	0.0682391	0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000	0.0682391	0.0341196	+	0.999418i	$0.489137\pi$
$860$	0	0
$861$	−18.0000	−0.613438
$862$	0	0
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.00000i	0.0679236i
$868$	0	0
$869$	55.0000	1.86575
$870$	0	0
$871$	40.0000	1.35535
$872$	0	0
$873$	− 16.0000i	− 0.541518i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	0	0
$879$	−56.0000	−1.88883
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 48.0000i	− 1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$	− 48.0000i	− 1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$	0	0
$889$	−30.0000	−1.00617
$890$	0	0
$891$	−55.0000	−1.84257
$892$	0	0
$893$	− 12.0000i	− 0.401565i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 10.0000i	− 0.333890i
$898$	0	0
$899$	18.0000	0.600334
$900$	0	0
$901$	−48.0000	−1.59911
$902$	0	0
$903$	42.0000i	1.39767i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 15.0000i	− 0.498067i	−0.968495	−	0.249033i	$-0.919887\pi$
$907$	− 15.0000i	− 0.498067i	0.968495	−	0.249033i	$-0.0801129\pi$
$908$	0	0
$909$	14.0000	0.464351
$910$	0	0
$911$	5.00000	0.165657	0.0828287	−	0.996564i	$-0.473605\pi$
$911$	5.00000	0.165657	0.0828287	+	0.996564i	$0.473605\pi$
$912$	0	0
$913$	45.0000i	1.48928i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.0000i	0.594412i
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	0	0
$923$	− 10.0000i	− 0.329154i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.00000i	0.295599i
$928$	0	0
$929$	13.0000	0.426516	0.213258	−	0.976996i	$-0.431592\pi$
$929$	13.0000	0.426516	0.213258	+	0.976996i	$0.431592\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	36.0000i	1.17859i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	44.0000i	1.43742i	0.695311	+	0.718709i	$0.255266\pi$
$937$	44.0000i	1.43742i	−0.695311	+	0.718709i	$0.744734\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	0	0
$943$	− 3.00000i	− 0.0976934i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	5.00000	0.162307
$950$	0	0
$951$	34.0000	1.10253
$952$	0	0
$953$	− 2.00000i	− 0.0647864i	−0.999475	−	0.0323932i	$-0.989687\pi$
$953$	− 2.00000i	− 0.0647864i	0.999475	−	0.0323932i	$-0.0103129\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 90.0000i	− 2.90929i
$958$	0	0
$959$	18.0000	0.581250
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	16.0000i	0.515593i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 12.0000i	− 0.385894i	−0.981209	−	0.192947i	$-0.938195\pi$
$967$	− 12.0000i	− 0.385894i	0.981209	−	0.192947i	$-0.0618045\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	−41.0000	−1.31575	−0.657876	−	0.753126i	$-0.728545\pi$
$971$	−41.0000	−1.31575	−0.657876	+	0.753126i	$0.728545\pi$
$972$	0	0
$973$	− 36.0000i	− 1.15411i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.0000i	0.639857i	0.947442	+	0.319928i	$0.103659\pi$
$977$	20.0000i	0.639857i	−0.947442	+	0.319928i	$0.896341\pi$
$978$	0	0
$979$	70.0000	2.23721
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	− 1.00000i	− 0.0318950i	−0.999873	−	0.0159475i	$-0.994924\pi$
$983$	− 1.00000i	− 0.0318950i	0.999873	−	0.0159475i	$-0.00507647\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 72.0000i	− 2.29179i
$988$	0	0
$989$	−7.00000	−0.222587
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	52.0000i	1.65017i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 5.00000i	− 0.158352i	−0.996861	−	0.0791758i	$-0.974771\pi$
$997$	− 5.00000i	− 0.158352i	0.996861	−	0.0791758i	$-0.0252288\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.d.4049.2		2
5.2	odd	4		4600.2.a.m.1.1	yes	1
5.3	odd	4		4600.2.a.d.1.1	✓	1
5.4	even	2	inner	4600.2.e.d.4049.1		2
20.3	even	4		9200.2.a.bc.1.1		1
20.7	even	4		9200.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.d.1.1	✓	1	5.3	odd	4
4600.2.a.m.1.1	yes	1	5.2	odd	4
4600.2.e.d.4049.1		2	5.4	even	2	inner
4600.2.e.d.4049.2		2	1.1	even	1	trivial
9200.2.a.j.1.1		1	20.7	even	4
9200.2.a.bc.1.1		1	20.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} -5.00000i q^{13} +4.00000i q^{17} -1.00000 q^{19} -6.00000 q^{21} -1.00000i q^{23} +4.00000i q^{27} -9.00000 q^{29} -2.00000 q^{31} +10.0000i q^{33} +2.00000i q^{37} +10.0000 q^{39} +3.00000 q^{41} -7.00000i q^{43} +12.0000i q^{47} -2.00000 q^{49} -8.00000 q^{51} +12.0000i q^{53} -2.00000i q^{57} +6.00000 q^{59} -10.0000 q^{61} -3.00000i q^{63} +8.00000i q^{67} +2.00000 q^{69} +2.00000 q^{71} +1.00000i q^{73} +15.0000i q^{77} +11.0000 q^{79} -11.0000 q^{81} +9.00000i q^{83} -18.0000i q^{87} +14.0000 q^{89} +15.0000 q^{91} -4.00000i q^{93} +16.0000i q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} + 10 q^{11} - 2 q^{19} - 12 q^{21} - 18 q^{29} - 4 q^{31} + 20 q^{39} + 6 q^{41} - 4 q^{49} - 16 q^{51} + 12 q^{59} - 20 q^{61} + 4 q^{69} + 4 q^{71} + 22 q^{79} - 22 q^{81} + 28 q^{89} + 30 q^{91}+ \cdots - 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 10 * q^11 - 2 * q^19 - 12 * q^21 - 18 * q^29 - 4 * q^31 + 20 * q^39 + 6 * q^41 - 4 * q^49 - 16 * q^51 + 12 * q^59 - 20 * q^61 + 4 * q^69 + 4 * q^71 + 22 * q^79 - 22 * q^81 + 28 * q^89 + 30 * q^91 - 10 * q^99

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