LMFDB - Embedded newform 4600.2.e.a.4049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$36.7311849298$
Analytic rank:	$1$
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:	no (minimal twist has level 184)
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.a.4049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} -5.00000i q^{13} +6.00000i q^{17} -6.00000 q^{19} -6.00000 q^{21} +1.00000i q^{23} -9.00000i q^{27} -9.00000 q^{29} +3.00000 q^{31} +8.00000i q^{37} +15.0000 q^{39} +3.00000 q^{41} -8.00000i q^{43} -7.00000i q^{47} +3.00000 q^{49} -18.0000 q^{51} -2.00000i q^{53} -18.0000i q^{57} -4.00000 q^{59} -10.0000 q^{61} -12.0000i q^{63} -8.00000i q^{67} -3.00000 q^{69} +7.00000 q^{71} +9.00000i q^{73} +6.00000 q^{79} +9.00000 q^{81} -14.0000i q^{83} -27.0000i q^{87} -16.0000 q^{89} +10.0000 q^{91} +9.00000i q^{93} -6.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{9}+O(q^{10}) $ 2 * q - 12 * q^9	$ 2 q - 12 q^{9} - 12 q^{19} - 12 q^{21} - 18 q^{29} + 6 q^{31} + 30 q^{39} + 6 q^{41} + 6 q^{49} - 36 q^{51} - 8 q^{59} - 20 q^{61} - 6 q^{69} + 14 q^{71} + 12 q^{79} + 18 q^{81} - 32 q^{89} + 20 q^{91}+O(q^{100}) $ 2 * q - 12 * q^9 - 12 * q^19 - 12 * q^21 - 18 * q^29 + 6 * q^31 + 30 * q^39 + 6 * q^41 + 6 * q^49 - 36 * q^51 - 8 * q^59 - 20 * q^61 - 6 * q^69 + 14 * q^71 + 12 * q^79 + 18 * q^81 - 32 * q^89 + 20 * q^91

Display 100 coefficients 10 coefficients

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
$3$	3.00000i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$3$	3.00000i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	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
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	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
$27$	− 9.00000i	− 1.73205i
$28$	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
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	15.0000	2.40192
$40$	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
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.00000i	− 1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$	− 7.00000i	− 1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−18.0000	−2.52050
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 18.0000i	− 2.38416i
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	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
$63$	− 12.0000i	− 1.51186i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	0	0
$73$	9.00000i	1.05337i	0.850060	+	0.526685i	$0.176565\pi$
$73$	9.00000i	1.05337i	−0.850060	+	0.526685i	$0.823435\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 14.0000i	− 1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$	− 14.0000i	− 1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 27.0000i	− 2.89470i
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	0	0
$93$	9.00000i	0.933257i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.0000i	− 1.35343i	−0.736245	−	0.676716i	$-0.763403\pi$
$107$	− 14.0000i	− 1.35343i	0.736245	−	0.676716i	$-0.236597\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−24.0000	−2.27798
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	30.0000i	2.77350i
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	9.00000i	0.811503i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 5.00000i	− 0.443678i	−0.975083	−	0.221839i	$-0.928794\pi$
$127$	− 5.00000i	− 0.443678i	0.975083	−	0.221839i	$-0.0712060\pi$
$128$	0	0
$129$	24.0000	2.11308
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.00000i	− 0.341743i	−0.985293	−	0.170872i	$-0.945342\pi$
$137$	− 4.00000i	− 0.341743i	0.985293	−	0.170872i	$-0.0546583\pi$
$138$	0	0
$139$	23.0000	1.95083	0.975417	−	0.220366i	$-0.0707252\pi$
$139$	23.0000	1.95083	0.975417	+	0.220366i	$0.0707252\pi$
$140$	0	0
$141$	21.0000	1.76852
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.00000i	0.742307i
$148$	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
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	0	0
$153$	− 36.0000i	− 2.91043i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 12.0000i	− 0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	− 12.0000i	− 0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	− 11.0000i	− 0.861586i	−0.902451	−	0.430793i	$-0.858234\pi$
$163$	− 11.0000i	− 0.861586i	0.902451	−	0.430793i	$-0.141766\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.00000i	− 0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$	− 4.00000i	− 0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	36.0000	2.75299
$172$	0	0
$173$	− 10.0000i	− 0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$	− 10.0000i	− 0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 12.0000i	− 0.901975i
$178$	0	0
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	− 30.0000i	− 2.21766i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	18.0000	1.30931
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	− 7.00000i	− 0.503871i	−0.967744	−	0.251936i	$-0.918933\pi$
$193$	− 7.00000i	− 0.503871i	0.967744	−	0.251936i	$-0.0810671\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	27.0000i	1.92367i	0.273629	+	0.961835i	$0.411776\pi$
$197$	27.0000i	1.92367i	−0.273629	+	0.961835i	$0.588224\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	24.0000	1.69283
$202$	0	0
$203$	− 18.0000i	− 1.26335i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 6.00000i	− 0.417029i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	21.0000i	1.43890i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.00000i	0.407307i
$218$	0	0
$219$	−27.0000	−1.82449
$220$	0	0
$221$	30.0000	2.01802
$222$	0	0
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.0000i	0.663723i	0.943328	+	0.331862i	$0.107677\pi$
$227$	10.0000i	0.663723i	−0.943328	+	0.331862i	$0.892323\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.0000i	0.982683i	0.870967	+	0.491341i	$0.163493\pi$
$233$	15.0000i	0.982683i	−0.870967	+	0.491341i	$0.836507\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	18.0000i	1.16923i
$238$	0	0
$239$	13.0000	0.840900	0.420450	−	0.907316i	$-0.361872\pi$
$239$	13.0000	0.840900	0.420450	+	0.907316i	$0.361872\pi$
$240$	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
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	30.0000i	1.90885i
$248$	0	0
$249$	42.0000	2.66164
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.0000i	0.810918i	0.914113	+	0.405459i	$0.132888\pi$
$257$	13.0000i	0.810918i	−0.914113	+	0.405459i	$0.867112\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	54.0000	3.34252
$262$	0	0
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 48.0000i	− 2.93755i
$268$	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
$271$	−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
$273$	30.0000i	1.81568i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.0000i	1.38194i	0.722885	+	0.690968i	$0.242815\pi$
$277$	23.0000i	1.38194i	−0.722885	+	0.690968i	$0.757185\pi$
$278$	0	0
$279$	−18.0000	−1.07763
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.00000i	0.354169i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	18.0000	1.05518
$292$	0	0
$293$	12.0000i	0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$	12.0000i	0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.00000	0.289157
$300$	0	0
$301$	16.0000	0.922225
$302$	0	0
$303$	18.0000i	1.03407i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	−42.0000	−2.38930
$310$	0	0
$311$	−17.0000	−0.963982	−0.481991	−	0.876176i	$-0.660086\pi$
$311$	−17.0000	−0.963982	−0.481991	+	0.876176i	$0.660086\pi$
$312$	0	0
$313$	− 20.0000i	− 1.13047i	−0.824931	−	0.565233i	$-0.808786\pi$
$313$	− 20.0000i	− 1.13047i	0.824931	−	0.565233i	$-0.191214\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	42.0000	2.34421
$322$	0	0
$323$	− 36.0000i	− 2.00309i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	14.0000	0.771845
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	0	0
$333$	− 48.0000i	− 2.63038i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8.00000i	− 0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$	− 8.00000i	− 0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	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
$351$	−45.0000	−2.40192
$352$	0	0
$353$	− 11.0000i	− 0.585471i	−0.956193	−	0.292735i	$-0.905434\pi$
$353$	− 11.0000i	− 0.585471i	0.956193	−	0.292735i	$-0.0945655\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 36.0000i	− 1.90532i
$358$	0	0
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	− 33.0000i	− 1.73205i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.0000i	0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$	10.0000i	0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	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
$381$	15.0000	0.768473
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	48.0000i	2.43998i
$388$	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
$391$	−6.00000	−0.303433
$392$	0	0
$393$	− 27.0000i	− 1.36197i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 31.0000i	− 1.55585i	−0.628360	−	0.777923i	$-0.716273\pi$
$397$	− 31.0000i	− 1.55585i	0.628360	−	0.777923i	$-0.283727\pi$
$398$	0	0
$399$	36.0000	1.80225
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	− 15.0000i	− 0.747203i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	69.0000i	3.37894i
$418$	0	0
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	0	0
$423$	42.0000i	2.04211i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.00000i	− 0.287019i
$438$	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
$443$	− 9.00000i	− 0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$	− 9.00000i	− 0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 42.0000i	− 1.98653i
$448$	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
$453$	21.0000i	0.986666i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	0	0
$459$	54.0000	2.52050
$460$	0	0
$461$	1.00000	0.0465746	0.0232873	−	0.999729i	$-0.492587\pi$
$461$	1.00000	0.0465746	0.0232873	+	0.999729i	$0.492587\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	36.0000	1.65879
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.0000i	0.549442i
$478$	0	0
$479$	−38.0000	−1.73626	−0.868132	−	0.496333i	$-0.834679\pi$
$479$	−38.0000	−1.73626	−0.868132	+	0.496333i	$0.834679\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	0	0
$483$	− 6.00000i	− 0.273009i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 17.0000i	− 0.770344i	−0.922845	−	0.385172i	$-0.874142\pi$
$487$	− 17.0000i	− 0.770344i	0.922845	−	0.385172i	$-0.125858\pi$
$488$	0	0
$489$	33.0000	1.49231
$490$	0	0
$491$	−7.00000	−0.315906	−0.157953	−	0.987447i	$-0.550489\pi$
$491$	−7.00000	−0.315906	−0.157953	+	0.987447i	$0.550489\pi$
$492$	0	0
$493$	− 54.0000i	− 2.43204i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.0000i	0.627986i
$498$	0	0
$499$	1.00000	0.0447661	0.0223831	−	0.999749i	$-0.492875\pi$
$499$	1.00000	0.0447661	0.0223831	+	0.999749i	$0.492875\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	10.0000i	0.445878i	0.974832	+	0.222939i	$0.0715651\pi$
$503$	10.0000i	0.445878i	−0.974832	+	0.222939i	$0.928435\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 36.0000i	− 1.59882i
$508$	0	0
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	0	0
$513$	54.0000i	2.38416i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	30.0000	1.31685
$520$	0	0
$521$	8.00000	0.350486	0.175243	−	0.984525i	$-0.443929\pi$
$521$	8.00000	0.350486	0.175243	+	0.984525i	$0.443929\pi$
$522$	0	0
$523$	18.0000i	0.787085i	0.919306	+	0.393543i	$0.128751\pi$
$523$	18.0000i	0.787085i	−0.919306	+	0.393543i	$0.871249\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.0000i	0.784092i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	− 15.0000i	− 0.649722i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.00000i	0.388379i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−43.0000	−1.84871	−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000	−1.84871	−0.924357	+	0.381528i	$0.875398\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.0000i	0.812381i	0.913788	+	0.406191i	$0.133143\pi$
$547$	19.0000i	0.812381i	−0.913788	+	0.406191i	$0.866857\pi$
$548$	0	0
$549$	60.0000	2.56074
$550$	0	0
$551$	54.0000	2.30048
$552$	0	0
$553$	12.0000i	0.510292i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8.00000i	− 0.338971i	−0.985533	−	0.169485i	$-0.945789\pi$
$557$	− 8.00000i	− 0.338971i	0.985533	−	0.169485i	$-0.0542106\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	44.0000i	1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$	44.0000i	1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	18.0000i	0.755929i
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	0	0
$573$	− 48.0000i	− 2.00523i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 33.0000i	− 1.37381i	−0.726748	−	0.686904i	$-0.758969\pi$
$577$	− 33.0000i	− 1.37381i	0.726748	−	0.686904i	$-0.241031\pi$
$578$	0	0
$579$	21.0000	0.872730
$580$	0	0
$581$	28.0000	1.16164
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15.0000i	− 0.619116i	−0.950881	−	0.309558i	$-0.899819\pi$
$587$	− 15.0000i	− 0.619116i	0.950881	−	0.309558i	$-0.100181\pi$
$588$	0	0
$589$	−18.0000	−0.741677
$590$	0	0
$591$	−81.0000	−3.33189
$592$	0	0
$593$	22.0000i	0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$	22.0000i	0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 60.0000i	− 2.45564i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	13.0000	0.530281	0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000	0.530281	0.265141	+	0.964210i	$0.414582\pi$
$602$	0	0
$603$	48.0000i	1.95471i
$604$	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
$611$	−35.0000	−1.41595
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	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
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	9.00000	0.361158
$622$	0	0
$623$	− 32.0000i	− 1.28205i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−48.0000	−1.91389
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	24.0000i	0.953914i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 15.0000i	− 0.594322i
$638$	0	0
$639$	−42.0000	−1.66149
$640$	0	0
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	0	0
$643$	− 44.0000i	− 1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$	− 44.0000i	− 1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.0000i	1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$647$	33.0000i	1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−18.0000	−0.705476
$652$	0	0
$653$	39.0000i	1.52619i	0.646288	+	0.763094i	$0.276321\pi$
$653$	39.0000i	1.52619i	−0.646288	+	0.763094i	$0.723679\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 54.0000i	− 2.10674i
$658$	0	0
$659$	−50.0000	−1.94772	−0.973862	−	0.227142i	$-0.927062\pi$
$659$	−50.0000	−1.94772	−0.973862	+	0.227142i	$0.927062\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	0	0
$663$	90.0000i	3.49531i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9.00000i	− 0.348481i
$668$	0	0
$669$	−24.0000	−0.927894
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 5.00000i	− 0.192736i	−0.995346	−	0.0963679i	$-0.969277\pi$
$673$	− 5.00000i	− 0.192736i	0.995346	−	0.0963679i	$-0.0307225\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	0	0
$681$	−30.0000	−1.14960
$682$	0	0
$683$	29.0000i	1.10965i	0.831966	+	0.554827i	$0.187216\pi$
$683$	29.0000i	1.10965i	−0.831966	+	0.554827i	$0.812784\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	60.0000i	2.28914i
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.0000i	0.681799i
$698$	0	0
$699$	−45.0000	−1.70206
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	− 48.0000i	− 1.81035i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.0000i	0.451306i
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	−36.0000	−1.35011
$712$	0	0
$713$	3.00000i	0.112351i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	39.0000i	1.45648i
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	0	0
$723$	− 12.0000i	− 0.446285i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	36.0000i	1.33517i	0.744535	+	0.667583i	$0.232671\pi$
$727$	36.0000i	1.33517i	−0.744535	+	0.667583i	$0.767329\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	48.0000	1.77534
$732$	0	0
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−29.0000	−1.06678	−0.533391	−	0.845869i	$-0.679083\pi$
$739$	−29.0000	−1.06678	−0.533391	+	0.845869i	$0.679083\pi$
$740$	0	0
$741$	−90.0000	−3.30623
$742$	0	0
$743$	− 6.00000i	− 0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$	− 6.00000i	− 0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	84.0000i	3.07340i
$748$	0	0
$749$	28.0000	1.02310
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	30.0000i	1.09326i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.0000i	0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$757$	12.0000i	0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41.0000	1.48625	0.743124	−	0.669153i	$-0.233343\pi$
$761$	41.0000	1.48625	0.743124	+	0.669153i	$0.233343\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.0000i	0.722158i
$768$	0	0
$769$	28.0000	1.00971	0.504853	−	0.863205i	$-0.331547\pi$
$769$	28.0000	1.00971	0.504853	+	0.863205i	$0.331547\pi$
$770$	0	0
$771$	−39.0000	−1.40455
$772$	0	0
$773$	8.00000i	0.287740i	0.989597	+	0.143870i	$0.0459547\pi$
$773$	8.00000i	0.287740i	−0.989597	+	0.143870i	$0.954045\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 48.0000i	− 1.72199i
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	0	0
$782$	0	0
$783$	81.0000i	2.89470i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 40.0000i	− 1.42585i	−0.701242	−	0.712923i	$-0.747371\pi$
$787$	− 40.0000i	− 1.42585i	0.701242	−	0.712923i	$-0.252629\pi$
$788$	0	0
$789$	−18.0000	−0.640817
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	50.0000i	1.77555i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	42.0000	1.48585
$800$	0	0
$801$	96.0000	3.39199
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 15.0000i	− 0.528025i
$808$	0	0
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	−5.00000	−0.175574	−0.0877869	−	0.996139i	$-0.527979\pi$
$811$	−5.00000	−0.175574	−0.0877869	+	0.996139i	$0.527979\pi$
$812$	0	0
$813$	− 24.0000i	− 0.841717i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	48.0000i	1.67931i
$818$	0	0
$819$	−60.0000	−2.09657
$820$	0	0
$821$	−54.0000	−1.88461	−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000	−1.88461	−0.942306	+	0.334751i	$0.891348\pi$
$822$	0	0
$823$	− 57.0000i	− 1.98690i	−0.114289	−	0.993448i	$-0.536459\pi$
$823$	− 57.0000i	− 1.98690i	0.114289	−	0.993448i	$-0.463541\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−54.0000	−1.87550	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−54.0000	−1.87550	−0.937749	+	0.347314i	$0.887094\pi$
$830$	0	0
$831$	−69.0000	−2.39358
$832$	0	0
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 27.0000i	− 0.933257i
$838$	0	0
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	− 54.0000i	− 1.85986i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 22.0000i	− 0.755929i
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 23.0000i	− 0.785665i	−0.919610	−	0.392833i	$-0.871495\pi$
$857$	− 23.0000i	− 0.785665i	0.919610	−	0.392833i	$-0.128505\pi$
$858$	0	0
$859$	17.0000	0.580033	0.290016	−	0.957022i	$-0.406339\pi$
$859$	17.0000	0.580033	0.290016	+	0.957022i	$0.406339\pi$
$860$	0	0
$861$	−18.0000	−0.613438
$862$	0	0
$863$	49.0000i	1.66798i	0.551780	+	0.833990i	$0.313949\pi$
$863$	49.0000i	1.66798i	−0.551780	+	0.833990i	$0.686051\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 57.0000i	− 1.93582i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−40.0000	−1.35535
$872$	0	0
$873$	36.0000i	1.21842i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	0	0
$879$	−36.0000	−1.21425
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	− 36.0000i	− 1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$	− 36.0000i	− 1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 27.0000i	− 0.906571i	−0.891365	−	0.453286i	$-0.850252\pi$
$887$	− 27.0000i	− 0.906571i	0.891365	−	0.453286i	$-0.149748\pi$
$888$	0	0
$889$	10.0000	0.335389
$890$	0	0
$891$	0	0
$892$	0	0
$893$	42.0000i	1.40548i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	15.0000i	0.500835i
$898$	0	0
$899$	−27.0000	−0.900500
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	48.0000i	1.59734i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	0	0
$909$	−36.0000	−1.19404
$910$	0	0
$911$	−50.0000	−1.65657	−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000	−1.65657	−0.828287	+	0.560304i	$0.810684\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 18.0000i	− 0.594412i
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	− 35.0000i	− 1.15204i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 84.0000i	− 2.75892i
$928$	0	0
$929$	−7.00000	−0.229663	−0.114831	−	0.993385i	$-0.536633\pi$
$929$	−7.00000	−0.229663	−0.114831	+	0.993385i	$0.536633\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	0	0
$933$	− 51.0000i	− 1.66967i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.00000i	− 0.130674i	−0.997863	−	0.0653372i	$-0.979188\pi$
$937$	− 4.00000i	− 0.130674i	0.997863	−	0.0653372i	$-0.0208123\pi$
$938$	0	0
$939$	60.0000	1.95803
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	3.00000i	0.0976934i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.00000i	− 0.227469i	−0.993511	−	0.113735i	$-0.963719\pi$
$947$	− 7.00000i	− 0.227469i	0.993511	−	0.113735i	$-0.0362814\pi$
$948$	0	0
$949$	45.0000	1.46076
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	42.0000i	1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$	42.0000i	1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	84.0000i	2.70686i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	47.0000i	1.51142i	0.654907	+	0.755709i	$0.272708\pi$
$967$	47.0000i	1.51142i	−0.654907	+	0.755709i	$0.727292\pi$
$968$	0	0
$969$	108.000	3.46946
$970$	0	0
$971$	−26.0000	−0.834380	−0.417190	−	0.908819i	$-0.636985\pi$
$971$	−26.0000	−0.834380	−0.417190	+	0.908819i	$0.636985\pi$
$972$	0	0
$973$	46.0000i	1.47469i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 14.0000i	− 0.446531i	−0.974758	−	0.223265i	$-0.928328\pi$
$983$	− 14.0000i	− 0.446531i	0.974758	−	0.223265i	$-0.0716716\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	42.0000i	1.33687i
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	− 57.0000i	− 1.80884i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	50.0000i	1.58352i	0.610835	+	0.791758i	$0.290834\pi$
$997$	50.0000i	1.58352i	−0.610835	+	0.791758i	$0.709166\pi$
$998$	0	0
$999$	72.0000	2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.a.4049.2		2
5.2	odd	4		184.2.a.d.1.1	✓	1
5.3	odd	4		4600.2.a.a.1.1		1
5.4	even	2	inner	4600.2.e.a.4049.1		2
15.2	even	4		1656.2.a.c.1.1		1
20.3	even	4		9200.2.a.bj.1.1		1
20.7	even	4		368.2.a.a.1.1		1
35.27	even	4		9016.2.a.b.1.1		1
40.27	even	4		1472.2.a.m.1.1		1
40.37	odd	4		1472.2.a.a.1.1		1
60.47	odd	4		3312.2.a.i.1.1		1
115.22	even	4		4232.2.a.j.1.1		1
460.367	odd	4		8464.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.a.d.1.1	✓	1	5.2	odd	4
368.2.a.a.1.1		1	20.7	even	4
1472.2.a.a.1.1		1	40.37	odd	4
1472.2.a.m.1.1		1	40.27	even	4
1656.2.a.c.1.1		1	15.2	even	4
3312.2.a.i.1.1		1	60.47	odd	4
4232.2.a.j.1.1		1	115.22	even	4
4600.2.a.a.1.1		1	5.3	odd	4
4600.2.e.a.4049.1		2	5.4	even	2	inner
4600.2.e.a.4049.2		2	1.1	even	1	trivial
8464.2.a.b.1.1		1	460.367	odd	4
9016.2.a.b.1.1		1	35.27	even	4
9200.2.a.bj.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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} -5.00000i q^{13} +6.00000i q^{17} -6.00000 q^{19} -6.00000 q^{21} +1.00000i q^{23} -9.00000i q^{27} -9.00000 q^{29} +3.00000 q^{31} +8.00000i q^{37} +15.0000 q^{39} +3.00000 q^{41} -8.00000i q^{43} -7.00000i q^{47} +3.00000 q^{49} -18.0000 q^{51} -2.00000i q^{53} -18.0000i q^{57} -4.00000 q^{59} -10.0000 q^{61} -12.0000i q^{63} -8.00000i q^{67} -3.00000 q^{69} +7.00000 q^{71} +9.00000i q^{73} +6.00000 q^{79} +9.00000 q^{81} -14.0000i q^{83} -27.0000i q^{87} -16.0000 q^{89} +10.0000 q^{91} +9.00000i q^{93} -6.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9}+O(q^{10}) \) 2 * q - 12 * q^9	\( 2 q - 12 q^{9} - 12 q^{19} - 12 q^{21} - 18 q^{29} + 6 q^{31} + 30 q^{39} + 6 q^{41} + 6 q^{49} - 36 q^{51} - 8 q^{59} - 20 q^{61} - 6 q^{69} + 14 q^{71} + 12 q^{79} + 18 q^{81} - 32 q^{89} + 20 q^{91}+O(q^{100}) \) 2 * q - 12 * q^9 - 12 * q^19 - 12 * q^21 - 18 * q^29 + 6 * q^31 + 30 * q^39 + 6 * q^41 + 6 * q^49 - 36 * q^51 - 8 * q^59 - 20 * q^61 - 6 * q^69 + 14 * q^71 + 12 * q^79 + 18 * q^81 - 32 * q^89 + 20 * q^91

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