LMFDB - Embedded newform 4600.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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, 0, 0]))

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

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

chi := DirichletCharacter("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.a (trivial)

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:	yes
Analytic conductor:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.15529.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 2 $ x^4 - x^3 - 6*x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.774457$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.43375 q^{3} -2.58245 q^{7} +8.79066 q^{9} +O(q^{10})$	$q-3.43375 q^{3} -2.58245 q^{7} +8.79066 q^{9} +1.03354 q^{11} -1.95669 q^{13} +1.54891 q^{17} -2.96646 q^{19} +8.86751 q^{21} -1.00000 q^{23} -19.8837 q^{27} +6.93936 q^{29} -8.05951 q^{31} -3.54891 q^{33} -5.70260 q^{37} +6.71881 q^{39} -2.85130 q^{41} +9.90105 q^{43} -0.625758 q^{47} -0.330937 q^{49} -5.31859 q^{51} +13.7350 q^{53} +10.1861 q^{57} +8.08549 q^{59} +8.80043 q^{61} -22.7015 q^{63} +10.8675 q^{67} +3.43375 q^{69} +2.69283 q^{71} -7.84153 q^{73} -2.66906 q^{77} +3.96812 q^{79} +41.9038 q^{81} +9.53657 q^{83} -23.8281 q^{87} -7.25152 q^{89} +5.05307 q^{91} +27.6744 q^{93} -13.6160 q^{97} +9.08549 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q + q^7 + 6 * q^9	$ 4 q + q^{7} + 6 q^{9} + q^{11} - 3 q^{13} - 2 q^{17} - 15 q^{19} + 8 q^{21} - 4 q^{23} - 27 q^{27} + q^{29} - 12 q^{31} - 6 q^{33} - 18 q^{37} - 3 q^{39} - 9 q^{41} + 9 q^{43} + 4 q^{47} - 3 q^{49} - 2 q^{51} - 6 q^{57} - 5 q^{59} + 14 q^{61} - 39 q^{63} + 16 q^{67} - 2 q^{71} - 21 q^{73} - 9 q^{77} - 21 q^{79} + 44 q^{81} + 9 q^{83} - 17 q^{87} - 16 q^{89} + 33 q^{91} + 29 q^{93} - 40 q^{97} - q^{99}+O(q^{100}) $ 4 * q + q^7 + 6 * q^9 + q^11 - 3 * q^13 - 2 * q^17 - 15 * q^19 + 8 * q^21 - 4 * q^23 - 27 * q^27 + q^29 - 12 * q^31 - 6 * q^33 - 18 * q^37 - 3 * q^39 - 9 * q^41 + 9 * q^43 + 4 * q^47 - 3 * q^49 - 2 * q^51 - 6 * q^57 - 5 * q^59 + 14 * q^61 - 39 * q^63 + 16 * q^67 - 2 * q^71 - 21 * q^73 - 9 * q^77 - 21 * q^79 + 44 * q^81 + 9 * q^83 - 17 * q^87 - 16 * q^89 + 33 * q^91 + 29 * q^93 - 40 * q^97 - q^99

Display 100 coefficients 10 coefficients

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
$3$	−3.43375	−1.98248	−0.991239	−	0.132078i	$-0.957835\pi$
$3$	−3.43375	−1.98248	−0.991239	+	0.132078i	$0.957835\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.58245	−0.976075	−0.488038	−	0.872823i	$-0.662287\pi$
$7$	−2.58245	−0.976075	−0.488038	+	0.872823i	$0.662287\pi$
$8$	0	0
$9$	8.79066	2.93022
$10$	0	0
$11$	1.03354	0.311623	0.155812	−	0.987787i	$-0.450201\pi$
$11$	1.03354	0.311623	0.155812	+	0.987787i	$0.450201\pi$
$12$	0	0
$13$	−1.95669	−0.542689	−0.271345	−	0.962482i	$-0.587468\pi$
$13$	−1.95669	−0.542689	−0.271345	+	0.962482i	$0.587468\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.54891	0.375667	0.187834	−	0.982201i	$-0.439853\pi$
$17$	1.54891	0.375667	0.187834	+	0.982201i	$0.439853\pi$
$18$	0	0
$19$	−2.96646	−0.680553	−0.340277	−	0.940325i	$-0.610521\pi$
$19$	−2.96646	−0.680553	−0.340277	+	0.940325i	$0.610521\pi$
$20$	0	0
$21$	8.86751	1.93505
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−19.8837	−3.82662
$28$	0	0
$29$	6.93936	1.28861	0.644304	−	0.764770i	$-0.277147\pi$
$29$	6.93936	1.28861	0.644304	+	0.764770i	$0.277147\pi$
$30$	0	0
$31$	−8.05951	−1.44753	−0.723766	−	0.690046i	$-0.757590\pi$
$31$	−8.05951	−1.44753	−0.723766	+	0.690046i	$0.757590\pi$
$32$	0	0
$33$	−3.54891	−0.617787
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.70260	−0.937502	−0.468751	−	0.883330i	$-0.655296\pi$
$37$	−5.70260	−0.937502	−0.468751	+	0.883330i	$0.655296\pi$
$38$	0	0
$39$	6.71881	1.07587
$40$	0	0
$41$	−2.85130	−0.445298	−0.222649	−	0.974899i	$-0.571470\pi$
$41$	−2.85130	−0.445298	−0.222649	+	0.974899i	$0.571470\pi$
$42$	0	0
$43$	9.90105	1.50990	0.754948	−	0.655785i	$-0.227662\pi$
$43$	9.90105	1.50990	0.754948	+	0.655785i	$0.227662\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.625758	−0.0912762	−0.0456381	−	0.998958i	$-0.514532\pi$
$47$	−0.625758	−0.0912762	−0.0456381	+	0.998958i	$0.514532\pi$
$48$	0	0
$49$	−0.330937	−0.0472767
$50$	0	0
$51$	−5.31859	−0.744752
$52$	0	0
$53$	13.7350	1.88665	0.943325	−	0.331871i	$-0.107680\pi$
$53$	13.7350	1.88665	0.943325	+	0.331871i	$0.107680\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10.1861	1.34918
$58$	0	0
$59$	8.08549	1.05264	0.526320	−	0.850286i	$-0.323571\pi$
$59$	8.08549	1.05264	0.526320	+	0.850286i	$0.323571\pi$
$60$	0	0
$61$	8.80043	1.12678	0.563390	−	0.826191i	$-0.309497\pi$
$61$	8.80043	1.12678	0.563390	+	0.826191i	$0.309497\pi$
$62$	0	0
$63$	−22.7015	−2.86012
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.8675	1.32768	0.663839	−	0.747876i	$-0.268926\pi$
$67$	10.8675	1.32768	0.663839	+	0.747876i	$0.268926\pi$
$68$	0	0
$69$	3.43375	0.413375
$70$	0	0
$71$	2.69283	0.319581	0.159790	−	0.987151i	$-0.448918\pi$
$71$	2.69283	0.319581	0.159790	+	0.987151i	$0.448918\pi$
$72$	0	0
$73$	−7.84153	−0.917782	−0.458891	−	0.888493i	$-0.651753\pi$
$73$	−7.84153	−0.917782	−0.458891	+	0.888493i	$0.651753\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.66906	−0.304168
$78$	0	0
$79$	3.96812	0.446449	0.223224	−	0.974767i	$-0.428342\pi$
$79$	3.96812	0.446449	0.223224	+	0.974767i	$0.428342\pi$
$80$	0	0
$81$	41.9038	4.65598
$82$	0	0
$83$	9.53657	1.04677	0.523387	−	0.852095i	$-0.324668\pi$
$83$	9.53657	1.04677	0.523387	+	0.852095i	$0.324668\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−23.8281	−2.55464
$88$	0	0
$89$	−7.25152	−0.768659	−0.384330	−	0.923196i	$-0.625567\pi$
$89$	−7.25152	−0.768659	−0.384330	+	0.923196i	$0.625567\pi$
$90$	0	0
$91$	5.05307	0.529706
$92$	0	0
$93$	27.6744	2.86970
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.6160	−1.38249	−0.691247	−	0.722618i	$-0.742938\pi$
$97$	−13.6160	−1.38249	−0.691247	+	0.722618i	$0.742938\pi$
$98$	0	0
$99$	9.08549	0.913126
$100$	0	0
$101$	−6.90105	−0.686680	−0.343340	−	0.939211i	$-0.611558\pi$
$101$	−6.90105	−0.686680	−0.343340	+	0.939211i	$0.611558\pi$
$102$	0	0
$103$	4.13137	0.407076	0.203538	−	0.979067i	$-0.434756\pi$
$103$	4.13137	0.407076	0.203538	+	0.979067i	$0.434756\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−8.26274	−0.791427	−0.395713	−	0.918374i	$-0.629503\pi$
$109$	−8.26274	−0.791427	−0.395713	+	0.918374i	$0.629503\pi$
$110$	0	0
$111$	19.5813	1.85858
$112$	0	0
$113$	12.1861	1.14637	0.573186	−	0.819425i	$-0.305707\pi$
$113$	12.1861	1.14637	0.573186	+	0.819425i	$0.305707\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−17.2006	−1.59020
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−9.93180	−0.902891
$122$	0	0
$123$	9.79066	0.882794
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.05951	−0.715166	−0.357583	−	0.933881i	$-0.616399\pi$
$127$	−8.05951	−0.715166	−0.357583	+	0.933881i	$0.616399\pi$
$128$	0	0
$129$	−33.9978	−2.99334
$130$	0	0
$131$	0.796569	0.0695966	0.0347983	−	0.999394i	$-0.488921\pi$
$131$	0.796569	0.0695966	0.0347983	+	0.999394i	$0.488921\pi$
$132$	0	0
$133$	7.66075	0.664271
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.16491	0.0995246	0.0497623	−	0.998761i	$-0.484154\pi$
$137$	1.16491	0.0995246	0.0497623	+	0.998761i	$0.484154\pi$
$138$	0	0
$139$	−12.6258	−1.07090	−0.535451	−	0.844566i	$-0.679859\pi$
$139$	−12.6258	−1.07090	−0.535451	+	0.844566i	$0.679859\pi$
$140$	0	0
$141$	2.14870	0.180953
$142$	0	0
$143$	−2.02232	−0.169115
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.13635	0.0937250
$148$	0	0
$149$	11.0783	0.907569	0.453785	−	0.891111i	$-0.350074\pi$
$149$	11.0783	0.907569	0.453785	+	0.891111i	$0.350074\pi$
$150$	0	0
$151$	−7.63698	−0.621488	−0.310744	−	0.950494i	$-0.600578\pi$
$151$	−7.63698	−0.621488	−0.310744	+	0.950494i	$0.600578\pi$
$152$	0	0
$153$	13.6160	1.10079
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.50303	0.518999	0.259499	−	0.965743i	$-0.416442\pi$
$157$	6.50303	0.518999	0.259499	+	0.965743i	$0.416442\pi$
$158$	0	0
$159$	−47.1627	−3.74024
$160$	0	0
$161$	2.58245	0.203526
$162$	0	0
$163$	−24.4564	−1.91557	−0.957787	−	0.287480i	$-0.907182\pi$
$163$	−24.4564	−1.91557	−0.957787	+	0.287480i	$0.907182\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.74736	−0.676891	−0.338445	−	0.940986i	$-0.609901\pi$
$167$	−8.74736	−0.676891	−0.338445	+	0.940986i	$0.609901\pi$
$168$	0	0
$169$	−9.17135	−0.705488
$170$	0	0
$171$	−26.0772	−1.99417
$172$	0	0
$173$	19.5255	1.48449	0.742247	−	0.670126i	$-0.233760\pi$
$173$	19.5255	1.48449	0.742247	+	0.670126i	$0.233760\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−27.7636	−2.08684
$178$	0	0
$179$	1.81922	0.135975	0.0679873	−	0.997686i	$-0.478342\pi$
$179$	1.81922	0.135975	0.0679873	+	0.997686i	$0.478342\pi$
$180$	0	0
$181$	22.8675	1.69973	0.849864	−	0.527002i	$-0.176684\pi$
$181$	22.8675	1.69973	0.849864	+	0.527002i	$0.176684\pi$
$182$	0	0
$183$	−30.2185	−2.23382
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.60086	0.117067
$188$	0	0
$189$	51.3488	3.73507
$190$	0	0
$191$	26.7490	1.93549	0.967746	−	0.251930i	$-0.0810652\pi$
$191$	26.7490	1.93549	0.967746	+	0.251930i	$0.0810652\pi$
$192$	0	0
$193$	20.5220	1.47721	0.738604	−	0.674140i	$-0.235485\pi$
$193$	20.5220	1.47721	0.738604	+	0.674140i	$0.235485\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.3153	−1.23366	−0.616831	−	0.787096i	$-0.711584\pi$
$197$	−17.3153	−1.23366	−0.616831	+	0.787096i	$0.711584\pi$
$198$	0	0
$199$	0.361691	0.0256396	0.0128198	−	0.999918i	$-0.495919\pi$
$199$	0.361691	0.0256396	0.0128198	+	0.999918i	$0.495919\pi$
$200$	0	0
$201$	−37.3163	−2.63209
$202$	0	0
$203$	−17.9206	−1.25778
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.79066	−0.610993
$208$	0	0
$209$	−3.06595	−0.212076
$210$	0	0
$211$	−23.6919	−1.63102	−0.815509	−	0.578744i	$-0.803543\pi$
$211$	−23.6919	−1.63102	−0.815509	+	0.578744i	$0.803543\pi$
$212$	0	0
$213$	−9.24653	−0.633562
$214$	0	0
$215$	0	0
$216$	0	0
$217$	20.8133	1.41290
$218$	0	0
$219$	26.9259	1.81948
$220$	0	0
$221$	−3.03075	−0.203871
$222$	0	0
$223$	14.1526	0.947726	0.473863	−	0.880599i	$-0.342859\pi$
$223$	14.1526	0.947726	0.473863	+	0.880599i	$0.342859\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.8021	0.916077	0.458038	−	0.888932i	$-0.348552\pi$
$227$	13.8021	0.916077	0.458038	+	0.888932i	$0.348552\pi$
$228$	0	0
$229$	−7.43762	−0.491491	−0.245746	−	0.969334i	$-0.579033\pi$
$229$	−7.43762	−0.491491	−0.245746	+	0.969334i	$0.579033\pi$
$230$	0	0
$231$	9.16491	0.603007
$232$	0	0
$233$	−19.3843	−1.26991	−0.634955	−	0.772549i	$-0.718981\pi$
$233$	−19.3843	−1.26991	−0.634955	+	0.772549i	$0.718981\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−13.6256	−0.885075
$238$	0	0
$239$	14.7943	0.956965	0.478482	−	0.878097i	$-0.341187\pi$
$239$	14.7943	0.956965	0.478482	+	0.878097i	$0.341187\pi$
$240$	0	0
$241$	−7.58133	−0.488356	−0.244178	−	0.969730i	$-0.578518\pi$
$241$	−7.58133	−0.488356	−0.244178	+	0.969730i	$0.578518\pi$
$242$	0	0
$243$	−84.2361	−5.40375
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.80446	0.369329
$248$	0	0
$249$	−32.7462	−2.07521
$250$	0	0
$251$	−23.3359	−1.47295	−0.736474	−	0.676466i	$-0.763511\pi$
$251$	−23.3359	−1.47295	−0.736474	+	0.676466i	$0.763511\pi$
$252$	0	0
$253$	−1.03354	−0.0649780
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.0233966	0.00145944	0.000729720	−	1.00000i	$-0.499768\pi$
$257$	0.0233966	0.00145944	0.000729720		1.00000i	$0.499768\pi$
$258$	0	0
$259$	14.7267	0.915073
$260$	0	0
$261$	61.0016	3.77591
$262$	0	0
$263$	−26.6792	−1.64511	−0.822554	−	0.568687i	$-0.807451\pi$
$263$	−26.6792	−1.64511	−0.822554	+	0.568687i	$0.807451\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	24.8999	1.52385
$268$	0	0
$269$	−1.49605	−0.0912158	−0.0456079	−	0.998959i	$-0.514522\pi$
$269$	−1.49605	−0.0912158	−0.0456079	+	0.998959i	$0.514522\pi$
$270$	0	0
$271$	2.01841	0.122610	0.0613048	−	0.998119i	$-0.480474\pi$
$271$	2.01841	0.122610	0.0613048	+	0.998119i	$0.480474\pi$
$272$	0	0
$273$	−17.3510	−1.05013
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.74956	0.405542	0.202771	−	0.979226i	$-0.435005\pi$
$277$	6.74956	0.405542	0.202771	+	0.979226i	$0.435005\pi$
$278$	0	0
$279$	−70.8485	−4.24159
$280$	0	0
$281$	−13.0212	−0.776779	−0.388390	−	0.921495i	$-0.626969\pi$
$281$	−13.0212	−0.776779	−0.388390	+	0.921495i	$0.626969\pi$
$282$	0	0
$283$	−33.0860	−1.96676	−0.983380	−	0.181560i	$-0.941885\pi$
$283$	−33.0860	−1.96676	−0.983380	+	0.181560i	$0.941885\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.36335	0.434645
$288$	0	0
$289$	−14.6009	−0.858874
$290$	0	0
$291$	46.7540	2.74077
$292$	0	0
$293$	−20.0519	−1.17145	−0.585724	−	0.810511i	$-0.699190\pi$
$293$	−20.0519	−1.17145	−0.585724	+	0.810511i	$0.699190\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−20.5506	−1.19247
$298$	0	0
$299$	1.95669	0.113159
$300$	0	0
$301$	−25.5690	−1.47377
$302$	0	0
$303$	23.6965	1.36133
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−14.1861	−0.807019
$310$	0	0
$311$	−12.3669	−0.701262	−0.350631	−	0.936514i	$-0.614033\pi$
$311$	−12.3669	−0.701262	−0.350631	+	0.936514i	$0.614033\pi$
$312$	0	0
$313$	4.24986	0.240216	0.120108	−	0.992761i	$-0.461676\pi$
$313$	4.24986	0.240216	0.120108	+	0.992761i	$0.461676\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−34.0720	−1.91368	−0.956838	−	0.290623i	$-0.906137\pi$
$317$	−34.0720	−1.91368	−0.956838	+	0.290623i	$0.906137\pi$
$318$	0	0
$319$	7.17210	0.401560
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.59480	−0.255661
$324$	0	0
$325$	0	0
$326$	0	0
$327$	28.3722	1.56899
$328$	0	0
$329$	1.61599	0.0890925
$330$	0	0
$331$	−26.5964	−1.46187	−0.730935	−	0.682447i	$-0.760916\pi$
$331$	−26.5964	−1.46187	−0.730935	+	0.682447i	$0.760916\pi$
$332$	0	0
$333$	−50.1297	−2.74709
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.4376	−1.60357	−0.801785	−	0.597613i	$-0.796116\pi$
$337$	−29.4376	−1.60357	−0.801785	+	0.597613i	$0.796116\pi$
$338$	0	0
$339$	−41.8441	−2.27266
$340$	0	0
$341$	−8.32981	−0.451085
$342$	0	0
$343$	18.9318	1.02222
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.8535	−0.582647	−0.291323	−	0.956625i	$-0.594096\pi$
$347$	−10.8535	−0.582647	−0.291323	+	0.956625i	$0.594096\pi$
$348$	0	0
$349$	20.5463	1.09982	0.549910	−	0.835224i	$-0.314662\pi$
$349$	20.5463	1.09982	0.549910	+	0.835224i	$0.314662\pi$
$350$	0	0
$351$	38.9064	2.07667
$352$	0	0
$353$	−21.3543	−1.13658	−0.568288	−	0.822830i	$-0.692394\pi$
$353$	−21.3543	−1.13658	−0.568288	+	0.822830i	$0.692394\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	13.7350	0.726934
$358$	0	0
$359$	−2.23751	−0.118091	−0.0590457	−	0.998255i	$-0.518806\pi$
$359$	−2.23751	−0.118091	−0.0590457	+	0.998255i	$0.518806\pi$
$360$	0	0
$361$	−10.2001	−0.536848
$362$	0	0
$363$	34.1034	1.78996
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.90711	−0.464947	−0.232474	−	0.972603i	$-0.574682\pi$
$367$	−8.90711	−0.464947	−0.232474	+	0.972603i	$0.574682\pi$
$368$	0	0
$369$	−25.0648	−1.30482
$370$	0	0
$371$	−35.4700	−1.84151
$372$	0	0
$373$	33.7870	1.74942	0.874711	−	0.484644i	$-0.161051\pi$
$373$	33.7870	1.74942	0.874711	+	0.484644i	$0.161051\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.5782	−0.699314
$378$	0	0
$379$	−12.5948	−0.646951	−0.323476	−	0.946236i	$-0.604851\pi$
$379$	−12.5948	−0.646951	−0.323476	+	0.946236i	$0.604851\pi$
$380$	0	0
$381$	27.6744	1.41780
$382$	0	0
$383$	8.11183	0.414495	0.207248	−	0.978289i	$-0.433549\pi$
$383$	8.11183	0.414495	0.207248	+	0.978289i	$0.433549\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	87.0368	4.42433
$388$	0	0
$389$	−26.0844	−1.32253	−0.661265	−	0.750153i	$-0.729980\pi$
$389$	−26.0844	−1.32253	−0.661265	+	0.750153i	$0.729980\pi$
$390$	0	0
$391$	−1.54891	−0.0783320
$392$	0	0
$393$	−2.73522	−0.137974
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−27.9231	−1.40142	−0.700710	−	0.713446i	$-0.747133\pi$
$397$	−27.9231	−1.40142	−0.700710	+	0.713446i	$0.747133\pi$
$398$	0	0
$399$	−26.3051	−1.31690
$400$	0	0
$401$	7.83119	0.391071	0.195535	−	0.980697i	$-0.437356\pi$
$401$	7.83119	0.391071	0.195535	+	0.980697i	$0.437356\pi$
$402$	0	0
$403$	15.7700	0.785560
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.89386	−0.292148
$408$	0	0
$409$	−17.6941	−0.874918	−0.437459	−	0.899238i	$-0.644121\pi$
$409$	−17.6941	−0.874918	−0.437459	+	0.899238i	$0.644121\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	−20.8804	−1.02746
$414$	0	0
$415$	0	0
$416$	0	0
$417$	43.3537	2.12304
$418$	0	0
$419$	23.3387	1.14017	0.570084	−	0.821586i	$-0.306911\pi$
$419$	23.3387	1.14017	0.570084	+	0.821586i	$0.306911\pi$
$420$	0	0
$421$	3.86144	0.188195	0.0940976	−	0.995563i	$-0.470003\pi$
$421$	3.86144	0.188195	0.0940976	+	0.995563i	$0.470003\pi$
$422$	0	0
$423$	−5.50083	−0.267459
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−22.7267	−1.09982
$428$	0	0
$429$	6.94414	0.335266
$430$	0	0
$431$	6.80043	0.327565	0.163783	−	0.986496i	$-0.447630\pi$
$431$	6.80043	0.327565	0.163783	+	0.986496i	$0.447630\pi$
$432$	0	0
$433$	−19.5618	−0.940080	−0.470040	−	0.882645i	$-0.655760\pi$
$433$	−19.5618	−0.940080	−0.470040	+	0.882645i	$0.655760\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.96646	0.141905
$438$	0	0
$439$	−19.1017	−0.911674	−0.455837	−	0.890063i	$-0.650660\pi$
$439$	−19.1017	−0.911674	−0.455837	+	0.890063i	$0.650660\pi$
$440$	0	0
$441$	−2.90915	−0.138531
$442$	0	0
$443$	30.7996	1.46333	0.731667	−	0.681662i	$-0.238742\pi$
$443$	30.7996	1.46333	0.731667	+	0.681662i	$0.238742\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−38.0401	−1.79924
$448$	0	0
$449$	30.0660	1.41890	0.709450	−	0.704756i	$-0.248943\pi$
$449$	30.0660	1.41890	0.709450	+	0.704756i	$0.248943\pi$
$450$	0	0
$451$	−2.94693	−0.138765
$452$	0	0
$453$	26.2235	1.23209
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.0860	1.36059	0.680293	−	0.732940i	$-0.261852\pi$
$457$	29.0860	1.36059	0.680293	+	0.732940i	$0.261852\pi$
$458$	0	0
$459$	−30.7982	−1.43754
$460$	0	0
$461$	2.50395	0.116621	0.0583103	−	0.998299i	$-0.481429\pi$
$461$	2.50395	0.116621	0.0583103	+	0.998299i	$0.481429\pi$
$462$	0	0
$463$	3.85076	0.178960	0.0894799	−	0.995989i	$-0.471480\pi$
$463$	3.85076	0.178960	0.0894799	+	0.995989i	$0.471480\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.24433	−0.242678	−0.121339	−	0.992611i	$-0.538719\pi$
$467$	−5.24433	−0.242678	−0.121339	+	0.992611i	$0.538719\pi$
$468$	0	0
$469$	−28.0648	−1.29591
$470$	0	0
$471$	−22.3298	−1.02890
$472$	0	0
$473$	10.2331	0.470519
$474$	0	0
$475$	0	0
$476$	0	0
$477$	120.740	5.52830
$478$	0	0
$479$	−13.7893	−0.630051	−0.315025	−	0.949083i	$-0.602013\pi$
$479$	−13.7893	−0.630051	−0.315025	+	0.949083i	$0.602013\pi$
$480$	0	0
$481$	11.1583	0.508772
$482$	0	0
$483$	−8.86751	−0.403486
$484$	0	0
$485$	0	0
$486$	0	0
$487$	24.6906	1.11884	0.559419	−	0.828885i	$-0.311024\pi$
$487$	24.6906	1.11884	0.559419	+	0.828885i	$0.311024\pi$
$488$	0	0
$489$	83.9773	3.79758
$490$	0	0
$491$	−23.3915	−1.05564	−0.527822	−	0.849355i	$-0.676991\pi$
$491$	−23.3915	−1.05564	−0.527822	+	0.849355i	$0.676991\pi$
$492$	0	0
$493$	10.7485	0.484087
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.95412	−0.311935
$498$	0	0
$499$	33.1241	1.48284	0.741420	−	0.671041i	$-0.234153\pi$
$499$	33.1241	1.48284	0.741420	+	0.671041i	$0.234153\pi$
$500$	0	0
$501$	30.0363	1.34192
$502$	0	0
$503$	21.8641	0.974874	0.487437	−	0.873158i	$-0.337932\pi$
$503$	21.8641	0.974874	0.487437	+	0.873158i	$0.337932\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	31.4921	1.39862
$508$	0	0
$509$	19.5628	0.867104	0.433552	−	0.901129i	$-0.357260\pi$
$509$	19.5628	0.867104	0.433552	+	0.901129i	$0.357260\pi$
$510$	0	0
$511$	20.2504	0.895825
$512$	0	0
$513$	58.9843	2.60422
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.646745	−0.0284438
$518$	0	0
$519$	−67.0457	−2.94298
$520$	0	0
$521$	−13.1800	−0.577428	−0.288714	−	0.957415i	$-0.593228\pi$
$521$	−13.1800	−0.577428	−0.288714	+	0.957415i	$0.593228\pi$
$522$	0	0
$523$	10.4041	0.454939	0.227469	−	0.973785i	$-0.426955\pi$
$523$	10.4041	0.454939	0.227469	+	0.973785i	$0.426955\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.4835	−0.543790
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	71.0768	3.08447
$532$	0	0
$533$	5.57913	0.241659
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6.24674	−0.269567
$538$	0	0
$539$	−0.342036	−0.0147325
$540$	0	0
$541$	18.3707	0.789818	0.394909	−	0.918720i	$-0.370776\pi$
$541$	18.3707	0.789818	0.394909	+	0.918720i	$0.370776\pi$
$542$	0	0
$543$	−78.5214	−3.36968
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13.9886	0.598108	0.299054	−	0.954236i	$-0.403329\pi$
$547$	13.9886	0.598108	0.299054	+	0.954236i	$0.403329\pi$
$548$	0	0
$549$	77.3616	3.30171
$550$	0	0
$551$	−20.5854	−0.876966
$552$	0	0
$553$	−10.2475	−0.435767
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−38.0800	−1.61350	−0.806750	−	0.590893i	$-0.798775\pi$
$557$	−38.0800	−1.61350	−0.806750	+	0.590893i	$0.798775\pi$
$558$	0	0
$559$	−19.3733	−0.819404
$560$	0	0
$561$	−5.49697	−0.232082
$562$	0	0
$563$	44.0473	1.85637	0.928187	−	0.372115i	$-0.121367\pi$
$563$	44.0473	1.85637	0.928187	+	0.372115i	$0.121367\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−108.215	−4.54458
$568$	0	0
$569$	−24.7116	−1.03596	−0.517981	−	0.855392i	$-0.673316\pi$
$569$	−24.7116	−1.03596	−0.517981	+	0.855392i	$0.673316\pi$
$570$	0	0
$571$	−30.3247	−1.26905	−0.634524	−	0.772903i	$-0.718804\pi$
$571$	−30.3247	−1.26905	−0.634524	+	0.772903i	$0.718804\pi$
$572$	0	0
$573$	−91.8495	−3.83707
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.72047	−0.196516	−0.0982578	−	0.995161i	$-0.531327\pi$
$577$	−4.72047	−0.196516	−0.0982578	+	0.995161i	$0.531327\pi$
$578$	0	0
$579$	−70.4676	−2.92853
$580$	0	0
$581$	−24.6277	−1.02173
$582$	0	0
$583$	14.1957	0.587924
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.5224	−0.558128	−0.279064	−	0.960273i	$-0.590024\pi$
$587$	−13.5224	−0.558128	−0.279064	+	0.960273i	$0.590024\pi$
$588$	0	0
$589$	23.9082	0.985122
$590$	0	0
$591$	59.4564	2.44571
$592$	0	0
$593$	0.827904	0.0339979	0.0169990	−	0.999856i	$-0.494589\pi$
$593$	0.827904	0.0339979	0.0169990	+	0.999856i	$0.494589\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.24196	−0.0508299
$598$	0	0
$599$	11.2526	0.459770	0.229885	−	0.973218i	$-0.426165\pi$
$599$	11.2526	0.459770	0.229885	+	0.973218i	$0.426165\pi$
$600$	0	0
$601$	14.3764	0.586427	0.293214	−	0.956047i	$-0.405275\pi$
$601$	14.3764	0.586427	0.293214	+	0.956047i	$0.405275\pi$
$602$	0	0
$603$	95.5326	3.89039
$604$	0	0
$605$	0	0
$606$	0	0
$607$	27.6432	1.12200	0.561002	−	0.827814i	$-0.310416\pi$
$607$	27.6432	1.12200	0.561002	+	0.827814i	$0.310416\pi$
$608$	0	0
$609$	61.5349	2.49352
$610$	0	0
$611$	1.22442	0.0495346
$612$	0	0
$613$	−21.4030	−0.864457	−0.432229	−	0.901764i	$-0.642273\pi$
$613$	−21.4030	−0.864457	−0.432229	+	0.901764i	$0.642273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−39.5471	−1.59211	−0.796053	−	0.605227i	$-0.793082\pi$
$617$	−39.5471	−1.59211	−0.796053	+	0.605227i	$0.793082\pi$
$618$	0	0
$619$	−23.3410	−0.938155	−0.469078	−	0.883157i	$-0.655414\pi$
$619$	−23.3410	−0.938155	−0.469078	+	0.883157i	$0.655414\pi$
$620$	0	0
$621$	19.8837	0.797906
$622$	0	0
$623$	18.7267	0.750269
$624$	0	0
$625$	0	0
$626$	0	0
$627$	10.5277	0.420437
$628$	0	0
$629$	−8.83284	−0.352189
$630$	0	0
$631$	−49.5439	−1.97231	−0.986155	−	0.165824i	$-0.946972\pi$
$631$	−49.5439	−1.97231	−0.986155	+	0.165824i	$0.946972\pi$
$632$	0	0
$633$	81.3522	3.23346
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.647542	0.0256566
$638$	0	0
$639$	23.6718	0.936442
$640$	0	0
$641$	−29.5566	−1.16742	−0.583709	−	0.811963i	$-0.698399\pi$
$641$	−29.5566	−1.16742	−0.583709	+	0.811963i	$0.698399\pi$
$642$	0	0
$643$	14.3274	0.565019	0.282510	−	0.959264i	$-0.408833\pi$
$643$	14.3274	0.565019	0.282510	+	0.959264i	$0.408833\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.16566	0.242397	0.121198	−	0.992628i	$-0.461326\pi$
$647$	6.16566	0.242397	0.121198	+	0.992628i	$0.461326\pi$
$648$	0	0
$649$	8.35666	0.328027
$650$	0	0
$651$	−71.4678	−2.80104
$652$	0	0
$653$	−39.4553	−1.54400	−0.772002	−	0.635620i	$-0.780745\pi$
$653$	−39.4553	−1.54400	−0.772002	+	0.635620i	$0.780745\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−68.9323	−2.68930
$658$	0	0
$659$	22.9123	0.892535	0.446268	−	0.894900i	$-0.352753\pi$
$659$	22.9123	0.892535	0.446268	+	0.894900i	$0.352753\pi$
$660$	0	0
$661$	18.1537	0.706097	0.353048	−	0.935605i	$-0.385145\pi$
$661$	18.1537	0.706097	0.353048	+	0.935605i	$0.385145\pi$
$662$	0	0
$663$	10.4069	0.404169
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.93936	−0.268693
$668$	0	0
$669$	−48.5964	−1.87885
$670$	0	0
$671$	9.09558	0.351131
$672$	0	0
$673$	−12.3807	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$673$	−12.3807	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.2146	1.43027	0.715137	−	0.698985i	$-0.246364\pi$
$677$	37.2146	1.43027	0.715137	+	0.698985i	$0.246364\pi$
$678$	0	0
$679$	35.1627	1.34942
$680$	0	0
$681$	−47.3930	−1.81610
$682$	0	0
$683$	−23.2915	−0.891224	−0.445612	−	0.895226i	$-0.647014\pi$
$683$	−23.2915	−0.891224	−0.445612	+	0.895226i	$0.647014\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	25.5389	0.974371
$688$	0	0
$689$	−26.8752	−1.02386
$690$	0	0
$691$	−25.0772	−0.953981	−0.476990	−	0.878909i	$-0.658272\pi$
$691$	−25.0772	−0.953981	−0.476990	+	0.878909i	$0.658272\pi$
$692$	0	0
$693$	−23.4628	−0.891280
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.41642	−0.167284
$698$	0	0
$699$	66.5610	2.51757
$700$	0	0
$701$	−37.1180	−1.40193	−0.700964	−	0.713197i	$-0.747247\pi$
$701$	−37.1180	−1.40193	−0.700964	+	0.713197i	$0.747247\pi$
$702$	0	0
$703$	16.9166	0.638020
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.8216	0.670251
$708$	0	0
$709$	−11.6731	−0.438392	−0.219196	−	0.975681i	$-0.570343\pi$
$709$	−11.6731	−0.438392	−0.219196	+	0.975681i	$0.570343\pi$
$710$	0	0
$711$	34.8824	1.30819
$712$	0	0
$713$	8.05951	0.301831
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−50.8000	−1.89716
$718$	0	0
$719$	20.4360	0.762133	0.381066	−	0.924548i	$-0.375557\pi$
$719$	20.4360	0.762133	0.381066	+	0.924548i	$0.375557\pi$
$720$	0	0
$721$	−10.6691	−0.397337
$722$	0	0
$723$	26.0324	0.968156
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−47.6211	−1.76617	−0.883084	−	0.469216i	$-0.844537\pi$
$727$	−47.6211	−1.76617	−0.883084	+	0.469216i	$0.844537\pi$
$728$	0	0
$729$	163.535	6.05684
$730$	0	0
$731$	15.3359	0.567218
$732$	0	0
$733$	26.8609	0.992128	0.496064	−	0.868286i	$-0.334778\pi$
$733$	26.8609	0.992128	0.496064	+	0.868286i	$0.334778\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.2320	0.413735
$738$	0	0
$739$	−20.9075	−0.769094	−0.384547	−	0.923105i	$-0.625642\pi$
$739$	−20.9075	−0.769094	−0.384547	+	0.923105i	$0.625642\pi$
$740$	0	0
$741$	−19.9311	−0.732187
$742$	0	0
$743$	−3.30143	−0.121118	−0.0605588	−	0.998165i	$-0.519288\pi$
$743$	−3.30143	−0.121118	−0.0605588	+	0.998165i	$0.519288\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	83.8328	3.06728
$748$	0	0
$749$	0	0
$750$	0	0
$751$	6.19619	0.226102	0.113051	−	0.993589i	$-0.463938\pi$
$751$	6.19619	0.226102	0.113051	+	0.993589i	$0.463938\pi$
$752$	0	0
$753$	80.1297	2.92009
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.39007	−0.0505231	−0.0252616	−	0.999681i	$-0.508042\pi$
$757$	−1.39007	−0.0505231	−0.0252616	+	0.999681i	$0.508042\pi$
$758$	0	0
$759$	3.54891	0.128817
$760$	0	0
$761$	46.7816	1.69583	0.847916	−	0.530131i	$-0.177857\pi$
$761$	46.7816	1.69583	0.847916	+	0.530131i	$0.177857\pi$
$762$	0	0
$763$	21.3381	0.772492
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15.8208	−0.571257
$768$	0	0
$769$	−45.3878	−1.63673	−0.818363	−	0.574701i	$-0.805118\pi$
$769$	−45.3878	−1.63673	−0.818363	+	0.574701i	$0.805118\pi$
$770$	0	0
$771$	−0.0803382	−0.00289331
$772$	0	0
$773$	−27.1454	−0.976351	−0.488176	−	0.872745i	$-0.662337\pi$
$773$	−27.1454	−0.976351	−0.488176	+	0.872745i	$0.662337\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−50.5679	−1.81411
$778$	0	0
$779$	8.45828	0.303049
$780$	0	0
$781$	2.78315	0.0995888
$782$	0	0
$783$	−137.980	−4.93101
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7.17168	−0.255643	−0.127821	−	0.991797i	$-0.540798\pi$
$787$	−7.17168	−0.255643	−0.127821	+	0.991797i	$0.540798\pi$
$788$	0	0
$789$	91.6097	3.26139
$790$	0	0
$791$	−31.4700	−1.11895
$792$	0	0
$793$	−17.2198	−0.611492
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−52.8859	−1.87331	−0.936657	−	0.350249i	$-0.886097\pi$
$797$	−52.8859	−1.87331	−0.936657	+	0.350249i	$0.886097\pi$
$798$	0	0
$799$	−0.969246	−0.0342895
$800$	0	0
$801$	−63.7456	−2.25234
$802$	0	0
$803$	−8.10452	−0.286002
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.13707	0.180833
$808$	0	0
$809$	11.7411	0.412794	0.206397	−	0.978468i	$-0.433826\pi$
$809$	11.7411	0.412794	0.206397	+	0.978468i	$0.433826\pi$
$810$	0	0
$811$	49.1122	1.72456	0.862281	−	0.506431i	$-0.169035\pi$
$811$	49.1122	1.72456	0.862281	+	0.506431i	$0.169035\pi$
$812$	0	0
$813$	−6.93072	−0.243071
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−29.3711	−1.02756
$818$	0	0
$819$	44.4199	1.55216
$820$	0	0
$821$	−27.1263	−0.946716	−0.473358	−	0.880870i	$-0.656958\pi$
$821$	−27.1263	−0.946716	−0.473358	+	0.880870i	$0.656958\pi$
$822$	0	0
$823$	−4.00386	−0.139566	−0.0697829	−	0.997562i	$-0.522231\pi$
$823$	−4.00386	−0.139566	−0.0697829	+	0.997562i	$0.522231\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.9200	−0.657912	−0.328956	−	0.944345i	$-0.606697\pi$
$827$	−18.9200	−0.657912	−0.328956	+	0.944345i	$0.606697\pi$
$828$	0	0
$829$	−5.02244	−0.174437	−0.0872183	−	0.996189i	$-0.527798\pi$
$829$	−5.02244	−0.174437	−0.0872183	+	0.996189i	$0.527798\pi$
$830$	0	0
$831$	−23.1763	−0.803978
$832$	0	0
$833$	−0.512593	−0.0177603
$834$	0	0
$835$	0	0
$836$	0	0
$837$	160.253	5.53915
$838$	0	0
$839$	−19.1621	−0.661550	−0.330775	−	0.943710i	$-0.607310\pi$
$839$	−19.1621	−0.661550	−0.330775	+	0.943710i	$0.607310\pi$
$840$	0	0
$841$	19.1548	0.660509
$842$	0	0
$843$	44.7116	1.53995
$844$	0	0
$845$	0	0
$846$	0	0
$847$	25.6484	0.881290
$848$	0	0
$849$	113.609	3.89906
$850$	0	0
$851$	5.70260	0.195483
$852$	0	0
$853$	16.6053	0.568555	0.284277	−	0.958742i	$-0.408246\pi$
$853$	16.6053	0.568555	0.284277	+	0.958742i	$0.408246\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.8213	0.984516	0.492258	−	0.870449i	$-0.336172\pi$
$857$	28.8213	0.984516	0.492258	+	0.870449i	$0.336172\pi$
$858$	0	0
$859$	44.3127	1.51193	0.755965	−	0.654612i	$-0.227168\pi$
$859$	44.3127	1.51193	0.755965	+	0.654612i	$0.227168\pi$
$860$	0	0
$861$	−25.2839	−0.861674
$862$	0	0
$863$	−39.0246	−1.32841	−0.664207	−	0.747549i	$-0.731231\pi$
$863$	−39.0246	−1.32841	−0.664207	+	0.747549i	$0.731231\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	50.1358	1.70270
$868$	0	0
$869$	4.10120	0.139124
$870$	0	0
$871$	−21.2644	−0.720517
$872$	0	0
$873$	−119.694	−4.05101
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.3534	1.02496	0.512480	−	0.858699i	$-0.328727\pi$
$877$	30.3534	1.02496	0.512480	+	0.858699i	$0.328727\pi$
$878$	0	0
$879$	68.8534	2.32237
$880$	0	0
$881$	−4.68366	−0.157796	−0.0788982	−	0.996883i	$-0.525140\pi$
$881$	−4.68366	−0.157796	−0.0788982	+	0.996883i	$0.525140\pi$
$882$	0	0
$883$	−9.64396	−0.324545	−0.162273	−	0.986746i	$-0.551882\pi$
$883$	−9.64396	−0.324545	−0.162273	+	0.986746i	$0.551882\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.8514	1.17020	0.585098	−	0.810963i	$-0.301056\pi$
$887$	34.8514	1.17020	0.585098	+	0.810963i	$0.301056\pi$
$888$	0	0
$889$	20.8133	0.698056
$890$	0	0
$891$	43.3092	1.45091
$892$	0	0
$893$	1.85629	0.0621183
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.71881	−0.224334
$898$	0	0
$899$	−55.9279	−1.86530
$900$	0	0
$901$	21.2744	0.708752
$902$	0	0
$903$	87.7976	2.92172
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.99763	0.0663302	0.0331651	−	0.999450i	$-0.489441\pi$
$907$	1.99763	0.0663302	0.0331651	+	0.999450i	$0.489441\pi$
$908$	0	0
$909$	−60.6648	−2.01212
$910$	0	0
$911$	−16.3393	−0.541344	−0.270672	−	0.962672i	$-0.587246\pi$
$911$	−16.3393	−0.541344	−0.270672	+	0.962672i	$0.587246\pi$
$912$	0	0
$913$	9.85641	0.326200
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.05710	−0.0679315
$918$	0	0
$919$	−13.4148	−0.442512	−0.221256	−	0.975216i	$-0.571016\pi$
$919$	−13.4148	−0.442512	−0.221256	+	0.975216i	$0.571016\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.26905	−0.173433
$924$	0	0
$925$	0	0
$926$	0	0
$927$	36.3175	1.19282
$928$	0	0
$929$	36.6648	1.20293	0.601467	−	0.798898i	$-0.294583\pi$
$929$	36.6648	1.20293	0.601467	+	0.798898i	$0.294583\pi$
$930$	0	0
$931$	0.981711	0.0321743
$932$	0	0
$933$	42.4648	1.39024
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−45.7427	−1.49435	−0.747175	−	0.664627i	$-0.768590\pi$
$937$	−45.7427	−1.49435	−0.747175	+	0.664627i	$0.768590\pi$
$938$	0	0
$939$	−14.5930	−0.476223
$940$	0	0
$941$	−36.3203	−1.18401	−0.592003	−	0.805936i	$-0.701663\pi$
$941$	−36.3203	−1.18401	−0.592003	+	0.805936i	$0.701663\pi$
$942$	0	0
$943$	2.85130	0.0928511
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.3926	1.31258	0.656292	−	0.754507i	$-0.272124\pi$
$947$	40.3926	1.31258	0.656292	+	0.754507i	$0.272124\pi$
$948$	0	0
$949$	15.3435	0.498071
$950$	0	0
$951$	116.995	3.79382
$952$	0	0
$953$	24.1286	0.781601	0.390801	−	0.920475i	$-0.372198\pi$
$953$	24.1286	0.781601	0.390801	+	0.920475i	$0.372198\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−24.6272	−0.796085
$958$	0	0
$959$	−3.00831	−0.0971436
$960$	0	0
$961$	33.9557	1.09535
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−51.7162	−1.66308	−0.831540	−	0.555465i	$-0.812540\pi$
$967$	−51.7162	−1.66308	−0.831540	+	0.555465i	$0.812540\pi$
$968$	0	0
$969$	15.7774	0.506843
$970$	0	0
$971$	20.0396	0.643101	0.321551	−	0.946892i	$-0.395796\pi$
$971$	20.0396	0.643101	0.321551	+	0.946892i	$0.395796\pi$
$972$	0	0
$973$	32.6054	1.04528
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.2983	0.425451	0.212725	−	0.977112i	$-0.431766\pi$
$977$	13.2983	0.425451	0.212725	+	0.977112i	$0.431766\pi$
$978$	0	0
$979$	−7.49472	−0.239532
$980$	0	0
$981$	−72.6349	−2.31906
$982$	0	0
$983$	45.6489	1.45598	0.727988	−	0.685590i	$-0.240456\pi$
$983$	45.6489	1.45598	0.727988	+	0.685590i	$0.240456\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.54891	−0.176624
$988$	0	0
$989$	−9.90105	−0.314835
$990$	0	0
$991$	11.0525	0.351094	0.175547	−	0.984471i	$-0.443831\pi$
$991$	11.0525	0.351094	0.175547	+	0.984471i	$0.443831\pi$
$992$	0	0
$993$	91.3255	2.89813
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−36.7081	−1.16256	−0.581278	−	0.813705i	$-0.697447\pi$
$997$	−36.7081	−1.16256	−0.581278	+	0.813705i	$0.697447\pi$
$998$	0	0
$999$	113.389	3.58747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.bb.1.1	yes	4
4.3	odd	2		9200.2.a.cn.1.4		4
5.2	odd	4		4600.2.e.t.4049.8		8
5.3	odd	4		4600.2.e.t.4049.1		8
5.4	even	2		4600.2.a.ba.1.4	✓	4
20.19	odd	2		9200.2.a.cp.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.ba.1.4	✓	4	5.4	even	2
4600.2.a.bb.1.1	yes	4	1.1	even	1	trivial
4600.2.e.t.4049.1		8	5.3	odd	4
4600.2.e.t.4049.8		8	5.2	odd	4
9200.2.a.cn.1.4		4	4.3	odd	2
9200.2.a.cp.1.1		4	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q-3.43375 q^{3} -2.58245 q^{7} +8.79066 q^{9} +O(q^{10})\)	\(q-3.43375 q^{3} -2.58245 q^{7} +8.79066 q^{9} +1.03354 q^{11} -1.95669 q^{13} +1.54891 q^{17} -2.96646 q^{19} +8.86751 q^{21} -1.00000 q^{23} -19.8837 q^{27} +6.93936 q^{29} -8.05951 q^{31} -3.54891 q^{33} -5.70260 q^{37} +6.71881 q^{39} -2.85130 q^{41} +9.90105 q^{43} -0.625758 q^{47} -0.330937 q^{49} -5.31859 q^{51} +13.7350 q^{53} +10.1861 q^{57} +8.08549 q^{59} +8.80043 q^{61} -22.7015 q^{63} +10.8675 q^{67} +3.43375 q^{69} +2.69283 q^{71} -7.84153 q^{73} -2.66906 q^{77} +3.96812 q^{79} +41.9038 q^{81} +9.53657 q^{83} -23.8281 q^{87} -7.25152 q^{89} +5.05307 q^{91} +27.6744 q^{93} -13.6160 q^{97} +9.08549 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q + q^7 + 6 * q^9	\( 4 q + q^{7} + 6 q^{9} + q^{11} - 3 q^{13} - 2 q^{17} - 15 q^{19} + 8 q^{21} - 4 q^{23} - 27 q^{27} + q^{29} - 12 q^{31} - 6 q^{33} - 18 q^{37} - 3 q^{39} - 9 q^{41} + 9 q^{43} + 4 q^{47} - 3 q^{49} - 2 q^{51} - 6 q^{57} - 5 q^{59} + 14 q^{61} - 39 q^{63} + 16 q^{67} - 2 q^{71} - 21 q^{73} - 9 q^{77} - 21 q^{79} + 44 q^{81} + 9 q^{83} - 17 q^{87} - 16 q^{89} + 33 q^{91} + 29 q^{93} - 40 q^{97} - q^{99}+O(q^{100}) \) 4 * q + q^7 + 6 * q^9 + q^11 - 3 * q^13 - 2 * q^17 - 15 * q^19 + 8 * q^21 - 4 * q^23 - 27 * q^27 + q^29 - 12 * q^31 - 6 * q^33 - 18 * q^37 - 3 * q^39 - 9 * q^41 + 9 * q^43 + 4 * q^47 - 3 * q^49 - 2 * q^51 - 6 * q^57 - 5 * q^59 + 14 * q^61 - 39 * q^63 + 16 * q^67 - 2 * q^71 - 21 * q^73 - 9 * q^77 - 21 * q^79 + 44 * q^81 + 9 * q^83 - 17 * q^87 - 16 * q^89 + 33 * q^91 + 29 * q^93 - 40 * q^97 - q^99

Introduction

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