LMFDB - Embedded newform 4508.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4508,2,Mod(1,4508)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4508, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4508.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4508 = 2^{2} \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4508.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:	$35.9965612312$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.6963152.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 $ x^5 - 2x^4 - 10x^3 + 10x^2 + 29x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 644)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.435854$ of defining polynomial
Character	$\chi$	$=$	4508.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43585 q^{3} -2.34236 q^{5} -0.938323 q^{9} +O(q^{10})$	$q-1.43585 q^{3} -2.34236 q^{5} -0.938323 q^{9} +5.38507 q^{11} +5.32339 q^{13} +3.36328 q^{15} -7.88754 q^{17} -6.06364 q^{19} -1.00000 q^{23} +0.486637 q^{25} +5.65486 q^{27} +1.87468 q^{29} +5.83096 q^{31} -7.73218 q^{33} +1.19193 q^{37} -7.64362 q^{39} -6.45169 q^{41} -6.93535 q^{43} +2.19789 q^{45} +10.8577 q^{47} +11.3254 q^{51} +7.94029 q^{53} -12.6138 q^{55} +8.70650 q^{57} +2.69561 q^{59} +3.27771 q^{61} -12.4693 q^{65} -1.82865 q^{67} +1.43585 q^{69} +7.30161 q^{71} +7.51039 q^{73} -0.698740 q^{75} -2.49158 q^{79} -5.30458 q^{81} +16.8120 q^{83} +18.4754 q^{85} -2.69177 q^{87} -13.2665 q^{89} -8.37240 q^{93} +14.2032 q^{95} -11.9624 q^{97} -5.05294 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9	$ 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9} + 2 q^{11} - 13 q^{13} + 4 q^{15} - 4 q^{17} - 12 q^{19} - 5 q^{23} + 19 q^{25} - 15 q^{27} + 13 q^{29} + 3 q^{31} - 24 q^{33} - 4 q^{37} + 3 q^{39} - q^{41} - 8 q^{43} + 16 q^{45} - 5 q^{47} - 16 q^{51} - 8 q^{53} + 2 q^{55} + 12 q^{57} - 12 q^{59} - 20 q^{61} - 12 q^{65} - 12 q^{67} + 3 q^{69} + 9 q^{71} + 9 q^{73} - 35 q^{75} - 8 q^{79} - 11 q^{81} + 28 q^{83} + 16 q^{85} + 15 q^{87} - 32 q^{89} - 15 q^{93} - 36 q^{95} - 4 q^{97} + 28 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9 + 2 * q^11 - 13 * q^13 + 4 * q^15 - 4 * q^17 - 12 * q^19 - 5 * q^23 + 19 * q^25 - 15 * q^27 + 13 * q^29 + 3 * q^31 - 24 * q^33 - 4 * q^37 + 3 * q^39 - q^41 - 8 * q^43 + 16 * q^45 - 5 * q^47 - 16 * q^51 - 8 * q^53 + 2 * q^55 + 12 * q^57 - 12 * q^59 - 20 * q^61 - 12 * q^65 - 12 * q^67 + 3 * q^69 + 9 * q^71 + 9 * q^73 - 35 * q^75 - 8 * q^79 - 11 * q^81 + 28 * q^83 + 16 * q^85 + 15 * q^87 - 32 * q^89 - 15 * q^93 - 36 * q^95 - 4 * q^97 + 28 * 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
$3$	−1.43585	−0.828991	−0.414495	−	0.910051i	$-0.636042\pi$
$3$	−1.43585	−0.828991	−0.414495	+	0.910051i	$0.636042\pi$
$4$	0	0
$5$	−2.34236	−1.04753	−0.523767	−	0.851862i	$-0.675474\pi$
$5$	−2.34236	−1.04753	−0.523767	+	0.851862i	$0.675474\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.938323	−0.312774
$10$	0	0
$11$	5.38507	1.62366	0.811830	−	0.583894i	$-0.198472\pi$
$11$	5.38507	1.62366	0.811830	+	0.583894i	$0.198472\pi$
$12$	0	0
$13$	5.32339	1.47644	0.738222	−	0.674558i	$-0.235666\pi$
$13$	5.32339	1.47644	0.738222	+	0.674558i	$0.235666\pi$
$14$	0	0
$15$	3.36328	0.868396
$16$	0	0
$17$	−7.88754	−1.91301	−0.956505	−	0.291717i	$-0.905773\pi$
$17$	−7.88754	−1.91301	−0.956505	+	0.291717i	$0.905773\pi$
$18$	0	0
$19$	−6.06364	−1.39109	−0.695547	−	0.718480i	$-0.744838\pi$
$19$	−6.06364	−1.39109	−0.695547	+	0.718480i	$0.744838\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0.486637	0.0973275
$26$	0	0
$27$	5.65486	1.08828
$28$	0	0
$29$	1.87468	0.348120	0.174060	−	0.984735i	$-0.444311\pi$
$29$	1.87468	0.348120	0.174060	+	0.984735i	$0.444311\pi$
$30$	0	0
$31$	5.83096	1.04727	0.523635	−	0.851942i	$-0.324575\pi$
$31$	5.83096	1.04727	0.523635	+	0.851942i	$0.324575\pi$
$32$	0	0
$33$	−7.73218	−1.34600
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.19193	0.195952	0.0979762	−	0.995189i	$-0.468763\pi$
$37$	1.19193	0.195952	0.0979762	+	0.995189i	$0.468763\pi$
$38$	0	0
$39$	−7.64362	−1.22396
$40$	0	0
$41$	−6.45169	−1.00758	−0.503792	−	0.863825i	$-0.668062\pi$
$41$	−6.45169	−1.00758	−0.503792	+	0.863825i	$0.668062\pi$
$42$	0	0
$43$	−6.93535	−1.05763	−0.528815	−	0.848737i	$-0.677364\pi$
$43$	−6.93535	−1.05763	−0.528815	+	0.848737i	$0.677364\pi$
$44$	0	0
$45$	2.19789	0.327642
$46$	0	0
$47$	10.8577	1.58376	0.791878	−	0.610679i	$-0.209104\pi$
$47$	10.8577	1.58376	0.791878	+	0.610679i	$0.209104\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	11.3254	1.58587
$52$	0	0
$53$	7.94029	1.09068	0.545341	−	0.838214i	$-0.316400\pi$
$53$	7.94029	1.09068	0.545341	+	0.838214i	$0.316400\pi$
$54$	0	0
$55$	−12.6138	−1.70084
$56$	0	0
$57$	8.70650	1.15320
$58$	0	0
$59$	2.69561	0.350938	0.175469	−	0.984485i	$-0.443856\pi$
$59$	2.69561	0.350938	0.175469	+	0.984485i	$0.443856\pi$
$60$	0	0
$61$	3.27771	0.419667	0.209834	−	0.977737i	$-0.432708\pi$
$61$	3.27771	0.419667	0.209834	+	0.977737i	$0.432708\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.4693	−1.54663
$66$	0	0
$67$	−1.82865	−0.223405	−0.111702	−	0.993742i	$-0.535630\pi$
$67$	−1.82865	−0.223405	−0.111702	+	0.993742i	$0.535630\pi$
$68$	0	0
$69$	1.43585	0.172857
$70$	0	0
$71$	7.30161	0.866541	0.433271	−	0.901264i	$-0.357359\pi$
$71$	7.30161	0.866541	0.433271	+	0.901264i	$0.357359\pi$
$72$	0	0
$73$	7.51039	0.879024	0.439512	−	0.898237i	$-0.355151\pi$
$73$	7.51039	0.879024	0.439512	+	0.898237i	$0.355151\pi$
$74$	0	0
$75$	−0.698740	−0.0806836
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.49158	−0.280324	−0.140162	−	0.990129i	$-0.544762\pi$
$79$	−2.49158	−0.280324	−0.140162	+	0.990129i	$0.544762\pi$
$80$	0	0
$81$	−5.30458	−0.589398
$82$	0	0
$83$	16.8120	1.84536	0.922678	−	0.385571i	$-0.125996\pi$
$83$	16.8120	1.84536	0.922678	+	0.385571i	$0.125996\pi$
$84$	0	0
$85$	18.4754	2.00394
$86$	0	0
$87$	−2.69177	−0.288588
$88$	0	0
$89$	−13.2665	−1.40624	−0.703121	−	0.711070i	$-0.748211\pi$
$89$	−13.2665	−1.40624	−0.703121	+	0.711070i	$0.748211\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.37240	−0.868178
$94$	0	0
$95$	14.2032	1.45722
$96$	0	0
$97$	−11.9624	−1.21460	−0.607300	−	0.794473i	$-0.707747\pi$
$97$	−11.9624	−1.21460	−0.607300	+	0.794473i	$0.707747\pi$
$98$	0	0
$99$	−5.05294	−0.507839
$100$	0	0
$101$	14.4918	1.44199	0.720993	−	0.692943i	$-0.243686\pi$
$101$	14.4918	1.44199	0.720993	+	0.692943i	$0.243686\pi$
$102$	0	0
$103$	−7.55642	−0.744556	−0.372278	−	0.928121i	$-0.621423\pi$
$103$	−7.55642	−0.744556	−0.372278	+	0.928121i	$0.621423\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.9353	−1.05716	−0.528580	−	0.848883i	$-0.677275\pi$
$107$	−10.9353	−1.05716	−0.528580	+	0.848883i	$0.677275\pi$
$108$	0	0
$109$	−12.6250	−1.20926	−0.604628	−	0.796508i	$-0.706678\pi$
$109$	−12.6250	−1.20926	−0.604628	+	0.796508i	$0.706678\pi$
$110$	0	0
$111$	−1.71144	−0.162443
$112$	0	0
$113$	−3.15008	−0.296335	−0.148167	−	0.988962i	$-0.547337\pi$
$113$	−3.15008	−0.296335	−0.148167	+	0.988962i	$0.547337\pi$
$114$	0	0
$115$	2.34236	0.218426
$116$	0	0
$117$	−4.99506	−0.461794
$118$	0	0
$119$	0	0
$120$	0	0
$121$	17.9990	1.63627
$122$	0	0
$123$	9.26368	0.835278
$124$	0	0
$125$	10.5719	0.945580
$126$	0	0
$127$	−9.25975	−0.821670	−0.410835	−	0.911710i	$-0.634763\pi$
$127$	−9.25975	−0.821670	−0.410835	+	0.911710i	$0.634763\pi$
$128$	0	0
$129$	9.95815	0.876766
$130$	0	0
$131$	−5.02289	−0.438852	−0.219426	−	0.975629i	$-0.570418\pi$
$131$	−5.02289	−0.438852	−0.219426	+	0.975629i	$0.570418\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−13.2457	−1.14001
$136$	0	0
$137$	−15.7114	−1.34232	−0.671159	−	0.741313i	$-0.734203\pi$
$137$	−15.7114	−1.34232	−0.671159	+	0.741313i	$0.734203\pi$
$138$	0	0
$139$	0.0569290	0.00482865	0.00241433	−	0.999997i	$-0.499231\pi$
$139$	0.0569290	0.00482865	0.00241433	+	0.999997i	$0.499231\pi$
$140$	0	0
$141$	−15.5900	−1.31292
$142$	0	0
$143$	28.6669	2.39724
$144$	0	0
$145$	−4.39118	−0.364667
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.55642	0.127507	0.0637536	−	0.997966i	$-0.479693\pi$
$149$	1.55642	0.127507	0.0637536	+	0.997966i	$0.479693\pi$
$150$	0	0
$151$	−3.44675	−0.280492	−0.140246	−	0.990117i	$-0.544789\pi$
$151$	−3.44675	−0.280492	−0.140246	+	0.990117i	$0.544789\pi$
$152$	0	0
$153$	7.40106	0.598340
$154$	0	0
$155$	−13.6582	−1.09705
$156$	0	0
$157$	−20.0366	−1.59910	−0.799548	−	0.600603i	$-0.794927\pi$
$157$	−20.0366	−1.59910	−0.799548	+	0.600603i	$0.794927\pi$
$158$	0	0
$159$	−11.4011	−0.904165
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.4556	−1.60221	−0.801104	−	0.598526i	$-0.795753\pi$
$163$	−20.4556	−1.60221	−0.801104	+	0.598526i	$0.795753\pi$
$164$	0	0
$165$	18.1115	1.40998
$166$	0	0
$167$	−18.3044	−1.41644	−0.708220	−	0.705992i	$-0.750501\pi$
$167$	−18.3044	−1.41644	−0.708220	+	0.705992i	$0.750501\pi$
$168$	0	0
$169$	15.3385	1.17989
$170$	0	0
$171$	5.68965	0.435099
$172$	0	0
$173$	−0.0636396	−0.00483843	−0.00241922	−	0.999997i	$-0.500770\pi$
$173$	−0.0636396	−0.00483843	−0.00241922	+	0.999997i	$0.500770\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.87050	−0.290925
$178$	0	0
$179$	−10.0500	−0.751169	−0.375585	−	0.926788i	$-0.622558\pi$
$179$	−10.0500	−0.751169	−0.375585	+	0.926788i	$0.622558\pi$
$180$	0	0
$181$	0.138173	0.0102703	0.00513516	−	0.999987i	$-0.498365\pi$
$181$	0.138173	0.0102703	0.00513516	+	0.999987i	$0.498365\pi$
$182$	0	0
$183$	−4.70631	−0.347900
$184$	0	0
$185$	−2.79193	−0.205267
$186$	0	0
$187$	−42.4750	−3.10608
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.55149	−0.618764	−0.309382	−	0.950938i	$-0.600122\pi$
$191$	−8.55149	−0.618764	−0.309382	+	0.950938i	$0.600122\pi$
$192$	0	0
$193$	−24.3765	−1.75466	−0.877330	−	0.479887i	$-0.840678\pi$
$193$	−24.3765	−1.75466	−0.877330	+	0.479887i	$0.840678\pi$
$194$	0	0
$195$	17.9041	1.28214
$196$	0	0
$197$	21.5421	1.53481	0.767404	−	0.641164i	$-0.221548\pi$
$197$	21.5421	1.53481	0.767404	+	0.641164i	$0.221548\pi$
$198$	0	0
$199$	−6.51951	−0.462156	−0.231078	−	0.972935i	$-0.574225\pi$
$199$	−6.51951	−0.462156	−0.231078	+	0.972935i	$0.574225\pi$
$200$	0	0
$201$	2.62567	0.185201
$202$	0	0
$203$	0	0
$204$	0	0
$205$	15.1122	1.05548
$206$	0	0
$207$	0.938323	0.0652180
$208$	0	0
$209$	−32.6531	−2.25866
$210$	0	0
$211$	−18.2352	−1.25536	−0.627681	−	0.778471i	$-0.715996\pi$
$211$	−18.2352	−1.25536	−0.627681	+	0.778471i	$0.715996\pi$
$212$	0	0
$213$	−10.4840	−0.718355
$214$	0	0
$215$	16.2451	1.10790
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−10.7838	−0.728703
$220$	0	0
$221$	−41.9885	−2.82445
$222$	0	0
$223$	19.8496	1.32923	0.664614	−	0.747187i	$-0.268596\pi$
$223$	19.8496	1.32923	0.664614	+	0.747187i	$0.268596\pi$
$224$	0	0
$225$	−0.456623	−0.0304415
$226$	0	0
$227$	0.579223	0.0384444	0.0192222	−	0.999815i	$-0.493881\pi$
$227$	0.579223	0.0384444	0.0192222	+	0.999815i	$0.493881\pi$
$228$	0	0
$229$	−14.6538	−0.968347	−0.484174	−	0.874972i	$-0.660880\pi$
$229$	−14.6538	−0.968347	−0.484174	+	0.874972i	$0.660880\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.5953	0.956170	0.478085	−	0.878314i	$-0.341331\pi$
$233$	14.5953	0.956170	0.478085	+	0.878314i	$0.341331\pi$
$234$	0	0
$235$	−25.4326	−1.65904
$236$	0	0
$237$	3.57754	0.232386
$238$	0	0
$239$	−12.9435	−0.837243	−0.418621	−	0.908161i	$-0.637487\pi$
$239$	−12.9435	−0.837243	−0.418621	+	0.908161i	$0.637487\pi$
$240$	0	0
$241$	−1.24075	−0.0799239	−0.0399619	−	0.999201i	$-0.512724\pi$
$241$	−1.24075	−0.0799239	−0.0399619	+	0.999201i	$0.512724\pi$
$242$	0	0
$243$	−9.34797	−0.599672
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−32.2791	−2.05387
$248$	0	0
$249$	−24.1396	−1.52978
$250$	0	0
$251$	−19.8766	−1.25460	−0.627301	−	0.778777i	$-0.715841\pi$
$251$	−19.8766	−1.25460	−0.627301	+	0.778777i	$0.715841\pi$
$252$	0	0
$253$	−5.38507	−0.338557
$254$	0	0
$255$	−26.5280	−1.66125
$256$	0	0
$257$	−4.85803	−0.303036	−0.151518	−	0.988455i	$-0.548416\pi$
$257$	−4.85803	−0.303036	−0.151518	+	0.988455i	$0.548416\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.75906	−0.108883
$262$	0	0
$263$	11.5760	0.713806	0.356903	−	0.934141i	$-0.383833\pi$
$263$	11.5760	0.713806	0.356903	+	0.934141i	$0.383833\pi$
$264$	0	0
$265$	−18.5990	−1.14253
$266$	0	0
$267$	19.0487	1.16576
$268$	0	0
$269$	4.38805	0.267544	0.133772	−	0.991012i	$-0.457291\pi$
$269$	4.38805	0.267544	0.133772	+	0.991012i	$0.457291\pi$
$270$	0	0
$271$	1.44396	0.0877145	0.0438572	−	0.999038i	$-0.486035\pi$
$271$	1.44396	0.0877145	0.0438572	+	0.999038i	$0.486035\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.62058	0.158027
$276$	0	0
$277$	0.281027	0.0168853	0.00844263	−	0.999964i	$-0.497313\pi$
$277$	0.281027	0.0168853	0.00844263	+	0.999964i	$0.497313\pi$
$278$	0	0
$279$	−5.47132	−0.327559
$280$	0	0
$281$	−19.5185	−1.16438	−0.582188	−	0.813054i	$-0.697803\pi$
$281$	−19.5185	−1.16438	−0.582188	+	0.813054i	$0.697803\pi$
$282$	0	0
$283$	−27.3097	−1.62339	−0.811697	−	0.584079i	$-0.801456\pi$
$283$	−27.3097	−1.62339	−0.811697	+	0.584079i	$0.801456\pi$
$284$	0	0
$285$	−20.3937	−1.20802
$286$	0	0
$287$	0	0
$288$	0	0
$289$	45.2133	2.65960
$290$	0	0
$291$	17.1763	1.00689
$292$	0	0
$293$	−20.8180	−1.21620	−0.608100	−	0.793860i	$-0.708068\pi$
$293$	−20.8180	−1.21620	−0.608100	+	0.793860i	$0.708068\pi$
$294$	0	0
$295$	−6.31408	−0.367620
$296$	0	0
$297$	30.4518	1.76699
$298$	0	0
$299$	−5.32339	−0.307860
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−20.8081	−1.19539
$304$	0	0
$305$	−7.67756	−0.439616
$306$	0	0
$307$	−18.1772	−1.03742	−0.518712	−	0.854949i	$-0.673588\pi$
$307$	−18.1772	−1.03742	−0.518712	+	0.854949i	$0.673588\pi$
$308$	0	0
$309$	10.8499	0.617230
$310$	0	0
$311$	12.6176	0.715478	0.357739	−	0.933822i	$-0.383548\pi$
$311$	12.6176	0.715478	0.357739	+	0.933822i	$0.383548\pi$
$312$	0	0
$313$	19.2510	1.08813	0.544066	−	0.839042i	$-0.316884\pi$
$313$	19.2510	1.08813	0.544066	+	0.839042i	$0.316884\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.9197	0.613312	0.306656	−	0.951820i	$-0.400790\pi$
$317$	10.9197	0.613312	0.306656	+	0.951820i	$0.400790\pi$
$318$	0	0
$319$	10.0953	0.565228
$320$	0	0
$321$	15.7016	0.876376
$322$	0	0
$323$	47.8272	2.66118
$324$	0	0
$325$	2.59056	0.143699
$326$	0	0
$327$	18.1277	1.00246
$328$	0	0
$329$	0	0
$330$	0	0
$331$	33.1364	1.82134	0.910671	−	0.413133i	$-0.135566\pi$
$331$	33.1364	1.82134	0.910671	+	0.413133i	$0.135566\pi$
$332$	0	0
$333$	−1.11842	−0.0612888
$334$	0	0
$335$	4.28335	0.234024
$336$	0	0
$337$	35.6458	1.94175	0.970875	−	0.239588i	$-0.0770125\pi$
$337$	35.6458	1.94175	0.970875	+	0.239588i	$0.0770125\pi$
$338$	0	0
$339$	4.52305	0.245659
$340$	0	0
$341$	31.4001	1.70041
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.36328	−0.181073
$346$	0	0
$347$	−0.117783	−0.00632293	−0.00316146	−	0.999995i	$-0.501006\pi$
$347$	−0.117783	−0.00632293	−0.00316146	+	0.999995i	$0.501006\pi$
$348$	0	0
$349$	12.7083	0.680258	0.340129	−	0.940379i	$-0.389529\pi$
$349$	12.7083	0.680258	0.340129	+	0.940379i	$0.389529\pi$
$350$	0	0
$351$	30.1030	1.60678
$352$	0	0
$353$	15.4791	0.823870	0.411935	−	0.911213i	$-0.364853\pi$
$353$	15.4791	0.823870	0.411935	+	0.911213i	$0.364853\pi$
$354$	0	0
$355$	−17.1030	−0.907731
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.66806	0.404705	0.202352	−	0.979313i	$-0.435141\pi$
$359$	7.66806	0.404705	0.202352	+	0.979313i	$0.435141\pi$
$360$	0	0
$361$	17.7677	0.935143
$362$	0	0
$363$	−25.8439	−1.35645
$364$	0	0
$365$	−17.5920	−0.920808
$366$	0	0
$367$	−10.1079	−0.527628	−0.263814	−	0.964574i	$-0.584981\pi$
$367$	−10.1079	−0.527628	−0.263814	+	0.964574i	$0.584981\pi$
$368$	0	0
$369$	6.05376	0.315146
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−28.8437	−1.49347	−0.746734	−	0.665123i	$-0.768379\pi$
$373$	−28.8437	−1.49347	−0.746734	+	0.665123i	$0.768379\pi$
$374$	0	0
$375$	−15.1797	−0.783877
$376$	0	0
$377$	9.97968	0.513980
$378$	0	0
$379$	27.2020	1.39727	0.698636	−	0.715477i	$-0.253790\pi$
$379$	27.2020	1.39727	0.698636	+	0.715477i	$0.253790\pi$
$380$	0	0
$381$	13.2957	0.681157
$382$	0	0
$383$	−19.6517	−1.00416	−0.502078	−	0.864822i	$-0.667431\pi$
$383$	−19.6517	−1.00416	−0.502078	+	0.864822i	$0.667431\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.50760	0.330800
$388$	0	0
$389$	12.9990	0.659075	0.329537	−	0.944143i	$-0.393107\pi$
$389$	12.9990	0.659075	0.329537	+	0.944143i	$0.393107\pi$
$390$	0	0
$391$	7.88754	0.398890
$392$	0	0
$393$	7.21214	0.363804
$394$	0	0
$395$	5.83616	0.293649
$396$	0	0
$397$	−27.2637	−1.36832	−0.684162	−	0.729330i	$-0.739832\pi$
$397$	−27.2637	−1.36832	−0.684162	+	0.729330i	$0.739832\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.3543	0.517069	0.258535	−	0.966002i	$-0.416760\pi$
$401$	10.3543	0.517069	0.258535	+	0.966002i	$0.416760\pi$
$402$	0	0
$403$	31.0405	1.54624
$404$	0	0
$405$	12.4252	0.617414
$406$	0	0
$407$	6.41863	0.318160
$408$	0	0
$409$	−14.4338	−0.713707	−0.356853	−	0.934160i	$-0.616150\pi$
$409$	−14.4338	−0.713707	−0.356853	+	0.934160i	$0.616150\pi$
$410$	0	0
$411$	22.5593	1.11277
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−39.3797	−1.93307
$416$	0	0
$417$	−0.0817417	−0.00400291
$418$	0	0
$419$	−38.7780	−1.89443	−0.947214	−	0.320601i	$-0.896115\pi$
$419$	−38.7780	−1.89443	−0.947214	+	0.320601i	$0.896115\pi$
$420$	0	0
$421$	33.3097	1.62342	0.811708	−	0.584063i	$-0.198538\pi$
$421$	33.3097	1.62342	0.811708	+	0.584063i	$0.198538\pi$
$422$	0	0
$423$	−10.1880	−0.495358
$424$	0	0
$425$	−3.83837	−0.186188
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−41.1614	−1.98729
$430$	0	0
$431$	−0.967325	−0.0465944	−0.0232972	−	0.999729i	$-0.507416\pi$
$431$	−0.967325	−0.0465944	−0.0232972	+	0.999729i	$0.507416\pi$
$432$	0	0
$433$	−11.4173	−0.548679	−0.274340	−	0.961633i	$-0.588459\pi$
$433$	−11.4173	−0.548679	−0.274340	+	0.961633i	$0.588459\pi$
$434$	0	0
$435$	6.30509	0.302306
$436$	0	0
$437$	6.06364	0.290063
$438$	0	0
$439$	10.3789	0.495357	0.247678	−	0.968842i	$-0.420332\pi$
$439$	10.3789	0.495357	0.247678	+	0.968842i	$0.420332\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−23.2218	−1.10330	−0.551651	−	0.834075i	$-0.686002\pi$
$443$	−23.2218	−1.10330	−0.551651	+	0.834075i	$0.686002\pi$
$444$	0	0
$445$	31.0748	1.47309
$446$	0	0
$447$	−2.23480	−0.105702
$448$	0	0
$449$	−10.6505	−0.502629	−0.251314	−	0.967905i	$-0.580863\pi$
$449$	−10.6505	−0.502629	−0.251314	+	0.967905i	$0.580863\pi$
$450$	0	0
$451$	−34.7428	−1.63597
$452$	0	0
$453$	4.94903	0.232526
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.379938	0.0177727	0.00888637	−	0.999961i	$-0.497171\pi$
$457$	0.379938	0.0177727	0.00888637	+	0.999961i	$0.497171\pi$
$458$	0	0
$459$	−44.6029	−2.08189
$460$	0	0
$461$	16.3880	0.763267	0.381634	−	0.924314i	$-0.375362\pi$
$461$	16.3880	0.763267	0.381634	+	0.924314i	$0.375362\pi$
$462$	0	0
$463$	−36.2349	−1.68398	−0.841989	−	0.539495i	$-0.818615\pi$
$463$	−36.2349	−1.68398	−0.841989	+	0.539495i	$0.818615\pi$
$464$	0	0
$465$	19.6112	0.909446
$466$	0	0
$467$	−17.1603	−0.794083	−0.397041	−	0.917801i	$-0.629963\pi$
$467$	−17.1603	−0.794083	−0.397041	+	0.917801i	$0.629963\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	28.7696	1.32564
$472$	0	0
$473$	−37.3473	−1.71723
$474$	0	0
$475$	−2.95079	−0.135392
$476$	0	0
$477$	−7.45055	−0.341137
$478$	0	0
$479$	16.8299	0.768976	0.384488	−	0.923130i	$-0.374378\pi$
$479$	16.8299	0.768976	0.384488	+	0.923130i	$0.374378\pi$
$480$	0	0
$481$	6.34512	0.289313
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28.0203	1.27233
$486$	0	0
$487$	−26.9276	−1.22021	−0.610103	−	0.792322i	$-0.708872\pi$
$487$	−26.9276	−1.22021	−0.610103	+	0.792322i	$0.708872\pi$
$488$	0	0
$489$	29.3713	1.32821
$490$	0	0
$491$	−21.3432	−0.963203	−0.481602	−	0.876390i	$-0.659945\pi$
$491$	−21.3432	−0.963203	−0.481602	+	0.876390i	$0.659945\pi$
$492$	0	0
$493$	−14.7866	−0.665957
$494$	0	0
$495$	11.8358	0.531979
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−19.5203	−0.873847	−0.436923	−	0.899499i	$-0.643932\pi$
$499$	−19.5203	−0.873847	−0.436923	+	0.899499i	$0.643932\pi$
$500$	0	0
$501$	26.2825	1.17422
$502$	0	0
$503$	−5.02673	−0.224131	−0.112065	−	0.993701i	$-0.535747\pi$
$503$	−5.02673	−0.224131	−0.112065	+	0.993701i	$0.535747\pi$
$504$	0	0
$505$	−33.9449	−1.51053
$506$	0	0
$507$	−22.0239	−0.978115
$508$	0	0
$509$	10.5354	0.466973	0.233486	−	0.972360i	$-0.424987\pi$
$509$	10.5354	0.466973	0.233486	+	0.972360i	$0.424987\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−34.2890	−1.51390
$514$	0	0
$515$	17.6998	0.779948
$516$	0	0
$517$	58.4694	2.57148
$518$	0	0
$519$	0.0913772	0.00401101
$520$	0	0
$521$	−10.8559	−0.475606	−0.237803	−	0.971313i	$-0.576427\pi$
$521$	−10.8559	−0.475606	−0.237803	+	0.971313i	$0.576427\pi$
$522$	0	0
$523$	−16.5175	−0.722259	−0.361130	−	0.932516i	$-0.617609\pi$
$523$	−16.5175	−0.722259	−0.361130	+	0.932516i	$0.617609\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−45.9919	−2.00344
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.52935	−0.109764
$532$	0	0
$533$	−34.3449	−1.48764
$534$	0	0
$535$	25.6145	1.10741
$536$	0	0
$537$	14.4303	0.622712
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.0983	−0.477152	−0.238576	−	0.971124i	$-0.576681\pi$
$541$	−11.0983	−0.477152	−0.238576	+	0.971124i	$0.576681\pi$
$542$	0	0
$543$	−0.198396	−0.00851400
$544$	0	0
$545$	29.5723	1.26674
$546$	0	0
$547$	−3.70757	−0.158524	−0.0792621	−	0.996854i	$-0.525256\pi$
$547$	−3.70757	−0.158524	−0.0792621	+	0.996854i	$0.525256\pi$
$548$	0	0
$549$	−3.07555	−0.131261
$550$	0	0
$551$	−11.3674	−0.484268
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.00880	0.170164
$556$	0	0
$557$	−14.5792	−0.617741	−0.308871	−	0.951104i	$-0.599951\pi$
$557$	−14.5792	−0.617741	−0.308871	+	0.951104i	$0.599951\pi$
$558$	0	0
$559$	−36.9196	−1.56153
$560$	0	0
$561$	60.9878	2.57491
$562$	0	0
$563$	15.2872	0.644280	0.322140	−	0.946692i	$-0.395598\pi$
$563$	15.2872	0.644280	0.322140	+	0.946692i	$0.395598\pi$
$564$	0	0
$565$	7.37861	0.310420
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.3853	0.728828	0.364414	−	0.931237i	$-0.381269\pi$
$569$	17.3853	0.728828	0.364414	+	0.931237i	$0.381269\pi$
$570$	0	0
$571$	−5.96802	−0.249754	−0.124877	−	0.992172i	$-0.539854\pi$
$571$	−5.96802	−0.249754	−0.124877	+	0.992172i	$0.539854\pi$
$572$	0	0
$573$	12.2787	0.512949
$574$	0	0
$575$	−0.486637	−0.0202942
$576$	0	0
$577$	22.1611	0.922579	0.461289	−	0.887250i	$-0.347387\pi$
$577$	22.1611	0.922579	0.461289	+	0.887250i	$0.347387\pi$
$578$	0	0
$579$	35.0011	1.45460
$580$	0	0
$581$	0	0
$582$	0	0
$583$	42.7590	1.77090
$584$	0	0
$585$	11.7002	0.483745
$586$	0	0
$587$	−9.96131	−0.411147	−0.205574	−	0.978642i	$-0.565906\pi$
$587$	−9.96131	−0.411147	−0.205574	+	0.978642i	$0.565906\pi$
$588$	0	0
$589$	−35.3568	−1.45685
$590$	0	0
$591$	−30.9312	−1.27234
$592$	0	0
$593$	−8.77014	−0.360147	−0.180073	−	0.983653i	$-0.557633\pi$
$593$	−8.77014	−0.360147	−0.180073	+	0.983653i	$0.557633\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.36106	0.383123
$598$	0	0
$599$	11.3276	0.462832	0.231416	−	0.972855i	$-0.425664\pi$
$599$	11.3276	0.462832	0.231416	+	0.972855i	$0.425664\pi$
$600$	0	0
$601$	−5.77090	−0.235400	−0.117700	−	0.993049i	$-0.537552\pi$
$601$	−5.77090	−0.235400	−0.117700	+	0.993049i	$0.537552\pi$
$602$	0	0
$603$	1.71586	0.0698753
$604$	0	0
$605$	−42.1601	−1.71405
$606$	0	0
$607$	−1.19400	−0.0484630	−0.0242315	−	0.999706i	$-0.507714\pi$
$607$	−1.19400	−0.0484630	−0.0242315	+	0.999706i	$0.507714\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	57.7997	2.33833
$612$	0	0
$613$	−42.3622	−1.71099	−0.855496	−	0.517810i	$-0.826747\pi$
$613$	−42.3622	−1.71099	−0.855496	+	0.517810i	$0.826747\pi$
$614$	0	0
$615$	−21.6988	−0.874982
$616$	0	0
$617$	33.9063	1.36502	0.682508	−	0.730878i	$-0.260889\pi$
$617$	33.9063	1.36502	0.682508	+	0.730878i	$0.260889\pi$
$618$	0	0
$619$	−27.6830	−1.11267	−0.556337	−	0.830957i	$-0.687794\pi$
$619$	−27.6830	−1.11267	−0.556337	+	0.830957i	$0.687794\pi$
$620$	0	0
$621$	−5.65486	−0.226922
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−27.1964	−1.08785
$626$	0	0
$627$	46.8851	1.87241
$628$	0	0
$629$	−9.40141	−0.374859
$630$	0	0
$631$	−0.801923	−0.0319240	−0.0159620	−	0.999873i	$-0.505081\pi$
$631$	−0.801923	−0.0319240	−0.0159620	+	0.999873i	$0.505081\pi$
$632$	0	0
$633$	26.1831	1.04068
$634$	0	0
$635$	21.6897	0.860728
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.85126	−0.271032
$640$	0	0
$641$	19.1287	0.755538	0.377769	−	0.925900i	$-0.376691\pi$
$641$	19.1287	0.755538	0.377769	+	0.925900i	$0.376691\pi$
$642$	0	0
$643$	43.8827	1.73056	0.865282	−	0.501286i	$-0.167139\pi$
$643$	43.8827	1.73056	0.865282	+	0.501286i	$0.167139\pi$
$644$	0	0
$645$	−23.3255	−0.918442
$646$	0	0
$647$	−41.0032	−1.61200	−0.806001	−	0.591914i	$-0.798373\pi$
$647$	−41.0032	−1.61200	−0.806001	+	0.591914i	$0.798373\pi$
$648$	0	0
$649$	14.5160	0.569804
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.1744	1.37648	0.688240	−	0.725483i	$-0.258383\pi$
$653$	35.1744	1.37648	0.688240	+	0.725483i	$0.258383\pi$
$654$	0	0
$655$	11.7654	0.459712
$656$	0	0
$657$	−7.04717	−0.274936
$658$	0	0
$659$	−24.7839	−0.965445	−0.482723	−	0.875773i	$-0.660352\pi$
$659$	−24.7839	−0.965445	−0.482723	+	0.875773i	$0.660352\pi$
$660$	0	0
$661$	−41.4545	−1.61239	−0.806197	−	0.591648i	$-0.798478\pi$
$661$	−41.4545	−1.61239	−0.806197	+	0.591648i	$0.798478\pi$
$662$	0	0
$663$	60.2893	2.34144
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.87468	−0.0725880
$668$	0	0
$669$	−28.5011	−1.10192
$670$	0	0
$671$	17.6507	0.681397
$672$	0	0
$673$	−36.0660	−1.39024	−0.695121	−	0.718893i	$-0.744649\pi$
$673$	−36.0660	−1.39024	−0.695121	+	0.718893i	$0.744649\pi$
$674$	0	0
$675$	2.75187	0.105919
$676$	0	0
$677$	−3.41266	−0.131159	−0.0655795	−	0.997847i	$-0.520890\pi$
$677$	−3.41266	−0.131159	−0.0655795	+	0.997847i	$0.520890\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−0.831679	−0.0318700
$682$	0	0
$683$	−24.2314	−0.927188	−0.463594	−	0.886048i	$-0.653440\pi$
$683$	−24.2314	−0.927188	−0.463594	+	0.886048i	$0.653440\pi$
$684$	0	0
$685$	36.8018	1.40612
$686$	0	0
$687$	21.0407	0.802751
$688$	0	0
$689$	42.2693	1.61033
$690$	0	0
$691$	−7.37573	−0.280586	−0.140293	−	0.990110i	$-0.544804\pi$
$691$	−7.37573	−0.280586	−0.140293	+	0.990110i	$0.544804\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.133348	−0.00505818
$696$	0	0
$697$	50.8879	1.92752
$698$	0	0
$699$	−20.9567	−0.792656
$700$	0	0
$701$	3.25950	0.123109	0.0615547	−	0.998104i	$-0.480394\pi$
$701$	3.25950	0.123109	0.0615547	+	0.998104i	$0.480394\pi$
$702$	0	0
$703$	−7.22744	−0.272588
$704$	0	0
$705$	36.5175	1.37533
$706$	0	0
$707$	0	0
$708$	0	0
$709$	43.0749	1.61771	0.808856	−	0.588006i	$-0.200087\pi$
$709$	43.0749	1.61771	0.808856	+	0.588006i	$0.200087\pi$
$710$	0	0
$711$	2.33790	0.0876782
$712$	0	0
$713$	−5.83096	−0.218371
$714$	0	0
$715$	−67.1480	−2.51119
$716$	0	0
$717$	18.5849	0.694066
$718$	0	0
$719$	−0.219003	−0.00816744	−0.00408372	−	0.999992i	$-0.501300\pi$
$719$	−0.219003	−0.00816744	−0.00408372	+	0.999992i	$0.501300\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.78154	0.0662562
$724$	0	0
$725$	0.912291	0.0338816
$726$	0	0
$727$	22.7045	0.842062	0.421031	−	0.907046i	$-0.361668\pi$
$727$	22.7045	0.842062	0.421031	+	0.907046i	$0.361668\pi$
$728$	0	0
$729$	29.3361	1.08652
$730$	0	0
$731$	54.7028	2.02326
$732$	0	0
$733$	−17.9129	−0.661628	−0.330814	−	0.943696i	$-0.607323\pi$
$733$	−17.9129	−0.661628	−0.330814	+	0.943696i	$0.607323\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.84740	−0.362734
$738$	0	0
$739$	−11.3452	−0.417339	−0.208670	−	0.977986i	$-0.566913\pi$
$739$	−11.3452	−0.417339	−0.208670	+	0.977986i	$0.566913\pi$
$740$	0	0
$741$	46.3481	1.70264
$742$	0	0
$743$	9.31548	0.341752	0.170876	−	0.985293i	$-0.445340\pi$
$743$	9.31548	0.341752	0.170876	+	0.985293i	$0.445340\pi$
$744$	0	0
$745$	−3.64570	−0.133568
$746$	0	0
$747$	−15.7751	−0.577180
$748$	0	0
$749$	0	0
$750$	0	0
$751$	5.43047	0.198161	0.0990803	−	0.995079i	$-0.468410\pi$
$751$	5.43047	0.198161	0.0990803	+	0.995079i	$0.468410\pi$
$752$	0	0
$753$	28.5400	1.04005
$754$	0	0
$755$	8.07351	0.293825
$756$	0	0
$757$	−9.50823	−0.345582	−0.172791	−	0.984958i	$-0.555279\pi$
$757$	−9.50823	−0.345582	−0.172791	+	0.984958i	$0.555279\pi$
$758$	0	0
$759$	7.73218	0.280660
$760$	0	0
$761$	−21.6974	−0.786529	−0.393265	−	0.919425i	$-0.628654\pi$
$761$	−21.6974	−0.786529	−0.393265	+	0.919425i	$0.628654\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−17.3359	−0.626782
$766$	0	0
$767$	14.3498	0.518141
$768$	0	0
$769$	10.7850	0.388916	0.194458	−	0.980911i	$-0.437705\pi$
$769$	10.7850	0.388916	0.194458	+	0.980911i	$0.437705\pi$
$770$	0	0
$771$	6.97542	0.251214
$772$	0	0
$773$	25.1118	0.903210	0.451605	−	0.892218i	$-0.350852\pi$
$773$	25.1118	0.903210	0.451605	+	0.892218i	$0.350852\pi$
$774$	0	0
$775$	2.83756	0.101928
$776$	0	0
$777$	0	0
$778$	0	0
$779$	39.1207	1.40164
$780$	0	0
$781$	39.3197	1.40697
$782$	0	0
$783$	10.6011	0.378851
$784$	0	0
$785$	46.9329	1.67511
$786$	0	0
$787$	−1.64538	−0.0586516	−0.0293258	−	0.999570i	$-0.509336\pi$
$787$	−1.64538	−0.0586516	−0.0293258	+	0.999570i	$0.509336\pi$
$788$	0	0
$789$	−16.6214	−0.591739
$790$	0	0
$791$	0	0
$792$	0	0
$793$	17.4485	0.619615
$794$	0	0
$795$	26.7054	0.947144
$796$	0	0
$797$	−12.0243	−0.425923	−0.212962	−	0.977061i	$-0.568311\pi$
$797$	−12.0243	−0.425923	−0.212962	+	0.977061i	$0.568311\pi$
$798$	0	0
$799$	−85.6404	−3.02974
$800$	0	0
$801$	12.4482	0.439837
$802$	0	0
$803$	40.4440	1.42724
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.30059	−0.221791
$808$	0	0
$809$	11.4285	0.401803	0.200901	−	0.979611i	$-0.435613\pi$
$809$	11.4285	0.401803	0.200901	+	0.979611i	$0.435613\pi$
$810$	0	0
$811$	−24.2552	−0.851714	−0.425857	−	0.904790i	$-0.640027\pi$
$811$	−24.2552	−0.851714	−0.425857	+	0.904790i	$0.640027\pi$
$812$	0	0
$813$	−2.07332	−0.0727145
$814$	0	0
$815$	47.9143	1.67837
$816$	0	0
$817$	42.0535	1.47126
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.1394	0.563268	0.281634	−	0.959522i	$-0.409124\pi$
$821$	16.1394	0.563268	0.281634	+	0.959522i	$0.409124\pi$
$822$	0	0
$823$	−5.72512	−0.199565	−0.0997825	−	0.995009i	$-0.531815\pi$
$823$	−5.72512	−0.199565	−0.0997825	+	0.995009i	$0.531815\pi$
$824$	0	0
$825$	−3.76277	−0.131003
$826$	0	0
$827$	−27.6141	−0.960237	−0.480118	−	0.877204i	$-0.659406\pi$
$827$	−27.6141	−0.960237	−0.480118	+	0.877204i	$0.659406\pi$
$828$	0	0
$829$	−27.3694	−0.950580	−0.475290	−	0.879829i	$-0.657657\pi$
$829$	−27.3694	−0.950580	−0.475290	+	0.879829i	$0.657657\pi$
$830$	0	0
$831$	−0.403514	−0.0139977
$832$	0	0
$833$	0	0
$834$	0	0
$835$	42.8755	1.48377
$836$	0	0
$837$	32.9732	1.13972
$838$	0	0
$839$	−9.24538	−0.319186	−0.159593	−	0.987183i	$-0.551018\pi$
$839$	−9.24538	−0.319186	−0.159593	+	0.987183i	$0.551018\pi$
$840$	0	0
$841$	−25.4856	−0.878813
$842$	0	0
$843$	28.0257	0.965257
$844$	0	0
$845$	−35.9283	−1.23597
$846$	0	0
$847$	0	0
$848$	0	0
$849$	39.2128	1.34578
$850$	0	0
$851$	−1.19193	−0.0408589
$852$	0	0
$853$	−48.3601	−1.65582	−0.827910	−	0.560862i	$-0.810470\pi$
$853$	−48.3601	−1.65582	−0.827910	+	0.560862i	$0.810470\pi$
$854$	0	0
$855$	−13.3272	−0.455781
$856$	0	0
$857$	1.42104	0.0485416	0.0242708	−	0.999705i	$-0.492274\pi$
$857$	1.42104	0.0485416	0.0242708	+	0.999705i	$0.492274\pi$
$858$	0	0
$859$	7.68890	0.262342	0.131171	−	0.991360i	$-0.458126\pi$
$859$	7.68890	0.262342	0.131171	+	0.991360i	$0.458126\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.91806	0.0652916	0.0326458	−	0.999467i	$-0.489607\pi$
$863$	1.91806	0.0652916	0.0326458	+	0.999467i	$0.489607\pi$
$864$	0	0
$865$	0.149067	0.00506842
$866$	0	0
$867$	−64.9197	−2.20479
$868$	0	0
$869$	−13.4173	−0.455151
$870$	0	0
$871$	−9.73461	−0.329845
$872$	0	0
$873$	11.2246	0.379896
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−18.2136	−0.615029	−0.307515	−	0.951543i	$-0.599497\pi$
$877$	−18.2136	−0.615029	−0.307515	+	0.951543i	$0.599497\pi$
$878$	0	0
$879$	29.8916	1.00822
$880$	0	0
$881$	42.2662	1.42399	0.711993	−	0.702187i	$-0.247793\pi$
$881$	42.2662	1.42399	0.711993	+	0.702187i	$0.247793\pi$
$882$	0	0
$883$	17.1758	0.578011	0.289006	−	0.957327i	$-0.406675\pi$
$883$	17.1758	0.578011	0.289006	+	0.957327i	$0.406675\pi$
$884$	0	0
$885$	9.06609	0.304753
$886$	0	0
$887$	−11.8742	−0.398696	−0.199348	−	0.979929i	$-0.563882\pi$
$887$	−11.8742	−0.398696	−0.199348	+	0.979929i	$0.563882\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−28.5655	−0.956982
$892$	0	0
$893$	−65.8371	−2.20315
$894$	0	0
$895$	23.5406	0.786875
$896$	0	0
$897$	7.64362	0.255213
$898$	0	0
$899$	10.9312	0.364576
$900$	0	0
$901$	−62.6293	−2.08649
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.323651	−0.0107585
$906$	0	0
$907$	16.7923	0.557579	0.278790	−	0.960352i	$-0.410067\pi$
$907$	16.7923	0.557579	0.278790	+	0.960352i	$0.410067\pi$
$908$	0	0
$909$	−13.5980	−0.451016
$910$	0	0
$911$	42.6355	1.41258	0.706288	−	0.707925i	$-0.250368\pi$
$911$	42.6355	1.41258	0.706288	+	0.707925i	$0.250368\pi$
$912$	0	0
$913$	90.5338	2.99623
$914$	0	0
$915$	11.0239	0.364437
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−19.9742	−0.658887	−0.329444	−	0.944175i	$-0.606861\pi$
$919$	−19.9742	−0.658887	−0.329444	+	0.944175i	$0.606861\pi$
$920$	0	0
$921$	26.0997	0.860016
$922$	0	0
$923$	38.8693	1.27940
$924$	0	0
$925$	0.580038	0.0190715
$926$	0	0
$927$	7.09036	0.232878
$928$	0	0
$929$	−6.02222	−0.197583	−0.0987914	−	0.995108i	$-0.531498\pi$
$929$	−6.02222	−0.197583	−0.0987914	+	0.995108i	$0.531498\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−18.1170	−0.593125
$934$	0	0
$935$	99.4915	3.25372
$936$	0	0
$937$	24.6553	0.805452	0.402726	−	0.915320i	$-0.368063\pi$
$937$	24.6553	0.805452	0.402726	+	0.915320i	$0.368063\pi$
$938$	0	0
$939$	−27.6417	−0.902051
$940$	0	0
$941$	−50.1049	−1.63337	−0.816686	−	0.577082i	$-0.804191\pi$
$941$	−50.1049	−1.63337	−0.816686	+	0.577082i	$0.804191\pi$
$942$	0	0
$943$	6.45169	0.210096
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.9519	0.648351	0.324175	−	0.945997i	$-0.394913\pi$
$947$	19.9519	0.648351	0.324175	+	0.945997i	$0.394913\pi$
$948$	0	0
$949$	39.9807	1.29783
$950$	0	0
$951$	−15.6791	−0.508430
$952$	0	0
$953$	−40.2873	−1.30503	−0.652517	−	0.757774i	$-0.726287\pi$
$953$	−40.2873	−1.30503	−0.652517	+	0.757774i	$0.726287\pi$
$954$	0	0
$955$	20.0306	0.648176
$956$	0	0
$957$	−14.4954	−0.468569
$958$	0	0
$959$	0	0
$960$	0	0
$961$	3.00006	0.0967762
$962$	0	0
$963$	10.2609	0.330652
$964$	0	0
$965$	57.0985	1.83807
$966$	0	0
$967$	−31.2689	−1.00554	−0.502771	−	0.864420i	$-0.667686\pi$
$967$	−31.2689	−1.00554	−0.502771	+	0.864420i	$0.667686\pi$
$968$	0	0
$969$	−68.6729	−2.20609
$970$	0	0
$971$	42.5037	1.36401	0.682005	−	0.731347i	$-0.261108\pi$
$971$	42.5037	1.36401	0.682005	+	0.731347i	$0.261108\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.71967	−0.119125
$976$	0	0
$977$	−33.2556	−1.06394	−0.531971	−	0.846762i	$-0.678549\pi$
$977$	−33.2556	−1.06394	−0.531971	+	0.846762i	$0.678549\pi$
$978$	0	0
$979$	−71.4409	−2.28326
$980$	0	0
$981$	11.8463	0.378224
$982$	0	0
$983$	−46.5016	−1.48317	−0.741586	−	0.670858i	$-0.765926\pi$
$983$	−46.5016	−1.48317	−0.741586	+	0.670858i	$0.765926\pi$
$984$	0	0
$985$	−50.4592	−1.60776
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.93535	0.220531
$990$	0	0
$991$	−31.1965	−0.990991	−0.495496	−	0.868610i	$-0.665014\pi$
$991$	−31.1965	−0.990991	−0.495496	+	0.868610i	$0.665014\pi$
$992$	0	0
$993$	−47.5790	−1.50988
$994$	0	0
$995$	15.2710	0.484124
$996$	0	0
$997$	−7.37962	−0.233715	−0.116857	−	0.993149i	$-0.537282\pi$
$997$	−7.37962	−0.233715	−0.116857	+	0.993149i	$0.537282\pi$
$998$	0	0
$999$	6.74020	0.213251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4508.2.a.f.1.3		5
7.6	odd	2		644.2.a.d.1.3	✓	5
21.20	even	2		5796.2.a.t.1.2		5
28.27	even	2		2576.2.a.bb.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
644.2.a.d.1.3	✓	5	7.6	odd	2
2576.2.a.bb.1.3		5	28.27	even	2
4508.2.a.f.1.3		5	1.1	even	1	trivial
5796.2.a.t.1.2		5	21.20	even	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:	\( N \)	\(=\)	\( 4508 = 2^{2} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4508.a (trivial)

\(f(q)\)	\(=\)	\(q-1.43585 q^{3} -2.34236 q^{5} -0.938323 q^{9} +O(q^{10})\)	\(q-1.43585 q^{3} -2.34236 q^{5} -0.938323 q^{9} +5.38507 q^{11} +5.32339 q^{13} +3.36328 q^{15} -7.88754 q^{17} -6.06364 q^{19} -1.00000 q^{23} +0.486637 q^{25} +5.65486 q^{27} +1.87468 q^{29} +5.83096 q^{31} -7.73218 q^{33} +1.19193 q^{37} -7.64362 q^{39} -6.45169 q^{41} -6.93535 q^{43} +2.19789 q^{45} +10.8577 q^{47} +11.3254 q^{51} +7.94029 q^{53} -12.6138 q^{55} +8.70650 q^{57} +2.69561 q^{59} +3.27771 q^{61} -12.4693 q^{65} -1.82865 q^{67} +1.43585 q^{69} +7.30161 q^{71} +7.51039 q^{73} -0.698740 q^{75} -2.49158 q^{79} -5.30458 q^{81} +16.8120 q^{83} +18.4754 q^{85} -2.69177 q^{87} -13.2665 q^{89} -8.37240 q^{93} +14.2032 q^{95} -11.9624 q^{97} -5.05294 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9	\( 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9} + 2 q^{11} - 13 q^{13} + 4 q^{15} - 4 q^{17} - 12 q^{19} - 5 q^{23} + 19 q^{25} - 15 q^{27} + 13 q^{29} + 3 q^{31} - 24 q^{33} - 4 q^{37} + 3 q^{39} - q^{41} - 8 q^{43} + 16 q^{45} - 5 q^{47} - 16 q^{51} - 8 q^{53} + 2 q^{55} + 12 q^{57} - 12 q^{59} - 20 q^{61} - 12 q^{65} - 12 q^{67} + 3 q^{69} + 9 q^{71} + 9 q^{73} - 35 q^{75} - 8 q^{79} - 11 q^{81} + 28 q^{83} + 16 q^{85} + 15 q^{87} - 32 q^{89} - 15 q^{93} - 36 q^{95} - 4 q^{97} + 28 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 - 2 * q^5 + 10 * q^9 + 2 * q^11 - 13 * q^13 + 4 * q^15 - 4 * q^17 - 12 * q^19 - 5 * q^23 + 19 * q^25 - 15 * q^27 + 13 * q^29 + 3 * q^31 - 24 * q^33 - 4 * q^37 + 3 * q^39 - q^41 - 8 * q^43 + 16 * q^45 - 5 * q^47 - 16 * q^51 - 8 * q^53 + 2 * q^55 + 12 * q^57 - 12 * q^59 - 20 * q^61 - 12 * q^65 - 12 * q^67 + 3 * q^69 + 9 * q^71 + 9 * q^73 - 35 * q^75 - 8 * q^79 - 11 * q^81 + 28 * q^83 + 16 * q^85 + 15 * q^87 - 32 * q^89 - 15 * q^93 - 36 * q^95 - 4 * q^97 + 28 * q^99

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