LMFDB - Embedded newform 4815.2.a.u.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4815,2,Mod(1,4815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4815, 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("4815.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4815 = 3^{2} \cdot 5 \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4815.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:	$38.4479685732$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 15 x^{10} + 49 x^{9} + 71 x^{8} - 278 x^{7} - 92 x^{6} + 649 x^{5} - 127 x^{4} + \cdots - 6 $ x^12 - 3x^11 - 15x^10 + 49x^9 + 71x^8 - 278x^7 - 92x^6 + 649x^5 - 127x^4 - 529x^3 + 267x^2 + 15*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1605)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.155707$ of defining polynomial
Character	$\chi$	$=$	4815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.155707 q^{2} -1.97576 q^{4} -1.00000 q^{5} +3.91791 q^{7} -0.619053 q^{8} +O(q^{10})$	\(q+0.155707 q^{2} -1.97576 q^{4} -1.00000 q^{5} +3.91791 q^{7} -0.619053 q^{8} -0.155707 q^{10} +4.20576 q^{11} +6.15258 q^{13} +0.610045 q^{14} +3.85512 q^{16} +4.53299 q^{17} +6.39031 q^{19} +1.97576 q^{20} +0.654866 q^{22} +1.55096 q^{23} +1.00000 q^{25} +0.957999 q^{26} -7.74083 q^{28} +3.34760 q^{29} -4.12813 q^{31} +1.83837 q^{32} +0.705817 q^{34} -3.91791 q^{35} +7.16219 q^{37} +0.995015 q^{38} +0.619053 q^{40} -7.18328 q^{41} -9.32016 q^{43} -8.30956 q^{44} +0.241495 q^{46} -11.9785 q^{47} +8.35000 q^{49} +0.155707 q^{50} -12.1560 q^{52} +12.4671 q^{53} -4.20576 q^{55} -2.42539 q^{56} +0.521244 q^{58} -6.02997 q^{59} +9.14686 q^{61} -0.642779 q^{62} -7.42399 q^{64} -6.15258 q^{65} +5.70713 q^{67} -8.95607 q^{68} -0.610045 q^{70} -7.85627 q^{71} -9.15535 q^{73} +1.11520 q^{74} -12.6257 q^{76} +16.4778 q^{77} +3.30344 q^{79} -3.85512 q^{80} -1.11849 q^{82} -3.87613 q^{83} -4.53299 q^{85} -1.45121 q^{86} -2.60359 q^{88} +6.68613 q^{89} +24.1052 q^{91} -3.06432 q^{92} -1.86514 q^{94} -6.39031 q^{95} -6.56241 q^{97} +1.30015 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8	$ 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8} + 3 q^{10} - 4 q^{11} + 13 q^{13} - 4 q^{14} + 13 q^{16} + 4 q^{17} + 14 q^{19} - 15 q^{20} + 15 q^{22} - 11 q^{23} + 12 q^{25} + 8 q^{26} + 16 q^{28} + 7 q^{29} + 4 q^{31} - 4 q^{32} + q^{34} - 7 q^{35} + 24 q^{37} + 11 q^{38} + 3 q^{40} - 13 q^{41} + 25 q^{43} - 10 q^{44} - 22 q^{46} - 19 q^{47} + 9 q^{49} - 3 q^{50} + 20 q^{52} - 11 q^{53} + 4 q^{55} + 37 q^{56} - 2 q^{58} - 8 q^{59} + 7 q^{61} + 11 q^{62} - 19 q^{64} - 13 q^{65} + 33 q^{67} + 24 q^{68} + 4 q^{70} + 34 q^{73} + 27 q^{74} - 9 q^{76} + 29 q^{77} - 13 q^{80} + q^{82} + 24 q^{83} - 4 q^{85} + 36 q^{86} - 6 q^{88} + 10 q^{89} + 30 q^{91} + 28 q^{92} - 8 q^{94} - 14 q^{95} + 16 q^{97} + 36 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8 + 3 * q^10 - 4 * q^11 + 13 * q^13 - 4 * q^14 + 13 * q^16 + 4 * q^17 + 14 * q^19 - 15 * q^20 + 15 * q^22 - 11 * q^23 + 12 * q^25 + 8 * q^26 + 16 * q^28 + 7 * q^29 + 4 * q^31 - 4 * q^32 + q^34 - 7 * q^35 + 24 * q^37 + 11 * q^38 + 3 * q^40 - 13 * q^41 + 25 * q^43 - 10 * q^44 - 22 * q^46 - 19 * q^47 + 9 * q^49 - 3 * q^50 + 20 * q^52 - 11 * q^53 + 4 * q^55 + 37 * q^56 - 2 * q^58 - 8 * q^59 + 7 * q^61 + 11 * q^62 - 19 * q^64 - 13 * q^65 + 33 * q^67 + 24 * q^68 + 4 * q^70 + 34 * q^73 + 27 * q^74 - 9 * q^76 + 29 * q^77 - 13 * q^80 + q^82 + 24 * q^83 - 4 * q^85 + 36 * q^86 - 6 * q^88 + 10 * q^89 + 30 * q^91 + 28 * q^92 - 8 * q^94 - 14 * q^95 + 16 * q^97 + 36 * q^98

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 $

$2$	0.155707	0.110101	0.0550507	−	0.998484i	$-0.482468\pi$
$2$	0.155707	0.110101	0.0550507	+	0.998484i	$0.482468\pi$
$3$	0	0
$3$	0	0
$4$	−1.97576	−0.987878
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.91791	1.48083	0.740415	−	0.672150i	$-0.234629\pi$
$7$	3.91791	1.48083	0.740415	+	0.672150i	$0.234629\pi$
$8$	−0.619053	−0.218868
$9$	0	0
$10$	−0.155707	−0.0492389
$11$	4.20576	1.26809	0.634043	−	0.773298i	$-0.281394\pi$
$11$	4.20576	1.26809	0.634043	+	0.773298i	$0.281394\pi$
$12$	0	0
$13$	6.15258	1.70642	0.853209	−	0.521568i	$-0.174653\pi$
$13$	6.15258	1.70642	0.853209	+	0.521568i	$0.174653\pi$
$14$	0.610045	0.163041
$15$	0	0
$16$	3.85512	0.963780
$17$	4.53299	1.09941	0.549705	−	0.835359i	$-0.314740\pi$
$17$	4.53299	1.09941	0.549705	+	0.835359i	$0.314740\pi$
$18$	0	0
$19$	6.39031	1.46604	0.733018	−	0.680209i	$-0.238111\pi$
$19$	6.39031	1.46604	0.733018	+	0.680209i	$0.238111\pi$
$20$	1.97576	0.441792
$21$	0	0
$22$	0.654866	0.139618
$23$	1.55096	0.323397	0.161699	−	0.986840i	$-0.448303\pi$
$23$	1.55096	0.323397	0.161699	+	0.986840i	$0.448303\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.957999	0.187879
$27$	0	0
$28$	−7.74083	−1.46288
$29$	3.34760	0.621634	0.310817	−	0.950470i	$-0.399397\pi$
$29$	3.34760	0.621634	0.310817	+	0.950470i	$0.399397\pi$
$30$	0	0
$31$	−4.12813	−0.741434	−0.370717	−	0.928746i	$-0.620888\pi$
$31$	−4.12813	−0.741434	−0.370717	+	0.928746i	$0.620888\pi$
$32$	1.83837	0.324982
$33$	0	0
$34$	0.705817	0.121047
$35$	−3.91791	−0.662247
$36$	0	0
$37$	7.16219	1.17746	0.588728	−	0.808331i	$-0.299629\pi$
$37$	7.16219	1.17746	0.588728	+	0.808331i	$0.299629\pi$
$38$	0.995015	0.161413
$39$	0	0
$40$	0.619053	0.0978808
$41$	−7.18328	−1.12184	−0.560920	−	0.827870i	$-0.689552\pi$
$41$	−7.18328	−1.12184	−0.560920	+	0.827870i	$0.689552\pi$
$42$	0	0
$43$	−9.32016	−1.42131	−0.710656	−	0.703540i	$-0.751602\pi$
$43$	−9.32016	−1.42131	−0.710656	+	0.703540i	$0.751602\pi$
$44$	−8.30956	−1.25271
$45$	0	0
$46$	0.241495	0.0356065
$47$	−11.9785	−1.74725	−0.873623	−	0.486603i	$-0.838236\pi$
$47$	−11.9785	−1.74725	−0.873623	+	0.486603i	$0.838236\pi$
$48$	0	0
$49$	8.35000	1.19286
$50$	0.155707	0.0220203
$51$	0	0
$52$	−12.1560	−1.68573
$53$	12.4671	1.71249	0.856247	−	0.516567i	$-0.172790\pi$
$53$	12.4671	1.71249	0.856247	+	0.516567i	$0.172790\pi$
$54$	0	0
$55$	−4.20576	−0.567105
$56$	−2.42539	−0.324107
$57$	0	0
$58$	0.521244	0.0684428
$59$	−6.02997	−0.785035	−0.392518	−	0.919744i	$-0.628396\pi$
$59$	−6.02997	−0.785035	−0.392518	+	0.919744i	$0.628396\pi$
$60$	0	0
$61$	9.14686	1.17113	0.585567	−	0.810624i	$-0.300872\pi$
$61$	9.14686	1.17113	0.585567	+	0.810624i	$0.300872\pi$
$62$	−0.642779	−0.0816330
$63$	0	0
$64$	−7.42399	−0.927999
$65$	−6.15258	−0.763134
$66$	0	0
$67$	5.70713	0.697237	0.348619	−	0.937265i	$-0.386651\pi$
$67$	5.70713	0.697237	0.348619	+	0.937265i	$0.386651\pi$
$68$	−8.95607	−1.08608
$69$	0	0
$70$	−0.610045	−0.0729144
$71$	−7.85627	−0.932368	−0.466184	−	0.884688i	$-0.654372\pi$
$71$	−7.85627	−0.932368	−0.466184	+	0.884688i	$0.654372\pi$
$72$	0	0
$73$	−9.15535	−1.07155	−0.535776	−	0.844360i	$-0.679981\pi$
$73$	−9.15535	−1.07155	−0.535776	+	0.844360i	$0.679981\pi$
$74$	1.11520	0.129640
$75$	0	0
$76$	−12.6257	−1.44827
$77$	16.4778	1.87782
$78$	0	0
$79$	3.30344	0.371666	0.185833	−	0.982581i	$-0.440502\pi$
$79$	3.30344	0.371666	0.185833	+	0.982581i	$0.440502\pi$
$80$	−3.85512	−0.431016
$81$	0	0
$82$	−1.11849	−0.123516
$83$	−3.87613	−0.425460	−0.212730	−	0.977111i	$-0.568236\pi$
$83$	−3.87613	−0.425460	−0.212730	+	0.977111i	$0.568236\pi$
$84$	0	0
$85$	−4.53299	−0.491671
$86$	−1.45121	−0.156488
$87$	0	0
$88$	−2.60359	−0.277543
$89$	6.68613	0.708728	0.354364	−	0.935107i	$-0.384697\pi$
$89$	6.68613	0.708728	0.354364	+	0.935107i	$0.384697\pi$
$90$	0	0
$91$	24.1052	2.52692
$92$	−3.06432	−0.319477
$93$	0	0
$94$	−1.86514	−0.192374
$95$	−6.39031	−0.655632
$96$	0	0
$97$	−6.56241	−0.666312	−0.333156	−	0.942872i	$-0.608114\pi$
$97$	−6.56241	−0.666312	−0.333156	+	0.942872i	$0.608114\pi$
$98$	1.30015	0.131335
$99$	0	0
$100$	−1.97576	−0.197576
$101$	−1.61730	−0.160927	−0.0804637	−	0.996758i	$-0.525640\pi$
$101$	−1.61730	−0.160927	−0.0804637	+	0.996758i	$0.525640\pi$
$102$	0	0
$103$	−2.47404	−0.243775	−0.121887	−	0.992544i	$-0.538895\pi$
$103$	−2.47404	−0.243775	−0.121887	+	0.992544i	$0.538895\pi$
$104$	−3.80877	−0.373481
$105$	0	0
$106$	1.94122	0.188548
$107$	−1.00000	−0.0966736
$107$	−1.00000	−0.0966736
$108$	0	0
$109$	16.1850	1.55024	0.775121	−	0.631812i	$-0.217689\pi$
$109$	16.1850	1.55024	0.775121	+	0.631812i	$0.217689\pi$
$110$	−0.654866	−0.0624391
$111$	0	0
$112$	15.1040	1.42719
$113$	−16.5476	−1.55667	−0.778334	−	0.627851i	$-0.783935\pi$
$113$	−16.5476	−1.55667	−0.778334	+	0.627851i	$0.783935\pi$
$114$	0	0
$115$	−1.55096	−0.144628
$116$	−6.61404	−0.614098
$117$	0	0
$118$	−0.938908	−0.0864335
$119$	17.7598	1.62804
$120$	0	0
$121$	6.68844	0.608040
$122$	1.42423	0.128944
$123$	0	0
$124$	8.15618	0.732447
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	21.3131	1.89123	0.945615	−	0.325288i	$-0.105461\pi$
$127$	21.3131	1.89123	0.945615	+	0.325288i	$0.105461\pi$
$128$	−4.83272	−0.427156
$129$	0	0
$130$	−0.957999	−0.0840221
$131$	−7.55127	−0.659758	−0.329879	−	0.944023i	$-0.607008\pi$
$131$	−7.55127	−0.659758	−0.329879	+	0.944023i	$0.607008\pi$
$132$	0	0
$133$	25.0366	2.17095
$134$	0.888640	0.0767668
$135$	0	0
$136$	−2.80616	−0.240626
$137$	13.2009	1.12783	0.563915	−	0.825833i	$-0.309294\pi$
$137$	13.2009	1.12783	0.563915	+	0.825833i	$0.309294\pi$
$138$	0	0
$139$	−13.2352	−1.12260	−0.561298	−	0.827614i	$-0.689698\pi$
$139$	−13.2352	−1.12260	−0.561298	+	0.827614i	$0.689698\pi$
$140$	7.74083	0.654219
$141$	0	0
$142$	−1.22328	−0.102655
$143$	25.8763	2.16388
$144$	0	0
$145$	−3.34760	−0.278003
$146$	−1.42555	−0.117979
$147$	0	0
$148$	−14.1507	−1.16318
$149$	0.813693	0.0666603	0.0333302	−	0.999444i	$-0.489389\pi$
$149$	0.813693	0.0666603	0.0333302	+	0.999444i	$0.489389\pi$
$150$	0	0
$151$	−5.50264	−0.447799	−0.223899	−	0.974612i	$-0.571879\pi$
$151$	−5.50264	−0.447799	−0.223899	+	0.974612i	$0.571879\pi$
$152$	−3.95594	−0.320869
$153$	0	0
$154$	2.56571	0.206750
$155$	4.12813	0.331580
$156$	0	0
$157$	−0.601189	−0.0479801	−0.0239900	−	0.999712i	$-0.507637\pi$
$157$	−0.601189	−0.0479801	−0.0239900	+	0.999712i	$0.507637\pi$
$158$	0.514369	0.0409210
$159$	0	0
$160$	−1.83837	−0.145336
$161$	6.07652	0.478897
$162$	0	0
$163$	−21.6686	−1.69721	−0.848607	−	0.529025i	$-0.822558\pi$
$163$	−21.6686	−1.69721	−0.848607	+	0.529025i	$0.822558\pi$
$164$	14.1924	1.10824
$165$	0	0
$166$	−0.603540	−0.0468438
$167$	−6.89307	−0.533402	−0.266701	−	0.963779i	$-0.585934\pi$
$167$	−6.89307	−0.533402	−0.266701	+	0.963779i	$0.585934\pi$
$168$	0	0
$169$	24.8543	1.91187
$170$	−0.705817	−0.0541337
$171$	0	0
$172$	18.4144	1.40408
$173$	−13.4977	−1.02621	−0.513107	−	0.858325i	$-0.671506\pi$
$173$	−13.4977	−1.02621	−0.513107	+	0.858325i	$0.671506\pi$
$174$	0	0
$175$	3.91791	0.296166
$176$	16.2137	1.22216
$177$	0	0
$178$	1.04108	0.0780320
$179$	23.0759	1.72478	0.862389	−	0.506247i	$-0.168968\pi$
$179$	23.0759	1.72478	0.862389	+	0.506247i	$0.168968\pi$
$180$	0	0
$181$	3.23219	0.240247	0.120123	−	0.992759i	$-0.461671\pi$
$181$	3.23219	0.240247	0.120123	+	0.992759i	$0.461671\pi$
$182$	3.75335	0.278217
$183$	0	0
$184$	−0.960125	−0.0707814
$185$	−7.16219	−0.526574
$186$	0	0
$187$	19.0647	1.39415
$188$	23.6666	1.72607
$189$	0	0
$190$	−0.995015	−0.0721860
$191$	−7.48523	−0.541612	−0.270806	−	0.962634i	$-0.587290\pi$
$191$	−7.48523	−0.541612	−0.270806	+	0.962634i	$0.587290\pi$
$192$	0	0
$193$	−23.6594	−1.70304	−0.851519	−	0.524324i	$-0.824318\pi$
$193$	−23.6594	−1.70304	−0.851519	+	0.524324i	$0.824318\pi$
$194$	−1.02181	−0.0733619
$195$	0	0
$196$	−16.4976	−1.17840
$197$	0.502435	0.0357970	0.0178985	−	0.999840i	$-0.494302\pi$
$197$	0.502435	0.0357970	0.0178985	+	0.999840i	$0.494302\pi$
$198$	0	0
$199$	12.0176	0.851906	0.425953	−	0.904745i	$-0.359939\pi$
$199$	12.0176	0.851906	0.425953	+	0.904745i	$0.359939\pi$
$200$	−0.619053	−0.0437736
$201$	0	0
$202$	−0.251825	−0.0177183
$203$	13.1156	0.920534
$204$	0	0
$205$	7.18328	0.501702
$206$	−0.385226	−0.0268399
$207$	0	0
$208$	23.7189	1.64461
$209$	26.8761	1.85906
$210$	0	0
$211$	−22.5450	−1.55206	−0.776032	−	0.630694i	$-0.782770\pi$
$211$	−22.5450	−1.55206	−0.776032	+	0.630694i	$0.782770\pi$
$212$	−24.6320	−1.69173
$213$	0	0
$214$	−0.155707	−0.0106439
$215$	9.32016	0.635630
$216$	0	0
$217$	−16.1736	−1.09794
$218$	2.52012	0.170684
$219$	0	0
$220$	8.30956	0.560230
$221$	27.8896	1.87606
$222$	0	0
$223$	−11.2208	−0.751399	−0.375700	−	0.926741i	$-0.622598\pi$
$223$	−11.2208	−0.751399	−0.375700	+	0.926741i	$0.622598\pi$
$224$	7.20258	0.481243
$225$	0	0
$226$	−2.57658	−0.171391
$227$	−3.79410	−0.251823	−0.125912	−	0.992041i	$-0.540186\pi$
$227$	−3.79410	−0.251823	−0.125912	+	0.992041i	$0.540186\pi$
$228$	0	0
$229$	−21.8196	−1.44188	−0.720941	−	0.692996i	$-0.756290\pi$
$229$	−21.8196	−1.44188	−0.720941	+	0.692996i	$0.756290\pi$
$230$	−0.241495	−0.0159237
$231$	0	0
$232$	−2.07234	−0.136056
$233$	−2.52926	−0.165698	−0.0828488	−	0.996562i	$-0.526402\pi$
$233$	−2.52926	−0.165698	−0.0828488	+	0.996562i	$0.526402\pi$
$234$	0	0
$235$	11.9785	0.781392
$236$	11.9137	0.775519
$237$	0	0
$238$	2.76533	0.179250
$239$	0.0952535	0.00616144	0.00308072	−	0.999995i	$-0.499019\pi$
$239$	0.0952535	0.00616144	0.00308072	+	0.999995i	$0.499019\pi$
$240$	0	0
$241$	−19.8084	−1.27597	−0.637984	−	0.770050i	$-0.720231\pi$
$241$	−19.8084	−1.27597	−0.637984	+	0.770050i	$0.720231\pi$
$242$	1.04144	0.0669461
$243$	0	0
$244$	−18.0719	−1.15694
$245$	−8.35000	−0.533462
$246$	0	0
$247$	39.3169	2.50167
$248$	2.55553	0.162276
$249$	0	0
$250$	−0.155707	−0.00984777
$251$	6.15527	0.388517	0.194259	−	0.980950i	$-0.437770\pi$
$251$	6.15527	0.388517	0.194259	+	0.980950i	$0.437770\pi$
$252$	0	0
$253$	6.52297	0.410095
$254$	3.31859	0.208227
$255$	0	0
$256$	14.0955	0.880969
$257$	−11.0517	−0.689384	−0.344692	−	0.938716i	$-0.612017\pi$
$257$	−11.0517	−0.689384	−0.344692	+	0.938716i	$0.612017\pi$
$258$	0	0
$259$	28.0608	1.74361
$260$	12.1560	0.753883
$261$	0	0
$262$	−1.17579	−0.0726403
$263$	−17.1372	−1.05672	−0.528361	−	0.849020i	$-0.677193\pi$
$263$	−17.1372	−1.05672	−0.528361	+	0.849020i	$0.677193\pi$
$264$	0	0
$265$	−12.4671	−0.765850
$266$	3.89838	0.239025
$267$	0	0
$268$	−11.2759	−0.688785
$269$	2.71376	0.165461	0.0827303	−	0.996572i	$-0.473636\pi$
$269$	2.71376	0.165461	0.0827303	+	0.996572i	$0.473636\pi$
$270$	0	0
$271$	−8.31668	−0.505202	−0.252601	−	0.967571i	$-0.581286\pi$
$271$	−8.31668	−0.505202	−0.252601	+	0.967571i	$0.581286\pi$
$272$	17.4752	1.05959
$273$	0	0
$274$	2.05547	0.124176
$275$	4.20576	0.253617
$276$	0	0
$277$	0.169617	0.0101913	0.00509565	−	0.999987i	$-0.498378\pi$
$277$	0.169617	0.0101913	0.00509565	+	0.999987i	$0.498378\pi$
$278$	−2.06081	−0.123599
$279$	0	0
$280$	2.42539	0.144945
$281$	29.6929	1.77133	0.885665	−	0.464324i	$-0.153703\pi$
$281$	29.6929	1.77133	0.885665	+	0.464324i	$0.153703\pi$
$282$	0	0
$283$	−11.3215	−0.672995	−0.336497	−	0.941684i	$-0.609242\pi$
$283$	−11.3215	−0.672995	−0.336497	+	0.941684i	$0.609242\pi$
$284$	15.5221	0.921066
$285$	0	0
$286$	4.02912	0.238247
$287$	−28.1434	−1.66125
$288$	0	0
$289$	3.54797	0.208704
$290$	−0.521244	−0.0306085
$291$	0	0
$292$	18.0887	1.05856
$293$	32.0481	1.87227	0.936134	−	0.351642i	$-0.114377\pi$
$293$	32.0481	1.87227	0.936134	+	0.351642i	$0.114377\pi$
$294$	0	0
$295$	6.02997	0.351078
$296$	−4.43377	−0.257708
$297$	0	0
$298$	0.126698	0.00733940
$299$	9.54240	0.551851
$300$	0	0
$301$	−36.5155	−2.10472
$302$	−0.856799	−0.0493033
$303$	0	0
$304$	24.6354	1.41294
$305$	−9.14686	−0.523747
$306$	0	0
$307$	−3.07724	−0.175627	−0.0878135	−	0.996137i	$-0.527988\pi$
$307$	−3.07724	−0.175627	−0.0878135	+	0.996137i	$0.527988\pi$
$308$	−32.5561	−1.85505
$309$	0	0
$310$	0.642779	0.0365074
$311$	−12.6191	−0.715562	−0.357781	−	0.933806i	$-0.616467\pi$
$311$	−12.6191	−0.715562	−0.357781	+	0.933806i	$0.616467\pi$
$312$	0	0
$313$	−24.2861	−1.37273	−0.686364	−	0.727258i	$-0.740794\pi$
$313$	−24.2861	−1.37273	−0.686364	+	0.727258i	$0.740794\pi$
$314$	−0.0936093	−0.00528268
$315$	0	0
$316$	−6.52679	−0.367161
$317$	−9.52720	−0.535101	−0.267550	−	0.963544i	$-0.586214\pi$
$317$	−9.52720	−0.535101	−0.267550	+	0.963544i	$0.586214\pi$
$318$	0	0
$319$	14.0792	0.788284
$320$	7.42399	0.415014
$321$	0	0
$322$	0.946156	0.0527272
$323$	28.9672	1.61178
$324$	0	0
$325$	6.15258	0.341284
$326$	−3.37395	−0.186866
$327$	0	0
$328$	4.44683	0.245535
$329$	−46.9307	−2.58737
$330$	0	0
$331$	−0.655437	−0.0360261	−0.0180130	−	0.999838i	$-0.505734\pi$
$331$	−0.655437	−0.0360261	−0.0180130	+	0.999838i	$0.505734\pi$
$332$	7.65828	0.420303
$333$	0	0
$334$	−1.07330	−0.0587283
$335$	−5.70713	−0.311814
$336$	0	0
$337$	−17.5012	−0.953352	−0.476676	−	0.879079i	$-0.658159\pi$
$337$	−17.5012	−0.953352	−0.476676	+	0.879079i	$0.658159\pi$
$338$	3.86998	0.210499
$339$	0	0
$340$	8.95607	0.485711
$341$	−17.3619	−0.940202
$342$	0	0
$343$	5.28919	0.285589
$344$	5.76967	0.311080
$345$	0	0
$346$	−2.10169	−0.112988
$347$	9.99897	0.536773	0.268386	−	0.963311i	$-0.413510\pi$
$347$	9.99897	0.536773	0.268386	+	0.963311i	$0.413510\pi$
$348$	0	0
$349$	3.77143	0.201880	0.100940	−	0.994893i	$-0.467815\pi$
$349$	3.77143	0.201880	0.100940	+	0.994893i	$0.467815\pi$
$350$	0.610045	0.0326083
$351$	0	0
$352$	7.73177	0.412104
$353$	−17.2527	−0.918267	−0.459133	−	0.888367i	$-0.651840\pi$
$353$	−17.2527	−0.918267	−0.459133	+	0.888367i	$0.651840\pi$
$354$	0	0
$355$	7.85627	0.416968
$356$	−13.2102	−0.700137
$357$	0	0
$358$	3.59308	0.189900
$359$	−20.3780	−1.07551	−0.537755	−	0.843101i	$-0.680728\pi$
$359$	−20.3780	−1.07551	−0.537755	+	0.843101i	$0.680728\pi$
$360$	0	0
$361$	21.8360	1.14926
$362$	0.503275	0.0264515
$363$	0	0
$364$	−47.6261	−2.49628
$365$	9.15535	0.479213
$366$	0	0
$367$	17.4162	0.909119	0.454560	−	0.890716i	$-0.349797\pi$
$367$	17.4162	0.909119	0.454560	+	0.890716i	$0.349797\pi$
$368$	5.97913	0.311684
$369$	0	0
$370$	−1.11520	−0.0579766
$371$	48.8451	2.53591
$372$	0	0
$373$	−0.0544363	−0.00281861	−0.00140930	−	0.999999i	$-0.500449\pi$
$373$	−0.0544363	−0.00281861	−0.00140930	+	0.999999i	$0.500449\pi$
$374$	2.96850	0.153498
$375$	0	0
$376$	7.41533	0.382417
$377$	20.5964	1.06077
$378$	0	0
$379$	5.64875	0.290157	0.145079	−	0.989420i	$-0.453657\pi$
$379$	5.64875	0.290157	0.145079	+	0.989420i	$0.453657\pi$
$380$	12.6257	0.647684
$381$	0	0
$382$	−1.16550	−0.0596323
$383$	33.3118	1.70216	0.851078	−	0.525040i	$-0.175950\pi$
$383$	33.3118	1.70216	0.851078	+	0.525040i	$0.175950\pi$
$384$	0	0
$385$	−16.4778	−0.839786
$386$	−3.68393	−0.187507
$387$	0	0
$388$	12.9657	0.658235
$389$	8.32421	0.422054	0.211027	−	0.977480i	$-0.432319\pi$
$389$	8.32421	0.422054	0.211027	+	0.977480i	$0.432319\pi$
$390$	0	0
$391$	7.03048	0.355547
$392$	−5.16909	−0.261079
$393$	0	0
$394$	0.0782325	0.00394130
$395$	−3.30344	−0.166214
$396$	0	0
$397$	7.56190	0.379521	0.189760	−	0.981830i	$-0.439229\pi$
$397$	7.56190	0.379521	0.189760	+	0.981830i	$0.439229\pi$
$398$	1.87123	0.0937961
$399$	0	0
$400$	3.85512	0.192756
$401$	13.2500	0.661676	0.330838	−	0.943688i	$-0.392669\pi$
$401$	13.2500	0.661676	0.330838	+	0.943688i	$0.392669\pi$
$402$	0	0
$403$	−25.3987	−1.26520
$404$	3.19539	0.158977
$405$	0	0
$406$	2.04219	0.101352
$407$	30.1225	1.49311
$408$	0	0
$409$	−26.6915	−1.31981	−0.659906	−	0.751348i	$-0.729404\pi$
$409$	−26.6915	−1.31981	−0.659906	+	0.751348i	$0.729404\pi$
$410$	1.11849	0.0552381
$411$	0	0
$412$	4.88810	0.240820
$413$	−23.6249	−1.16250
$414$	0	0
$415$	3.87613	0.190272
$416$	11.3107	0.554555
$417$	0	0
$418$	4.18480	0.204685
$419$	26.2407	1.28194	0.640971	−	0.767565i	$-0.278532\pi$
$419$	26.2407	1.28194	0.640971	+	0.767565i	$0.278532\pi$
$420$	0	0
$421$	23.0876	1.12522	0.562609	−	0.826723i	$-0.309798\pi$
$421$	23.0876	1.12522	0.562609	+	0.826723i	$0.309798\pi$
$422$	−3.51042	−0.170884
$423$	0	0
$424$	−7.71782	−0.374810
$425$	4.53299	0.219882
$426$	0	0
$427$	35.8365	1.73425
$428$	1.97576	0.0955017
$429$	0	0
$430$	1.45121	0.0699838
$431$	22.4525	1.08150	0.540748	−	0.841184i	$-0.318141\pi$
$431$	22.4525	1.08150	0.540748	+	0.841184i	$0.318141\pi$
$432$	0	0
$433$	−10.3947	−0.499538	−0.249769	−	0.968305i	$-0.580355\pi$
$433$	−10.3947	−0.499538	−0.249769	+	0.968305i	$0.580355\pi$
$434$	−2.51835	−0.120885
$435$	0	0
$436$	−31.9776	−1.53145
$437$	9.91111	0.474112
$438$	0	0
$439$	−31.2659	−1.49224	−0.746120	−	0.665812i	$-0.768085\pi$
$439$	−31.2659	−1.49224	−0.746120	+	0.665812i	$0.768085\pi$
$440$	2.60359	0.124121
$441$	0	0
$442$	4.34260	0.206556
$443$	3.36078	0.159676	0.0798378	−	0.996808i	$-0.474560\pi$
$443$	3.36078	0.159676	0.0798378	+	0.996808i	$0.474560\pi$
$444$	0	0
$445$	−6.68613	−0.316953
$446$	−1.74715	−0.0827301
$447$	0	0
$448$	−29.0865	−1.37421
$449$	−5.41796	−0.255689	−0.127845	−	0.991794i	$-0.540806\pi$
$449$	−5.41796	−0.255689	−0.127845	+	0.991794i	$0.540806\pi$
$450$	0	0
$451$	−30.2112	−1.42259
$452$	32.6940	1.53780
$453$	0	0
$454$	−0.590768	−0.0277261
$455$	−24.1052	−1.13007
$456$	0	0
$457$	14.4160	0.674354	0.337177	−	0.941441i	$-0.390528\pi$
$457$	14.4160	0.674354	0.337177	+	0.941441i	$0.390528\pi$
$458$	−3.39747	−0.158753
$459$	0	0
$460$	3.06432	0.142874
$461$	−28.3177	−1.31889	−0.659444	−	0.751754i	$-0.729208\pi$
$461$	−28.3177	−1.31889	−0.659444	+	0.751754i	$0.729208\pi$
$462$	0	0
$463$	17.7568	0.825227	0.412614	−	0.910906i	$-0.364616\pi$
$463$	17.7568	0.825227	0.412614	+	0.910906i	$0.364616\pi$
$464$	12.9054	0.599118
$465$	0	0
$466$	−0.393824	−0.0182435
$467$	32.4812	1.50305	0.751525	−	0.659705i	$-0.229319\pi$
$467$	32.4812	1.50305	0.751525	+	0.659705i	$0.229319\pi$
$468$	0	0
$469$	22.3600	1.03249
$470$	1.86514	0.0860324
$471$	0	0
$472$	3.73287	0.171819
$473$	−39.1984	−1.80234
$474$	0	0
$475$	6.39031	0.293207
$476$	−35.0891	−1.60830
$477$	0	0
$478$	0.0148316	0.000678383 0
$479$	−32.5182	−1.48579	−0.742897	−	0.669405i	$-0.766549\pi$
$479$	−32.5182	−1.48579	−0.742897	+	0.669405i	$0.766549\pi$
$480$	0	0
$481$	44.0659	2.00923
$482$	−3.08430	−0.140486
$483$	0	0
$484$	−13.2147	−0.600669
$485$	6.56241	0.297984
$486$	0	0
$487$	−34.8343	−1.57849	−0.789245	−	0.614078i	$-0.789528\pi$
$487$	−34.8343	−1.57849	−0.789245	+	0.614078i	$0.789528\pi$
$488$	−5.66238	−0.256324
$489$	0	0
$490$	−1.30015	−0.0587349
$491$	9.42858	0.425506	0.212753	−	0.977106i	$-0.431757\pi$
$491$	9.42858	0.425506	0.212753	+	0.977106i	$0.431757\pi$
$492$	0	0
$493$	15.1746	0.683431
$494$	6.12191	0.275438
$495$	0	0
$496$	−15.9144	−0.714580
$497$	−30.7802	−1.38068
$498$	0	0
$499$	11.1288	0.498192	0.249096	−	0.968479i	$-0.419866\pi$
$499$	11.1288	0.498192	0.249096	+	0.968479i	$0.419866\pi$
$500$	1.97576	0.0883585
$501$	0	0
$502$	0.958418	0.0427763
$503$	5.12488	0.228507	0.114253	−	0.993452i	$-0.463552\pi$
$503$	5.12488	0.228507	0.114253	+	0.993452i	$0.463552\pi$
$504$	0	0
$505$	1.61730	0.0719689
$506$	1.01567	0.0451521
$507$	0	0
$508$	−42.1094	−1.86830
$509$	32.5419	1.44240	0.721198	−	0.692729i	$-0.243592\pi$
$509$	32.5419	1.44240	0.721198	+	0.692729i	$0.243592\pi$
$510$	0	0
$511$	−35.8698	−1.58679
$512$	11.8602	0.524152
$513$	0	0
$514$	−1.72082	−0.0759021
$515$	2.47404	0.109019
$516$	0	0
$517$	−50.3788	−2.21566
$518$	4.36926	0.191974
$519$	0	0
$520$	3.80877	0.167026
$521$	29.6282	1.29803	0.649017	−	0.760774i	$-0.275180\pi$
$521$	29.6282	1.29803	0.649017	+	0.760774i	$0.275180\pi$
$522$	0	0
$523$	18.7648	0.820528	0.410264	−	0.911967i	$-0.365437\pi$
$523$	18.7648	0.820528	0.410264	+	0.911967i	$0.365437\pi$
$524$	14.9195	0.651760
$525$	0	0
$526$	−2.66837	−0.116347
$527$	−18.7128	−0.815141
$528$	0	0
$529$	−20.5945	−0.895414
$530$	−1.94122	−0.0843212
$531$	0	0
$532$	−49.4663	−2.14463
$533$	−44.1957	−1.91433
$534$	0	0
$535$	1.00000	0.0432338
$536$	−3.53302	−0.152603
$537$	0	0
$538$	0.422551	0.0182175
$539$	35.1181	1.51264
$540$	0	0
$541$	−33.0478	−1.42084	−0.710419	−	0.703779i	$-0.751494\pi$
$541$	−33.0478	−1.42084	−0.710419	+	0.703779i	$0.751494\pi$
$542$	−1.29496	−0.0556235
$543$	0	0
$544$	8.33333	0.357288
$545$	−16.1850	−0.693290
$546$	0	0
$547$	20.6974	0.884958	0.442479	−	0.896779i	$-0.354099\pi$
$547$	20.6974	0.884958	0.442479	+	0.896779i	$0.354099\pi$
$548$	−26.0818	−1.11416
$549$	0	0
$550$	0.654866	0.0279236
$551$	21.3922	0.911338
$552$	0	0
$553$	12.9426	0.550375
$554$	0.0264105	0.00112208
$555$	0	0
$556$	26.1495	1.10899
$557$	−14.0895	−0.596991	−0.298496	−	0.954411i	$-0.596485\pi$
$557$	−14.0895	−0.596991	−0.298496	+	0.954411i	$0.596485\pi$
$558$	0	0
$559$	−57.3431	−2.42535
$560$	−15.1040	−0.638261
$561$	0	0
$562$	4.62339	0.195026
$563$	−39.1610	−1.65044	−0.825221	−	0.564811i	$-0.808949\pi$
$563$	−39.1610	−1.65044	−0.825221	+	0.564811i	$0.808949\pi$
$564$	0	0
$565$	16.5476	0.696163
$566$	−1.76284	−0.0740977
$567$	0	0
$568$	4.86345	0.204066
$569$	−16.7962	−0.704134	−0.352067	−	0.935975i	$-0.614521\pi$
$569$	−16.7962	−0.704134	−0.352067	+	0.935975i	$0.614521\pi$
$570$	0	0
$571$	−7.40957	−0.310081	−0.155040	−	0.987908i	$-0.549551\pi$
$571$	−7.40957	−0.310081	−0.155040	+	0.987908i	$0.549551\pi$
$572$	−51.1252	−2.13765
$573$	0	0
$574$	−4.38213	−0.182906
$575$	1.55096	0.0646795
$576$	0	0
$577$	−4.43326	−0.184559	−0.0922795	−	0.995733i	$-0.529415\pi$
$577$	−4.43326	−0.184559	−0.0922795	+	0.995733i	$0.529415\pi$
$578$	0.552444	0.0229786
$579$	0	0
$580$	6.61404	0.274633
$581$	−15.1863	−0.630034
$582$	0	0
$583$	52.4338	2.17159
$584$	5.66764	0.234529
$585$	0	0
$586$	4.99011	0.206139
$587$	33.3266	1.37554	0.687769	−	0.725930i	$-0.258590\pi$
$587$	33.3266	1.37554	0.687769	+	0.725930i	$0.258590\pi$
$588$	0	0
$589$	−26.3800	−1.08697
$590$	0.938908	0.0386542
$591$	0	0
$592$	27.6111	1.13481
$593$	23.9282	0.982612	0.491306	−	0.870987i	$-0.336520\pi$
$593$	23.9282	0.982612	0.491306	+	0.870987i	$0.336520\pi$
$594$	0	0
$595$	−17.7598	−0.728082
$596$	−1.60766	−0.0658522
$597$	0	0
$598$	1.48582	0.0607596
$599$	−11.5448	−0.471707	−0.235854	−	0.971789i	$-0.575789\pi$
$599$	−11.5448	−0.471707	−0.235854	+	0.971789i	$0.575789\pi$
$600$	0	0
$601$	16.2357	0.662268	0.331134	−	0.943584i	$-0.392569\pi$
$601$	16.2357	0.662268	0.331134	+	0.943584i	$0.392569\pi$
$602$	−5.68572	−0.231733
$603$	0	0
$604$	10.8719	0.442370
$605$	−6.68844	−0.271924
$606$	0	0
$607$	−10.3573	−0.420391	−0.210196	−	0.977659i	$-0.567410\pi$
$607$	−10.3573	−0.420391	−0.210196	+	0.977659i	$0.567410\pi$
$608$	11.7478	0.476435
$609$	0	0
$610$	−1.42423	−0.0576653
$611$	−73.6988	−2.98153
$612$	0	0
$613$	5.74382	0.231991	0.115995	−	0.993250i	$-0.462994\pi$
$613$	5.74382	0.231991	0.115995	+	0.993250i	$0.462994\pi$
$614$	−0.479147	−0.0193368
$615$	0	0
$616$	−10.2006	−0.410995
$617$	−11.0518	−0.444929	−0.222465	−	0.974941i	$-0.571410\pi$
$617$	−11.0518	−0.444929	−0.222465	+	0.974941i	$0.571410\pi$
$618$	0	0
$619$	11.5390	0.463792	0.231896	−	0.972741i	$-0.425507\pi$
$619$	11.5390	0.463792	0.231896	+	0.972741i	$0.425507\pi$
$620$	−8.15618	−0.327560
$621$	0	0
$622$	−1.96488	−0.0787844
$623$	26.1956	1.04951
$624$	0	0
$625$	1.00000	0.0400000
$626$	−3.78151	−0.151139
$627$	0	0
$628$	1.18780	0.0473985
$629$	32.4661	1.29451
$630$	0	0
$631$	17.9927	0.716278	0.358139	−	0.933668i	$-0.383411\pi$
$631$	17.9927	0.716278	0.358139	+	0.933668i	$0.383411\pi$
$632$	−2.04500	−0.0813459
$633$	0	0
$634$	−1.48345	−0.0589154
$635$	−21.3131	−0.845784
$636$	0	0
$637$	51.3741	2.03551
$638$	2.19223	0.0867912
$639$	0	0
$640$	4.83272	0.191030
$641$	−12.1989	−0.481826	−0.240913	−	0.970547i	$-0.577447\pi$
$641$	−12.1989	−0.481826	−0.240913	+	0.970547i	$0.577447\pi$
$642$	0	0
$643$	32.4279	1.27883	0.639416	−	0.768861i	$-0.279176\pi$
$643$	32.4279	1.27883	0.639416	+	0.768861i	$0.279176\pi$
$644$	−12.0057	−0.473091
$645$	0	0
$646$	4.51039	0.177459
$647$	−24.5804	−0.966354	−0.483177	−	0.875523i	$-0.660517\pi$
$647$	−24.5804	−0.966354	−0.483177	+	0.875523i	$0.660517\pi$
$648$	0	0
$649$	−25.3606	−0.995492
$650$	0.957999	0.0375758
$651$	0	0
$652$	42.8118	1.67664
$653$	48.6767	1.90487	0.952433	−	0.304749i	$-0.0985727\pi$
$653$	48.6767	1.90487	0.952433	+	0.304749i	$0.0985727\pi$
$654$	0	0
$655$	7.55127	0.295053
$656$	−27.6924	−1.08121
$657$	0	0
$658$	−7.30744	−0.284874
$659$	−41.9601	−1.63453	−0.817267	−	0.576259i	$-0.804512\pi$
$659$	−41.9601	−1.63453	−0.817267	+	0.576259i	$0.804512\pi$
$660$	0	0
$661$	−39.9051	−1.55213	−0.776064	−	0.630654i	$-0.782787\pi$
$661$	−39.9051	−1.55213	−0.776064	+	0.630654i	$0.782787\pi$
$662$	−0.102056	−0.00396652
$663$	0	0
$664$	2.39953	0.0931197
$665$	−25.0366	−0.970879
$666$	0	0
$667$	5.19199	0.201035
$668$	13.6190	0.526936
$669$	0	0
$670$	−0.888640	−0.0343312
$671$	38.4695	1.48510
$672$	0	0
$673$	29.9856	1.15586	0.577930	−	0.816086i	$-0.303861\pi$
$673$	29.9856	1.15586	0.577930	+	0.816086i	$0.303861\pi$
$674$	−2.72506	−0.104965
$675$	0	0
$676$	−49.1059	−1.88869
$677$	−27.3960	−1.05291	−0.526456	−	0.850202i	$-0.676480\pi$
$677$	−27.3960	−1.05291	−0.526456	+	0.850202i	$0.676480\pi$
$678$	0	0
$679$	−25.7109	−0.986695
$680$	2.80616	0.107611
$681$	0	0
$682$	−2.70338	−0.103518
$683$	−3.71237	−0.142050	−0.0710249	−	0.997475i	$-0.522627\pi$
$683$	−3.71237	−0.142050	−0.0710249	+	0.997475i	$0.522627\pi$
$684$	0	0
$685$	−13.2009	−0.504381
$686$	0.823563	0.0314438
$687$	0	0
$688$	−35.9303	−1.36983
$689$	76.7051	2.92223
$690$	0	0
$691$	−29.2686	−1.11343	−0.556714	−	0.830704i	$-0.687938\pi$
$691$	−29.2686	−1.11343	−0.556714	+	0.830704i	$0.687938\pi$
$692$	26.6682	1.01377
$693$	0	0
$694$	1.55691	0.0590994
$695$	13.2352	0.502040
$696$	0	0
$697$	−32.5617	−1.23336
$698$	0.587237	0.0222273
$699$	0	0
$700$	−7.74083	−0.292576
$701$	−39.3801	−1.48736	−0.743682	−	0.668534i	$-0.766922\pi$
$701$	−39.3801	−1.48736	−0.743682	+	0.668534i	$0.766922\pi$
$702$	0	0
$703$	45.7686	1.72619
$704$	−31.2235	−1.17678
$705$	0	0
$706$	−2.68636	−0.101102
$707$	−6.33644	−0.238306
$708$	0	0
$709$	24.6721	0.926582	0.463291	−	0.886206i	$-0.346669\pi$
$709$	24.6721	0.926582	0.463291	+	0.886206i	$0.346669\pi$
$710$	1.22328	0.0459087
$711$	0	0
$712$	−4.13907	−0.155118
$713$	−6.40257	−0.239778
$714$	0	0
$715$	−25.8763	−0.967719
$716$	−45.5924	−1.70387
$717$	0	0
$718$	−3.17300	−0.118415
$719$	23.8360	0.888933	0.444467	−	0.895795i	$-0.353393\pi$
$719$	23.8360	0.888933	0.444467	+	0.895795i	$0.353393\pi$
$720$	0	0
$721$	−9.69307	−0.360989
$722$	3.40002	0.126536
$723$	0	0
$724$	−6.38602	−0.237335
$725$	3.34760	0.124327
$726$	0	0
$727$	47.3237	1.75514	0.877570	−	0.479449i	$-0.159163\pi$
$727$	47.3237	1.75514	0.877570	+	0.479449i	$0.159163\pi$
$728$	−14.9224	−0.553062
$729$	0	0
$730$	1.42555	0.0527620
$731$	−42.2482	−1.56261
$732$	0	0
$733$	39.1381	1.44560	0.722799	−	0.691058i	$-0.242855\pi$
$733$	39.1381	1.44560	0.722799	+	0.691058i	$0.242855\pi$
$734$	2.71183	0.100095
$735$	0	0
$736$	2.85124	0.105098
$737$	24.0028	0.884156
$738$	0	0
$739$	20.0503	0.737564	0.368782	−	0.929516i	$-0.379775\pi$
$739$	20.0503	0.737564	0.368782	+	0.929516i	$0.379775\pi$
$740$	14.1507	0.520191
$741$	0	0
$742$	7.60552	0.279208
$743$	−52.0682	−1.91020	−0.955099	−	0.296287i	$-0.904251\pi$
$743$	−52.0682	−1.91020	−0.955099	+	0.296287i	$0.904251\pi$
$744$	0	0
$745$	−0.813693	−0.0298114
$746$	−0.00847611	−0.000310332 0
$747$	0	0
$748$	−37.6671	−1.37725
$749$	−3.91791	−0.143157
$750$	0	0
$751$	−18.5212	−0.675848	−0.337924	−	0.941173i	$-0.609725\pi$
$751$	−18.5212	−0.675848	−0.337924	+	0.941173i	$0.609725\pi$
$752$	−46.1786	−1.68396
$753$	0	0
$754$	3.20700	0.116792
$755$	5.50264	0.200262
$756$	0	0
$757$	−12.1790	−0.442655	−0.221327	−	0.975200i	$-0.571039\pi$
$757$	−12.1790	−0.442655	−0.221327	+	0.975200i	$0.571039\pi$
$758$	0.879550	0.0319467
$759$	0	0
$760$	3.95594	0.143497
$761$	38.4420	1.39352	0.696760	−	0.717304i	$-0.254624\pi$
$761$	38.4420	1.39352	0.696760	+	0.717304i	$0.254624\pi$
$762$	0	0
$763$	63.4114	2.29565
$764$	14.7890	0.535047
$765$	0	0
$766$	5.18688	0.187410
$767$	−37.0999	−1.33960
$768$	0	0
$769$	16.9516	0.611290	0.305645	−	0.952145i	$-0.401128\pi$
$769$	16.9516	0.611290	0.305645	+	0.952145i	$0.401128\pi$
$770$	−2.56571	−0.0924616
$771$	0	0
$772$	46.7451	1.68239
$773$	43.7652	1.57412	0.787062	−	0.616874i	$-0.211601\pi$
$773$	43.7652	1.57412	0.787062	+	0.616874i	$0.211601\pi$
$774$	0	0
$775$	−4.12813	−0.148287
$776$	4.06248	0.145835
$777$	0	0
$778$	1.29614	0.0464688
$779$	−45.9033	−1.64466
$780$	0	0
$781$	−33.0416	−1.18232
$782$	1.09469	0.0391462
$783$	0	0
$784$	32.1903	1.14965
$785$	0.601189	0.0214573
$786$	0	0
$787$	−37.0068	−1.31915	−0.659574	−	0.751639i	$-0.729263\pi$
$787$	−37.0068	−1.31915	−0.659574	+	0.751639i	$0.729263\pi$
$788$	−0.992688	−0.0353630
$789$	0	0
$790$	−0.514369	−0.0183004
$791$	−64.8320	−2.30516
$792$	0	0
$793$	56.2768	1.99845
$794$	1.17744	0.0417858
$795$	0	0
$796$	−23.7439	−0.841579
$797$	−1.30055	−0.0460680	−0.0230340	−	0.999735i	$-0.507333\pi$
$797$	−1.30055	−0.0460680	−0.0230340	+	0.999735i	$0.507333\pi$
$798$	0	0
$799$	−54.2985	−1.92094
$800$	1.83837	0.0649963
$801$	0	0
$802$	2.06312	0.0728515
$803$	−38.5052	−1.35882
$804$	0	0
$805$	−6.07652	−0.214169
$806$	−3.95475	−0.139300
$807$	0	0
$808$	1.00119	0.0352219
$809$	−18.8423	−0.662459	−0.331230	−	0.943550i	$-0.607463\pi$
$809$	−18.8423	−0.662459	−0.331230	+	0.943550i	$0.607463\pi$
$810$	0	0
$811$	−10.6066	−0.372448	−0.186224	−	0.982507i	$-0.559625\pi$
$811$	−10.6066	−0.372448	−0.186224	+	0.982507i	$0.559625\pi$
$812$	−25.9132	−0.909375
$813$	0	0
$814$	4.69027	0.164394
$815$	21.6686	0.759017
$816$	0	0
$817$	−59.5587	−2.08370
$818$	−4.15606	−0.145313
$819$	0	0
$820$	−14.1924	−0.495620
$821$	−13.1312	−0.458281	−0.229141	−	0.973393i	$-0.573592\pi$
$821$	−13.1312	−0.458281	−0.229141	+	0.973393i	$0.573592\pi$
$822$	0	0
$823$	−27.6426	−0.963561	−0.481781	−	0.876292i	$-0.660010\pi$
$823$	−27.6426	−0.963561	−0.481781	+	0.876292i	$0.660010\pi$
$824$	1.53156	0.0533545
$825$	0	0
$826$	−3.67856	−0.127993
$827$	−33.2253	−1.15536	−0.577679	−	0.816264i	$-0.696041\pi$
$827$	−33.2253	−1.15536	−0.577679	+	0.816264i	$0.696041\pi$
$828$	0	0
$829$	23.1934	0.805540	0.402770	−	0.915301i	$-0.368047\pi$
$829$	23.1934	0.805540	0.402770	+	0.915301i	$0.368047\pi$
$830$	0.603540	0.0209492
$831$	0	0
$832$	−45.6767	−1.58356
$833$	37.8505	1.31144
$834$	0	0
$835$	6.89307	0.238545
$836$	−53.1006	−1.83652
$837$	0	0
$838$	4.08586	0.141144
$839$	−22.1976	−0.766347	−0.383174	−	0.923676i	$-0.625169\pi$
$839$	−22.1976	−0.766347	−0.383174	+	0.923676i	$0.625169\pi$
$840$	0	0
$841$	−17.7936	−0.613572
$842$	3.59489	0.123888
$843$	0	0
$844$	44.5434	1.53325
$845$	−24.8543	−0.855012
$846$	0	0
$847$	26.2047	0.900404
$848$	48.0623	1.65047
$849$	0	0
$850$	0.705817	0.0242093
$851$	11.1083	0.380786
$852$	0	0
$853$	−33.7654	−1.15611	−0.578053	−	0.815999i	$-0.696187\pi$
$853$	−33.7654	−1.15611	−0.578053	+	0.815999i	$0.696187\pi$
$854$	5.58000	0.190944
$855$	0	0
$856$	0.619053	0.0211588
$857$	8.43718	0.288209	0.144104	−	0.989562i	$-0.453970\pi$
$857$	8.43718	0.288209	0.144104	+	0.989562i	$0.453970\pi$
$858$	0	0
$859$	10.2973	0.351338	0.175669	−	0.984449i	$-0.443791\pi$
$859$	10.2973	0.351338	0.175669	+	0.984449i	$0.443791\pi$
$860$	−18.4144	−0.627925
$861$	0	0
$862$	3.49600	0.119074
$863$	42.9481	1.46197	0.730985	−	0.682394i	$-0.239061\pi$
$863$	42.9481	1.46197	0.730985	+	0.682394i	$0.239061\pi$
$864$	0	0
$865$	13.4977	0.458937
$866$	−1.61853	−0.0549999
$867$	0	0
$868$	31.9552	1.08463
$869$	13.8935	0.471304
$870$	0	0
$871$	35.1136	1.18978
$872$	−10.0194	−0.339299
$873$	0	0
$874$	1.54323	0.0522005
$875$	−3.91791	−0.132449
$876$	0	0
$877$	39.5482	1.33545	0.667724	−	0.744409i	$-0.267269\pi$
$877$	39.5482	1.33545	0.667724	+	0.744409i	$0.267269\pi$
$878$	−4.86832	−0.164298
$879$	0	0
$880$	−16.2137	−0.546564
$881$	−0.331700	−0.0111752	−0.00558762	−	0.999984i	$-0.501779\pi$
$881$	−0.331700	−0.0111752	−0.00558762	+	0.999984i	$0.501779\pi$
$882$	0	0
$883$	−3.97236	−0.133680	−0.0668402	−	0.997764i	$-0.521292\pi$
$883$	−3.97236	−0.133680	−0.0668402	+	0.997764i	$0.521292\pi$
$884$	−55.1030	−1.85331
$885$	0	0
$886$	0.523297	0.0175805
$887$	−41.9397	−1.40820	−0.704099	−	0.710102i	$-0.748649\pi$
$887$	−41.9397	−1.40820	−0.704099	+	0.710102i	$0.748649\pi$
$888$	0	0
$889$	83.5027	2.80059
$890$	−1.04108	−0.0348970
$891$	0	0
$892$	22.1695	0.742291
$893$	−76.5464	−2.56153
$894$	0	0
$895$	−23.0759	−0.771344
$896$	−18.9341	−0.632545
$897$	0	0
$898$	−0.843613	−0.0281517
$899$	−13.8193	−0.460901
$900$	0	0
$901$	56.5134	1.88273
$902$	−4.70409	−0.156629
$903$	0	0
$904$	10.2438	0.340705
$905$	−3.23219	−0.107442
$906$	0	0
$907$	19.3926	0.643922	0.321961	−	0.946753i	$-0.395658\pi$
$907$	19.3926	0.643922	0.321961	+	0.946753i	$0.395658\pi$
$908$	7.49622	0.248771
$909$	0	0
$910$	−3.75335	−0.124422
$911$	14.3443	0.475248	0.237624	−	0.971357i	$-0.423631\pi$
$911$	14.3443	0.475248	0.237624	+	0.971357i	$0.423631\pi$
$912$	0	0
$913$	−16.3021	−0.539520
$914$	2.24468	0.0742473
$915$	0	0
$916$	43.1103	1.42440
$917$	−29.5852	−0.976989
$918$	0	0
$919$	−44.0493	−1.45305	−0.726526	−	0.687138i	$-0.758867\pi$
$919$	−44.0493	−1.45305	−0.726526	+	0.687138i	$0.758867\pi$
$920$	0.960125	0.0316544
$921$	0	0
$922$	−4.40926	−0.145211
$923$	−48.3364	−1.59101
$924$	0	0
$925$	7.16219	0.235491
$926$	2.76485	0.0908587
$927$	0	0
$928$	6.15414	0.202020
$929$	11.7163	0.384400	0.192200	−	0.981356i	$-0.438438\pi$
$929$	11.7163	0.384400	0.192200	+	0.981356i	$0.438438\pi$
$930$	0	0
$931$	53.3591	1.74877
$932$	4.99721	0.163689
$933$	0	0
$934$	5.05755	0.165488
$935$	−19.0647	−0.623481
$936$	0	0
$937$	−14.2892	−0.466808	−0.233404	−	0.972380i	$-0.574987\pi$
$937$	−14.2892	−0.466808	−0.233404	+	0.972380i	$0.574987\pi$
$938$	3.48161	0.113679
$939$	0	0
$940$	−23.6666	−0.771920
$941$	11.1338	0.362950	0.181475	−	0.983396i	$-0.441913\pi$
$941$	11.1338	0.362950	0.181475	+	0.983396i	$0.441913\pi$
$942$	0	0
$943$	−11.1410	−0.362800
$944$	−23.2463	−0.756601
$945$	0	0
$946$	−6.10346	−0.198441
$947$	41.4663	1.34747	0.673737	−	0.738971i	$-0.264688\pi$
$947$	41.4663	1.34747	0.673737	+	0.738971i	$0.264688\pi$
$948$	0	0
$949$	−56.3290	−1.82852
$950$	0.995015	0.0322825
$951$	0	0
$952$	−10.9943	−0.356326
$953$	33.3251	1.07951	0.539753	−	0.841824i	$-0.318518\pi$
$953$	33.3251	1.07951	0.539753	+	0.841824i	$0.318518\pi$
$954$	0	0
$955$	7.48523	0.242216
$956$	−0.188198	−0.00608674
$957$	0	0
$958$	−5.06331	−0.163588
$959$	51.7200	1.67013
$960$	0	0
$961$	−13.9585	−0.450275
$962$	6.86137	0.221219
$963$	0	0
$964$	39.1365	1.26050
$965$	23.6594	0.761622
$966$	0	0
$967$	50.1545	1.61286	0.806431	−	0.591329i	$-0.201396\pi$
$967$	50.1545	1.61286	0.806431	+	0.591329i	$0.201396\pi$
$968$	−4.14050	−0.133081
$969$	0	0
$970$	1.02181	0.0328084
$971$	−1.04282	−0.0334657	−0.0167329	−	0.999860i	$-0.505326\pi$
$971$	−1.04282	−0.0334657	−0.0167329	+	0.999860i	$0.505326\pi$
$972$	0	0
$973$	−51.8543	−1.66237
$974$	−5.42393	−0.173794
$975$	0	0
$976$	35.2622	1.12872
$977$	59.7517	1.91163	0.955813	−	0.293976i	$-0.0949785\pi$
$977$	59.7517	1.91163	0.955813	+	0.293976i	$0.0949785\pi$
$978$	0	0
$979$	28.1203	0.898728
$980$	16.4976	0.526995
$981$	0	0
$982$	1.46809	0.0468488
$983$	−19.2009	−0.612413	−0.306207	−	0.951965i	$-0.599060\pi$
$983$	−19.2009	−0.612413	−0.306207	+	0.951965i	$0.599060\pi$
$984$	0	0
$985$	−0.502435	−0.0160089
$986$	2.36279	0.0752467
$987$	0	0
$988$	−77.6805	−2.47135
$989$	−14.4552	−0.459648
$990$	0	0
$991$	−26.6856	−0.847695	−0.423848	−	0.905734i	$-0.639321\pi$
$991$	−26.6856	−0.847695	−0.423848	+	0.905734i	$0.639321\pi$
$992$	−7.58905	−0.240953
$993$	0	0
$994$	−4.79268	−0.152015
$995$	−12.0176	−0.380984
$996$	0	0
$997$	10.7703	0.341099	0.170550	−	0.985349i	$-0.445446\pi$
$997$	10.7703	0.341099	0.170550	+	0.985349i	$0.445446\pi$
$998$	1.73283	0.0548517
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4815.2.a.u.1.8		12
3.2	odd	2		1605.2.a.n.1.5	✓	12
15.14	odd	2		8025.2.a.bf.1.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1605.2.a.n.1.5	✓	12	3.2	odd	2
4815.2.a.u.1.8		12	1.1	even	1	trivial
8025.2.a.bf.1.8		12	15.14	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:	\( N \)	\(=\)	\( 4815 = 3^{2} \cdot 5 \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4815.a (trivial)

\(f(q)\)	\(=\)	\(q+0.155707 q^{2} -1.97576 q^{4} -1.00000 q^{5} +3.91791 q^{7} -0.619053 q^{8} +O(q^{10})\)	\(q+0.155707 q^{2} -1.97576 q^{4} -1.00000 q^{5} +3.91791 q^{7} -0.619053 q^{8} -0.155707 q^{10} +4.20576 q^{11} +6.15258 q^{13} +0.610045 q^{14} +3.85512 q^{16} +4.53299 q^{17} +6.39031 q^{19} +1.97576 q^{20} +0.654866 q^{22} +1.55096 q^{23} +1.00000 q^{25} +0.957999 q^{26} -7.74083 q^{28} +3.34760 q^{29} -4.12813 q^{31} +1.83837 q^{32} +0.705817 q^{34} -3.91791 q^{35} +7.16219 q^{37} +0.995015 q^{38} +0.619053 q^{40} -7.18328 q^{41} -9.32016 q^{43} -8.30956 q^{44} +0.241495 q^{46} -11.9785 q^{47} +8.35000 q^{49} +0.155707 q^{50} -12.1560 q^{52} +12.4671 q^{53} -4.20576 q^{55} -2.42539 q^{56} +0.521244 q^{58} -6.02997 q^{59} +9.14686 q^{61} -0.642779 q^{62} -7.42399 q^{64} -6.15258 q^{65} +5.70713 q^{67} -8.95607 q^{68} -0.610045 q^{70} -7.85627 q^{71} -9.15535 q^{73} +1.11520 q^{74} -12.6257 q^{76} +16.4778 q^{77} +3.30344 q^{79} -3.85512 q^{80} -1.11849 q^{82} -3.87613 q^{83} -4.53299 q^{85} -1.45121 q^{86} -2.60359 q^{88} +6.68613 q^{89} +24.1052 q^{91} -3.06432 q^{92} -1.86514 q^{94} -6.39031 q^{95} -6.56241 q^{97} +1.30015 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8	\( 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8} + 3 q^{10} - 4 q^{11} + 13 q^{13} - 4 q^{14} + 13 q^{16} + 4 q^{17} + 14 q^{19} - 15 q^{20} + 15 q^{22} - 11 q^{23} + 12 q^{25} + 8 q^{26} + 16 q^{28} + 7 q^{29} + 4 q^{31} - 4 q^{32} + q^{34} - 7 q^{35} + 24 q^{37} + 11 q^{38} + 3 q^{40} - 13 q^{41} + 25 q^{43} - 10 q^{44} - 22 q^{46} - 19 q^{47} + 9 q^{49} - 3 q^{50} + 20 q^{52} - 11 q^{53} + 4 q^{55} + 37 q^{56} - 2 q^{58} - 8 q^{59} + 7 q^{61} + 11 q^{62} - 19 q^{64} - 13 q^{65} + 33 q^{67} + 24 q^{68} + 4 q^{70} + 34 q^{73} + 27 q^{74} - 9 q^{76} + 29 q^{77} - 13 q^{80} + q^{82} + 24 q^{83} - 4 q^{85} + 36 q^{86} - 6 q^{88} + 10 q^{89} + 30 q^{91} + 28 q^{92} - 8 q^{94} - 14 q^{95} + 16 q^{97} + 36 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8 + 3 * q^10 - 4 * q^11 + 13 * q^13 - 4 * q^14 + 13 * q^16 + 4 * q^17 + 14 * q^19 - 15 * q^20 + 15 * q^22 - 11 * q^23 + 12 * q^25 + 8 * q^26 + 16 * q^28 + 7 * q^29 + 4 * q^31 - 4 * q^32 + q^34 - 7 * q^35 + 24 * q^37 + 11 * q^38 + 3 * q^40 - 13 * q^41 + 25 * q^43 - 10 * q^44 - 22 * q^46 - 19 * q^47 + 9 * q^49 - 3 * q^50 + 20 * q^52 - 11 * q^53 + 4 * q^55 + 37 * q^56 - 2 * q^58 - 8 * q^59 + 7 * q^61 + 11 * q^62 - 19 * q^64 - 13 * q^65 + 33 * q^67 + 24 * q^68 + 4 * q^70 + 34 * q^73 + 27 * q^74 - 9 * q^76 + 29 * q^77 - 13 * q^80 + q^82 + 24 * q^83 - 4 * q^85 + 36 * q^86 - 6 * q^88 + 10 * q^89 + 30 * q^91 + 28 * q^92 - 8 * q^94 - 14 * q^95 + 16 * q^97 + 36 * q^98

Introduction

L-functions

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Database

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