LMFDB - Embedded newform 4815.2.a.u.1.3

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			1.3
Root			$2.36308$ 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)

$f(q)$	$=$	$q-2.36308 q^{2} +3.58415 q^{4} -1.00000 q^{5} -1.44739 q^{7} -3.74348 q^{8} +O(q^{10})$	\(q-2.36308 q^{2} +3.58415 q^{4} -1.00000 q^{5} -1.44739 q^{7} -3.74348 q^{8} +2.36308 q^{10} -4.67457 q^{11} -0.873126 q^{13} +3.42031 q^{14} +1.67784 q^{16} -2.24884 q^{17} -5.64861 q^{19} -3.58415 q^{20} +11.0464 q^{22} -5.37771 q^{23} +1.00000 q^{25} +2.06327 q^{26} -5.18767 q^{28} -2.30410 q^{29} -1.09517 q^{31} +3.52208 q^{32} +5.31420 q^{34} +1.44739 q^{35} -6.05796 q^{37} +13.3481 q^{38} +3.74348 q^{40} -10.1973 q^{41} -4.88520 q^{43} -16.7544 q^{44} +12.7080 q^{46} -0.209041 q^{47} -4.90506 q^{49} -2.36308 q^{50} -3.12942 q^{52} +8.83811 q^{53} +4.67457 q^{55} +5.41829 q^{56} +5.44477 q^{58} -2.79345 q^{59} -8.08566 q^{61} +2.58797 q^{62} -11.6786 q^{64} +0.873126 q^{65} +10.5805 q^{67} -8.06019 q^{68} -3.42031 q^{70} +15.7811 q^{71} +0.698172 q^{73} +14.3154 q^{74} -20.2455 q^{76} +6.76594 q^{77} -11.6878 q^{79} -1.67784 q^{80} +24.0971 q^{82} -14.4572 q^{83} +2.24884 q^{85} +11.5441 q^{86} +17.4992 q^{88} -2.10490 q^{89} +1.26376 q^{91} -19.2745 q^{92} +0.493981 q^{94} +5.64861 q^{95} -3.68546 q^{97} +11.5910 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.36308	−1.67095	−0.835475	−	0.549528i	$-0.814808\pi$
$2$	−2.36308	−1.67095	−0.835475	+	0.549528i	$0.814808\pi$
$3$	0	0
$3$	0	0
$4$	3.58415	1.79208
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.44739	−0.547063	−0.273531	−	0.961863i	$-0.588192\pi$
$7$	−1.44739	−0.547063	−0.273531	+	0.961863i	$0.588192\pi$
$8$	−3.74348	−1.32352
$9$	0	0
$10$	2.36308	0.747272
$11$	−4.67457	−1.40944	−0.704718	−	0.709487i	$-0.748927\pi$
$11$	−4.67457	−1.40944	−0.704718	+	0.709487i	$0.748927\pi$
$12$	0	0
$13$	−0.873126	−0.242161	−0.121081	−	0.992643i	$-0.538636\pi$
$13$	−0.873126	−0.242161	−0.121081	+	0.992643i	$0.538636\pi$
$14$	3.42031	0.914115
$15$	0	0
$16$	1.67784	0.419461
$17$	−2.24884	−0.545424	−0.272712	−	0.962096i	$-0.587921\pi$
$17$	−2.24884	−0.545424	−0.272712	+	0.962096i	$0.587921\pi$
$18$	0	0
$19$	−5.64861	−1.29588	−0.647940	−	0.761691i	$-0.724369\pi$
$19$	−5.64861	−1.29588	−0.647940	+	0.761691i	$0.724369\pi$
$20$	−3.58415	−0.801441
$21$	0	0
$22$	11.0464	2.35510
$23$	−5.37771	−1.12133	−0.560665	−	0.828043i	$-0.689454\pi$
$23$	−5.37771	−1.12133	−0.560665	+	0.828043i	$0.689454\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.06327	0.404640
$27$	0	0
$28$	−5.18767	−0.980378
$29$	−2.30410	−0.427860	−0.213930	−	0.976849i	$-0.568626\pi$
$29$	−2.30410	−0.427860	−0.213930	+	0.976849i	$0.568626\pi$
$30$	0	0
$31$	−1.09517	−0.196698	−0.0983491	−	0.995152i	$-0.531356\pi$
$31$	−1.09517	−0.196698	−0.0983491	+	0.995152i	$0.531356\pi$
$32$	3.52208	0.622622
$33$	0	0
$34$	5.31420	0.911377
$35$	1.44739	0.244654
$36$	0	0
$37$	−6.05796	−0.995922	−0.497961	−	0.867199i	$-0.665918\pi$
$37$	−6.05796	−0.995922	−0.497961	+	0.867199i	$0.665918\pi$
$38$	13.3481	2.16535
$39$	0	0
$40$	3.74348	0.591896
$41$	−10.1973	−1.59255	−0.796277	−	0.604933i	$-0.793200\pi$
$41$	−10.1973	−1.59255	−0.796277	+	0.604933i	$0.793200\pi$
$42$	0	0
$43$	−4.88520	−0.744987	−0.372493	−	0.928035i	$-0.621497\pi$
$43$	−4.88520	−0.744987	−0.372493	+	0.928035i	$0.621497\pi$
$44$	−16.7544	−2.52582
$45$	0	0
$46$	12.7080	1.87369
$47$	−0.209041	−0.0304918	−0.0152459	−	0.999884i	$-0.504853\pi$
$47$	−0.209041	−0.0304918	−0.0152459	+	0.999884i	$0.504853\pi$
$48$	0	0
$49$	−4.90506	−0.700722
$50$	−2.36308	−0.334190
$51$	0	0
$52$	−3.12942	−0.433972
$53$	8.83811	1.21401	0.607004	−	0.794699i	$-0.292371\pi$
$53$	8.83811	1.21401	0.607004	+	0.794699i	$0.292371\pi$
$54$	0	0
$55$	4.67457	0.630319
$56$	5.41829	0.724049
$57$	0	0
$58$	5.44477	0.714933
$59$	−2.79345	−0.363677	−0.181838	−	0.983328i	$-0.558205\pi$
$59$	−2.79345	−0.363677	−0.181838	+	0.983328i	$0.558205\pi$
$60$	0	0
$61$	−8.08566	−1.03526	−0.517631	−	0.855604i	$-0.673186\pi$
$61$	−8.08566	−1.03526	−0.517631	+	0.855604i	$0.673186\pi$
$62$	2.58797	0.328673
$63$	0	0
$64$	−11.6786	−1.45983
$65$	0.873126	0.108298
$66$	0	0
$67$	10.5805	1.29262	0.646308	−	0.763077i	$-0.276312\pi$
$67$	10.5805	1.29262	0.646308	+	0.763077i	$0.276312\pi$
$68$	−8.06019	−0.977442
$69$	0	0
$70$	−3.42031	−0.408805
$71$	15.7811	1.87287	0.936437	−	0.350837i	$-0.114103\pi$
$71$	15.7811	1.87287	0.936437	+	0.350837i	$0.114103\pi$
$72$	0	0
$73$	0.698172	0.0817149	0.0408574	−	0.999165i	$-0.486991\pi$
$73$	0.698172	0.0817149	0.0408574	+	0.999165i	$0.486991\pi$
$74$	14.3154	1.66414
$75$	0	0
$76$	−20.2455	−2.32232
$77$	6.76594	0.771051
$78$	0	0
$79$	−11.6878	−1.31498	−0.657488	−	0.753465i	$-0.728381\pi$
$79$	−11.6878	−1.31498	−0.657488	+	0.753465i	$0.728381\pi$
$80$	−1.67784	−0.187589
$81$	0	0
$82$	24.0971	2.66108
$83$	−14.4572	−1.58688	−0.793441	−	0.608647i	$-0.791713\pi$
$83$	−14.4572	−1.58688	−0.793441	+	0.608647i	$0.791713\pi$
$84$	0	0
$85$	2.24884	0.243921
$86$	11.5441	1.24484
$87$	0	0
$88$	17.4992	1.86542
$89$	−2.10490	−0.223119	−0.111559	−	0.993758i	$-0.535585\pi$
$89$	−2.10490	−0.223119	−0.111559	+	0.993758i	$0.535585\pi$
$90$	0	0
$91$	1.26376	0.132478
$92$	−19.2745	−2.00951
$93$	0	0
$94$	0.493981	0.0509503
$95$	5.64861	0.579535
$96$	0	0
$97$	−3.68546	−0.374201	−0.187101	−	0.982341i	$-0.559909\pi$
$97$	−3.68546	−0.374201	−0.187101	+	0.982341i	$0.559909\pi$
$98$	11.5910	1.17087
$99$	0	0
$100$	3.58415	0.358415
$101$	0.924119	0.0919533	0.0459767	−	0.998943i	$-0.485360\pi$
$101$	0.924119	0.0919533	0.0459767	+	0.998943i	$0.485360\pi$
$102$	0	0
$103$	17.7880	1.75271	0.876354	−	0.481668i	$-0.159969\pi$
$103$	17.7880	1.75271	0.876354	+	0.481668i	$0.159969\pi$
$104$	3.26853	0.320506
$105$	0	0
$106$	−20.8852	−2.02855
$107$	−1.00000	−0.0966736
$107$	−1.00000	−0.0966736
$108$	0	0
$109$	−5.32028	−0.509590	−0.254795	−	0.966995i	$-0.582008\pi$
$109$	−5.32028	−0.509590	−0.254795	+	0.966995i	$0.582008\pi$
$110$	−11.0464	−1.05323
$111$	0	0
$112$	−2.42850	−0.229471
$113$	−1.33883	−0.125947	−0.0629734	−	0.998015i	$-0.520058\pi$
$113$	−1.33883	−0.125947	−0.0629734	+	0.998015i	$0.520058\pi$
$114$	0	0
$115$	5.37771	0.501474
$116$	−8.25823	−0.766758
$117$	0	0
$118$	6.60116	0.607686
$119$	3.25496	0.298381
$120$	0	0
$121$	10.8516	0.986512
$122$	19.1071	1.72987
$123$	0	0
$124$	−3.92525	−0.352498
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.6810	−1.74641	−0.873203	−	0.487357i	$-0.837961\pi$
$127$	−19.6810	−1.74641	−0.873203	+	0.487357i	$0.837961\pi$
$128$	20.5534	1.81668
$129$	0	0
$130$	−2.06327	−0.180960
$131$	2.85846	0.249745	0.124872	−	0.992173i	$-0.460148\pi$
$131$	2.85846	0.249745	0.124872	+	0.992173i	$0.460148\pi$
$132$	0	0
$133$	8.17575	0.708928
$134$	−25.0026	−2.15990
$135$	0	0
$136$	8.41850	0.721880
$137$	16.2725	1.39026	0.695129	−	0.718885i	$-0.255347\pi$
$137$	16.2725	1.39026	0.695129	+	0.718885i	$0.255347\pi$
$138$	0	0
$139$	−12.1709	−1.03233	−0.516163	−	0.856490i	$-0.672640\pi$
$139$	−12.1709	−1.03233	−0.516163	+	0.856490i	$0.672640\pi$
$140$	5.18767	0.438439
$141$	0	0
$142$	−37.2920	−3.12948
$143$	4.08149	0.341311
$144$	0	0
$145$	2.30410	0.191345
$146$	−1.64984	−0.136542
$147$	0	0
$148$	−21.7126	−1.78477
$149$	7.27573	0.596051	0.298026	−	0.954558i	$-0.403672\pi$
$149$	7.27573	0.596051	0.298026	+	0.954558i	$0.403672\pi$
$150$	0	0
$151$	−0.767327	−0.0624442	−0.0312221	−	0.999512i	$-0.509940\pi$
$151$	−0.767327	−0.0624442	−0.0312221	+	0.999512i	$0.509940\pi$
$152$	21.1455	1.71512
$153$	0	0
$154$	−15.9885	−1.28839
$155$	1.09517	0.0879661
$156$	0	0
$157$	12.9677	1.03493	0.517466	−	0.855704i	$-0.326875\pi$
$157$	12.9677	1.03493	0.517466	+	0.855704i	$0.326875\pi$
$158$	27.6191	2.19726
$159$	0	0
$160$	−3.52208	−0.278445
$161$	7.78366	0.613438
$162$	0	0
$163$	5.60955	0.439374	0.219687	−	0.975570i	$-0.429496\pi$
$163$	5.60955	0.439374	0.219687	+	0.975570i	$0.429496\pi$
$164$	−36.5487	−2.85398
$165$	0	0
$166$	34.1635	2.65160
$167$	16.0465	1.24172	0.620859	−	0.783923i	$-0.286784\pi$
$167$	16.0465	1.24172	0.620859	+	0.783923i	$0.286784\pi$
$168$	0	0
$169$	−12.2377	−0.941358
$170$	−5.31420	−0.407580
$171$	0	0
$172$	−17.5093	−1.33507
$173$	−3.75154	−0.285225	−0.142612	−	0.989779i	$-0.545550\pi$
$173$	−3.75154	−0.285225	−0.142612	+	0.989779i	$0.545550\pi$
$174$	0	0
$175$	−1.44739	−0.109413
$176$	−7.84320	−0.591204
$177$	0	0
$178$	4.97405	0.372821
$179$	−4.26811	−0.319014	−0.159507	−	0.987197i	$-0.550990\pi$
$179$	−4.26811	−0.319014	−0.159507	+	0.987197i	$0.550990\pi$
$180$	0	0
$181$	8.56462	0.636603	0.318301	−	0.947990i	$-0.396888\pi$
$181$	8.56462	0.636603	0.318301	+	0.947990i	$0.396888\pi$
$182$	−2.98636	−0.221363
$183$	0	0
$184$	20.1314	1.48410
$185$	6.05796	0.445390
$186$	0	0
$187$	10.5124	0.768741
$188$	−0.749235	−0.0546436
$189$	0	0
$190$	−13.3481	−0.968375
$191$	−20.6622	−1.49507	−0.747533	−	0.664224i	$-0.768762\pi$
$191$	−20.6622	−1.49507	−0.747533	+	0.664224i	$0.768762\pi$
$192$	0	0
$193$	2.01647	0.145149	0.0725744	−	0.997363i	$-0.476879\pi$
$193$	2.01647	0.145149	0.0725744	+	0.997363i	$0.476879\pi$
$194$	8.70903	0.625272
$195$	0	0
$196$	−17.5805	−1.25575
$197$	−16.0640	−1.14451	−0.572257	−	0.820075i	$-0.693932\pi$
$197$	−16.0640	−1.14451	−0.572257	+	0.820075i	$0.693932\pi$
$198$	0	0
$199$	−3.08539	−0.218718	−0.109359	−	0.994002i	$-0.534880\pi$
$199$	−3.08539	−0.218718	−0.109359	+	0.994002i	$0.534880\pi$
$200$	−3.74348	−0.264704
$201$	0	0
$202$	−2.18377	−0.153649
$203$	3.33493	0.234066
$204$	0	0
$205$	10.1973	0.712211
$206$	−42.0346	−2.92869
$207$	0	0
$208$	−1.46497	−0.101577
$209$	26.4048	1.82646
$210$	0	0
$211$	−11.1915	−0.770453	−0.385226	−	0.922822i	$-0.625877\pi$
$211$	−11.1915	−0.770453	−0.385226	+	0.922822i	$0.625877\pi$
$212$	31.6771	2.17559
$213$	0	0
$214$	2.36308	0.161537
$215$	4.88520	0.333168
$216$	0	0
$217$	1.58514	0.107606
$218$	12.5722	0.851500
$219$	0	0
$220$	16.7544	1.12958
$221$	1.96352	0.132081
$222$	0	0
$223$	11.7627	0.787691	0.393846	−	0.919177i	$-0.371144\pi$
$223$	11.7627	0.787691	0.393846	+	0.919177i	$0.371144\pi$
$224$	−5.09783	−0.340613
$225$	0	0
$226$	3.16377	0.210451
$227$	−7.91558	−0.525375	−0.262688	−	0.964881i	$-0.584609\pi$
$227$	−7.91558	−0.525375	−0.262688	+	0.964881i	$0.584609\pi$
$228$	0	0
$229$	5.70800	0.377195	0.188598	−	0.982054i	$-0.439606\pi$
$229$	5.70800	0.377195	0.188598	+	0.982054i	$0.439606\pi$
$230$	−12.7080	−0.837938
$231$	0	0
$232$	8.62534	0.566281
$233$	13.5436	0.887272	0.443636	−	0.896207i	$-0.353688\pi$
$233$	13.5436	0.887272	0.443636	+	0.896207i	$0.353688\pi$
$234$	0	0
$235$	0.209041	0.0136363
$236$	−10.0122	−0.651736
$237$	0	0
$238$	−7.69173	−0.498581
$239$	3.96445	0.256439	0.128220	−	0.991746i	$-0.459074\pi$
$239$	3.96445	0.256439	0.128220	+	0.991746i	$0.459074\pi$
$240$	0	0
$241$	15.7066	1.01175	0.505876	−	0.862606i	$-0.331169\pi$
$241$	15.7066	1.01175	0.505876	+	0.862606i	$0.331169\pi$
$242$	−25.6433	−1.64841
$243$	0	0
$244$	−28.9802	−1.85527
$245$	4.90506	0.313372
$246$	0	0
$247$	4.93195	0.313812
$248$	4.09975	0.260334
$249$	0	0
$250$	2.36308	0.149454
$251$	−28.2574	−1.78359	−0.891795	−	0.452440i	$-0.850554\pi$
$251$	−28.2574	−1.78359	−0.891795	+	0.452440i	$0.850554\pi$
$252$	0	0
$253$	25.1385	1.58044
$254$	46.5078	2.91816
$255$	0	0
$256$	−25.2121	−1.57576
$257$	−26.0172	−1.62291	−0.811454	−	0.584416i	$-0.801324\pi$
$257$	−26.0172	−1.62291	−0.811454	+	0.584416i	$0.801324\pi$
$258$	0	0
$259$	8.76824	0.544832
$260$	3.12942	0.194078
$261$	0	0
$262$	−6.75477	−0.417311
$263$	−30.8359	−1.90142	−0.950712	−	0.310074i	$-0.899646\pi$
$263$	−30.8359	−1.90142	−0.950712	+	0.310074i	$0.899646\pi$
$264$	0	0
$265$	−8.83811	−0.542921
$266$	−19.3200	−1.18458
$267$	0	0
$268$	37.9222	2.31646
$269$	−11.4911	−0.700623	−0.350311	−	0.936633i	$-0.613924\pi$
$269$	−11.4911	−0.700623	−0.350311	+	0.936633i	$0.613924\pi$
$270$	0	0
$271$	−1.84758	−0.112233	−0.0561164	−	0.998424i	$-0.517872\pi$
$271$	−1.84758	−0.112233	−0.0561164	+	0.998424i	$0.517872\pi$
$272$	−3.77320	−0.228784
$273$	0	0
$274$	−38.4533	−2.32305
$275$	−4.67457	−0.281887
$276$	0	0
$277$	1.59420	0.0957863	0.0478932	−	0.998852i	$-0.484749\pi$
$277$	1.59420	0.0957863	0.0478932	+	0.998852i	$0.484749\pi$
$278$	28.7609	1.72497
$279$	0	0
$280$	−5.41829	−0.323804
$281$	11.4596	0.683624	0.341812	−	0.939768i	$-0.388959\pi$
$281$	11.4596	0.683624	0.341812	+	0.939768i	$0.388959\pi$
$282$	0	0
$283$	−21.9390	−1.30414	−0.652068	−	0.758160i	$-0.726098\pi$
$283$	−21.9390	−1.30414	−0.652068	+	0.758160i	$0.726098\pi$
$284$	56.5619	3.35633
$285$	0	0
$286$	−9.64489	−0.570314
$287$	14.7595	0.871227
$288$	0	0
$289$	−11.9427	−0.702512
$290$	−5.44477	−0.319728
$291$	0	0
$292$	2.50236	0.146439
$293$	27.9899	1.63519	0.817594	−	0.575796i	$-0.195308\pi$
$293$	27.9899	1.63519	0.817594	+	0.575796i	$0.195308\pi$
$294$	0	0
$295$	2.79345	0.162641
$296$	22.6778	1.31812
$297$	0	0
$298$	−17.1931	−0.995972
$299$	4.69542	0.271543
$300$	0	0
$301$	7.07081	0.407555
$302$	1.81326	0.104341
$303$	0	0
$304$	−9.47748	−0.543571
$305$	8.08566	0.462983
$306$	0	0
$307$	11.6608	0.665514	0.332757	−	0.943013i	$-0.392021\pi$
$307$	11.6608	0.665514	0.332757	+	0.943013i	$0.392021\pi$
$308$	24.2502	1.38178
$309$	0	0
$310$	−2.58797	−0.146987
$311$	−28.9140	−1.63957	−0.819783	−	0.572675i	$-0.805906\pi$
$311$	−28.9140	−1.63957	−0.819783	+	0.572675i	$0.805906\pi$
$312$	0	0
$313$	−17.3774	−0.982230	−0.491115	−	0.871095i	$-0.663410\pi$
$313$	−17.3774	−0.982230	−0.491115	+	0.871095i	$0.663410\pi$
$314$	−30.6436	−1.72932
$315$	0	0
$316$	−41.8907	−2.35654
$317$	−21.6442	−1.21566	−0.607829	−	0.794068i	$-0.707959\pi$
$317$	−21.6442	−1.21566	−0.607829	+	0.794068i	$0.707959\pi$
$318$	0	0
$319$	10.7707	0.603041
$320$	11.6786	0.652856
$321$	0	0
$322$	−18.3934	−1.02502
$323$	12.7028	0.706804
$324$	0	0
$325$	−0.873126	−0.0484323
$326$	−13.2558	−0.734172
$327$	0	0
$328$	38.1735	2.10778
$329$	0.302565	0.0166809
$330$	0	0
$331$	11.2133	0.616336	0.308168	−	0.951332i	$-0.400284\pi$
$331$	11.2133	0.616336	0.308168	+	0.951332i	$0.400284\pi$
$332$	−51.8168	−2.84381
$333$	0	0
$334$	−37.9192	−2.07485
$335$	−10.5805	−0.578075
$336$	0	0
$337$	1.11702	0.0608482	0.0304241	−	0.999537i	$-0.490314\pi$
$337$	1.11702	0.0608482	0.0304241	+	0.999537i	$0.490314\pi$
$338$	28.9186	1.57296
$339$	0	0
$340$	8.06019	0.437125
$341$	5.11945	0.277234
$342$	0	0
$343$	17.2313	0.930402
$344$	18.2877	0.986005
$345$	0	0
$346$	8.86520	0.476596
$347$	−14.5839	−0.782907	−0.391454	−	0.920198i	$-0.628028\pi$
$347$	−14.5839	−0.782907	−0.391454	+	0.920198i	$0.628028\pi$
$348$	0	0
$349$	−27.2044	−1.45622	−0.728109	−	0.685461i	$-0.759601\pi$
$349$	−27.2044	−1.45622	−0.728109	+	0.685461i	$0.759601\pi$
$350$	3.42031	0.182823
$351$	0	0
$352$	−16.4642	−0.877546
$353$	4.59066	0.244336	0.122168	−	0.992509i	$-0.461015\pi$
$353$	4.59066	0.244336	0.122168	+	0.992509i	$0.461015\pi$
$354$	0	0
$355$	−15.7811	−0.837574
$356$	−7.54428	−0.399846
$357$	0	0
$358$	10.0859	0.533056
$359$	0.640646	0.0338120	0.0169060	−	0.999857i	$-0.494618\pi$
$359$	0.640646	0.0338120	0.0169060	+	0.999857i	$0.494618\pi$
$360$	0	0
$361$	12.9068	0.679305
$362$	−20.2389	−1.06373
$363$	0	0
$364$	4.52949	0.237410
$365$	−0.698172	−0.0365440
$366$	0	0
$367$	−20.2023	−1.05455	−0.527276	−	0.849694i	$-0.676787\pi$
$367$	−20.2023	−1.05455	−0.527276	+	0.849694i	$0.676787\pi$
$368$	−9.02295	−0.470354
$369$	0	0
$370$	−14.3154	−0.744225
$371$	−12.7922	−0.664139
$372$	0	0
$373$	37.2391	1.92817	0.964085	−	0.265594i	$-0.0855680\pi$
$373$	37.2391	1.92817	0.964085	+	0.265594i	$0.0855680\pi$
$374$	−24.8416	−1.28453
$375$	0	0
$376$	0.782542	0.0403565
$377$	2.01177	0.103611
$378$	0	0
$379$	−16.1652	−0.830350	−0.415175	−	0.909742i	$-0.636280\pi$
$379$	−16.1652	−0.830350	−0.415175	+	0.909742i	$0.636280\pi$
$380$	20.2455	1.03857
$381$	0	0
$382$	48.8265	2.49818
$383$	22.5153	1.15048	0.575239	−	0.817986i	$-0.304909\pi$
$383$	22.5153	1.15048	0.575239	+	0.817986i	$0.304909\pi$
$384$	0	0
$385$	−6.76594	−0.344824
$386$	−4.76509	−0.242536
$387$	0	0
$388$	−13.2092	−0.670598
$389$	34.8276	1.76583	0.882914	−	0.469534i	$-0.155578\pi$
$389$	34.8276	1.76583	0.882914	+	0.469534i	$0.155578\pi$
$390$	0	0
$391$	12.0936	0.611601
$392$	18.3620	0.927420
$393$	0	0
$394$	37.9606	1.91243
$395$	11.6878	0.588075
$396$	0	0
$397$	−4.98338	−0.250109	−0.125054	−	0.992150i	$-0.539911\pi$
$397$	−4.98338	−0.250109	−0.125054	+	0.992150i	$0.539911\pi$
$398$	7.29103	0.365466
$399$	0	0
$400$	1.67784	0.0838922
$401$	18.6735	0.932513	0.466256	−	0.884650i	$-0.345602\pi$
$401$	18.6735	0.932513	0.466256	+	0.884650i	$0.345602\pi$
$402$	0	0
$403$	0.956220	0.0476327
$404$	3.31218	0.164787
$405$	0	0
$406$	−7.88071	−0.391113
$407$	28.3184	1.40369
$408$	0	0
$409$	25.8370	1.27756	0.638779	−	0.769391i	$-0.279440\pi$
$409$	25.8370	1.27756	0.638779	+	0.769391i	$0.279440\pi$
$410$	−24.0971	−1.19007
$411$	0	0
$412$	63.7550	3.14099
$413$	4.04322	0.198954
$414$	0	0
$415$	14.4572	0.709676
$416$	−3.07522	−0.150775
$417$	0	0
$418$	−62.3968	−3.05193
$419$	−20.3519	−0.994253	−0.497127	−	0.867678i	$-0.665612\pi$
$419$	−20.3519	−0.994253	−0.497127	+	0.867678i	$0.665612\pi$
$420$	0	0
$421$	11.0249	0.537320	0.268660	−	0.963235i	$-0.413419\pi$
$421$	11.0249	0.537320	0.268660	+	0.963235i	$0.413419\pi$
$422$	26.4464	1.28739
$423$	0	0
$424$	−33.0853	−1.60676
$425$	−2.24884	−0.109085
$426$	0	0
$427$	11.7031	0.566353
$428$	−3.58415	−0.173247
$429$	0	0
$430$	−11.5441	−0.556708
$431$	−4.18429	−0.201550	−0.100775	−	0.994909i	$-0.532132\pi$
$431$	−4.18429	−0.201550	−0.100775	+	0.994909i	$0.532132\pi$
$432$	0	0
$433$	−1.66788	−0.0801530	−0.0400765	−	0.999197i	$-0.512760\pi$
$433$	−1.66788	−0.0801530	−0.0400765	+	0.999197i	$0.512760\pi$
$434$	−3.74581	−0.179805
$435$	0	0
$436$	−19.0687	−0.913224
$437$	30.3766	1.45311
$438$	0	0
$439$	0.146208	0.00697812	0.00348906	−	0.999994i	$-0.498889\pi$
$439$	0.146208	0.00697812	0.00348906	+	0.999994i	$0.498889\pi$
$440$	−17.4992	−0.834240
$441$	0	0
$442$	−4.63996	−0.220700
$443$	−1.57764	−0.0749561	−0.0374780	−	0.999297i	$-0.511932\pi$
$443$	−1.57764	−0.0749561	−0.0374780	+	0.999297i	$0.511932\pi$
$444$	0	0
$445$	2.10490	0.0997818
$446$	−27.7963	−1.31619
$447$	0	0
$448$	16.9036	0.798619
$449$	12.7601	0.602185	0.301093	−	0.953595i	$-0.402649\pi$
$449$	12.7601	0.602185	0.301093	+	0.953595i	$0.402649\pi$
$450$	0	0
$451$	47.6681	2.24460
$452$	−4.79858	−0.225706
$453$	0	0
$454$	18.7051	0.877876
$455$	−1.26376	−0.0592458
$456$	0	0
$457$	26.7828	1.25285	0.626424	−	0.779483i	$-0.284518\pi$
$457$	26.7828	1.25285	0.626424	+	0.779483i	$0.284518\pi$
$458$	−13.4885	−0.630274
$459$	0	0
$460$	19.2745	0.898680
$461$	8.95079	0.416880	0.208440	−	0.978035i	$-0.433161\pi$
$461$	8.95079	0.416880	0.208440	+	0.978035i	$0.433161\pi$
$462$	0	0
$463$	41.2338	1.91630	0.958148	−	0.286274i	$-0.0924168\pi$
$463$	41.2338	1.91630	0.958148	+	0.286274i	$0.0924168\pi$
$464$	−3.86591	−0.179470
$465$	0	0
$466$	−32.0047	−1.48259
$467$	32.0527	1.48322	0.741611	−	0.670831i	$-0.234062\pi$
$467$	32.0527	1.48322	0.741611	+	0.670831i	$0.234062\pi$
$468$	0	0
$469$	−15.3142	−0.707142
$470$	−0.493981	−0.0227857
$471$	0	0
$472$	10.4572	0.481333
$473$	22.8362	1.05001
$474$	0	0
$475$	−5.64861	−0.259176
$476$	11.6663	0.534722
$477$	0	0
$478$	−9.36833	−0.428497
$479$	−4.32250	−0.197500	−0.0987500	−	0.995112i	$-0.531484\pi$
$479$	−4.32250	−0.197500	−0.0987500	+	0.995112i	$0.531484\pi$
$480$	0	0
$481$	5.28936	0.241174
$482$	−37.1160	−1.69059
$483$	0	0
$484$	38.8939	1.76790
$485$	3.68546	0.167348
$486$	0	0
$487$	−4.78685	−0.216913	−0.108456	−	0.994101i	$-0.534591\pi$
$487$	−4.78685	−0.216913	−0.108456	+	0.994101i	$0.534591\pi$
$488$	30.2685	1.37019
$489$	0	0
$490$	−11.5910	−0.523630
$491$	−2.84131	−0.128226	−0.0641132	−	0.997943i	$-0.520422\pi$
$491$	−2.84131	−0.128226	−0.0641132	+	0.997943i	$0.520422\pi$
$492$	0	0
$493$	5.18155	0.233365
$494$	−11.6546	−0.524365
$495$	0	0
$496$	−1.83752	−0.0825072
$497$	−22.8415	−1.02458
$498$	0	0
$499$	−4.14553	−0.185579	−0.0927897	−	0.995686i	$-0.529578\pi$
$499$	−4.14553	−0.185579	−0.0927897	+	0.995686i	$0.529578\pi$
$500$	−3.58415	−0.160288
$501$	0	0
$502$	66.7745	2.98029
$503$	−19.5052	−0.869692	−0.434846	−	0.900505i	$-0.643197\pi$
$503$	−19.5052	−0.869692	−0.434846	+	0.900505i	$0.643197\pi$
$504$	0	0
$505$	−0.924119	−0.0411228
$506$	−59.4043	−2.64084
$507$	0	0
$508$	−70.5397	−3.12969
$509$	−11.2511	−0.498698	−0.249349	−	0.968414i	$-0.580217\pi$
$509$	−11.2511	−0.498698	−0.249349	+	0.968414i	$0.580217\pi$
$510$	0	0
$511$	−1.01053	−0.0447032
$512$	18.4714	0.816330
$513$	0	0
$514$	61.4808	2.71180
$515$	−17.7880	−0.783835
$516$	0	0
$517$	0.977178	0.0429762
$518$	−20.7201	−0.910387
$519$	0	0
$520$	−3.26853	−0.143334
$521$	−2.34748	−0.102845	−0.0514224	−	0.998677i	$-0.516375\pi$
$521$	−2.34748	−0.102845	−0.0514224	+	0.998677i	$0.516375\pi$
$522$	0	0
$523$	29.0453	1.27006	0.635032	−	0.772486i	$-0.280987\pi$
$523$	29.0453	1.27006	0.635032	+	0.772486i	$0.280987\pi$
$524$	10.2451	0.447561
$525$	0	0
$526$	72.8678	3.17719
$527$	2.46286	0.107284
$528$	0	0
$529$	5.91976	0.257381
$530$	20.8852	0.907194
$531$	0	0
$532$	29.3032	1.27045
$533$	8.90354	0.385655
$534$	0	0
$535$	1.00000	0.0432338
$536$	−39.6079	−1.71080
$537$	0	0
$538$	27.1543	1.17071
$539$	22.9290	0.987624
$540$	0	0
$541$	−28.2067	−1.21270	−0.606351	−	0.795197i	$-0.707367\pi$
$541$	−28.2067	−1.21270	−0.606351	+	0.795197i	$0.707367\pi$
$542$	4.36599	0.187535
$543$	0	0
$544$	−7.92060	−0.339593
$545$	5.32028	0.227896
$546$	0	0
$547$	11.6167	0.496694	0.248347	−	0.968671i	$-0.420113\pi$
$547$	11.6167	0.496694	0.248347	+	0.968671i	$0.420113\pi$
$548$	58.3233	2.49145
$549$	0	0
$550$	11.0464	0.471020
$551$	13.0149	0.554455
$552$	0	0
$553$	16.9168	0.719375
$554$	−3.76723	−0.160054
$555$	0	0
$556$	−43.6225	−1.85001
$557$	11.7725	0.498815	0.249408	−	0.968399i	$-0.419764\pi$
$557$	11.7725	0.498815	0.249408	+	0.968399i	$0.419764\pi$
$558$	0	0
$559$	4.26540	0.180407
$560$	2.42850	0.102623
$561$	0	0
$562$	−27.0800	−1.14230
$563$	−9.33586	−0.393460	−0.196730	−	0.980458i	$-0.563032\pi$
$563$	−9.33586	−0.393460	−0.196730	+	0.980458i	$0.563032\pi$
$564$	0	0
$565$	1.33883	0.0563251
$566$	51.8436	2.17915
$567$	0	0
$568$	−59.0763	−2.47879
$569$	10.8231	0.453729	0.226864	−	0.973926i	$-0.427153\pi$
$569$	10.8231	0.453729	0.226864	+	0.973926i	$0.427153\pi$
$570$	0	0
$571$	35.8249	1.49923	0.749613	−	0.661876i	$-0.230239\pi$
$571$	35.8249	1.49923	0.749613	+	0.661876i	$0.230239\pi$
$572$	14.6287	0.611656
$573$	0	0
$574$	−34.8779	−1.45578
$575$	−5.37771	−0.224266
$576$	0	0
$577$	−0.290697	−0.0121019	−0.00605093	−	0.999982i	$-0.501926\pi$
$577$	−0.290697	−0.0121019	−0.00605093	+	0.999982i	$0.501926\pi$
$578$	28.2216	1.17386
$579$	0	0
$580$	8.25823	0.342904
$581$	20.9252	0.868125
$582$	0	0
$583$	−41.3144	−1.71107
$584$	−2.61359	−0.108151
$585$	0	0
$586$	−66.1424	−2.73232
$587$	−1.59450	−0.0658119	−0.0329059	−	0.999458i	$-0.510476\pi$
$587$	−1.59450	−0.0658119	−0.0329059	+	0.999458i	$0.510476\pi$
$588$	0	0
$589$	6.18618	0.254897
$590$	−6.60116	−0.271765
$591$	0	0
$592$	−10.1643	−0.417750
$593$	9.88819	0.406059	0.203030	−	0.979173i	$-0.434921\pi$
$593$	9.88819	0.406059	0.203030	+	0.979173i	$0.434921\pi$
$594$	0	0
$595$	−3.25496	−0.133440
$596$	26.0773	1.06817
$597$	0	0
$598$	−11.0956	−0.453735
$599$	38.6178	1.57788	0.788940	−	0.614470i	$-0.210630\pi$
$599$	38.6178	1.57788	0.788940	+	0.614470i	$0.210630\pi$
$600$	0	0
$601$	47.5013	1.93762	0.968810	−	0.247805i	$-0.0797092\pi$
$601$	47.5013	1.93762	0.968810	+	0.247805i	$0.0797092\pi$
$602$	−16.7089	−0.681004
$603$	0	0
$604$	−2.75022	−0.111905
$605$	−10.8516	−0.441182
$606$	0	0
$607$	21.8153	0.885454	0.442727	−	0.896656i	$-0.354011\pi$
$607$	21.8153	0.885454	0.442727	+	0.896656i	$0.354011\pi$
$608$	−19.8949	−0.806843
$609$	0	0
$610$	−19.1071	−0.773622
$611$	0.182519	0.00738394
$612$	0	0
$613$	20.6800	0.835256	0.417628	−	0.908618i	$-0.362861\pi$
$613$	20.6800	0.835256	0.417628	+	0.908618i	$0.362861\pi$
$614$	−27.5553	−1.11204
$615$	0	0
$616$	−25.3282	−1.02050
$617$	−17.3073	−0.696764	−0.348382	−	0.937353i	$-0.613269\pi$
$617$	−17.3073	−0.696764	−0.348382	+	0.937353i	$0.613269\pi$
$618$	0	0
$619$	−36.6184	−1.47182	−0.735909	−	0.677081i	$-0.763245\pi$
$619$	−36.6184	−1.47182	−0.735909	+	0.677081i	$0.763245\pi$
$620$	3.92525	0.157642
$621$	0	0
$622$	68.3262	2.73963
$623$	3.04662	0.122060
$624$	0	0
$625$	1.00000	0.0400000
$626$	41.0642	1.64126
$627$	0	0
$628$	46.4781	1.85468
$629$	13.6234	0.543200
$630$	0	0
$631$	21.4040	0.852079	0.426039	−	0.904705i	$-0.359909\pi$
$631$	21.4040	0.852079	0.426039	+	0.904705i	$0.359909\pi$
$632$	43.7529	1.74040
$633$	0	0
$634$	51.1469	2.03130
$635$	19.6810	0.781016
$636$	0	0
$637$	4.28273	0.169688
$638$	−25.4520	−1.00765
$639$	0	0
$640$	−20.5534	−0.812446
$641$	−36.9750	−1.46042	−0.730212	−	0.683221i	$-0.760579\pi$
$641$	−36.9750	−1.46042	−0.730212	+	0.683221i	$0.760579\pi$
$642$	0	0
$643$	11.1122	0.438224	0.219112	−	0.975700i	$-0.429684\pi$
$643$	11.1122	0.438224	0.219112	+	0.975700i	$0.429684\pi$
$644$	27.8978	1.09933
$645$	0	0
$646$	−30.0178	−1.18104
$647$	−23.8168	−0.936336	−0.468168	−	0.883639i	$-0.655086\pi$
$647$	−23.8168	−0.936336	−0.468168	+	0.883639i	$0.655086\pi$
$648$	0	0
$649$	13.0582	0.512579
$650$	2.06327	0.0809280
$651$	0	0
$652$	20.1055	0.787391
$653$	48.0214	1.87922	0.939612	−	0.342241i	$-0.111186\pi$
$653$	48.0214	1.87922	0.939612	+	0.342241i	$0.111186\pi$
$654$	0	0
$655$	−2.85846	−0.111689
$656$	−17.1095	−0.668014
$657$	0	0
$658$	−0.714985	−0.0278730
$659$	−23.4923	−0.915131	−0.457565	−	0.889176i	$-0.651278\pi$
$659$	−23.4923	−0.915131	−0.457565	+	0.889176i	$0.651278\pi$
$660$	0	0
$661$	5.94768	0.231338	0.115669	−	0.993288i	$-0.463099\pi$
$661$	5.94768	0.231338	0.115669	+	0.993288i	$0.463099\pi$
$662$	−26.4978	−1.02987
$663$	0	0
$664$	54.1202	2.10027
$665$	−8.17575	−0.317042
$666$	0	0
$667$	12.3908	0.479772
$668$	57.5132	2.22525
$669$	0	0
$670$	25.0026	0.965935
$671$	37.7970	1.45914
$672$	0	0
$673$	10.8664	0.418868	0.209434	−	0.977823i	$-0.432838\pi$
$673$	10.8664	0.418868	0.209434	+	0.977823i	$0.432838\pi$
$674$	−2.63962	−0.101674
$675$	0	0
$676$	−43.8616	−1.68698
$677$	−27.9593	−1.07456	−0.537281	−	0.843403i	$-0.680549\pi$
$677$	−27.9593	−1.07456	−0.537281	+	0.843403i	$0.680549\pi$
$678$	0	0
$679$	5.33430	0.204712
$680$	−8.41850	−0.322835
$681$	0	0
$682$	−12.0977	−0.463244
$683$	−35.4747	−1.35740	−0.678700	−	0.734415i	$-0.737456\pi$
$683$	−35.4747	−1.35740	−0.678700	+	0.734415i	$0.737456\pi$
$684$	0	0
$685$	−16.2725	−0.621742
$686$	−40.7189	−1.55466
$687$	0	0
$688$	−8.19661	−0.312493
$689$	−7.71678	−0.293986
$690$	0	0
$691$	−10.2516	−0.389988	−0.194994	−	0.980804i	$-0.562469\pi$
$691$	−10.2516	−0.389988	−0.194994	+	0.980804i	$0.562469\pi$
$692$	−13.4461	−0.511144
$693$	0	0
$694$	34.4630	1.30820
$695$	12.1709	0.461670
$696$	0	0
$697$	22.9322	0.868617
$698$	64.2862	2.43327
$699$	0	0
$700$	−5.18767	−0.196076
$701$	16.5070	0.623462	0.311731	−	0.950170i	$-0.399091\pi$
$701$	16.5070	0.623462	0.311731	+	0.950170i	$0.399091\pi$
$702$	0	0
$703$	34.2190	1.29060
$704$	54.5927	2.05754
$705$	0	0
$706$	−10.8481	−0.408274
$707$	−1.33756	−0.0503042
$708$	0	0
$709$	38.4846	1.44532	0.722659	−	0.691204i	$-0.242920\pi$
$709$	38.4846	1.44532	0.722659	+	0.691204i	$0.242920\pi$
$710$	37.2920	1.39955
$711$	0	0
$712$	7.87965	0.295303
$713$	5.88950	0.220564
$714$	0	0
$715$	−4.08149	−0.152639
$716$	−15.2976	−0.571697
$717$	0	0
$718$	−1.51390	−0.0564981
$719$	−17.1250	−0.638653	−0.319327	−	0.947645i	$-0.603457\pi$
$719$	−17.1250	−0.638653	−0.319327	+	0.947645i	$0.603457\pi$
$720$	0	0
$721$	−25.7463	−0.958841
$722$	−30.4998	−1.13509
$723$	0	0
$724$	30.6969	1.14084
$725$	−2.30410	−0.0855720
$726$	0	0
$727$	−19.1313	−0.709542	−0.354771	−	0.934953i	$-0.615441\pi$
$727$	−19.1313	−0.709542	−0.354771	+	0.934953i	$0.615441\pi$
$728$	−4.73084	−0.175337
$729$	0	0
$730$	1.64984	0.0610632
$731$	10.9861	0.406334
$732$	0	0
$733$	−35.0229	−1.29360	−0.646800	−	0.762660i	$-0.723893\pi$
$733$	−35.0229	−1.29360	−0.646800	+	0.762660i	$0.723893\pi$
$734$	47.7397	1.76210
$735$	0	0
$736$	−18.9407	−0.698165
$737$	−49.4594	−1.82186
$738$	0	0
$739$	29.0459	1.06847	0.534235	−	0.845336i	$-0.320600\pi$
$739$	29.0459	1.06847	0.534235	+	0.845336i	$0.320600\pi$
$740$	21.7126	0.798173
$741$	0	0
$742$	30.2290	1.10974
$743$	1.22055	0.0447777	0.0223888	−	0.999749i	$-0.492873\pi$
$743$	1.22055	0.0447777	0.0223888	+	0.999749i	$0.492873\pi$
$744$	0	0
$745$	−7.27573	−0.266562
$746$	−87.9991	−3.22188
$747$	0	0
$748$	37.6780	1.37764
$749$	1.44739	0.0528866
$750$	0	0
$751$	−0.853157	−0.0311321	−0.0155661	−	0.999879i	$-0.504955\pi$
$751$	−0.853157	−0.0311321	−0.0155661	+	0.999879i	$0.504955\pi$
$752$	−0.350738	−0.0127901
$753$	0	0
$754$	−4.75396	−0.173129
$755$	0.767327	0.0279259
$756$	0	0
$757$	−31.4896	−1.14451	−0.572255	−	0.820076i	$-0.693931\pi$
$757$	−31.4896	−1.14451	−0.572255	+	0.820076i	$0.693931\pi$
$758$	38.1997	1.38747
$759$	0	0
$760$	−21.1455	−0.767027
$761$	−52.5069	−1.90337	−0.951687	−	0.307070i	$-0.900651\pi$
$761$	−52.5069	−1.90337	−0.951687	+	0.307070i	$0.900651\pi$
$762$	0	0
$763$	7.70053	0.278778
$764$	−74.0566	−2.67927
$765$	0	0
$766$	−53.2054	−1.92239
$767$	2.43904	0.0880685
$768$	0	0
$769$	20.2190	0.729115	0.364557	−	0.931181i	$-0.381220\pi$
$769$	20.2190	0.729115	0.364557	+	0.931181i	$0.381220\pi$
$770$	15.9885	0.576184
$771$	0	0
$772$	7.22734	0.260118
$773$	−25.0425	−0.900715	−0.450357	−	0.892848i	$-0.648703\pi$
$773$	−25.0425	−0.900715	−0.450357	+	0.892848i	$0.648703\pi$
$774$	0	0
$775$	−1.09517	−0.0393396
$776$	13.7964	0.495263
$777$	0	0
$778$	−82.3004	−2.95061
$779$	57.6007	2.06376
$780$	0	0
$781$	−73.7699	−2.63970
$782$	−28.5782	−1.02195
$783$	0	0
$784$	−8.22991	−0.293926
$785$	−12.9677	−0.462835
$786$	0	0
$787$	−1.40150	−0.0499580	−0.0249790	−	0.999688i	$-0.507952\pi$
$787$	−1.40150	−0.0499580	−0.0249790	+	0.999688i	$0.507952\pi$
$788$	−57.5759	−2.05105
$789$	0	0
$790$	−27.6191	−0.982644
$791$	1.93782	0.0689009
$792$	0	0
$793$	7.05979	0.250701
$794$	11.7761	0.417919
$795$	0	0
$796$	−11.0585	−0.391959
$797$	−21.5005	−0.761587	−0.380794	−	0.924660i	$-0.624349\pi$
$797$	−21.5005	−0.761587	−0.380794	+	0.924660i	$0.624349\pi$
$798$	0	0
$799$	0.470101	0.0166310
$800$	3.52208	0.124524
$801$	0	0
$802$	−44.1271	−1.55818
$803$	−3.26366	−0.115172
$804$	0	0
$805$	−7.78366	−0.274338
$806$	−2.25963	−0.0795919
$807$	0	0
$808$	−3.45942	−0.121702
$809$	21.6260	0.760331	0.380165	−	0.924919i	$-0.375867\pi$
$809$	21.6260	0.760331	0.380165	+	0.924919i	$0.375867\pi$
$810$	0	0
$811$	−38.6557	−1.35738	−0.678692	−	0.734423i	$-0.737453\pi$
$811$	−38.6557	−1.35738	−0.678692	+	0.734423i	$0.737453\pi$
$812$	11.9529	0.419465
$813$	0	0
$814$	−66.9186	−2.34550
$815$	−5.60955	−0.196494
$816$	0	0
$817$	27.5946	0.965413
$818$	−61.0549	−2.13474
$819$	0	0
$820$	36.5487	1.27634
$821$	22.5086	0.785555	0.392777	−	0.919634i	$-0.371514\pi$
$821$	22.5086	0.785555	0.392777	+	0.919634i	$0.371514\pi$
$822$	0	0
$823$	2.92883	0.102092	0.0510462	−	0.998696i	$-0.483744\pi$
$823$	2.92883	0.102092	0.0510462	+	0.998696i	$0.483744\pi$
$824$	−66.5892	−2.31974
$825$	0	0
$826$	−9.55447	−0.332442
$827$	−30.0926	−1.04642	−0.523211	−	0.852203i	$-0.675266\pi$
$827$	−30.0926	−1.04642	−0.523211	+	0.852203i	$0.675266\pi$
$828$	0	0
$829$	23.4175	0.813323	0.406661	−	0.913579i	$-0.366693\pi$
$829$	23.4175	0.813323	0.406661	+	0.913579i	$0.366693\pi$
$830$	−34.1635	−1.18583
$831$	0	0
$832$	10.1969	0.353515
$833$	11.0307	0.382191
$834$	0	0
$835$	−16.0465	−0.555313
$836$	94.6390	3.27316
$837$	0	0
$838$	48.0931	1.66135
$839$	43.5906	1.50491	0.752457	−	0.658641i	$-0.228868\pi$
$839$	43.5906	1.50491	0.752457	+	0.658641i	$0.228868\pi$
$840$	0	0
$841$	−23.6911	−0.816936
$842$	−26.0527	−0.897836
$843$	0	0
$844$	−40.1119	−1.38071
$845$	12.2377	0.420988
$846$	0	0
$847$	−15.7066	−0.539684
$848$	14.8290	0.509229
$849$	0	0
$850$	5.31420	0.182275
$851$	32.5779	1.11676
$852$	0	0
$853$	−12.6420	−0.432855	−0.216428	−	0.976299i	$-0.569441\pi$
$853$	−12.6420	−0.432855	−0.216428	+	0.976299i	$0.569441\pi$
$854$	−27.6554	−0.946349
$855$	0	0
$856$	3.74348	0.127950
$857$	54.2198	1.85211	0.926056	−	0.377385i	$-0.123177\pi$
$857$	54.2198	1.85211	0.926056	+	0.377385i	$0.123177\pi$
$858$	0	0
$859$	−11.8641	−0.404798	−0.202399	−	0.979303i	$-0.564874\pi$
$859$	−11.8641	−0.404798	−0.202399	+	0.979303i	$0.564874\pi$
$860$	17.5093	0.597063
$861$	0	0
$862$	9.88781	0.336780
$863$	−32.9713	−1.12236	−0.561178	−	0.827695i	$-0.689652\pi$
$863$	−32.9713	−1.12236	−0.561178	+	0.827695i	$0.689652\pi$
$864$	0	0
$865$	3.75154	0.127556
$866$	3.94133	0.133932
$867$	0	0
$868$	5.68138	0.192839
$869$	54.6353	1.85338
$870$	0	0
$871$	−9.23812	−0.313022
$872$	19.9164	0.674453
$873$	0	0
$874$	−71.7823	−2.42807
$875$	1.44739	0.0489308
$876$	0	0
$877$	−16.5623	−0.559268	−0.279634	−	0.960107i	$-0.590213\pi$
$877$	−16.5623	−0.559268	−0.279634	+	0.960107i	$0.590213\pi$
$878$	−0.345501	−0.0116601
$879$	0	0
$880$	7.84320	0.264394
$881$	−4.29022	−0.144541	−0.0722706	−	0.997385i	$-0.523025\pi$
$881$	−4.29022	−0.144541	−0.0722706	+	0.997385i	$0.523025\pi$
$882$	0	0
$883$	6.93590	0.233412	0.116706	−	0.993167i	$-0.462767\pi$
$883$	6.93590	0.233412	0.116706	+	0.993167i	$0.462767\pi$
$884$	7.03756	0.236699
$885$	0	0
$886$	3.72810	0.125248
$887$	−34.1013	−1.14501	−0.572504	−	0.819902i	$-0.694028\pi$
$887$	−34.1013	−1.14501	−0.572504	+	0.819902i	$0.694028\pi$
$888$	0	0
$889$	28.4861	0.955394
$890$	−4.97405	−0.166731
$891$	0	0
$892$	42.1594	1.41160
$893$	1.18079	0.0395137
$894$	0	0
$895$	4.26811	0.142667
$896$	−29.7489	−0.993840
$897$	0	0
$898$	−30.1531	−1.00622
$899$	2.52338	0.0841593
$900$	0	0
$901$	−19.8755	−0.662149
$902$	−112.644	−3.75062
$903$	0	0
$904$	5.01190	0.166693
$905$	−8.56462	−0.284697
$906$	0	0
$907$	29.9049	0.992975	0.496488	−	0.868044i	$-0.334623\pi$
$907$	29.9049	0.992975	0.496488	+	0.868044i	$0.334623\pi$
$908$	−28.3706	−0.941512
$909$	0	0
$910$	2.98636	0.0989968
$911$	−16.1032	−0.533524	−0.266762	−	0.963762i	$-0.585954\pi$
$911$	−16.1032	−0.533524	−0.266762	+	0.963762i	$0.585954\pi$
$912$	0	0
$913$	67.5812	2.23661
$914$	−63.2900	−2.09345
$915$	0	0
$916$	20.4583	0.675962
$917$	−4.13731	−0.136626
$918$	0	0
$919$	12.1131	0.399574	0.199787	−	0.979839i	$-0.435975\pi$
$919$	12.1131	0.399574	0.199787	+	0.979839i	$0.435975\pi$
$920$	−20.1314	−0.663711
$921$	0	0
$922$	−21.1514	−0.696585
$923$	−13.7789	−0.453538
$924$	0	0
$925$	−6.05796	−0.199184
$926$	−97.4387	−3.20204
$927$	0	0
$928$	−8.11521	−0.266395
$929$	47.2640	1.55068	0.775340	−	0.631544i	$-0.217578\pi$
$929$	47.2640	1.55068	0.775340	+	0.631544i	$0.217578\pi$
$930$	0	0
$931$	27.7067	0.908052
$932$	48.5424	1.59006
$933$	0	0
$934$	−75.7431	−2.47839
$935$	−10.5124	−0.343791
$936$	0	0
$937$	−21.0685	−0.688278	−0.344139	−	0.938919i	$-0.611829\pi$
$937$	−21.0685	−0.688278	−0.344139	+	0.938919i	$0.611829\pi$
$938$	36.1886	1.18160
$939$	0	0
$940$	0.749235	0.0244374
$941$	23.4389	0.764086	0.382043	−	0.924145i	$-0.375221\pi$
$941$	23.4389	0.764086	0.382043	+	0.924145i	$0.375221\pi$
$942$	0	0
$943$	54.8382	1.78578
$944$	−4.68698	−0.152548
$945$	0	0
$946$	−53.9639	−1.75452
$947$	−15.4007	−0.500456	−0.250228	−	0.968187i	$-0.580506\pi$
$947$	−15.4007	−0.500456	−0.250228	+	0.968187i	$0.580506\pi$
$948$	0	0
$949$	−0.609592	−0.0197882
$950$	13.3481	0.433070
$951$	0	0
$952$	−12.1849	−0.394914
$953$	34.2377	1.10907	0.554535	−	0.832161i	$-0.312896\pi$
$953$	34.2377	1.10907	0.554535	+	0.832161i	$0.312896\pi$
$954$	0	0
$955$	20.6622	0.668614
$956$	14.2092	0.459559
$957$	0	0
$958$	10.2144	0.330013
$959$	−23.5528	−0.760558
$960$	0	0
$961$	−29.8006	−0.961310
$962$	−12.4992	−0.402990
$963$	0	0
$964$	56.2949	1.81314
$965$	−2.01647	−0.0649125
$966$	0	0
$967$	−54.3298	−1.74713	−0.873564	−	0.486709i	$-0.838197\pi$
$967$	−54.3298	−1.74713	−0.873564	+	0.486709i	$0.838197\pi$
$968$	−40.6229	−1.30567
$969$	0	0
$970$	−8.70903	−0.279630
$971$	40.4098	1.29681	0.648407	−	0.761294i	$-0.275436\pi$
$971$	40.4098	1.29681	0.648407	+	0.761294i	$0.275436\pi$
$972$	0	0
$973$	17.6161	0.564747
$974$	11.3117	0.362451
$975$	0	0
$976$	−13.5665	−0.434252
$977$	33.3177	1.06593	0.532963	−	0.846138i	$-0.321078\pi$
$977$	33.3177	1.06593	0.532963	+	0.846138i	$0.321078\pi$
$978$	0	0
$979$	9.83951	0.314472
$980$	17.5805	0.561587
$981$	0	0
$982$	6.71424	0.214260
$983$	−3.45352	−0.110150	−0.0550751	−	0.998482i	$-0.517540\pi$
$983$	−3.45352	−0.110150	−0.0550751	+	0.998482i	$0.517540\pi$
$984$	0	0
$985$	16.0640	0.511842
$986$	−12.2444	−0.389942
$987$	0	0
$988$	17.6768	0.562375
$989$	26.2712	0.835376
$990$	0	0
$991$	−5.35863	−0.170222	−0.0851112	−	0.996371i	$-0.527125\pi$
$991$	−5.35863	−0.170222	−0.0851112	+	0.996371i	$0.527125\pi$
$992$	−3.85728	−0.122469
$993$	0	0
$994$	53.9762	1.71202
$995$	3.08539	0.0978135
$996$	0	0
$997$	−60.5247	−1.91684	−0.958419	−	0.285365i	$-0.907885\pi$
$997$	−60.5247	−1.91684	−0.958419	+	0.285365i	$0.907885\pi$
$998$	9.79623	0.310094
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4815.2.a.u.1.3		12
3.2	odd	2		1605.2.a.n.1.10	✓	12
15.14	odd	2		8025.2.a.bf.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1605.2.a.n.1.10	✓	12	3.2	odd	2
4815.2.a.u.1.3		12	1.1	even	1	trivial
8025.2.a.bf.1.3		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)

\(f(q)\)	\(=\)	\(q-2.36308 q^{2} +3.58415 q^{4} -1.00000 q^{5} -1.44739 q^{7} -3.74348 q^{8} +O(q^{10})\)	\(q-2.36308 q^{2} +3.58415 q^{4} -1.00000 q^{5} -1.44739 q^{7} -3.74348 q^{8} +2.36308 q^{10} -4.67457 q^{11} -0.873126 q^{13} +3.42031 q^{14} +1.67784 q^{16} -2.24884 q^{17} -5.64861 q^{19} -3.58415 q^{20} +11.0464 q^{22} -5.37771 q^{23} +1.00000 q^{25} +2.06327 q^{26} -5.18767 q^{28} -2.30410 q^{29} -1.09517 q^{31} +3.52208 q^{32} +5.31420 q^{34} +1.44739 q^{35} -6.05796 q^{37} +13.3481 q^{38} +3.74348 q^{40} -10.1973 q^{41} -4.88520 q^{43} -16.7544 q^{44} +12.7080 q^{46} -0.209041 q^{47} -4.90506 q^{49} -2.36308 q^{50} -3.12942 q^{52} +8.83811 q^{53} +4.67457 q^{55} +5.41829 q^{56} +5.44477 q^{58} -2.79345 q^{59} -8.08566 q^{61} +2.58797 q^{62} -11.6786 q^{64} +0.873126 q^{65} +10.5805 q^{67} -8.06019 q^{68} -3.42031 q^{70} +15.7811 q^{71} +0.698172 q^{73} +14.3154 q^{74} -20.2455 q^{76} +6.76594 q^{77} -11.6878 q^{79} -1.67784 q^{80} +24.0971 q^{82} -14.4572 q^{83} +2.24884 q^{85} +11.5441 q^{86} +17.4992 q^{88} -2.10490 q^{89} +1.26376 q^{91} -19.2745 q^{92} +0.493981 q^{94} +5.64861 q^{95} -3.68546 q^{97} +11.5910 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8}+O(q^{10}) \) 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

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