LMFDB - Embedded newform 6030.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(1,6030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, 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("6030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.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:	$48.1497924188$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.11324.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 4x + 2 $ x^4 - x^3 - 5x^2 + 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2010)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.25619$ of defining polynomial
Character	$\chi$	$=$	6030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.51238 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.51238 q^{7} +1.00000 q^{8} -1.00000 q^{10} -5.29433 q^{11} +4.18078 q^{13} -4.51238 q^{14} +1.00000 q^{16} +7.39883 q^{17} +6.92026 q^{19} -1.00000 q^{20} -5.29433 q^{22} -1.26052 q^{23} +1.00000 q^{25} +4.18078 q^{26} -4.51238 q^{28} +5.43264 q^{29} -7.47512 q^{31} +1.00000 q^{32} +7.39883 q^{34} +4.51238 q^{35} -9.06723 q^{37} +6.92026 q^{38} -1.00000 q^{40} -3.29433 q^{41} +2.10450 q^{43} -5.29433 q^{44} -1.26052 q^{46} -8.51238 q^{47} +13.3616 q^{49} +1.00000 q^{50} +4.18078 q^{52} -8.17212 q^{53} +5.29433 q^{55} -4.51238 q^{56} +5.43264 q^{58} -6.33160 q^{59} -10.0338 q^{61} -7.47512 q^{62} +1.00000 q^{64} -4.18078 q^{65} -1.00000 q^{67} +7.39883 q^{68} +4.51238 q^{70} +0.0762887 q^{71} +11.0924 q^{73} -9.06723 q^{74} +6.92026 q^{76} +23.8900 q^{77} +10.7270 q^{79} -1.00000 q^{80} -3.29433 q^{82} -4.47857 q^{83} -7.39883 q^{85} +2.10450 q^{86} -5.29433 q^{88} +4.58867 q^{89} -18.8653 q^{91} -1.26052 q^{92} -8.51238 q^{94} -6.92026 q^{95} +10.4661 q^{97} +13.3616 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 2 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 - 2 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 2 q^{7} + 4 q^{8} - 4 q^{10} - 10 q^{11} - 2 q^{13} - 2 q^{14} + 4 q^{16} + 6 q^{17} + 8 q^{19} - 4 q^{20} - 10 q^{22} - 6 q^{23} + 4 q^{25} - 2 q^{26} - 2 q^{28} - 14 q^{29} + 4 q^{32} + 6 q^{34} + 2 q^{35} - 10 q^{37} + 8 q^{38} - 4 q^{40} - 2 q^{41} - 4 q^{43} - 10 q^{44} - 6 q^{46} - 18 q^{47} + 16 q^{49} + 4 q^{50} - 2 q^{52} + 4 q^{53} + 10 q^{55} - 2 q^{56} - 14 q^{58} - 28 q^{59} - 28 q^{61} + 4 q^{64} + 2 q^{65} - 4 q^{67} + 6 q^{68} + 2 q^{70} - 6 q^{71} - 12 q^{73} - 10 q^{74} + 8 q^{76} - 8 q^{77} - 4 q^{79} - 4 q^{80} - 2 q^{82} - 14 q^{83} - 6 q^{85} - 4 q^{86} - 10 q^{88} - 4 q^{89} - 4 q^{91} - 6 q^{92} - 18 q^{94} - 8 q^{95} - 8 q^{97} + 16 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 - 2 * q^7 + 4 * q^8 - 4 * q^10 - 10 * q^11 - 2 * q^13 - 2 * q^14 + 4 * q^16 + 6 * q^17 + 8 * q^19 - 4 * q^20 - 10 * q^22 - 6 * q^23 + 4 * q^25 - 2 * q^26 - 2 * q^28 - 14 * q^29 + 4 * q^32 + 6 * q^34 + 2 * q^35 - 10 * q^37 + 8 * q^38 - 4 * q^40 - 2 * q^41 - 4 * q^43 - 10 * q^44 - 6 * q^46 - 18 * q^47 + 16 * q^49 + 4 * q^50 - 2 * q^52 + 4 * q^53 + 10 * q^55 - 2 * q^56 - 14 * q^58 - 28 * q^59 - 28 * q^61 + 4 * q^64 + 2 * q^65 - 4 * q^67 + 6 * q^68 + 2 * q^70 - 6 * q^71 - 12 * q^73 - 10 * q^74 + 8 * q^76 - 8 * q^77 - 4 * q^79 - 4 * q^80 - 2 * q^82 - 14 * q^83 - 6 * q^85 - 4 * q^86 - 10 * q^88 - 4 * q^89 - 4 * q^91 - 6 * q^92 - 18 * q^94 - 8 * q^95 - 8 * q^97 + 16 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.51238	−1.70552	−0.852760	−	0.522304i	$-0.825073\pi$
$7$	−4.51238	−1.70552	−0.852760	+	0.522304i	$0.825073\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−5.29433	−1.59630	−0.798151	−	0.602458i	$-0.794188\pi$
$11$	−5.29433	−1.59630	−0.798151	+	0.602458i	$0.794188\pi$
$12$	0	0
$13$	4.18078	1.15954	0.579770	−	0.814780i	$-0.303142\pi$
$13$	4.18078	1.15954	0.579770	+	0.814780i	$0.303142\pi$
$14$	−4.51238	−1.20598
$15$	0	0
$16$	1.00000	0.250000
$17$	7.39883	1.79448	0.897240	−	0.441543i	$-0.145569\pi$
$17$	7.39883	1.79448	0.897240	+	0.441543i	$0.145569\pi$
$18$	0	0
$19$	6.92026	1.58762	0.793809	−	0.608168i	$-0.208095\pi$
$19$	6.92026	1.58762	0.793809	+	0.608168i	$0.208095\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−5.29433	−1.12876
$23$	−1.26052	−0.262837	−0.131418	−	0.991327i	$-0.541953\pi$
$23$	−1.26052	−0.262837	−0.131418	+	0.991327i	$0.541953\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.18078	0.819919
$27$	0	0
$28$	−4.51238	−0.852760
$29$	5.43264	1.00882	0.504408	−	0.863465i	$-0.331711\pi$
$29$	5.43264	1.00882	0.504408	+	0.863465i	$0.331711\pi$
$30$	0	0
$31$	−7.47512	−1.34257	−0.671285	−	0.741199i	$-0.734258\pi$
$31$	−7.47512	−1.34257	−0.671285	+	0.741199i	$0.734258\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.39883	1.26889
$35$	4.51238	0.762731
$36$	0	0
$37$	−9.06723	−1.49064	−0.745322	−	0.666705i	$-0.767704\pi$
$37$	−9.06723	−1.49064	−0.745322	+	0.666705i	$0.767704\pi$
$38$	6.92026	1.12261
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−3.29433	−0.514489	−0.257244	−	0.966346i	$-0.582814\pi$
$41$	−3.29433	−0.514489	−0.257244	+	0.966346i	$0.582814\pi$
$42$	0	0
$43$	2.10450	0.320933	0.160466	−	0.987041i	$-0.448700\pi$
$43$	2.10450	0.320933	0.160466	+	0.987041i	$0.448700\pi$
$44$	−5.29433	−0.798151
$45$	0	0
$46$	−1.26052	−0.185854
$47$	−8.51238	−1.24166	−0.620829	−	0.783946i	$-0.713204\pi$
$47$	−8.51238	−1.24166	−0.620829	+	0.783946i	$0.713204\pi$
$48$	0	0
$49$	13.3616	1.90880
$50$	1.00000	0.141421
$51$	0	0
$52$	4.18078	0.579770
$53$	−8.17212	−1.12253	−0.561264	−	0.827637i	$-0.689685\pi$
$53$	−8.17212	−1.12253	−0.561264	+	0.827637i	$0.689685\pi$
$54$	0	0
$55$	5.29433	0.713888
$56$	−4.51238	−0.602992
$57$	0	0
$58$	5.43264	0.713341
$59$	−6.33160	−0.824303	−0.412152	−	0.911115i	$-0.635223\pi$
$59$	−6.33160	−0.824303	−0.412152	+	0.911115i	$0.635223\pi$
$60$	0	0
$61$	−10.0338	−1.28470	−0.642349	−	0.766412i	$-0.722040\pi$
$61$	−10.0338	−1.28470	−0.642349	+	0.766412i	$0.722040\pi$
$62$	−7.47512	−0.949341
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.18078	−0.518562
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	7.39883	0.897240
$69$	0	0
$70$	4.51238	0.539333
$71$	0.0762887	0.00905380	0.00452690	−	0.999990i	$-0.498559\pi$
$71$	0.0762887	0.00905380	0.00452690	+	0.999990i	$0.498559\pi$
$72$	0	0
$73$	11.0924	1.29827	0.649133	−	0.760675i	$-0.275132\pi$
$73$	11.0924	1.29827	0.649133	+	0.760675i	$0.275132\pi$
$74$	−9.06723	−1.05404
$75$	0	0
$76$	6.92026	0.793809
$77$	23.8900	2.72252
$78$	0	0
$79$	10.7270	1.20688	0.603440	−	0.797409i	$-0.293796\pi$
$79$	10.7270	1.20688	0.603440	+	0.797409i	$0.293796\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−3.29433	−0.363798
$83$	−4.47857	−0.491587	−0.245793	−	0.969322i	$-0.579048\pi$
$83$	−4.47857	−0.491587	−0.245793	+	0.969322i	$0.579048\pi$
$84$	0	0
$85$	−7.39883	−0.802516
$86$	2.10450	0.226934
$87$	0	0
$88$	−5.29433	−0.564378
$89$	4.58867	0.486398	0.243199	−	0.969976i	$-0.421803\pi$
$89$	4.58867	0.486398	0.243199	+	0.969976i	$0.421803\pi$
$90$	0	0
$91$	−18.8653	−1.97762
$92$	−1.26052	−0.131418
$93$	0	0
$94$	−8.51238	−0.877985
$95$	−6.92026	−0.710004
$96$	0	0
$97$	10.4661	1.06267	0.531334	−	0.847162i	$-0.321691\pi$
$97$	10.4661	1.06267	0.531334	+	0.847162i	$0.321691\pi$
$98$	13.3616	1.34972
$99$	0	0
$100$	1.00000	0.100000
$101$	−13.2519	−1.31861	−0.659305	−	0.751876i	$-0.729149\pi$
$101$	−13.2519	−1.31861	−0.659305	+	0.751876i	$0.729149\pi$
$102$	0	0
$103$	−12.9078	−1.27184	−0.635920	−	0.771755i	$-0.719379\pi$
$103$	−12.9078	−1.27184	−0.635920	+	0.771755i	$0.719379\pi$
$104$	4.18078	0.409960
$105$	0	0
$106$	−8.17212	−0.793747
$107$	−12.5462	−1.21289	−0.606443	−	0.795127i	$-0.707404\pi$
$107$	−12.5462	−1.21289	−0.606443	+	0.795127i	$0.707404\pi$
$108$	0	0
$109$	−18.3104	−1.75382	−0.876911	−	0.480654i	$-0.840399\pi$
$109$	−18.3104	−1.75382	−0.876911	+	0.480654i	$0.840399\pi$
$110$	5.29433	0.504795
$111$	0	0
$112$	−4.51238	−0.426380
$113$	−8.50372	−0.799962	−0.399981	−	0.916523i	$-0.630983\pi$
$113$	−8.50372	−0.799962	−0.399981	+	0.916523i	$0.630983\pi$
$114$	0	0
$115$	1.26052	0.117544
$116$	5.43264	0.504408
$117$	0	0
$118$	−6.33160	−0.582871
$119$	−33.3863	−3.06052
$120$	0	0
$121$	17.0300	1.54818
$122$	−10.0338	−0.908419
$123$	0	0
$124$	−7.47512	−0.671285
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.3439	1.71649	0.858245	−	0.513241i	$-0.171555\pi$
$127$	19.3439	1.71649	0.858245	+	0.513241i	$0.171555\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.18078	−0.366679
$131$	−9.73603	−0.850641	−0.425320	−	0.905043i	$-0.639839\pi$
$131$	−9.73603	−0.850641	−0.425320	+	0.905043i	$0.639839\pi$
$132$	0	0
$133$	−31.2269	−2.70771
$134$	−1.00000	−0.0863868
$135$	0	0
$136$	7.39883	0.634444
$137$	−19.0924	−1.63117	−0.815586	−	0.578635i	$-0.803585\pi$
$137$	−19.0924	−1.63117	−0.815586	+	0.578635i	$0.803585\pi$
$138$	0	0
$139$	−7.71792	−0.654626	−0.327313	−	0.944916i	$-0.606143\pi$
$139$	−7.71792	−0.654626	−0.327313	+	0.944916i	$0.606143\pi$
$140$	4.51238	0.381366
$141$	0	0
$142$	0.0762887	0.00640201
$143$	−22.1345	−1.85098
$144$	0	0
$145$	−5.43264	−0.451156
$146$	11.0924	0.918012
$147$	0	0
$148$	−9.06723	−0.745322
$149$	13.4626	1.10290	0.551450	−	0.834208i	$-0.314075\pi$
$149$	13.4626	1.10290	0.551450	+	0.834208i	$0.314075\pi$
$150$	0	0
$151$	−6.52104	−0.530675	−0.265337	−	0.964156i	$-0.585483\pi$
$151$	−6.52104	−0.530675	−0.265337	+	0.964156i	$0.585483\pi$
$152$	6.92026	0.561307
$153$	0	0
$154$	23.8900	1.92511
$155$	7.47512	0.600416
$156$	0	0
$157$	−1.81538	−0.144883	−0.0724414	−	0.997373i	$-0.523079\pi$
$157$	−1.81538	−0.144883	−0.0724414	+	0.997373i	$0.523079\pi$
$158$	10.7270	0.853392
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	5.68795	0.448273
$162$	0	0
$163$	13.0924	1.02547	0.512737	−	0.858546i	$-0.328631\pi$
$163$	13.0924	1.02547	0.512737	+	0.858546i	$0.328631\pi$
$164$	−3.29433	−0.257244
$165$	0	0
$166$	−4.47857	−0.347604
$167$	−19.3950	−1.50083	−0.750415	−	0.660967i	$-0.770146\pi$
$167$	−19.3950	−1.50083	−0.750415	+	0.660967i	$0.770146\pi$
$168$	0	0
$169$	4.47896	0.344535
$170$	−7.39883	−0.567464
$171$	0	0
$172$	2.10450	0.160466
$173$	−11.7266	−0.891556	−0.445778	−	0.895144i	$-0.647073\pi$
$173$	−11.7266	−0.891556	−0.445778	+	0.895144i	$0.647073\pi$
$174$	0	0
$175$	−4.51238	−0.341104
$176$	−5.29433	−0.399075
$177$	0	0
$178$	4.58867	0.343935
$179$	4.09199	0.305850	0.152925	−	0.988238i	$-0.451131\pi$
$179$	4.09199	0.305850	0.152925	+	0.988238i	$0.451131\pi$
$180$	0	0
$181$	6.20899	0.461511	0.230755	−	0.973012i	$-0.425880\pi$
$181$	6.20899	0.461511	0.230755	+	0.973012i	$0.425880\pi$
$182$	−18.8653	−1.39839
$183$	0	0
$184$	−1.26052	−0.0929268
$185$	9.06723	0.666636
$186$	0	0
$187$	−39.1719	−2.86453
$188$	−8.51238	−0.620829
$189$	0	0
$190$	−6.92026	−0.502049
$191$	−11.9150	−0.862143	−0.431071	−	0.902318i	$-0.641864\pi$
$191$	−11.9150	−0.862143	−0.431071	+	0.902318i	$0.641864\pi$
$192$	0	0
$193$	−15.0924	−1.08637	−0.543187	−	0.839612i	$-0.682782\pi$
$193$	−15.0924	−1.08637	−0.543187	+	0.839612i	$0.682782\pi$
$194$	10.4661	0.751420
$195$	0	0
$196$	13.3616	0.954398
$197$	−6.33160	−0.451107	−0.225554	−	0.974231i	$-0.572419\pi$
$197$	−6.33160	−0.451107	−0.225554	+	0.974231i	$0.572419\pi$
$198$	0	0
$199$	5.41133	0.383599	0.191800	−	0.981434i	$-0.438568\pi$
$199$	5.41133	0.383599	0.191800	+	0.981434i	$0.438568\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−13.2519	−0.932398
$203$	−24.5141	−1.72056
$204$	0	0
$205$	3.29433	0.230086
$206$	−12.9078	−0.899326
$207$	0	0
$208$	4.18078	0.289885
$209$	−36.6382	−2.53432
$210$	0	0
$211$	14.6563	1.00898	0.504490	−	0.863417i	$-0.331680\pi$
$211$	14.6563	1.00898	0.504490	+	0.863417i	$0.331680\pi$
$212$	−8.17212	−0.561264
$213$	0	0
$214$	−12.5462	−0.857640
$215$	−2.10450	−0.143525
$216$	0	0
$217$	33.7306	2.28978
$218$	−18.3104	−1.24014
$219$	0	0
$220$	5.29433	0.356944
$221$	30.9329	2.08077
$222$	0	0
$223$	2.49667	0.167190	0.0835948	−	0.996500i	$-0.473360\pi$
$223$	2.49667	0.167190	0.0835948	+	0.996500i	$0.473360\pi$
$224$	−4.51238	−0.301496
$225$	0	0
$226$	−8.50372	−0.565659
$227$	−10.9996	−0.730070	−0.365035	−	0.930994i	$-0.618943\pi$
$227$	−10.9996	−0.730070	−0.365035	+	0.930994i	$0.618943\pi$
$228$	0	0
$229$	9.57961	0.633039	0.316519	−	0.948586i	$-0.397486\pi$
$229$	9.57961	0.633039	0.316519	+	0.948586i	$0.397486\pi$
$230$	1.26052	0.0831163
$231$	0	0
$232$	5.43264	0.356670
$233$	29.2269	1.91471	0.957357	−	0.288906i	$-0.0932915\pi$
$233$	29.2269	1.91471	0.957357	+	0.288906i	$0.0932915\pi$
$234$	0	0
$235$	8.51238	0.555286
$236$	−6.33160	−0.412152
$237$	0	0
$238$	−33.3863	−2.16411
$239$	−23.9074	−1.54644	−0.773220	−	0.634138i	$-0.781355\pi$
$239$	−23.9074	−1.54644	−0.773220	+	0.634138i	$0.781355\pi$
$240$	0	0
$241$	−24.4111	−1.57246	−0.786228	−	0.617936i	$-0.787969\pi$
$241$	−24.4111	−1.57246	−0.786228	+	0.617936i	$0.787969\pi$
$242$	17.0300	1.09473
$243$	0	0
$244$	−10.0338	−0.642349
$245$	−13.3616	−0.853639
$246$	0	0
$247$	28.9321	1.84091
$248$	−7.47512	−0.474670
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−26.5281	−1.67444	−0.837219	−	0.546868i	$-0.815820\pi$
$251$	−26.5281	−1.67444	−0.837219	+	0.546868i	$0.815820\pi$
$252$	0	0
$253$	6.67362	0.419567
$254$	19.3439	1.21374
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.6259	0.849962	0.424981	−	0.905202i	$-0.360281\pi$
$257$	13.6259	0.849962	0.424981	+	0.905202i	$0.360281\pi$
$258$	0	0
$259$	40.9148	2.54232
$260$	−4.18078	−0.259281
$261$	0	0
$262$	−9.73603	−0.601494
$263$	−11.8319	−0.729584	−0.364792	−	0.931089i	$-0.618860\pi$
$263$	−11.8319	−0.729584	−0.364792	+	0.931089i	$0.618860\pi$
$264$	0	0
$265$	8.17212	0.502009
$266$	−31.2269	−1.91464
$267$	0	0
$268$	−1.00000	−0.0610847
$269$	4.43786	0.270581	0.135290	−	0.990806i	$-0.456803\pi$
$269$	4.43786	0.270581	0.135290	+	0.990806i	$0.456803\pi$
$270$	0	0
$271$	−22.1878	−1.34781	−0.673907	−	0.738816i	$-0.735385\pi$
$271$	−22.1878	−1.34781	−0.673907	+	0.738816i	$0.735385\pi$
$272$	7.39883	0.448620
$273$	0	0
$274$	−19.0924	−1.15341
$275$	−5.29433	−0.319260
$276$	0	0
$277$	8.68066	0.521570	0.260785	−	0.965397i	$-0.416019\pi$
$277$	8.68066	0.521570	0.260785	+	0.965397i	$0.416019\pi$
$278$	−7.71792	−0.462890
$279$	0	0
$280$	4.51238	0.269666
$281$	3.68091	0.219584	0.109792	−	0.993955i	$-0.464981\pi$
$281$	3.68091	0.219584	0.109792	+	0.993955i	$0.464981\pi$
$282$	0	0
$283$	2.07452	0.123318	0.0616588	−	0.998097i	$-0.480361\pi$
$283$	2.07452	0.123318	0.0616588	+	0.998097i	$0.480361\pi$
$284$	0.0762887	0.00452690
$285$	0	0
$286$	−22.1345	−1.30884
$287$	14.8653	0.877470
$288$	0	0
$289$	37.7427	2.22016
$290$	−5.43264	−0.319016
$291$	0	0
$292$	11.0924	0.649133
$293$	4.92893	0.287951	0.143975	−	0.989581i	$-0.454011\pi$
$293$	4.92893	0.287951	0.143975	+	0.989581i	$0.454011\pi$
$294$	0	0
$295$	6.33160	0.368640
$296$	−9.06723	−0.527022
$297$	0	0
$298$	13.4626	0.779868
$299$	−5.26997	−0.304770
$300$	0	0
$301$	−9.49628	−0.547357
$302$	−6.52104	−0.375244
$303$	0	0
$304$	6.92026	0.396904
$305$	10.0338	0.574534
$306$	0	0
$307$	−13.4713	−0.768847	−0.384423	−	0.923157i	$-0.625600\pi$
$307$	−13.4713	−0.768847	−0.384423	+	0.923157i	$0.625600\pi$
$308$	23.8900	1.36126
$309$	0	0
$310$	7.47512	0.424558
$311$	−4.36157	−0.247322	−0.123661	−	0.992325i	$-0.539464\pi$
$311$	−4.36157	−0.247322	−0.123661	+	0.992325i	$0.539464\pi$
$312$	0	0
$313$	25.6629	1.45056	0.725278	−	0.688456i	$-0.241711\pi$
$313$	25.6629	1.45056	0.725278	+	0.688456i	$0.241711\pi$
$314$	−1.81538	−0.102448
$315$	0	0
$316$	10.7270	0.603440
$317$	3.73426	0.209737	0.104869	−	0.994486i	$-0.466558\pi$
$317$	3.73426	0.209737	0.104869	+	0.994486i	$0.466558\pi$
$318$	0	0
$319$	−28.7622	−1.61038
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	5.68795	0.316977
$323$	51.2019	2.84895
$324$	0	0
$325$	4.18078	0.231908
$326$	13.0924	0.725120
$327$	0	0
$328$	−3.29433	−0.181899
$329$	38.4111	2.11767
$330$	0	0
$331$	22.4592	1.23447	0.617234	−	0.786780i	$-0.288253\pi$
$331$	22.4592	1.23447	0.617234	+	0.786780i	$0.288253\pi$
$332$	−4.47857	−0.245793
$333$	0	0
$334$	−19.3950	−1.06125
$335$	1.00000	0.0546358
$336$	0	0
$337$	−28.6682	−1.56165	−0.780827	−	0.624747i	$-0.785202\pi$
$337$	−28.6682	−1.56165	−0.780827	+	0.624747i	$0.785202\pi$
$338$	4.47896	0.243623
$339$	0	0
$340$	−7.39883	−0.401258
$341$	39.5758	2.14315
$342$	0	0
$343$	−28.7058	−1.54997
$344$	2.10450	0.113467
$345$	0	0
$346$	−11.7266	−0.630425
$347$	30.4284	1.63348	0.816741	−	0.577004i	$-0.195778\pi$
$347$	30.4284	1.63348	0.816741	+	0.577004i	$0.195778\pi$
$348$	0	0
$349$	5.54580	0.296860	0.148430	−	0.988923i	$-0.452578\pi$
$349$	5.54580	0.296860	0.148430	+	0.988923i	$0.452578\pi$
$350$	−4.51238	−0.241197
$351$	0	0
$352$	−5.29433	−0.282189
$353$	−23.4713	−1.24925	−0.624625	−	0.780925i	$-0.714748\pi$
$353$	−23.4713	−1.24925	−0.624625	+	0.780925i	$0.714748\pi$
$354$	0	0
$355$	−0.0762887	−0.00404898
$356$	4.58867	0.243199
$357$	0	0
$358$	4.09199	0.216269
$359$	5.57401	0.294185	0.147092	−	0.989123i	$-0.453009\pi$
$359$	5.57401	0.294185	0.147092	+	0.989123i	$0.453009\pi$
$360$	0	0
$361$	28.8900	1.52053
$362$	6.20899	0.326337
$363$	0	0
$364$	−18.8653	−0.988810
$365$	−11.0924	−0.580602
$366$	0	0
$367$	−33.3596	−1.74135	−0.870677	−	0.491855i	$-0.836319\pi$
$367$	−33.3596	−1.74135	−0.870677	+	0.491855i	$0.836319\pi$
$368$	−1.26052	−0.0657092
$369$	0	0
$370$	9.06723	0.471383
$371$	36.8757	1.91449
$372$	0	0
$373$	−3.97179	−0.205652	−0.102826	−	0.994699i	$-0.532788\pi$
$373$	−3.97179	−0.205652	−0.102826	+	0.994699i	$0.532788\pi$
$374$	−39.1719	−2.02553
$375$	0	0
$376$	−8.51238	−0.438992
$377$	22.7127	1.16976
$378$	0	0
$379$	1.87740	0.0964354	0.0482177	−	0.998837i	$-0.484646\pi$
$379$	1.87740	0.0964354	0.0482177	+	0.998837i	$0.484646\pi$
$380$	−6.92026	−0.355002
$381$	0	0
$382$	−11.9150	−0.609627
$383$	13.8955	0.710027	0.355014	−	0.934861i	$-0.384476\pi$
$383$	13.8955	0.710027	0.355014	+	0.934861i	$0.384476\pi$
$384$	0	0
$385$	−23.8900	−1.21755
$386$	−15.0924	−0.768182
$387$	0	0
$388$	10.4661	0.531334
$389$	0.948470	0.0480894	0.0240447	−	0.999711i	$-0.492346\pi$
$389$	0.948470	0.0480894	0.0240447	+	0.999711i	$0.492346\pi$
$390$	0	0
$391$	−9.32638	−0.471655
$392$	13.3616	0.674861
$393$	0	0
$394$	−6.33160	−0.318981
$395$	−10.7270	−0.539733
$396$	0	0
$397$	20.3191	1.01979	0.509893	−	0.860238i	$-0.329685\pi$
$397$	20.3191	1.01979	0.509893	+	0.860238i	$0.329685\pi$
$398$	5.41133	0.271246
$399$	0	0
$400$	1.00000	0.0500000
$401$	2.93277	0.146455	0.0732277	−	0.997315i	$-0.476670\pi$
$401$	2.93277	0.146455	0.0732277	+	0.997315i	$0.476670\pi$
$402$	0	0
$403$	−31.2519	−1.55677
$404$	−13.2519	−0.659305
$405$	0	0
$406$	−24.5141	−1.21662
$407$	48.0050	2.37952
$408$	0	0
$409$	−11.5389	−0.570562	−0.285281	−	0.958444i	$-0.592087\pi$
$409$	−11.5389	−0.570562	−0.285281	+	0.958444i	$0.592087\pi$
$410$	3.29433	0.162696
$411$	0	0
$412$	−12.9078	−0.635920
$413$	28.5706	1.40587
$414$	0	0
$415$	4.47857	0.219844
$416$	4.18078	0.204980
$417$	0	0
$418$	−36.6382	−1.79203
$419$	−35.4163	−1.73020	−0.865100	−	0.501600i	$-0.832745\pi$
$419$	−35.4163	−1.73020	−0.865100	+	0.501600i	$0.832745\pi$
$420$	0	0
$421$	−22.2939	−1.08654	−0.543270	−	0.839558i	$-0.682814\pi$
$421$	−22.2939	−1.08654	−0.543270	+	0.839558i	$0.682814\pi$
$422$	14.6563	0.713457
$423$	0	0
$424$	−8.17212	−0.396873
$425$	7.39883	0.358896
$426$	0	0
$427$	45.2764	2.19108
$428$	−12.5462	−0.606443
$429$	0	0
$430$	−2.10450	−0.101488
$431$	−19.7761	−0.952581	−0.476291	−	0.879288i	$-0.658019\pi$
$431$	−19.7761	−0.952581	−0.476291	+	0.879288i	$0.658019\pi$
$432$	0	0
$433$	21.6629	1.04105	0.520527	−	0.853845i	$-0.325735\pi$
$433$	21.6629	1.04105	0.520527	+	0.853845i	$0.325735\pi$
$434$	33.7306	1.61912
$435$	0	0
$436$	−18.3104	−0.876911
$437$	−8.72314	−0.417284
$438$	0	0
$439$	16.0599	0.766499	0.383250	−	0.923645i	$-0.374805\pi$
$439$	16.0599	0.766499	0.383250	+	0.923645i	$0.374805\pi$
$440$	5.29433	0.252397
$441$	0	0
$442$	30.9329	1.47133
$443$	24.6382	1.17060	0.585298	−	0.810818i	$-0.300978\pi$
$443$	24.6382	1.17060	0.585298	+	0.810818i	$0.300978\pi$
$444$	0	0
$445$	−4.58867	−0.217524
$446$	2.49667	0.118221
$447$	0	0
$448$	−4.51238	−0.213190
$449$	−9.39322	−0.443294	−0.221647	−	0.975127i	$-0.571143\pi$
$449$	−9.39322	−0.443294	−0.221647	+	0.975127i	$0.571143\pi$
$450$	0	0
$451$	17.4413	0.821279
$452$	−8.50372	−0.399981
$453$	0	0
$454$	−10.9996	−0.516237
$455$	18.8653	0.884418
$456$	0	0
$457$	14.5879	0.682392	0.341196	−	0.939992i	$-0.389168\pi$
$457$	14.5879	0.682392	0.341196	+	0.939992i	$0.389168\pi$
$458$	9.57961	0.447626
$459$	0	0
$460$	1.26052	0.0587721
$461$	11.0642	0.515310	0.257655	−	0.966237i	$-0.417050\pi$
$461$	11.0642	0.515310	0.257655	+	0.966237i	$0.417050\pi$
$462$	0	0
$463$	−9.71472	−0.451481	−0.225741	−	0.974187i	$-0.572480\pi$
$463$	−9.71472	−0.451481	−0.225741	+	0.974187i	$0.572480\pi$
$464$	5.43264	0.252204
$465$	0	0
$466$	29.2269	1.35391
$467$	2.52808	0.116986	0.0584929	−	0.998288i	$-0.481370\pi$
$467$	2.52808	0.116986	0.0584929	+	0.998288i	$0.481370\pi$
$468$	0	0
$469$	4.51238	0.208362
$470$	8.51238	0.392647
$471$	0	0
$472$	−6.33160	−0.291435
$473$	−11.1419	−0.512305
$474$	0	0
$475$	6.92026	0.317523
$476$	−33.3863	−1.53026
$477$	0	0
$478$	−23.9074	−1.09350
$479$	10.1258	0.462660	0.231330	−	0.972875i	$-0.425692\pi$
$479$	10.1258	0.462660	0.231330	+	0.972875i	$0.425692\pi$
$480$	0	0
$481$	−37.9082	−1.72846
$482$	−24.4111	−1.11189
$483$	0	0
$484$	17.0300	0.774090
$485$	−10.4661	−0.475240
$486$	0	0
$487$	16.4874	0.747114	0.373557	−	0.927607i	$-0.378138\pi$
$487$	16.4874	0.747114	0.373557	+	0.927607i	$0.378138\pi$
$488$	−10.0338	−0.454209
$489$	0	0
$490$	−13.3616	−0.603614
$491$	30.6082	1.38133	0.690665	−	0.723175i	$-0.257318\pi$
$491$	30.6082	1.38133	0.690665	+	0.723175i	$0.257318\pi$
$492$	0	0
$493$	40.1952	1.81030
$494$	28.9321	1.30172
$495$	0	0
$496$	−7.47512	−0.335643
$497$	−0.344244	−0.0154414
$498$	0	0
$499$	−42.6005	−1.90706	−0.953531	−	0.301295i	$-0.902581\pi$
$499$	−42.6005	−1.90706	−0.953531	+	0.301295i	$0.902581\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−26.5281	−1.18401
$503$	20.5156	0.914745	0.457372	−	0.889275i	$-0.348791\pi$
$503$	20.5156	0.914745	0.457372	+	0.889275i	$0.348791\pi$
$504$	0	0
$505$	13.2519	0.589700
$506$	6.67362	0.296679
$507$	0	0
$508$	19.3439	0.858245
$509$	−0.948470	−0.0420402	−0.0210201	−	0.999779i	$-0.506691\pi$
$509$	−0.948470	−0.0420402	−0.0210201	+	0.999779i	$0.506691\pi$
$510$	0	0
$511$	−50.0531	−2.21422
$512$	1.00000	0.0441942
$513$	0	0
$514$	13.6259	0.601014
$515$	12.9078	0.568784
$516$	0	0
$517$	45.0674	1.98206
$518$	40.9148	1.79769
$519$	0	0
$520$	−4.18078	−0.183340
$521$	−26.3122	−1.15276	−0.576379	−	0.817182i	$-0.695535\pi$
$521$	−26.3122	−1.15276	−0.576379	+	0.817182i	$0.695535\pi$
$522$	0	0
$523$	36.7231	1.60579	0.802895	−	0.596120i	$-0.203292\pi$
$523$	36.7231	1.60579	0.802895	+	0.596120i	$0.203292\pi$
$524$	−9.73603	−0.425320
$525$	0	0
$526$	−11.8319	−0.515894
$527$	−55.3071	−2.40922
$528$	0	0
$529$	−21.4111	−0.930917
$530$	8.17212	0.354974
$531$	0	0
$532$	−31.2269	−1.35386
$533$	−13.7729	−0.596571
$534$	0	0
$535$	12.5462	0.542419
$536$	−1.00000	−0.0431934
$537$	0	0
$538$	4.43786	0.191330
$539$	−70.7406	−3.04701
$540$	0	0
$541$	−22.8134	−0.980823	−0.490412	−	0.871491i	$-0.663154\pi$
$541$	−22.8134	−0.980823	−0.490412	+	0.871491i	$0.663154\pi$
$542$	−22.1878	−0.953049
$543$	0	0
$544$	7.39883	0.317222
$545$	18.3104	0.784333
$546$	0	0
$547$	−33.7306	−1.44222	−0.721108	−	0.692823i	$-0.756367\pi$
$547$	−33.7306	−1.44222	−0.721108	+	0.692823i	$0.756367\pi$
$548$	−19.0924	−0.815586
$549$	0	0
$550$	−5.29433	−0.225751
$551$	37.5953	1.60161
$552$	0	0
$553$	−48.4042	−2.05836
$554$	8.68066	0.368806
$555$	0	0
$556$	−7.71792	−0.327313
$557$	−20.7513	−0.879263	−0.439631	−	0.898178i	$-0.644891\pi$
$557$	−20.7513	−0.879263	−0.439631	+	0.898178i	$0.644891\pi$
$558$	0	0
$559$	8.79844	0.372134
$560$	4.51238	0.190683
$561$	0	0
$562$	3.68091	0.155270
$563$	26.9217	1.13461	0.567307	−	0.823506i	$-0.307985\pi$
$563$	26.9217	1.13461	0.567307	+	0.823506i	$0.307985\pi$
$564$	0	0
$565$	8.50372	0.357754
$566$	2.07452	0.0871987
$567$	0	0
$568$	0.0762887	0.00320100
$569$	−23.9082	−1.00228	−0.501141	−	0.865366i	$-0.667086\pi$
$569$	−23.9082	−1.00228	−0.501141	+	0.865366i	$0.667086\pi$
$570$	0	0
$571$	24.9134	1.04259	0.521296	−	0.853376i	$-0.325449\pi$
$571$	24.9134	1.04259	0.521296	+	0.853376i	$0.325449\pi$
$572$	−22.1345	−0.925489
$573$	0	0
$574$	14.8653	0.620465
$575$	−1.26052	−0.0525674
$576$	0	0
$577$	16.1240	0.671253	0.335626	−	0.941995i	$-0.391052\pi$
$577$	16.1240	0.671253	0.335626	+	0.941995i	$0.391052\pi$
$578$	37.7427	1.56989
$579$	0	0
$580$	−5.43264	−0.225578
$581$	20.2090	0.838410
$582$	0	0
$583$	43.2659	1.79189
$584$	11.0924	0.459006
$585$	0	0
$586$	4.92893	0.203612
$587$	25.8155	1.06552	0.532760	−	0.846266i	$-0.321155\pi$
$587$	25.8155	1.06552	0.532760	+	0.846266i	$0.321155\pi$
$588$	0	0
$589$	−51.7298	−2.13149
$590$	6.33160	0.260668
$591$	0	0
$592$	−9.06723	−0.372661
$593$	−32.0495	−1.31612	−0.658058	−	0.752967i	$-0.728622\pi$
$593$	−32.0495	−1.31612	−0.658058	+	0.752967i	$0.728622\pi$
$594$	0	0
$595$	33.3863	1.36871
$596$	13.4626	0.551450
$597$	0	0
$598$	−5.26997	−0.215505
$599$	−10.7300	−0.438417	−0.219209	−	0.975678i	$-0.570348\pi$
$599$	−10.7300	−0.438417	−0.219209	+	0.975678i	$0.570348\pi$
$600$	0	0
$601$	−25.1405	−1.02550	−0.512751	−	0.858538i	$-0.671373\pi$
$601$	−25.1405	−1.02550	−0.512751	+	0.858538i	$0.671373\pi$
$602$	−9.49628	−0.387040
$603$	0	0
$604$	−6.52104	−0.265337
$605$	−17.0300	−0.692367
$606$	0	0
$607$	−41.1844	−1.67162	−0.835811	−	0.549017i	$-0.815002\pi$
$607$	−41.1844	−1.67162	−0.835811	+	0.549017i	$0.815002\pi$
$608$	6.92026	0.280654
$609$	0	0
$610$	10.0338	0.406257
$611$	−35.5884	−1.43975
$612$	0	0
$613$	−28.7483	−1.16113	−0.580566	−	0.814213i	$-0.697169\pi$
$613$	−28.7483	−1.16113	−0.580566	+	0.814213i	$0.697169\pi$
$614$	−13.4713	−0.543657
$615$	0	0
$616$	23.8900	0.962557
$617$	5.04769	0.203212	0.101606	−	0.994825i	$-0.467602\pi$
$617$	5.04769	0.203212	0.101606	+	0.994825i	$0.467602\pi$
$618$	0	0
$619$	−29.9381	−1.20331	−0.601657	−	0.798754i	$-0.705493\pi$
$619$	−29.9381	−1.20331	−0.601657	+	0.798754i	$0.705493\pi$
$620$	7.47512	0.300208
$621$	0	0
$622$	−4.36157	−0.174883
$623$	−20.7058	−0.829561
$624$	0	0
$625$	1.00000	0.0400000
$626$	25.6629	1.02570
$627$	0	0
$628$	−1.81538	−0.0724414
$629$	−67.0869	−2.67493
$630$	0	0
$631$	−16.2736	−0.647840	−0.323920	−	0.946084i	$-0.605001\pi$
$631$	−16.2736	−0.647840	−0.323920	+	0.946084i	$0.605001\pi$
$632$	10.7270	0.426696
$633$	0	0
$634$	3.73426	0.148307
$635$	−19.3439	−0.767637
$636$	0	0
$637$	55.8618	2.21333
$638$	−28.7622	−1.13871
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	7.97485	0.314988	0.157494	−	0.987520i	$-0.449659\pi$
$641$	7.97485	0.314988	0.157494	+	0.987520i	$0.449659\pi$
$642$	0	0
$643$	18.5887	0.733066	0.366533	−	0.930405i	$-0.380545\pi$
$643$	18.5887	0.733066	0.366533	+	0.930405i	$0.380545\pi$
$644$	5.68795	0.224137
$645$	0	0
$646$	51.2019	2.01451
$647$	36.8757	1.44973	0.724867	−	0.688889i	$-0.241901\pi$
$647$	36.8757	1.44973	0.724867	+	0.688889i	$0.241901\pi$
$648$	0	0
$649$	33.5216	1.31584
$650$	4.18078	0.163984
$651$	0	0
$652$	13.0924	0.512737
$653$	−4.70006	−0.183928	−0.0919638	−	0.995762i	$-0.529314\pi$
$653$	−4.70006	−0.183928	−0.0919638	+	0.995762i	$0.529314\pi$
$654$	0	0
$655$	9.73603	0.380418
$656$	−3.29433	−0.128622
$657$	0	0
$658$	38.4111	1.49742
$659$	−49.5584	−1.93052	−0.965262	−	0.261286i	$-0.915854\pi$
$659$	−49.5584	−1.93052	−0.965262	+	0.261286i	$0.915854\pi$
$660$	0	0
$661$	30.0825	1.17008	0.585038	−	0.811006i	$-0.301080\pi$
$661$	30.0825	1.17008	0.585038	+	0.811006i	$0.301080\pi$
$662$	22.4592	0.872900
$663$	0	0
$664$	−4.47857	−0.173802
$665$	31.2269	1.21093
$666$	0	0
$667$	−6.84796	−0.265154
$668$	−19.3950	−0.750415
$669$	0	0
$670$	1.00000	0.0386334
$671$	53.1224	2.05077
$672$	0	0
$673$	0.870745	0.0335648	0.0167824	−	0.999859i	$-0.494658\pi$
$673$	0.870745	0.0335648	0.0167824	+	0.999859i	$0.494658\pi$
$674$	−28.6682	−1.10426
$675$	0	0
$676$	4.47896	0.172268
$677$	16.7789	0.644865	0.322433	−	0.946592i	$-0.395499\pi$
$677$	16.7789	0.644865	0.322433	+	0.946592i	$0.395499\pi$
$678$	0	0
$679$	−47.2269	−1.81240
$680$	−7.39883	−0.283732
$681$	0	0
$682$	39.5758	1.51543
$683$	31.9150	1.22120	0.610598	−	0.791941i	$-0.290929\pi$
$683$	31.9150	1.22120	0.610598	+	0.791941i	$0.290929\pi$
$684$	0	0
$685$	19.0924	0.729483
$686$	−28.7058	−1.09599
$687$	0	0
$688$	2.10450	0.0802331
$689$	−34.1659	−1.30162
$690$	0	0
$691$	10.7127	0.407531	0.203765	−	0.979020i	$-0.434682\pi$
$691$	10.7127	0.407531	0.203765	+	0.979020i	$0.434682\pi$
$692$	−11.7266	−0.445778
$693$	0	0
$694$	30.4284	1.15505
$695$	7.71792	0.292758
$696$	0	0
$697$	−24.3742	−0.923239
$698$	5.54580	0.209912
$699$	0	0
$700$	−4.51238	−0.170552
$701$	23.6030	0.891473	0.445736	−	0.895164i	$-0.352942\pi$
$701$	23.6030	0.891473	0.445736	+	0.895164i	$0.352942\pi$
$702$	0	0
$703$	−62.7477	−2.36657
$704$	−5.29433	−0.199538
$705$	0	0
$706$	−23.4713	−0.883353
$707$	59.7974	2.24891
$708$	0	0
$709$	−0.656294	−0.0246477	−0.0123238	−	0.999924i	$-0.503923\pi$
$709$	−0.656294	−0.0246477	−0.0123238	+	0.999924i	$0.503923\pi$
$710$	−0.0762887	−0.00286306
$711$	0	0
$712$	4.58867	0.171968
$713$	9.42254	0.352877
$714$	0	0
$715$	22.1345	0.827782
$716$	4.09199	0.152925
$717$	0	0
$718$	5.57401	0.208020
$719$	−21.9718	−0.819410	−0.409705	−	0.912218i	$-0.634368\pi$
$719$	−21.9718	−0.819410	−0.409705	+	0.912218i	$0.634368\pi$
$720$	0	0
$721$	58.2447	2.16915
$722$	28.8900	1.07518
$723$	0	0
$724$	6.20899	0.230755
$725$	5.43264	0.201763
$726$	0	0
$727$	9.71472	0.360299	0.180150	−	0.983639i	$-0.442342\pi$
$727$	9.71472	0.360299	0.180150	+	0.983639i	$0.442342\pi$
$728$	−18.8653	−0.699194
$729$	0	0
$730$	−11.0924	−0.410548
$731$	15.5708	0.575907
$732$	0	0
$733$	−1.05297	−0.0388922	−0.0194461	−	0.999811i	$-0.506190\pi$
$733$	−1.05297	−0.0388922	−0.0194461	+	0.999811i	$0.506190\pi$
$734$	−33.3596	−1.23132
$735$	0	0
$736$	−1.26052	−0.0464634
$737$	5.29433	0.195019
$738$	0	0
$739$	20.1359	0.740711	0.370356	−	0.928890i	$-0.379236\pi$
$739$	20.1359	0.740711	0.370356	+	0.928890i	$0.379236\pi$
$740$	9.06723	0.333318
$741$	0	0
$742$	36.8757	1.35375
$743$	27.9671	1.02601	0.513007	−	0.858384i	$-0.328531\pi$
$743$	27.9671	1.02601	0.513007	+	0.858384i	$0.328531\pi$
$744$	0	0
$745$	−13.4626	−0.493232
$746$	−3.97179	−0.145418
$747$	0	0
$748$	−39.1719	−1.43227
$749$	56.6132	2.06860
$750$	0	0
$751$	50.7727	1.85272	0.926360	−	0.376639i	$-0.122920\pi$
$751$	50.7727	1.85272	0.926360	+	0.376639i	$0.122920\pi$
$752$	−8.51238	−0.310415
$753$	0	0
$754$	22.7127	0.827148
$755$	6.52104	0.237325
$756$	0	0
$757$	25.1318	0.913431	0.456715	−	0.889613i	$-0.349026\pi$
$757$	25.1318	0.913431	0.456715	+	0.889613i	$0.349026\pi$
$758$	1.87740	0.0681901
$759$	0	0
$760$	−6.92026	−0.251024
$761$	23.0248	0.834647	0.417323	−	0.908758i	$-0.362968\pi$
$761$	23.0248	0.834647	0.417323	+	0.908758i	$0.362968\pi$
$762$	0	0
$763$	82.6236	2.99118
$764$	−11.9150	−0.431071
$765$	0	0
$766$	13.8955	0.502065
$767$	−26.4710	−0.955814
$768$	0	0
$769$	30.7908	1.11034	0.555172	−	0.831736i	$-0.312653\pi$
$769$	30.7908	1.11034	0.555172	+	0.831736i	$0.312653\pi$
$770$	−23.8900	−0.860937
$771$	0	0
$772$	−15.0924	−0.543187
$773$	−13.1741	−0.473841	−0.236920	−	0.971529i	$-0.576138\pi$
$773$	−13.1741	−0.473841	−0.236920	+	0.971529i	$0.576138\pi$
$774$	0	0
$775$	−7.47512	−0.268514
$776$	10.4661	0.375710
$777$	0	0
$778$	0.948470	0.0340043
$779$	−22.7977	−0.816811
$780$	0	0
$781$	−0.403898	−0.0144526
$782$	−9.32638	−0.333511
$783$	0	0
$784$	13.3616	0.477199
$785$	1.81538	0.0647936
$786$	0	0
$787$	42.8103	1.52602	0.763011	−	0.646385i	$-0.223720\pi$
$787$	42.8103	1.52602	0.763011	+	0.646385i	$0.223720\pi$
$788$	−6.33160	−0.225554
$789$	0	0
$790$	−10.7270	−0.381649
$791$	38.3720	1.36435
$792$	0	0
$793$	−41.9492	−1.48966
$794$	20.3191	0.721098
$795$	0	0
$796$	5.41133	0.191800
$797$	4.19811	0.148705	0.0743523	−	0.997232i	$-0.476311\pi$
$797$	4.19811	0.148705	0.0743523	+	0.997232i	$0.476311\pi$
$798$	0	0
$799$	−62.9816	−2.22813
$800$	1.00000	0.0353553
$801$	0	0
$802$	2.93277	0.103560
$803$	−58.7268	−2.07242
$804$	0	0
$805$	−5.68795	−0.200474
$806$	−31.2519	−1.10080
$807$	0	0
$808$	−13.2519	−0.466199
$809$	23.0568	0.810634	0.405317	−	0.914176i	$-0.367161\pi$
$809$	23.0568	0.810634	0.405317	+	0.914176i	$0.367161\pi$
$810$	0	0
$811$	−24.4661	−0.859120	−0.429560	−	0.903038i	$-0.641331\pi$
$811$	−24.4661	−0.859120	−0.429560	+	0.903038i	$0.641331\pi$
$812$	−24.5141	−0.860278
$813$	0	0
$814$	48.0050	1.68257
$815$	−13.0924	−0.458606
$816$	0	0
$817$	14.5637	0.509518
$818$	−11.5389	−0.403448
$819$	0	0
$820$	3.29433	0.115043
$821$	−22.0694	−0.770227	−0.385114	−	0.922869i	$-0.625838\pi$
$821$	−22.0694	−0.770227	−0.385114	+	0.922869i	$0.625838\pi$
$822$	0	0
$823$	28.1346	0.980711	0.490356	−	0.871522i	$-0.336867\pi$
$823$	28.1346	0.980711	0.490356	+	0.871522i	$0.336867\pi$
$824$	−12.9078	−0.449663
$825$	0	0
$826$	28.5706	0.994097
$827$	7.34385	0.255371	0.127685	−	0.991815i	$-0.459245\pi$
$827$	7.34385	0.255371	0.127685	+	0.991815i	$0.459245\pi$
$828$	0	0
$829$	−28.9893	−1.00684	−0.503420	−	0.864042i	$-0.667925\pi$
$829$	−28.9893	−1.00684	−0.503420	+	0.864042i	$0.667925\pi$
$830$	4.47857	0.155453
$831$	0	0
$832$	4.18078	0.144943
$833$	98.8600	3.42530
$834$	0	0
$835$	19.3950	0.671191
$836$	−36.6382	−1.26716
$837$	0	0
$838$	−35.4163	−1.22344
$839$	−37.0084	−1.27767	−0.638836	−	0.769343i	$-0.720584\pi$
$839$	−37.0084	−1.27767	−0.638836	+	0.769343i	$0.720584\pi$
$840$	0	0
$841$	0.513608	0.0177106
$842$	−22.2939	−0.768300
$843$	0	0
$844$	14.6563	0.504490
$845$	−4.47896	−0.154081
$846$	0	0
$847$	−76.8457	−2.64045
$848$	−8.17212	−0.280632
$849$	0	0
$850$	7.39883	0.253778
$851$	11.4294	0.391796
$852$	0	0
$853$	15.7485	0.539220	0.269610	−	0.962970i	$-0.413105\pi$
$853$	15.7485	0.539220	0.269610	+	0.962970i	$0.413105\pi$
$854$	45.2764	1.54933
$855$	0	0
$856$	−12.5462	−0.428820
$857$	1.19191	0.0407149	0.0203575	−	0.999793i	$-0.493520\pi$
$857$	1.19191	0.0407149	0.0203575	+	0.999793i	$0.493520\pi$
$858$	0	0
$859$	1.91283	0.0652649	0.0326324	−	0.999467i	$-0.489611\pi$
$859$	1.91283	0.0652649	0.0326324	+	0.999467i	$0.489611\pi$
$860$	−2.10450	−0.0717627
$861$	0	0
$862$	−19.7761	−0.673577
$863$	−9.39577	−0.319836	−0.159918	−	0.987130i	$-0.551123\pi$
$863$	−9.39577	−0.319836	−0.159918	+	0.987130i	$0.551123\pi$
$864$	0	0
$865$	11.7266	0.398716
$866$	21.6629	0.736137
$867$	0	0
$868$	33.7306	1.14489
$869$	−56.7922	−1.92654
$870$	0	0
$871$	−4.18078	−0.141660
$872$	−18.3104	−0.620069
$873$	0	0
$874$	−8.72314	−0.295065
$875$	4.51238	0.152546
$876$	0	0
$877$	−33.5278	−1.13215	−0.566077	−	0.824352i	$-0.691540\pi$
$877$	−33.5278	−1.13215	−0.566077	+	0.824352i	$0.691540\pi$
$878$	16.0599	0.541997
$879$	0	0
$880$	5.29433	0.178472
$881$	−39.8155	−1.34142	−0.670710	−	0.741720i	$-0.734010\pi$
$881$	−39.8155	−1.34142	−0.670710	+	0.741720i	$0.734010\pi$
$882$	0	0
$883$	−31.2645	−1.05213	−0.526067	−	0.850443i	$-0.676334\pi$
$883$	−31.2645	−1.05213	−0.526067	+	0.850443i	$0.676334\pi$
$884$	30.9329	1.04039
$885$	0	0
$886$	24.6382	0.827736
$887$	−23.9168	−0.803048	−0.401524	−	0.915848i	$-0.631519\pi$
$887$	−23.9168	−0.803048	−0.401524	+	0.915848i	$0.631519\pi$
$888$	0	0
$889$	−87.2868	−2.92751
$890$	−4.58867	−0.153813
$891$	0	0
$892$	2.49667	0.0835948
$893$	−58.9079	−1.97128
$894$	0	0
$895$	−4.09199	−0.136780
$896$	−4.51238	−0.150748
$897$	0	0
$898$	−9.39322	−0.313456
$899$	−40.6096	−1.35441
$900$	0	0
$901$	−60.4641	−2.01435
$902$	17.4413	0.580732
$903$	0	0
$904$	−8.50372	−0.282829
$905$	−6.20899	−0.206394
$906$	0	0
$907$	−38.9929	−1.29474	−0.647368	−	0.762177i	$-0.724130\pi$
$907$	−38.9929	−1.29474	−0.647368	+	0.762177i	$0.724130\pi$
$908$	−10.9996	−0.365035
$909$	0	0
$910$	18.8653	0.625378
$911$	30.3160	1.00442	0.502208	−	0.864747i	$-0.332521\pi$
$911$	30.3160	1.00442	0.502208	+	0.864747i	$0.332521\pi$
$912$	0	0
$913$	23.7110	0.784720
$914$	14.5879	0.482524
$915$	0	0
$916$	9.57961	0.316519
$917$	43.9327	1.45078
$918$	0	0
$919$	19.2299	0.634336	0.317168	−	0.948369i	$-0.397268\pi$
$919$	19.2299	0.634336	0.317168	+	0.948369i	$0.397268\pi$
$920$	1.26052	0.0415581
$921$	0	0
$922$	11.0642	0.364379
$923$	0.318947	0.0104983
$924$	0	0
$925$	−9.06723	−0.298129
$926$	−9.71472	−0.319246
$927$	0	0
$928$	5.43264	0.178335
$929$	−13.6740	−0.448630	−0.224315	−	0.974517i	$-0.572014\pi$
$929$	−13.6740	−0.448630	−0.224315	+	0.974517i	$0.572014\pi$
$930$	0	0
$931$	92.4656	3.03044
$932$	29.2269	0.957357
$933$	0	0
$934$	2.52808	0.0827214
$935$	39.1719	1.28106
$936$	0	0
$937$	−30.8967	−1.00935	−0.504676	−	0.863309i	$-0.668388\pi$
$937$	−30.8967	−1.00935	−0.504676	+	0.863309i	$0.668388\pi$
$938$	4.51238	0.147334
$939$	0	0
$940$	8.51238	0.277643
$941$	22.4180	0.730805	0.365403	−	0.930850i	$-0.380931\pi$
$941$	22.4180	0.730805	0.365403	+	0.930850i	$0.380931\pi$
$942$	0	0
$943$	4.15258	0.135227
$944$	−6.33160	−0.206076
$945$	0	0
$946$	−11.1419	−0.362255
$947$	46.8473	1.52233	0.761167	−	0.648556i	$-0.224627\pi$
$947$	46.8473	1.52233	0.761167	+	0.648556i	$0.224627\pi$
$948$	0	0
$949$	46.3749	1.50539
$950$	6.92026	0.224523
$951$	0	0
$952$	−33.3863	−1.08206
$953$	−15.1536	−0.490874	−0.245437	−	0.969413i	$-0.578931\pi$
$953$	−15.1536	−0.490874	−0.245437	+	0.969413i	$0.578931\pi$
$954$	0	0
$955$	11.9150	0.385562
$956$	−23.9074	−0.773220
$957$	0	0
$958$	10.1258	0.327150
$959$	86.1521	2.78200
$960$	0	0
$961$	24.8774	0.802497
$962$	−37.9082	−1.22221
$963$	0	0
$964$	−24.4111	−0.786228
$965$	15.0924	0.485841
$966$	0	0
$967$	13.0853	0.420796	0.210398	−	0.977616i	$-0.432524\pi$
$967$	13.0853	0.420796	0.210398	+	0.977616i	$0.432524\pi$
$968$	17.0300	0.547364
$969$	0	0
$970$	−10.4661	−0.336045
$971$	−53.4831	−1.71636	−0.858178	−	0.513352i	$-0.828403\pi$
$971$	−53.4831	−1.71636	−0.858178	+	0.513352i	$0.828403\pi$
$972$	0	0
$973$	34.8262	1.11648
$974$	16.4874	0.528290
$975$	0	0
$976$	−10.0338	−0.321175
$977$	41.9875	1.34330	0.671650	−	0.740869i	$-0.265586\pi$
$977$	41.9875	1.34330	0.671650	+	0.740869i	$0.265586\pi$
$978$	0	0
$979$	−24.2939	−0.776438
$980$	−13.3616	−0.426820
$981$	0	0
$982$	30.6082	0.976747
$983$	6.48339	0.206788	0.103394	−	0.994640i	$-0.467030\pi$
$983$	6.48339	0.206788	0.103394	+	0.994640i	$0.467030\pi$
$984$	0	0
$985$	6.33160	0.201741
$986$	40.1952	1.28008
$987$	0	0
$988$	28.9321	0.920454
$989$	−2.65276	−0.0843529
$990$	0	0
$991$	2.72698	0.0866253	0.0433126	−	0.999062i	$-0.486209\pi$
$991$	2.72698	0.0866253	0.0433126	+	0.999062i	$0.486209\pi$
$992$	−7.47512	−0.237335
$993$	0	0
$994$	−0.344244	−0.0109187
$995$	−5.41133	−0.171551
$996$	0	0
$997$	−2.84014	−0.0899480	−0.0449740	−	0.998988i	$-0.514320\pi$
$997$	−2.84014	−0.0899480	−0.0449740	+	0.998988i	$0.514320\pi$
$998$	−42.6005	−1.34850
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.a.bs.1.1		4
3.2	odd	2		2010.2.a.s.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2010.2.a.s.1.1	✓	4	3.2	odd	2
6030.2.a.bs.1.1		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.51238 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.51238 q^{7} +1.00000 q^{8} -1.00000 q^{10} -5.29433 q^{11} +4.18078 q^{13} -4.51238 q^{14} +1.00000 q^{16} +7.39883 q^{17} +6.92026 q^{19} -1.00000 q^{20} -5.29433 q^{22} -1.26052 q^{23} +1.00000 q^{25} +4.18078 q^{26} -4.51238 q^{28} +5.43264 q^{29} -7.47512 q^{31} +1.00000 q^{32} +7.39883 q^{34} +4.51238 q^{35} -9.06723 q^{37} +6.92026 q^{38} -1.00000 q^{40} -3.29433 q^{41} +2.10450 q^{43} -5.29433 q^{44} -1.26052 q^{46} -8.51238 q^{47} +13.3616 q^{49} +1.00000 q^{50} +4.18078 q^{52} -8.17212 q^{53} +5.29433 q^{55} -4.51238 q^{56} +5.43264 q^{58} -6.33160 q^{59} -10.0338 q^{61} -7.47512 q^{62} +1.00000 q^{64} -4.18078 q^{65} -1.00000 q^{67} +7.39883 q^{68} +4.51238 q^{70} +0.0762887 q^{71} +11.0924 q^{73} -9.06723 q^{74} +6.92026 q^{76} +23.8900 q^{77} +10.7270 q^{79} -1.00000 q^{80} -3.29433 q^{82} -4.47857 q^{83} -7.39883 q^{85} +2.10450 q^{86} -5.29433 q^{88} +4.58867 q^{89} -18.8653 q^{91} -1.26052 q^{92} -8.51238 q^{94} -6.92026 q^{95} +10.4661 q^{97} +13.3616 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 2 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 - 2 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 2 q^{7} + 4 q^{8} - 4 q^{10} - 10 q^{11} - 2 q^{13} - 2 q^{14} + 4 q^{16} + 6 q^{17} + 8 q^{19} - 4 q^{20} - 10 q^{22} - 6 q^{23} + 4 q^{25} - 2 q^{26} - 2 q^{28} - 14 q^{29} + 4 q^{32} + 6 q^{34} + 2 q^{35} - 10 q^{37} + 8 q^{38} - 4 q^{40} - 2 q^{41} - 4 q^{43} - 10 q^{44} - 6 q^{46} - 18 q^{47} + 16 q^{49} + 4 q^{50} - 2 q^{52} + 4 q^{53} + 10 q^{55} - 2 q^{56} - 14 q^{58} - 28 q^{59} - 28 q^{61} + 4 q^{64} + 2 q^{65} - 4 q^{67} + 6 q^{68} + 2 q^{70} - 6 q^{71} - 12 q^{73} - 10 q^{74} + 8 q^{76} - 8 q^{77} - 4 q^{79} - 4 q^{80} - 2 q^{82} - 14 q^{83} - 6 q^{85} - 4 q^{86} - 10 q^{88} - 4 q^{89} - 4 q^{91} - 6 q^{92} - 18 q^{94} - 8 q^{95} - 8 q^{97} + 16 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 - 2 * q^7 + 4 * q^8 - 4 * q^10 - 10 * q^11 - 2 * q^13 - 2 * q^14 + 4 * q^16 + 6 * q^17 + 8 * q^19 - 4 * q^20 - 10 * q^22 - 6 * q^23 + 4 * q^25 - 2 * q^26 - 2 * q^28 - 14 * q^29 + 4 * q^32 + 6 * q^34 + 2 * q^35 - 10 * q^37 + 8 * q^38 - 4 * q^40 - 2 * q^41 - 4 * q^43 - 10 * q^44 - 6 * q^46 - 18 * q^47 + 16 * q^49 + 4 * q^50 - 2 * q^52 + 4 * q^53 + 10 * q^55 - 2 * q^56 - 14 * q^58 - 28 * q^59 - 28 * q^61 + 4 * q^64 + 2 * q^65 - 4 * q^67 + 6 * q^68 + 2 * q^70 - 6 * q^71 - 12 * q^73 - 10 * q^74 + 8 * q^76 - 8 * q^77 - 4 * q^79 - 4 * q^80 - 2 * q^82 - 14 * q^83 - 6 * q^85 - 4 * q^86 - 10 * q^88 - 4 * q^89 - 4 * q^91 - 6 * q^92 - 18 * q^94 - 8 * q^95 - 8 * q^97 + 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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