LMFDB - Embedded newform 8085.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8085,2,Mod(1,8085)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8085, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("8085.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8085.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:	$64.5590500342$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	8085.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{2} -1.00000 q^{3} +4.00000 q^{4} -1.00000 q^{5} -2.44949 q^{6} +4.89898 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.44949 q^{2} -1.00000 q^{3} +4.00000 q^{4} -1.00000 q^{5} -2.44949 q^{6} +4.89898 q^{8} +1.00000 q^{9} -2.44949 q^{10} -1.00000 q^{11} -4.00000 q^{12} +0.449490 q^{13} +1.00000 q^{15} +4.00000 q^{16} +3.00000 q^{17} +2.44949 q^{18} +1.00000 q^{19} -4.00000 q^{20} -2.44949 q^{22} -1.89898 q^{23} -4.89898 q^{24} +1.00000 q^{25} +1.10102 q^{26} -1.00000 q^{27} -0.550510 q^{29} +2.44949 q^{30} +5.34847 q^{31} +1.00000 q^{33} +7.34847 q^{34} +4.00000 q^{36} +4.44949 q^{37} +2.44949 q^{38} -0.449490 q^{39} -4.89898 q^{40} -2.44949 q^{41} +2.55051 q^{43} -4.00000 q^{44} -1.00000 q^{45} -4.65153 q^{46} +8.44949 q^{47} -4.00000 q^{48} +2.44949 q^{50} -3.00000 q^{51} +1.79796 q^{52} +3.00000 q^{53} -2.44949 q^{54} +1.00000 q^{55} -1.00000 q^{57} -1.34847 q^{58} -0.550510 q^{59} +4.00000 q^{60} +2.10102 q^{61} +13.1010 q^{62} -8.00000 q^{64} -0.449490 q^{65} +2.44949 q^{66} +2.00000 q^{67} +12.0000 q^{68} +1.89898 q^{69} +12.2474 q^{71} +4.89898 q^{72} -3.10102 q^{73} +10.8990 q^{74} -1.00000 q^{75} +4.00000 q^{76} -1.10102 q^{78} +3.34847 q^{79} -4.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +9.00000 q^{83} -3.00000 q^{85} +6.24745 q^{86} +0.550510 q^{87} -4.89898 q^{88} +10.3485 q^{89} -2.44949 q^{90} -7.59592 q^{92} -5.34847 q^{93} +20.6969 q^{94} -1.00000 q^{95} +3.44949 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 8 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 8 * q^4 - 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} + 8 q^{4} - 2 q^{5} + 2 q^{9} - 2 q^{11} - 8 q^{12} - 4 q^{13} + 2 q^{15} + 8 q^{16} + 6 q^{17} + 2 q^{19} - 8 q^{20} + 6 q^{23} + 2 q^{25} + 12 q^{26} - 2 q^{27} - 6 q^{29} - 4 q^{31} + 2 q^{33} + 8 q^{36} + 4 q^{37} + 4 q^{39} + 10 q^{43} - 8 q^{44} - 2 q^{45} - 24 q^{46} + 12 q^{47} - 8 q^{48} - 6 q^{51} - 16 q^{52} + 6 q^{53} + 2 q^{55} - 2 q^{57} + 12 q^{58} - 6 q^{59} + 8 q^{60} + 14 q^{61} + 36 q^{62} - 16 q^{64} + 4 q^{65} + 4 q^{67} + 24 q^{68} - 6 q^{69} - 16 q^{73} + 12 q^{74} - 2 q^{75} + 8 q^{76} - 12 q^{78} - 8 q^{79} - 8 q^{80} + 2 q^{81} - 12 q^{82} + 18 q^{83} - 6 q^{85} - 12 q^{86} + 6 q^{87} + 6 q^{89} + 24 q^{92} + 4 q^{93} + 12 q^{94} - 2 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 8 * q^4 - 2 * q^5 + 2 * q^9 - 2 * q^11 - 8 * q^12 - 4 * q^13 + 2 * q^15 + 8 * q^16 + 6 * q^17 + 2 * q^19 - 8 * q^20 + 6 * q^23 + 2 * q^25 + 12 * q^26 - 2 * q^27 - 6 * q^29 - 4 * q^31 + 2 * q^33 + 8 * q^36 + 4 * q^37 + 4 * q^39 + 10 * q^43 - 8 * q^44 - 2 * q^45 - 24 * q^46 + 12 * q^47 - 8 * q^48 - 6 * q^51 - 16 * q^52 + 6 * q^53 + 2 * q^55 - 2 * q^57 + 12 * q^58 - 6 * q^59 + 8 * q^60 + 14 * q^61 + 36 * q^62 - 16 * q^64 + 4 * q^65 + 4 * q^67 + 24 * q^68 - 6 * q^69 - 16 * q^73 + 12 * q^74 - 2 * q^75 + 8 * q^76 - 12 * q^78 - 8 * q^79 - 8 * q^80 + 2 * q^81 - 12 * q^82 + 18 * q^83 - 6 * q^85 - 12 * q^86 + 6 * q^87 + 6 * q^89 + 24 * q^92 + 4 * q^93 + 12 * q^94 - 2 * q^95 + 2 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.44949	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$2$	2.44949	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.00000	2.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.44949	−1.00000
$7$	0	0
$7$	0	0
$8$	4.89898	1.73205
$9$	1.00000	0.333333
$10$	−2.44949	−0.774597
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−4.00000	−1.15470
$13$	0.449490	0.124666	0.0623330	−	0.998055i	$-0.480146\pi$
$13$	0.449490	0.124666	0.0623330	+	0.998055i	$0.480146\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	4.00000	1.00000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	2.44949	0.577350
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	−4.00000	−0.894427
$21$	0	0
$22$	−2.44949	−0.522233
$23$	−1.89898	−0.395965	−0.197982	−	0.980206i	$-0.563439\pi$
$23$	−1.89898	−0.395965	−0.197982	+	0.980206i	$0.563439\pi$
$24$	−4.89898	−1.00000
$25$	1.00000	0.200000
$26$	1.10102	0.215928
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.550510	−0.102227	−0.0511136	−	0.998693i	$-0.516277\pi$
$29$	−0.550510	−0.102227	−0.0511136	+	0.998693i	$0.516277\pi$
$30$	2.44949	0.447214
$31$	5.34847	0.960613	0.480307	−	0.877101i	$-0.340525\pi$
$31$	5.34847	0.960613	0.480307	+	0.877101i	$0.340525\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	7.34847	1.26025
$35$	0	0
$36$	4.00000	0.666667
$37$	4.44949	0.731492	0.365746	−	0.930715i	$-0.380814\pi$
$37$	4.44949	0.731492	0.365746	+	0.930715i	$0.380814\pi$
$38$	2.44949	0.397360
$39$	−0.449490	−0.0719760
$40$	−4.89898	−0.774597
$41$	−2.44949	−0.382546	−0.191273	−	0.981537i	$-0.561262\pi$
$41$	−2.44949	−0.382546	−0.191273	+	0.981537i	$0.561262\pi$
$42$	0	0
$43$	2.55051	0.388949	0.194475	−	0.980908i	$-0.437700\pi$
$43$	2.55051	0.388949	0.194475	+	0.980908i	$0.437700\pi$
$44$	−4.00000	−0.603023
$45$	−1.00000	−0.149071
$46$	−4.65153	−0.685831
$47$	8.44949	1.23248	0.616242	−	0.787557i	$-0.288654\pi$
$47$	8.44949	1.23248	0.616242	+	0.787557i	$0.288654\pi$
$48$	−4.00000	−0.577350
$49$	0	0
$50$	2.44949	0.346410
$51$	−3.00000	−0.420084
$52$	1.79796	0.249332
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−2.44949	−0.333333
$55$	1.00000	0.134840
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−1.34847	−0.177063
$59$	−0.550510	−0.0716703	−0.0358352	−	0.999358i	$-0.511409\pi$
$59$	−0.550510	−0.0716703	−0.0358352	+	0.999358i	$0.511409\pi$
$60$	4.00000	0.516398
$61$	2.10102	0.269008	0.134504	−	0.990913i	$-0.457056\pi$
$61$	2.10102	0.269008	0.134504	+	0.990913i	$0.457056\pi$
$62$	13.1010	1.66383
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−0.449490	−0.0557523
$66$	2.44949	0.301511
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	12.0000	1.45521
$69$	1.89898	0.228610
$70$	0	0
$71$	12.2474	1.45350	0.726752	−	0.686900i	$-0.241029\pi$
$71$	12.2474	1.45350	0.726752	+	0.686900i	$0.241029\pi$
$72$	4.89898	0.577350
$73$	−3.10102	−0.362947	−0.181473	−	0.983396i	$-0.558087\pi$
$73$	−3.10102	−0.362947	−0.181473	+	0.983396i	$0.558087\pi$
$74$	10.8990	1.26698
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	−1.10102	−0.124666
$79$	3.34847	0.376732	0.188366	−	0.982099i	$-0.439681\pi$
$79$	3.34847	0.376732	0.188366	+	0.982099i	$0.439681\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	6.24745	0.673680
$87$	0.550510	0.0590209
$88$	−4.89898	−0.522233
$89$	10.3485	1.09694	0.548468	−	0.836172i	$-0.315211\pi$
$89$	10.3485	1.09694	0.548468	+	0.836172i	$0.315211\pi$
$90$	−2.44949	−0.258199
$91$	0	0
$92$	−7.59592	−0.791929
$93$	−5.34847	−0.554610
$94$	20.6969	2.13473
$95$	−1.00000	−0.102598
$96$	0	0
$97$	3.44949	0.350243	0.175121	−	0.984547i	$-0.443968\pi$
$97$	3.44949	0.350243	0.175121	+	0.984547i	$0.443968\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	4.00000	0.400000
$101$	3.79796	0.377911	0.188956	−	0.981986i	$-0.439490\pi$
$101$	3.79796	0.377911	0.188956	+	0.981986i	$0.439490\pi$
$102$	−7.34847	−0.727607
$103$	−18.3485	−1.80793	−0.903964	−	0.427608i	$-0.859356\pi$
$103$	−18.3485	−1.80793	−0.903964	+	0.427608i	$0.859356\pi$
$104$	2.20204	0.215928
$105$	0	0
$106$	7.34847	0.713746
$107$	−1.34847	−0.130361	−0.0651807	−	0.997873i	$-0.520762\pi$
$107$	−1.34847	−0.130361	−0.0651807	+	0.997873i	$0.520762\pi$
$108$	−4.00000	−0.384900
$109$	−0.449490	−0.0430533	−0.0215267	−	0.999768i	$-0.506853\pi$
$109$	−0.449490	−0.0430533	−0.0215267	+	0.999768i	$0.506853\pi$
$110$	2.44949	0.233550
$111$	−4.44949	−0.422327
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	−2.44949	−0.229416
$115$	1.89898	0.177081
$116$	−2.20204	−0.204454
$117$	0.449490	0.0415553
$118$	−1.34847	−0.124137
$119$	0	0
$120$	4.89898	0.447214
$121$	1.00000	0.0909091
$122$	5.14643	0.465936
$123$	2.44949	0.220863
$124$	21.3939	1.92123
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.5505	−0.936206	−0.468103	−	0.883674i	$-0.655062\pi$
$127$	−10.5505	−0.936206	−0.468103	+	0.883674i	$0.655062\pi$
$128$	−19.5959	−1.73205
$129$	−2.55051	−0.224560
$130$	−1.10102	−0.0965659
$131$	3.55051	0.310210	0.155105	−	0.987898i	$-0.450428\pi$
$131$	3.55051	0.310210	0.155105	+	0.987898i	$0.450428\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	4.89898	0.423207
$135$	1.00000	0.0860663
$136$	14.6969	1.26025
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	4.65153	0.395965
$139$	7.79796	0.661414	0.330707	−	0.943733i	$-0.392713\pi$
$139$	7.79796	0.661414	0.330707	+	0.943733i	$0.392713\pi$
$140$	0	0
$141$	−8.44949	−0.711575
$142$	30.0000	2.51754
$143$	−0.449490	−0.0375882
$144$	4.00000	0.333333
$145$	0.550510	0.0457174
$146$	−7.59592	−0.628643
$147$	0	0
$148$	17.7980	1.46298
$149$	−8.69694	−0.712481	−0.356240	−	0.934394i	$-0.615942\pi$
$149$	−8.69694	−0.712481	−0.356240	+	0.934394i	$0.615942\pi$
$150$	−2.44949	−0.200000
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	4.89898	0.397360
$153$	3.00000	0.242536
$154$	0	0
$155$	−5.34847	−0.429599
$156$	−1.79796	−0.143952
$157$	12.1464	0.969391	0.484695	−	0.874683i	$-0.338931\pi$
$157$	12.1464	0.969391	0.484695	+	0.874683i	$0.338931\pi$
$158$	8.20204	0.652519
$159$	−3.00000	−0.237915
$160$	0	0
$161$	0	0
$162$	2.44949	0.192450
$163$	−12.4495	−0.975119	−0.487560	−	0.873090i	$-0.662113\pi$
$163$	−12.4495	−0.975119	−0.487560	+	0.873090i	$0.662113\pi$
$164$	−9.79796	−0.765092
$165$	−1.00000	−0.0778499
$166$	22.0454	1.71106
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	−12.7980	−0.984458
$170$	−7.34847	−0.563602
$171$	1.00000	0.0764719
$172$	10.2020	0.777898
$173$	2.20204	0.167418	0.0837090	−	0.996490i	$-0.473323\pi$
$173$	2.20204	0.167418	0.0837090	+	0.996490i	$0.473323\pi$
$174$	1.34847	0.102227
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0.550510	0.0413789
$178$	25.3485	1.89995
$179$	14.4495	1.08001	0.540003	−	0.841663i	$-0.318423\pi$
$179$	14.4495	1.08001	0.540003	+	0.841663i	$0.318423\pi$
$180$	−4.00000	−0.298142
$181$	−23.7980	−1.76889	−0.884444	−	0.466646i	$-0.845462\pi$
$181$	−23.7980	−1.76889	−0.884444	+	0.466646i	$0.845462\pi$
$182$	0	0
$183$	−2.10102	−0.155312
$184$	−9.30306	−0.685831
$185$	−4.44949	−0.327133
$186$	−13.1010	−0.960613
$187$	−3.00000	−0.219382
$188$	33.7980	2.46497
$189$	0	0
$190$	−2.44949	−0.177705
$191$	−2.20204	−0.159334	−0.0796670	−	0.996822i	$-0.525386\pi$
$191$	−2.20204	−0.159334	−0.0796670	+	0.996822i	$0.525386\pi$
$192$	8.00000	0.577350
$193$	26.4949	1.90714	0.953572	−	0.301164i	$-0.0973753\pi$
$193$	26.4949	1.90714	0.953572	+	0.301164i	$0.0973753\pi$
$194$	8.44949	0.606638
$195$	0.449490	0.0321886
$196$	0	0
$197$	16.8990	1.20400	0.602001	−	0.798495i	$-0.294370\pi$
$197$	16.8990	1.20400	0.602001	+	0.798495i	$0.294370\pi$
$198$	−2.44949	−0.174078
$199$	25.7980	1.82877	0.914384	−	0.404847i	$-0.132675\pi$
$199$	25.7980	1.82877	0.914384	+	0.404847i	$0.132675\pi$
$200$	4.89898	0.346410
$201$	−2.00000	−0.141069
$202$	9.30306	0.654561
$203$	0	0
$204$	−12.0000	−0.840168
$205$	2.44949	0.171080
$206$	−44.9444	−3.13142
$207$	−1.89898	−0.131988
$208$	1.79796	0.124666
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	6.89898	0.474945	0.237473	−	0.971394i	$-0.423681\pi$
$211$	6.89898	0.474945	0.237473	+	0.971394i	$0.423681\pi$
$212$	12.0000	0.824163
$213$	−12.2474	−0.839181
$214$	−3.30306	−0.225793
$215$	−2.55051	−0.173943
$216$	−4.89898	−0.333333
$217$	0	0
$218$	−1.10102	−0.0745705
$219$	3.10102	0.209548
$220$	4.00000	0.269680
$221$	1.34847	0.0907079
$222$	−10.8990	−0.731492
$223$	−24.3485	−1.63049	−0.815247	−	0.579113i	$-0.803399\pi$
$223$	−24.3485	−1.63049	−0.815247	+	0.579113i	$0.803399\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	22.0454	1.46644
$227$	27.4949	1.82490	0.912450	−	0.409188i	$-0.134188\pi$
$227$	27.4949	1.82490	0.912450	+	0.409188i	$0.134188\pi$
$228$	−4.00000	−0.264906
$229$	2.65153	0.175218	0.0876090	−	0.996155i	$-0.472077\pi$
$229$	2.65153	0.175218	0.0876090	+	0.996155i	$0.472077\pi$
$230$	4.65153	0.306713
$231$	0	0
$232$	−2.69694	−0.177063
$233$	−22.0454	−1.44424	−0.722121	−	0.691766i	$-0.756833\pi$
$233$	−22.0454	−1.44424	−0.722121	+	0.691766i	$0.756833\pi$
$234$	1.10102	0.0719760
$235$	−8.44949	−0.551184
$236$	−2.20204	−0.143341
$237$	−3.34847	−0.217506
$238$	0	0
$239$	−0.550510	−0.0356095	−0.0178048	−	0.999841i	$-0.505668\pi$
$239$	−0.550510	−0.0356095	−0.0178048	+	0.999841i	$0.505668\pi$
$240$	4.00000	0.258199
$241$	−10.6969	−0.689050	−0.344525	−	0.938777i	$-0.611960\pi$
$241$	−10.6969	−0.689050	−0.344525	+	0.938777i	$0.611960\pi$
$242$	2.44949	0.157459
$243$	−1.00000	−0.0641500
$244$	8.40408	0.538016
$245$	0	0
$246$	6.00000	0.382546
$247$	0.449490	0.0286003
$248$	26.2020	1.66383
$249$	−9.00000	−0.570352
$250$	−2.44949	−0.154919
$251$	−21.7980	−1.37587	−0.687937	−	0.725770i	$-0.741484\pi$
$251$	−21.7980	−1.37587	−0.687937	+	0.725770i	$0.741484\pi$
$252$	0	0
$253$	1.89898	0.119388
$254$	−25.8434	−1.62156
$255$	3.00000	0.187867
$256$	−32.0000	−2.00000
$257$	−14.4495	−0.901334	−0.450667	−	0.892692i	$-0.648814\pi$
$257$	−14.4495	−0.901334	−0.450667	+	0.892692i	$0.648814\pi$
$258$	−6.24745	−0.388949
$259$	0	0
$260$	−1.79796	−0.111505
$261$	−0.550510	−0.0340757
$262$	8.69694	0.537299
$263$	−16.8990	−1.04204	−0.521018	−	0.853546i	$-0.674448\pi$
$263$	−16.8990	−1.04204	−0.521018	+	0.853546i	$0.674448\pi$
$264$	4.89898	0.301511
$265$	−3.00000	−0.184289
$266$	0	0
$267$	−10.3485	−0.633316
$268$	8.00000	0.488678
$269$	11.4495	0.698088	0.349044	−	0.937106i	$-0.386506\pi$
$269$	11.4495	0.698088	0.349044	+	0.937106i	$0.386506\pi$
$270$	2.44949	0.149071
$271$	1.00000	0.0607457	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	1.00000	0.0607457	0.0303728	+	0.999539i	$0.490331\pi$
$272$	12.0000	0.727607
$273$	0	0
$274$	29.3939	1.77575
$275$	−1.00000	−0.0603023
$276$	7.59592	0.457221
$277$	10.6969	0.642717	0.321358	−	0.946958i	$-0.395861\pi$
$277$	10.6969	0.642717	0.321358	+	0.946958i	$0.395861\pi$
$278$	19.1010	1.14560
$279$	5.34847	0.320204
$280$	0	0
$281$	16.8990	1.00811	0.504054	−	0.863672i	$-0.331841\pi$
$281$	16.8990	1.00811	0.504054	+	0.863672i	$0.331841\pi$
$282$	−20.6969	−1.23248
$283$	14.6515	0.870943	0.435472	−	0.900202i	$-0.356582\pi$
$283$	14.6515	0.870943	0.435472	+	0.900202i	$0.356582\pi$
$284$	48.9898	2.90701
$285$	1.00000	0.0592349
$286$	−1.10102	−0.0651047
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	1.34847	0.0791848
$291$	−3.44949	−0.202213
$292$	−12.4041	−0.725894
$293$	16.5959	0.969544	0.484772	−	0.874641i	$-0.338903\pi$
$293$	16.5959	0.969544	0.484772	+	0.874641i	$0.338903\pi$
$294$	0	0
$295$	0.550510	0.0320519
$296$	21.7980	1.26698
$297$	1.00000	0.0580259
$298$	−21.3031	−1.23405
$299$	−0.853572	−0.0493633
$300$	−4.00000	−0.230940
$301$	0	0
$302$	34.2929	1.97333
$303$	−3.79796	−0.218187
$304$	4.00000	0.229416
$305$	−2.10102	−0.120304
$306$	7.34847	0.420084
$307$	5.10102	0.291131	0.145565	−	0.989349i	$-0.453500\pi$
$307$	5.10102	0.291131	0.145565	+	0.989349i	$0.453500\pi$
$308$	0	0
$309$	18.3485	1.04381
$310$	−13.1010	−0.744088
$311$	−4.89898	−0.277796	−0.138898	−	0.990307i	$-0.544356\pi$
$311$	−4.89898	−0.277796	−0.138898	+	0.990307i	$0.544356\pi$
$312$	−2.20204	−0.124666
$313$	−23.2474	−1.31402	−0.657012	−	0.753880i	$-0.728180\pi$
$313$	−23.2474	−1.31402	−0.657012	+	0.753880i	$0.728180\pi$
$314$	29.7526	1.67903
$315$	0	0
$316$	13.3939	0.753464
$317$	2.69694	0.151475	0.0757376	−	0.997128i	$-0.475869\pi$
$317$	2.69694	0.151475	0.0757376	+	0.997128i	$0.475869\pi$
$318$	−7.34847	−0.412082
$319$	0.550510	0.0308227
$320$	8.00000	0.447214
$321$	1.34847	0.0752642
$322$	0	0
$323$	3.00000	0.166924
$324$	4.00000	0.222222
$325$	0.449490	0.0249332
$326$	−30.4949	−1.68896
$327$	0.449490	0.0248568
$328$	−12.0000	−0.662589
$329$	0	0
$330$	−2.44949	−0.134840
$331$	31.6969	1.74222	0.871111	−	0.491087i	$-0.163400\pi$
$331$	31.6969	1.74222	0.871111	+	0.491087i	$0.163400\pi$
$332$	36.0000	1.97576
$333$	4.44949	0.243831
$334$	14.6969	0.804181
$335$	−2.00000	−0.109272
$336$	0	0
$337$	−2.34847	−0.127929	−0.0639646	−	0.997952i	$-0.520374\pi$
$337$	−2.34847	−0.127929	−0.0639646	+	0.997952i	$0.520374\pi$
$338$	−31.3485	−1.70513
$339$	−9.00000	−0.488813
$340$	−12.0000	−0.650791
$341$	−5.34847	−0.289636
$342$	2.44949	0.132453
$343$	0	0
$344$	12.4949	0.673680
$345$	−1.89898	−0.102238
$346$	5.39388	0.289977
$347$	12.4949	0.670761	0.335381	−	0.942083i	$-0.391135\pi$
$347$	12.4949	0.670761	0.335381	+	0.942083i	$0.391135\pi$
$348$	2.20204	0.118042
$349$	−15.8990	−0.851053	−0.425526	−	0.904946i	$-0.639911\pi$
$349$	−15.8990	−0.851053	−0.425526	+	0.904946i	$0.639911\pi$
$350$	0	0
$351$	−0.449490	−0.0239920
$352$	0	0
$353$	8.69694	0.462891	0.231446	−	0.972848i	$-0.425654\pi$
$353$	8.69694	0.462891	0.231446	+	0.972848i	$0.425654\pi$
$354$	1.34847	0.0716703
$355$	−12.2474	−0.650027
$356$	41.3939	2.19387
$357$	0	0
$358$	35.3939	1.87062
$359$	−21.2474	−1.12140	−0.560699	−	0.828020i	$-0.689467\pi$
$359$	−21.2474	−1.12140	−0.560699	+	0.828020i	$0.689467\pi$
$360$	−4.89898	−0.258199
$361$	−18.0000	−0.947368
$362$	−58.2929	−3.06380
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	3.10102	0.162315
$366$	−5.14643	−0.269008
$367$	−15.0454	−0.785364	−0.392682	−	0.919674i	$-0.628453\pi$
$367$	−15.0454	−0.785364	−0.392682	+	0.919674i	$0.628453\pi$
$368$	−7.59592	−0.395965
$369$	−2.44949	−0.127515
$370$	−10.8990	−0.566611
$371$	0	0
$372$	−21.3939	−1.10922
$373$	16.1464	0.836030	0.418015	−	0.908440i	$-0.362726\pi$
$373$	16.1464	0.836030	0.418015	+	0.908440i	$0.362726\pi$
$374$	−7.34847	−0.379980
$375$	1.00000	0.0516398
$376$	41.3939	2.13473
$377$	−0.247449	−0.0127443
$378$	0	0
$379$	35.4949	1.82325	0.911625	−	0.411022i	$-0.134828\pi$
$379$	35.4949	1.82325	0.911625	+	0.411022i	$0.134828\pi$
$380$	−4.00000	−0.205196
$381$	10.5505	0.540519
$382$	−5.39388	−0.275975
$383$	−2.44949	−0.125163	−0.0625815	−	0.998040i	$-0.519933\pi$
$383$	−2.44949	−0.125163	−0.0625815	+	0.998040i	$0.519933\pi$
$384$	19.5959	1.00000
$385$	0	0
$386$	64.8990	3.30327
$387$	2.55051	0.129650
$388$	13.7980	0.700485
$389$	−6.24745	−0.316758	−0.158379	−	0.987378i	$-0.550627\pi$
$389$	−6.24745	−0.316758	−0.158379	+	0.987378i	$0.550627\pi$
$390$	1.10102	0.0557523
$391$	−5.69694	−0.288107
$392$	0	0
$393$	−3.55051	−0.179100
$394$	41.3939	2.08539
$395$	−3.34847	−0.168480
$396$	−4.00000	−0.201008
$397$	−22.6969	−1.13913	−0.569563	−	0.821947i	$-0.692888\pi$
$397$	−22.6969	−1.13913	−0.569563	+	0.821947i	$0.692888\pi$
$398$	63.1918	3.16752
$399$	0	0
$400$	4.00000	0.200000
$401$	−31.3485	−1.56547	−0.782734	−	0.622356i	$-0.786175\pi$
$401$	−31.3485	−1.56547	−0.782734	+	0.622356i	$0.786175\pi$
$402$	−4.89898	−0.244339
$403$	2.40408	0.119756
$404$	15.1918	0.755822
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−4.44949	−0.220553
$408$	−14.6969	−0.727607
$409$	−12.8990	−0.637813	−0.318907	−	0.947786i	$-0.603316\pi$
$409$	−12.8990	−0.637813	−0.318907	+	0.947786i	$0.603316\pi$
$410$	6.00000	0.296319
$411$	−12.0000	−0.591916
$412$	−73.3939	−3.61586
$413$	0	0
$414$	−4.65153	−0.228610
$415$	−9.00000	−0.441793
$416$	0	0
$417$	−7.79796	−0.381868
$418$	−2.44949	−0.119808
$419$	28.3485	1.38491	0.692457	−	0.721459i	$-0.256528\pi$
$419$	28.3485	1.38491	0.692457	+	0.721459i	$0.256528\pi$
$420$	0	0
$421$	14.7980	0.721208	0.360604	−	0.932719i	$-0.382571\pi$
$421$	14.7980	0.721208	0.360604	+	0.932719i	$0.382571\pi$
$422$	16.8990	0.822629
$423$	8.44949	0.410828
$424$	14.6969	0.713746
$425$	3.00000	0.145521
$426$	−30.0000	−1.45350
$427$	0	0
$428$	−5.39388	−0.260723
$429$	0.449490	0.0217016
$430$	−6.24745	−0.301279
$431$	−36.4949	−1.75790	−0.878949	−	0.476916i	$-0.841754\pi$
$431$	−36.4949	−1.75790	−0.878949	+	0.476916i	$0.841754\pi$
$432$	−4.00000	−0.192450
$433$	−38.4949	−1.84995	−0.924973	−	0.380032i	$-0.875913\pi$
$433$	−38.4949	−1.84995	−0.924973	+	0.380032i	$0.875913\pi$
$434$	0	0
$435$	−0.550510	−0.0263949
$436$	−1.79796	−0.0861066
$437$	−1.89898	−0.0908405
$438$	7.59592	0.362947
$439$	−15.8990	−0.758817	−0.379408	−	0.925229i	$-0.623872\pi$
$439$	−15.8990	−0.758817	−0.379408	+	0.925229i	$0.623872\pi$
$440$	4.89898	0.233550
$441$	0	0
$442$	3.30306	0.157111
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	−17.7980	−0.844654
$445$	−10.3485	−0.490564
$446$	−59.6413	−2.82410
$447$	8.69694	0.411351
$448$	0	0
$449$	7.34847	0.346796	0.173398	−	0.984852i	$-0.444525\pi$
$449$	7.34847	0.346796	0.173398	+	0.984852i	$0.444525\pi$
$450$	2.44949	0.115470
$451$	2.44949	0.115342
$452$	36.0000	1.69330
$453$	−14.0000	−0.657777
$454$	67.3485	3.16082
$455$	0	0
$456$	−4.89898	−0.229416
$457$	31.4495	1.47115	0.735573	−	0.677446i	$-0.236913\pi$
$457$	31.4495	1.47115	0.735573	+	0.677446i	$0.236913\pi$
$458$	6.49490	0.303487
$459$	−3.00000	−0.140028
$460$	7.59592	0.354162
$461$	−1.10102	−0.0512796	−0.0256398	−	0.999671i	$-0.508162\pi$
$461$	−1.10102	−0.0512796	−0.0256398	+	0.999671i	$0.508162\pi$
$462$	0	0
$463$	4.69694	0.218285	0.109143	−	0.994026i	$-0.465189\pi$
$463$	4.69694	0.218285	0.109143	+	0.994026i	$0.465189\pi$
$464$	−2.20204	−0.102227
$465$	5.34847	0.248029
$466$	−54.0000	−2.50150
$467$	−12.4949	−0.578195	−0.289097	−	0.957300i	$-0.593355\pi$
$467$	−12.4949	−0.578195	−0.289097	+	0.957300i	$0.593355\pi$
$468$	1.79796	0.0831107
$469$	0	0
$470$	−20.6969	−0.954679
$471$	−12.1464	−0.559678
$472$	−2.69694	−0.124137
$473$	−2.55051	−0.117273
$474$	−8.20204	−0.376732
$475$	1.00000	0.0458831
$476$	0	0
$477$	3.00000	0.137361
$478$	−1.34847	−0.0616775
$479$	11.1464	0.509293	0.254647	−	0.967034i	$-0.418041\pi$
$479$	11.1464	0.509293	0.254647	+	0.967034i	$0.418041\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	−26.2020	−1.19347
$483$	0	0
$484$	4.00000	0.181818
$485$	−3.44949	−0.156633
$486$	−2.44949	−0.111111
$487$	9.10102	0.412407	0.206203	−	0.978509i	$-0.433889\pi$
$487$	9.10102	0.412407	0.206203	+	0.978509i	$0.433889\pi$
$488$	10.2929	0.465936
$489$	12.4495	0.562985
$490$	0	0
$491$	−6.55051	−0.295620	−0.147810	−	0.989016i	$-0.547222\pi$
$491$	−6.55051	−0.295620	−0.147810	+	0.989016i	$0.547222\pi$
$492$	9.79796	0.441726
$493$	−1.65153	−0.0743812
$494$	1.10102	0.0495373
$495$	1.00000	0.0449467
$496$	21.3939	0.960613
$497$	0	0
$498$	−22.0454	−0.987878
$499$	−9.20204	−0.411940	−0.205970	−	0.978558i	$-0.566035\pi$
$499$	−9.20204	−0.411940	−0.205970	+	0.978558i	$0.566035\pi$
$500$	−4.00000	−0.178885
$501$	−6.00000	−0.268060
$502$	−53.3939	−2.38309
$503$	26.3939	1.17684	0.588422	−	0.808554i	$-0.299749\pi$
$503$	26.3939	1.17684	0.588422	+	0.808554i	$0.299749\pi$
$504$	0	0
$505$	−3.79796	−0.169007
$506$	4.65153	0.206786
$507$	12.7980	0.568377
$508$	−42.2020	−1.87241
$509$	−7.65153	−0.339148	−0.169574	−	0.985517i	$-0.554239\pi$
$509$	−7.65153	−0.339148	−0.169574	+	0.985517i	$0.554239\pi$
$510$	7.34847	0.325396
$511$	0	0
$512$	−39.1918	−1.73205
$513$	−1.00000	−0.0441511
$514$	−35.3939	−1.56116
$515$	18.3485	0.808530
$516$	−10.2020	−0.449120
$517$	−8.44949	−0.371608
$518$	0	0
$519$	−2.20204	−0.0966589
$520$	−2.20204	−0.0965659
$521$	−11.9444	−0.523293	−0.261647	−	0.965164i	$-0.584265\pi$
$521$	−11.9444	−0.523293	−0.261647	+	0.965164i	$0.584265\pi$
$522$	−1.34847	−0.0590209
$523$	−44.4949	−1.94563	−0.972813	−	0.231592i	$-0.925607\pi$
$523$	−44.4949	−1.94563	−0.972813	+	0.231592i	$0.925607\pi$
$524$	14.2020	0.620419
$525$	0	0
$526$	−41.3939	−1.80486
$527$	16.0454	0.698949
$528$	4.00000	0.174078
$529$	−19.3939	−0.843212
$530$	−7.34847	−0.319197
$531$	−0.550510	−0.0238901
$532$	0	0
$533$	−1.10102	−0.0476905
$534$	−25.3485	−1.09694
$535$	1.34847	0.0582994
$536$	9.79796	0.423207
$537$	−14.4495	−0.623542
$538$	28.0454	1.20912
$539$	0	0
$540$	4.00000	0.172133
$541$	−8.04541	−0.345899	−0.172950	−	0.984931i	$-0.555330\pi$
$541$	−8.04541	−0.345899	−0.172950	+	0.984931i	$0.555330\pi$
$542$	2.44949	0.105215
$543$	23.7980	1.02127
$544$	0	0
$545$	0.449490	0.0192540
$546$	0	0
$547$	−26.3485	−1.12658	−0.563290	−	0.826260i	$-0.690464\pi$
$547$	−26.3485	−1.12658	−0.563290	+	0.826260i	$0.690464\pi$
$548$	48.0000	2.05046
$549$	2.10102	0.0896694
$550$	−2.44949	−0.104447
$551$	−0.550510	−0.0234525
$552$	9.30306	0.395965
$553$	0	0
$554$	26.2020	1.11322
$555$	4.44949	0.188870
$556$	31.1918	1.32283
$557$	−28.2929	−1.19881	−0.599403	−	0.800447i	$-0.704595\pi$
$557$	−28.2929	−1.19881	−0.599403	+	0.800447i	$0.704595\pi$
$558$	13.1010	0.554610
$559$	1.14643	0.0484887
$560$	0	0
$561$	3.00000	0.126660
$562$	41.3939	1.74610
$563$	15.3031	0.644947	0.322474	−	0.946578i	$-0.395486\pi$
$563$	15.3031	0.644947	0.322474	+	0.946578i	$0.395486\pi$
$564$	−33.7980	−1.42315
$565$	−9.00000	−0.378633
$566$	35.8888	1.50852
$567$	0	0
$568$	60.0000	2.51754
$569$	−35.9444	−1.50687	−0.753434	−	0.657524i	$-0.771604\pi$
$569$	−35.9444	−1.50687	−0.753434	+	0.657524i	$0.771604\pi$
$570$	2.44949	0.102598
$571$	−28.7423	−1.20283	−0.601415	−	0.798937i	$-0.705396\pi$
$571$	−28.7423	−1.20283	−0.601415	+	0.798937i	$0.705396\pi$
$572$	−1.79796	−0.0751764
$573$	2.20204	0.0919916
$574$	0	0
$575$	−1.89898	−0.0791929
$576$	−8.00000	−0.333333
$577$	17.5959	0.732528	0.366264	−	0.930511i	$-0.380637\pi$
$577$	17.5959	0.732528	0.366264	+	0.930511i	$0.380637\pi$
$578$	−19.5959	−0.815083
$579$	−26.4949	−1.10109
$580$	2.20204	0.0914348
$581$	0	0
$582$	−8.44949	−0.350243
$583$	−3.00000	−0.124247
$584$	−15.1918	−0.628643
$585$	−0.449490	−0.0185841
$586$	40.6515	1.67930
$587$	6.85357	0.282877	0.141439	−	0.989947i	$-0.454827\pi$
$587$	6.85357	0.282877	0.141439	+	0.989947i	$0.454827\pi$
$588$	0	0
$589$	5.34847	0.220380
$590$	1.34847	0.0555156
$591$	−16.8990	−0.695131
$592$	17.7980	0.731492
$593$	−36.4949	−1.49867	−0.749333	−	0.662193i	$-0.769626\pi$
$593$	−36.4949	−1.49867	−0.749333	+	0.662193i	$0.769626\pi$
$594$	2.44949	0.100504
$595$	0	0
$596$	−34.7878	−1.42496
$597$	−25.7980	−1.05584
$598$	−2.09082	−0.0854998
$599$	20.6969	0.845654	0.422827	−	0.906210i	$-0.361038\pi$
$599$	20.6969	0.845654	0.422827	+	0.906210i	$0.361038\pi$
$600$	−4.89898	−0.200000
$601$	−28.3939	−1.15821	−0.579105	−	0.815253i	$-0.696598\pi$
$601$	−28.3939	−1.15821	−0.579105	+	0.815253i	$0.696598\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	56.0000	2.27861
$605$	−1.00000	−0.0406558
$606$	−9.30306	−0.377911
$607$	14.8990	0.604731	0.302365	−	0.953192i	$-0.402224\pi$
$607$	14.8990	0.604731	0.302365	+	0.953192i	$0.402224\pi$
$608$	0	0
$609$	0	0
$610$	−5.14643	−0.208373
$611$	3.79796	0.153649
$612$	12.0000	0.485071
$613$	−7.79796	−0.314957	−0.157478	−	0.987522i	$-0.550336\pi$
$613$	−7.79796	−0.314957	−0.157478	+	0.987522i	$0.550336\pi$
$614$	12.4949	0.504253
$615$	−2.44949	−0.0987730
$616$	0	0
$617$	12.4949	0.503026	0.251513	−	0.967854i	$-0.419072\pi$
$617$	12.4949	0.503026	0.251513	+	0.967854i	$0.419072\pi$
$618$	44.9444	1.80793
$619$	31.7980	1.27807	0.639034	−	0.769179i	$-0.279334\pi$
$619$	31.7980	1.27807	0.639034	+	0.769179i	$0.279334\pi$
$620$	−21.3939	−0.859199
$621$	1.89898	0.0762034
$622$	−12.0000	−0.481156
$623$	0	0
$624$	−1.79796	−0.0719760
$625$	1.00000	0.0400000
$626$	−56.9444	−2.27596
$627$	1.00000	0.0399362
$628$	48.5857	1.93878
$629$	13.3485	0.532238
$630$	0	0
$631$	−10.3031	−0.410158	−0.205079	−	0.978745i	$-0.565745\pi$
$631$	−10.3031	−0.410158	−0.205079	+	0.978745i	$0.565745\pi$
$632$	16.4041	0.652519
$633$	−6.89898	−0.274210
$634$	6.60612	0.262363
$635$	10.5505	0.418684
$636$	−12.0000	−0.475831
$637$	0	0
$638$	1.34847	0.0533864
$639$	12.2474	0.484502
$640$	19.5959	0.774597
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	3.30306	0.130361
$643$	−41.7423	−1.64616	−0.823079	−	0.567927i	$-0.807745\pi$
$643$	−41.7423	−1.64616	−0.823079	+	0.567927i	$0.807745\pi$
$644$	0	0
$645$	2.55051	0.100426
$646$	7.34847	0.289122
$647$	7.10102	0.279170	0.139585	−	0.990210i	$-0.455423\pi$
$647$	7.10102	0.279170	0.139585	+	0.990210i	$0.455423\pi$
$648$	4.89898	0.192450
$649$	0.550510	0.0216094
$650$	1.10102	0.0431856
$651$	0	0
$652$	−49.7980	−1.95024
$653$	48.1918	1.88589	0.942946	−	0.332945i	$-0.108042\pi$
$653$	48.1918	1.88589	0.942946	+	0.332945i	$0.108042\pi$
$654$	1.10102	0.0430533
$655$	−3.55051	−0.138730
$656$	−9.79796	−0.382546
$657$	−3.10102	−0.120982
$658$	0	0
$659$	−31.0454	−1.20936	−0.604679	−	0.796470i	$-0.706698\pi$
$659$	−31.0454	−1.20936	−0.604679	+	0.796470i	$0.706698\pi$
$660$	−4.00000	−0.155700
$661$	22.2474	0.865325	0.432663	−	0.901556i	$-0.357574\pi$
$661$	22.2474	0.865325	0.432663	+	0.901556i	$0.357574\pi$
$662$	77.6413	3.01762
$663$	−1.34847	−0.0523702
$664$	44.0908	1.71106
$665$	0	0
$666$	10.8990	0.422327
$667$	1.04541	0.0404783
$668$	24.0000	0.928588
$669$	24.3485	0.941366
$670$	−4.89898	−0.189264
$671$	−2.10102	−0.0811090
$672$	0	0
$673$	24.3485	0.938565	0.469282	−	0.883048i	$-0.344513\pi$
$673$	24.3485	0.938565	0.469282	+	0.883048i	$0.344513\pi$
$674$	−5.75255	−0.221580
$675$	−1.00000	−0.0384900
$676$	−51.1918	−1.96892
$677$	−36.1918	−1.39097	−0.695483	−	0.718543i	$-0.744809\pi$
$677$	−36.1918	−1.39097	−0.695483	+	0.718543i	$0.744809\pi$
$678$	−22.0454	−0.846649
$679$	0	0
$680$	−14.6969	−0.563602
$681$	−27.4949	−1.05361
$682$	−13.1010	−0.501664
$683$	31.5959	1.20898	0.604492	−	0.796611i	$-0.293376\pi$
$683$	31.5959	1.20898	0.604492	+	0.796611i	$0.293376\pi$
$684$	4.00000	0.152944
$685$	−12.0000	−0.458496
$686$	0	0
$687$	−2.65153	−0.101162
$688$	10.2020	0.388949
$689$	1.34847	0.0513726
$690$	−4.65153	−0.177081
$691$	34.0000	1.29342	0.646710	−	0.762736i	$-0.276144\pi$
$691$	34.0000	1.29342	0.646710	+	0.762736i	$0.276144\pi$
$692$	8.80816	0.334836
$693$	0	0
$694$	30.6061	1.16179
$695$	−7.79796	−0.295793
$696$	2.69694	0.102227
$697$	−7.34847	−0.278343
$698$	−38.9444	−1.47407
$699$	22.0454	0.833834
$700$	0	0
$701$	22.3485	0.844090	0.422045	−	0.906575i	$-0.361312\pi$
$701$	22.3485	0.844090	0.422045	+	0.906575i	$0.361312\pi$
$702$	−1.10102	−0.0415553
$703$	4.44949	0.167816
$704$	8.00000	0.301511
$705$	8.44949	0.318226
$706$	21.3031	0.801751
$707$	0	0
$708$	2.20204	0.0827578
$709$	−1.00000	−0.0375558	−0.0187779	−	0.999824i	$-0.505978\pi$
$709$	−1.00000	−0.0375558	−0.0187779	+	0.999824i	$0.505978\pi$
$710$	−30.0000	−1.12588
$711$	3.34847	0.125577
$712$	50.6969	1.89995
$713$	−10.1566	−0.380369
$714$	0	0
$715$	0.449490	0.0168100
$716$	57.7980	2.16001
$717$	0.550510	0.0205592
$718$	−52.0454	−1.94232
$719$	33.2474	1.23992	0.619960	−	0.784633i	$-0.287149\pi$
$719$	33.2474	1.23992	0.619960	+	0.784633i	$0.287149\pi$
$720$	−4.00000	−0.149071
$721$	0	0
$722$	−44.0908	−1.64089
$723$	10.6969	0.397823
$724$	−95.1918	−3.53778
$725$	−0.550510	−0.0204454
$726$	−2.44949	−0.0909091
$727$	−1.94439	−0.0721133	−0.0360567	−	0.999350i	$-0.511480\pi$
$727$	−1.94439	−0.0721133	−0.0360567	+	0.999350i	$0.511480\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	7.59592	0.281138
$731$	7.65153	0.283002
$732$	−8.40408	−0.310624
$733$	−42.8990	−1.58451	−0.792255	−	0.610190i	$-0.791093\pi$
$733$	−42.8990	−1.58451	−0.792255	+	0.610190i	$0.791093\pi$
$734$	−36.8536	−1.36029
$735$	0	0
$736$	0	0
$737$	−2.00000	−0.0736709
$738$	−6.00000	−0.220863
$739$	−48.6969	−1.79135	−0.895673	−	0.444713i	$-0.853306\pi$
$739$	−48.6969	−1.79135	−0.895673	+	0.444713i	$0.853306\pi$
$740$	−17.7980	−0.654266
$741$	−0.449490	−0.0165124
$742$	0	0
$743$	52.2929	1.91844	0.959219	−	0.282663i	$-0.0912177\pi$
$743$	52.2929	1.91844	0.959219	+	0.282663i	$0.0912177\pi$
$744$	−26.2020	−0.960613
$745$	8.69694	0.318631
$746$	39.5505	1.44805
$747$	9.00000	0.329293
$748$	−12.0000	−0.438763
$749$	0	0
$750$	2.44949	0.0894427
$751$	−31.4949	−1.14927	−0.574633	−	0.818412i	$-0.694855\pi$
$751$	−31.4949	−1.14927	−0.574633	+	0.818412i	$0.694855\pi$
$752$	33.7980	1.23248
$753$	21.7980	0.794362
$754$	−0.606123	−0.0220737
$755$	−14.0000	−0.509512
$756$	0	0
$757$	−20.2929	−0.737556	−0.368778	−	0.929517i	$-0.620224\pi$
$757$	−20.2929	−0.737556	−0.368778	+	0.929517i	$0.620224\pi$
$758$	86.9444	3.15796
$759$	−1.89898	−0.0689286
$760$	−4.89898	−0.177705
$761$	−3.79796	−0.137676	−0.0688380	−	0.997628i	$-0.521929\pi$
$761$	−3.79796	−0.137676	−0.0688380	+	0.997628i	$0.521929\pi$
$762$	25.8434	0.936206
$763$	0	0
$764$	−8.80816	−0.318668
$765$	−3.00000	−0.108465
$766$	−6.00000	−0.216789
$767$	−0.247449	−0.00893486
$768$	32.0000	1.15470
$769$	32.1010	1.15759	0.578796	−	0.815472i	$-0.303523\pi$
$769$	32.1010	1.15759	0.578796	+	0.815472i	$0.303523\pi$
$770$	0	0
$771$	14.4495	0.520386
$772$	105.980	3.81429
$773$	−1.10102	−0.0396010	−0.0198005	−	0.999804i	$-0.506303\pi$
$773$	−1.10102	−0.0396010	−0.0198005	+	0.999804i	$0.506303\pi$
$774$	6.24745	0.224560
$775$	5.34847	0.192123
$776$	16.8990	0.606638
$777$	0	0
$778$	−15.3031	−0.548641
$779$	−2.44949	−0.0877621
$780$	1.79796	0.0643773
$781$	−12.2474	−0.438248
$782$	−13.9546	−0.499015
$783$	0.550510	0.0196736
$784$	0	0
$785$	−12.1464	−0.433525
$786$	−8.69694	−0.310210
$787$	21.3939	0.762609	0.381305	−	0.924449i	$-0.375475\pi$
$787$	21.3939	0.762609	0.381305	+	0.924449i	$0.375475\pi$
$788$	67.5959	2.40800
$789$	16.8990	0.601620
$790$	−8.20204	−0.291816
$791$	0	0
$792$	−4.89898	−0.174078
$793$	0.944387	0.0335362
$794$	−55.5959	−1.97303
$795$	3.00000	0.106399
$796$	103.192	3.65754
$797$	−40.2929	−1.42725	−0.713623	−	0.700530i	$-0.752947\pi$
$797$	−40.2929	−1.42725	−0.713623	+	0.700530i	$0.752947\pi$
$798$	0	0
$799$	25.3485	0.896764
$800$	0	0
$801$	10.3485	0.365645
$802$	−76.7878	−2.71147
$803$	3.10102	0.109433
$804$	−8.00000	−0.282138
$805$	0	0
$806$	5.88877	0.207423
$807$	−11.4495	−0.403041
$808$	18.6061	0.654561
$809$	8.69694	0.305768	0.152884	−	0.988244i	$-0.451144\pi$
$809$	8.69694	0.305768	0.152884	+	0.988244i	$0.451144\pi$
$810$	−2.44949	−0.0860663
$811$	−25.3939	−0.891700	−0.445850	−	0.895108i	$-0.647098\pi$
$811$	−25.3939	−0.891700	−0.445850	+	0.895108i	$0.647098\pi$
$812$	0	0
$813$	−1.00000	−0.0350715
$814$	−10.8990	−0.382009
$815$	12.4495	0.436087
$816$	−12.0000	−0.420084
$817$	2.55051	0.0892311
$818$	−31.5959	−1.10473
$819$	0	0
$820$	9.79796	0.342160
$821$	30.5505	1.06622	0.533110	−	0.846046i	$-0.321023\pi$
$821$	30.5505	1.06622	0.533110	+	0.846046i	$0.321023\pi$
$822$	−29.3939	−1.02523
$823$	38.2474	1.33322	0.666611	−	0.745406i	$-0.267744\pi$
$823$	38.2474	1.33322	0.666611	+	0.745406i	$0.267744\pi$
$824$	−89.8888	−3.13142
$825$	1.00000	0.0348155
$826$	0	0
$827$	2.69694	0.0937817	0.0468909	−	0.998900i	$-0.485069\pi$
$827$	2.69694	0.0937817	0.0468909	+	0.998900i	$0.485069\pi$
$828$	−7.59592	−0.263976
$829$	−46.0908	−1.60080	−0.800400	−	0.599466i	$-0.795380\pi$
$829$	−46.0908	−1.60080	−0.800400	+	0.599466i	$0.795380\pi$
$830$	−22.0454	−0.765207
$831$	−10.6969	−0.371073
$832$	−3.59592	−0.124666
$833$	0	0
$834$	−19.1010	−0.661414
$835$	−6.00000	−0.207639
$836$	−4.00000	−0.138343
$837$	−5.34847	−0.184870
$838$	69.4393	2.39874
$839$	−14.1464	−0.488389	−0.244194	−	0.969726i	$-0.578524\pi$
$839$	−14.1464	−0.488389	−0.244194	+	0.969726i	$0.578524\pi$
$840$	0	0
$841$	−28.6969	−0.989550
$842$	36.2474	1.24917
$843$	−16.8990	−0.582032
$844$	27.5959	0.949891
$845$	12.7980	0.440263
$846$	20.6969	0.711575
$847$	0	0
$848$	12.0000	0.412082
$849$	−14.6515	−0.502839
$850$	7.34847	0.252050
$851$	−8.44949	−0.289645
$852$	−48.9898	−1.67836
$853$	14.0454	0.480906	0.240453	−	0.970661i	$-0.422704\pi$
$853$	14.0454	0.480906	0.240453	+	0.970661i	$0.422704\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	−6.60612	−0.225793
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	1.10102	0.0375882
$859$	−11.5505	−0.394098	−0.197049	−	0.980394i	$-0.563136\pi$
$859$	−11.5505	−0.394098	−0.197049	+	0.980394i	$0.563136\pi$
$860$	−10.2020	−0.347887
$861$	0	0
$862$	−89.3939	−3.04477
$863$	1.40408	0.0477955	0.0238978	−	0.999714i	$-0.492392\pi$
$863$	1.40408	0.0477955	0.0238978	+	0.999714i	$0.492392\pi$
$864$	0	0
$865$	−2.20204	−0.0748716
$866$	−94.2929	−3.20420
$867$	8.00000	0.271694
$868$	0	0
$869$	−3.34847	−0.113589
$870$	−1.34847	−0.0457174
$871$	0.898979	0.0304608
$872$	−2.20204	−0.0745705
$873$	3.44949	0.116748
$874$	−4.65153	−0.157340
$875$	0	0
$876$	12.4041	0.419095
$877$	−17.6515	−0.596050	−0.298025	−	0.954558i	$-0.596328\pi$
$877$	−17.6515	−0.596050	−0.298025	+	0.954558i	$0.596328\pi$
$878$	−38.9444	−1.31431
$879$	−16.5959	−0.559766
$880$	4.00000	0.134840
$881$	1.04541	0.0352207	0.0176103	−	0.999845i	$-0.494394\pi$
$881$	1.04541	0.0352207	0.0176103	+	0.999845i	$0.494394\pi$
$882$	0	0
$883$	−14.2929	−0.480993	−0.240496	−	0.970650i	$-0.577310\pi$
$883$	−14.2929	−0.480993	−0.240496	+	0.970650i	$0.577310\pi$
$884$	5.39388	0.181416
$885$	−0.550510	−0.0185052
$886$	14.6969	0.493753
$887$	−21.4949	−0.721728	−0.360864	−	0.932618i	$-0.617518\pi$
$887$	−21.4949	−0.721728	−0.360864	+	0.932618i	$0.617518\pi$
$888$	−21.7980	−0.731492
$889$	0	0
$890$	−25.3485	−0.849683
$891$	−1.00000	−0.0335013
$892$	−97.3939	−3.26099
$893$	8.44949	0.282751
$894$	21.3031	0.712481
$895$	−14.4495	−0.482993
$896$	0	0
$897$	0.853572	0.0284999
$898$	18.0000	0.600668
$899$	−2.94439	−0.0982008
$900$	4.00000	0.133333
$901$	9.00000	0.299833
$902$	6.00000	0.199778
$903$	0	0
$904$	44.0908	1.46644
$905$	23.7980	0.791071
$906$	−34.2929	−1.13930
$907$	−20.0454	−0.665597	−0.332798	−	0.942998i	$-0.607993\pi$
$907$	−20.0454	−0.665597	−0.332798	+	0.942998i	$0.607993\pi$
$908$	109.980	3.64980
$909$	3.79796	0.125970
$910$	0	0
$911$	−58.2929	−1.93133	−0.965664	−	0.259793i	$-0.916346\pi$
$911$	−58.2929	−1.93133	−0.965664	+	0.259793i	$0.916346\pi$
$912$	−4.00000	−0.132453
$913$	−9.00000	−0.297857
$914$	77.0352	2.54810
$915$	2.10102	0.0694576
$916$	10.6061	0.350436
$917$	0	0
$918$	−7.34847	−0.242536
$919$	26.0000	0.857661	0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000	0.857661	0.428830	+	0.903385i	$0.358926\pi$
$920$	9.30306	0.306713
$921$	−5.10102	−0.168084
$922$	−2.69694	−0.0888189
$923$	5.50510	0.181203
$924$	0	0
$925$	4.44949	0.146298
$926$	11.5051	0.378081
$927$	−18.3485	−0.602643
$928$	0	0
$929$	7.10102	0.232977	0.116488	−	0.993192i	$-0.462836\pi$
$929$	7.10102	0.232977	0.116488	+	0.993192i	$0.462836\pi$
$930$	13.1010	0.429599
$931$	0	0
$932$	−88.1816	−2.88849
$933$	4.89898	0.160385
$934$	−30.6061	−1.00146
$935$	3.00000	0.0981105
$936$	2.20204	0.0719760
$937$	−0.651531	−0.0212846	−0.0106423	−	0.999943i	$-0.503388\pi$
$937$	−0.651531	−0.0212846	−0.0106423	+	0.999943i	$0.503388\pi$
$938$	0	0
$939$	23.2474	0.758652
$940$	−33.7980	−1.10237
$941$	−16.0454	−0.523065	−0.261533	−	0.965195i	$-0.584228\pi$
$941$	−16.0454	−0.523065	−0.261533	+	0.965195i	$0.584228\pi$
$942$	−29.7526	−0.969391
$943$	4.65153	0.151475
$944$	−2.20204	−0.0716703
$945$	0	0
$946$	−6.24745	−0.203122
$947$	−36.1918	−1.17608	−0.588038	−	0.808833i	$-0.700100\pi$
$947$	−36.1918	−1.17608	−0.588038	+	0.808833i	$0.700100\pi$
$948$	−13.3939	−0.435013
$949$	−1.39388	−0.0452472
$950$	2.44949	0.0794719
$951$	−2.69694	−0.0874542
$952$	0	0
$953$	23.3939	0.757802	0.378901	−	0.925437i	$-0.376302\pi$
$953$	23.3939	0.757802	0.378901	+	0.925437i	$0.376302\pi$
$954$	7.34847	0.237915
$955$	2.20204	0.0712564
$956$	−2.20204	−0.0712191
$957$	−0.550510	−0.0177955
$958$	27.3031	0.882122
$959$	0	0
$960$	−8.00000	−0.258199
$961$	−2.39388	−0.0772218
$962$	4.89898	0.157949
$963$	−1.34847	−0.0434538
$964$	−42.7878	−1.37810
$965$	−26.4949	−0.852901
$966$	0	0
$967$	−14.8434	−0.477330	−0.238665	−	0.971102i	$-0.576710\pi$
$967$	−14.8434	−0.477330	−0.238665	+	0.971102i	$0.576710\pi$
$968$	4.89898	0.157459
$969$	−3.00000	−0.0963739
$970$	−8.44949	−0.271297
$971$	22.3485	0.717197	0.358598	−	0.933492i	$-0.383255\pi$
$971$	22.3485	0.717197	0.358598	+	0.933492i	$0.383255\pi$
$972$	−4.00000	−0.128300
$973$	0	0
$974$	22.2929	0.714309
$975$	−0.449490	−0.0143952
$976$	8.40408	0.269008
$977$	19.4041	0.620792	0.310396	−	0.950607i	$-0.399538\pi$
$977$	19.4041	0.620792	0.310396	+	0.950607i	$0.399538\pi$
$978$	30.4949	0.975119
$979$	−10.3485	−0.330739
$980$	0	0
$981$	−0.449490	−0.0143511
$982$	−16.0454	−0.512030
$983$	30.4949	0.972636	0.486318	−	0.873782i	$-0.338340\pi$
$983$	30.4949	0.972636	0.486318	+	0.873782i	$0.338340\pi$
$984$	12.0000	0.382546
$985$	−16.8990	−0.538446
$986$	−4.04541	−0.128832
$987$	0	0
$988$	1.79796	0.0572007
$989$	−4.84337	−0.154010
$990$	2.44949	0.0778499
$991$	6.59592	0.209526	0.104763	−	0.994497i	$-0.466592\pi$
$991$	6.59592	0.209526	0.104763	+	0.994497i	$0.466592\pi$
$992$	0	0
$993$	−31.6969	−1.00587
$994$	0	0
$995$	−25.7980	−0.817850
$996$	−36.0000	−1.14070
$997$	−7.39388	−0.234166	−0.117083	−	0.993122i	$-0.537354\pi$
$997$	−7.39388	−0.234166	−0.117083	+	0.993122i	$0.537354\pi$
$998$	−22.5403	−0.713501
$999$	−4.44949	−0.140776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8085.2.a.be.1.2		2
7.6	odd	2		1155.2.a.q.1.2	✓	2
21.20	even	2		3465.2.a.y.1.1		2
35.34	odd	2		5775.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.q.1.2	✓	2	7.6	odd	2
3465.2.a.y.1.1		2	21.20	even	2
5775.2.a.bh.1.1		2	35.34	odd	2
8085.2.a.be.1.2		2	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:	\( N \)	\(=\)	\( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8085.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{2} -1.00000 q^{3} +4.00000 q^{4} -1.00000 q^{5} -2.44949 q^{6} +4.89898 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.44949 q^{2} -1.00000 q^{3} +4.00000 q^{4} -1.00000 q^{5} -2.44949 q^{6} +4.89898 q^{8} +1.00000 q^{9} -2.44949 q^{10} -1.00000 q^{11} -4.00000 q^{12} +0.449490 q^{13} +1.00000 q^{15} +4.00000 q^{16} +3.00000 q^{17} +2.44949 q^{18} +1.00000 q^{19} -4.00000 q^{20} -2.44949 q^{22} -1.89898 q^{23} -4.89898 q^{24} +1.00000 q^{25} +1.10102 q^{26} -1.00000 q^{27} -0.550510 q^{29} +2.44949 q^{30} +5.34847 q^{31} +1.00000 q^{33} +7.34847 q^{34} +4.00000 q^{36} +4.44949 q^{37} +2.44949 q^{38} -0.449490 q^{39} -4.89898 q^{40} -2.44949 q^{41} +2.55051 q^{43} -4.00000 q^{44} -1.00000 q^{45} -4.65153 q^{46} +8.44949 q^{47} -4.00000 q^{48} +2.44949 q^{50} -3.00000 q^{51} +1.79796 q^{52} +3.00000 q^{53} -2.44949 q^{54} +1.00000 q^{55} -1.00000 q^{57} -1.34847 q^{58} -0.550510 q^{59} +4.00000 q^{60} +2.10102 q^{61} +13.1010 q^{62} -8.00000 q^{64} -0.449490 q^{65} +2.44949 q^{66} +2.00000 q^{67} +12.0000 q^{68} +1.89898 q^{69} +12.2474 q^{71} +4.89898 q^{72} -3.10102 q^{73} +10.8990 q^{74} -1.00000 q^{75} +4.00000 q^{76} -1.10102 q^{78} +3.34847 q^{79} -4.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +9.00000 q^{83} -3.00000 q^{85} +6.24745 q^{86} +0.550510 q^{87} -4.89898 q^{88} +10.3485 q^{89} -2.44949 q^{90} -7.59592 q^{92} -5.34847 q^{93} +20.6969 q^{94} -1.00000 q^{95} +3.44949 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 8 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 8 * q^4 - 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} + 8 q^{4} - 2 q^{5} + 2 q^{9} - 2 q^{11} - 8 q^{12} - 4 q^{13} + 2 q^{15} + 8 q^{16} + 6 q^{17} + 2 q^{19} - 8 q^{20} + 6 q^{23} + 2 q^{25} + 12 q^{26} - 2 q^{27} - 6 q^{29} - 4 q^{31} + 2 q^{33} + 8 q^{36} + 4 q^{37} + 4 q^{39} + 10 q^{43} - 8 q^{44} - 2 q^{45} - 24 q^{46} + 12 q^{47} - 8 q^{48} - 6 q^{51} - 16 q^{52} + 6 q^{53} + 2 q^{55} - 2 q^{57} + 12 q^{58} - 6 q^{59} + 8 q^{60} + 14 q^{61} + 36 q^{62} - 16 q^{64} + 4 q^{65} + 4 q^{67} + 24 q^{68} - 6 q^{69} - 16 q^{73} + 12 q^{74} - 2 q^{75} + 8 q^{76} - 12 q^{78} - 8 q^{79} - 8 q^{80} + 2 q^{81} - 12 q^{82} + 18 q^{83} - 6 q^{85} - 12 q^{86} + 6 q^{87} + 6 q^{89} + 24 q^{92} + 4 q^{93} + 12 q^{94} - 2 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 8 * q^4 - 2 * q^5 + 2 * q^9 - 2 * q^11 - 8 * q^12 - 4 * q^13 + 2 * q^15 + 8 * q^16 + 6 * q^17 + 2 * q^19 - 8 * q^20 + 6 * q^23 + 2 * q^25 + 12 * q^26 - 2 * q^27 - 6 * q^29 - 4 * q^31 + 2 * q^33 + 8 * q^36 + 4 * q^37 + 4 * q^39 + 10 * q^43 - 8 * q^44 - 2 * q^45 - 24 * q^46 + 12 * q^47 - 8 * q^48 - 6 * q^51 - 16 * q^52 + 6 * q^53 + 2 * q^55 - 2 * q^57 + 12 * q^58 - 6 * q^59 + 8 * q^60 + 14 * q^61 + 36 * q^62 - 16 * q^64 + 4 * q^65 + 4 * q^67 + 24 * q^68 - 6 * q^69 - 16 * q^73 + 12 * q^74 - 2 * q^75 + 8 * q^76 - 12 * q^78 - 8 * q^79 - 8 * q^80 + 2 * q^81 - 12 * q^82 + 18 * q^83 - 6 * q^85 - 12 * q^86 + 6 * q^87 + 6 * q^89 + 24 * q^92 + 4 * q^93 + 12 * q^94 - 2 * q^95 + 2 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more